r"""This module implements differential operators on spherical grids.
.. autosummary::
:nosignatures:
SphericalLaplacian
SphericalGradient
SphericalGradientSquared
SphericalDivergence
.. codeauthor:: David Zwicker <david.zwicker@ds.mpg.de>
"""
from __future__ import annotations
from typing import TYPE_CHECKING, Literal
import torch
from .... import config
from ....grids import GridBase, SphericalSymGrid
from ....tools.docstrings import fill_in_docstring
from .. import torch_backend
from .common import TorchDifferentialOperator
if TYPE_CHECKING:
import numpy as np
from torch import Tensor
from ....grids.boundaries import BoundariesList
[docs]
@torch_backend.register_operator(SphericalSymGrid, "laplace", rank_in=0, rank_out=0)
@fill_in_docstring
class SphericalLaplacian(TorchDifferentialOperator):
"""Spherical Laplace using torch.
{DESCR_SPHERICAL_GRID}
"""
rank_in = 0
def __init__(
self,
grid: GridBase,
bcs: BoundariesList | None,
*,
dtype: np.dtype,
conservative: bool | None = None,
):
"""Initialize the Spherical Laplacian operator.
Args:
grid (:class:`~pde.grids.base.GridBase`):
The grid on which the operator acts
bcs (:class:`~pde.grids.boundaries.axes.BoundariesList` or None):
The boundary conditions applied to the field. If `None`, no boundary
conditions are enforced.
dtype:
The data type of the field
conservative (bool):
Flag indicating whether the Laplace operator should be conservative
(which results in slightly slower computations). Conservative operators
ensure mass conservation. If `None`, the value is read from the
configuration option `operators.conservative_stencil`.
"""
super().__init__(grid, bcs, dtype=dtype)
if conservative is None:
conservative = config["operators.conservative_stencil"]
self.conservative = conservative
# calculate preliminary quantities
dr = grid.discretization[0]
self.dr = dr
rs = grid.axes_coords[0]
self.dr_2 = 1 / dr**2
if self.conservative:
# create a conservative spherical laplace operator
rl = rs - dr / 2 # inner radii of spherical shells
rh = rs + dr / 2 # outer radii
volumes = (rh**3 - rl**3) / 3 # volume of the spherical shells
self.register_array("factor_l", rl**2 / (dr * volumes))
self.register_array("factor_h", rh**2 / (dr * volumes))
else:
self.register_array("factor", 1 / (rs * dr))
[docs]
def forward(self, arr: Tensor, args=None) -> Tensor:
"""Fill internal data array, apply operator, and return valid data."""
data_full = self.get_full_data(arr, args=args)
if self.conservative:
term_h = self.factor_h * (arr[2:] - arr[1:-1]) # type: ignore
term_l = self.factor_l * (arr[1:-1] - arr[:-2]) # type: ignore
return term_h - term_l
term1 = (data_full[2:] - 2 * data_full[1:-1] + data_full[:-2]) * self.dr_2
term2 = self.factor * (data_full[2:] - data_full[:-2]) # type: ignore
return term1 + term2 # type: ignore
[docs]
@torch_backend.register_operator(SphericalSymGrid, "gradient", rank_in=0, rank_out=1)
@fill_in_docstring
class SphericalGradient(TorchDifferentialOperator):
"""Spherical gradient operator using torch.
{DESCR_SPHERICAL_GRID}
"""
rank_in = 0
def __init__(
self,
grid: GridBase,
bcs: BoundariesList | None,
*,
dtype: np.dtype,
method: Literal["central", "forward", "backward"] = "central",
):
"""Initialize the Spherical gradient operator.
Args:
grid (:class:`~pde.grids.base.GridBase`):
The grid on which the operator acts
bcs (:class:`~pde.grids.boundaries.axes.BoundariesList` or None):
The boundary conditions applied to the field. If `None`, no boundary
conditions are enforced.
dtype:
The data type of the field
method (str):
The method for calculating the derivative. Possible values are
'central', 'forward', and 'backward'.
