"""
Common functions that are used by many operators
.. codeauthor:: David Zwicker <david.zwicker@ds.mpg.de>
"""
from __future__ import annotations
import logging
import warnings
from typing import Callable, Literal
import numpy as np
from ...tools.numba import jit
from ...tools.typing import OperatorType
from ..base import GridBase
logger = logging.getLogger(__name__)
[docs]
def make_derivative(
grid: GridBase,
axis: int = 0,
method: Literal["central", "forward", "backward"] = "central",
) -> OperatorType:
"""make a derivative operator along a single axis using numba compilation
Args:
grid (:class:`~pde.grids.base.GridBase`):
The grid for which the operator is created
axis (int):
The axis along which the derivative will be taken
method (str):
The method for calculating the derivative. Possible values are
'central', 'forward', and 'backward'.
Returns:
A function that can be applied to an full array of values including those at
ghost cells. The result will be an array of the same shape containing the actual
derivatives at the valid (interior) grid points.
"""
if method not in {"central", "forward", "backward"}:
raise ValueError(f"Unknown derivative type `{method}`")
shape = grid.shape
num_axes = len(shape)
dx = grid.discretization[axis]
if axis == 0:
di, dj, dk = 1, 0, 0
elif axis == 1:
di, dj, dk = 0, 1, 0
elif axis == 2:
di, dj, dk = 0, 0, 1
else:
raise NotImplementedError(f"Derivative for {axis:d} dimensions")
if num_axes == 1:
@jit
def diff(arr: np.ndarray, out: np.ndarray) -> None:
"""calculate derivative of 1d array `arr`"""
for i in range(1, shape[0] + 1):
if method == "central":
out[i - 1] = (arr[i + 1] - arr[i - 1]) / (2 * dx)
elif method == "forward":
out[i - 1] = (arr[i + 1] - arr[i]) / dx
elif method == "backward":
out[i - 1] = (arr[i] - arr[i - 1]) / dx
elif num_axes == 2:
@jit
def diff(arr: np.ndarray, out: np.ndarray) -> None:
"""calculate derivative of 2d array `arr`"""
for i in range(1, shape[0] + 1):
for j in range(1, shape[1] + 1):
arr_l = arr[i - di, j - dj]
arr_r = arr[i + di, j + dj]
if method == "central":
out[i - 1, j - 1] = (arr_r - arr_l) / (2 * dx)
elif method == "forward":
out[i - 1, j - 1] = (arr_r - arr[i, j]) / dx
elif method == "backward":
out[i - 1, j - 1] = (arr[i, j] - arr_l) / dx
elif num_axes == 3:
@jit
def diff(arr: np.ndarray, out: np.ndarray) -> None:
"""calculate derivative of 3d array `arr`"""
for i in range(1, shape[0] + 1):
for j in range(1, shape[1] + 1):
for k in range(1, shape[2] + 1):
arr_l = arr[i - di, j - dj, k - dk]
arr_r = arr[i + di, j + dj, k + dk]
if method == "central":
out[i - 1, j - 1, k - 1] = (arr_r - arr_l) / (2 * dx)
elif method == "forward":
out[i - 1, j - 1, k - 1] = (arr_r - arr[i, j, k]) / dx
elif method == "backward":
out[i - 1, j - 1, k - 1] = (arr[i, j, k] - arr_l) / dx
else:
raise NotImplementedError(
f"Numba derivative operator not implemented for {num_axes:d} axes"
)
return diff # type: ignore
[docs]
def make_derivative2(grid: GridBase, axis: int = 0) -> OperatorType:
"""make a second-order derivative operator along a single axis
Args:
grid (:class:`~pde.grids.base.GridBase`):
The grid for which the operator is created
axis (int):
The axis along which the derivative will be taken
Returns:
A function that can be applied to an full array of values including those at
ghost cells. The result will be an array of the same shape containing the actual
derivatives at the valid (interior) grid points.
