Source code for gemseo.utils.derivatives.complex_step

# Copyright 2021 IRT Saint Exupéry,
# This program is free software; you can redistribute it and/or
# modify it under the terms of the GNU Lesser General Public
# License version 3 as published by the Free Software Foundation.
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# Lesser General Public License for more details.
# You should have received a copy of the GNU Lesser General Public License
# along with this program; if not, write to the Free Software Foundation,
# Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301, USA.
# Contributors:
#    INITIAL AUTHORS - API and implementation and/or documentation
#       :author : Francois Gallard
"""Gradient approximation by complex step."""

from __future__ import annotations

from typing import TYPE_CHECKING
from typing import Any
from typing import ClassVar

from numpy import bool_
from numpy import complex128
from numpy import dtype
from numpy import ndarray
from numpy import where
from numpy import zeros
from numpy.linalg import norm

from gemseo.core.parallel_execution.callable_parallel_execution import (
from gemseo.utils.derivatives.approximation_modes import ApproximationMode
from gemseo.utils.derivatives.gradient_approximator import GradientApproximator

    from import Sequence

[docs] class ComplexStep(GradientApproximator): r"""Complex step approximator, performing a second-order gradient calculation. Enable a much lower step than real finite differences, typically 1e-30, since there is no cancellation error due to a difference calculation. .. math:: \frac{df(x)}{dx} \approx Im\left( \frac{f(x+j*\\delta x)}{\\delta x} \right) See Martins, Joaquim RRA, Peter Sturdza, and Juan J. Alonso. "The complex-step derivative approximation." ACM Transactions on Mathematical Software (TOMS) 29.3 (2003): 245-262. """ _APPROXIMATION_MODE = ApproximationMode.COMPLEX_STEP _DEFAULT_STEP: ClassVar[complex] = 1e-20 @GradientApproximator.step.setter def step(self, value) -> None: # noqa:D102 if value.imag != 0: self._step = value.imag else: self._step = value
[docs] def f_gradient( # noqa:D102 self, x_vect: ndarray, step: complex | None = None, x_indices: Sequence[int] | None = None, **kwargs: Any, ) -> ndarray: if norm(x_vect.imag) != 0.0: msg = ( "Impossible to check the gradient at a complex " "point using the complex step method." ) raise ValueError(msg) return super().f_gradient(x_vect, step=step, x_indices=x_indices, **kwargs)
def _compute_parallel_grad( self, input_values: ndarray, n_perturbations: int, input_perturbations: ndarray, step: float, **kwargs: Any, ) -> list[ndarray]: self._function_kwargs = kwargs functions = [self._wrap_function] * n_perturbations parallel_execution = CallableParallelExecution(functions, **self._parallel_args) perturbed_inputs: list[ndarray[Any, dtype[bool_]]] = [ input_values + input_perturbations[:, perturbation_index] for perturbation_index in range(n_perturbations) ] perturbed_outputs = parallel_execution.execute(perturbed_inputs) return [ perturbed_outputs[perturbation_index].imag / input_perturbations[perturbation_index, perturbation_index].imag for perturbation_index in range(n_perturbations) ] def _compute_grad( self, input_values: ndarray, n_perturbations: int, input_perturbations: ndarray, step: float, **kwargs: Any, ) -> ndarray: gradient = [] for perturbation_index in range(n_perturbations): perturbated_input = ( input_values + input_perturbations[:, perturbation_index] ) perturbated_output = self.f_pointer(perturbated_input, **kwargs) gradient.append( perturbated_output.imag / input_perturbations[perturbation_index, perturbation_index].imag ) return gradient def _generate_perturbations( self, input_values: ndarray, input_indices: list[int], step: float, ) -> tuple[ndarray, float | ndarray]: input_dimension = len(input_values) n_indices = len(input_indices) input_perturbations = zeros((input_dimension, n_indices), dtype=complex128) x_nnz = where(input_values == 0.0, 1.0, input_values)[input_indices] input_perturbations[input_indices, range(n_indices)] = 1j * x_nnz * step return input_perturbations, step