"""
super().__init__(grid, bcs, dtype=dtype)
# calculate preliminary quantities
self.method = method
if method == "central":
self.scale_r = 0.5 / grid.discretization[0]
elif method in {"forward", "backward"}:
self.scale_r = 1 / grid.discretization[0]
else:
msg = f"Unknown derivative type `{method}`"
raise ValueError(msg)
[docs]
def forward(self, arr: Tensor, args=None) -> Tensor:
"""Fill internal data array, apply operator, and return valid data."""
data_full = self.get_full_data(arr, args=args)
if self.method == "central":
r = (data_full[2:] - data_full[:-2]) * self.scale_r
elif self.method == "forward":
r = (data_full[2:] - data_full[1:-1]) * self.scale_r
elif self.method == "backward":
r = (data_full[1:-1] - data_full[:-2]) * self.scale_r
# no angular dependence by definition
return torch.stack((r, torch.zeros_like(r), torch.zeros_like(r)))
[docs]
@torch_backend.register_operator(
SphericalSymGrid, "gradient_squared", rank_in=0, rank_out=0
)
@fill_in_docstring
class SphericalGradientSquared(TorchDifferentialOperator):
"""Spherical gradient squared operator using torch.
{DESCR_SPHERICAL_GRID}
"""
rank_in = 0
def __init__(
self,
grid: GridBase,
bcs: BoundariesList | None,
*,
central: bool = True,
dtype: np.dtype,
):
"""Initialize the Spherical gradient squared operator.
Args:
grid (:class:`~pde.grids.base.GridBase`):
The grid on which the operator acts
bcs (:class:`~pde.grids.boundaries.axes.BoundariesList` or None):
The boundary conditions applied to the field. If `None`, no boundary
conditions are enforced.
central (bool):
Whether to use central differences. If `False`, forward and backward
differences are used.
dtype:
The data type of the field
"""
super().__init__(grid, bcs, dtype=dtype)
self.central = central
dr = grid.discretization[0]
if self.central:
self.scale = 0.25 / dr**2
else:
self.scale = 0.5 / dr**2
[docs]
def forward(self, arr: Tensor, args=None) -> Tensor:
"""Fill internal data array, apply operator, and return valid data."""
data_full = self.get_full_data(arr, args=args)
if self.central:
# simple squared sum of central differences
return (data_full[2:] - data_full[:-2]) ** 2 * self.scale # type: ignore
term1 = (data_full[2:] - data_full[1:-1]) ** 2
term2 = (data_full[1:-1] - data_full[:-2]) ** 2
return (term1 + term2) * self.scale # type: ignore
[docs]
@torch_backend.register_operator(SphericalSymGrid, "divergence", rank_in=1, rank_out=0)
@fill_in_docstring
class SphericalDivergence(TorchDifferentialOperator):
"""Spherical divergence operator using torch.
{DESCR_SPHERICAL_GRID}
"""
rank_in = 1
def __init__(
self,
grid: GridBase,
bcs: BoundariesList | None,
*,
dtype: np.dtype,
conservative: bool | None = None,
method: Literal["central", "forward", "backward"] = "central",
):
"""Initialize the Spherical divergence operator.
Args:
grid (:class:`~pde.grids.base.GridBase`):
The grid on which the operator acts
bcs (:class:`~pde.grids.boundaries.axes.BoundariesList` or None):
The boundary conditions applied to the field. If `None`, no boundary
conditions are enforced.
dtype:
The data type of the field
conservative (bool):
Flag indicating whether the operator should be conservative (which
results in slightly slower computations). Conservative operators ensure
mass conservation. If `None`, the value is read from the configuration
option `operators.conservative_stencil`.
method (str):
The method for calculating the derivative. Possible values are
'central', 'forward', and 'backward'.
