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API Reference

This page is generated from the Python docstrings. For task-oriented usage, start with the component pages and examples.

gbmsc_pde

Grid-Based Multinode Shepard Collocation solvers for PDEs.

BoundaryConditions

Boundary-condition container for rectangular source-node rows.

Conditions are registered by flattened source-node id. Dirichlet rows replace the corresponding PDE row by u_i = value. Neumann and Robin rows use the derivative matrices assembled by the PDE solver. Scalars are broadcast to all supplied nodes; arrays must have the same length as nodes.

Source code in src/gbmsc_pde/boundary/conditions.py 
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class BoundaryConditions:
    """
    Boundary-condition container for rectangular source-node rows.

    Conditions are registered by flattened source-node id.  Dirichlet rows
    replace the corresponding PDE row by ``u_i = value``.  Neumann and Robin
    rows use the derivative matrices assembled by the PDE solver.  Scalars are
    broadcast to all supplied nodes; arrays must have the same length as
    ``nodes``.
    """

    def __init__(self, grid: SourceGrid) -> None:
        self.grid = grid
        self.dirichlet: Dict[int, float] = {}
        self.neumann: Dict[int, Dict[str, Union[Tuple[float, float], float]]] = {}
        self.robin: Dict[int, Dict[str, Union[Tuple[float, float], float]]] = {}
        self.second_neumann: Dict[int, Dict[str, Union[Tuple[float, float], float, int]]] = {}

    def add_dirichlet(
        self,
        nodes: Union[int, np.ndarray],
        values: Union[float, np.ndarray],
    ) -> None:
        """
        Add Dirichlet conditions ``u_i = value`` on rectangular grid nodes.

        Parameters
        ----------
        nodes : int or ndarray
            Flattened source-node ids.
        values : float or ndarray
            Scalar value or one value per node.
        """
        nodes_arr = np.atleast_1d(nodes).astype(int)
        vals_arr = np.atleast_1d(values).astype(float)
        if vals_arr.size == 1:
            vals_arr = np.full(nodes_arr.shape, vals_arr.item(), dtype=float)
        if nodes_arr.size != vals_arr.size:
            raise ValueError(f"Mismatched nodes ({nodes_arr.size}) and values ({vals_arr.size}).")

        n_nodes = getattr(self.grid, "N", None)
        if n_nodes is None:
            raise AttributeError("SourceGrid must define total node count 'N'.")
        if np.any(nodes_arr < 0) or np.any(nodes_arr >= n_nodes):
            raise ValueError(f"Dirichlet indices out of [0, {n_nodes}): {nodes_arr}")

        for node, val in zip(nodes_arr, vals_arr):
            self.dirichlet[int(node)] = float(val)

    def add_neumann(
        self,
        nodes: Union[int, np.ndarray],
        outwardnormal: Union[Tuple[float, float], Tuple[np.ndarray, np.ndarray]],
        fluxes: Union[float, np.ndarray],
    ) -> None:
        """
        Add Neumann conditions ``du/dn = flux`` on grid nodes.

        Parameters
        ----------
        nodes : int or ndarray
            Flattened source-node ids.
        outwardnormal : tuple
            Normal components ``(n_x, n_y)`` as scalars or arrays.
        fluxes : float or ndarray
            Scalar flux or one flux per node.
        """
        nodes_arr = np.atleast_1d(nodes).astype(int)
        k = nodes_arr.size

        if not (isinstance(outwardnormal, tuple) and len(outwardnormal) == 2):
            raise TypeError("`outwardnormal` must be a tuple of length 2.")
        nx, ny = outwardnormal

        if np.isscalar(nx) and np.isscalar(ny):
            nx_arr = np.full(k, float(nx))
            ny_arr = np.full(k, float(ny))
        else:
            nx_arr = np.atleast_1d(nx).astype(float)
            ny_arr = np.atleast_1d(ny).astype(float)
        if nx_arr.shape != (k,) or ny_arr.shape != (k,):
            raise ValueError("`outwardnormal` components must match length of `nodes`.")

        flux_arr = np.atleast_1d(fluxes).astype(float)
        if flux_arr.size == 1:
            flux_arr = np.full(k, flux_arr.item())
        if flux_arr.shape != (k,):
            raise ValueError("`fluxes` must be scalar or match length of `nodes`.")

        n_nodes = getattr(self.grid, "N", None)
        if n_nodes is None:
            raise AttributeError("BoundaryConditions requires `self.grid.N`.")
        if np.any(nodes_arr < 0) or np.any(nodes_arr >= n_nodes):
            raise ValueError(f"Node indices must be in [0, {n_nodes}); got {nodes_arr}.")

        for node, nx_val, ny_val, flux_val in zip(nodes_arr, nx_arr, ny_arr, flux_arr):
            self.neumann[int(node)] = {
                "normal": (float(nx_val), float(ny_val)),
                "flux": float(flux_val),
            }

    def add_robin(
        self,
        nodes: Union[int, np.ndarray],
        outwardnormal: Union[Tuple[float, float], Tuple[np.ndarray, np.ndarray]],
        alphas: Union[float, np.ndarray],
        betas: Union[float, np.ndarray],
        values: Union[float, np.ndarray],
    ) -> None:
        """
        Add Robin conditions ``alpha*u + beta*du/dn = value`` on grid nodes.

        ``alphas``, ``betas``, and ``values`` may be scalars or arrays with
        one entry per node.
        """
        nodes_arr = np.atleast_1d(nodes).astype(int)
        k = nodes_arr.size

        if not (isinstance(outwardnormal, tuple) and len(outwardnormal) == 2):
            raise TypeError("`outwardnormal` must be a tuple (nx, ny).")
        nx, ny = outwardnormal
        nx_arr = np.full(k, nx, dtype=float) if np.isscalar(nx) else np.atleast_1d(nx).astype(float)
        ny_arr = np.full(k, ny, dtype=float) if np.isscalar(ny) else np.atleast_1d(ny).astype(float)
        if nx_arr.shape != (k,) or ny_arr.shape != (k,):
            raise ValueError("`outwardnormal` components must match length of `nodes`.")

        alpha_arr = np.atleast_1d(alphas).astype(float)
        beta_arr = np.atleast_1d(betas).astype(float)
        value_arr = np.atleast_1d(values).astype(float)

        if alpha_arr.size == 1:
            alpha_arr = np.full(k, alpha_arr.item(), dtype=float)
        if beta_arr.size == 1:
            beta_arr = np.full(k, beta_arr.item(), dtype=float)
        if value_arr.size == 1:
            value_arr = np.full(k, value_arr.item(), dtype=float)

        if not (alpha_arr.size == beta_arr.size == value_arr.size == k):
            raise ValueError(
                f"Sizes must match: nodes({k}), alphas({alpha_arr.size}), "
                f"betas({beta_arr.size}), values({value_arr.size})."
            )

        n_nodes = getattr(self.grid, "N", None)
        if n_nodes is None:
            raise AttributeError("SourceGrid must define total node count 'N'.")
        if np.any(nodes_arr < 0) or np.any(nodes_arr >= n_nodes):
            raise ValueError(f"Node indices out of [0, {n_nodes}): {nodes_arr}")

        for node, nx_val, ny_val, alpha, beta, value in zip(
            nodes_arr, nx_arr, ny_arr, alpha_arr, beta_arr, value_arr
        ):
            self.robin[int(node)] = {
                "normal": (float(nx_val), float(ny_val)),
                "alpha": float(alpha),
                "beta": float(beta),
                "value": float(value),
            }

    def add_neumann_as_second_condition(
        self,
        nodes: Union[int, np.ndarray],
        outwardnormal: Union[Tuple[float, float], Tuple[np.ndarray, np.ndarray]],
        fluxes: Union[float, np.ndarray],
        rows: Union[int, np.ndarray],
    ) -> None:
        """
        Add Neumann equations on user-selected matrix rows.

        This compatibility method is for augmented systems where a Neumann
        boundary equation should be imposed as an additional/second condition
        rather than replacing the row associated with ``node``.
        """
        if not hasattr(self.grid, "N"):
            raise AttributeError("BoundaryConditions.grid must define integer `.N`")

        nodes_arr = np.atleast_1d(nodes).astype(int)
        rows_arr = np.atleast_1d(rows).astype(int)

        if nodes_arr.ndim != 1 or rows_arr.ndim != 1:
            raise TypeError("`nodes` and `rows` must be scalars or 1D sequences")

        n = nodes_arr.size
        if rows_arr.size != n:
            raise ValueError(f"`rows` length ({rows_arr.size}) != `nodes` length ({n})")

        if not (isinstance(outwardnormal, tuple) and len(outwardnormal) == 2):
            raise TypeError("`outwardnormal` must be a tuple of two elements")
        nx_in, ny_in = outwardnormal

        try:
            nx_arr = np.broadcast_to(nx_in, (n,)).astype(float)
            ny_arr = np.broadcast_to(ny_in, (n,)).astype(float)
        except Exception as exc:
            raise ValueError(
                "Cannot broadcast `outwardnormal` components to match `nodes` length"
            ) from exc

        try:
            flux_arr = np.broadcast_to(fluxes, (n,)).astype(float)
        except Exception as exc:
            raise ValueError("Cannot broadcast `fluxes` to match `nodes` length") from exc

        if np.any(nodes_arr < 0) or np.any(nodes_arr >= self.grid.N):
            raise ValueError(f"All `nodes` must be in [0, {self.grid.N})")

        for node, row, nx_val, ny_val, flux in zip(nodes_arr, rows_arr, nx_arr, ny_arr, flux_arr):
            self.second_neumann[int(node)] = {
                "normal": (float(nx_val), float(ny_val)),
                "flux": float(flux),
                "row": int(row),
            }

    def apply(
        self,
        A: spmatrix,
        b: np.ndarray,
        Mx: spmatrix,
        My: spmatrix,
    ) -> tuple[spmatrix, np.ndarray]:
        """Apply all registered conditions to an assembled rectangular system."""
        return apply_boundary_conditions(self, A, b, Mx, My)

add_dirichlet(nodes, values)

Add Dirichlet conditions u_i = value on rectangular grid nodes.

Parameters:

Name Type Description Default
nodes int or ndarray

Flattened source-node ids.

 required
values float or ndarray

Scalar value or one value per node.

 required
Source code in src/gbmsc_pde/boundary/conditions.py 
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def add_dirichlet(
    self,
    nodes: Union[int, np.ndarray],
    values: Union[float, np.ndarray],
) -> None:
    """
    Add Dirichlet conditions ``u_i = value`` on rectangular grid nodes.

    Parameters
    ----------
    nodes : int or ndarray
        Flattened source-node ids.
    values : float or ndarray
        Scalar value or one value per node.
    """
    nodes_arr = np.atleast_1d(nodes).astype(int)
    vals_arr = np.atleast_1d(values).astype(float)
    if vals_arr.size == 1:
        vals_arr = np.full(nodes_arr.shape, vals_arr.item(), dtype=float)
    if nodes_arr.size != vals_arr.size:
        raise ValueError(f"Mismatched nodes ({nodes_arr.size}) and values ({vals_arr.size}).")

    n_nodes = getattr(self.grid, "N", None)
    if n_nodes is None:
        raise AttributeError("SourceGrid must define total node count 'N'.")
    if np.any(nodes_arr < 0) or np.any(nodes_arr >= n_nodes):
        raise ValueError(f"Dirichlet indices out of [0, {n_nodes}): {nodes_arr}")

    for node, val in zip(nodes_arr, vals_arr):
        self.dirichlet[int(node)] = float(val)

add_neumann(nodes, outwardnormal, fluxes)

Add Neumann conditions du/dn = flux on grid nodes.

Parameters:

Name Type Description Default
nodes int or ndarray

Flattened source-node ids.

 required
outwardnormal tuple

Normal components (n_x, n_y) as scalars or arrays.

 required
fluxes float or ndarray

Scalar flux or one flux per node.

 required
Source code in src/gbmsc_pde/boundary/conditions.py 
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def add_neumann(
    self,
    nodes: Union[int, np.ndarray],
    outwardnormal: Union[Tuple[float, float], Tuple[np.ndarray, np.ndarray]],
    fluxes: Union[float, np.ndarray],
) -> None:
    """
    Add Neumann conditions ``du/dn = flux`` on grid nodes.

    Parameters
    ----------
    nodes : int or ndarray
        Flattened source-node ids.
    outwardnormal : tuple
        Normal components ``(n_x, n_y)`` as scalars or arrays.
    fluxes : float or ndarray
        Scalar flux or one flux per node.
    """
    nodes_arr = np.atleast_1d(nodes).astype(int)
    k = nodes_arr.size

    if not (isinstance(outwardnormal, tuple) and len(outwardnormal) == 2):
        raise TypeError("`outwardnormal` must be a tuple of length 2.")
    nx, ny = outwardnormal

    if np.isscalar(nx) and np.isscalar(ny):
        nx_arr = np.full(k, float(nx))
        ny_arr = np.full(k, float(ny))
    else:
        nx_arr = np.atleast_1d(nx).astype(float)
        ny_arr = np.atleast_1d(ny).astype(float)
    if nx_arr.shape != (k,) or ny_arr.shape != (k,):
        raise ValueError("`outwardnormal` components must match length of `nodes`.")

    flux_arr = np.atleast_1d(fluxes).astype(float)
    if flux_arr.size == 1:
        flux_arr = np.full(k, flux_arr.item())
    if flux_arr.shape != (k,):
        raise ValueError("`fluxes` must be scalar or match length of `nodes`.")

    n_nodes = getattr(self.grid, "N", None)
    if n_nodes is None:
        raise AttributeError("BoundaryConditions requires `self.grid.N`.")
    if np.any(nodes_arr < 0) or np.any(nodes_arr >= n_nodes):
        raise ValueError(f"Node indices must be in [0, {n_nodes}); got {nodes_arr}.")

    for node, nx_val, ny_val, flux_val in zip(nodes_arr, nx_arr, ny_arr, flux_arr):
        self.neumann[int(node)] = {
            "normal": (float(nx_val), float(ny_val)),
            "flux": float(flux_val),
        }

add_neumann_as_second_condition(nodes, outwardnormal, fluxes, rows)

Add Neumann equations on user-selected matrix rows.

This compatibility method is for augmented systems where a Neumann boundary equation should be imposed as an additional/second condition rather than replacing the row associated with node.

Source code in src/gbmsc_pde/boundary/conditions.py 
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def add_neumann_as_second_condition(
    self,
    nodes: Union[int, np.ndarray],
    outwardnormal: Union[Tuple[float, float], Tuple[np.ndarray, np.ndarray]],
    fluxes: Union[float, np.ndarray],
    rows: Union[int, np.ndarray],
) -> None:
    """
    Add Neumann equations on user-selected matrix rows.

    This compatibility method is for augmented systems where a Neumann
    boundary equation should be imposed as an additional/second condition
    rather than replacing the row associated with ``node``.
    """
    if not hasattr(self.grid, "N"):
        raise AttributeError("BoundaryConditions.grid must define integer `.N`")

    nodes_arr = np.atleast_1d(nodes).astype(int)
    rows_arr = np.atleast_1d(rows).astype(int)

    if nodes_arr.ndim != 1 or rows_arr.ndim != 1:
        raise TypeError("`nodes` and `rows` must be scalars or 1D sequences")

    n = nodes_arr.size
    if rows_arr.size != n:
        raise ValueError(f"`rows` length ({rows_arr.size}) != `nodes` length ({n})")

    if not (isinstance(outwardnormal, tuple) and len(outwardnormal) == 2):
        raise TypeError("`outwardnormal` must be a tuple of two elements")
    nx_in, ny_in = outwardnormal

    try:
        nx_arr = np.broadcast_to(nx_in, (n,)).astype(float)
        ny_arr = np.broadcast_to(ny_in, (n,)).astype(float)
    except Exception as exc:
        raise ValueError(
            "Cannot broadcast `outwardnormal` components to match `nodes` length"
        ) from exc

    try:
        flux_arr = np.broadcast_to(fluxes, (n,)).astype(float)
    except Exception as exc:
        raise ValueError("Cannot broadcast `fluxes` to match `nodes` length") from exc

    if np.any(nodes_arr < 0) or np.any(nodes_arr >= self.grid.N):
        raise ValueError(f"All `nodes` must be in [0, {self.grid.N})")

    for node, row, nx_val, ny_val, flux in zip(nodes_arr, rows_arr, nx_arr, ny_arr, flux_arr):
        self.second_neumann[int(node)] = {
            "normal": (float(nx_val), float(ny_val)),
            "flux": float(flux),
            "row": int(row),
        }

add_robin(nodes, outwardnormal, alphas, betas, values)

Add Robin conditions alpha*u + beta*du/dn = value on grid nodes.

alphas, betas, and values may be scalars or arrays with one entry per node.

Source code in src/gbmsc_pde/boundary/conditions.py 
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def add_robin(
    self,
    nodes: Union[int, np.ndarray],
    outwardnormal: Union[Tuple[float, float], Tuple[np.ndarray, np.ndarray]],
    alphas: Union[float, np.ndarray],
    betas: Union[float, np.ndarray],
    values: Union[float, np.ndarray],
) -> None:
    """
    Add Robin conditions ``alpha*u + beta*du/dn = value`` on grid nodes.

    ``alphas``, ``betas``, and ``values`` may be scalars or arrays with
    one entry per node.
    """
    nodes_arr = np.atleast_1d(nodes).astype(int)
    k = nodes_arr.size

    if not (isinstance(outwardnormal, tuple) and len(outwardnormal) == 2):
        raise TypeError("`outwardnormal` must be a tuple (nx, ny).")
    nx, ny = outwardnormal
    nx_arr = np.full(k, nx, dtype=float) if np.isscalar(nx) else np.atleast_1d(nx).astype(float)
    ny_arr = np.full(k, ny, dtype=float) if np.isscalar(ny) else np.atleast_1d(ny).astype(float)
    if nx_arr.shape != (k,) or ny_arr.shape != (k,):
        raise ValueError("`outwardnormal` components must match length of `nodes`.")

    alpha_arr = np.atleast_1d(alphas).astype(float)
    beta_arr = np.atleast_1d(betas).astype(float)
    value_arr = np.atleast_1d(values).astype(float)

    if alpha_arr.size == 1:
        alpha_arr = np.full(k, alpha_arr.item(), dtype=float)
    if beta_arr.size == 1:
        beta_arr = np.full(k, beta_arr.item(), dtype=float)
    if value_arr.size == 1:
        value_arr = np.full(k, value_arr.item(), dtype=float)

    if not (alpha_arr.size == beta_arr.size == value_arr.size == k):
        raise ValueError(
            f"Sizes must match: nodes({k}), alphas({alpha_arr.size}), "
            f"betas({beta_arr.size}), values({value_arr.size})."
        )

    n_nodes = getattr(self.grid, "N", None)
    if n_nodes is None:
        raise AttributeError("SourceGrid must define total node count 'N'.")
    if np.any(nodes_arr < 0) or np.any(nodes_arr >= n_nodes):
        raise ValueError(f"Node indices out of [0, {n_nodes}): {nodes_arr}")

    for node, nx_val, ny_val, alpha, beta, value in zip(
        nodes_arr, nx_arr, ny_arr, alpha_arr, beta_arr, value_arr
    ):
        self.robin[int(node)] = {
            "normal": (float(nx_val), float(ny_val)),
            "alpha": float(alpha),
            "beta": float(beta),
            "value": float(value),
        }

apply(A, b, Mx, My)

Apply all registered conditions to an assembled rectangular system.

Source code in src/gbmsc_pde/boundary/conditions.py 
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def apply(
    self,
    A: spmatrix,
    b: np.ndarray,
    Mx: spmatrix,
    My: spmatrix,
) -> tuple[spmatrix, np.ndarray]:
    """Apply all registered conditions to an assembled rectangular system."""
    return apply_boundary_conditions(self, A, b, Mx, My)

GBMSCApproximation

Grid-Based Shepard approximation on a structured grid.

The approximation blends local tensor-product Lagrange polynomials defined on sampled subgrids of a :class:~gbmsc_pde.source_grid.grid.SourceGrid. It is used both for interpolation at arbitrary coordinates and for sparse nodal differential operators in the rectangular-domain PDE solver.

For nodal PDE operators, support_mode controls the normalization:

"active" Use only sampled subgrids that contain the evaluated source node. This is the original sparse rectangular-domain PDE mode. "all" Keep the sparse active contributors at source nodes, but normalize the Shepard denominator over all sampled subgrids. "nodal_limit" Use the source-node limiting formula. The common singular factor is cancelled from all active subgrids containing the source node before evaluating the weight values and derivatives.

Interpolation at arbitrary query coordinates uses all sampled subgrids.

Attributes:

Name Type Description
grid SourceGrid

Structured source grid.
step tuple[int, int]

Sampling stride for local subgrids.
subgrids_step_row (ndarray, shape(S, m))

Flattened sampled local subgrid node ids, where m = n_x * n_y.
support_mode {active, all, nodal_limit}

Nodal operator normalization mode.
Source code in src/gbmsc_pde/approximation/shepard.py 
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class GBMSCApproximation:
    """
    Grid-Based Shepard approximation on a structured grid.

    The approximation blends local tensor-product Lagrange polynomials defined
    on sampled subgrids of a :class:`~gbmsc_pde.source_grid.grid.SourceGrid`.  It is used
    both for interpolation at arbitrary coordinates and for sparse nodal
    differential operators in the rectangular-domain PDE solver.

    For nodal PDE operators, ``support_mode`` controls the normalization:

    ``"active"``
        Use only sampled subgrids that contain the evaluated source node.  This
        is the original sparse rectangular-domain PDE mode.
    ``"all"``
        Keep the sparse active contributors at source nodes, but normalize the
        Shepard denominator over all sampled subgrids.
    ``"nodal_limit"``
        Use the source-node limiting formula.  The common singular factor is
        cancelled from all active subgrids containing the source node before
        evaluating the weight values and derivatives.

    Interpolation at arbitrary query coordinates uses all sampled subgrids.

    Attributes
    ----------
    grid : SourceGrid
        Structured source grid.
    step : tuple[int, int]
        Sampling stride for local subgrids.
    subgrids_step_row : ndarray, shape (S, m)
        Flattened sampled local subgrid node ids, where
        ``m = n_x * n_y``.
    support_mode : {"active", "all", "nodal_limit"}
        Nodal operator normalization mode.
    """

    def __init__(
        self,
        grid: SourceGrid,
        step: Tuple[int, int] = (1, 1),
        mu: float = 2.005,
        eps: float = 1e-12,
        support_mode: str = "active",
    ) -> None:
        """
        Initialize the Grid-Based Shepard approximation.

        Parameters
        ----------
        grid : SourceGrid
            Structured source grid providing source-node coordinates, local subgrids,
            and pairwise distances.
        step : tuple[int, int], default=(1, 1)
            Sampling stride for local subgrid starts.
        mu : float, default=2.005
            Shepard exponent parameter.  The implementation stores ``2*mu`` so
            formulas based on squared distances recover the requested power.
        eps : float, default=1e-12
            Distance floor used to avoid division by zero and logarithms of
            zero.
        support_mode : {"active", "all", "nodal_limit"}, default="active"
            Nodal PDE-operator support mode.

        Raises
        ------
        TypeError
            If ``grid`` does not provide local subgrids.
        ValueError
            If ``step`` is invalid, ``mu <= 2``, or ``support_mode`` is
            unknown.
        """
        # Input validation
        if not hasattr(grid, 'subgrids'):
            raise TypeError("grid must have attribute 'subgrids'.")
        if any(s < 1 or not isinstance(s, int) for s in step):
            raise ValueError("step must be a tuple of positive integers.")

        if mu <= 2:
            raise ValueError("mu parameter must be greater than 2.")
        if support_mode not in ("active", "all", "nodal_limit"):
            raise ValueError("support_mode must be 'active', 'all', or 'nodal_limit'.")

        self.grid = grid
        self.step = step
        self.mu = float(2 * mu)
        self.eps = float(eps)
        self.support_mode = support_mode

        # Unpack shapes
        n_x, n_y = grid.subgrid_shape

        # Sample subgrids at the given stride, then flatten for vectorized ops
        subgrids_step, self.subgrids_x_step, self.subgrids_y_step = grid.subgrids_with_step(self.step)
        self.subgrids_step_grid = subgrids_step.reshape(-1, n_x, n_y)    # (S, n_x, n_y)
        self.subgrids_step_row = self.subgrids_step_grid.reshape(-1, n_x * n_y)  # (S, m)
        self._subgrid_x_coords = self.grid.x_coords_flat[self.subgrids_step_row]
        self._subgrid_y_coords = self.grid.y_coords_flat[self.subgrids_step_row]
        self._x_lines = self.grid.X[self.subgrids_x_step, 0]
        self._y_lines = self.grid.Y[0, self.subgrids_y_step]
        self._x_lagrange_weights = _barycentric_weights_batch(self._x_lines)
        self._y_lagrange_weights = _barycentric_weights_batch(self._y_lines)
        self._lagrange_derivative_cache = {}
        self._node_subgrid_ids, self._node_subgrid_counts = self._build_node_subgrid_lookup()
        self._safe_dist = None
        self._safe_logd = None
        self._nodal_limit_data = None
        self._all_node_subgrid_metric_cache = {}

    def _build_node_subgrid_lookup(self) -> Tuple[np.ndarray, np.ndarray]:
        """
        Build a padded lookup from global node id to sampled subgrids containing it.

        Returns
        -------
        node_subgrid_ids : (N, C) ndarray(int)
            Padded subgrid ids for each global node. Unused slots are ``-1``.
        node_subgrid_counts : (N,) ndarray(int)
            Number of valid sampled subgrids per global node.
        """
        flat_nodes = self.subgrids_step_row.ravel()
        node_subgrid_counts = np.bincount(flat_nodes, minlength=self.grid.N)
        max_count = int(node_subgrid_counts.max()) if node_subgrid_counts.size else 0
        node_subgrid_ids = np.full((self.grid.N, max_count), -1, dtype=int)
        offsets = np.zeros(self.grid.N, dtype=int)

        subgrid_ids = np.repeat(np.arange(self.subgrids_step_row.shape[0]), self.subgrids_step_row.shape[1])
        for node_id, subgrid_id in zip(flat_nodes, subgrid_ids):
            slot = offsets[node_id]
            node_subgrid_ids[node_id, slot] = subgrid_id
            offsets[node_id] = slot + 1

        return node_subgrid_ids, node_subgrid_counts

    def _safe_pairwise_data(self) -> Tuple[np.ndarray, np.ndarray]:
        """
        Return cached pairwise squared distances and their logarithm with epsilon floor.
        """
        if self._safe_dist is None or self._safe_logd is None:
            safe_dist = np.maximum(self.grid.dist, self.eps)
            self._safe_dist = safe_dist
            self._safe_logd = np.log(safe_dist)
        return self._safe_dist, self._safe_logd

    def _nodal_limit_pairwise_data(self) -> Tuple[np.ndarray, ...]:
        """
        Return pairwise data for source-node limiting weights.

        At a source node, every active subgrid contains the common singular
        factor associated with the collocation node.  The limiting formula
        cancels that factor before differentiating the normalized weight.  In
        vectorized pairwise arrays this is represented by a neutral diagonal:
        ``dist_ii = 1`` and zero diagonal derivative contributions.
        """
        if self._nodal_limit_data is not None:
            return self._nodal_limit_data

        dist = self.grid.dist.copy()
        np.fill_diagonal(dist, 1.0)
        logd = np.log(dist)

        diff_x = self.grid.diff_x
        diff_y = self.grid.diff_y
        A_x_data = 2.0 * diff_x / dist
        A_y_data = 2.0 * diff_y / dist
        A_xx_data = 2.0 / dist - 4.0 * (diff_x / dist) ** 2
        A_yy_data = 2.0 / dist - 4.0 * (diff_y / dist) ** 2
        np.fill_diagonal(A_xx_data, 0.0)
        np.fill_diagonal(A_yy_data, 0.0)

        self._nodal_limit_data = (
            dist,
            logd,
            A_x_data,
            A_y_data,
            A_xx_data,
            A_yy_data,
        )
        return self._nodal_limit_data

    def _all_node_subgrid_metric_sums(self, second_derivatives: bool = True):
        """
        Precompute node-to-subgrid metric sums for all-denominator nodal rows.

        The active-denominator path only needs the subgrids containing each
        row node.  The all-denominator path needs every sampled subgrid in the
        normalizing sum; using these ``(N, S)`` tables avoids repeatedly
        building much larger ``(block, S, m)`` distance tensors.
        """
        cache_key = bool(second_derivatives)
        cached = self._all_node_subgrid_metric_cache.get(cache_key)
        if cached is not None:
            return cached

        dist, logd = self._safe_pairwise_data()
        idx_rows = self.subgrids_step_row
        S, m = idx_rows.shape

        indicator = np.zeros((self.grid.N, S), dtype=float)
        indicator[
            idx_rows.ravel(),
            np.repeat(np.arange(S, dtype=int), m),
        ] = 1.0

        diff_x = self.grid.diff_x
        diff_y = self.grid.diff_y
        A = logd @ indicator
        A_x = (2.0 * diff_x / dist) @ indicator
        A_y = (2.0 * diff_y / dist) @ indicator

        if not second_derivatives:
            result = (A, A_x, A_y)
            self._all_node_subgrid_metric_cache[cache_key] = result
            return result

        A_xx = (2.0 / dist - 4.0 * (diff_x / dist) ** 2) @ indicator
        A_yy = (2.0 / dist - 4.0 * (diff_y / dist) ** 2) @ indicator
        result = (A, A_x, A_y, A_xx, A_yy)
        self._all_node_subgrid_metric_cache[cache_key] = result
        return result

    def eval_weight_functions_at_nodes_all_denominator_minmax_shift(
        self,
        second_derivatives=True,
        block_s: int = 64,
    ) -> Union[Tuple[np.ndarray, np.ndarray, np.ndarray, np.ndarray, np.ndarray],
               Tuple[np.ndarray, np.ndarray, np.ndarray]
               ]:
        """
        Compute nodal Shepard weights with active contributors and all denominator.

        The returned arrays have the same ``(S, m)`` shape as
        :meth:`eval_weight_functions_at_nodes_data_minmax_shift`: one row for
        each retained/sampled subgrid and one column for each local node.  Only
        the normalization denominator is enlarged to all sampled subgrids.

        Parameters
        ----------
        second_derivatives : bool, default=True
            Whether to also return second derivatives of the weights.
        block_s : int, default=64
            Number of sampled subgrids processed per vectorized block.

        Returns
        -------
        tuple[ndarray, ...]
            ``(f, f_x, f_y, f_xx, f_yy)`` when ``second_derivatives`` is true,
            otherwise ``(f, f_x, f_y)``.  Each array has shape ``(S, m)``.
        """
        mu = self.mu * 0.5
        idx_rows = self.subgrids_step_row
        S, m = idx_rows.shape

        metrics = self._all_node_subgrid_metric_sums(second_derivatives=second_derivatives)
        if second_derivatives:
            A, A_x, A_y, A_xx, A_yy = metrics
        else:
            A, A_x, A_y = metrics

        f_val = np.empty((S, m))
        f_x = np.empty_like(f_val)
        f_y = np.empty_like(f_val)
        if second_derivatives:
            f_xx = np.empty_like(f_val)
            f_yy = np.empty_like(f_val)

        for s0 in range(0, S, block_s):
            s1 = min(s0 + block_s, S)
            blk = slice(s0, s1)
            B = s1 - s0

            flat_eval_nodes = idx_rows[blk].reshape(-1)
            current_subgrid_ids = np.repeat(np.arange(s0, s1, dtype=int), m)

            A_current = A[flat_eval_nodes, current_subgrid_ids]
            A_candidates = A[flat_eval_nodes]
            DeltaA = A_current[:, None] - A_candidates

            z = mu * DeltaA
            z_max = np.max(z, axis=1, keepdims=True)
            with np.errstate(under="ignore"):
                R_shift = np.exp(z - z_max)
            R_shift_sum = R_shift.sum(axis=1)
            normalized_R = R_shift / R_shift_sum[:, None]

            f_val_block = np.exp(-z_max.ravel()) / R_shift_sum

            A_x_current = A_x[flat_eval_nodes, current_subgrid_ids]
            A_y_current = A_y[flat_eval_nodes, current_subgrid_ids]
            beta_x = mu * (A_x_current[:, None] - A_x[flat_eval_nodes])
            beta_y = mu * (A_y_current[:, None] - A_y[flat_eval_nodes])
            log_derivative_x = (normalized_R * beta_x).sum(axis=1)
            log_derivative_y = (normalized_R * beta_y).sum(axis=1)

            f_val[blk] = f_val_block.reshape(B, m)
            f_x[blk] = (-f_val_block * log_derivative_x).reshape(B, m)
            f_y[blk] = (-f_val_block * log_derivative_y).reshape(B, m)

            if second_derivatives:
                A_xx_current = A_xx[flat_eval_nodes, current_subgrid_ids]
                A_yy_current = A_yy[flat_eval_nodes, current_subgrid_ids]
                beta_xx = mu * (A_xx_current[:, None] - A_xx[flat_eval_nodes])
                beta_yy = mu * (A_yy_current[:, None] - A_yy[flat_eval_nodes])
                curvature_x = (normalized_R * (beta_xx + beta_x ** 2)).sum(axis=1)
                curvature_y = (normalized_R * (beta_yy + beta_y ** 2)).sum(axis=1)

                f_xx[blk] = (
                    f_val_block * (2.0 * log_derivative_x ** 2 - curvature_x)
                ).reshape(B, m)
                f_yy[blk] = (
                    f_val_block * (2.0 * log_derivative_y ** 2 - curvature_y)
                ).reshape(B, m)

        if second_derivatives:
            return f_val, f_x, f_y, f_xx, f_yy

        return f_val, f_x, f_y

    def _normalized_weights_at_queries_stable(
        self,
        xq: np.ndarray,
        yq: np.ndarray,
        *,
        return_derivatives: bool = False,
        regularized: bool = True,
    ) -> np.ndarray:
        """
        Stable normalized Shepard weights computed in log-space.

        This variant never forms the raw weights ``W = exp(log_w)`` directly.
        Instead it applies a max-shift per query so the largest exponent is 0,
        then normalizes the shifted weights.

        Parameters
        ----------
        xq, yq : (l,) 1-D arrays
            Query coordinates.
        return_derivatives : bool, default False
            If True, also return derivatives of the normalized weights.

        Returns
        -------
        P : (l, S) ndarray
            Normalized Shepard weights for each query and sampled sub-grid.
        Px, Py : (l, S) ndarray
            First derivatives of the normalized weights (only if requested).
        """
        xq = np.asarray(xq, dtype=float).ravel()
        yq = np.asarray(yq, dtype=float).ravel()
        if xq.size != yq.size:
            raise ValueError("x and y arrays must have same length.")

