Source code for openmdao.approximation_schemes.finite_difference

"""Finite difference derivative approximations."""
from __future__ import division, print_function

from collections import namedtuple, defaultdict
from six import iteritems
from six.moves import range, zip

import numpy as np

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

FDForm = namedtuple('FDForm', ['deltas', 'coeffs', 'current_coeff'])

DEFAULT_ORDER = {
    'forward': 1,
    'backward': 1,
    'central': 2,
}

FD_COEFFS = {
    ('forward', 1): FDForm(deltas=np.array([1.0]),
                           coeffs=np.array([1.0]),
                           current_coeff=-1.0),
    ('backward', 1): FDForm(deltas=np.array([-1.0]),
                            coeffs=np.array([-1.0]),
                            current_coeff=1.0),
    ('central', 2): FDForm(deltas=np.array([1.0, -1.0]),
                           coeffs=np.array([0.5, -0.5]),
                           current_coeff=0.),
}

_full_slice = slice(None)


def _generate_fd_coeff(form, order, system):
    """
    Create an FDForm namedtuple containing the deltas, coefficients, and current coefficient.

    Parameters
    ----------
    form : str
        Requested form of FD (e.g. 'forward', 'central', 'backward').
    order : int
        The order of accuracy of the requested FD scheme.
    system : System
        Containing system.

    Returns
    -------
    FDForm
        namedtuple containing the 'deltas', 'coeffs', and 'current_coeff'. These deltas and
        coefficients need to be scaled by the step size.
    """
    try:
        fd_form = FD_COEFFS[form, order]
    except KeyError:
        # TODO: Automatically generate requested form and store in dict.
        raise ValueError('{}: Finite Difference form="{}" and order={} are not '
                         'supported'.format(system.msginfo, form, order))
    return fd_form


[docs]class FiniteDifference(ApproximationScheme): r""" Approximation scheme using finite differences to estimate derivatives. For example, using the 'forward' form with a step size of 'h' will approximate the derivative in the following way: .. math:: f'(x) = \frac{f(x+h) - f(x)}{h} + O(h). Attributes ---------- _starting_outs : ndarray A copy of the starting outputs array used to restore the outputs to original values. _starting_ins : ndarray A copy of the starting inputs array used to restore the inputs to original values. _results_tmp : ndarray An array the same size as the system outputs. Used to store the results temporarily. """ DEFAULT_OPTIONS = { 'step': 1e-6, 'form': 'forward', 'order': None, 'step_calc': 'abs', 'directional': False, }
[docs] def __init__(self): """ Initialize the ApproximationScheme. """ super(FiniteDifference, self).__init__() self._starting_ins = self._starting_outs = self._results_tmp = 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) if options['order'] is None: form = options['form'] if form in DEFAULT_ORDER: options['order'] = DEFAULT_ORDER[options['form']] else: raise ValueError("{}: '{}' is not a valid form of finite difference; must be " "one of {}".format(system.msginfo, form, list(DEFAULT_ORDER.keys()))) key = (abs_key[1], options['form'], options['order'], options['step'], options['step_calc'], 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 deltas and coefficients. Parameters ---------- system : System System whose derivatives are being approximated. data : tuple Tuple of the form (wrt, form, order, step, step_calc, directional) Returns ------- tuple Tuple of the form (deltas, coeffs, current_coeff) """ wrt, form, order, step, step_calc, _ = data # FD forms are written as a collection of changes to inputs (deltas) and the associated # coefficients (coeffs). Since we do not need to (re)evaluate the current step, its # coefficient is stored seperately (current_coeff). For example, # f'(x) = (f(x+h) - f(x))/h + O(h) = 1/h * f(x+h) + (-1/h) * f(x) + O(h) # would be stored as deltas = [h], coeffs = [1/h], and current_coeff = -1/h. # A central second order accurate approximation for the first derivative would be stored # as deltas = [-2, -1, 1, 2] * h, coeffs = [1/12, -2/3, 2/3 , -1/12] * 1/h, # current_coeff = 0. fd_form = _generate_fd_coeff(form, order, system) if step_calc == 'rel': if wrt in system._outputs._views_flat: step *= np.linalg.norm(system._outputs._views_flat[wrt]) elif wrt in system._inputs._views_flat: step *= np.linalg.norm(system._inputs._views_flat[wrt]) deltas = fd_form.deltas * step coeffs = fd_form.coeffs / step current_coeff = fd_form.current_coeff / step return deltas, coeffs, current_coeff
[docs] def compute_approximations(self, system, jac=None, total=False): """ Execute the system to compute the approximate sub-Jacobians. Parameters ---------- system : System System on which the execution is run. jac : None or dict-like If None, update system with the approximated sub-Jacobians. Otherwise, store the approximations in the given dict-like object. total : bool If True total derivatives are being approximated, else partials. """ if not self._exec_dict: return if jac is None: jac = system._jacobian self._starting_outs = system._outputs._data.copy() self._starting_resids = system._residuals._data.copy() self._starting_ins = system._inputs._data.copy() if total: self._results_tmp = self._starting_outs.copy() else: self._results_tmp = self._starting_resids.copy() self._compute_approximations(system, jac, total, system._outputs._under_complex_step) # reclaim some memory self._starting_ins = self._starting_outs = self._results_tmp = None
def _get_multiplier(self, data): """ Return a multiplier to be applied to the jacobian. Always returns 1.0 for finite difference. Parameters ---------- data : tuple Not used. Returns ------- float 1.0 """ return 1.0 def _transform_result(self, array): """ Return the given array. Parameters ---------- array : ndarray Result array after doing a finite difference. Returns ------- array The givan array, unchanged. """ return array def _run_point(self, system, idx_info, data, results_array, total): """ Alter the specified inputs by the given deltas, run the system, 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. data : tuple of float Tuple of the form (deltas, coeffs, current_coeff) results_array : ndarray Where the results will be stored. total : bool If True total derivatives are being approximated, else partials. Returns ------- ndarray The results from running the perturbed system. """ deltas, coeffs, current_coeff = data if current_coeff: current_vec = system._outputs if total else system._residuals # copy data from outputs (if doing total derivs) or residuals (if doing partials) results_array[:] = current_vec._data results_array *= current_coeff else: results_array[:] = 0. # Run the Finite Difference for delta, coeff in zip(deltas, coeffs): results = self._run_sub_point(system, idx_info, delta, total) results *= coeff results_array += results return results_array def _run_sub_point(self, system, idx_info, delta, total): """ Alter the specified inputs by the given delta, run the system, 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 : float Perturbation amount. total : bool If True total derivatives are being approximated, else partials. Returns ------- ndarray 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() self._results_tmp[:] = system._outputs._data system._outputs._data[:] = self._starting_outs else: system.run_apply_nonlinear() self._results_tmp[:] = system._residuals._data system._residuals._data[:] = self._starting_resids # save results and restore starting inputs/outputs system._inputs._data[:] = self._starting_ins # if results_vec are the residuals then we need to remove the delta's we added earlier # to the outputs if not total: for arr, idxs in idx_info: if arr is system._outputs: arr._data[idxs] -= delta return self._results_tmp