"""
super().__init__(grid, bcs, dtype=dtype)
if conservative is None:
conservative = config["operators.conservative_stencil"]
self.conservative = conservative
self.method = method
dr = self.grid.discretization[0]
self.dr = dr
rs = self.grid.axes_coords[0]
self.register_array("rs", rs)
self.scale_r = 1 / (2 * dr)
# create a conservative spherical divergence operator
if self.conservative:
rl = rs - dr / 2 # inner radii of spherical shells
rh = rs + dr / 2 # outer radii
volumes = (rh**3 - rl**3) / 3 # volume of the spherical shells
self.register_array("factor_l", rl**2 / (2 * volumes))
self.register_array("factor_h", rh**2 / (2 * volumes))
else:
self.register_array("factor", 1 / (rs * dr))
[docs]
def forward(self, arr: Tensor, args=None) -> Tensor:
"""Fill internal data array, apply operator, and return valid data."""
data_full = self.get_full_data(arr, args=args)
arr_r = data_full[0]
if self.conservative:
if self.method == "central":
term_h = self.factor_h * (arr_r[1:-1] + arr_r[2:]) # type: ignore
term_l = self.factor_l * (arr_r[:-2] + arr_r[1:-1]) # type: ignore
elif self.method == "forward":
term_h = 2 * self.factor_h * arr_r[2:] # type: ignore
term_l = 2 * self.factor_l * arr_r[1:-1] # type: ignore
elif self.method == "backward":
term_h = 2 * self.factor_h * arr_r[1:-1] # type: ignore
term_l = 2 * self.factor_l * arr_r[:-2] # type: ignore
return term_h - term_l
# non-conservative implementation
if self.method == "central":
diff_r = (arr_r[2:] - arr_r[:-2]) / (2 * self.dr)
elif self.method == "forward":
diff_r = (arr_r[2:] - arr_r[1:-1]) / self.dr
elif self.method == "backward":
diff_r = (arr_r[1:-1] - arr_r[:-2]) / self.dr
return diff_r + self.factor * arr_r[1:-1] # type: ignore
# @torch_backend.register_operator(
# SphericalSymGrid, "vector_gradient", rank_in=1, rank_out=2
# )
# class SphericalVectorGradient(TorchDifferentialOperator):
# """Spherical vector gradient operator using torch."""
# rank_in = 1
# def __init__(
# self,
# grid: GridBase,
# bcs: BoundariesList | None,
# *,
# dtype: np.dtype,
# ):
# """Initialize the Spherical divergence operator.
# Args:
# grid (:class:`~pde.grids.base.GridBase`):
# The grid on which the operator acts
# bcs (:class:`~pde.grids.boundaries.axes.BoundariesList` or None):
# The boundary conditions applied to the field. If `None`, no boundary
# conditions are enforced.
# dtype:
# The data type of the field
# """
# super().__init__(grid, bcs, dtype=dtype)
# dr = self.grid.discretization[0]
# self.register_array("rs", self.grid.axes_coords[0])
# self.scale_r = 1 / (2 * dr)
# def forward(self, arr: Tensor, args=None) -> Tensor:
# """Fill internal data array, apply operator, and return valid data."""
# data_full = self.get_full_data(arr, args=args)
# # assign aliases
# arr_r, arr_φ = arr
# out_rr, out_rφ = out[0, 0, :], out[0, 1, :]
# out_φr, out_φφ = out[1, 0, :], out[1, 1, :]
# for i in range(1, dim_r + 1): # iterate radial points
# out_rr[i - 1] = (arr_r[i + 1] - arr_r[i - 1]) * scale_r
# out_rφ[i - 1] = -arr_φ[i] / rs[i - 1]
# out_φr[i - 1] = (arr_φ[i + 1] - arr_φ[i - 1]) * scale_r
# out_φφ[i - 1] = arr_r[i] / rs[i - 1]
# term1 = (data_full[0, 2:] - data_full[0, :-2]) * self.scale_r
# term2 = data_full[0, 1:-1] / self.rs
# return term1 + term2
__all__ = [
"SphericalDivergence",
"SphericalGradient",
"SphericalGradientSquared",
"SphericalLaplacian",
]