        Xn = self._subgrid_x_coords
        Yn = self._subgrid_y_coords

        # Broadcasted differences (l, S, m)
        dx = xq[:, None, None] - Xn[None, :, :]
        dy = yq[:, None, None] - Yn[None, :, :]

        # Squared distance.  The legacy path uses an epsilon floor; the
        # exact-nodal interpolation path passes only non-source queries and
        # therefore uses true distances.
        dist2 = dx * dx + dy * dy
        if regularized:
            dist2[dist2 < self.eps] = self.eps
        elif np.any(dist2 <= 0.0):
            raise ZeroDivisionError(
                "Exact non-regularized interpolation received a source-node query. "
                "Use interpolation_mode='exact_nodal' so node hits are handled separately."
            )

        log_w = (-self.mu * 0.5) * np.log(dist2).sum(axis=2)  # (l, S)

        # Stable normalization: exp(log_w - max(log_w))
        log_w_shift = log_w - np.max(log_w, axis=1, keepdims=True)
        W_shift = np.exp(log_w_shift)
        W_sum = W_shift.sum(axis=1, keepdims=True)
        P = W_shift / W_sum

        if not return_derivatives:
            return (P,)

        # alpha_x / alpha_y are derivatives of log_w
        alpha_x = -self.mu * (dx / dist2).sum(axis=2)  # (l, S)
        alpha_y = -self.mu * (dy / dist2).sum(axis=2)  # (l, S)
        mean_alpha_x = (P * alpha_x).sum(axis=1, keepdims=True)
        mean_alpha_y = (P * alpha_y).sum(axis=1, keepdims=True)

        # Derivatives of normalized weights:
        # p_i,x = p_i * (alpha_i,x - sum_j p_j alpha_j,x)
        Px = P * (alpha_x - mean_alpha_x)
        Py = P * (alpha_y - mean_alpha_y)

        return P, Px, Py

    def _source_node_hits(
        self,
        xq: np.ndarray,
        yq: np.ndarray,
        *,
        tolerance: float,
        chunk: int,
    ) -> np.ndarray:
        """Return source-node ids for query points within tolerance, else -1."""
        hits = np.full(xq.shape, -1, dtype=int)
        coords = self.grid.coords
        tol2 = float(tolerance) ** 2
        chunk = max(int(chunk), 1)

        for q0 in range(0, xq.size, chunk):
            q1 = min(q0 + chunk, xq.size)
            dx = xq[q0:q1, None] - coords[None, :, 0]
            dy = yq[q0:q1, None] - coords[None, :, 1]
            dist2 = dx * dx + dy * dy
            nearest = np.argmin(dist2, axis=1)
            nearest_dist2 = dist2[np.arange(q1 - q0), nearest]
            mask = nearest_dist2 <= tol2
            hits[q0:q1][mask] = nearest[mask]

        return hits

    def interpolator(
        self,
        x: np.ndarray,
        y: np.ndarray,
        u: np.ndarray,
        *,
        return_derivatives: bool = False,
        chunk: int = 1000,
        interpolation_mode: str = "exact_nodal",
        tolerance: float = 1e-12,
        Mx=None,
        My=None,
    ) -> Union[
        Tuple[np.ndarray],
        Tuple[np.ndarray, np.ndarray, np.ndarray]
    ]:
        """
        Interpolate a source-node field at arbitrary coordinates.

        Query interpolation uses all sampled subgrids as contributors and
        normalizes the Shepard weights over all sampled subgrids.  Computation
        is chunked and performed in log space to avoid overflow in the raw
        weights.

        ``interpolation_mode="exact_nodal"`` returns exact nodal values when a
        query point coincides with a source node within ``tolerance``.  Other
        query points use true distances without an epsilon floor.

        ``interpolation_mode="regularized"`` preserves the legacy epsilon-floor
        behavior for all query points.

        Parameters
        ----------
        x, y : array_like, shape (n_eval,)
            Query coordinates.
        u : array_like, shape (grid.N,)
            Source-node values ordered like ``grid.coords``.
        return_derivatives : bool, default=False
            If true, also return first derivatives ``du/dx`` and ``du/dy``.
        chunk : int, default=1000
            Number of query points processed per block.
        interpolation_mode : {"exact_nodal", "regularized"}, default="exact_nodal"
            Query distance policy.
        tolerance : float, default=1e-12
            Source-node hit tolerance for ``"exact_nodal"`` mode.
        Mx, My : sparse matrices, optional
            Existing first-derivative operators.  When
            ``return_derivatives=True`` and source-node hits are present, these
            matrices are reused instead of rebuilding operators.

        Returns
        -------
        tuple
            ``(u_eval,)`` or ``(u_eval, u_x, u_y)``.
        """
        xq = np.asarray(x, dtype=float).ravel()
        yq = np.asarray(y, dtype=float).ravel()
        u = np.asarray(u, dtype=float).ravel()
        if xq.size != yq.size:
            raise ValueError("x and y arrays must have same length.")
        if u.size != self.grid.N:
            raise ValueError("`u` must have length grid.N")
        if interpolation_mode not in ("exact_nodal", "regularized"):
            raise ValueError("interpolation_mode must be 'exact_nodal' or 'regularized'.")
        if tolerance < 0:
            raise ValueError("tolerance must be non-negative.")

        l = xq.size
        chunk = max(chunk, 10)

        val = np.empty(l)
        if return_derivatives:
            ux = np.empty_like(val)
            uy = np.empty_like(val)

        hit_nodes = np.full(l, -1, dtype=int)
        if interpolation_mode == "exact_nodal":
            hit_nodes = self._source_node_hits(
                xq,
                yq,
                tolerance=tolerance,
                chunk=chunk,
            )
            hit_mask = hit_nodes != -1
            if np.any(hit_mask):
                val[hit_mask] = u[hit_nodes[hit_mask]]
                if return_derivatives:
                    if Mx is None or My is None:
                        from ..operators.differential import build_differential_operators

                        Mx_built, My_built, _, _ = build_differential_operators(self)
                        if Mx is None:
                            Mx = Mx_built
                        if My is None:
                            My = My_built
                    ux[hit_mask] = Mx.tocsr()[hit_nodes[hit_mask]].dot(u)
                    uy[hit_mask] = My.tocsr()[hit_nodes[hit_mask]].dot(u)

        # Nodal values per sub-grid
        U = u[self.subgrids_step_row]  # (S, m)

        x_lines = self._x_lines
        y_lines = self._y_lines
        x_weights = self._x_lagrange_weights
        y_weights = self._y_lagrange_weights

        Sx, nx = x_lines.shape
        Sy, ny = y_lines.shape
        m = nx * ny
        S = Sx * Sy

        for q0 in range(0, l, chunk):
            q1 = min(q0 + chunk, l)
            block_indices = np.arange(q0, q1)
            if interpolation_mode == "exact_nodal":
                block_indices = block_indices[hit_nodes[q0:q1] == -1]
                if block_indices.size == 0:
                    continue

            xs = xq[block_indices]
            ys = yq[block_indices]
            l_b = block_indices.size

            P_parts = self._normalized_weights_at_queries_stable(
                xs,
                ys,
                return_derivatives=return_derivatives,
                regularized=interpolation_mode == "regularized",
            )
            P = P_parts[0]
            if return_derivatives:
                Px, Py = P_parts[1], P_parts[2]

            Lx = np.empty((l_b, Sx, nx))
            Ly = np.empty((l_b, Sy, ny))
            if return_derivatives:
                dLx = np.empty_like(Lx)
                dLy = np.empty_like(Ly)

            for s in range(Sx):
                if return_derivatives:
                    Lx[:, s], dLx[:, s] = lagrange_1d(
                        x_lines[s],
                        xs,
                        with_first_derivatives=True,
                        barycentric_weights=x_weights[s],
                    )
                else:
                    Lx[:, s] = lagrange_1d(
                        x_lines[s],
                        xs,
                        with_first_derivatives=False,
                        barycentric_weights=x_weights[s],
                    )[0]

            for s in range(Sy):
                if return_derivatives:
                    Ly[:, s], dLy[:, s] = lagrange_1d(
                        y_lines[s],
                        ys,
                        with_first_derivatives=True,
                        barycentric_weights=y_weights[s],
                    )
                else:
                    Ly[:, s] = lagrange_1d(
                        y_lines[s],
                        ys,
                        with_first_derivatives=False,
                        barycentric_weights=y_weights[s],
                    )[0]

            Phi = (Lx[:, :, None, :, None] * Ly[:, None, :, None, :]).reshape(l_b, S, m)

            Li = np.einsum('qsm,sm->qs', Phi, U, optimize=True)
            val[block_indices] = (P * Li).sum(axis=1)

            if return_derivatives:
                Phi_x = (dLx[:, :, None, :, None] * Ly[:, None, :, None, :]).reshape(l_b, S, m)
                Phi_y = (Lx[:, :, None, :, None] * dLy[:, None, :, None, :]).reshape(l_b, S, m)
                Li_x = np.einsum('qsm,sm->qs', Phi_x, U, optimize=True)
                Li_y = np.einsum('qsm,sm->qs', Phi_y, U, optimize=True)
                ux[block_indices] = (P * Li_x + Px * Li).sum(axis=1)
                uy[block_indices] = (P * Li_y + Py * Li).sum(axis=1)

        if return_derivatives:
            return val, ux, uy
        return (val,)

    def lagrange_derivatives_x_direction(self):
        """Return repeated 1D Lagrange derivative matrices in the x direction."""
        l = len(self.subgrids_x_step) 
        (I, L1, L2) = self._lagrange_derivatives_one_subgrids_1dim(self.grid.subgrids_x_coords[0])
        return (I[None, :, :].repeat(l, axis=0), L1[None, :, :].repeat(l, axis=0), L2[None, :, :].repeat(l, axis=0))

    def lagrange_derivatives_y_direction(self):
        """Return repeated 1D Lagrange derivative matrices in the y direction."""
        (I, L1, L2) = self._lagrange_derivatives_one_subgrids_1dim(self.grid.subgrids_y_coords[0])
        l = len(self.subgrids_y_step)
        return (I[None, :, :].repeat(l, axis=0), L1[None, :, :].repeat(l, axis=0), L2[None, :, :].repeat(l, axis=0))

    def _lagrange_derivatives_one_subgrids_1dim(self, X: np.ndarray) -> Tuple[np.ndarray, np.ndarray, np.ndarray]:
        """
        Compute nodal 1D Lagrange derivative matrices.

        Parameters
        ----------
        X : ndarray, shape (m,)
            Distinct interpolation nodes.

        Returns
        -------
        I, L1, L2 : tuple[ndarray, ndarray, ndarray]
            Basis, first-derivative, and second-derivative matrices evaluated
            at the nodes ``X``.
        """
        X = np.asarray(X, dtype=float)
        cache_key = tuple(X.tolist())
        cached = self._lagrange_derivative_cache.get(cache_key)
        if cached is not None:
            return cached

        m = X.size
        diff = X[:, None] - X[None, :]
        diag = np.eye(m, dtype=bool)

        # Barycentric weights
        diff[diag] = 1.0
        w = 1.0 / np.prod(diff, axis=1)

        # First derivative
        with np.errstate(divide='ignore', invalid='ignore'):
            L1 = np.where(
                diag, 
                0.0, 
                (w[None, :] / (w[:, None] * diff))
            )
        L1[diag] = -np.sum(L1, axis=1)

        # Compute S_vector for second derivative
        offdiag = ~diag
        with np.errstate(divide='ignore', invalid='ignore'):
            inv_diff = np.where(offdiag, 1.0 / diff, 0.0)
        S_vec = np.sum(inv_diff, axis=1)

        # Second derivative
        with np.errstate(divide='ignore', invalid='ignore'):
            L2 = np.where(
                offdiag,
                2 * L1 * (S_vec[:, None] - 1.0 / diff),
                0.0
            )
        L2[diag] = -np.sum(L2, axis=1)

        result = (np.eye(m), L1, L2)
        self._lagrange_derivative_cache[cache_key] = result
        return result

    def eval_weight_functions_at_nodes_data_minmax_shift(
        self,
        second_derivatives=True,
        block_s: int = 128
    ) -> Union[Tuple[np.ndarray, np.ndarray, np.ndarray, np.ndarray, np.ndarray],
               Tuple[np.ndarray, np.ndarray, np.ndarray]
               ]:
        """
        Compute active-mode nodal Shepard weights and derivatives.

        For each source-node row, only subgrids containing that source node
        participate in both the contributors and the denominator.  The
        implementation uses shifted log-ratios,

            f_i = 1 / sum_j exp(z_j),  z_j = mu * (A_i - A_j)

        and evaluates ``exp(z_j - max(z))`` instead of directly forming
        ``exp(z_j)``.

        Parameters
        ----------
        second_derivatives : bool, default=True
            Whether to also return second derivatives of the weights.
        block_s : int, default=128
            Number of sampled subgrids processed per vectorized block.

        Returns
        -------
        tuple[ndarray, ...]
            ``(f, f_x, f_y, f_xx, f_yy)`` when ``second_derivatives`` is true,
            otherwise ``(f, f_x, f_y)``.  Each array has shape ``(S, m)``.
        """
        mu = self.mu * 0.5
        diff_x, diff_y = self.grid.diff_x, self.grid.diff_y
        dist, logd = self._safe_pairwise_data()

        idx_rows = self.subgrids_step_row
        S, m = idx_rows.shape

        f_val = np.empty((S, m))
        f_x = np.empty_like(f_val)
        f_y = np.empty_like(f_val)
        if second_derivatives:
            f_xx = np.empty_like(f_val)
            f_yy = np.empty_like(f_val)

        for s0 in range(0, S, block_s):
            s1 = min(s0 + block_s, S)
            blk = slice(s0, s1)
            B = s1 - s0

            rows = idx_rows[blk]
            flat_eval_nodes = rows.reshape(-1)
            current_subgrid_ids = np.repeat(np.arange(s0, s1), m)
            current_subgrid_nodes = idx_rows[current_subgrid_ids]

            current_row_index = flat_eval_nodes[:, None]
            current_logd = logd[current_row_index, current_subgrid_nodes]
            current_dx = diff_x[current_row_index, current_subgrid_nodes]
            current_dy = diff_y[current_row_index, current_subgrid_nodes]
            current_dist = dist[current_row_index, current_subgrid_nodes]

            A_current = current_logd.sum(axis=1)
            A_x_current = (2.0 * current_dx / current_dist).sum(axis=1)
            A_y_current = (2.0 * current_dy / current_dist).sum(axis=1)

            candidate_subgrid_ids = self._node_subgrid_ids[flat_eval_nodes]
            candidate_mask = candidate_subgrid_ids != -1
            safe_candidate_ids = np.where(candidate_mask, candidate_subgrid_ids, 0)
            candidate_nodes = idx_rows[safe_candidate_ids]

            row_index = flat_eval_nodes[:, None, None]
            candidate_logd = logd[row_index, candidate_nodes]
            candidate_dx = diff_x[row_index, candidate_nodes]
            candidate_dy = diff_y[row_index, candidate_nodes]
            candidate_dist = dist[row_index, candidate_nodes]

            A_candidates = candidate_logd.sum(axis=2)
            A_x_candidates = (2.0 * candidate_dx / candidate_dist).sum(axis=2)
            A_y_candidates = (2.0 * candidate_dy / candidate_dist).sum(axis=2)

            DeltaA = A_current[:, None] - A_candidates
            DeltaA_x = A_x_current[:, None] - A_x_candidates
            DeltaA_y = A_y_current[:, None] - A_y_candidates

            if second_derivatives:
                A_xx_current = (2.0 / current_dist - 4.0 * (current_dx / current_dist) ** 2).sum(axis=1)
                A_yy_current = (2.0 / current_dist - 4.0 * (current_dy / current_dist) ** 2).sum(axis=1)
                A_xx_candidates = (2.0 / candidate_dist - 4.0 * (candidate_dx / candidate_dist) ** 2).sum(axis=2)
                A_yy_candidates = (2.0 / candidate_dist - 4.0 * (candidate_dy / candidate_dist) ** 2).sum(axis=2)
                DeltaA_xx = A_xx_current[:, None] - A_xx_candidates
                DeltaA_yy = A_yy_current[:, None] - A_yy_candidates

            z = np.where(candidate_mask, mu * DeltaA, -np.inf)
            z_max = np.max(z, axis=1, keepdims=True)

            with np.errstate(under="ignore"):
                R_shift = np.exp(z - z_max)
            R_shift *= candidate_mask

            R_shift_sum = R_shift.sum(axis=1)
            normalized_R = R_shift / R_shift_sum[:, None]

            f_val_block = np.exp(-z_max.ravel()) / R_shift_sum

            beta_x = np.where(candidate_mask, mu * DeltaA_x, 0.0)
            beta_y = np.where(candidate_mask, mu * DeltaA_y, 0.0)
            log_derivative_x = (normalized_R * beta_x).sum(axis=1)
            log_derivative_y = (normalized_R * beta_y).sum(axis=1)

            f_x_block = -f_val_block * log_derivative_x
            f_y_block = -f_val_block * log_derivative_y

            f_val[blk] = f_val_block.reshape(B, m)
            f_x[blk] = f_x_block.reshape(B, m)
            f_y[blk] = f_y_block.reshape(B, m)

            if second_derivatives:
                beta_xx = np.where(candidate_mask, mu * DeltaA_xx, 0.0)
                beta_yy = np.where(candidate_mask, mu * DeltaA_yy, 0.0)
                curvature_x = (normalized_R * (beta_xx + beta_x ** 2)).sum(axis=1)
                curvature_y = (normalized_R * (beta_yy + beta_y ** 2)).sum(axis=1)

                f_xx[blk] = (
                    f_val_block * (2.0 * log_derivative_x ** 2 - curvature_x)
                ).reshape(B, m)
                f_yy[blk] = (
                    f_val_block * (2.0 * log_derivative_y ** 2 - curvature_y)
                ).reshape(B, m)

        if second_derivatives:
            return f_val, f_x, f_y, f_xx, f_yy

        return f_val, f_x, f_y

    def eval_weight_functions_at_nodes_nodal_limit(
        self,
        second_derivatives=True,
        block_s: int = 128
    ) -> Union[Tuple[np.ndarray, np.ndarray, np.ndarray, np.ndarray, np.ndarray],
               Tuple[np.ndarray, np.ndarray, np.ndarray]
               ]:
        """
        Compute source-node limiting Shepard weights and derivatives.

        For each source-node row, only subgrids containing that source node
        participate.  The common singular factor is cancelled analytically
        before forming the normalized Shepard fraction.  In the vectorized
        metric sums this is equivalent to using a neutral self-distance and
        zero self-contributions for first and second metric derivatives.
        """
        mu = self.mu * 0.5
        _, logd, A_x_data, A_y_data, A_xx_data, A_yy_data = self._nodal_limit_pairwise_data()

        idx_rows = self.subgrids_step_row
        S, m = idx_rows.shape

        f_val = np.empty((S, m))
        f_x = np.empty_like(f_val)
        f_y = np.empty_like(f_val)
        if second_derivatives:
            f_xx = np.empty_like(f_val)
            f_yy = np.empty_like(f_val)

        for s0 in range(0, S, block_s):
            s1 = min(s0 + block_s, S)
            blk = slice(s0, s1)
            B = s1 - s0

            rows = idx_rows[blk]
            flat_eval_nodes = rows.reshape(-1)
            current_subgrid_ids = np.repeat(np.arange(s0, s1), m)
            current_subgrid_nodes = idx_rows[current_subgrid_ids]

            current_row_index = flat_eval_nodes[:, None]
            current_logd = logd[current_row_index, current_subgrid_nodes]
            current_A_x = A_x_data[current_row_index, current_subgrid_nodes]
            current_A_y = A_y_data[current_row_index, current_subgrid_nodes]

            A_current = current_logd.sum(axis=1)
            A_x_current = current_A_x.sum(axis=1)
            A_y_current = current_A_y.sum(axis=1)

            candidate_subgrid_ids = self._node_subgrid_ids[flat_eval_nodes]
            candidate_mask = candidate_subgrid_ids != -1
            safe_candidate_ids = np.where(candidate_mask, candidate_subgrid_ids, 0)
            candidate_nodes = idx_rows[safe_candidate_ids]

            row_index = flat_eval_nodes[:, None, None]
            candidate_logd = logd[row_index, candidate_nodes]
            candidate_A_x = A_x_data[row_index, candidate_nodes]
            candidate_A_y = A_y_data[row_index, candidate_nodes]

            A_candidates = candidate_logd.sum(axis=2)
            A_x_candidates = candidate_A_x.sum(axis=2)
            A_y_candidates = candidate_A_y.sum(axis=2)

            DeltaA = A_current[:, None] - A_candidates
            DeltaA_x = A_x_current[:, None] - A_x_candidates
            DeltaA_y = A_y_current[:, None] - A_y_candidates

            if second_derivatives:
                current_A_xx = A_xx_data[current_row_index, current_subgrid_nodes]
                current_A_yy = A_yy_data[current_row_index, current_subgrid_nodes]
                candidate_A_xx = A_xx_data[row_index, candidate_nodes]
                candidate_A_yy = A_yy_data[row_index, candidate_nodes]
                A_xx_current = current_A_xx.sum(axis=1)
                A_yy_current = current_A_yy.sum(axis=1)
                A_xx_candidates = candidate_A_xx.sum(axis=2)
                A_yy_candidates = candidate_A_yy.sum(axis=2)
                DeltaA_xx = A_xx_current[:, None] - A_xx_candidates
                DeltaA_yy = A_yy_current[:, None] - A_yy_candidates

            z = np.where(candidate_mask, mu * DeltaA, -np.inf)
            z_max = np.max(z, axis=1, keepdims=True)

            with np.errstate(under="ignore"):
                R_shift = np.exp(z - z_max)
            R_shift *= candidate_mask

            R_shift_sum = R_shift.sum(axis=1)
            normalized_R = R_shift / R_shift_sum[:, None]

            f_val_block = np.exp(-z_max.ravel()) / R_shift_sum

            beta_x = np.where(candidate_mask, mu * DeltaA_x, 0.0)
            beta_y = np.where(candidate_mask, mu * DeltaA_y, 0.0)
            log_derivative_x = (normalized_R * beta_x).sum(axis=1)
            log_derivative_y = (normalized_R * beta_y).sum(axis=1)

            f_x_block = -f_val_block * log_derivative_x
            f_y_block = -f_val_block * log_derivative_y

            f_val[blk] = f_val_block.reshape(B, m)
            f_x[blk] = f_x_block.reshape(B, m)
            f_y[blk] = f_y_block.reshape(B, m)

            if second_derivatives:
                beta_xx = np.where(candidate_mask, mu * DeltaA_xx, 0.0)
                beta_yy = np.where(candidate_mask, mu * DeltaA_yy, 0.0)
                curvature_x = (normalized_R * (beta_xx + beta_x ** 2)).sum(axis=1)
                curvature_y = (normalized_R * (beta_yy + beta_y ** 2)).sum(axis=1)

                f_xx[blk] = (
                    f_val_block * (2.0 * log_derivative_x ** 2 - curvature_x)
                ).reshape(B, m)
                f_yy[blk] = (
                    f_val_block * (2.0 * log_derivative_y ** 2 - curvature_y)
                ).reshape(B, m)

        if second_derivatives:
            return f_val, f_x, f_y, f_xx, f_yy

        return f_val, f_x, f_y

__init__(grid, step=(1, 1), mu=2.005, eps=1e-12, support_mode='active')

Initialize the Grid-Based Shepard approximation.

Parameters:

Name Type Description Default
grid SourceGrid

Structured source grid providing source-node coordinates, local subgrids, and pairwise distances.

 required
step tuple[int, int]

Sampling stride for local subgrid starts.

 (1, 1)
mu float

Shepard exponent parameter. The implementation stores 2*mu so formulas based on squared distances recover the requested power.

 2.005
eps float

Distance floor used to avoid division by zero and logarithms of zero.

 1e-12
support_mode (active, all, nodal_limit)

Nodal PDE-operator support mode.

 "active"

Raises:

Type Description
TypeError

If grid does not provide local subgrids.
ValueError

If step is invalid, mu <= 2, or support_mode is unknown.
Source code in src/gbmsc_pde/approximation/shepard.py 
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def __init__(
    self,
    grid: SourceGrid,
    step: Tuple[int, int] = (1, 1),
    mu: float = 2.005,
    eps: float = 1e-12,
    support_mode: str = "active",
) -> None:
    """
    Initialize the Grid-Based Shepard approximation.

    Parameters
    ----------
    grid : SourceGrid
        Structured source grid providing source-node coordinates, local subgrids,
        and pairwise distances.
    step : tuple[int, int], default=(1, 1)
        Sampling stride for local subgrid starts.
    mu : float, default=2.005
        Shepard exponent parameter.  The implementation stores ``2*mu`` so
        formulas based on squared distances recover the requested power.
    eps : float, default=1e-12
        Distance floor used to avoid division by zero and logarithms of
        zero.
    support_mode : {"active", "all", "nodal_limit"}, default="active"
        Nodal PDE-operator support mode.

    Raises
    ------
    TypeError
        If ``grid`` does not provide local subgrids.
    ValueError
        If ``step`` is invalid, ``mu <= 2``, or ``support_mode`` is
        unknown.
    """
    # Input validation
    if not hasattr(grid, 'subgrids'):
        raise TypeError("grid must have attribute 'subgrids'.")
    if any(s < 1 or not isinstance(s, int) for s in step):
        raise ValueError("step must be a tuple of positive integers.")

    if mu <= 2:
        raise ValueError("mu parameter must be greater than 2.")
    if support_mode not in ("active", "all", "nodal_limit"):
        raise ValueError("support_mode must be 'active', 'all', or 'nodal_limit'.")

    self.grid = grid
    self.step = step
    self.mu = float(2 * mu)
    self.eps = float(eps)
    self.support_mode = support_mode

    # Unpack shapes
    n_x, n_y = grid.subgrid_shape

    # Sample subgrids at the given stride, then flatten for vectorized ops
    subgrids_step, self.subgrids_x_step, self.subgrids_y_step = grid.subgrids_with_step(self.step)
    self.subgrids_step_grid = subgrids_step.reshape(-1, n_x, n_y)    # (S, n_x, n_y)
    self.subgrids_step_row = self.subgrids_step_grid.reshape(-1, n_x * n_y)  # (S, m)
    self._subgrid_x_coords = self.grid.x_coords_flat[self.subgrids_step_row]
    self._subgrid_y_coords = self.grid.y_coords_flat[self.subgrids_step_row]
    self._x_lines = self.grid.X[self.subgrids_x_step, 0]
    self._y_lines = self.grid.Y[0, self.subgrids_y_step]
    self._x_lagrange_weights = _barycentric_weights_batch(self._x_lines)
    self._y_lagrange_weights = _barycentric_weights_batch(self._y_lines)
    self._lagrange_derivative_cache = {}
    self._node_subgrid_ids, self._node_subgrid_counts = self._build_node_subgrid_lookup()
    self._safe_dist = None
    self._safe_logd = None
    self._nodal_limit_data = None
    self._all_node_subgrid_metric_cache = {}

eval_weight_functions_at_nodes_all_denominator_minmax_shift(second_derivatives=True, block_s=64)

Compute nodal Shepard weights with active contributors and all denominator.

The returned arrays have the same (S, m) shape as :meth:eval_weight_functions_at_nodes_data_minmax_shift: one row for each retained/sampled subgrid and one column for each local node. Only the normalization denominator is enlarged to all sampled subgrids.

Parameters:

Name Type Description Default
second_derivatives bool

Whether to also return second derivatives of the weights.

 True
block_s int

Number of sampled subgrids processed per vectorized block.

 64

Returns:

Type Description
tuple[ndarray, ...]

(f, f_x, f_y, f_xx, f_yy) when second_derivatives is true, otherwise (f, f_x, f_y). Each array has shape (S, m).
Source code in src/gbmsc_pde/approximation/shepard.py 
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def eval_weight_functions_at_nodes_all_denominator_minmax_shift(
    self,
    second_derivatives=True,
    block_s: int = 64,
) -> Union[Tuple[np.ndarray, np.ndarray, np.ndarray, np.ndarray, np.ndarray],
           Tuple[np.ndarray, np.ndarray, np.ndarray]
           ]:
    """
    Compute nodal Shepard weights with active contributors and all denominator.

    The returned arrays have the same ``(S, m)`` shape as
    :meth:`eval_weight_functions_at_nodes_data_minmax_shift`: one row for
    each retained/sampled subgrid and one column for each local node.  Only
    the normalization denominator is enlarged to all sampled subgrids.

    Parameters
    ----------
    second_derivatives : bool, default=True
        Whether to also return second derivatives of the weights.
    block_s : int, default=64
        Number of sampled subgrids processed per vectorized block.

    Returns
    -------
    tuple[ndarray, ...]
        ``(f, f_x, f_y, f_xx, f_yy)`` when ``second_derivatives`` is true,
        otherwise ``(f, f_x, f_y)``.  Each array has shape ``(S, m)``.
    """
    mu = self.mu * 0.5
    idx_rows = self.subgrids_step_row
    S, m = idx_rows.shape

    metrics = self._all_node_subgrid_metric_sums(second_derivatives=second_derivatives)
    if second_derivatives:
        A, A_x, A_y, A_xx, A_yy = metrics
    else:
        A, A_x, A_y = metrics

    f_val = np.empty((S, m))
    f_x = np.empty_like(f_val)
    f_y = np.empty_like(f_val)
    if second_derivatives:
        f_xx = np.empty_like(f_val)
        f_yy = np.empty_like(f_val)

    for s0 in range(0, S, block_s):
        s1 = min(s0 + block_s, S)
        blk = slice(s0, s1)
        B = s1 - s0

        flat_eval_nodes = idx_rows[blk].reshape(-1)
        current_subgrid_ids = np.repeat(np.arange(s0, s1, dtype=int), m)

        A_current = A[flat_eval_nodes, current_subgrid_ids]
        A_candidates = A[flat_eval_nodes]
        DeltaA = A_current[:, None] - A_candidates

        z = mu * DeltaA
        z_max = np.max(z, axis=1, keepdims=True)
        with np.errstate(under="ignore"):
            R_shift = np.exp(z - z_max)
        R_shift_sum = R_shift.sum(axis=1)
        normalized_R = R_shift / R_shift_sum[:, None]

        f_val_block = np.exp(-z_max.ravel()) / R_shift_sum

        A_x_current = A_x[flat_eval_nodes, current_subgrid_ids]
        A_y_current = A_y[flat_eval_nodes, current_subgrid_ids]
        beta_x = mu * (A_x_current[:, None] - A_x[flat_eval_nodes])
        beta_y = mu * (A_y_current[:, None] - A_y[flat_eval_nodes])
        log_derivative_x = (normalized_R * beta_x).sum(axis=1)
        log_derivative_y = (normalized_R * beta_y).sum(axis=1)

        f_val[blk] = f_val_block.reshape(B, m)
        f_x[blk] = (-f_val_block * log_derivative_x).reshape(B, m)
        f_y[blk] = (-f_val_block * log_derivative_y).reshape(B, m)

        if second_derivatives:
            A_xx_current = A_xx[flat_eval_nodes, current_subgrid_ids]
            A_yy_current = A_yy[flat_eval_nodes, current_subgrid_ids]
            beta_xx = mu * (A_xx_current[:, None] - A_xx[flat_eval_nodes])
            beta_yy = mu * (A_yy_current[:, None] - A_yy[flat_eval_nodes])
            curvature_x = (normalized_R * (beta_xx + beta_x ** 2)).sum(axis=1)
            curvature_y = (normalized_R * (beta_yy + beta_y ** 2)).sum(axis=1)

            f_xx[blk] = (
                f_val_block * (2.0 * log_derivative_x ** 2 - curvature_x)
            ).reshape(B, m)
            f_yy[blk] = (
                f_val_block * (2.0 * log_derivative_y ** 2 - curvature_y)
            ).reshape(B, m)

    if second_derivatives:
        return f_val, f_x, f_y, f_xx, f_yy

    return f_val, f_x, f_y

eval_weight_functions_at_nodes_data_minmax_shift(second_derivatives=True, block_s=128)

Compute active-mode nodal Shepard weights and derivatives.

For each source-node row, only subgrids containing that source node participate in both the contributors and the denominator. The implementation uses shifted log-ratios,

f_i = 1 / sum_j exp(z_j),  z_j = mu * (A_i - A_j)

and evaluates exp(z_j - max(z)) instead of directly forming exp(z_j).

Parameters:

Name Type Description Default
second_derivatives bool

Whether to also return second derivatives of the weights.

 True
block_s int

Number of sampled subgrids processed per vectorized block.

 128

Returns:

Type Description
tuple[ndarray, ...]

(f, f_x, f_y, f_xx, f_yy) when second_derivatives is true, otherwise (f, f_x, f_y). Each array has shape (S, m).
Source code in src/gbmsc_pde/approximation/shepard.py 
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def eval_weight_functions_at_nodes_data_minmax_shift(
    self,
    second_derivatives=True,
    block_s: int = 128
) -> Union[Tuple[np.ndarray, np.ndarray, np.ndarray, np.ndarray, np.ndarray],
           Tuple[np.ndarray, np.ndarray, np.ndarray]
           ]:
    """
    Compute active-mode nodal Shepard weights and derivatives.

    For each source-node row, only subgrids containing that source node
    participate in both the contributors and the denominator.  The
    implementation uses shifted log-ratios,

        f_i = 1 / sum_j exp(z_j),  z_j = mu * (A_i - A_j)

    and evaluates ``exp(z_j - max(z))`` instead of directly forming
    ``exp(z_j)``.

    Parameters
    ----------
    second_derivatives : bool, default=True
        Whether to also return second derivatives of the weights.
    block_s : int, default=128
        Number of sampled subgrids processed per vectorized block.

