Source code for openmdao.approximation_schemes.complex_step

"""Complex Step derivative approximations."""
from __future__ import division, print_function

from six import iteritems, itervalues
from six.moves import range
from collections import defaultdict

import numpy as np

from openmdao.approximation_schemes.approximation_scheme import ApproximationScheme, \
    _gather_jac_results, _get_wrt_subjacs
from openmdao.utils.general_utils import simple_warning
from openmdao.utils.array_utils import sub2full_indices
from openmdao.utils.coloring import Coloring


_full_slice = slice(None)


[docs]class ComplexStep(ApproximationScheme): r""" Approximation scheme using complex step to calculate derivatives. For example, using a step size of 'h' will approximate the derivative in the following way: .. math:: f'(x) = \Im{\frac{f(x+ih)}{h}}. Attributes ---------- _fd : <FiniteDifference> When nested complex step is detected, we switch to Finite Difference. """ DEFAULT_OPTIONS = { 'step': 1e-40, 'directional': False, }
[docs] def __init__(self): """ Initialize the ApproximationScheme. """ super(ComplexStep, self).__init__() # Only used when nested under complex step. self._fd = None
[docs] def add_approximation(self, abs_key, system, kwargs): """ Use this approximation scheme to approximate the derivative d(of)/d(wrt). Parameters ---------- abs_key : tuple(str,str) Absolute name pairing of (of, wrt) for the derivative. system : System Containing System. kwargs : dict Additional keyword arguments, to be interpreted by sub-classes. """ options = self.DEFAULT_OPTIONS.copy() options.update(kwargs) key = (abs_key[1], options['step'], options['directional']) self._exec_dict[key].append((abs_key, options)) self._reset() # force later regen of approx_groups
def _get_approx_data(self, system, data): """ Given approximation metadata, compute necessary delta for complex step. Parameters ---------- system : System System whose derivatives are being approximated. data : tuple Tuple of the form (wrt, delta, directional) Returns ------- float Delta needed for complex step perturbation. """ _, delta, _ = data delta *= 1j return delta
[docs] def compute_approximations(self, system, jac, total=False): """ Execute the system to compute the approximate sub-Jacobians. Parameters ---------- system : System System on which the execution is run. jac : dict-like Approximations are stored in the given dict-like object. total : bool If True total derivatives are being approximated, else partials. """ if not self._exec_dict: return if system.under_complex_step: # If we are nested under another complex step, then warn and swap to FD. if not self._fd: from openmdao.approximation_schemes.finite_difference import FiniteDifference msg = "Nested complex step detected. Finite difference will be used for '%s'." simple_warning(msg % system.pathname) fd = self._fd = FiniteDifference() empty = {} for lst in itervalues(self._exec_dict): for apprx in lst: fd.add_approximation(apprx[0], system, empty) self._fd.compute_approximations(system, jac, total=total) return # Turn on complex step. system._set_complex_step_mode(True) self._compute_approximations(system, jac, total, under_cs=True) # Turn off complex step. system._set_complex_step_mode(False)
def _get_multiplier(self, delta): """ Return a multiplier to be applied to the jacobian. Parameters ---------- delta : complex Complex number used to compute the multiplier. Returns ------- float multiplier to apply to the jacobian. """ return (1.0 / delta * 1j).real def _transform_result(self, array): """ Return the imaginary part of the given array. Parameters ---------- array : ndarray of complex Result array after doing a complex step. Returns ------- ndarray Imaginary part of the result array. """ return array.imag def _run_point(self, system, idx_info, delta, result_array, total): """ Perturb the system inputs with a complex step, run, and return the results. Parameters ---------- system : System The system having its derivs approximated. idx_info : tuple of (ndarray of int, ndarray of float) Tuple of wrt indices and corresponding data array to perturb. delta : complex Perturbation amount. result_array : ndarray An array used to store the results. total : bool If True total derivatives are being approximated, else partials. Returns ------- Vector Copy of the results from running the perturbed system. """ for arr, idxs in idx_info: if arr is not None: arr._data[idxs] += delta if total: system.run_solve_nonlinear() results_vec = system._outputs else: system.run_apply_nonlinear() results_vec = system._residuals result_array[:] = results_vec._data for arr, idxs in idx_info: if arr is not None: arr._data[idxs] -= delta return result_array