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1219 | @_escalate_complex_warning
def function_solve_VIE_2(*, kernel, g=None, mesh_breakpoints,
coll_divs: int = None, coll_choices: list[int] = None,
coll_nodes=None,
kernel_singularity=None,
return_function: bool = False,
show_warnings: bool = True,
_smooth_gl_order: int = 6):
r"""Solve the scalar Volterra integral equation of the second kind
$$y(t) = g(t) + \int_0^t K(t-s)\,y(s)\,ds$$
with callable kernel and right-hand side, on an arbitrary mesh.
Parameters
----------
kernel : callable
``kernel(u)`` returns $K(u)$ for scalar $u > 0$: a scalar for scalar
equations, or a square $(d, d)$ matrix for $d$-dimensional vector and
matrix-valued equations.
g : callable, optional
``g(t)`` returns the forcing term $g(t)$. Defaults to zero. Return a
scalar for scalar equations, a $(d,)$ array for vector equations, or a
$(d, m)$ array to solve $m$ right-hand sides simultaneously (the
matrix-valued case). A 2-D return signals the matrix case; matrix-valued
problems require a $(d, d)$ matrix kernel.
mesh_breakpoints : array_like
Strictly increasing 1-D array starting at 0. Defines the integration
intervals directly.
coll_divs, coll_choices : int, list of int, optional
Collocation node positions: nodes lie at
``coll_choices[k] / coll_divs`` in each interval. Unlike the array-
based solvers, ``coll_divs`` does *not* sub-divide intervals. Defaults
to ``coll_divs=2, coll_choices=[0, 1, 2]`` when neither these nor
``coll_nodes`` are given. Mutually exclusive with ``coll_nodes``.
coll_nodes : array_like, optional
Collocation node positions given directly as floats in $[0, 1]$, for
node sets that are not rational multiples of $1/c$ (e.g. Gauss-Legendre
or Radau IIA points; see ``gauss_legendre_nodes``, ``radau_iia_nodes``,
``lobatto_nodes``). Mutually exclusive with
``coll_divs``/``coll_choices``. Convergence of the chosen node set is
the caller's responsibility.
kernel_singularity : None, float, list of float, or callable
Declare the singularity structure of the kernel.
- ``None``: kernel is smooth everywhere.
- ``float`` or list of float: convolution-style singularity locations
(in $u = t - s$); e.g. ``0.0`` for $K(u) \sim u^{-\alpha}$.
- callable ``f(t) -> list[float]``: returns the singular $s$-locations
for collocation point $t$. Forward-compatible with non-convolution
$K(s, t)$.
return_function : bool, optional
If True, also return a callable solution wrapper.
show_warnings : bool, optional
Currently unused; reserved for the graded-mesh / mesh-uniformity hint.
Returns
-------
soln_values : ndarray
Solution values at collocation nodes, where M is the number of
intervals and p = ``len(coll_choices)``. Shape ``(M, p)`` for scalar
equations, ``(M, p, d)`` for vector equations, and ``(M, p, d, m)`` for
matrix-valued equations.
(soln_values, y_callable) : tuple
When ``return_function=True``. ``y_callable(t)`` evaluates the
piecewise polynomial at any time t, returning a scalar / ``(d,)`` /
``(d, m)`` value for scalar t (with a leading time axis for array t).
Notes
-----
See the module docstring for the mesh-convention difference from the
array-based solvers.
Matrix-valued problems share the kernel weight tensor across all $m$
right-hand sides (it is built once), so they are substantially cheaper than
$m$ separate calls.
"""
mesh_breakpoints = np.asarray(mesh_breakpoints, dtype=float)
if mesh_breakpoints.ndim != 1 or len(mesh_breakpoints) < 2:
raise ValueError("mesh_breakpoints must be 1-D with at least two entries")
if not np.all(np.diff(mesh_breakpoints) > 0):
raise ValueError("mesh_breakpoints must be strictly increasing")
if mesh_breakpoints[0] != 0.0:
raise ValueError("mesh_breakpoints[0] must be 0")
node_pos, _, _ = _resolve_node_pos(
coll_nodes, coll_divs, coll_choices,
default_divs=2, default_choices=[0, 1, 2],
exclude_zero=False, fname="function_solve_VIE_2")
if not callable(kernel):
raise TypeError("kernel must be callable")
if g is not None and not callable(g):
raise TypeError("g must be callable or None")