    Returns
    -------
    tuple[ndarray, ...]
        ``(f, f_x, f_y, f_xx, f_yy)`` when ``second_derivatives`` is true,
        otherwise ``(f, f_x, f_y)``.  Each array has shape ``(S, m)``.
    """
    mu = self.mu * 0.5
    diff_x, diff_y = self.grid.diff_x, self.grid.diff_y
    dist, logd = self._safe_pairwise_data()

    idx_rows = self.subgrids_step_row
    S, m = idx_rows.shape

    f_val = np.empty((S, m))
    f_x = np.empty_like(f_val)
    f_y = np.empty_like(f_val)
    if second_derivatives:
        f_xx = np.empty_like(f_val)
        f_yy = np.empty_like(f_val)

    for s0 in range(0, S, block_s):
        s1 = min(s0 + block_s, S)
        blk = slice(s0, s1)
        B = s1 - s0

        rows = idx_rows[blk]
        flat_eval_nodes = rows.reshape(-1)
        current_subgrid_ids = np.repeat(np.arange(s0, s1), m)
        current_subgrid_nodes = idx_rows[current_subgrid_ids]

        current_row_index = flat_eval_nodes[:, None]
        current_logd = logd[current_row_index, current_subgrid_nodes]
        current_dx = diff_x[current_row_index, current_subgrid_nodes]
        current_dy = diff_y[current_row_index, current_subgrid_nodes]
        current_dist = dist[current_row_index, current_subgrid_nodes]

        A_current = current_logd.sum(axis=1)
        A_x_current = (2.0 * current_dx / current_dist).sum(axis=1)
        A_y_current = (2.0 * current_dy / current_dist).sum(axis=1)

        candidate_subgrid_ids = self._node_subgrid_ids[flat_eval_nodes]
        candidate_mask = candidate_subgrid_ids != -1
        safe_candidate_ids = np.where(candidate_mask, candidate_subgrid_ids, 0)
        candidate_nodes = idx_rows[safe_candidate_ids]

        row_index = flat_eval_nodes[:, None, None]
        candidate_logd = logd[row_index, candidate_nodes]
        candidate_dx = diff_x[row_index, candidate_nodes]
        candidate_dy = diff_y[row_index, candidate_nodes]
        candidate_dist = dist[row_index, candidate_nodes]

        A_candidates = candidate_logd.sum(axis=2)
        A_x_candidates = (2.0 * candidate_dx / candidate_dist).sum(axis=2)
        A_y_candidates = (2.0 * candidate_dy / candidate_dist).sum(axis=2)

        DeltaA = A_current[:, None] - A_candidates
        DeltaA_x = A_x_current[:, None] - A_x_candidates
        DeltaA_y = A_y_current[:, None] - A_y_candidates

        if second_derivatives:
            A_xx_current = (2.0 / current_dist - 4.0 * (current_dx / current_dist) ** 2).sum(axis=1)
            A_yy_current = (2.0 / current_dist - 4.0 * (current_dy / current_dist) ** 2).sum(axis=1)
            A_xx_candidates = (2.0 / candidate_dist - 4.0 * (candidate_dx / candidate_dist) ** 2).sum(axis=2)
            A_yy_candidates = (2.0 / candidate_dist - 4.0 * (candidate_dy / candidate_dist) ** 2).sum(axis=2)
            DeltaA_xx = A_xx_current[:, None] - A_xx_candidates
            DeltaA_yy = A_yy_current[:, None] - A_yy_candidates

        z = np.where(candidate_mask, mu * DeltaA, -np.inf)
        z_max = np.max(z, axis=1, keepdims=True)

        with np.errstate(under="ignore"):
            R_shift = np.exp(z - z_max)
        R_shift *= candidate_mask

        R_shift_sum = R_shift.sum(axis=1)
        normalized_R = R_shift / R_shift_sum[:, None]

        f_val_block = np.exp(-z_max.ravel()) / R_shift_sum

        beta_x = np.where(candidate_mask, mu * DeltaA_x, 0.0)
        beta_y = np.where(candidate_mask, mu * DeltaA_y, 0.0)
        log_derivative_x = (normalized_R * beta_x).sum(axis=1)
        log_derivative_y = (normalized_R * beta_y).sum(axis=1)

        f_x_block = -f_val_block * log_derivative_x
        f_y_block = -f_val_block * log_derivative_y

        f_val[blk] = f_val_block.reshape(B, m)
        f_x[blk] = f_x_block.reshape(B, m)
        f_y[blk] = f_y_block.reshape(B, m)

        if second_derivatives:
            beta_xx = np.where(candidate_mask, mu * DeltaA_xx, 0.0)
            beta_yy = np.where(candidate_mask, mu * DeltaA_yy, 0.0)
            curvature_x = (normalized_R * (beta_xx + beta_x ** 2)).sum(axis=1)
            curvature_y = (normalized_R * (beta_yy + beta_y ** 2)).sum(axis=1)

            f_xx[blk] = (
                f_val_block * (2.0 * log_derivative_x ** 2 - curvature_x)
            ).reshape(B, m)
            f_yy[blk] = (
                f_val_block * (2.0 * log_derivative_y ** 2 - curvature_y)
            ).reshape(B, m)

    if second_derivatives:
        return f_val, f_x, f_y, f_xx, f_yy

    return f_val, f_x, f_y

eval_weight_functions_at_nodes_nodal_limit(second_derivatives=True, block_s=128)

Compute source-node limiting Shepard weights and derivatives.

For each source-node row, only subgrids containing that source node participate. The common singular factor is cancelled analytically before forming the normalized Shepard fraction. In the vectorized metric sums this is equivalent to using a neutral self-distance and zero self-contributions for first and second metric derivatives.

Source code in src/gbmsc_pde/approximation/shepard.py 
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def eval_weight_functions_at_nodes_nodal_limit(
    self,
    second_derivatives=True,
    block_s: int = 128
) -> Union[Tuple[np.ndarray, np.ndarray, np.ndarray, np.ndarray, np.ndarray],
           Tuple[np.ndarray, np.ndarray, np.ndarray]
           ]:
    """
    Compute source-node limiting Shepard weights and derivatives.

    For each source-node row, only subgrids containing that source node
    participate.  The common singular factor is cancelled analytically
    before forming the normalized Shepard fraction.  In the vectorized
    metric sums this is equivalent to using a neutral self-distance and
    zero self-contributions for first and second metric derivatives.
    """
    mu = self.mu * 0.5
    _, logd, A_x_data, A_y_data, A_xx_data, A_yy_data = self._nodal_limit_pairwise_data()

    idx_rows = self.subgrids_step_row
    S, m = idx_rows.shape

    f_val = np.empty((S, m))
    f_x = np.empty_like(f_val)
    f_y = np.empty_like(f_val)
    if second_derivatives:
        f_xx = np.empty_like(f_val)
        f_yy = np.empty_like(f_val)

    for s0 in range(0, S, block_s):
        s1 = min(s0 + block_s, S)
        blk = slice(s0, s1)
        B = s1 - s0

        rows = idx_rows[blk]
        flat_eval_nodes = rows.reshape(-1)
        current_subgrid_ids = np.repeat(np.arange(s0, s1), m)
        current_subgrid_nodes = idx_rows[current_subgrid_ids]

        current_row_index = flat_eval_nodes[:, None]
        current_logd = logd[current_row_index, current_subgrid_nodes]
        current_A_x = A_x_data[current_row_index, current_subgrid_nodes]
        current_A_y = A_y_data[current_row_index, current_subgrid_nodes]

        A_current = current_logd.sum(axis=1)
        A_x_current = current_A_x.sum(axis=1)
        A_y_current = current_A_y.sum(axis=1)

        candidate_subgrid_ids = self._node_subgrid_ids[flat_eval_nodes]
        candidate_mask = candidate_subgrid_ids != -1
        safe_candidate_ids = np.where(candidate_mask, candidate_subgrid_ids, 0)
        candidate_nodes = idx_rows[safe_candidate_ids]

        row_index = flat_eval_nodes[:, None, None]
        candidate_logd = logd[row_index, candidate_nodes]
        candidate_A_x = A_x_data[row_index, candidate_nodes]
        candidate_A_y = A_y_data[row_index, candidate_nodes]

        A_candidates = candidate_logd.sum(axis=2)
        A_x_candidates = candidate_A_x.sum(axis=2)
        A_y_candidates = candidate_A_y.sum(axis=2)

        DeltaA = A_current[:, None] - A_candidates
        DeltaA_x = A_x_current[:, None] - A_x_candidates
        DeltaA_y = A_y_current[:, None] - A_y_candidates

        if second_derivatives:
            current_A_xx = A_xx_data[current_row_index, current_subgrid_nodes]
            current_A_yy = A_yy_data[current_row_index, current_subgrid_nodes]
            candidate_A_xx = A_xx_data[row_index, candidate_nodes]
            candidate_A_yy = A_yy_data[row_index, candidate_nodes]
            A_xx_current = current_A_xx.sum(axis=1)
            A_yy_current = current_A_yy.sum(axis=1)
            A_xx_candidates = candidate_A_xx.sum(axis=2)
            A_yy_candidates = candidate_A_yy.sum(axis=2)
            DeltaA_xx = A_xx_current[:, None] - A_xx_candidates
            DeltaA_yy = A_yy_current[:, None] - A_yy_candidates

        z = np.where(candidate_mask, mu * DeltaA, -np.inf)
        z_max = np.max(z, axis=1, keepdims=True)

        with np.errstate(under="ignore"):
            R_shift = np.exp(z - z_max)
        R_shift *= candidate_mask

        R_shift_sum = R_shift.sum(axis=1)
        normalized_R = R_shift / R_shift_sum[:, None]

        f_val_block = np.exp(-z_max.ravel()) / R_shift_sum

        beta_x = np.where(candidate_mask, mu * DeltaA_x, 0.0)
        beta_y = np.where(candidate_mask, mu * DeltaA_y, 0.0)
        log_derivative_x = (normalized_R * beta_x).sum(axis=1)
        log_derivative_y = (normalized_R * beta_y).sum(axis=1)

        f_x_block = -f_val_block * log_derivative_x
        f_y_block = -f_val_block * log_derivative_y

        f_val[blk] = f_val_block.reshape(B, m)
        f_x[blk] = f_x_block.reshape(B, m)
        f_y[blk] = f_y_block.reshape(B, m)

        if second_derivatives:
            beta_xx = np.where(candidate_mask, mu * DeltaA_xx, 0.0)
            beta_yy = np.where(candidate_mask, mu * DeltaA_yy, 0.0)
            curvature_x = (normalized_R * (beta_xx + beta_x ** 2)).sum(axis=1)
            curvature_y = (normalized_R * (beta_yy + beta_y ** 2)).sum(axis=1)

            f_xx[blk] = (
                f_val_block * (2.0 * log_derivative_x ** 2 - curvature_x)
            ).reshape(B, m)
            f_yy[blk] = (
                f_val_block * (2.0 * log_derivative_y ** 2 - curvature_y)
            ).reshape(B, m)

    if second_derivatives:
        return f_val, f_x, f_y, f_xx, f_yy

    return f_val, f_x, f_y

interpolator(x, y, u, *, return_derivatives=False, chunk=1000, interpolation_mode='exact_nodal', tolerance=1e-12, Mx=None, My=None)

Interpolate a source-node field at arbitrary coordinates.

Query interpolation uses all sampled subgrids as contributors and normalizes the Shepard weights over all sampled subgrids. Computation is chunked and performed in log space to avoid overflow in the raw weights.

interpolation_mode="exact_nodal" returns exact nodal values when a query point coincides with a source node within tolerance. Other query points use true distances without an epsilon floor.

interpolation_mode="regularized" preserves the legacy epsilon-floor behavior for all query points.

Parameters:

Name Type Description Default
x (array_like, shape(n_eval))

Query coordinates.

 required
y (array_like, shape(n_eval))

Query coordinates.

 required
u (array_like, shape(N))

Source-node values ordered like grid.coords.

 required
return_derivatives bool

If true, also return first derivatives du/dx and du/dy.

 False
chunk int

Number of query points processed per block.

 1000
interpolation_mode (exact_nodal, regularized)

Query distance policy.

 "exact_nodal"
tolerance float

Source-node hit tolerance for "exact_nodal" mode.

 1e-12
Mx sparse matrices

Existing first-derivative operators. When return_derivatives=True and source-node hits are present, these matrices are reused instead of rebuilding operators.

 None
My sparse matrices

Existing first-derivative operators. When return_derivatives=True and source-node hits are present, these matrices are reused instead of rebuilding operators.

 None

Returns:

Type Description
tuple

(u_eval,) or (u_eval, u_x, u_y).
Source code in src/gbmsc_pde/approximation/shepard.py 
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def interpolator(
    self,
    x: np.ndarray,
    y: np.ndarray,
    u: np.ndarray,
    *,
    return_derivatives: bool = False,
    chunk: int = 1000,
    interpolation_mode: str = "exact_nodal",
    tolerance: float = 1e-12,
    Mx=None,
    My=None,
) -> Union[
    Tuple[np.ndarray],
    Tuple[np.ndarray, np.ndarray, np.ndarray]
]:
    """
    Interpolate a source-node field at arbitrary coordinates.

    Query interpolation uses all sampled subgrids as contributors and
    normalizes the Shepard weights over all sampled subgrids.  Computation
    is chunked and performed in log space to avoid overflow in the raw
    weights.

    ``interpolation_mode="exact_nodal"`` returns exact nodal values when a
    query point coincides with a source node within ``tolerance``.  Other
    query points use true distances without an epsilon floor.

    ``interpolation_mode="regularized"`` preserves the legacy epsilon-floor
    behavior for all query points.

    Parameters
    ----------
    x, y : array_like, shape (n_eval,)
        Query coordinates.
    u : array_like, shape (grid.N,)
        Source-node values ordered like ``grid.coords``.
    return_derivatives : bool, default=False
        If true, also return first derivatives ``du/dx`` and ``du/dy``.
    chunk : int, default=1000
        Number of query points processed per block.
    interpolation_mode : {"exact_nodal", "regularized"}, default="exact_nodal"
        Query distance policy.
    tolerance : float, default=1e-12
        Source-node hit tolerance for ``"exact_nodal"`` mode.
    Mx, My : sparse matrices, optional
        Existing first-derivative operators.  When
        ``return_derivatives=True`` and source-node hits are present, these
        matrices are reused instead of rebuilding operators.

    Returns
    -------
    tuple
        ``(u_eval,)`` or ``(u_eval, u_x, u_y)``.
    """
    xq = np.asarray(x, dtype=float).ravel()
    yq = np.asarray(y, dtype=float).ravel()
    u = np.asarray(u, dtype=float).ravel()
    if xq.size != yq.size:
        raise ValueError("x and y arrays must have same length.")
    if u.size != self.grid.N:
        raise ValueError("`u` must have length grid.N")
    if interpolation_mode not in ("exact_nodal", "regularized"):
        raise ValueError("interpolation_mode must be 'exact_nodal' or 'regularized'.")
    if tolerance < 0:
        raise ValueError("tolerance must be non-negative.")

    l = xq.size
    chunk = max(chunk, 10)

    val = np.empty(l)
    if return_derivatives:
        ux = np.empty_like(val)
        uy = np.empty_like(val)

    hit_nodes = np.full(l, -1, dtype=int)
    if interpolation_mode == "exact_nodal":
        hit_nodes = self._source_node_hits(
            xq,
            yq,
            tolerance=tolerance,
            chunk=chunk,
        )
        hit_mask = hit_nodes != -1
        if np.any(hit_mask):
            val[hit_mask] = u[hit_nodes[hit_mask]]
            if return_derivatives:
                if Mx is None or My is None:
                    from ..operators.differential import build_differential_operators

                    Mx_built, My_built, _, _ = build_differential_operators(self)
                    if Mx is None:
                        Mx = Mx_built
                    if My is None:
                        My = My_built
                ux[hit_mask] = Mx.tocsr()[hit_nodes[hit_mask]].dot(u)
                uy[hit_mask] = My.tocsr()[hit_nodes[hit_mask]].dot(u)

    # Nodal values per sub-grid
    U = u[self.subgrids_step_row]  # (S, m)

    x_lines = self._x_lines
    y_lines = self._y_lines
    x_weights = self._x_lagrange_weights
    y_weights = self._y_lagrange_weights

    Sx, nx = x_lines.shape
    Sy, ny = y_lines.shape
    m = nx * ny
    S = Sx * Sy

    for q0 in range(0, l, chunk):
        q1 = min(q0 + chunk, l)
        block_indices = np.arange(q0, q1)
        if interpolation_mode == "exact_nodal":
            block_indices = block_indices[hit_nodes[q0:q1] == -1]
            if block_indices.size == 0:
                continue

        xs = xq[block_indices]
        ys = yq[block_indices]
        l_b = block_indices.size

        P_parts = self._normalized_weights_at_queries_stable(
            xs,
            ys,
            return_derivatives=return_derivatives,
            regularized=interpolation_mode == "regularized",
        )
        P = P_parts[0]
        if return_derivatives:
            Px, Py = P_parts[1], P_parts[2]

        Lx = np.empty((l_b, Sx, nx))
        Ly = np.empty((l_b, Sy, ny))
        if return_derivatives:
            dLx = np.empty_like(Lx)
            dLy = np.empty_like(Ly)

        for s in range(Sx):
            if return_derivatives:
                Lx[:, s], dLx[:, s] = lagrange_1d(
                    x_lines[s],
                    xs,
                    with_first_derivatives=True,
                    barycentric_weights=x_weights[s],
                )
            else:
                Lx[:, s] = lagrange_1d(
                    x_lines[s],
                    xs,
                    with_first_derivatives=False,
                    barycentric_weights=x_weights[s],
                )[0]

        for s in range(Sy):
            if return_derivatives:
                Ly[:, s], dLy[:, s] = lagrange_1d(
                    y_lines[s],
                    ys,
                    with_first_derivatives=True,
                    barycentric_weights=y_weights[s],
                )
            else:
                Ly[:, s] = lagrange_1d(
                    y_lines[s],
                    ys,
                    with_first_derivatives=False,
                    barycentric_weights=y_weights[s],
                )[0]

        Phi = (Lx[:, :, None, :, None] * Ly[:, None, :, None, :]).reshape(l_b, S, m)

        Li = np.einsum('qsm,sm->qs', Phi, U, optimize=True)
        val[block_indices] = (P * Li).sum(axis=1)

        if return_derivatives:
            Phi_x = (dLx[:, :, None, :, None] * Ly[:, None, :, None, :]).reshape(l_b, S, m)
            Phi_y = (Lx[:, :, None, :, None] * dLy[:, None, :, None, :]).reshape(l_b, S, m)
            Li_x = np.einsum('qsm,sm->qs', Phi_x, U, optimize=True)
            Li_y = np.einsum('qsm,sm->qs', Phi_y, U, optimize=True)
            ux[block_indices] = (P * Li_x + Px * Li).sum(axis=1)
            uy[block_indices] = (P * Li_y + Py * Li).sum(axis=1)

    if return_derivatives:
        return val, ux, uy
    return (val,)

lagrange_derivatives_x_direction()

Return repeated 1D Lagrange derivative matrices in the x direction.

Source code in src/gbmsc_pde/approximation/shepard.py 
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def lagrange_derivatives_x_direction(self):
    """Return repeated 1D Lagrange derivative matrices in the x direction."""
    l = len(self.subgrids_x_step) 
    (I, L1, L2) = self._lagrange_derivatives_one_subgrids_1dim(self.grid.subgrids_x_coords[0])
    return (I[None, :, :].repeat(l, axis=0), L1[None, :, :].repeat(l, axis=0), L2[None, :, :].repeat(l, axis=0))

lagrange_derivatives_y_direction()

Return repeated 1D Lagrange derivative matrices in the y direction.

Source code in src/gbmsc_pde/approximation/shepard.py 
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def lagrange_derivatives_y_direction(self):
    """Return repeated 1D Lagrange derivative matrices in the y direction."""
    (I, L1, L2) = self._lagrange_derivatives_one_subgrids_1dim(self.grid.subgrids_y_coords[0])
    l = len(self.subgrids_y_step)
    return (I[None, :, :].repeat(l, axis=0), L1[None, :, :].repeat(l, axis=0), L2[None, :, :].repeat(l, axis=0))

LinearPDE

Linear PDE builder on a structured rectangular source grid.

The assembled equation has the form

div(D grad u) + v . grad u + r u = f

where diffusion and convection coefficients may be scalar, two-component constants, or field callables. Boundary conditions are applied separately by the solver/boundary-condition layer.

Source code in src/gbmsc_pde/pde/problem.py 
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class LinearPDE:
    """
    Linear PDE builder on a structured rectangular source grid.

    The assembled equation has the form

    ``div(D grad u) + v . grad u + r u = f``

    where diffusion and convection coefficients may be scalar, two-component
    constants, or field callables.  Boundary conditions are applied separately
    by the solver/boundary-condition layer.
    """

    def __init__(self, approximation: GBMSCApproximation) -> None:
        self.approximation = approximation
        self.terms: Dict[str, object] = {
            "diffusion": None,
            "convection": None,
            "reaction": None,
            "source": None,
        }

    def add_diffusion_term(self, coeff) -> None:
        """Add diffusion coefficients for ``div(D grad u)``."""
        self.terms["diffusion"] = normalize_vector_coefficient(coeff)

    def add_convection_term(self, coeff) -> None:
        """Add convection coefficients for ``v . grad u``."""
        self.terms["convection"] = normalize_vector_coefficient(coeff)

    def add_reaction_term(self, coeff) -> None:
        """Add reaction coefficient ``r`` for ``r*u``."""
        self.terms["reaction"] = normalize_scalar_coefficient(coeff, name="coeff")

    def add_source_term(self, source) -> None:
        """Add right-hand side source field ``f``."""
        self.terms["source"] = normalize_scalar_coefficient(source, name="source")

    def assemble(self) -> tuple[spmatrix, np.ndarray, spmatrix, spmatrix, spmatrix, spmatrix]:
        """
        Assemble ``A u = b`` on all rectangular source nodes.

        Returns
        -------
        A, b, Mx, My, Mxx, Myy
            Sparse PDE matrix, right-hand side, first-derivative matrices, and
            second-derivative matrices.
        """
        return assemble_linear_system(self.approximation, self.terms)

add_convection_term(coeff)

Add convection coefficients for v . grad u.

Source code in src/gbmsc_pde/pde/problem.py 
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def add_convection_term(self, coeff) -> None:
    """Add convection coefficients for ``v . grad u``."""
    self.terms["convection"] = normalize_vector_coefficient(coeff)

add_diffusion_term(coeff)

Add diffusion coefficients for div(D grad u).

Source code in src/gbmsc_pde/pde/problem.py 
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def add_diffusion_term(self, coeff) -> None:
    """Add diffusion coefficients for ``div(D grad u)``."""
    self.terms["diffusion"] = normalize_vector_coefficient(coeff)

add_reaction_term(coeff)

Add reaction coefficient r for r*u.

Source code in src/gbmsc_pde/pde/problem.py 
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def add_reaction_term(self, coeff) -> None:
    """Add reaction coefficient ``r`` for ``r*u``."""
    self.terms["reaction"] = normalize_scalar_coefficient(coeff, name="coeff")

add_source_term(source)

Add right-hand side source field f.

Source code in src/gbmsc_pde/pde/problem.py 
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def add_source_term(self, source) -> None:
    """Add right-hand side source field ``f``."""
    self.terms["source"] = normalize_scalar_coefficient(source, name="source")

assemble()

Assemble A u = b on all rectangular source nodes.

Returns:

Type Description
(A, b, Mx, My, Mxx, Myy)

Sparse PDE matrix, right-hand side, first-derivative matrices, and second-derivative matrices.
Source code in src/gbmsc_pde/pde/problem.py 
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def assemble(self) -> tuple[spmatrix, np.ndarray, spmatrix, spmatrix, spmatrix, spmatrix]:
    """
    Assemble ``A u = b`` on all rectangular source nodes.

    Returns
    -------
    A, b, Mx, My, Mxx, Myy
        Sparse PDE matrix, right-hand side, first-derivative matrices, and
        second-derivative matrices.
    """
    return assemble_linear_system(self.approximation, self.terms)

LinearSolver

Linear-system solver for rectangular-domain PDE problems.

LinearSolver assembles the PDE rows from :class:LinearPDE, applies node-indexed boundary conditions, and solves the resulting square sparse system for the source-node vector.

Parameters:

Name Type Description Default
pde LinearPDE

Configured rectangular linear PDE builder.

 required
bcs BoundaryConditions

Boundary conditions registered on flattened grid nodes.

 required
Source code in src/gbmsc_pde/solvers/linear.py 
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class LinearSolver:
    """
    Linear-system solver for rectangular-domain PDE problems.

    ``LinearSolver`` assembles the PDE rows from :class:`LinearPDE`, applies
    node-indexed boundary conditions, and solves the resulting square sparse
    system for the source-node vector.

    Parameters
    ----------
    pde : LinearPDE
        Configured rectangular linear PDE builder.
    bcs : BoundaryConditions
        Boundary conditions registered on flattened grid nodes.
    """
    def __init__(self, pde: LinearPDE, bcs: BoundaryConditions) -> None:
        if not isinstance(pde, LinearPDE):
            raise TypeError("pde must be an instance of LinearPDE")
        if not isinstance(bcs, BoundaryConditions):
            raise TypeError("bcs must be an instance of BoundaryConditions")

        self.pde = pde
        self.bcs = bcs
        self.A = None
        self.b = None
        self.Mx = None
        self.My = None
        self.Mxx = None
        self.Myy = None
        self.solution = None
        self.last_data = None

    def assemble(self) -> Tuple[spmatrix, np.ndarray, spmatrix, spmatrix, spmatrix, spmatrix]:
        """
        Assemble and boundary-modify the rectangular collocation system.

        Returns
        -------
        A, b, Mx, My, Mxx, Myy
            Boundary-modified sparse system, right-hand side, and first
            and second derivative matrices.  The same objects are stored on
            ``self`` as ``A``, ``b``, ``Mx``, ``My``, ``Mxx``, and ``Myy`` for
            later inspection.
        """
        A, b, Mx, My, Mxx, Myy = self.pde.assemble()
        A, b = BoundaryConditionApplier(self.bcs).apply(A, b, Mx, My)
        self.A = A
        self.b = b
        self.Mx = Mx
        self.My = My
        self.Mxx = Mxx
        self.Myy = Myy
        return A, b, Mx, My, Mxx, Myy

    @staticmethod
    def _as_linear_operator(preconditioner: Optional[Any]):
        if preconditioner is None:
            return None
        if isinstance(preconditioner, spmatrix):
            return spla.LinearOperator(preconditioner.shape, preconditioner.dot)
        return preconditioner

    @staticmethod
    def _jacobi_preconditioner(A: spmatrix):
        diagonal = A.diagonal()
        if np.any(np.isclose(diagonal, 0.0)):
            raise ValueError("Jacobi preconditioner requires a nonzero matrix diagonal.")
        inv_diagonal = 1.0 / diagonal
        return spla.LinearOperator(A.shape, matvec=lambda x: inv_diagonal * x)

    @staticmethod
    def _iterative_call(method, A, b, *, tol, maxiter, M):
        try:
            return method(A, b, rtol=tol, atol=0.0, maxiter=maxiter, M=M)
        except TypeError:
            return method(A, b, tol=tol, maxiter=maxiter, M=M)

    def solve(
        self,
        solver: str = 'direct',
        tol: float = 1e-8,
        maxiter: Optional[int] = None,
        preconditioner: Optional[Any] = None,
        return_data: bool = False,
    ) -> Union[
        np.ndarray,
        Tuple[np.ndarray, Dict[str, object]],
    ]:
        """
        Assemble, apply boundary conditions, and solve the linear system.

        Parameters
        ----------
        solver : {'direct', 'cg', 'gmres', 'bicgstab'}
            Linear solver. ``"direct"`` uses sparse direct solve; ``"cg"`` and
            ``"gmres"`` and ``"bicgstab"`` use SciPy iterative solvers.
        tol : float, optional
            Tolerance for iterative solvers.
        maxiter : int, optional
            Maximum iterations (defaults to N).
        preconditioner : object, optional
            Preconditioner LinearOperator, sparse matrix, or ``"jacobi"``.
        return_data : bool
            If true, return ``(u, data)`` where ``data`` contains ``A``, ``b``,
            derivative matrices, ``nnz``, ``sparsity``, ``solve_time``,
            ``assembly_time``, and ``solver_info``.

        Returns
        -------
        ndarray or tuple
            Solution vector ``u`` with shape ``(grid.N,)``.  If
            ``return_data=True``, returns ``(u, data)``.
        """
        assembly_start = perf_counter()
        A, b, Mx, My, Mxx, Myy = self.assemble()
        assembly_time = perf_counter() - assembly_start

        if A.shape[0] != A.shape[1]:
            raise ValueError(f"Linear system must be square; got {A.shape}.")
        if not np.all(np.isfinite(A.data)):
            raise ValueError("Linear system matrix contains non-finite values.")
        if not np.all(np.isfinite(b)):
            raise ValueError("Right-hand side contains non-finite values.")

        solve_start = perf_counter()
        if solver == 'direct':
            sol = spla.spsolve(A, b)
            info = 0
        else:
            N = A.shape[0]
            maxiter = maxiter or N
            if preconditioner == "jacobi":
                M = self._jacobi_preconditioner(A)
            else:
                M = self._as_linear_operator(preconditioner)
            methods = {'cg': spla.cg, 'gmres': spla.gmres, 'bicgstab': spla.bicgstab}
            if solver not in methods:
                raise ValueError(
                    f"Unsupported solver '{solver}'. Choose 'direct', 'cg', 'gmres', or 'bicgstab'."
                )
            sol, info = self._iterative_call(
                methods[solver],
                A,
                b,
                tol=tol,
                maxiter=maxiter,
                M=M,
            )
            if info != 0:
                raise RuntimeError(f"Solver '{solver}' failed to converge (info={info}).")
        solve_time = perf_counter() - solve_start

        self.solution = sol

        if return_data:
            data: Dict[str, object] = {
                "A": A,
                "b": b,
                "first_derivatives": (Mx, My),
                "second_derivatives": (Mxx, Myy),
                "nnz": int(A.nnz),
                "sparsity": float((1.0 - A.nnz / (A.shape[0] * A.shape[1])) * 100.0),
                "solve_time": float(solve_time),
                "assembly_time": float(assembly_time),
                "solver_info": int(info),
            }
            self.last_data = data
            return sol, data

        return sol

assemble()

Assemble and boundary-modify the rectangular collocation system.

Returns:

Type Description
(A, b, Mx, My, Mxx, Myy)

Boundary-modified sparse system, right-hand side, and first and second derivative matrices. The same objects are stored on self as A, b, Mx, My, Mxx, and Myy for later inspection.
Source code in src/gbmsc_pde/solvers/linear.py 
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def assemble(self) -> Tuple[spmatrix, np.ndarray, spmatrix, spmatrix, spmatrix, spmatrix]:
    """
    Assemble and boundary-modify the rectangular collocation system.

    Returns
    -------
    A, b, Mx, My, Mxx, Myy
        Boundary-modified sparse system, right-hand side, and first
        and second derivative matrices.  The same objects are stored on
        ``self`` as ``A``, ``b``, ``Mx``, ``My``, ``Mxx``, and ``Myy`` for
        later inspection.
    """
    A, b, Mx, My, Mxx, Myy = self.pde.assemble()
    A, b = BoundaryConditionApplier(self.bcs).apply(A, b, Mx, My)
    self.A = A
    self.b = b
    self.Mx = Mx
    self.My = My
    self.Mxx = Mxx
    self.Myy = Myy
    return A, b, Mx, My, Mxx, Myy

solve(solver='direct', tol=1e-08, maxiter=None, preconditioner=None, return_data=False)

Assemble, apply boundary conditions, and solve the linear system.

Parameters:

Name Type Description Default
solver (direct, cg, gmres, bicgstab)

Linear solver. "direct" uses sparse direct solve; "cg" and "gmres" and "bicgstab" use SciPy iterative solvers.

 'direct'
tol float

Tolerance for iterative solvers.

 1e-08
maxiter int

Maximum iterations (defaults to N).

 None
preconditioner object

Preconditioner LinearOperator, sparse matrix, or "jacobi".

 None
return_data bool

If true, return (u, data) where data contains A, b, derivative matrices, nnz, sparsity, solve_time, assembly_time, and solver_info.

 False

Returns:

Type Description
ndarray or tuple

Solution vector u with shape (grid.N,). If return_data=True, returns (u, data).
Source code in src/gbmsc_pde/solvers/linear.py 
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def solve(
    self,
    solver: str = 'direct',
    tol: float = 1e-8,
    maxiter: Optional[int] = None,
    preconditioner: Optional[Any] = None,
    return_data: bool = False,
) -> Union[
    np.ndarray,
    Tuple[np.ndarray, Dict[str, object]],
]:
    """
    Assemble, apply boundary conditions, and solve the linear system.

    Parameters
    ----------
    solver : {'direct', 'cg', 'gmres', 'bicgstab'}
        Linear solver. ``"direct"`` uses sparse direct solve; ``"cg"`` and
        ``"gmres"`` and ``"bicgstab"`` use SciPy iterative solvers.
    tol : float, optional
        Tolerance for iterative solvers.
    maxiter : int, optional
        Maximum iterations (defaults to N).
    preconditioner : object, optional
        Preconditioner LinearOperator, sparse matrix, or ``"jacobi"``.
    return_data : bool
        If true, return ``(u, data)`` where ``data`` contains ``A``, ``b``,
        derivative matrices, ``nnz``, ``sparsity``, ``solve_time``,
        ``assembly_time``, and ``solver_info``.

    Returns
    -------
    ndarray or tuple
        Solution vector ``u`` with shape ``(grid.N,)``.  If
        ``return_data=True``, returns ``(u, data)``.
    """
    assembly_start = perf_counter()
    A, b, Mx, My, Mxx, Myy = self.assemble()
    assembly_time = perf_counter() - assembly_start

    if A.shape[0] != A.shape[1]:
        raise ValueError(f"Linear system must be square; got {A.shape}.")
    if not np.all(np.isfinite(A.data)):
        raise ValueError("Linear system matrix contains non-finite values.")
    if not np.all(np.isfinite(b)):
        raise ValueError("Right-hand side contains non-finite values.")

    solve_start = perf_counter()
    if solver == 'direct':
        sol = spla.spsolve(A, b)
        info = 0
    else:
        N = A.shape[0]
        maxiter = maxiter or N
        if preconditioner == "jacobi":
            M = self._jacobi_preconditioner(A)
        else:
            M = self._as_linear_operator(preconditioner)
        methods = {'cg': spla.cg, 'gmres': spla.gmres, 'bicgstab': spla.bicgstab}
        if solver not in methods:
            raise ValueError(
                f"Unsupported solver '{solver}'. Choose 'direct', 'cg', 'gmres', or 'bicgstab'."
            )
        sol, info = self._iterative_call(
            methods[solver],
            A,
            b,
            tol=tol,
            maxiter=maxiter,
            M=M,
        )
        if info != 0:
            raise RuntimeError(f"Solver '{solver}' failed to converge (info={info}).")
    solve_time = perf_counter() - solve_start

    self.solution = sol

    if return_data:
        data: Dict[str, object] = {
            "A": A,
            "b": b,
            "first_derivatives": (Mx, My),
            "second_derivatives": (Mxx, Myy),
            "nnz": int(A.nnz),
            "sparsity": float((1.0 - A.nnz / (A.shape[0] * A.shape[1])) * 100.0),
            "solve_time": float(solve_time),
            "assembly_time": float(assembly_time),
            "solver_info": int(info),
        }
        self.last_data = data
        return sol, data

    return sol

SourceGrid

Structured two-dimensional tensor-product grid.

SourceGrid is the rectangular-domain source-node container used by the Grid-Based Multinode Shepard Collocation approximation. It stores the global source nodes, all sliding local tensor-product subgrids, flattened node coordinates, and pairwise coordinate differences used by the weight formulas.