"""
shape = grid.shape
num_axes = len(shape)
scale = 1 / grid.discretization[axis] ** 2
if axis == 0:
di, dj, dk = 1, 0, 0
elif axis == 1:
di, dj, dk = 0, 1, 0
elif axis == 2:
di, dj, dk = 0, 0, 1
else:
raise NotImplementedError(f"Derivative for {axis:d} dimensions")
if num_axes == 1:
@jit
def diff(arr: np.ndarray, out: np.ndarray) -> None:
"""calculate derivative of 1d array `arr`"""
for i in range(1, shape[0] + 1):
out[i - 1] = (arr[i + 1] - 2 * arr[i] + arr[i - 1]) * scale
elif num_axes == 2:
@jit
def diff(arr: np.ndarray, out: np.ndarray) -> None:
"""calculate derivative of 2d array `arr`"""
for i in range(1, shape[0] + 1):
for j in range(1, shape[1] + 1):
arr_l = arr[i - di, j - dj]
arr_r = arr[i + di, j + dj]
out[i - 1, j - 1] = (arr_r - 2 * arr[i, j] + arr_l) * scale
elif num_axes == 3:
@jit
def diff(arr: np.ndarray, out: np.ndarray) -> None:
"""calculate derivative of 3d array `arr`"""
for i in range(1, shape[0] + 1):
for j in range(1, shape[1] + 1):
for k in range(1, shape[2] + 1):
arr_l = arr[i - di, j - dj, k - dk]
arr_r = arr[i + di, j + dj, k + dk]
out[i - 1, j - 1, k - 1] = (
arr_r - 2 * arr[i, j, k] + arr_l
) * scale
else:
raise NotImplementedError(
f"Numba derivative operator not implemented for {num_axes:d} axes"
)
return diff # type: ignore
[docs]
def make_laplace_from_matrix(
matrix, vector
) -> Callable[[np.ndarray, np.ndarray | None], np.ndarray]:
"""make a Laplace operator using matrix vector products
Args:
matrix:
(Sparse) matrix representing the laplace operator on the given grid
vector:
Constant part representing the boundary conditions of the Laplace operator
Returns:
A function that can be applied to an array of values to obtain the result of
applying the linear operator `matrix` and the offset given by `vector`.
"""
mat = matrix.tocsc()
vec = vector.toarray()[:, 0]
def laplace(arr: np.ndarray, out: np.ndarray | None = None) -> np.ndarray:
"""apply the laplace operator to `arr`"""
result = mat.dot(arr.flat) + vec
if out is None:
out = result.reshape(arr.shape)
else:
out[:] = result.reshape(arr.shape)
return out
return laplace
[docs]
def make_general_poisson_solver(
matrix, vector, method: Literal["auto", "scipy"] = "auto"
) -> OperatorType:
"""make an operator that solves Poisson's problem
Args:
matrix:
The (sparse) matrix representing the laplace operator on the given grid.
vector:
The constant part representing the boundary conditions of the Laplace
operator.
method (str):
The chosen method for implementing the operator
Returns:
A function that can be applied to an array of values to obtain the solution to
Poisson's equation where the array is used as the right hand side
"""
from scipy import sparse
try:
from scipy.sparse.linalg import MatrixRankWarning
except ImportError:
from scipy.sparse.linalg.dsolve.linsolve import MatrixRankWarning
if method not in {"auto", "scipy"}:
raise ValueError(f"Method {method} is not available")
# prepare the matrix representing the operator
mat = matrix.tocsc()
vec = vector.toarray()[:, 0]
def solve_poisson(arr: np.ndarray, out: np.ndarray) -> None:
"""solves Poisson's equation using sparse linear algebra"""
# prepare the right hand side vector
rhs = np.ravel(arr) - vec
# solve the linear problem using a sparse solver
try:
with warnings.catch_warnings():
warnings.simplefilter("error") # enable warning catching
result = sparse.linalg.spsolve(mat, rhs)
except MatrixRankWarning:
# this can happen for singular laplace matrix, e.g. when pure
# Neumann conditions are considered. In this case, a solution is
# obtained using least squares
logger.warning(
"Poisson problem seems to be under-determined and "
"could not be solved using sparse.linalg.spsolve"
)
use_leastsquares = True
else:
# test whether the solution is good enough
if np.allclose(mat.dot(result), rhs, rtol=1e-5, atol=1e-5):
logger.info("Solved Poisson problem with sparse.linalg.spsolve")
use_leastsquares = False
else:
logger.warning(
"Poisson problem was not solved using sparse.linalg.spsolve"
)
use_leastsquares = True
if use_leastsquares:
# use least squares to solve an underdetermined problem
result = sparse.linalg.lsmr(mat, rhs)[0]
if not np.allclose(mat.dot(result), rhs, rtol=1e-5, atol=1e-5):
residual = np.linalg.norm(mat.dot(result) - rhs)
raise RuntimeError(
f"Poisson problem could not be solved (Residual: {residual})"
)
logger.info("Solved Poisson problem with sparse.linalg.lsmr")
# convert the result to the correct format
out[:] = result.reshape(arr.shape)
return solve_poisson