# Complex dispatch: block-decompose to a real problem of doubled dimension.
if _samples_indicate_complex([kernel, g], mesh_breakpoints, None):
sample_u = float(np.diff(mesh_breakpoints)[0]) * 0.5
d_orig = _detect_complex_d_orig(kernel, sample_u)
result = function_solve_VIE_2(
kernel=_block_wrap_kernel(kernel, d_orig),
g=_block_wrap_g(g, d_orig),
mesh_breakpoints=mesh_breakpoints,
coll_nodes=node_pos,
kernel_singularity=kernel_singularity,
return_function=return_function, show_warnings=show_warnings,
_smooth_gl_order=_smooth_gl_order)
if return_function:
y_real, y_func_real = result
return (_recombine_complex_y(y_real, d_orig),
_ComplexSolutionFunction(y_func_real, d_orig))
return _recombine_complex_y(result, d_orig)
M = len(mesh_breakpoints) - 1
p = len(node_pos)
widths = np.diff(mesh_breakpoints)
# Detect scalar vs vector kernel by sampling at a non-singular point.
sample_u = float(widths[0]) * 0.5
is_vector, d = _detect_kernel_shape(kernel, sample_u)
_maybe_warn_mesh_uniform_with_singularity(
mesh_breakpoints, kernel_singularity, show_warnings)
from . import _dlang as _dlang_module
max_p = _dlang_module.function_solve_max_p_d()
if p > max_p:
raise ValueError(
f"number of collocation nodes p = {p} exceeds the maximum compiled "
f"into the D extension ({max_p}). Use fewer collocation nodes.")
if not is_vector:
W = _build_W_scalar(kernel, mesh_breakpoints, node_pos,
kernel_singularity, _smooth_gl_order)
g_arr = np.zeros((M, p), dtype=np.float64)
if g is not None:
for n in range(M):
t_n = mesh_breakpoints[n]
h_n = widths[n]
for i in range(p):
g_arr[n, i] = float(g(t_n + node_pos[i] * h_n))
y = _dlang_module.function_solve_vie2_d(W, g_arr)
if return_function:
polys = _build_polynomials(y, mesh_breakpoints, node_pos)
y_func = _SolutionFunction(polys, mesh_breakpoints, d=0)
return y, y_func
return y
# ----- Vector / matrix path -----
max_d = _dlang_module.function_solve_max_d_d()
if d > max_d:
raise ValueError(
f"kernel dimension d = {d} exceeds the maximum compiled into the "
f"D extension ({max_d}).")
# The weight tensor depends only on the kernel, so it is built once and
# shared across all right-hand sides in the matrix case.
W = _build_W_vector(kernel, mesh_breakpoints, node_pos,
kernel_singularity, _smooth_gl_order, d)
sample_t = float(mesh_breakpoints[0] + node_pos[0] * widths[0])
m = _detect_g_matrix_cols(g, d, sample_t)
if m is not None:
# Matrix-valued problem: m simultaneous right-hand sides sharing W.
G = _sample_g_at_coll_matrix(g, mesh_breakpoints, node_pos, widths,
M, p, d, m)
def _col_solve(j):
g_arr_j = np.ascontiguousarray(G[:, :, :, j])
return _dlang_module.function_solve_vie2_vec_d(W, g_arr_j)
with ThreadPoolExecutor(max_workers=m) as ex:
cols = list(ex.map(_col_solve, range(m)))
y = np.stack(cols, axis=3) # (M, p, d, m)
if return_function:
polys = _build_polynomials_matrix(y, mesh_breakpoints, node_pos,
d, m)
return y, _SolutionFunction(polys, mesh_breakpoints, d=d, m=m)
return y
g_arr = _sample_g_at_coll_vec(g, mesh_breakpoints, node_pos, widths, M, p, d)
y = _dlang_module.function_solve_vie2_vec_d(W, g_arr)
if return_function:
polys = _build_polynomials_vector(y, mesh_breakpoints, node_pos, d)
y_func = _SolutionFunction(polys, mesh_breakpoints, d=d)
return y, y_func
return y
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