Attributes:

Name Type Description
grid_shape tuple[int, int]

Number of grid nodes (N_x, N_y).
subgrid_shape tuple[int, int]

Number of nodes in each local interpolation window (n_x, n_y).
N int

Total number of source nodes, equal to N_x * N_y.
coords (ndarray, shape(N, 2))

Flattened source-node coordinates ordered consistently with grid.ravel().
subgrids (ndarray, shape(N_x - n_x + 1, N_y - n_y + 1, n_x, n_y))

Sliding-window node-index view.
diff_x, diff_y, dist (ndarray, shape(N, N))

Pairwise coordinate differences and squared Euclidean distances.
Source code in src/gbmsc_pde/source_grid/grid.py 
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class SourceGrid:
    """
    Structured two-dimensional tensor-product grid.

    ``SourceGrid`` is the rectangular-domain source-node container used by the
    Grid-Based Multinode Shepard Collocation approximation.  It stores the
    global source nodes, all sliding local tensor-product subgrids, flattened
    node coordinates, and pairwise coordinate differences used by the weight
    formulas.

    Attributes
    ----------
    grid_shape : tuple[int, int]
        Number of grid nodes ``(N_x, N_y)``.
    subgrid_shape : tuple[int, int]
        Number of nodes in each local interpolation window ``(n_x, n_y)``.
    N : int
        Total number of source nodes, equal to ``N_x * N_y``.
    coords : ndarray, shape (N, 2)
        Flattened source-node coordinates ordered consistently with
        ``grid.ravel()``.
    subgrids : ndarray, shape (N_x-n_x+1, N_y-n_y+1, n_x, n_y)
        Sliding-window node-index view.
    diff_x, diff_y, dist : ndarray, shape (N, N)
        Pairwise coordinate differences and squared Euclidean distances.
    """

    def __init__(self,
                 grid_shape: Tuple[int, int] = (10, 10),
                 subgrid_shape: Tuple[int, int] = (3, 3),
                 xlim: Tuple[float, float] = (0., 1.),
                 ylim: Tuple[float, float] = (0., 1.),
                 limits: Optional[Tuple[Tuple[float, float], Tuple[float, float]]] = None):
        """
        Initialize a rectangular structured grid.

        Parameters
        ----------
        grid_shape : tuple[int, int], default=(10, 10)
            Number of source nodes in the x and y directions.
        subgrid_shape : tuple[int, int], default=(3, 3)
            Size of each local tensor-product interpolation window.
        xlim, ylim : tuple[float, float]
            Coordinate limits of the rectangular domain.
        limits : tuple[tuple[float, float], tuple[float, float]], optional
            Compatibility alias for ``(xlim, ylim)``.

        Raises
        ------
        ValueError
            If either shape is not two-dimensional, if a local subgrid is
            larger than the global grid, or if ``limits`` is malformed.
        """
        if len(grid_shape) != 2 or len(subgrid_shape) != 2:
            raise ValueError("grid_shape and subgrid_shape must be of length 2.")
        if limits is not None:
            if len(limits) != 2:
                raise ValueError("limits must be ((x_min, x_max), (y_min, y_max)).")
            xlim, ylim = limits
        N_x, N_y = grid_shape
        self.N = N_x * N_y
        n_x, n_y = subgrid_shape
        if n_x > N_x or n_y > N_y:
            raise ValueError("subgrid_shape must be <= grid_shape in both dimensions.")

        self.grid_shape = (int(N_x), int(N_y))
        self.subgrid_shape = (int(n_x), int(n_y))
        self.xlim = tuple(float(v) for v in xlim)
        self.ylim = tuple(float(v) for v in ylim)

        # Generate grid indices and sliding subgrids
        self.grid, self.subgrids, self.subgrids_x, self.subgrids_y = self._generate_sliding_subgrids()

        # Generate coordinates
        (self.subgrids_x_coords,
         self.subgrids_y_coords,
         self.X, self.Y,
         self.x_coords_flat,
         self.y_coords_flat) = self._generate_coordinates()
        self.coords = np.column_stack((self.x_coords_flat, self.y_coords_flat))
        self.axes = (self.X[:, 0].copy(), self.Y[0, :].copy())
        self.dim = 2

        # Precompute pairwise differences and distances
        self.diff_x = self.x_coords_flat[:, None] - self.x_coords_flat[None, :]
        self.diff_y = self.y_coords_flat[:, None] - self.y_coords_flat[None, :]
        self.dist = self.diff_x**2 + self.diff_y**2

    def _generate_coordinates(self) -> Tuple[np.ndarray, ...]:
        """
        Compute global and subgrid coordinates.

        Returns:
            tuple containing:
             - subgrids_x_coords: x values for subgrid windows
             - subgrids_y_coords: y values for subgrid windows
             - X, Y: meshgrid arrays of shape (N_x, N_y)
             - x_coords_flat, y_coords_flat: flattened coords arrays of length N_x*N_y
        """
        N_x, N_y = self.grid_shape
        x_vals = np.linspace(self.xlim[0], self.xlim[1], N_x, dtype=float)
        y_vals = np.linspace(self.ylim[0], self.ylim[1], N_y, dtype=float)
        X, Y = np.meshgrid(x_vals, y_vals, indexing='ij')
        coords = np.vstack([X.ravel(), Y.ravel()]).T
        # Subgrid coordinate windows

        subgrids_x_coords = np.lib.stride_tricks.sliding_window_view(x_vals, self.subgrid_shape[0])
        subgrids_y_coords = np.lib.stride_tricks.sliding_window_view(y_vals, self.subgrid_shape[1])
        return subgrids_x_coords, subgrids_y_coords, X, Y, coords[:, 0], coords[:, 1]

    def _generate_sliding_subgrids(self) -> Tuple[np.ndarray, np.ndarray, np.ndarray, np.ndarray]:
        """
        Create sliding-window index subgrids using stride_tricks.

        Returns:
            grid: index grid of shape (N_x, N_y)
            subgrids: view of shape (N_x - n_x + 1, N_y - n_y + 1, n_x, n_y)
            subgrids_x: x-index windows of shape (N_x - n_x + 1, n_x)
            subgrids_y: y-index windows of shape (N_y - n_y + 1, n_y)
        """
        N_x, N_y = self.grid_shape
        n_x, n_y = self.subgrid_shape
        # index grid
        grid = np.arange(N_x * N_y).reshape(N_x, N_y)
        # sliding windows
        subgrids = np.lib.stride_tricks.sliding_window_view(grid, (n_x, n_y))
        subgrids_x = np.lib.stride_tricks.sliding_window_view(np.arange(N_x), n_x)
        subgrids_y = np.lib.stride_tricks.sliding_window_view(np.arange(N_y), n_y)
        return grid, subgrids, subgrids_x, subgrids_y

    def subgrids_with_step(self, step: Tuple[int, int] = (1, 1)) -> Tuple[np.ndarray, np.ndarray, np.ndarray]:
        """
        Return sampled local subgrids at a fixed stride.

        Parameters
        ----------
        step : tuple[int, int], default=(1, 1)
            Sampling stride ``(step_x, step_y)`` over the sliding subgrid
            starts.  The stride must be compatible with the grid so the last
            sampled window reaches the upper/right boundary.

        Returns
        -------
        subgrids, subgrids_x, subgrids_y : tuple[ndarray, ndarray, ndarray]
            Sampled two-dimensional node windows and their one-dimensional
            x/y index windows.

        Raises
        ------
        ValueError
            If the stride is not positive or is incompatible with the grid and
            local-window sizes.
        """
        dx, dy = step
        N_x, N_y = self.grid_shape
        n_x, n_y = self.subgrid_shape

        if not isinstance(dx, int) or not isinstance(dy, int) or dx < 1 or dy < 1 or dx >= n_x or dy >= n_y:
            raise ValueError("step values must be positive integers.")

        if (N_x - n_x) % dx != 0 or (N_y - n_y) % dy != 0:
            raise ValueError(
                "step must satisfy N_x = step_x * k_x + n_x and "
                "N_y = step_y * k_y + n_y for integer k_x, k_y."
            )

        return (
            self.subgrids[::dx, ::dy, :, :],
            self.subgrids_x[::dx, :],
            self.subgrids_y[::dy, :]
        )

    def boundary_indices(self) -> Dict[str, np.ndarray]:
            """
            Return flattened node indices for the rectangular boundary.

            The returned dictionary includes both geometric names
            ``"left"``, ``"right"``, ``"bottom"``, ``"top"`` and coordinate
            aliases ``"x_min"``, ``"x_max"``, ``"y_min"``, ``"y_max"``.
            ``"all"`` contains the unique union of all four sides.
            """
            N_x, N_y = self.grid_shape
            total = N_x * N_y

            bottom = np.arange(0, total, N_y)

            top = np.arange(N_y - 1, total, N_y)

            left = np.arange(0, N_y)

            right = np.arange((N_x - 1) * N_y, total)

            return {
                'left':   left,
                'right':  right,
                'bottom': bottom,
                'top':    top,
                "x_min": left,
                "x_max": right,
                "y_min": bottom,
                "y_max": top,
                "all": np.unique(np.concatenate([left, right, bottom, top])),
            }

    def plot(
        self,
        solution: np.ndarray,
        view: str = "3d",
        cmap: str = "viridis",
        contour_levels=20,
        show_colorbar: bool = True,
        xlabel: str = "X",
        ylabel: str = "Y",
        zlabel: str = "",
        title: str = "",
        figsize: Tuple[int, int] = (5, 4),
        show: bool = True,
    ):
        """
        Plot a discrete scalar solution on this structured grid.

        Parameters
        ----------
        solution : ndarray
            Solution values as either a flat ``(N_x * N_y,)`` array or a
            ``(N_x, N_y)`` grid-shaped array.
        view : {"2d", "contour", "contourf", "3d"}
            Plot style for the discrete solution.
        show : bool
            Whether to display the plot with ``matplotlib.pyplot.show``.

        Returns
        -------
        fig, ax
            Matplotlib figure and axes objects.
        """
        nx, ny = self.grid_shape
        sol = np.asarray(solution)
        if sol.ndim == 1:
            if sol.size != nx * ny:
                raise ValueError(
                    f"1D solution length {sol.size} does not match grid size {nx * ny}"
                )
            sol = sol.reshape(nx, ny)
        elif sol.shape != (nx, ny):
            raise ValueError(f"solution array shape {sol.shape} != grid.grid_shape {(nx, ny)}")

        try:
            import matplotlib.pyplot as plt
        except ImportError as exc:
            raise ImportError(
                "SourceGrid.plot requires matplotlib. "
                "Install it with `pip install matplotlib` or `pip install .[plot]`."
            ) from exc

        fig = plt.figure(figsize=figsize)
        if view == "3d":
            ax = fig.add_subplot(111, projection="3d")
            surf = ax.plot_surface(
                self.X,
                self.Y,
                sol,
                cmap=cmap,
                edgecolor="none",
                antialiased=True,
            )
            if show_colorbar:
                fig.colorbar(surf, ax=ax, shrink=0.5)
            ax.set_zlabel(zlabel)
        else:
            ax = fig.add_subplot(111)
            if view == "2d":
                mappable = ax.imshow(
                    sol.T,
                    origin="lower",
                    extent=(self.xlim[0], self.xlim[1], self.ylim[0], self.ylim[1]),
                    cmap=cmap,
                    aspect="auto",
                )
            elif view in ("contour", "contourf"):
                levels = contour_levels or 10
                plot_fn = ax.contourf if view == "contourf" else ax.contour
                mappable = plot_fn(self.X, self.Y, sol, levels=levels, cmap=cmap)
            else:
                raise ValueError(
                    f"Unknown view '{view}'. Choose from '2d', 'contour', 'contourf', '3d'."
                )
            if show_colorbar:
                fig.colorbar(mappable, ax=ax)

        ax.set_xlabel(xlabel)
        ax.set_ylabel(ylabel)
        if title:
            ax.set_title(title)
        fig.tight_layout()

        if show:
            plt.show()

        return fig, ax

    def eval_function_at_nodes(self, f: Callable[[np.ndarray, np.ndarray], np.ndarray]) -> np.ndarray:
            """
            Evaluate a scalar field at every source node.

            Parameters
            ----------
            f : callable
                Function with signature ``f(x, y)`` where ``x`` and ``y`` are
                one-dimensional coordinate arrays of length ``N``.

            Returns
            -------
            ndarray, shape (N,)
                Field values ordered like ``x_coords_flat`` and
                ``y_coords_flat``.

            Raises
            ------
            TypeError
                If ``f`` is not callable.
            RuntimeError
                If ``f`` raises during evaluation.
            ValueError
                If the result cannot be converted to an array with shape
                ``(N,)``.
            """
            # 1. Check that f is callable
            if not callable(f):
                raise TypeError("`f` must be a callable taking two numpy arrays.")

            # 2. Grab the node coordinates
            try:
                x = self.x_coords_flat
                y = self.y_coords_flat
            except AttributeError as e:
                raise AttributeError(
                    "Grid node coordinates not found on `self`. "
                    "Expected `self.x_coords` and `self.y_coords`. "
                    f"Original error: {e}"
                )

            # 3. Attempt evaluation
            try:
                values = f(x, y)
            except Exception as e:
                raise RuntimeError(f"Error while evaluating `f` at nodes: {e}") from e

            # 4. Convert to numpy array if needed
            if not isinstance(values, np.ndarray):
                try:
                    values = np.asarray(values)
                except Exception:
                    raise ValueError(
                        "Output of `f` is not array-like or cannot be converted to numpy.ndarray."
                    )

            # 5. Validate shape
            if values.ndim != 1 or values.shape[0] != x.shape[0]:
                raise ValueError(
                    f"Function output has shape {values.shape}, "
                    f"but expected (N,) where N={x.shape[0]}."
                )

            return values

__init__(grid_shape=(10, 10), subgrid_shape=(3, 3), xlim=(0.0, 1.0), ylim=(0.0, 1.0), limits=None)

Initialize a rectangular structured grid.

Parameters:

Name Type Description Default
grid_shape tuple[int, int]

Number of source nodes in the x and y directions.

 (10, 10)
subgrid_shape tuple[int, int]

Size of each local tensor-product interpolation window.

 (3, 3)
xlim tuple[float, float]

Coordinate limits of the rectangular domain.

 (0.0, 1.0)
ylim tuple[float, float]

Coordinate limits of the rectangular domain.

 (0.0, 1.0)
limits tuple[tuple[float, float], tuple[float, float]]

Compatibility alias for (xlim, ylim).

 None

Raises:

Type Description
ValueError

If either shape is not two-dimensional, if a local subgrid is larger than the global grid, or if limits is malformed.
Source code in src/gbmsc_pde/source_grid/grid.py 
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def __init__(self,
             grid_shape: Tuple[int, int] = (10, 10),
             subgrid_shape: Tuple[int, int] = (3, 3),
             xlim: Tuple[float, float] = (0., 1.),
             ylim: Tuple[float, float] = (0., 1.),
             limits: Optional[Tuple[Tuple[float, float], Tuple[float, float]]] = None):
    """
    Initialize a rectangular structured grid.

    Parameters
    ----------
    grid_shape : tuple[int, int], default=(10, 10)
        Number of source nodes in the x and y directions.
    subgrid_shape : tuple[int, int], default=(3, 3)
        Size of each local tensor-product interpolation window.
    xlim, ylim : tuple[float, float]
        Coordinate limits of the rectangular domain.
    limits : tuple[tuple[float, float], tuple[float, float]], optional
        Compatibility alias for ``(xlim, ylim)``.

    Raises
    ------
    ValueError
        If either shape is not two-dimensional, if a local subgrid is
        larger than the global grid, or if ``limits`` is malformed.
    """
    if len(grid_shape) != 2 or len(subgrid_shape) != 2:
        raise ValueError("grid_shape and subgrid_shape must be of length 2.")
    if limits is not None:
        if len(limits) != 2:
            raise ValueError("limits must be ((x_min, x_max), (y_min, y_max)).")
        xlim, ylim = limits
    N_x, N_y = grid_shape
    self.N = N_x * N_y
    n_x, n_y = subgrid_shape
    if n_x > N_x or n_y > N_y:
        raise ValueError("subgrid_shape must be <= grid_shape in both dimensions.")

    self.grid_shape = (int(N_x), int(N_y))
    self.subgrid_shape = (int(n_x), int(n_y))
    self.xlim = tuple(float(v) for v in xlim)
    self.ylim = tuple(float(v) for v in ylim)

    # Generate grid indices and sliding subgrids
    self.grid, self.subgrids, self.subgrids_x, self.subgrids_y = self._generate_sliding_subgrids()

    # Generate coordinates
    (self.subgrids_x_coords,
     self.subgrids_y_coords,
     self.X, self.Y,
     self.x_coords_flat,
     self.y_coords_flat) = self._generate_coordinates()
    self.coords = np.column_stack((self.x_coords_flat, self.y_coords_flat))
    self.axes = (self.X[:, 0].copy(), self.Y[0, :].copy())
    self.dim = 2

    # Precompute pairwise differences and distances
    self.diff_x = self.x_coords_flat[:, None] - self.x_coords_flat[None, :]
    self.diff_y = self.y_coords_flat[:, None] - self.y_coords_flat[None, :]
    self.dist = self.diff_x**2 + self.diff_y**2

boundary_indices()

Return flattened node indices for the rectangular boundary.

The returned dictionary includes both geometric names "left", "right", "bottom", "top" and coordinate aliases "x_min", "x_max", "y_min", "y_max". "all" contains the unique union of all four sides.

Source code in src/gbmsc_pde/source_grid/grid.py 
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def boundary_indices(self) -> Dict[str, np.ndarray]:
        """
        Return flattened node indices for the rectangular boundary.

        The returned dictionary includes both geometric names
        ``"left"``, ``"right"``, ``"bottom"``, ``"top"`` and coordinate
        aliases ``"x_min"``, ``"x_max"``, ``"y_min"``, ``"y_max"``.
        ``"all"`` contains the unique union of all four sides.
        """
        N_x, N_y = self.grid_shape
        total = N_x * N_y

        bottom = np.arange(0, total, N_y)

        top = np.arange(N_y - 1, total, N_y)

        left = np.arange(0, N_y)

        right = np.arange((N_x - 1) * N_y, total)

        return {
            'left':   left,
            'right':  right,
            'bottom': bottom,
            'top':    top,
            "x_min": left,
            "x_max": right,
            "y_min": bottom,
            "y_max": top,
            "all": np.unique(np.concatenate([left, right, bottom, top])),
        }

eval_function_at_nodes(f)

Evaluate a scalar field at every source node.

Parameters:

Name Type Description Default
f callable

Function with signature f(x, y) where x and y are one-dimensional coordinate arrays of length N.

 required

Returns:

Type Description
(ndarray, shape(N))

Field values ordered like x_coords_flat and y_coords_flat.

Raises:

Type Description
TypeError

If f is not callable.
RuntimeError

If f raises during evaluation.
ValueError

If the result cannot be converted to an array with shape (N,).
Source code in src/gbmsc_pde/source_grid/grid.py 
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def eval_function_at_nodes(self, f: Callable[[np.ndarray, np.ndarray], np.ndarray]) -> np.ndarray:
        """
        Evaluate a scalar field at every source node.

        Parameters
        ----------
        f : callable
            Function with signature ``f(x, y)`` where ``x`` and ``y`` are
            one-dimensional coordinate arrays of length ``N``.

        Returns
        -------
        ndarray, shape (N,)
            Field values ordered like ``x_coords_flat`` and
            ``y_coords_flat``.

        Raises
        ------
        TypeError
            If ``f`` is not callable.
        RuntimeError
            If ``f`` raises during evaluation.
        ValueError
            If the result cannot be converted to an array with shape
            ``(N,)``.
        """
        # 1. Check that f is callable
        if not callable(f):
            raise TypeError("`f` must be a callable taking two numpy arrays.")

        # 2. Grab the node coordinates
        try:
            x = self.x_coords_flat
            y = self.y_coords_flat
        except AttributeError as e:
            raise AttributeError(
                "Grid node coordinates not found on `self`. "
                "Expected `self.x_coords` and `self.y_coords`. "
                f"Original error: {e}"
            )

        # 3. Attempt evaluation
        try:
            values = f(x, y)
        except Exception as e:
            raise RuntimeError(f"Error while evaluating `f` at nodes: {e}") from e

        # 4. Convert to numpy array if needed
        if not isinstance(values, np.ndarray):
            try:
                values = np.asarray(values)
            except Exception:
                raise ValueError(
                    "Output of `f` is not array-like or cannot be converted to numpy.ndarray."
                )

        # 5. Validate shape
        if values.ndim != 1 or values.shape[0] != x.shape[0]:
            raise ValueError(
                f"Function output has shape {values.shape}, "
                f"but expected (N,) where N={x.shape[0]}."
            )

        return values

plot(solution, view='3d', cmap='viridis', contour_levels=20, show_colorbar=True, xlabel='X', ylabel='Y', zlabel='', title='', figsize=(5, 4), show=True)

Plot a discrete scalar solution on this structured grid.

Parameters:

Name Type Description Default
solution ndarray

Solution values as either a flat (N_x * N_y,) array or a (N_x, N_y) grid-shaped array.

 required
view ('2d', contour, contourf, '3d')

Plot style for the discrete solution.

 "2d"
show bool

Whether to display the plot with matplotlib.pyplot.show.

 True

Returns:

Type Description
(fig, ax)

Matplotlib figure and axes objects.
Source code in src/gbmsc_pde/source_grid/grid.py 
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def plot(
    self,
    solution: np.ndarray,
    view: str = "3d",
    cmap: str = "viridis",
    contour_levels=20,
    show_colorbar: bool = True,
    xlabel: str = "X",
    ylabel: str = "Y",
    zlabel: str = "",
    title: str = "",
    figsize: Tuple[int, int] = (5, 4),
    show: bool = True,
):
    """
    Plot a discrete scalar solution on this structured grid.

    Parameters
    ----------
    solution : ndarray
        Solution values as either a flat ``(N_x * N_y,)`` array or a
        ``(N_x, N_y)`` grid-shaped array.
    view : {"2d", "contour", "contourf", "3d"}
        Plot style for the discrete solution.
    show : bool
        Whether to display the plot with ``matplotlib.pyplot.show``.

    Returns
    -------
    fig, ax
        Matplotlib figure and axes objects.
    """
    nx, ny = self.grid_shape
    sol = np.asarray(solution)
    if sol.ndim == 1:
        if sol.size != nx * ny:
            raise ValueError(
                f"1D solution length {sol.size} does not match grid size {nx * ny}"
            )
        sol = sol.reshape(nx, ny)
    elif sol.shape != (nx, ny):
        raise ValueError(f"solution array shape {sol.shape} != grid.grid_shape {(nx, ny)}")

    try:
        import matplotlib.pyplot as plt
    except ImportError as exc:
        raise ImportError(
            "SourceGrid.plot requires matplotlib. "
            "Install it with `pip install matplotlib` or `pip install .[plot]`."
        ) from exc

    fig = plt.figure(figsize=figsize)
    if view == "3d":
        ax = fig.add_subplot(111, projection="3d")
        surf = ax.plot_surface(
            self.X,
            self.Y,
            sol,
            cmap=cmap,
            edgecolor="none",
            antialiased=True,
        )
        if show_colorbar:
            fig.colorbar(surf, ax=ax, shrink=0.5)
        ax.set_zlabel(zlabel)
    else:
        ax = fig.add_subplot(111)
        if view == "2d":
            mappable = ax.imshow(
                sol.T,
                origin="lower",
                extent=(self.xlim[0], self.xlim[1], self.ylim[0], self.ylim[1]),
                cmap=cmap,
                aspect="auto",
            )
        elif view in ("contour", "contourf"):
            levels = contour_levels or 10
            plot_fn = ax.contourf if view == "contourf" else ax.contour
            mappable = plot_fn(self.X, self.Y, sol, levels=levels, cmap=cmap)
        else:
            raise ValueError(
                f"Unknown view '{view}'. Choose from '2d', 'contour', 'contourf', '3d'."
            )
        if show_colorbar:
            fig.colorbar(mappable, ax=ax)

    ax.set_xlabel(xlabel)
    ax.set_ylabel(ylabel)
    if title:
        ax.set_title(title)
    fig.tight_layout()

    if show:
        plt.show()

    return fig, ax

subgrids_with_step(step=(1, 1))

Return sampled local subgrids at a fixed stride.

Parameters:

Name Type Description Default
step tuple[int, int]

Sampling stride (step_x, step_y) over the sliding subgrid starts. The stride must be compatible with the grid so the last sampled window reaches the upper/right boundary.

 (1, 1)

Returns:

Type Description
subgrids, subgrids_x, subgrids_y : tuple[ndarray, ndarray, ndarray]

Sampled two-dimensional node windows and their one-dimensional x/y index windows.

Raises:

Type Description
ValueError

If the stride is not positive or is incompatible with the grid and local-window sizes.
Source code in src/gbmsc_pde/source_grid/grid.py 
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def subgrids_with_step(self, step: Tuple[int, int] = (1, 1)) -> Tuple[np.ndarray, np.ndarray, np.ndarray]:
    """
    Return sampled local subgrids at a fixed stride.

    Parameters
    ----------
    step : tuple[int, int], default=(1, 1)
        Sampling stride ``(step_x, step_y)`` over the sliding subgrid
        starts.  The stride must be compatible with the grid so the last
        sampled window reaches the upper/right boundary.

    Returns
    -------
    subgrids, subgrids_x, subgrids_y : tuple[ndarray, ndarray, ndarray]
        Sampled two-dimensional node windows and their one-dimensional
        x/y index windows.

    Raises
    ------
    ValueError
        If the stride is not positive or is incompatible with the grid and
        local-window sizes.
    """
    dx, dy = step
    N_x, N_y = self.grid_shape
    n_x, n_y = self.subgrid_shape

    if not isinstance(dx, int) or not isinstance(dy, int) or dx < 1 or dy < 1 or dx >= n_x or dy >= n_y:
        raise ValueError("step values must be positive integers.")

    if (N_x - n_x) % dx != 0 or (N_y - n_y) % dy != 0:
        raise ValueError(
            "step must satisfy N_x = step_x * k_x + n_x and "
            "N_y = step_y * k_y + n_y for integer k_x, k_y."
        )

    return (
        self.subgrids[::dx, ::dy, :, :],
        self.subgrids_x[::dx, :],
        self.subgrids_y[::dy, :]
    )

gbmsc_pde.source_grid.grid.SourceGrid

Structured two-dimensional tensor-product grid.

SourceGrid is the rectangular-domain source-node container used by the Grid-Based Multinode Shepard Collocation approximation. It stores the global source nodes, all sliding local tensor-product subgrids, flattened node coordinates, and pairwise coordinate differences used by the weight formulas.

Attributes:

Name Type Description
grid_shape tuple[int, int]

Number of grid nodes (N_x, N_y).
subgrid_shape tuple[int, int]

Number of nodes in each local interpolation window (n_x, n_y).
N int

Total number of source nodes, equal to N_x * N_y.
coords (ndarray, shape(N, 2))

Flattened source-node coordinates ordered consistently with grid.ravel().
subgrids (ndarray, shape(N_x - n_x + 1, N_y - n_y + 1, n_x, n_y))

Sliding-window node-index view.
diff_x, diff_y, dist (ndarray, shape(N, N))

Pairwise coordinate differences and squared Euclidean distances.
Source code in src/gbmsc_pde/source_grid/grid.py 
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class SourceGrid:
    """
    Structured two-dimensional tensor-product grid.

    ``SourceGrid`` is the rectangular-domain source-node container used by the
    Grid-Based Multinode Shepard Collocation approximation.  It stores the
    global source nodes, all sliding local tensor-product subgrids, flattened
    node coordinates, and pairwise coordinate differences used by the weight
    formulas.

    Attributes
    ----------
    grid_shape : tuple[int, int]
        Number of grid nodes ``(N_x, N_y)``.
    subgrid_shape : tuple[int, int]
        Number of nodes in each local interpolation window ``(n_x, n_y)``.
    N : int
        Total number of source nodes, equal to ``N_x * N_y``.
    coords : ndarray, shape (N, 2)
        Flattened source-node coordinates ordered consistently with
        ``grid.ravel()``.
    subgrids : ndarray, shape (N_x-n_x+1, N_y-n_y+1, n_x, n_y)
        Sliding-window node-index view.
    diff_x, diff_y, dist : ndarray, shape (N, N)
        Pairwise coordinate differences and squared Euclidean distances.
    """

    def __init__(self,
                 grid_shape: Tuple[int, int] = (10, 10),
                 subgrid_shape: Tuple[int, int] = (3, 3),
                 xlim: Tuple[float, float] = (0., 1.),
                 ylim: Tuple[float, float] = (0., 1.),
                 limits: Optional[Tuple[Tuple[float, float], Tuple[float, float]]] = None):
        """
        Initialize a rectangular structured grid.

        Parameters
        ----------
        grid_shape : tuple[int, int], default=(10, 10)
            Number of source nodes in the x and y directions.
        subgrid_shape : tuple[int, int], default=(3, 3)
            Size of each local tensor-product interpolation window.
        xlim, ylim : tuple[float, float]
            Coordinate limits of the rectangular domain.
        limits : tuple[tuple[float, float], tuple[float, float]], optional
            Compatibility alias for ``(xlim, ylim)``.

        Raises
        ------
        ValueError
            If either shape is not two-dimensional, if a local subgrid is
            larger than the global grid, or if ``limits`` is malformed.
        """
        if len(grid_shape) != 2 or len(subgrid_shape) != 2:
            raise ValueError("grid_shape and subgrid_shape must be of length 2.")
        if limits is not None:
            if len(limits) != 2:
                raise ValueError("limits must be ((x_min, x_max), (y_min, y_max)).")
            xlim, ylim = limits
        N_x, N_y = grid_shape
        self.N = N_x * N_y
        n_x, n_y = subgrid_shape
        if n_x > N_x or n_y > N_y:
            raise ValueError("subgrid_shape must be <= grid_shape in both dimensions.")

        self.grid_shape = (int(N_x), int(N_y))
        self.subgrid_shape = (int(n_x), int(n_y))
        self.xlim = tuple(float(v) for v in xlim)
        self.ylim = tuple(float(v) for v in ylim)

        # Generate grid indices and sliding subgrids
        self.grid, self.subgrids, self.subgrids_x, self.subgrids_y = self._generate_sliding_subgrids()

        # Generate coordinates
        (self.subgrids_x_coords,
         self.subgrids_y_coords,
         self.X, self.Y,
         self.x_coords_flat,
         self.y_coords_flat) = self._generate_coordinates()
        self.coords = np.column_stack((self.x_coords_flat, self.y_coords_flat))
        self.axes = (self.X[:, 0].copy(), self.Y[0, :].copy())
        self.dim = 2

        # Precompute pairwise differences and distances
        self.diff_x = self.x_coords_flat[:, None] - self.x_coords_flat[None, :]
        self.diff_y = self.y_coords_flat[:, None] - self.y_coords_flat[None, :]
        self.dist = self.diff_x**2 + self.diff_y**2

    def _generate_coordinates(self) -> Tuple[np.ndarray, ...]:
        """
        Compute global and subgrid coordinates.

        Returns:
            tuple containing:
             - subgrids_x_coords: x values for subgrid windows
             - subgrids_y_coords: y values for subgrid windows
             - X, Y: meshgrid arrays of shape (N_x, N_y)
             - x_coords_flat, y_coords_flat: flattened coords arrays of length N_x*N_y
        """
        N_x, N_y = self.grid_shape
        x_vals = np.linspace(self.xlim[0], self.xlim[1], N_x, dtype=float)
        y_vals = np.linspace(self.ylim[0], self.ylim[1], N_y, dtype=float)
        X, Y = np.meshgrid(x_vals, y_vals, indexing='ij')
        coords = np.vstack([X.ravel(), Y.ravel()]).T
        # Subgrid coordinate windows

        subgrids_x_coords = np.lib.stride_tricks.sliding_window_view(x_vals, self.subgrid_shape[0])
        subgrids_y_coords = np.lib.stride_tricks.sliding_window_view(y_vals, self.subgrid_shape[1])
        return subgrids_x_coords, subgrids_y_coords, X, Y, coords[:, 0], coords[:, 1]

    def _generate_sliding_subgrids(self) -> Tuple[np.ndarray, np.ndarray, np.ndarray, np.ndarray]:
        """
        Create sliding-window index subgrids using stride_tricks.

        Returns:
            grid: index grid of shape (N_x, N_y)
            subgrids: view of shape (N_x - n_x + 1, N_y - n_y + 1, n_x, n_y)
            subgrids_x: x-index windows of shape (N_x - n_x + 1, n_x)
            subgrids_y: y-index windows of shape (N_y - n_y + 1, n_y)
        """
        N_x, N_y = self.grid_shape
        n_x, n_y = self.subgrid_shape
        # index grid
        grid = np.arange(N_x * N_y).reshape(N_x, N_y)
        # sliding windows
        subgrids = np.lib.stride_tricks.sliding_window_view(grid, (n_x, n_y))
        subgrids_x = np.lib.stride_tricks.sliding_window_view(np.arange(N_x), n_x)
        subgrids_y = np.lib.stride_tricks.sliding_window_view(np.arange(N_y), n_y)
        return grid, subgrids, subgrids_x, subgrids_y

    def subgrids_with_step(self, step: Tuple[int, int] = (1, 1)) -> Tuple[np.ndarray, np.ndarray, np.ndarray]:
        """
        Return sampled local subgrids at a fixed stride.

        Parameters
        ----------
        step : tuple[int, int], default=(1, 1)
            Sampling stride ``(step_x, step_y)`` over the sliding subgrid
            starts.  The stride must be compatible with the grid so the last
            sampled window reaches the upper/right boundary.

        Returns
        -------
        subgrids, subgrids_x, subgrids_y : tuple[ndarray, ndarray, ndarray]
            Sampled two-dimensional node windows and their one-dimensional
            x/y index windows.

        Raises
        ------
        ValueError
            If the stride is not positive or is incompatible with the grid and
            local-window sizes.
        """
        dx, dy = step
        N_x, N_y = self.grid_shape
        n_x, n_y = self.subgrid_shape

        if not isinstance(dx, int) or not isinstance(dy, int) or dx < 1 or dy < 1 or dx >= n_x or dy >= n_y:
            raise ValueError("step values must be positive integers.")

        if (N_x - n_x) % dx != 0 or (N_y - n_y) % dy != 0:
            raise ValueError(
                "step must satisfy N_x = step_x * k_x + n_x and "
                "N_y = step_y * k_y + n_y for integer k_x, k_y."
            )

        return (
            self.subgrids[::dx, ::dy, :, :],
            self.subgrids_x[::dx, :],
            self.subgrids_y[::dy, :]
        )

    def boundary_indices(self) -> Dict[str, np.ndarray]:
            """
            Return flattened node indices for the rectangular boundary.

            The returned dictionary includes both geometric names
            ``"left"``, ``"right"``, ``"bottom"``, ``"top"`` and coordinate
            aliases ``"x_min"``, ``"x_max"``, ``"y_min"``, ``"y_max"``.
            ``"all"`` contains the unique union of all four sides.
            """
            N_x, N_y = self.grid_shape
            total = N_x * N_y

            bottom = np.arange(0, total, N_y)

            top = np.arange(N_y - 1, total, N_y)

            left = np.arange(0, N_y)

            right = np.arange((N_x - 1) * N_y, total)

            return {
                'left':   left,
                'right':  right,
                'bottom': bottom,
                'top':    top,
                "x_min": left,
                "x_max": right,
                "y_min": bottom,
                "y_max": top,
                "all": np.unique(np.concatenate([left, right, bottom, top])),
            }

    def plot(
        self,
        solution: np.ndarray,
        view: str = "3d",
        cmap: str = "viridis",
        contour_levels=20,
        show_colorbar: bool = True,
        xlabel: str = "X",
        ylabel: str = "Y",
        zlabel: str = "",
        title: str = "",
        figsize: Tuple[int, int] = (5, 4),
        show: bool = True,
    ):
        """
        Plot a discrete scalar solution on this structured grid.

        Parameters
        ----------
        solution : ndarray
            Solution values as either a flat ``(N_x * N_y,)`` array or a
            ``(N_x, N_y)`` grid-shaped array.
        view : {"2d", "contour", "contourf", "3d"}
            Plot style for the discrete solution.
        show : bool
            Whether to display the plot with ``matplotlib.pyplot.show``.

        Returns
        -------
        fig, ax
            Matplotlib figure and axes objects.
        """
        nx, ny = self.grid_shape
        sol = np.asarray(solution)
        if sol.ndim == 1:
            if sol.size != nx * ny:
                raise ValueError(
                    f"1D solution length {sol.size} does not match grid size {nx * ny}"
                )
            sol = sol.reshape(nx, ny)
        elif sol.shape != (nx, ny):
            raise ValueError(f"solution array shape {sol.shape} != grid.grid_shape {(nx, ny)}")

        try:
            import matplotlib.pyplot as plt
        except ImportError as exc:
            raise ImportError(
                "SourceGrid.plot requires matplotlib. "
                "Install it with `pip install matplotlib` or `pip install .[plot]`."
            ) from exc

        fig = plt.figure(figsize=figsize)
        if view == "3d":
            ax = fig.add_subplot(111, projection="3d")
            surf = ax.plot_surface(
                self.X,
                self.Y,
                sol,
                cmap=cmap,
                edgecolor="none",
                antialiased=True,
            )
            if show_colorbar:
                fig.colorbar(surf, ax=ax, shrink=0.5)
            ax.set_zlabel(zlabel)
        else:
            ax = fig.add_subplot(111)
            if view == "2d":
                mappable = ax.imshow(
                    sol.T,
                    origin="lower",
                    extent=(self.xlim[0], self.xlim[1], self.ylim[0], self.ylim[1]),
                    cmap=cmap,
                    aspect="auto",
                )
            elif view in ("contour", "contourf"):
                levels = contour_levels or 10
                plot_fn = ax.contourf if view == "contourf" else ax.contour
                mappable = plot_fn(self.X, self.Y, sol, levels=levels, cmap=cmap)
            else:
                raise ValueError(
                    f"Unknown view '{view}'. Choose from '2d', 'contour', 'contourf', '3d'."
                )
            if show_colorbar:
                fig.colorbar(mappable, ax=ax)

        ax.set_xlabel(xlabel)
        ax.set_ylabel(ylabel)
        if title:
            ax.set_title(title)
        fig.tight_layout()

        if show:
            plt.show()

        return fig, ax

    def eval_function_at_nodes(self, f: Callable[[np.ndarray, np.ndarray], np.ndarray]) -> np.ndarray:
            """
            Evaluate a scalar field at every source node.

            Parameters
            ----------
            f : callable
                Function with signature ``f(x, y)`` where ``x`` and ``y`` are
                one-dimensional coordinate arrays of length ``N``.

            Returns
            -------
            ndarray, shape (N,)
                Field values ordered like ``x_coords_flat`` and
                ``y_coords_flat``.

            Raises
            ------
            TypeError
                If ``f`` is not callable.
            RuntimeError
                If ``f`` raises during evaluation.
            ValueError
                If the result cannot be converted to an array with shape
                ``(N,)``.
            """
            # 1. Check that f is callable
            if not callable(f):
                raise TypeError("`f` must be a callable taking two numpy arrays.")

            # 2. Grab the node coordinates
            try:
                x = self.x_coords_flat
                y = self.y_coords_flat
            except AttributeError as e:
                raise AttributeError(
                    "Grid node coordinates not found on `self`. "
                    "Expected `self.x_coords` and `self.y_coords`. "
                    f"Original error: {e}"
                )

            # 3. Attempt evaluation
            try:
                values = f(x, y)
            except Exception as e:
                raise RuntimeError(f"Error while evaluating `f` at nodes: {e}") from e

            # 4. Convert to numpy array if needed
            if not isinstance(values, np.ndarray):
                try:
                    values = np.asarray(values)
                except Exception:
                    raise ValueError(
                        "Output of `f` is not array-like or cannot be converted to numpy.ndarray."
                    )

            # 5. Validate shape
            if values.ndim != 1 or values.shape[0] != x.shape[0]:
                raise ValueError(
                    f"Function output has shape {values.shape}, "
                    f"but expected (N,) where N={x.shape[0]}."
                )

            return values

__init__(grid_shape=(10, 10), subgrid_shape=(3, 3), xlim=(0.0, 1.0), ylim=(0.0, 1.0), limits=None)

Initialize a rectangular structured grid.

Parameters:

Name Type Description Default
grid_shape tuple[int, int]

Number of source nodes in the x and y directions.

 (10, 10)
subgrid_shape tuple[int, int]

Size of each local tensor-product interpolation window.

 (3, 3)
xlim tuple[float, float]

Coordinate limits of the rectangular domain.

 (0.0, 1.0)
ylim tuple[float, float]

Coordinate limits of the rectangular domain.

 (0.0, 1.0)
limits tuple[tuple[float, float], tuple[float, float]]

Compatibility alias for (xlim, ylim).

 None

Raises:

Type Description
ValueError

If either shape is not two-dimensional, if a local subgrid is larger than the global grid, or if limits is malformed.
Source code in src/gbmsc_pde/source_grid/grid.py 
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def __init__(self,
             grid_shape: Tuple[int, int] = (10, 10),
             subgrid_shape: Tuple[int, int] = (3, 3),
             xlim: Tuple[float, float] = (0., 1.),
             ylim: Tuple[float, float] = (0., 1.),
             limits: Optional[Tuple[Tuple[float, float], Tuple[float, float]]] = None):
    """
    Initialize a rectangular structured grid.

    Parameters
    ----------
    grid_shape : tuple[int, int], default=(10, 10)
        Number of source nodes in the x and y directions.
    subgrid_shape : tuple[int, int], default=(3, 3)
        Size of each local tensor-product interpolation window.
    xlim, ylim : tuple[float, float]
        Coordinate limits of the rectangular domain.
    limits : tuple[tuple[float, float], tuple[float, float]], optional
        Compatibility alias for ``(xlim, ylim)``.

    Raises
    ------
    ValueError
        If either shape is not two-dimensional, if a local subgrid is
        larger than the global grid, or if ``limits`` is malformed.
    """
    if len(grid_shape) != 2 or len(subgrid_shape) != 2:
        raise ValueError("grid_shape and subgrid_shape must be of length 2.")
    if limits is not None:
        if len(limits) != 2:
            raise ValueError("limits must be ((x_min, x_max), (y_min, y_max)).")
        xlim, ylim = limits
    N_x, N_y = grid_shape
    self.N = N_x * N_y
    n_x, n_y = subgrid_shape
    if n_x > N_x or n_y > N_y:
        raise ValueError("subgrid_shape must be <= grid_shape in both dimensions.")

    self.grid_shape = (int(N_x), int(N_y))
    self.subgrid_shape = (int(n_x), int(n_y))
    self.xlim = tuple(float(v) for v in xlim)
    self.ylim = tuple(float(v) for v in ylim)

    # Generate grid indices and sliding subgrids
    self.grid, self.subgrids, self.subgrids_x, self.subgrids_y = self._generate_sliding_subgrids()

    # Generate coordinates
    (self.subgrids_x_coords,
     self.subgrids_y_coords,
     self.X, self.Y,
     self.x_coords_flat,
     self.y_coords_flat) = self._generate_coordinates()
    self.coords = np.column_stack((self.x_coords_flat, self.y_coords_flat))
    self.axes = (self.X[:, 0].copy(), self.Y[0, :].copy())
    self.dim = 2

    # Precompute pairwise differences and distances
    self.diff_x = self.x_coords_flat[:, None] - self.x_coords_flat[None, :]
    self.diff_y = self.y_coords_flat[:, None] - self.y_coords_flat[None, :]
    self.dist = self.diff_x**2 + self.diff_y**2

boundary_indices()

Return flattened node indices for the rectangular boundary.

The returned dictionary includes both geometric names "left", "right", "bottom", "top" and coordinate aliases "x_min", "x_max", "y_min", "y_max". "all" contains the unique union of all four sides.

Source code in src/gbmsc_pde/source_grid/grid.py 
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def boundary_indices(self) -> Dict[str, np.ndarray]:
        """
        Return flattened node indices for the rectangular boundary.

        The returned dictionary includes both geometric names
        ``"left"``, ``"right"``, ``"bottom"``, ``"top"`` and coordinate
        aliases ``"x_min"``, ``"x_max"``, ``"y_min"``, ``"y_max"``.
        ``"all"`` contains the unique union of all four sides.
        """
        N_x, N_y = self.grid_shape
        total = N_x * N_y

        bottom = np.arange(0, total, N_y)

        top = np.arange(N_y - 1, total, N_y)

        left = np.arange(0, N_y)

        right = np.arange((N_x - 1) * N_y, total)

        return {
            'left':   left,
            'right':  right,
            'bottom': bottom,
            'top':    top,
            "x_min": left,
            "x_max": right,
            "y_min": bottom,
            "y_max": top,
            "all": np.unique(np.concatenate([left, right, bottom, top])),
        }

eval_function_at_nodes(f)

Evaluate a scalar field at every source node.

Parameters:

Name Type Description Default
f callable

Function with signature f(x, y) where x and y are one-dimensional coordinate arrays of length N.

 required

Returns:

Type Description
(ndarray, shape(N))

Field values ordered like x_coords_flat and y_coords_flat.

Raises:

Type Description
TypeError

If f is not callable.
RuntimeError

If f raises during evaluation.
ValueError

If the result cannot be converted to an array with shape (N,).
Source code in src/gbmsc_pde/source_grid/grid.py 
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def eval_function_at_nodes(self, f: Callable[[np.ndarray, np.ndarray], np.ndarray]) -> np.ndarray:
        """
        Evaluate a scalar field at every source node.

        Parameters
        ----------
        f : callable
            Function with signature ``f(x, y)`` where ``x`` and ``y`` are
            one-dimensional coordinate arrays of length ``N``.

        Returns
        -------
        ndarray, shape (N,)
            Field values ordered like ``x_coords_flat`` and
            ``y_coords_flat``.

        Raises
        ------
        TypeError
            If ``f`` is not callable.
        RuntimeError
            If ``f`` raises during evaluation.
        ValueError
            If the result cannot be converted to an array with shape
            ``(N,)``.
        """
        # 1. Check that f is callable
        if not callable(f):
            raise TypeError("`f` must be a callable taking two numpy arrays.")

        # 2. Grab the node coordinates
        try:
            x = self.x_coords_flat
            y = self.y_coords_flat
        except AttributeError as e:
            raise AttributeError(
                "Grid node coordinates not found on `self`. "
                "Expected `self.x_coords` and `self.y_coords`. "
                f"Original error: {e}"
            )

        # 3. Attempt evaluation
        try:
            values = f(x, y)
        except Exception as e:
            raise RuntimeError(f"Error while evaluating `f` at nodes: {e}") from e

        # 4. Convert to numpy array if needed
        if not isinstance(values, np.ndarray):
            try:
                values = np.asarray(values)
            except Exception:
                raise ValueError(
                    "Output of `f` is not array-like or cannot be converted to numpy.ndarray."
                )

        # 5. Validate shape
        if values.ndim != 1 or values.shape[0] != x.shape[0]:
            raise ValueError(
                f"Function output has shape {values.shape}, "
                f"but expected (N,) where N={x.shape[0]}."
            )

        return values

plot(solution, view='3d', cmap='viridis', contour_levels=20, show_colorbar=True, xlabel='X', ylabel='Y', zlabel='', title='', figsize=(5, 4), show=True)

Plot a discrete scalar solution on this structured grid.

Parameters:

Name Type Description Default
solution ndarray

Solution values as either a flat (N_x * N_y,) array or a (N_x, N_y) grid-shaped array.

 required
view ('2d', contour, contourf, '3d')

Plot style for the discrete solution.

 "2d"
show bool

Whether to display the plot with matplotlib.pyplot.show.

 True

Returns:

Type Description
(fig, ax)

Matplotlib figure and axes objects.
Source code in src/gbmsc_pde/source_grid/grid.py 
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def plot(
    self,
    solution: np.ndarray,
    view: str = "3d",
    cmap: str = "viridis",
    contour_levels=20,
    show_colorbar: bool = True,
    xlabel: str = "X",
    ylabel: str = "Y",
    zlabel: str = "",
    title: str = "",
    figsize: Tuple[int, int] = (5, 4),
    show: bool = True,
):
    """
    Plot a discrete scalar solution on this structured grid.

    Parameters
    ----------
    solution : ndarray
        Solution values as either a flat ``(N_x * N_y,)`` array or a
        ``(N_x, N_y)`` grid-shaped array.
    view : {"2d", "contour", "contourf", "3d"}
        Plot style for the discrete solution.
    show : bool
        Whether to display the plot with ``matplotlib.pyplot.show``.

    Returns
    -------
    fig, ax
        Matplotlib figure and axes objects.
    """
    nx, ny = self.grid_shape
    sol = np.asarray(solution)
    if sol.ndim == 1:
        if sol.size != nx * ny:
            raise ValueError(
                f"1D solution length {sol.size} does not match grid size {nx * ny}"
            )
        sol = sol.reshape(nx, ny)
    elif sol.shape != (nx, ny):
        raise ValueError(f"solution array shape {sol.shape} != grid.grid_shape {(nx, ny)}")

    try:
        import matplotlib.pyplot as plt
    except ImportError as exc:
        raise ImportError(
            "SourceGrid.plot requires matplotlib. "
            "Install it with `pip install matplotlib` or `pip install .[plot]`."
        ) from exc

    fig = plt.figure(figsize=figsize)
    if view == "3d":
        ax = fig.add_subplot(111, projection="3d")
        surf = ax.plot_surface(
            self.X,
            self.Y,
            sol,
            cmap=cmap,
            edgecolor="none",
            antialiased=True,
        )
        if show_colorbar:
            fig.colorbar(surf, ax=ax, shrink=0.5)
        ax.set_zlabel(zlabel)
    else:
        ax = fig.add_subplot(111)
        if view == "2d":
            mappable = ax.imshow(
                sol.T,
                origin="lower",
                extent=(self.xlim[0], self.xlim[1], self.ylim[0], self.ylim[1]),
                cmap=cmap,
                aspect="auto",
            )
        elif view in ("contour", "contourf"):
            levels = contour_levels or 10
            plot_fn = ax.contourf if view == "contourf" else ax.contour
            mappable = plot_fn(self.X, self.Y, sol, levels=levels, cmap=cmap)
        else:
            raise ValueError(
                f"Unknown view '{view}'. Choose from '2d', 'contour', 'contourf', '3d'."
            )
        if show_colorbar:
            fig.colorbar(mappable, ax=ax)

    ax.set_xlabel(xlabel)
    ax.set_ylabel(ylabel)
    if title:
        ax.set_title(title)
    fig.tight_layout()

    if show:
        plt.show()

    return fig, ax

subgrids_with_step(step=(1, 1))

Return sampled local subgrids at a fixed stride.

Parameters:

Name Type Description Default
step tuple[int, int]

Sampling stride (step_x, step_y) over the sliding subgrid starts. The stride must be compatible with the grid so the last sampled window reaches the upper/right boundary.

 (1, 1)

Returns:

Type Description
subgrids, subgrids_x, subgrids_y : tuple[ndarray, ndarray, ndarray]

Sampled two-dimensional node windows and their one-dimensional x/y index windows.

Raises:

Type Description
ValueError

If the stride is not positive or is incompatible with the grid and local-window sizes.
Source code in src/gbmsc_pde/source_grid/grid.py 
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def subgrids_with_step(self, step: Tuple[int, int] = (1, 1)) -> Tuple[np.ndarray, np.ndarray, np.ndarray]:
    """
    Return sampled local subgrids at a fixed stride.

    Parameters
    ----------
    step : tuple[int, int], default=(1, 1)
        Sampling stride ``(step_x, step_y)`` over the sliding subgrid
        starts.  The stride must be compatible with the grid so the last
        sampled window reaches the upper/right boundary.

    Returns
    -------
    subgrids, subgrids_x, subgrids_y : tuple[ndarray, ndarray, ndarray]
        Sampled two-dimensional node windows and their one-dimensional
        x/y index windows.

    Raises
    ------
    ValueError
        If the stride is not positive or is incompatible with the grid and
        local-window sizes.
    """
    dx, dy = step
    N_x, N_y = self.grid_shape
    n_x, n_y = self.subgrid_shape

    if not isinstance(dx, int) or not isinstance(dy, int) or dx < 1 or dy < 1 or dx >= n_x or dy >= n_y:
        raise ValueError("step values must be positive integers.")

    if (N_x - n_x) % dx != 0 or (N_y - n_y) % dy != 0:
        raise ValueError(
            "step must satisfy N_x = step_x * k_x + n_x and "
            "N_y = step_y * k_y + n_y for integer k_x, k_y."
        )

    return (
        self.subgrids[::dx, ::dy, :, :],
        self.subgrids_x[::dx, :],
        self.subgrids_y[::dy, :]
    )

gbmsc_pde.approximation.shepard.GBMSCApproximation

Grid-Based Shepard approximation on a structured grid.

The approximation blends local tensor-product Lagrange polynomials defined on sampled subgrids of a :class:~gbmsc_pde.source_grid.grid.SourceGrid. It is used both for interpolation at arbitrary coordinates and for sparse nodal differential operators in the rectangular-domain PDE solver.

For nodal PDE operators, support_mode controls the normalization:

"active" Use only sampled subgrids that contain the evaluated source node. This is the original sparse rectangular-domain PDE mode. "all" Keep the sparse active contributors at source nodes, but normalize the Shepard denominator over all sampled subgrids. "nodal_limit" Use the source-node limiting formula. The common singular factor is cancelled from all active subgrids containing the source node before evaluating the weight values and derivatives.

Interpolation at arbitrary query coordinates uses all sampled subgrids.

Attributes:

Name Type Description
grid SourceGrid

Structured source grid.
step tuple[int, int]

Sampling stride for local subgrids.
subgrids_step_row (ndarray, shape(S, m))

Flattened sampled local subgrid node ids, where m = n_x * n_y.
support_mode {active, all, nodal_limit}

Nodal operator normalization mode.
Source code in src/gbmsc_pde/approximation/shepard.py 
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class GBMSCApproximation:
    """
    Grid-Based Shepard approximation on a structured grid.

    The approximation blends local tensor-product Lagrange polynomials defined
    on sampled subgrids of a :class:`~gbmsc_pde.source_grid.grid.SourceGrid`.  It is used
    both for interpolation at arbitrary coordinates and for sparse nodal
    differential operators in the rectangular-domain PDE solver.

    For nodal PDE operators, ``support_mode`` controls the normalization:

    ``"active"``
        Use only sampled subgrids that contain the evaluated source node.  This
        is the original sparse rectangular-domain PDE mode.
    ``"all"``
        Keep the sparse active contributors at source nodes, but normalize the
        Shepard denominator over all sampled subgrids.
    ``"nodal_limit"``
        Use the source-node limiting formula.  The common singular factor is
        cancelled from all active subgrids containing the source node before
        evaluating the weight values and derivatives.

    Interpolation at arbitrary query coordinates uses all sampled subgrids.

    Attributes
    ----------
    grid : SourceGrid
        Structured source grid.
    step : tuple[int, int]
        Sampling stride for local subgrids.
    subgrids_step_row : ndarray, shape (S, m)
        Flattened sampled local subgrid node ids, where
        ``m = n_x * n_y``.
    support_mode : {"active", "all", "nodal_limit"}
        Nodal operator normalization mode.
    """

    def __init__(
        self,
        grid: SourceGrid,
        step: Tuple[int, int] = (1, 1),
        mu: float = 2.005,
        eps: float = 1e-12,
        support_mode: str = "active",
    ) -> None:
        """
        Initialize the Grid-Based Shepard approximation.

        Parameters
        ----------
        grid : SourceGrid
            Structured source grid providing source-node coordinates, local subgrids,
            and pairwise distances.
        step : tuple[int, int], default=(1, 1)
            Sampling stride for local subgrid starts.
        mu : float, default=2.005
            Shepard exponent parameter.  The implementation stores ``2*mu`` so
            formulas based on squared distances recover the requested power.
        eps : float, default=1e-12
            Distance floor used to avoid division by zero and logarithms of
            zero.
        support_mode : {"active", "all", "nodal_limit"}, default="active"
            Nodal PDE-operator support mode.

        Raises
        ------
        TypeError
            If ``grid`` does not provide local subgrids.
        ValueError
            If ``step`` is invalid, ``mu <= 2``, or ``support_mode`` is
            unknown.
        """
        # Input validation
        if not hasattr(grid, 'subgrids'):
            raise TypeError("grid must have attribute 'subgrids'.")
        if any(s < 1 or not isinstance(s, int) for s in step):
            raise ValueError("step must be a tuple of positive integers.")

        if mu <= 2:
            raise ValueError("mu parameter must be greater than 2.")
        if support_mode not in ("active", "all", "nodal_limit"):
            raise ValueError("support_mode must be 'active', 'all', or 'nodal_limit'.")

        self.grid = grid
        self.step = step
        self.mu = float(2 * mu)
        self.eps = float(eps)
        self.support_mode = support_mode

        # Unpack shapes
        n_x, n_y = grid.subgrid_shape

        # Sample subgrids at the given stride, then flatten for vectorized ops
        subgrids_step, self.subgrids_x_step, self.subgrids_y_step = grid.subgrids_with_step(self.step)
        self.subgrids_step_grid = subgrids_step.reshape(-1, n_x, n_y)    # (S, n_x, n_y)
        self.subgrids_step_row = self.subgrids_step_grid.reshape(-1, n_x * n_y)  # (S, m)
        self._subgrid_x_coords = self.grid.x_coords_flat[self.subgrids_step_row]
        self._subgrid_y_coords = self.grid.y_coords_flat[self.subgrids_step_row]
        self._x_lines = self.grid.X[self.subgrids_x_step, 0]
        self._y_lines = self.grid.Y[0, self.subgrids_y_step]
        self._x_lagrange_weights = _barycentric_weights_batch(self._x_lines)
        self._y_lagrange_weights = _barycentric_weights_batch(self._y_lines)
        self._lagrange_derivative_cache = {}
        self._node_subgrid_ids, self._node_subgrid_counts = self._build_node_subgrid_lookup()
        self._safe_dist = None
        self._safe_logd = None
        self._nodal_limit_data = None
        self._all_node_subgrid_metric_cache = {}

    def _build_node_subgrid_lookup(self) -> Tuple[np.ndarray, np.ndarray]:
        """
        Build a padded lookup from global node id to sampled subgrids containing it.

        Returns
        -------
        node_subgrid_ids : (N, C) ndarray(int)
            Padded subgrid ids for each global node. Unused slots are ``-1``.
        node_subgrid_counts : (N,) ndarray(int)
            Number of valid sampled subgrids per global node.
        """
        flat_nodes = self.subgrids_step_row.ravel()
        node_subgrid_counts = np.bincount(flat_nodes, minlength=self.grid.N)
        max_count = int(node_subgrid_counts.max()) if node_subgrid_counts.size else 0
        node_subgrid_ids = np.full((self.grid.N, max_count), -1, dtype=int)
        offsets = np.zeros(self.grid.N, dtype=int)

        subgrid_ids = np.repeat(np.arange(self.subgrids_step_row.shape[0]), self.subgrids_step_row.shape[1])
        for node_id, subgrid_id in zip(flat_nodes, subgrid_ids):
            slot = offsets[node_id]
            node_subgrid_ids[node_id, slot] = subgrid_id
            offsets[node_id] = slot + 1

        return node_subgrid_ids, node_subgrid_counts

    def _safe_pairwise_data(self) -> Tuple[np.ndarray, np.ndarray]:
        """
        Return cached pairwise squared distances and their logarithm with epsilon floor.
        """
        if self._safe_dist is None or self._safe_logd is None:
            safe_dist = np.maximum(self.grid.dist, self.eps)
            self._safe_dist = safe_dist
            self._safe_logd = np.log(safe_dist)
        return self._safe_dist, self._safe_logd

    def _nodal_limit_pairwise_data(self) -> Tuple[np.ndarray, ...]:
        """
        Return pairwise data for source-node limiting weights.

        At a source node, every active subgrid contains the common singular
        factor associated with the collocation node.  The limiting formula
        cancels that factor before differentiating the normalized weight.  In
        vectorized pairwise arrays this is represented by a neutral diagonal:
        ``dist_ii = 1`` and zero diagonal derivative contributions.
        """
        if self._nodal_limit_data is not None:
            return self._nodal_limit_data

        dist = self.grid.dist.copy()
        np.fill_diagonal(dist, 1.0)
        logd = np.log(dist)

        diff_x = self.grid.diff_x
        diff_y = self.grid.diff_y
        A_x_data = 2.0 * diff_x / dist
        A_y_data = 2.0 * diff_y / dist
        A_xx_data = 2.0 / dist - 4.0 * (diff_x / dist) ** 2
        A_yy_data = 2.0 / dist - 4.0 * (diff_y / dist) ** 2
        np.fill_diagonal(A_xx_data, 0.0)
        np.fill_diagonal(A_yy_data, 0.0)

        self._nodal_limit_data = (
            dist,
            logd,
            A_x_data,
            A_y_data,
            A_xx_data,
            A_yy_data,
        )
        return self._nodal_limit_data

    def _all_node_subgrid_metric_sums(self, second_derivatives: bool = True):
        """
        Precompute node-to-subgrid metric sums for all-denominator nodal rows.

        The active-denominator path only needs the subgrids containing each
        row node.  The all-denominator path needs every sampled subgrid in the
        normalizing sum; using these ``(N, S)`` tables avoids repeatedly
        building much larger ``(block, S, m)`` distance tensors.
        """
        cache_key = bool(second_derivatives)
        cached = self._all_node_subgrid_metric_cache.get(cache_key)
        if cached is not None:
            return cached

        dist, logd = self._safe_pairwise_data()
        idx_rows = self.subgrids_step_row
        S, m = idx_rows.shape

        indicator = np.zeros((self.grid.N, S), dtype=float)
        indicator[
            idx_rows.ravel(),
            np.repeat(np.arange(S, dtype=int), m),
        ] = 1.0

        diff_x = self.grid.diff_x
        diff_y = self.grid.diff_y
        A = logd @ indicator
        A_x = (2.0 * diff_x / dist) @ indicator
        A_y = (2.0 * diff_y / dist) @ indicator

        if not second_derivatives:
            result = (A, A_x, A_y)
            self._all_node_subgrid_metric_cache[cache_key] = result
            return result

        A_xx = (2.0 / dist - 4.0 * (diff_x / dist) ** 2) @ indicator
        A_yy = (2.0 / dist - 4.0 * (diff_y / dist) ** 2) @ indicator
        result = (A, A_x, A_y, A_xx, A_yy)
        self._all_node_subgrid_metric_cache[cache_key] = result
        return result

    def eval_weight_functions_at_nodes_all_denominator_minmax_shift(
        self,
        second_derivatives=True,
        block_s: int = 64,
    ) -> Union[Tuple[np.ndarray, np.ndarray, np.ndarray, np.ndarray, np.ndarray],
               Tuple[np.ndarray, np.ndarray, np.ndarray]
               ]:
        """
        Compute nodal Shepard weights with active contributors and all denominator.

        The returned arrays have the same ``(S, m)`` shape as
        :meth:`eval_weight_functions_at_nodes_data_minmax_shift`: one row for
        each retained/sampled subgrid and one column for each local node.  Only
        the normalization denominator is enlarged to all sampled subgrids.

        Parameters
        ----------
        second_derivatives : bool, default=True
            Whether to also return second derivatives of the weights.
        block_s : int, default=64
            Number of sampled subgrids processed per vectorized block.

        Returns
        -------
        tuple[ndarray, ...]
            ``(f, f_x, f_y, f_xx, f_yy)`` when ``second_derivatives`` is true,
            otherwise ``(f, f_x, f_y)``.  Each array has shape ``(S, m)``.
        """
        mu = self.mu * 0.5
        idx_rows = self.subgrids_step_row
        S, m = idx_rows.shape

        metrics = self._all_node_subgrid_metric_sums(second_derivatives=second_derivatives)
        if second_derivatives:
            A, A_x, A_y, A_xx, A_yy = metrics
        else:
            A, A_x, A_y = metrics

        f_val = np.empty((S, m))
        f_x = np.empty_like(f_val)
        f_y = np.empty_like(f_val)
        if second_derivatives:
            f_xx = np.empty_like(f_val)
            f_yy = np.empty_like(f_val)

        for s0 in range(0, S, block_s):
            s1 = min(s0 + block_s, S)
            blk = slice(s0, s1)
            B = s1 - s0

            flat_eval_nodes = idx_rows[blk].reshape(-1)
            current_subgrid_ids = np.repeat(np.arange(s0, s1, dtype=int), m)

            A_current = A[flat_eval_nodes, current_subgrid_ids]
            A_candidates = A[flat_eval_nodes]
            DeltaA = A_current[:, None] - A_candidates

            z = mu * DeltaA
            z_max = np.max(z, axis=1, keepdims=True)
            with np.errstate(under="ignore"):
                R_shift = np.exp(z - z_max)
            R_shift_sum = R_shift.sum(axis=1)
            normalized_R = R_shift / R_shift_sum[:, None]

            f_val_block = np.exp(-z_max.ravel()) / R_shift_sum

            A_x_current = A_x[flat_eval_nodes, current_subgrid_ids]
            A_y_current = A_y[flat_eval_nodes, current_subgrid_ids]
            beta_x = mu * (A_x_current[:, None] - A_x[flat_eval_nodes])
            beta_y = mu * (A_y_current[:, None] - A_y[flat_eval_nodes])
            log_derivative_x = (normalized_R * beta_x).sum(axis=1)
            log_derivative_y = (normalized_R * beta_y).sum(axis=1)

            f_val[blk] = f_val_block.reshape(B, m)
            f_x[blk] = (-f_val_block * log_derivative_x).reshape(B, m)
            f_y[blk] = (-f_val_block * log_derivative_y).reshape(B, m)

            if second_derivatives:
                A_xx_current = A_xx[flat_eval_nodes, current_subgrid_ids]
                A_yy_current = A_yy[flat_eval_nodes, current_subgrid_ids]
                beta_xx = mu * (A_xx_current[:, None] - A_xx[flat_eval_nodes])
                beta_yy = mu * (A_yy_current[:, None] - A_yy[flat_eval_nodes])
                curvature_x = (normalized_R * (beta_xx + beta_x ** 2)).sum(axis=1)
                curvature_y = (normalized_R * (beta_yy + beta_y ** 2)).sum(axis=1)

                f_xx[blk] = (
                    f_val_block * (2.0 * log_derivative_x ** 2 - curvature_x)
                ).reshape(B, m)
                f_yy[blk] = (
                    f_val_block * (2.0 * log_derivative_y ** 2 - curvature_y)
                ).reshape(B, m)

        if second_derivatives:
            return f_val, f_x, f_y, f_xx, f_yy

        return f_val, f_x, f_y

    def _normalized_weights_at_queries_stable(
        self,
        xq: np.ndarray,
        yq: np.ndarray,
        *,
        return_derivatives: bool = False,
        regularized: bool = True,
    ) -> np.ndarray:
        """
        Stable normalized Shepard weights computed in log-space.

        This variant never forms the raw weights ``W = exp(log_w)`` directly.
        Instead it applies a max-shift per query so the largest exponent is 0,
        then normalizes the shifted weights.

        Parameters
        ----------
        xq, yq : (l,) 1-D arrays
            Query coordinates.
        return_derivatives : bool, default False
            If True, also return derivatives of the normalized weights.

        Returns
        -------
        P : (l, S) ndarray
            Normalized Shepard weights for each query and sampled sub-grid.
        Px, Py : (l, S) ndarray
            First derivatives of the normalized weights (only if requested).
        """
        xq = np.asarray(xq, dtype=float).ravel()
        yq = np.asarray(yq, dtype=float).ravel()
        if xq.size != yq.size:
            raise ValueError("x and y arrays must have same length.")

        Xn = self._subgrid_x_coords
        Yn = self._subgrid_y_coords

        # Broadcasted differences (l, S, m)
        dx = xq[:, None, None] - Xn[None, :, :]
        dy = yq[:, None, None] - Yn[None, :, :]

        # Squared distance.  The legacy path uses an epsilon floor; the
        # exact-nodal interpolation path passes only non-source queries and
        # therefore uses true distances.
        dist2 = dx * dx + dy * dy
        if regularized:
            dist2[dist2 < self.eps] = self.eps
        elif np.any(dist2 <= 0.0):
            raise ZeroDivisionError(
                "Exact non-regularized interpolation received a source-node query. "
                "Use interpolation_mode='exact_nodal' so node hits are handled separately."
            )

        log_w = (-self.mu * 0.5) * np.log(dist2).sum(axis=2)  # (l, S)

        # Stable normalization: exp(log_w - max(log_w))
        log_w_shift = log_w - np.max(log_w, axis=1, keepdims=True)
        W_shift = np.exp(log_w_shift)
        W_sum = W_shift.sum(axis=1, keepdims=True)
        P = W_shift / W_sum

        if not return_derivatives:
            return (P,)

        # alpha_x / alpha_y are derivatives of log_w
        alpha_x = -self.mu * (dx / dist2).sum(axis=2)  # (l, S)
        alpha_y = -self.mu * (dy / dist2).sum(axis=2)  # (l, S)
        mean_alpha_x = (P * alpha_x).sum(axis=1, keepdims=True)
        mean_alpha_y = (P * alpha_y).sum(axis=1, keepdims=True)

        # Derivatives of normalized weights:
        # p_i,x = p_i * (alpha_i,x - sum_j p_j alpha_j,x)
        Px = P * (alpha_x - mean_alpha_x)
        Py = P * (alpha_y - mean_alpha_y)

        return P, Px, Py

    def _source_node_hits(
        self,
        xq: np.ndarray,
        yq: np.ndarray,
        *,
        tolerance: float,
        chunk: int,
    ) -> np.ndarray:
        """Return source-node ids for query points within tolerance, else -1."""
        hits = np.full(xq.shape, -1, dtype=int)
        coords = self.grid.coords
        tol2 = float(tolerance) ** 2
        chunk = max(int(chunk), 1)

        for q0 in range(0, xq.size, chunk):
            q1 = min(q0 + chunk, xq.size)
            dx = xq[q0:q1, None] - coords[None, :, 0]
            dy = yq[q0:q1, None] - coords[None, :, 1]
            dist2 = dx * dx + dy * dy
            nearest = np.argmin(dist2, axis=1)
            nearest_dist2 = dist2[np.arange(q1 - q0), nearest]
            mask = nearest_dist2 <= tol2
            hits[q0:q1][mask] = nearest[mask]

        return hits

    def interpolator(
        self,
        x: np.ndarray,
        y: np.ndarray,
        u: np.ndarray,
        *,
        return_derivatives: bool = False,
        chunk: int = 1000,
        interpolation_mode: str = "exact_nodal",
        tolerance: float = 1e-12,
        Mx=None,
        My=None,
    ) -> Union[
        Tuple[np.ndarray],
        Tuple[np.ndarray, np.ndarray, np.ndarray]
    ]:
        """
        Interpolate a source-node field at arbitrary coordinates.

        Query interpolation uses all sampled subgrids as contributors and
        normalizes the Shepard weights over all sampled subgrids.  Computation
        is chunked and performed in log space to avoid overflow in the raw
        weights.

        ``interpolation_mode="exact_nodal"`` returns exact nodal values when a
        query point coincides with a source node within ``tolerance``.  Other
        query points use true distances without an epsilon floor.

        ``interpolation_mode="regularized"`` preserves the legacy epsilon-floor
        behavior for all query points.

        Parameters
        ----------
        x, y : array_like, shape (n_eval,)
            Query coordinates.
        u : array_like, shape (grid.N,)
            Source-node values ordered like ``grid.coords``.
        return_derivatives : bool, default=False
            If true, also return first derivatives ``du/dx`` and ``du/dy``.
        chunk : int, default=1000
            Number of query points processed per block.
        interpolation_mode : {"exact_nodal", "regularized"}, default="exact_nodal"
            Query distance policy.
        tolerance : float, default=1e-12
            Source-node hit tolerance for ``"exact_nodal"`` mode.
        Mx, My : sparse matrices, optional
            Existing first-derivative operators.  When
            ``return_derivatives=True`` and source-node hits are present, these
            matrices are reused instead of rebuilding operators.

        Returns
        -------
        tuple
            ``(u_eval,)`` or ``(u_eval, u_x, u_y)``.
        """
        xq = np.asarray(x, dtype=float).ravel()
        yq = np.asarray(y, dtype=float).ravel()
        u = np.asarray(u, dtype=float).ravel()
        if xq.size != yq.size:
            raise ValueError("x and y arrays must have same length.")
        if u.size != self.grid.N:
            raise ValueError("`u` must have length grid.N")
        if interpolation_mode not in ("exact_nodal", "regularized"):
            raise ValueError("interpolation_mode must be 'exact_nodal' or 'regularized'.")
        if tolerance < 0:
            raise ValueError("tolerance must be non-negative.")

        l = xq.size
        chunk = max(chunk, 10)

        val = np.empty(l)
        if return_derivatives:
            ux = np.empty_like(val)
            uy = np.empty_like(val)

        hit_nodes = np.full(l, -1, dtype=int)
        if interpolation_mode == "exact_nodal":
            hit_nodes = self._source_node_hits(
                xq,
                yq,
                tolerance=tolerance,
                chunk=chunk,
            )
            hit_mask = hit_nodes != -1
            if np.any(hit_mask):
                val[hit_mask] = u[hit_nodes[hit_mask]]
                if return_derivatives:
                    if Mx is None or My is None:
                        from ..operators.differential import build_differential_operators

                        Mx_built, My_built, _, _ = build_differential_operators(self)
                        if Mx is None:
                            Mx = Mx_built
                        if My is None:
                            My = My_built
                    ux[hit_mask] = Mx.tocsr()[hit_nodes[hit_mask]].dot(u)
                    uy[hit_mask] = My.tocsr()[hit_nodes[hit_mask]].dot(u)

        # Nodal values per sub-grid
        U = u[self.subgrids_step_row]  # (S, m)

        x_lines = self._x_lines
        y_lines = self._y_lines
        x_weights = self._x_lagrange_weights
        y_weights = self._y_lagrange_weights

        Sx, nx = x_lines.shape
        Sy, ny = y_lines.shape
        m = nx * ny
        S = Sx * Sy

        for q0 in range(0, l, chunk):
            q1 = min(q0 + chunk, l)
            block_indices = np.arange(q0, q1)
            if interpolation_mode == "exact_nodal":
                block_indices = block_indices[hit_nodes[q0:q1] == -1]
                if block_indices.size == 0:
                    continue

            xs = xq[block_indices]
            ys = yq[block_indices]
            l_b = block_indices.size

            P_parts = self._normalized_weights_at_queries_stable(
                xs,
                ys,
                return_derivatives=return_derivatives,
                regularized=interpolation_mode == "regularized",
            )
            P = P_parts[0]
            if return_derivatives:
                Px, Py = P_parts[1], P_parts[2]

            Lx = np.empty((l_b, Sx, nx))
            Ly = np.empty((l_b, Sy, ny))
            if return_derivatives:
                dLx = np.empty_like(Lx)
                dLy = np.empty_like(Ly)

            for s in range(Sx):
                if return_derivatives:
                    Lx[:, s], dLx[:, s] = lagrange_1d(
                        x_lines[s],
                        xs,
                        with_first_derivatives=True,
                        barycentric_weights=x_weights[s],
                    )
                else:
                    Lx[:, s] = lagrange_1d(
                        x_lines[s],
                        xs,
                        with_first_derivatives=False,
                        barycentric_weights=x_weights[s],
                    )[0]

            for s in range(Sy):
                if return_derivatives:
                    Ly[:, s], dLy[:, s] = lagrange_1d(
                        y_lines[s],
                        ys,
                        with_first_derivatives=True,
                        barycentric_weights=y_weights[s],
                    )
                else:
                    Ly[:, s] = lagrange_1d(
                        y_lines[s],
                        ys,
                        with_first_derivatives=False,
                        barycentric_weights=y_weights[s],
                    )[0]

            Phi = (Lx[:, :, None, :, None] * Ly[:, None, :, None, :]).reshape(l_b, S, m)

            Li = np.einsum('qsm,sm->qs', Phi, U, optimize=True)
            val[block_indices] = (P * Li).sum(axis=1)

            if return_derivatives:
                Phi_x = (dLx[:, :, None, :, None] * Ly[:, None, :, None, :]).reshape(l_b, S, m)
                Phi_y = (Lx[:, :, None, :, None] * dLy[:, None, :, None, :]).reshape(l_b, S, m)
                Li_x = np.einsum('qsm,sm->qs', Phi_x, U, optimize=True)
                Li_y = np.einsum('qsm,sm->qs', Phi_y, U, optimize=True)
                ux[block_indices] = (P * Li_x + Px * Li).sum(axis=1)
                uy[block_indices] = (P * Li_y + Py * Li).sum(axis=1)

        if return_derivatives:
            return val, ux, uy
        return (val,)

    def lagrange_derivatives_x_direction(self):
        """Return repeated 1D Lagrange derivative matrices in the x direction."""
        l = len(self.subgrids_x_step) 
        (I, L1, L2) = self._lagrange_derivatives_one_subgrids_1dim(self.grid.subgrids_x_coords[0])
        return (I[None, :, :].repeat(l, axis=0), L1[None, :, :].repeat(l, axis=0), L2[None, :, :].repeat(l, axis=0))

    def lagrange_derivatives_y_direction(self):
        """Return repeated 1D Lagrange derivative matrices in the y direction."""
        (I, L1, L2) = self._lagrange_derivatives_one_subgrids_1dim(self.grid.subgrids_y_coords[0])
        l = len(self.subgrids_y_step)
        return (I[None, :, :].repeat(l, axis=0), L1[None, :, :].repeat(l, axis=0), L2[None, :, :].repeat(l, axis=0))

    def _lagrange_derivatives_one_subgrids_1dim(self, X: np.ndarray) -> Tuple[np.ndarray, np.ndarray, np.ndarray]:
        """
        Compute nodal 1D Lagrange derivative matrices.

        Parameters
        ----------
        X : ndarray, shape (m,)
            Distinct interpolation nodes.

        Returns
        -------
        I, L1, L2 : tuple[ndarray, ndarray, ndarray]
            Basis, first-derivative, and second-derivative matrices evaluated
            at the nodes ``X``.
        """
        X = np.asarray(X, dtype=float)
        cache_key = tuple(X.tolist())
        cached = self._lagrange_derivative_cache.get(cache_key)
        if cached is not None:
            return cached

        m = X.size
        diff = X[:, None] - X[None, :]
        diag = np.eye(m, dtype=bool)

        # Barycentric weights
        diff[diag] = 1.0
        w = 1.0 / np.prod(diff, axis=1)

        # First derivative
        with np.errstate(divide='ignore', invalid='ignore'):
            L1 = np.where(
                diag, 
                0.0, 
                (w[None, :] / (w[:, None] * diff))
            )
        L1[diag] = -np.sum(L1, axis=1)

        # Compute S_vector for second derivative
        offdiag = ~diag
        with np.errstate(divide='ignore', invalid='ignore'):
            inv_diff = np.where(offdiag, 1.0 / diff, 0.0)
        S_vec = np.sum(inv_diff, axis=1)

        # Second derivative
        with np.errstate(divide='ignore', invalid='ignore'):
            L2 = np.where(
                offdiag,
                2 * L1 * (S_vec[:, None] - 1.0 / diff),
                0.0
            )
        L2[diag] = -np.sum(L2, axis=1)

        result = (np.eye(m), L1, L2)
        self._lagrange_derivative_cache[cache_key] = result
        return result

    def eval_weight_functions_at_nodes_data_minmax_shift(
        self,
        second_derivatives=True,
        block_s: int = 128
    ) -> Union[Tuple[np.ndarray, np.ndarray, np.ndarray, np.ndarray, np.ndarray],
               Tuple[np.ndarray, np.ndarray, np.ndarray]
               ]:
        """
        Compute active-mode nodal Shepard weights and derivatives.

        For each source-node row, only subgrids containing that source node
        participate in both the contributors and the denominator.  The
        implementation uses shifted log-ratios,

            f_i = 1 / sum_j exp(z_j),  z_j = mu * (A_i - A_j)

        and evaluates ``exp(z_j - max(z))`` instead of directly forming
        ``exp(z_j)``.

        Parameters
        ----------
        second_derivatives : bool, default=True
            Whether to also return second derivatives of the weights.
        block_s : int, default=128
            Number of sampled subgrids processed per vectorized block.

        Returns
        -------
        tuple[ndarray, ...]
            ``(f, f_x, f_y, f_xx, f_yy)`` when ``second_derivatives`` is true,
            otherwise ``(f, f_x, f_y)``.  Each array has shape ``(S, m)``.
        """
        mu = self.mu * 0.5
        diff_x, diff_y = self.grid.diff_x, self.grid.diff_y
        dist, logd = self._safe_pairwise_data()

        idx_rows = self.subgrids_step_row
        S, m = idx_rows.shape

        f_val = np.empty((S, m))
        f_x = np.empty_like(f_val)
        f_y = np.empty_like(f_val)
        if second_derivatives:
            f_xx = np.empty_like(f_val)
            f_yy = np.empty_like(f_val)

        for s0 in range(0, S, block_s):
            s1 = min(s0 + block_s, S)
            blk = slice(s0, s1)
            B = s1 - s0

            rows = idx_rows[blk]
            flat_eval_nodes = rows.reshape(-1)
            current_subgrid_ids = np.repeat(np.arange(s0, s1), m)
            current_subgrid_nodes = idx_rows[current_subgrid_ids]

            current_row_index = flat_eval_nodes[:, None]
            current_logd = logd[current_row_index, current_subgrid_nodes]
            current_dx = diff_x[current_row_index, current_subgrid_nodes]
            current_dy = diff_y[current_row_index, current_subgrid_nodes]
            current_dist = dist[current_row_index, current_subgrid_nodes]

            A_current = current_logd.sum(axis=1)
            A_x_current = (2.0 * current_dx / current_dist).sum(axis=1)
            A_y_current = (2.0 * current_dy / current_dist).sum(axis=1)

            candidate_subgrid_ids = self._node_subgrid_ids[flat_eval_nodes]
            candidate_mask = candidate_subgrid_ids != -1
            safe_candidate_ids = np.where(candidate_mask, candidate_subgrid_ids, 0)
            candidate_nodes = idx_rows[safe_candidate_ids]

            row_index = flat_eval_nodes[:, None, None]
            candidate_logd = logd[row_index, candidate_nodes]
            candidate_dx = diff_x[row_index, candidate_nodes]
            candidate_dy = diff_y[row_index, candidate_nodes]
            candidate_dist = dist[row_index, candidate_nodes]

            A_candidates = candidate_logd.sum(axis=2)
            A_x_candidates = (2.0 * candidate_dx / candidate_dist).sum(axis=2)
            A_y_candidates = (2.0 * candidate_dy / candidate_dist).sum(axis=2)

            DeltaA = A_current[:, None] - A_candidates
            DeltaA_x = A_x_current[:, None] - A_x_candidates
            DeltaA_y = A_y_current[:, None] - A_y_candidates

            if second_derivatives:
                A_xx_current = (2.0 / current_dist - 4.0 * (current_dx / current_dist) ** 2).sum(axis=1)
                A_yy_current = (2.0 / current_dist - 4.0 * (current_dy / current_dist) ** 2).sum(axis=1)
                A_xx_candidates = (2.0 / candidate_dist - 4.0 * (candidate_dx / candidate_dist) ** 2).sum(axis=2)
                A_yy_candidates = (2.0 / candidate_dist - 4.0 * (candidate_dy / candidate_dist) ** 2).sum(axis=2)
                DeltaA_xx = A_xx_current[:, None] - A_xx_candidates
                DeltaA_yy = A_yy_current[:, None] - A_yy_candidates

            z = np.where(candidate_mask, mu * DeltaA, -np.inf)
            z_max = np.max(z, axis=1, keepdims=True)

            with np.errstate(under="ignore"):
                R_shift = np.exp(z - z_max)
            R_shift *= candidate_mask

            R_shift_sum = R_shift.sum(axis=1)
            normalized_R = R_shift / R_shift_sum[:, None]

            f_val_block = np.exp(-z_max.ravel()) / R_shift_sum

            beta_x = np.where(candidate_mask, mu * DeltaA_x, 0.0)
            beta_y = np.where(candidate_mask, mu * DeltaA_y, 0.0)
            log_derivative_x = (normalized_R * beta_x).sum(axis=1)
            log_derivative_y = (normalized_R * beta_y).sum(axis=1)

            f_x_block = -f_val_block * log_derivative_x
            f_y_block = -f_val_block * log_derivative_y

            f_val[blk] = f_val_block.reshape(B, m)
            f_x[blk] = f_x_block.reshape(B, m)
            f_y[blk] = f_y_block.reshape(B, m)

            if second_derivatives:
                beta_xx = np.where(candidate_mask, mu * DeltaA_xx, 0.0)
                beta_yy = np.where(candidate_mask, mu * DeltaA_yy, 0.0)
                curvature_x = (normalized_R * (beta_xx + beta_x ** 2)).sum(axis=1)
                curvature_y = (normalized_R * (beta_yy + beta_y ** 2)).sum(axis=1)

                f_xx[blk] = (
                    f_val_block * (2.0 * log_derivative_x ** 2 - curvature_x)
                ).reshape(B, m)
                f_yy[blk] = (
                    f_val_block * (2.0 * log_derivative_y ** 2 - curvature_y)
                ).reshape(B, m)

        if second_derivatives:
            return f_val, f_x, f_y, f_xx, f_yy

        return f_val, f_x, f_y

    def eval_weight_functions_at_nodes_nodal_limit(
        self,
        second_derivatives=True,
        block_s: int = 128
    ) -> Union[Tuple[np.ndarray, np.ndarray, np.ndarray, np.ndarray, np.ndarray],
               Tuple[np.ndarray, np.ndarray, np.ndarray]
               ]:
        """
        Compute source-node limiting Shepard weights and derivatives.

        For each source-node row, only subgrids containing that source node
        participate.  The common singular factor is cancelled analytically
        before forming the normalized Shepard fraction.  In the vectorized
        metric sums this is equivalent to using a neutral self-distance and
        zero self-contributions for first and second metric derivatives.
        """
        mu = self.mu * 0.5
        _, logd, A_x_data, A_y_data, A_xx_data, A_yy_data = self._nodal_limit_pairwise_data()

        idx_rows = self.subgrids_step_row
        S, m = idx_rows.shape

        f_val = np.empty((S, m))
        f_x = np.empty_like(f_val)
        f_y = np.empty_like(f_val)
        if second_derivatives:
            f_xx = np.empty_like(f_val)
            f_yy = np.empty_like(f_val)

        for s0 in range(0, S, block_s):
            s1 = min(s0 + block_s, S)
            blk = slice(s0, s1)
            B = s1 - s0

            rows = idx_rows[blk]
            flat_eval_nodes = rows.reshape(-1)
            current_subgrid_ids = np.repeat(np.arange(s0, s1), m)
            current_subgrid_nodes = idx_rows[current_subgrid_ids]

            current_row_index = flat_eval_nodes[:, None]
            current_logd = logd[current_row_index, current_subgrid_nodes]
            current_A_x = A_x_data[current_row_index, current_subgrid_nodes]
            current_A_y = A_y_data[current_row_index, current_subgrid_nodes]

            A_current = current_logd.sum(axis=1)
            A_x_current = current_A_x.sum(axis=1)
            A_y_current = current_A_y.sum(axis=1)

            candidate_subgrid_ids = self._node_subgrid_ids[flat_eval_nodes]
            candidate_mask = candidate_subgrid_ids != -1
            safe_candidate_ids = np.where(candidate_mask, candidate_subgrid_ids, 0)
            candidate_nodes = idx_rows[safe_candidate_ids]

            row_index = flat_eval_nodes[:, None, None]
            candidate_logd = logd[row_index, candidate_nodes]
            candidate_A_x = A_x_data[row_index, candidate_nodes]
            candidate_A_y = A_y_data[row_index, candidate_nodes]

            A_candidates = candidate_logd.sum(axis=2)
            A_x_candidates = candidate_A_x.sum(axis=2)
            A_y_candidates = candidate_A_y.sum(axis=2)

            DeltaA = A_current[:, None] - A_candidates
            DeltaA_x = A_x_current[:, None] - A_x_candidates
            DeltaA_y = A_y_current[:, None] - A_y_candidates

            if second_derivatives:
                current_A_xx = A_xx_data[current_row_index, current_subgrid_nodes]
                current_A_yy = A_yy_data[current_row_index, current_subgrid_nodes]
                candidate_A_xx = A_xx_data[row_index, candidate_nodes]
                candidate_A_yy = A_yy_data[row_index, candidate_nodes]
                A_xx_current = current_A_xx.sum(axis=1)
                A_yy_current = current_A_yy.sum(axis=1)
                A_xx_candidates = candidate_A_xx.sum(axis=2)
                A_yy_candidates = candidate_A_yy.sum(axis=2)
                DeltaA_xx = A_xx_current[:, None] - A_xx_candidates
                DeltaA_yy = A_yy_current[:, None] - A_yy_candidates

            z = np.where(candidate_mask, mu * DeltaA, -np.inf)
            z_max = np.max(z, axis=1, keepdims=True)

            with np.errstate(under="ignore"):
                R_shift = np.exp(z - z_max)
            R_shift *= candidate_mask

            R_shift_sum = R_shift.sum(axis=1)
            normalized_R = R_shift / R_shift_sum[:, None]

            f_val_block = np.exp(-z_max.ravel()) / R_shift_sum

            beta_x = np.where(candidate_mask, mu * DeltaA_x, 0.0)
            beta_y = np.where(candidate_mask, mu * DeltaA_y, 0.0)
            log_derivative_x = (normalized_R * beta_x).sum(axis=1)
            log_derivative_y = (normalized_R * beta_y).sum(axis=1)

            f_x_block = -f_val_block * log_derivative_x
            f_y_block = -f_val_block * log_derivative_y

            f_val[blk] = f_val_block.reshape(B, m)
            f_x[blk] = f_x_block.reshape(B, m)
            f_y[blk] = f_y_block.reshape(B, m)

            if second_derivatives:
                beta_xx = np.where(candidate_mask, mu * DeltaA_xx, 0.0)
                beta_yy = np.where(candidate_mask, mu * DeltaA_yy, 0.0)
                curvature_x = (normalized_R * (beta_xx + beta_x ** 2)).sum(axis=1)
                curvature_y = (normalized_R * (beta_yy + beta_y ** 2)).sum(axis=1)

                f_xx[blk] = (
                    f_val_block * (2.0 * log_derivative_x ** 2 - curvature_x)
                ).reshape(B, m)
                f_yy[blk] = (
                    f_val_block * (2.0 * log_derivative_y ** 2 - curvature_y)
                ).reshape(B, m)

        if second_derivatives:
            return f_val, f_x, f_y, f_xx, f_yy

        return f_val, f_x, f_y

__init__(grid, step=(1, 1), mu=2.005, eps=1e-12, support_mode='active')

Initialize the Grid-Based Shepard approximation.

Parameters:

Name Type Description Default
grid SourceGrid

Structured source grid providing source-node coordinates, local subgrids, and pairwise distances.

 required
step tuple[int, int]

Sampling stride for local subgrid starts.

 (1, 1)
mu float

Shepard exponent parameter. The implementation stores 2*mu so formulas based on squared distances recover the requested power.

 2.005
eps float

Distance floor used to avoid division by zero and logarithms of zero.

 1e-12
support_mode (active, all, nodal_limit)

Nodal PDE-operator support mode.

 "active"

Raises:

Type Description
TypeError

If grid does not provide local subgrids.
ValueError

If step is invalid, mu <= 2, or support_mode is unknown.
Source code in src/gbmsc_pde/approximation/shepard.py 
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def __init__(
    self,
    grid: SourceGrid,
    step: Tuple[int, int] = (1, 1),
    mu: float = 2.005,
    eps: float = 1e-12,
    support_mode: str = "active",
) -> None:
    """
    Initialize the Grid-Based Shepard approximation.

    Parameters
    ----------
    grid : SourceGrid
        Structured source grid providing source-node coordinates, local subgrids,
        and pairwise distances.
    step : tuple[int, int], default=(1, 1)
        Sampling stride for local subgrid starts.
    mu : float, default=2.005
        Shepard exponent parameter.  The implementation stores ``2*mu`` so
        formulas based on squared distances recover the requested power.
    eps : float, default=1e-12
        Distance floor used to avoid division by zero and logarithms of
        zero.
    support_mode : {"active", "all", "nodal_limit"}, default="active"
        Nodal PDE-operator support mode.

    Raises
    ------
    TypeError
        If ``grid`` does not provide local subgrids.
    ValueError
        If ``step`` is invalid, ``mu <= 2``, or ``support_mode`` is
        unknown.
    """
    # Input validation
    if not hasattr(grid, 'subgrids'):
        raise TypeError("grid must have attribute 'subgrids'.")
    if any(s < 1 or not isinstance(s, int) for s in step):
        raise ValueError("step must be a tuple of positive integers.")

    if mu <= 2:
        raise ValueError("mu parameter must be greater than 2.")
    if support_mode not in ("active", "all", "nodal_limit"):
        raise ValueError("support_mode must be 'active', 'all', or 'nodal_limit'.")

    self.grid = grid
    self.step = step
    self.mu = float(2 * mu)
    self.eps = float(eps)
    self.support_mode = support_mode

    # Unpack shapes
    n_x, n_y = grid.subgrid_shape

    # Sample subgrids at the given stride, then flatten for vectorized ops
    subgrids_step, self.subgrids_x_step, self.subgrids_y_step = grid.subgrids_with_step(self.step)
    self.subgrids_step_grid = subgrids_step.reshape(-1, n_x, n_y)    # (S, n_x, n_y)
    self.subgrids_step_row = self.subgrids_step_grid.reshape(-1, n_x * n_y)  # (S, m)
    self._subgrid_x_coords = self.grid.x_coords_flat[self.subgrids_step_row]
    self._subgrid_y_coords = self.grid.y_coords_flat[self.subgrids_step_row]
    self._x_lines = self.grid.X[self.subgrids_x_step, 0]
    self._y_lines = self.grid.Y[0, self.subgrids_y_step]
    self._x_lagrange_weights = _barycentric_weights_batch(self._x_lines)
    self._y_lagrange_weights = _barycentric_weights_batch(self._y_lines)
    self._lagrange_derivative_cache = {}
    self._node_subgrid_ids, self._node_subgrid_counts = self._build_node_subgrid_lookup()
    self._safe_dist = None
    self._safe_logd = None
    self._nodal_limit_data = None
    self._all_node_subgrid_metric_cache = {}

eval_weight_functions_at_nodes_all_denominator_minmax_shift(second_derivatives=True, block_s=64)

Compute nodal Shepard weights with active contributors and all denominator.

The returned arrays have the same (S, m) shape as :meth:eval_weight_functions_at_nodes_data_minmax_shift: one row for each retained/sampled subgrid and one column for each local node. Only the normalization denominator is enlarged to all sampled subgrids.

Parameters:

Name Type Description Default
second_derivatives bool

Whether to also return second derivatives of the weights.

 True
block_s int

Number of sampled subgrids processed per vectorized block.

 64

Returns:

Type Description
tuple[ndarray, ...]

(f, f_x, f_y, f_xx, f_yy) when second_derivatives is true, otherwise (f, f_x, f_y). Each array has shape (S, m).
Source code in src/gbmsc_pde/approximation/shepard.py 
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def eval_weight_functions_at_nodes_all_denominator_minmax_shift(
    self,
    second_derivatives=True,
    block_s: int = 64,
) -> Union[Tuple[np.ndarray, np.ndarray, np.ndarray, np.ndarray, np.ndarray],
           Tuple[np.ndarray, np.ndarray, np.ndarray]
           ]:
    """
    Compute nodal Shepard weights with active contributors and all denominator.

    The returned arrays have the same ``(S, m)`` shape as
    :meth:`eval_weight_functions_at_nodes_data_minmax_shift`: one row for
    each retained/sampled subgrid and one column for each local node.  Only
    the normalization denominator is enlarged to all sampled subgrids.

    Parameters
    ----------
    second_derivatives : bool, default=True
        Whether to also return second derivatives of the weights.
    block_s : int, default=64
        Number of sampled subgrids processed per vectorized block.

    Returns
    -------
    tuple[ndarray, ...]
        ``(f, f_x, f_y, f_xx, f_yy)`` when ``second_derivatives`` is true,
        otherwise ``(f, f_x, f_y)``.  Each array has shape ``(S, m)``.
    """
    mu = self.mu * 0.5
    idx_rows = self.subgrids_step_row
    S, m = idx_rows.shape

    metrics = self._all_node_subgrid_metric_sums(second_derivatives=second_derivatives)
    if second_derivatives:
        A, A_x, A_y, A_xx, A_yy = metrics
    else:
        A, A_x, A_y = metrics

    f_val = np.empty((S, m))
    f_x = np.empty_like(f_val)
    f_y = np.empty_like(f_val)
    if second_derivatives:
        f_xx = np.empty_like(f_val)
        f_yy = np.empty_like(f_val)

    for s0 in range(0, S, block_s):
        s1 = min(s0 + block_s, S)
        blk = slice(s0, s1)
        B = s1 - s0

        flat_eval_nodes = idx_rows[blk].reshape(-1)
        current_subgrid_ids = np.repeat(np.arange(s0, s1, dtype=int), m)

        A_current = A[flat_eval_nodes, current_subgrid_ids]
        A_candidates = A[flat_eval_nodes]
        DeltaA = A_current[:, None] - A_candidates

        z = mu * DeltaA
        z_max = np.max(z, axis=1, keepdims=True)
        with np.errstate(under="ignore"):
            R_shift = np.exp(z - z_max)
        R_shift_sum = R_shift.sum(axis=1)
        normalized_R = R_shift / R_shift_sum[:, None]

        f_val_block = np.exp(-z_max.ravel()) / R_shift_sum

        A_x_current = A_x[flat_eval_nodes, current_subgrid_ids]
        A_y_current = A_y[flat_eval_nodes, current_subgrid_ids]
        beta_x = mu * (A_x_current[:, None] - A_x[flat_eval_nodes])
        beta_y = mu * (A_y_current[:, None] - A_y[flat_eval_nodes])
        log_derivative_x = (normalized_R * beta_x).sum(axis=1)
        log_derivative_y = (normalized_R * beta_y).sum(axis=1)

        f_val[blk] = f_val_block.reshape(B, m)
        f_x[blk] = (-f_val_block * log_derivative_x).reshape(B, m)
        f_y[blk] = (-f_val_block * log_derivative_y).reshape(B, m)

        if second_derivatives:
            A_xx_current = A_xx[flat_eval_nodes, current_subgrid_ids]
            A_yy_current = A_yy[flat_eval_nodes, current_subgrid_ids]
            beta_xx = mu * (A_xx_current[:, None] - A_xx[flat_eval_nodes])
            beta_yy = mu * (A_yy_current[:, None] - A_yy[flat_eval_nodes])
            curvature_x = (normalized_R * (beta_xx + beta_x ** 2)).sum(axis=1)
            curvature_y = (normalized_R * (beta_yy + beta_y ** 2)).sum(axis=1)

            f_xx[blk] = (
                f_val_block * (2.0 * log_derivative_x ** 2 - curvature_x)
            ).reshape(B, m)
            f_yy[blk] = (
                f_val_block * (2.0 * log_derivative_y ** 2 - curvature_y)
            ).reshape(B, m)

    if second_derivatives:
        return f_val, f_x, f_y, f_xx, f_yy

    return f_val, f_x, f_y

eval_weight_functions_at_nodes_data_minmax_shift(second_derivatives=True, block_s=128)

Compute active-mode nodal Shepard weights and derivatives.

For each source-node row, only subgrids containing that source node participate in both the contributors and the denominator. The implementation uses shifted log-ratios,

f_i = 1 / sum_j exp(z_j),  z_j = mu * (A_i - A_j)

and evaluates exp(z_j - max(z)) instead of directly forming exp(z_j).

Parameters:

Name Type Description Default
second_derivatives bool

Whether to also return second derivatives of the weights.

 True
block_s int

Number of sampled subgrids processed per vectorized block.

 128

Returns:

Type Description
tuple[ndarray, ...]

(f, f_x, f_y, f_xx, f_yy) when second_derivatives is true, otherwise (f, f_x, f_y). Each array has shape (S, m).
Source code in src/gbmsc_pde/approximation/shepard.py 
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def eval_weight_functions_at_nodes_data_minmax_shift(
    self,
    second_derivatives=True,
    block_s: int = 128
) -> Union[Tuple[np.ndarray, np.ndarray, np.ndarray, np.ndarray, np.ndarray],
           Tuple[np.ndarray, np.ndarray, np.ndarray]
           ]:
    """
    Compute active-mode nodal Shepard weights and derivatives.

    For each source-node row, only subgrids containing that source node
    participate in both the contributors and the denominator.  The
    implementation uses shifted log-ratios,

        f_i = 1 / sum_j exp(z_j),  z_j = mu * (A_i - A_j)

    and evaluates ``exp(z_j - max(z))`` instead of directly forming
    ``exp(z_j)``.

    Parameters
    ----------
    second_derivatives : bool, default=True
        Whether to also return second derivatives of the weights.
    block_s : int, default=128
        Number of sampled subgrids processed per vectorized block.

    Returns
    -------
    tuple[ndarray, ...]
        ``(f, f_x, f_y, f_xx, f_yy)`` when ``second_derivatives`` is true,
        otherwise ``(f, f_x, f_y)``.  Each array has shape ``(S, m)``.
    """
    mu = self.mu * 0.5
    diff_x, diff_y = self.grid.diff_x, self.grid.diff_y
    dist, logd = self._safe_pairwise_data()

    idx_rows = self.subgrids_step_row
    S, m = idx_rows.shape

    f_val = np.empty((S, m))
    f_x = np.empty_like(f_val)
    f_y = np.empty_like(f_val)
    if second_derivatives:
        f_xx = np.empty_like(f_val)
        f_yy = np.empty_like(f_val)

    for s0 in range(0, S, block_s):
        s1 = min(s0 + block_s, S)
        blk = slice(s0, s1)
        B = s1 - s0

        rows = idx_rows[blk]
        flat_eval_nodes = rows.reshape(-1)
        current_subgrid_ids = np.repeat(np.arange(s0, s1), m)
        current_subgrid_nodes = idx_rows[current_subgrid_ids]

        current_row_index = flat_eval_nodes[:, None]
        current_logd = logd[current_row_index, current_subgrid_nodes]
        current_dx = diff_x[current_row_index, current_subgrid_nodes]
        current_dy = diff_y[current_row_index, current_subgrid_nodes]
        current_dist = dist[current_row_index, current_subgrid_nodes]

        A_current = current_logd.sum(axis=1)
        A_x_current = (2.0 * current_dx / current_dist).sum(axis=1)
        A_y_current = (2.0 * current_dy / current_dist).sum(axis=1)

        candidate_subgrid_ids = self._node_subgrid_ids[flat_eval_nodes]
        candidate_mask = candidate_subgrid_ids != -1
        safe_candidate_ids = np.where(candidate_mask, candidate_subgrid_ids, 0)
        candidate_nodes = idx_rows[safe_candidate_ids]

        row_index = flat_eval_nodes[:, None, None]
        candidate_logd = logd[row_index, candidate_nodes]
        candidate_dx = diff_x[row_index, candidate_nodes]
        candidate_dy = diff_y[row_index, candidate_nodes]
        candidate_dist = dist[row_index, candidate_nodes]

        A_candidates = candidate_logd.sum(axis=2)
        A_x_candidates = (2.0 * candidate_dx / candidate_dist).sum(axis=2)
        A_y_candidates = (2.0 * candidate_dy / candidate_dist).sum(axis=2)

        DeltaA = A_current[:, None] - A_candidates
        DeltaA_x = A_x_current[:, None] - A_x_candidates
        DeltaA_y = A_y_current[:, None] - A_y_candidates

        if second_derivatives:
            A_xx_current = (2.0 / current_dist - 4.0 * (current_dx / current_dist) ** 2).sum(axis=1)
            A_yy_current = (2.0 / current_dist - 4.0 * (current_dy / current_dist) ** 2).sum(axis=1)
            A_xx_candidates = (2.0 / candidate_dist - 4.0 * (candidate_dx / candidate_dist) ** 2).sum(axis=2)
            A_yy_candidates = (2.0 / candidate_dist - 4.0 * (candidate_dy / candidate_dist) ** 2).sum(axis=2)
            DeltaA_xx = A_xx_current[:, None] - A_xx_candidates
            DeltaA_yy = A_yy_current[:, None] - A_yy_candidates

        z = np.where(candidate_mask, mu * DeltaA, -np.inf)
        z_max = np.max(z, axis=1, keepdims=True)

        with np.errstate(under="ignore"):
            R_shift = np.exp(z - z_max)
        R_shift *= candidate_mask

        R_shift_sum = R_shift.sum(axis=1)
        normalized_R = R_shift / R_shift_sum[:, None]

        f_val_block = np.exp(-z_max.ravel()) / R_shift_sum

        beta_x = np.where(candidate_mask, mu * DeltaA_x, 0.0)
        beta_y = np.where(candidate_mask, mu * DeltaA_y, 0.0)
        log_derivative_x = (normalized_R * beta_x).sum(axis=1)
        log_derivative_y = (normalized_R * beta_y).sum(axis=1)

        f_x_block = -f_val_block * log_derivative_x
        f_y_block = -f_val_block * log_derivative_y

        f_val[blk] = f_val_block.reshape(B, m)
        f_x[blk] = f_x_block.reshape(B, m)
        f_y[blk] = f_y_block.reshape(B, m)

        if second_derivatives:
            beta_xx = np.where(candidate_mask, mu * DeltaA_xx, 0.0)
            beta_yy = np.where(candidate_mask, mu * DeltaA_yy, 0.0)
            curvature_x = (normalized_R * (beta_xx + beta_x ** 2)).sum(axis=1)
            curvature_y = (normalized_R * (beta_yy + beta_y ** 2)).sum(axis=1)

            f_xx[blk] = (
                f_val_block * (2.0 * log_derivative_x ** 2 - curvature_x)
            ).reshape(B, m)
            f_yy[blk] = (
                f_val_block * (2.0 * log_derivative_y ** 2 - curvature_y)
            ).reshape(B, m)

    if second_derivatives:
        return f_val, f_x, f_y, f_xx, f_yy

    return f_val, f_x, f_y

eval_weight_functions_at_nodes_nodal_limit(second_derivatives=True, block_s=128)

Compute source-node limiting Shepard weights and derivatives.

For each source-node row, only subgrids containing that source node participate. The common singular factor is cancelled analytically before forming the normalized Shepard fraction. In the vectorized metric sums this is equivalent to using a neutral self-distance and zero self-contributions for first and second metric derivatives.

Source code in src/gbmsc_pde/approximation/shepard.py 
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def eval_weight_functions_at_nodes_nodal_limit(
    self,
    second_derivatives=True,
    block_s: int = 128
) -> Union[Tuple[np.ndarray, np.ndarray, np.ndarray, np.ndarray, np.ndarray],
           Tuple[np.ndarray, np.ndarray, np.ndarray]
           ]:
    """
    Compute source-node limiting Shepard weights and derivatives.

    For each source-node row, only subgrids containing that source node
    participate.  The common singular factor is cancelled analytically
    before forming the normalized Shepard fraction.  In the vectorized
    metric sums this is equivalent to using a neutral self-distance and
    zero self-contributions for first and second metric derivatives.
    """
    mu = self.mu * 0.5
    _, logd, A_x_data, A_y_data, A_xx_data, A_yy_data = self._nodal_limit_pairwise_data()

    idx_rows = self.subgrids_step_row
    S, m = idx_rows.shape

    f_val = np.empty((S, m))
    f_x = np.empty_like(f_val)
    f_y = np.empty_like(f_val)
    if second_derivatives:
        f_xx = np.empty_like(f_val)
        f_yy = np.empty_like(f_val)

    for s0 in range(0, S, block_s):
        s1 = min(s0 + block_s, S)
        blk = slice(s0, s1)
        B = s1 - s0

        rows = idx_rows[blk]
        flat_eval_nodes = rows.reshape(-1)
        current_subgrid_ids = np.repeat(np.arange(s0, s1), m)
        current_subgrid_nodes = idx_rows[current_subgrid_ids]

        current_row_index = flat_eval_nodes[:, None]
        current_logd = logd[current_row_index, current_subgrid_nodes]
        current_A_x = A_x_data[current_row_index, current_subgrid_nodes]
        current_A_y = A_y_data[current_row_index, current_subgrid_nodes]

        A_current = current_logd.sum(axis=1)
        A_x_current = current_A_x.sum(axis=1)
        A_y_current = current_A_y.sum(axis=1)

        candidate_subgrid_ids = self._node_subgrid_ids[flat_eval_nodes]
        candidate_mask = candidate_subgrid_ids != -1
        safe_candidate_ids = np.where(candidate_mask, candidate_subgrid_ids, 0)
        candidate_nodes = idx_rows[safe_candidate_ids]

        row_index = flat_eval_nodes[:, None, None]
        candidate_logd = logd[row_index, candidate_nodes]
        candidate_A_x = A_x_data[row_index, candidate_nodes]
        candidate_A_y = A_y_data[row_index, candidate_nodes]

        A_candidates = candidate_logd.sum(axis=2)
        A_x_candidates = candidate_A_x.sum(axis=2)
        A_y_candidates = candidate_A_y.sum(axis=2)

        DeltaA = A_current[:, None] - A_candidates
        DeltaA_x = A_x_current[:, None] - A_x_candidates
        DeltaA_y = A_y_current[:, None] - A_y_candidates

        if second_derivatives:
            current_A_xx = A_xx_data[current_row_index, current_subgrid_nodes]
            current_A_yy = A_yy_data[current_row_index, current_subgrid_nodes]
            candidate_A_xx = A_xx_data[row_index, candidate_nodes]
            candidate_A_yy = A_yy_data[row_index, candidate_nodes]
            A_xx_current = current_A_xx.sum(axis=1)
            A_yy_current = current_A_yy.sum(axis=1)
            A_xx_candidates = candidate_A_xx.sum(axis=2)
            A_yy_candidates = candidate_A_yy.sum(axis=2)
            DeltaA_xx = A_xx_current[:, None] - A_xx_candidates
            DeltaA_yy = A_yy_current[:, None] - A_yy_candidates

        z = np.where(candidate_mask, mu * DeltaA, -np.inf)
        z_max = np.max(z, axis=1, keepdims=True)

        with np.errstate(under="ignore"):
            R_shift = np.exp(z - z_max)
        R_shift *= candidate_mask

        R_shift_sum = R_shift.sum(axis=1)
        normalized_R = R_shift / R_shift_sum[:, None]

        f_val_block = np.exp(-z_max.ravel()) / R_shift_sum

        beta_x = np.where(candidate_mask, mu * DeltaA_x, 0.0)
        beta_y = np.where(candidate_mask, mu * DeltaA_y, 0.0)
        log_derivative_x = (normalized_R * beta_x).sum(axis=1)
        log_derivative_y = (normalized_R * beta_y).sum(axis=1)

        f_x_block = -f_val_block * log_derivative_x
        f_y_block = -f_val_block * log_derivative_y

        f_val[blk] = f_val_block.reshape(B, m)
        f_x[blk] = f_x_block.reshape(B, m)
        f_y[blk] = f_y_block.reshape(B, m)

        if second_derivatives:
            beta_xx = np.where(candidate_mask, mu * DeltaA_xx, 0.0)
            beta_yy = np.where(candidate_mask, mu * DeltaA_yy, 0.0)
            curvature_x = (normalized_R * (beta_xx + beta_x ** 2)).sum(axis=1)
            curvature_y = (normalized_R * (beta_yy + beta_y ** 2)).sum(axis=1)

            f_xx[blk] = (
                f_val_block * (2.0 * log_derivative_x ** 2 - curvature_x)
            ).reshape(B, m)
            f_yy[blk] = (
                f_val_block * (2.0 * log_derivative_y ** 2 - curvature_y)
            ).reshape(B, m)

    if second_derivatives:
        return f_val, f_x, f_y, f_xx, f_yy

    return f_val, f_x, f_y

interpolator(x, y, u, *, return_derivatives=False, chunk=1000, interpolation_mode='exact_nodal', tolerance=1e-12, Mx=None, My=None)

Interpolate a source-node field at arbitrary coordinates.

Query interpolation uses all sampled subgrids as contributors and normalizes the Shepard weights over all sampled subgrids. Computation is chunked and performed in log space to avoid overflow in the raw weights.

interpolation_mode="exact_nodal" returns exact nodal values when a query point coincides with a source node within tolerance. Other query points use true distances without an epsilon floor.

interpolation_mode="regularized" preserves the legacy epsilon-floor behavior for all query points.

Parameters:

Name Type Description Default
x (array_like, shape(n_eval))

Query coordinates.

 required
y (array_like, shape(n_eval))

Query coordinates.

 required
u (array_like, shape(N))

Source-node values ordered like grid.coords.

 required
return_derivatives bool

If true, also return first derivatives du/dx and du/dy.

 False
chunk int

Number of query points processed per block.

 1000
interpolation_mode (exact_nodal, regularized)

Query distance policy.

 "exact_nodal"
tolerance float

Source-node hit tolerance for "exact_nodal" mode.

 1e-12
Mx sparse matrices

Existing first-derivative operators. When return_derivatives=True and source-node hits are present, these matrices are reused instead of rebuilding operators.

 None
My sparse matrices

Existing first-derivative operators. When return_derivatives=True and source-node hits are present, these matrices are reused instead of rebuilding operators.

 None

Returns:

Type Description
tuple

(u_eval,) or (u_eval, u_x, u_y).
Source code in src/gbmsc_pde/approximation/shepard.py 
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def interpolator(
    self,
    x: np.ndarray,
    y: np.ndarray,
    u: np.ndarray,
    *,
    return_derivatives: bool = False,
    chunk: int = 1000,
    interpolation_mode: str = "exact_nodal",
    tolerance: float = 1e-12,
    Mx=None,
    My=None,
) -> Union[
    Tuple[np.ndarray],
    Tuple[np.ndarray, np.ndarray, np.ndarray]
]:
    """
    Interpolate a source-node field at arbitrary coordinates.

    Query interpolation uses all sampled subgrids as contributors and
    normalizes the Shepard weights over all sampled subgrids.  Computation
    is chunked and performed in log space to avoid overflow in the raw
    weights.

    ``interpolation_mode="exact_nodal"`` returns exact nodal values when a
    query point coincides with a source node within ``tolerance``.  Other
    query points use true distances without an epsilon floor.

    ``interpolation_mode="regularized"`` preserves the legacy epsilon-floor
    behavior for all query points.

    Parameters
    ----------
    x, y : array_like, shape (n_eval,)
        Query coordinates.
    u : array_like, shape (grid.N,)
        Source-node values ordered like ``grid.coords``.
    return_derivatives : bool, default=False
        If true, also return first derivatives ``du/dx`` and ``du/dy``.
    chunk : int, default=1000
        Number of query points processed per block.
    interpolation_mode : {"exact_nodal", "regularized"}, default="exact_nodal"
        Query distance policy.
    tolerance : float, default=1e-12
        Source-node hit tolerance for ``"exact_nodal"`` mode.
    Mx, My : sparse matrices, optional
        Existing first-derivative operators.  When
        ``return_derivatives=True`` and source-node hits are present, these
        matrices are reused instead of rebuilding operators.

    Returns
    -------
    tuple
        ``(u_eval,)`` or ``(u_eval, u_x, u_y)``.
    """
    xq = np.asarray(x, dtype=float).ravel()
    yq = np.asarray(y, dtype=float).ravel()
    u = np.asarray(u, dtype=float).ravel()
    if xq.size != yq.size:
        raise ValueError("x and y arrays must have same length.")
    if u.size != self.grid.N:
        raise ValueError("`u` must have length grid.N")
    if interpolation_mode not in ("exact_nodal", "regularized"):
        raise ValueError("interpolation_mode must be 'exact_nodal' or 'regularized'.")
    if tolerance < 0:
        raise ValueError("tolerance must be non-negative.")

    l = xq.size
    chunk = max(chunk, 10)

    val = np.empty(l)
    if return_derivatives:
        ux = np.empty_like(val)
        uy = np.empty_like(val)

    hit_nodes = np.full(l, -1, dtype=int)
    if interpolation_mode == "exact_nodal":
        hit_nodes = self._source_node_hits(
            xq,
            yq,
            tolerance=tolerance,
            chunk=chunk,
        )
        hit_mask = hit_nodes != -1
        if np.any(hit_mask):
            val[hit_mask] = u[hit_nodes[hit_mask]]
            if return_derivatives:
                if Mx is None or My is None:
                    from ..operators.differential import build_differential_operators

                    Mx_built, My_built, _, _ = build_differential_operators(self)
                    if Mx is None:
                        Mx = Mx_built
                    if My is None:
                        My = My_built
                ux[hit_mask] = Mx.tocsr()[hit_nodes[hit_mask]].dot(u)
                uy[hit_mask] = My.tocsr()[hit_nodes[hit_mask]].dot(u)

    # Nodal values per sub-grid
    U = u[self.subgrids_step_row]  # (S, m)

    x_lines = self._x_lines
    y_lines = self._y_lines
    x_weights = self._x_lagrange_weights
    y_weights = self._y_lagrange_weights

    Sx, nx = x_lines.shape
    Sy, ny = y_lines.shape
    m = nx * ny
    S = Sx * Sy

    for q0 in range(0, l, chunk):
        q1 = min(q0 + chunk, l)
        block_indices = np.arange(q0, q1)
        if interpolation_mode == "exact_nodal":
            block_indices = block_indices[hit_nodes[q0:q1] == -1]
            if block_indices.size == 0:
                continue

        xs = xq[block_indices]
        ys = yq[block_indices]
        l_b = block_indices.size

        P_parts = self._normalized_weights_at_queries_stable(
            xs,
            ys,
            return_derivatives=return_derivatives,
            regularized=interpolation_mode == "regularized",
        )
        P = P_parts[0]
        if return_derivatives:
            Px, Py = P_parts[1], P_parts[2]

        Lx = np.empty((l_b, Sx, nx))
        Ly = np.empty((l_b, Sy, ny))
        if return_derivatives:
            dLx = np.empty_like(Lx)
            dLy = np.empty_like(Ly)

        for s in range(Sx):
            if return_derivatives:
                Lx[:, s], dLx[:, s] = lagrange_1d(
                    x_lines[s],
                    xs,
                    with_first_derivatives=True,
                    barycentric_weights=x_weights[s],
                )
            else:
                Lx[:, s] = lagrange_1d(
                    x_lines[s],
                    xs,
                    with_first_derivatives=False,
                    barycentric_weights=x_weights[s],
                )[0]

        for s in range(Sy):
            if return_derivatives:
                Ly[:, s], dLy[:, s] = lagrange_1d(
                    y_lines[s],
                    ys,
                    with_first_derivatives=True,
                    barycentric_weights=y_weights[s],
                )
            else:
                Ly[:, s] = lagrange_1d(
                    y_lines[s],
                    ys,
                    with_first_derivatives=False,
                    barycentric_weights=y_weights[s],
                )[0]

        Phi = (Lx[:, :, None, :, None] * Ly[:, None, :, None, :]).reshape(l_b, S, m)

        Li = np.einsum('qsm,sm->qs', Phi, U, optimize=True)
        val[block_indices] = (P * Li).sum(axis=1)

        if return_derivatives:
            Phi_x = (dLx[:, :, None, :, None] * Ly[:, None, :, None, :]).reshape(l_b, S, m)
            Phi_y = (Lx[:, :, None, :, None] * dLy[:, None, :, None, :]).reshape(l_b, S, m)
            Li_x = np.einsum('qsm,sm->qs', Phi_x, U, optimize=True)
            Li_y = np.einsum('qsm,sm->qs', Phi_y, U, optimize=True)
            ux[block_indices] = (P * Li_x + Px * Li).sum(axis=1)
            uy[block_indices] = (P * Li_y + Py * Li).sum(axis=1)

    if return_derivatives:
        return val, ux, uy
    return (val,)

lagrange_derivatives_x_direction()

Return repeated 1D Lagrange derivative matrices in the x direction.

Source code in src/gbmsc_pde/approximation/shepard.py 
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def lagrange_derivatives_x_direction(self):
    """Return repeated 1D Lagrange derivative matrices in the x direction."""
    l = len(self.subgrids_x_step) 
    (I, L1, L2) = self._lagrange_derivatives_one_subgrids_1dim(self.grid.subgrids_x_coords[0])
    return (I[None, :, :].repeat(l, axis=0), L1[None, :, :].repeat(l, axis=0), L2[None, :, :].repeat(l, axis=0))

lagrange_derivatives_y_direction()

Return repeated 1D Lagrange derivative matrices in the y direction.

Source code in src/gbmsc_pde/approximation/shepard.py 
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def lagrange_derivatives_y_direction(self):
    """Return repeated 1D Lagrange derivative matrices in the y direction."""
    (I, L1, L2) = self._lagrange_derivatives_one_subgrids_1dim(self.grid.subgrids_y_coords[0])
    l = len(self.subgrids_y_step)
    return (I[None, :, :].repeat(l, axis=0), L1[None, :, :].repeat(l, axis=0), L2[None, :, :].repeat(l, axis=0))

gbmsc_pde.approximation.shepard.lagrange_1d(X_nodes, xq=None, *, with_first_derivatives=True, with_second_derivatives=False, barycentric_weights=None)

Barycentric Lagrange basis — log-stable and unified.

Parameters:

Name Type Description Default
X_nodes (m,) ndarray(float)

Distinct node coordinates.

 required
xq Optional[ndarray]

Query points. If None, the function treats xq = X_nodes (nodal mode). Returned L is then the identity matrix I.

 None
with_first_derivatives bool

Return first derivatives L' as well.

 True
with_second_derivatives bool

Return second derivatives L'' in addition (only valid if with_first_derivatives is True).

 False

Returns:

Type Description
Query mode (`xq` not None)

L : (l, m) L, Lp : if with_first_derivatives L, Lp, Lpp : if with_second_derivatives
Nodal mode (`xq` is None → treated as `xq=X_nodes`)

I, L1 : if with_first_derivatives I, L1, L2 : if with_second_derivatives I : if both derivative flags are False
Source code in src/gbmsc_pde/approximation/shepard.py 
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def lagrange_1d(
        X_nodes: np.ndarray,                                # (m,) nodal abscissae
        xq: Optional[np.ndarray] = None,                    # (l,) query points
        *,
        with_first_derivatives: bool  = True,
        with_second_derivatives: bool = False,
        barycentric_weights: Optional[np.ndarray] = None,
    ) -> Union[
        Tuple[np.ndarray],                               # L
        Tuple[np.ndarray, np.ndarray],                   # L,  L'
        Tuple[np.ndarray, np.ndarray, np.ndarray]        # L,  L', L''
    ]:
        """
        Barycentric Lagrange basis — **log-stable** and unified.

        Parameters
        ----------
        X_nodes : (m,) ndarray(float)
            Distinct node coordinates.
        xq      : (l,) ndarray or *None*
            Query points.  If *None*, the function **treats `xq = X_nodes`**
            (nodal mode).  Returned `L` is then the identity matrix `I`.
        with_first_derivatives  : bool, default False
            Return first derivatives  L'  as well.
        with_second_derivatives : bool, default False
            Return second derivatives L''  in addition (only valid if
            `with_first_derivatives` is True).

        Returns
        -------
        Query mode  (`xq` not None)
            L            : (l, m)
            L, Lp        : if `with_first_derivatives`
            L, Lp, Lpp   : if `with_second_derivatives`

        Nodal mode   (`xq` is None → treated as `xq=X_nodes`)
            I, L1                : if `with_first_derivatives`
            I, L1, L2            : if `with_second_derivatives`
            I                    : if both derivative flags are False
        """
        X_nodes = np.asarray(X_nodes, dtype=float)
        m = X_nodes.size
        nodal_mode = xq is None

        if barycentric_weights is None:
            w = _barycentric_weights_1d(X_nodes)
        else:
            w = np.asarray(barycentric_weights, dtype=float)
            if w.shape != (m,):
                raise ValueError("barycentric_weights must have shape (len(X_nodes),).")

        # ============================================================
        # === nodal mode  (use X_nodes as queries) ===================
        # ============================================================
        if nodal_mode:
            diff_mat = X_nodes[:, None] - X_nodes[None, :]
            np.fill_diagonal(diff_mat, 1.0)
            diag = np.eye(m, dtype=bool)
            L = np.eye(m)                                  

            if not with_first_derivatives:
                return L

            # First derivative
            with np.errstate(divide='ignore', invalid='ignore'):
                L1 = np.where(
                    diag, 
                    0.0, 
                    (w[None, :] / (w[:, None] * diff_mat))
                )
            L1[diag] = -L1.sum(axis=1)

            if not with_second_derivatives:
                return (L, L1)

            # Second derivative
            inv  = 1.0/diff_mat
            np.fill_diagonal(inv, 0.0)
            Svec = inv.sum(axis=1)

            with np.errstate(divide="ignore", invalid="ignore"):
                L2 = np.where(
                    diag, 
                    0.0,
                    2 * L1 * (Svec[:, None] - inv)
                )
            L2[diag] = -L2.sum(axis=1)

            return L, L1, L2
        else:
            xq = np.asarray(xq, dtype=float).ravel()
            l  = xq.size
            L  = np.zeros((l, m))
            tol   = 1e-14 * max(1.0, np.abs(X_nodes).max())      # scale-aware tol
            # ------------------------------------------------------------
            # Basis values at queries
            # ------------------------------------------------------------
            diff_q = xq[:, None] - X_nodes[None, :]                 # (l,m)
            hits = np.isclose(diff_q, 0.0, atol=tol)               # (l,m) bool mask

            if hits.any():
                rows, cols = np.nonzero(hits)                    # indices of coincidences

                # ----- pre-compute nodal derivative matrices once ----------
                _, L1, L2 = lagrange_1d(
                    X_nodes,
                    xq=None,
                    with_first_derivatives=True,
                    with_second_derivatives=True,
                    barycentric_weights=w,
                )

                # overwrite rows that contain a hit with exact values
                # value basis:
                L[rows, cols] = 1.0

                # first derivative if needed
                if with_first_derivatives:
                    Lp            = np.zeros_like(L)
                    Lp[rows, :]   = L1[cols, :]

                # second derivative if needed
                if with_second_derivatives:
                    Lpp            = np.zeros_like(L)
                    Lpp[rows, :]   = L2[cols, :]

                # for all **other** rows we can safely compute with the standard formula
                safe_mask = ~hits.any(axis=1)        # rows without any zero diff
                if safe_mask.any():
                    i_ok = np.nonzero(safe_mask)[0]          # row indices
                    diff_ok = diff_q[i_ok]                   # (l_ok, m)
                    bary_ok  = w / diff_ok
                    denom   = bary_ok.sum(axis=1, keepdims=True)
                    L[i_ok] = bary_ok / denom

                    if with_first_derivatives:
                        B = (bary_ok / diff_ok).sum(axis=1, keepdims=True)
                        A = B / denom
                        Lp[i_ok] = L[i_ok] * (A - 1.0 / diff_ok)

                        if with_second_derivatives:
                            C = (bary_ok / diff_ok**2).sum(axis=1, keepdims=True)
                            Lpp[i_ok] = L[i_ok] * (
                                2.0 / diff_ok**2
                                - 2.0 * C / denom
                                - 2.0 * A / diff_ok
                                + 2.0 * A**2
                            )
                #
                if not with_first_derivatives:
                    return (L,)

                if not with_second_derivatives:
                    return (L, Lp)
                return L, Lp, Lpp
            else:
                # no coincidences – original fast path
                bary_q  = w / diff_q                                    # (l,m)
                denom  = bary_q.sum(axis=1, keepdims=True)              # (l,1)
                L      = bary_q / denom                                 # (l,m)

                if not with_first_derivatives:
                    return (L,)

                # ------------------------------------------------------------
                # First derivatives
                # ------------------------------------------------------------
                B = (bary_q / diff_q).sum(axis=1, keepdims=True)
                A = B / denom
                Lp = L * (A - 1.0 / diff_q)                             # (l,m)

                if not with_second_derivatives:
                    return (L, Lp)

                # ------------------------------------------------------------
                # Second derivatives
                # ------------------------------------------------------------
                C = (bary_q / diff_q**2).sum(axis=1, keepdims=True)
                Lpp = L * (
                    2.0 / diff_q**2
                    - 2.0 * C / denom
                    - 2.0 * A / diff_q
                    + 2.0 * A**2
                )
                return L, Lp, Lpp

gbmsc_pde.operators.differential.build_differential_operators(approximation)

Build sparse nodal derivative operators on the source grid.

The returned matrices act on flattened source-node vectors ordered like approximation.grid.coords. Rows are evaluated at source nodes. The row support is determined by approximation.support_mode:

"active" Active contributors and active denominator; this is the original rectangular sparse PDE mode. "all" Active contributors with an all-subgrid denominator for nodal rows. "nodal_limit" Active source-node contributors with the common singular factor cancelled before weight derivatives are evaluated.

Parameters:

Name Type Description Default
approximation GBMSCApproximation

Prepared Grid-Based Shepard approximation.

 required

Returns:

Type Description
Mx, My, Mxx, Myy : tuple[spmatrix, spmatrix, spmatrix, spmatrix]

Sparse first- and second-derivative matrices.
Source code in src/gbmsc_pde/operators/differential.py 
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def build_differential_operators(
    approximation: GBMSCApproximation,
) -> Tuple[spmatrix, spmatrix, spmatrix, spmatrix]:
    """
    Build sparse nodal derivative operators on the source grid.

    The returned matrices act on flattened source-node vectors ordered like
    ``approximation.grid.coords``.  Rows are evaluated at source nodes.  The
    row support is determined by ``approximation.support_mode``:

    ``"active"``
        Active contributors and active denominator; this is the original
        rectangular sparse PDE mode.
    ``"all"``
        Active contributors with an all-subgrid denominator for nodal rows.
    ``"nodal_limit"``
        Active source-node contributors with the common singular factor
        cancelled before weight derivatives are evaluated.

    Parameters
    ----------
    approximation : GBMSCApproximation
        Prepared Grid-Based Shepard approximation.

    Returns
    -------
    Mx, My, Mxx, Myy : tuple[spmatrix, spmatrix, spmatrix, spmatrix]
        Sparse first- and second-derivative matrices.
    """
    support_mode = getattr(approximation, "support_mode", "active")
    cache_name = f"_differential_operator_cache_{support_mode}"
    cached = getattr(approximation, cache_name, None)
    if cached is not None:
        return cached

    grid = approximation.grid
    n_global_x, n_global_y = grid.grid_shape
    n_local_x, n_local_y = grid.subgrid_shape

    L0_x, L1_x, L2_x = approximation._lagrange_derivatives_one_subgrids_1dim(
        grid.subgrids_x_coords[0]
    )
    L0_y, L1_y, L2_y = approximation._lagrange_derivatives_one_subgrids_1dim(
        grid.subgrids_y_coords[0]
    )

    L0_x = np.tile(L0_x, (n_local_y, 1, 1)).reshape(-1, n_local_x)[None, :]
    L1_x = np.tile(L1_x, (n_local_y, 1, 1)).reshape(-1, n_local_x)[None, :]
    L2_x = np.tile(L2_x, (n_local_y, 1, 1)).reshape(-1, n_local_x)[None, :]

    L0_y = np.tile(L0_y, (n_local_x, 1, 1)).reshape(-1, n_local_y)[None, :]
    L1_y = np.tile(L1_y, (n_local_x, 1, 1)).reshape(-1, n_local_y)[None, :]
    L2_y = np.tile(L2_y, (n_local_x, 1, 1)).reshape(-1, n_local_y)[None, :]

    idx0 = approximation.subgrids_step_row
    if support_mode == "nodal_limit":
        f_val, f_x, f_y, f_xx, f_yy = approximation.eval_weight_functions_at_nodes_nodal_limit()
    elif support_mode == "all":
        f_val, f_x, f_y, f_xx, f_yy = (
            approximation.eval_weight_functions_at_nodes_all_denominator_minmax_shift()
        )
    else:
        f_val, f_x, f_y, f_xx, f_yy = approximation.eval_weight_functions_at_nodes_data_minmax_shift()

    idxy = idx0.reshape(-1, n_local_x, n_local_y)
    idxy_row, idxy_column = _construct_3d_from_list_of_matrices(idxy)
    f_val_toy = f_val.reshape(-1, n_local_y * n_local_x)[:, :, None]
    f_y = f_y.reshape(-1, n_local_y * n_local_x)[:, :, None]
    f_yy = f_yy.reshape(-1, n_local_y * n_local_x)[:, :, None]
    M_values_yy = f_yy * L0_y + 2 * f_y * L1_y + f_val_toy * L2_y
    M_values_y = f_y * L0_y + f_val_toy * L1_y

    Myy = coo_matrix(
        (
            M_values_yy.ravel(),
            (idxy_row.ravel(), idxy_column.ravel()),
        ),
        shape=(n_global_x * n_global_y, n_global_x * n_global_y),
    ).tocsr()
    My = coo_matrix(
        (
            M_values_y.ravel(),
            (idxy_row.ravel(), idxy_column.ravel()),
        ),
        shape=(n_global_x * n_global_y, n_global_x * n_global_y),
    ).tocsr()

    idxx = idx0.reshape(-1, n_local_y, n_local_x, order="F")
    idxx_row, idxx_column = _construct_3d_from_list_of_matrices(idxx)
    f_val_tox = f_val.reshape(-1, n_local_y, n_local_x, order="F").reshape(-1, n_local_y * n_local_x)[
        :, :, None
    ]
    f_x = f_x.reshape(-1, n_local_y, n_local_x, order="F").reshape(-1, n_local_y * n_local_x)[:, :, None]
    f_xx = f_xx.reshape(-1, n_local_y, n_local_x, order="F").reshape(-1, n_local_y * n_local_x)[:, :, None]
    M_values_xx = f_xx * L0_x + 2 * f_x * L1_x + f_val_tox * L2_x
    M_values_x = f_x * L0_x + f_val_tox * L1_x

    Mxx = coo_matrix(
        (
            M_values_xx.ravel(),
            (idxx_row.ravel(), idxx_column.ravel()),
        ),
        shape=(n_global_x * n_global_y, n_global_x * n_global_y),
    ).tocsr()
    Mx = coo_matrix(
        (
            M_values_x.ravel(),
            (idxx_row.ravel(), idxx_column.ravel()),
        ),
        shape=(n_global_x * n_global_y, n_global_x * n_global_y),
    ).tocsr()

    setattr(approximation, cache_name, (Mx, My, Mxx, Myy))
    return getattr(approximation, cache_name)

gbmsc_pde.pde.problem.LinearPDE

Linear PDE builder on a structured rectangular source grid.

The assembled equation has the form

div(D grad u) + v . grad u + r u = f

where diffusion and convection coefficients may be scalar, two-component constants, or field callables. Boundary conditions are applied separately by the solver/boundary-condition layer.

Source code in src/gbmsc_pde/pde/problem.py 
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class LinearPDE:
    """
    Linear PDE builder on a structured rectangular source grid.

    The assembled equation has the form

    ``div(D grad u) + v . grad u + r u = f``

    where diffusion and convection coefficients may be scalar, two-component
    constants, or field callables.  Boundary conditions are applied separately
    by the solver/boundary-condition layer.
    """

    def __init__(self, approximation: GBMSCApproximation) -> None:
        self.approximation = approximation
        self.terms: Dict[str, object] = {
            "diffusion": None,
            "convection": None,
            "reaction": None,
            "source": None,
        }

    def add_diffusion_term(self, coeff) -> None:
        """Add diffusion coefficients for ``div(D grad u)``."""
        self.terms["diffusion"] = normalize_vector_coefficient(coeff)

    def add_convection_term(self, coeff) -> None:
        """Add convection coefficients for ``v . grad u``."""
        self.terms["convection"] = normalize_vector_coefficient(coeff)

    def add_reaction_term(self, coeff) -> None:
        """Add reaction coefficient ``r`` for ``r*u``."""
        self.terms["reaction"] = normalize_scalar_coefficient(coeff, name="coeff")

    def add_source_term(self, source) -> None:
        """Add right-hand side source field ``f``."""
        self.terms["source"] = normalize_scalar_coefficient(source, name="source")

    def assemble(self) -> tuple[spmatrix, np.ndarray, spmatrix, spmatrix, spmatrix, spmatrix]:
        """
        Assemble ``A u = b`` on all rectangular source nodes.

        Returns
        -------
        A, b, Mx, My, Mxx, Myy
            Sparse PDE matrix, right-hand side, first-derivative matrices, and
            second-derivative matrices.
        """
        return assemble_linear_system(self.approximation, self.terms)

add_convection_term(coeff)

Add convection coefficients for v . grad u.

Source code in src/gbmsc_pde/pde/problem.py 
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def add_convection_term(self, coeff) -> None:
    """Add convection coefficients for ``v . grad u``."""
    self.terms["convection"] = normalize_vector_coefficient(coeff)

add_diffusion_term(coeff)

Add diffusion coefficients for div(D grad u).

Source code in src/gbmsc_pde/pde/problem.py 
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def add_diffusion_term(self, coeff) -> None:
    """Add diffusion coefficients for ``div(D grad u)``."""
    self.terms["diffusion"] = normalize_vector_coefficient(coeff)

add_reaction_term(coeff)

Add reaction coefficient r for r*u.

Source code in src/gbmsc_pde/pde/problem.py 
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def add_reaction_term(self, coeff) -> None:
    """Add reaction coefficient ``r`` for ``r*u``."""
    self.terms["reaction"] = normalize_scalar_coefficient(coeff, name="coeff")

add_source_term(source)

Add right-hand side source field f.

Source code in src/gbmsc_pde/pde/problem.py 
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def add_source_term(self, source) -> None:
    """Add right-hand side source field ``f``."""
    self.terms["source"] = normalize_scalar_coefficient(source, name="source")

assemble()

Assemble A u = b on all rectangular source nodes.

Returns:

Type Description
(A, b, Mx, My, Mxx, Myy)

Sparse PDE matrix, right-hand side, first-derivative matrices, and second-derivative matrices.
Source code in src/gbmsc_pde/pde/problem.py 
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def assemble(self) -> tuple[spmatrix, np.ndarray, spmatrix, spmatrix, spmatrix, spmatrix]:
    """
    Assemble ``A u = b`` on all rectangular source nodes.

    Returns
    -------
    A, b, Mx, My, Mxx, Myy
        Sparse PDE matrix, right-hand side, first-derivative matrices, and
        second-derivative matrices.
    """
    return assemble_linear_system(self.approximation, self.terms)

gbmsc_pde.boundary.conditions.BoundaryConditions

Boundary-condition container for rectangular source-node rows.

Conditions are registered by flattened source-node id. Dirichlet rows replace the corresponding PDE row by u_i = value. Neumann and Robin rows use the derivative matrices assembled by the PDE solver. Scalars are broadcast to all supplied nodes; arrays must have the same length as nodes.

Source code in src/gbmsc_pde/boundary/conditions.py 
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class BoundaryConditions:
    """
    Boundary-condition container for rectangular source-node rows.

    Conditions are registered by flattened source-node id.  Dirichlet rows
    replace the corresponding PDE row by ``u_i = value``.  Neumann and Robin
    rows use the derivative matrices assembled by the PDE solver.  Scalars are
    broadcast to all supplied nodes; arrays must have the same length as
    ``nodes``.
    """

    def __init__(self, grid: SourceGrid) -> None:
        self.grid = grid
        self.dirichlet: Dict[int, float] = {}
        self.neumann: Dict[int, Dict[str, Union[Tuple[float, float], float]]] = {}
        self.robin: Dict[int, Dict[str, Union[Tuple[float, float], float]]] = {}
        self.second_neumann: Dict[int, Dict[str, Union[Tuple[float, float], float, int]]] = {}

    def add_dirichlet(
        self,
        nodes: Union[int, np.ndarray],
        values: Union[float, np.ndarray],
    ) -> None:
        """
        Add Dirichlet conditions ``u_i = value`` on rectangular grid nodes.

        Parameters
        ----------
        nodes : int or ndarray
            Flattened source-node ids.
        values : float or ndarray
            Scalar value or one value per node.
        """
        nodes_arr = np.atleast_1d(nodes).astype(int)
        vals_arr = np.atleast_1d(values).astype(float)
        if vals_arr.size == 1:
            vals_arr = np.full(nodes_arr.shape, vals_arr.item(), dtype=float)
        if nodes_arr.size != vals_arr.size:
            raise ValueError(f"Mismatched nodes ({nodes_arr.size}) and values ({vals_arr.size}).")

        n_nodes = getattr(self.grid, "N", None)
        if n_nodes is None:
            raise AttributeError("SourceGrid must define total node count 'N'.")
        if np.any(nodes_arr < 0) or np.any(nodes_arr >= n_nodes):
            raise ValueError(f"Dirichlet indices out of [0, {n_nodes}): {nodes_arr}")

        for node, val in zip(nodes_arr, vals_arr):
            self.dirichlet[int(node)] = float(val)

    def add_neumann(
        self,
        nodes: Union[int, np.ndarray],
        outwardnormal: Union[Tuple[float, float], Tuple[np.ndarray, np.ndarray]],
        fluxes: Union[float, np.ndarray],
    ) -> None:
        """
        Add Neumann conditions ``du/dn = flux`` on grid nodes.

        Parameters
        ----------
        nodes : int or ndarray
            Flattened source-node ids.
        outwardnormal : tuple
            Normal components ``(n_x, n_y)`` as scalars or arrays.
        fluxes : float or ndarray
            Scalar flux or one flux per node.
        """
        nodes_arr = np.atleast_1d(nodes).astype(int)
        k = nodes_arr.size

        if not (isinstance(outwardnormal, tuple) and len(outwardnormal) == 2):
            raise TypeError("`outwardnormal` must be a tuple of length 2.")
        nx, ny = outwardnormal

        if np.isscalar(nx) and np.isscalar(ny):
            nx_arr = np.full(k, float(nx))
            ny_arr = np.full(k, float(ny))
        else:
            nx_arr = np.atleast_1d(nx).astype(float)
            ny_arr = np.atleast_1d(ny).astype(float)
        if nx_arr.shape != (k,) or ny_arr.shape != (k,):
            raise ValueError("`outwardnormal` components must match length of `nodes`.")

        flux_arr = np.atleast_1d(fluxes).astype(float)
        if flux_arr.size == 1:
            flux_arr = np.full(k, flux_arr.item())
        if flux_arr.shape != (k,):
            raise ValueError("`fluxes` must be scalar or match length of `nodes`.")

        n_nodes = getattr(self.grid, "N", None)
        if n_nodes is None:
            raise AttributeError("BoundaryConditions requires `self.grid.N`.")
        if np.any(nodes_arr < 0) or np.any(nodes_arr >= n_nodes):
            raise ValueError(f"Node indices must be in [0, {n_nodes}); got {nodes_arr}.")

        for node, nx_val, ny_val, flux_val in zip(nodes_arr, nx_arr, ny_arr, flux_arr):
            self.neumann[int(node)] = {
                "normal": (float(nx_val), float(ny_val)),
                "flux": float(flux_val),
            }

    def add_robin(
        self,
        nodes: Union[int, np.ndarray],
        outwardnormal: Union[Tuple[float, float], Tuple[np.ndarray, np.ndarray]],
        alphas: Union[float, np.ndarray],
        betas: Union[float, np.ndarray],
        values: Union[float, np.ndarray],
    ) -> None:
        """
        Add Robin conditions ``alpha*u + beta*du/dn = value`` on grid nodes.

        ``alphas``, ``betas``, and ``values`` may be scalars or arrays with
        one entry per node.
        """
        nodes_arr = np.atleast_1d(nodes).astype(int)
        k = nodes_arr.size

        if not (isinstance(outwardnormal, tuple) and len(outwardnormal) == 2):
            raise TypeError("`outwardnormal` must be a tuple (nx, ny).")
        nx, ny = outwardnormal
        nx_arr = np.full(k, nx, dtype=float) if np.isscalar(nx) else np.atleast_1d(nx).astype(float)
        ny_arr = np.full(k, ny, dtype=float) if np.isscalar(ny) else np.atleast_1d(ny).astype(float)
        if nx_arr.shape != (k,) or ny_arr.shape != (k,):
            raise ValueError("`outwardnormal` components must match length of `nodes`.")

        alpha_arr = np.atleast_1d(alphas).astype(float)
        beta_arr = np.atleast_1d(betas).astype(float)
        value_arr = np.atleast_1d(values).astype(float)

        if alpha_arr.size == 1:
            alpha_arr = np.full(k, alpha_arr.item(), dtype=float)
        if beta_arr.size == 1:
            beta_arr = np.full(k, beta_arr.item(), dtype=float)
        if value_arr.size == 1:
            value_arr = np.full(k, value_arr.item(), dtype=float)

        if not (alpha_arr.size == beta_arr.size == value_arr.size == k):
            raise ValueError(
                f"Sizes must match: nodes({k}), alphas({alpha_arr.size}), "
                f"betas({beta_arr.size}), values({value_arr.size})."
            )

        n_nodes = getattr(self.grid, "N", None)
        if n_nodes is None:
            raise AttributeError("SourceGrid must define total node count 'N'.")
        if np.any(nodes_arr < 0) or np.any(nodes_arr >= n_nodes):
            raise ValueError(f"Node indices out of [0, {n_nodes}): {nodes_arr}")

        for node, nx_val, ny_val, alpha, beta, value in zip(
            nodes_arr, nx_arr, ny_arr, alpha_arr, beta_arr, value_arr
        ):
            self.robin[int(node)] = {
                "normal": (float(nx_val), float(ny_val)),
                "alpha": float(alpha),
                "beta": float(beta),
                "value": float(value),
            }

    def add_neumann_as_second_condition(
        self,
        nodes: Union[int, np.ndarray],
        outwardnormal: Union[Tuple[float, float], Tuple[np.ndarray, np.ndarray]],
        fluxes: Union[float, np.ndarray],
        rows: Union[int, np.ndarray],
    ) -> None:
        """
        Add Neumann equations on user-selected matrix rows.

        This compatibility method is for augmented systems where a Neumann
        boundary equation should be imposed as an additional/second condition
        rather than replacing the row associated with ``node``.
        """
        if not hasattr(self.grid, "N"):
            raise AttributeError("BoundaryConditions.grid must define integer `.N`")

        nodes_arr = np.atleast_1d(nodes).astype(int)
        rows_arr = np.atleast_1d(rows).astype(int)

        if nodes_arr.ndim != 1 or rows_arr.ndim != 1:
            raise TypeError("`nodes` and `rows` must be scalars or 1D sequences")

        n = nodes_arr.size
        if rows_arr.size != n:
            raise ValueError(f"`rows` length ({rows_arr.size}) != `nodes` length ({n})")

        if not (isinstance(outwardnormal, tuple) and len(outwardnormal) == 2):
            raise TypeError("`outwardnormal` must be a tuple of two elements")
        nx_in, ny_in = outwardnormal

        try:
            nx_arr = np.broadcast_to(nx_in, (n,)).astype(float)
            ny_arr = np.broadcast_to(ny_in, (n,)).astype(float)
        except Exception as exc:
            raise ValueError(
                "Cannot broadcast `outwardnormal` components to match `nodes` length"
            ) from exc

        try:
            flux_arr = np.broadcast_to(fluxes, (n,)).astype(float)
        except Exception as exc:
            raise ValueError("Cannot broadcast `fluxes` to match `nodes` length") from exc

        if np.any(nodes_arr < 0) or np.any(nodes_arr >= self.grid.N):
            raise ValueError(f"All `nodes` must be in [0, {self.grid.N})")

        for node, row, nx_val, ny_val, flux in zip(nodes_arr, rows_arr, nx_arr, ny_arr, flux_arr):
            self.second_neumann[int(node)] = {
                "normal": (float(nx_val), float(ny_val)),
                "flux": float(flux),
                "row": int(row),
            }

    def apply(
        self,
        A: spmatrix,
        b: np.ndarray,
        Mx: spmatrix,
        My: spmatrix,
    ) -> tuple[spmatrix, np.ndarray]:
        """Apply all registered conditions to an assembled rectangular system."""
        return apply_boundary_conditions(self, A, b, Mx, My)

add_dirichlet(nodes, values)

Add Dirichlet conditions u_i = value on rectangular grid nodes.

Parameters:

Name Type Description Default
nodes int or ndarray

Flattened source-node ids.

 required
values float or ndarray

Scalar value or one value per node.

 required
Source code in src/gbmsc_pde/boundary/conditions.py 
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def add_dirichlet(
    self,
    nodes: Union[int, np.ndarray],
    values: Union[float, np.ndarray],
) -> None:
    """
    Add Dirichlet conditions ``u_i = value`` on rectangular grid nodes.

    Parameters
    ----------
    nodes : int or ndarray
        Flattened source-node ids.
    values : float or ndarray
        Scalar value or one value per node.
    """
    nodes_arr = np.atleast_1d(nodes).astype(int)
    vals_arr = np.atleast_1d(values).astype(float)
    if vals_arr.size == 1:
        vals_arr = np.full(nodes_arr.shape, vals_arr.item(), dtype=float)
    if nodes_arr.size != vals_arr.size:
        raise ValueError(f"Mismatched nodes ({nodes_arr.size}) and values ({vals_arr.size}).")

    n_nodes = getattr(self.grid, "N", None)
    if n_nodes is None:
        raise AttributeError("SourceGrid must define total node count 'N'.")
    if np.any(nodes_arr < 0) or np.any(nodes_arr >= n_nodes):
        raise ValueError(f"Dirichlet indices out of [0, {n_nodes}): {nodes_arr}")

    for node, val in zip(nodes_arr, vals_arr):
        self.dirichlet[int(node)] = float(val)

add_neumann(nodes, outwardnormal, fluxes)

Add Neumann conditions du/dn = flux on grid nodes.

Parameters:

Name Type Description Default
nodes int or ndarray

Flattened source-node ids.

 required
outwardnormal tuple

Normal components (n_x, n_y) as scalars or arrays.

 required
fluxes float or ndarray

Scalar flux or one flux per node.

 required
Source code in src/gbmsc_pde/boundary/conditions.py 
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def add_neumann(
    self,
    nodes: Union[int, np.ndarray],
    outwardnormal: Union[Tuple[float, float], Tuple[np.ndarray, np.ndarray]],
    fluxes: Union[float, np.ndarray],
) -> None:
    """
    Add Neumann conditions ``du/dn = flux`` on grid nodes.

    Parameters
    ----------
    nodes : int or ndarray
        Flattened source-node ids.
    outwardnormal : tuple
        Normal components ``(n_x, n_y)`` as scalars or arrays.
    fluxes : float or ndarray
        Scalar flux or one flux per node.
    """
    nodes_arr = np.atleast_1d(nodes).astype(int)
    k = nodes_arr.size

    if not (isinstance(outwardnormal, tuple) and len(outwardnormal) == 2):
        raise TypeError("`outwardnormal` must be a tuple of length 2.")
    nx, ny = outwardnormal

    if np.isscalar(nx) and np.isscalar(ny):
        nx_arr = np.full(k, float(nx))
        ny_arr = np.full(k, float(ny))
    else:
        nx_arr = np.atleast_1d(nx).astype(float)
        ny_arr = np.atleast_1d(ny).astype(float)
    if nx_arr.shape != (k,) or ny_arr.shape != (k,):
        raise ValueError("`outwardnormal` components must match length of `nodes`.")

    flux_arr = np.atleast_1d(fluxes).astype(float)
    if flux_arr.size == 1:
        flux_arr = np.full(k, flux_arr.item())
    if flux_arr.shape != (k,):
        raise ValueError("`fluxes` must be scalar or match length of `nodes`.")

    n_nodes = getattr(self.grid, "N", None)
    if n_nodes is None:
        raise AttributeError("BoundaryConditions requires `self.grid.N`.")
    if np.any(nodes_arr < 0) or np.any(nodes_arr >= n_nodes):
        raise ValueError(f"Node indices must be in [0, {n_nodes}); got {nodes_arr}.")

    for node, nx_val, ny_val, flux_val in zip(nodes_arr, nx_arr, ny_arr, flux_arr):
        self.neumann[int(node)] = {
            "normal": (float(nx_val), float(ny_val)),
            "flux": float(flux_val),
        }

add_neumann_as_second_condition(nodes, outwardnormal, fluxes, rows)

Add Neumann equations on user-selected matrix rows.

This compatibility method is for augmented systems where a Neumann boundary equation should be imposed as an additional/second condition rather than replacing the row associated with node.

Source code in src/gbmsc_pde/boundary/conditions.py 
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def add_neumann_as_second_condition(
    self,
    nodes: Union[int, np.ndarray],
    outwardnormal: Union[Tuple[float, float], Tuple[np.ndarray, np.ndarray]],
    fluxes: Union[float, np.ndarray],
    rows: Union[int, np.ndarray],
) -> None:
    """
    Add Neumann equations on user-selected matrix rows.

    This compatibility method is for augmented systems where a Neumann
    boundary equation should be imposed as an additional/second condition
    rather than replacing the row associated with ``node``.
    """
    if not hasattr(self.grid, "N"):
        raise AttributeError("BoundaryConditions.grid must define integer `.N`")

    nodes_arr = np.atleast_1d(nodes).astype(int)
    rows_arr = np.atleast_1d(rows).astype(int)

    if nodes_arr.ndim != 1 or rows_arr.ndim != 1:
        raise TypeError("`nodes` and `rows` must be scalars or 1D sequences")

    n = nodes_arr.size
    if rows_arr.size != n:
        raise ValueError(f"`rows` length ({rows_arr.size}) != `nodes` length ({n})")

    if not (isinstance(outwardnormal, tuple) and len(outwardnormal) == 2):
        raise TypeError("`outwardnormal` must be a tuple of two elements")
    nx_in, ny_in = outwardnormal

    try:
        nx_arr = np.broadcast_to(nx_in, (n,)).astype(float)
        ny_arr = np.broadcast_to(ny_in, (n,)).astype(float)
    except Exception as exc:
        raise ValueError(
            "Cannot broadcast `outwardnormal` components to match `nodes` length"
        ) from exc

    try:
        flux_arr = np.broadcast_to(fluxes, (n,)).astype(float)
    except Exception as exc:
        raise ValueError("Cannot broadcast `fluxes` to match `nodes` length") from exc

    if np.any(nodes_arr < 0) or np.any(nodes_arr >= self.grid.N):
        raise ValueError(f"All `nodes` must be in [0, {self.grid.N})")

    for node, row, nx_val, ny_val, flux in zip(nodes_arr, rows_arr, nx_arr, ny_arr, flux_arr):
        self.second_neumann[int(node)] = {
            "normal": (float(nx_val), float(ny_val)),
            "flux": float(flux),
            "row": int(row),
        }

add_robin(nodes, outwardnormal, alphas, betas, values)

Add Robin conditions alpha*u + beta*du/dn = value on grid nodes.

alphas, betas, and values may be scalars or arrays with one entry per node.

Source code in src/gbmsc_pde/boundary/conditions.py 
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def add_robin(
    self,
    nodes: Union[int, np.ndarray],
    outwardnormal: Union[Tuple[float, float], Tuple[np.ndarray, np.ndarray]],
    alphas: Union[float, np.ndarray],
    betas: Union[float, np.ndarray],
    values: Union[float, np.ndarray],
) -> None:
    """
    Add Robin conditions ``alpha*u + beta*du/dn = value`` on grid nodes.

    ``alphas``, ``betas``, and ``values`` may be scalars or arrays with
    one entry per node.
    """
    nodes_arr = np.atleast_1d(nodes).astype(int)
    k = nodes_arr.size

    if not (isinstance(outwardnormal, tuple) and len(outwardnormal) == 2):
        raise TypeError("`outwardnormal` must be a tuple (nx, ny).")
    nx, ny = outwardnormal
    nx_arr = np.full(k, nx, dtype=float) if np.isscalar(nx) else np.atleast_1d(nx).astype(float)
    ny_arr = np.full(k, ny, dtype=float) if np.isscalar(ny) else np.atleast_1d(ny).astype(float)
    if nx_arr.shape != (k,) or ny_arr.shape != (k,):
        raise ValueError("`outwardnormal` components must match length of `nodes`.")

    alpha_arr = np.atleast_1d(alphas).astype(float)
    beta_arr = np.atleast_1d(betas).astype(float)
    value_arr = np.atleast_1d(values).astype(float)

    if alpha_arr.size == 1:
        alpha_arr = np.full(k, alpha_arr.item(), dtype=float)
    if beta_arr.size == 1:
        beta_arr = np.full(k, beta_arr.item(), dtype=float)
    if value_arr.size == 1:
        value_arr = np.full(k, value_arr.item(), dtype=float)

    if not (alpha_arr.size == beta_arr.size == value_arr.size == k):
        raise ValueError(
            f"Sizes must match: nodes({k}), alphas({alpha_arr.size}), "
            f"betas({beta_arr.size}), values({value_arr.size})."
        )

    n_nodes = getattr(self.grid, "N", None)
    if n_nodes is None:
        raise AttributeError("SourceGrid must define total node count 'N'.")
    if np.any(nodes_arr < 0) or np.any(nodes_arr >= n_nodes):
        raise ValueError(f"Node indices out of [0, {n_nodes}): {nodes_arr}")

    for node, nx_val, ny_val, alpha, beta, value in zip(
        nodes_arr, nx_arr, ny_arr, alpha_arr, beta_arr, value_arr
    ):
        self.robin[int(node)] = {
            "normal": (float(nx_val), float(ny_val)),
            "alpha": float(alpha),
            "beta": float(beta),
            "value": float(value),
        }

apply(A, b, Mx, My)

Apply all registered conditions to an assembled rectangular system.

Source code in src/gbmsc_pde/boundary/conditions.py 
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def apply(
    self,
    A: spmatrix,
    b: np.ndarray,
    Mx: spmatrix,
    My: spmatrix,
) -> tuple[spmatrix, np.ndarray]:
    """Apply all registered conditions to an assembled rectangular system."""
    return apply_boundary_conditions(self, A, b, Mx, My)

gbmsc_pde.boundary.applier.BoundaryConditionApplier

Apply a boundary-condition container to a linear system.

Source code in src/gbmsc_pde/boundary/applier.py 
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class BoundaryConditionApplier:
    """Apply a boundary-condition container to a linear system."""

    def __init__(self, boundary_conditions) -> None:
        self.boundary_conditions = boundary_conditions

    def apply(
        self,
        A: spmatrix,
        b: np.ndarray,
        Mx: spmatrix,
        My: spmatrix,
    ) -> Tuple[spmatrix, np.ndarray]:
        """Return a system with all registered boundary conditions enforced."""
        bcs = self.boundary_conditions
        n_nodes = getattr(bcs.grid, "N", None)
        if n_nodes is None:
            raise AttributeError("SourceGrid must define total node count 'N'.")

        has_derivative_bcs = bool(bcs.neumann or bcs.robin or bcs.second_neumann)
        if not has_derivative_bcs and not bcs.dirichlet:
            return A.tocsr(), b.copy()

        A_bc = A.tolil()
        b_bc = b.copy()

        if has_derivative_bcs:
            Mx_csr = Mx.tocsr()
            My_csr = My.tocsr()

        for node, bc in bcs.neumann.items():
            nx, ny = bc["normal"]
            flux = bc["flux"]
            row_n = nx * Mx_csr.getrow(node) + ny * My_csr.getrow(node)
            A_bc[node, :] = row_n
            b_bc[node] = flux

        for node, bc in bcs.robin.items():
            nx, ny = bc["normal"]
            alpha = bc["alpha"]
            beta = bc["beta"]
            value = bc["value"]
            row_n = nx * Mx_csr.getrow(node) + ny * My_csr.getrow(node)
            row_r = (beta * row_n).tolil()
            row_r[0, node] += alpha
            A_bc[node, :] = row_r
            b_bc[node] = value

        for node, info in bcs.second_neumann.items():
            row = info["row"]
            nx, ny = info["normal"]
            flux = info["flux"]
            row_n = nx * Mx_csr.getrow(node) + ny * My_csr.getrow(node)
            A_bc[row, :] = row_n
            b_bc[row] = flux

        for node, value in bcs.dirichlet.items():
            A_bc.rows[node] = [node]
            A_bc.data[node] = [1.0]
            b_bc[node] = value

        return A_bc.tocsr(), b_bc

apply(A, b, Mx, My)

Return a system with all registered boundary conditions enforced.

Source code in src/gbmsc_pde/boundary/applier.py 
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def apply(
    self,
    A: spmatrix,
    b: np.ndarray,
    Mx: spmatrix,
    My: spmatrix,
) -> Tuple[spmatrix, np.ndarray]:
    """Return a system with all registered boundary conditions enforced."""
    bcs = self.boundary_conditions
    n_nodes = getattr(bcs.grid, "N", None)
    if n_nodes is None:
        raise AttributeError("SourceGrid must define total node count 'N'.")

    has_derivative_bcs = bool(bcs.neumann or bcs.robin or bcs.second_neumann)
    if not has_derivative_bcs and not bcs.dirichlet:
        return A.tocsr(), b.copy()

    A_bc = A.tolil()
    b_bc = b.copy()

    if has_derivative_bcs:
        Mx_csr = Mx.tocsr()
        My_csr = My.tocsr()

    for node, bc in bcs.neumann.items():
        nx, ny = bc["normal"]
        flux = bc["flux"]
        row_n = nx * Mx_csr.getrow(node) + ny * My_csr.getrow(node)
        A_bc[node, :] = row_n
        b_bc[node] = flux

    for node, bc in bcs.robin.items():
        nx, ny = bc["normal"]
        alpha = bc["alpha"]
        beta = bc["beta"]
        value = bc["value"]
        row_n = nx * Mx_csr.getrow(node) + ny * My_csr.getrow(node)
        row_r = (beta * row_n).tolil()
        row_r[0, node] += alpha
        A_bc[node, :] = row_r
        b_bc[node] = value

    for node, info in bcs.second_neumann.items():
        row = info["row"]
        nx, ny = info["normal"]
        flux = info["flux"]
        row_n = nx * Mx_csr.getrow(node) + ny * My_csr.getrow(node)
        A_bc[row, :] = row_n
        b_bc[row] = flux

    for node, value in bcs.dirichlet.items():
        A_bc.rows[node] = [node]
        A_bc.data[node] = [1.0]
        b_bc[node] = value

    return A_bc.tocsr(), b_bc

gbmsc_pde.solvers.linear.LinearSolver

Linear-system solver for rectangular-domain PDE problems.

LinearSolver assembles the PDE rows from :class:LinearPDE, applies node-indexed boundary conditions, and solves the resulting square sparse system for the source-node vector.

Parameters:

Name Type Description Default
pde LinearPDE

Configured rectangular linear PDE builder.

 required
bcs BoundaryConditions

Boundary conditions registered on flattened grid nodes.

 required
Source code in src/gbmsc_pde/solvers/linear.py 
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class LinearSolver:
    """
    Linear-system solver for rectangular-domain PDE problems.

    ``LinearSolver`` assembles the PDE rows from :class:`LinearPDE`, applies
    node-indexed boundary conditions, and solves the resulting square sparse
    system for the source-node vector.

    Parameters
    ----------
    pde : LinearPDE
        Configured rectangular linear PDE builder.
    bcs : BoundaryConditions
        Boundary conditions registered on flattened grid nodes.
    """
    def __init__(self, pde: LinearPDE, bcs: BoundaryConditions) -> None:
        if not isinstance(pde, LinearPDE):
            raise TypeError("pde must be an instance of LinearPDE")
        if not isinstance(bcs, BoundaryConditions):
            raise TypeError("bcs must be an instance of BoundaryConditions")

        self.pde = pde
        self.bcs = bcs
        self.A = None
        self.b = None
        self.Mx = None
        self.My = None
        self.Mxx = None
        self.Myy = None
        self.solution = None
        self.last_data = None

    def assemble(self) -> Tuple[spmatrix, np.ndarray, spmatrix, spmatrix, spmatrix, spmatrix]:
        """
        Assemble and boundary-modify the rectangular collocation system.

        Returns
        -------
        A, b, Mx, My, Mxx, Myy
            Boundary-modified sparse system, right-hand side, and first
            and second derivative matrices.  The same objects are stored on
            ``self`` as ``A``, ``b``, ``Mx``, ``My``, ``Mxx``, and ``Myy`` for
            later inspection.
        """
        A, b, Mx, My, Mxx, Myy = self.pde.assemble()
        A, b = BoundaryConditionApplier(self.bcs).apply(A, b, Mx, My)
        self.A = A
        self.b = b
        self.Mx = Mx
        self.My = My
        self.Mxx = Mxx
        self.Myy = Myy
        return A, b, Mx, My, Mxx, Myy

    @staticmethod
    def _as_linear_operator(preconditioner: Optional[Any]):
        if preconditioner is None:
            return None
        if isinstance(preconditioner, spmatrix):
            return spla.LinearOperator(preconditioner.shape, preconditioner.dot)
        return preconditioner

    @staticmethod
    def _jacobi_preconditioner(A: spmatrix):
        diagonal = A.diagonal()
        if np.any(np.isclose(diagonal, 0.0)):
            raise ValueError("Jacobi preconditioner requires a nonzero matrix diagonal.")
        inv_diagonal = 1.0 / diagonal
        return spla.LinearOperator(A.shape, matvec=lambda x: inv_diagonal * x)

    @staticmethod
    def _iterative_call(method, A, b, *, tol, maxiter, M):
        try:
            return method(A, b, rtol=tol, atol=0.0, maxiter=maxiter, M=M)
        except TypeError:
            return method(A, b, tol=tol, maxiter=maxiter, M=M)

    def solve(
        self,
        solver: str = 'direct',
        tol: float = 1e-8,
        maxiter: Optional[int] = None,
        preconditioner: Optional[Any] = None,
        return_data: bool = False,
    ) -> Union[
        np.ndarray,
        Tuple[np.ndarray, Dict[str, object]],
    ]:
        """
        Assemble, apply boundary conditions, and solve the linear system.

        Parameters
        ----------
        solver : {'direct', 'cg', 'gmres', 'bicgstab'}
            Linear solver. ``"direct"`` uses sparse direct solve; ``"cg"`` and
            ``"gmres"`` and ``"bicgstab"`` use SciPy iterative solvers.
        tol : float, optional
            Tolerance for iterative solvers.
        maxiter : int, optional
            Maximum iterations (defaults to N).
        preconditioner : object, optional
            Preconditioner LinearOperator, sparse matrix, or ``"jacobi"``.
        return_data : bool
            If true, return ``(u, data)`` where ``data`` contains ``A``, ``b``,
            derivative matrices, ``nnz``, ``sparsity``, ``solve_time``,
            ``assembly_time``, and ``solver_info``.

        Returns
        -------
        ndarray or tuple
            Solution vector ``u`` with shape ``(grid.N,)``.  If
            ``return_data=True``, returns ``(u, data)``.
        """
        assembly_start = perf_counter()
        A, b, Mx, My, Mxx, Myy = self.assemble()
        assembly_time = perf_counter() - assembly_start

        if A.shape[0] != A.shape[1]:
            raise ValueError(f"Linear system must be square; got {A.shape}.")
        if not np.all(np.isfinite(A.data)):
            raise ValueError("Linear system matrix contains non-finite values.")
        if not np.all(np.isfinite(b)):
            raise ValueError("Right-hand side contains non-finite values.")

        solve_start = perf_counter()
        if solver == 'direct':
            sol = spla.spsolve(A, b)
            info = 0
        else:
            N = A.shape[0]
            maxiter = maxiter or N
            if preconditioner == "jacobi":
                M = self._jacobi_preconditioner(A)
            else:
                M = self._as_linear_operator(preconditioner)
            methods = {'cg': spla.cg, 'gmres': spla.gmres, 'bicgstab': spla.bicgstab}
            if solver not in methods:
                raise ValueError(
                    f"Unsupported solver '{solver}'. Choose 'direct', 'cg', 'gmres', or 'bicgstab'."
                )
            sol, info = self._iterative_call(
                methods[solver],
                A,
                b,
                tol=tol,
                maxiter=maxiter,
                M=M,
            )
            if info != 0:
                raise RuntimeError(f"Solver '{solver}' failed to converge (info={info}).")
        solve_time = perf_counter() - solve_start

        self.solution = sol

        if return_data:
            data: Dict[str, object] = {
                "A": A,
                "b": b,
                "first_derivatives": (Mx, My),
                "second_derivatives": (Mxx, Myy),
                "nnz": int(A.nnz),
                "sparsity": float((1.0 - A.nnz / (A.shape[0] * A.shape[1])) * 100.0),
                "solve_time": float(solve_time),
                "assembly_time": float(assembly_time),
                "solver_info": int(info),
            }
            self.last_data = data
            return sol, data

        return sol

assemble()

Assemble and boundary-modify the rectangular collocation system.

Returns:

Type Description
(A, b, Mx, My, Mxx, Myy)

Boundary-modified sparse system, right-hand side, and first and second derivative matrices. The same objects are stored on self as A, b, Mx, My, Mxx, and Myy for later inspection.
Source code in src/gbmsc_pde/solvers/linear.py 
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def assemble(self) -> Tuple[spmatrix, np.ndarray, spmatrix, spmatrix, spmatrix, spmatrix]:
    """
    Assemble and boundary-modify the rectangular collocation system.

    Returns
    -------
    A, b, Mx, My, Mxx, Myy
        Boundary-modified sparse system, right-hand side, and first
        and second derivative matrices.  The same objects are stored on
        ``self`` as ``A``, ``b``, ``Mx``, ``My``, ``Mxx``, and ``Myy`` for
        later inspection.
    """
    A, b, Mx, My, Mxx, Myy = self.pde.assemble()
    A, b = BoundaryConditionApplier(self.bcs).apply(A, b, Mx, My)
    self.A = A
    self.b = b
    self.Mx = Mx
    self.My = My
    self.Mxx = Mxx
    self.Myy = Myy
    return A, b, Mx, My, Mxx, Myy

solve(solver='direct', tol=1e-08, maxiter=None, preconditioner=None, return_data=False)

Assemble, apply boundary conditions, and solve the linear system.

Parameters:

Name Type Description Default
solver (direct, cg, gmres, bicgstab)

Linear solver. "direct" uses sparse direct solve; "cg" and "gmres" and "bicgstab" use SciPy iterative solvers.

 'direct'
tol float

Tolerance for iterative solvers.

 1e-08
maxiter int

Maximum iterations (defaults to N).

 None
preconditioner object

Preconditioner LinearOperator, sparse matrix, or "jacobi".

 None
return_data bool

If true, return (u, data) where data contains A, b, derivative matrices, nnz, sparsity, solve_time, assembly_time, and solver_info.

 False

Returns:

Type Description
ndarray or tuple

Solution vector u with shape (grid.N,). If return_data=True, returns (u, data).
Source code in src/gbmsc_pde/solvers/linear.py 
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def solve(
    self,
    solver: str = 'direct',
    tol: float = 1e-8,
    maxiter: Optional[int] = None,
    preconditioner: Optional[Any] = None,
    return_data: bool = False,
) -> Union[
    np.ndarray,
    Tuple[np.ndarray, Dict[str, object]],
]:
    """
    Assemble, apply boundary conditions, and solve the linear system.

    Parameters
    ----------
    solver : {'direct', 'cg', 'gmres', 'bicgstab'}
        Linear solver. ``"direct"`` uses sparse direct solve; ``"cg"`` and
        ``"gmres"`` and ``"bicgstab"`` use SciPy iterative solvers.
    tol : float, optional
        Tolerance for iterative solvers.
    maxiter : int, optional
        Maximum iterations (defaults to N).
    preconditioner : object, optional
        Preconditioner LinearOperator, sparse matrix, or ``"jacobi"``.
    return_data : bool
        If true, return ``(u, data)`` where ``data`` contains ``A``, ``b``,
        derivative matrices, ``nnz``, ``sparsity``, ``solve_time``,
        ``assembly_time``, and ``solver_info``.

    Returns
    -------
    ndarray or tuple
        Solution vector ``u`` with shape ``(grid.N,)``.  If
        ``return_data=True``, returns ``(u, data)``.
    """
    assembly_start = perf_counter()
    A, b, Mx, My, Mxx, Myy = self.assemble()
    assembly_time = perf_counter() - assembly_start

    if A.shape[0] != A.shape[1]:
        raise ValueError(f"Linear system must be square; got {A.shape}.")
    if not np.all(np.isfinite(A.data)):
        raise ValueError("Linear system matrix contains non-finite values.")
    if not np.all(np.isfinite(b)):
        raise ValueError("Right-hand side contains non-finite values.")

    solve_start = perf_counter()
    if solver == 'direct':
        sol = spla.spsolve(A, b)
        info = 0
    else:
        N = A.shape[0]
        maxiter = maxiter or N
        if preconditioner == "jacobi":
            M = self._jacobi_preconditioner(A)
        else:
            M = self._as_linear_operator(preconditioner)
        methods = {'cg': spla.cg, 'gmres': spla.gmres, 'bicgstab': spla.bicgstab}
        if solver not in methods:
            raise ValueError(
                f"Unsupported solver '{solver}'. Choose 'direct', 'cg', 'gmres', or 'bicgstab'."
            )
        sol, info = self._iterative_call(
            methods[solver],
            A,
            b,
            tol=tol,
            maxiter=maxiter,
            M=M,
        )
        if info != 0:
            raise RuntimeError(f"Solver '{solver}' failed to converge (info={info}).")
    solve_time = perf_counter() - solve_start

    self.solution = sol

    if return_data:
        data: Dict[str, object] = {
            "A": A,
            "b": b,
            "first_derivatives": (Mx, My),
            "second_derivatives": (Mxx, Myy),
            "nnz": int(A.nnz),
            "sparsity": float((1.0 - A.nnz / (A.shape[0] * A.shape[1])) * 100.0),
            "solve_time": float(solve_time),
            "assembly_time": float(assembly_time),
            "solver_info": int(info),
        }
        self.last_data = data
        return sol, data

    return sol