Source code for pymor.operators.ei

# This file is part of the pyMOR project (http://www.pymor.org).
# Copyright 2013-2019 pyMOR developers and contributors. All rights reserved.
# License: BSD 2-Clause License (http://opensource.org/licenses/BSD-2-Clause)

import weakref

import numpy as np
from scipy.linalg import solve, solve_triangular


from pymor.operators.basic import OperatorBase
from pymor.operators.constructions import VectorArrayOperator, Concatenation, ComponentProjection, ZeroOperator
from pymor.operators.interfaces import OperatorInterface
from pymor.operators.numpy import NumpyMatrixOperator
from pymor.vectorarrays.interfaces import VectorArrayInterface
from pymor.vectorarrays.numpy import NumpyVectorSpace


[docs]class EmpiricalInterpolatedOperator(OperatorBase): """Interpolate an |Operator| using Empirical Operator Interpolation. Let `L` be an |Operator|, `0 <= c_1, ..., c_M < L.range.dim` indices of interpolation DOFs and let `b_1, ..., b_M in R^(L.range.dim)` be collateral basis vectors. If moreover `ψ_j(U)` denotes the j-th component of `U`, the empirical interpolation `L_EI` of `L` w.r.t. the given data is given by :: | M | L_EI(U, μ) = ∑ b_i⋅λ_i such that | i=1 | | ψ_(c_i)(L_EI(U, μ)) = ψ_(c_i)(L(U, μ)) for i=0,...,M Since the original operator only has to be evaluated at the given interpolation DOFs, |EmpiricalInterpolatedOperator| calls :meth:`~pymor.operators.interfaces.OperatorInterface.restricted` to obtain a restricted version of the operator which is used to quickly obtain the required evaluations. If the `restricted` method, is not implemented, the full operator will be evaluated (which will lead to the same result, but without any speedup). The interpolation DOFs and the collateral basis can be generated using the algorithms provided in the :mod:`pymor.algorithms.ei` module. Parameters ---------- operator The |Operator| to interpolate. interpolation_dofs List or 1D |NumPy array| of the interpolation DOFs `c_1, ..., c_M`. collateral_basis |VectorArray| containing the collateral basis `b_1, ..., b_M`. triangular If `True`, assume that ψ_(c_i)(b_j) = 0 for i < j, which means that the interpolation matrix is triangular. solver_options The |solver_options| for the operator. name Name of the operator. """ def __init__(self, operator, interpolation_dofs, collateral_basis, triangular, solver_options=None, name=None): assert isinstance(operator, OperatorInterface) assert isinstance(collateral_basis, VectorArrayInterface) assert collateral_basis in operator.range assert len(interpolation_dofs) == len(collateral_basis) self.build_parameter_type(operator) self.source = operator.source self.range = operator.range self.linear = operator.linear self.solver_options = solver_options self.name = name or f'{operator.name}_interpolated' self._operator = weakref.ref(operator) interpolation_dofs = np.array(interpolation_dofs, dtype=np.int32) self.interpolation_dofs = interpolation_dofs self.triangular = triangular if len(interpolation_dofs) > 0: try: self.restricted_operator, self.source_dofs = operator.restricted(interpolation_dofs) except NotImplementedError: self.logger.warning('Operator has no "restricted" method. The full operator will be evaluated.') self.operator = operator interpolation_matrix = collateral_basis.dofs(interpolation_dofs).T self.interpolation_matrix = interpolation_matrix self.collateral_basis = collateral_basis.copy() @property def operator(self): return self._operator()
[docs] def apply(self, U, mu=None): mu = self.parse_parameter(mu) if len(self.interpolation_dofs) == 0: return self.range.zeros(len(U)) if hasattr(self, 'restricted_operator'): U_dofs = NumpyVectorSpace.make_array(U.dofs(self.source_dofs)) AU = self.restricted_operator.apply(U_dofs, mu=mu) else: AU = NumpyVectorSpace.make_array(self.operator.apply(U, mu=mu).dofs(self.interpolation_dofs)) try: if self.triangular: interpolation_coefficients = solve_triangular(self.interpolation_matrix, AU.to_numpy().T, lower=True, unit_diagonal=True).T else: interpolation_coefficients = solve(self.interpolation_matrix, AU.to_numpy().T).T except ValueError: # this exception occurs when AU contains NaNs ... interpolation_coefficients = np.empty((len(AU), len(self.collateral_basis))) + np.nan return self.collateral_basis.lincomb(interpolation_coefficients)
[docs] def jacobian(self, U, mu=None): mu = self.parse_parameter(mu) options = self.solver_options.get('jacobian') if self.solver_options else None if len(self.interpolation_dofs) == 0: if isinstance(self.source, NumpyVectorSpace) and isinstance(self.range, NumpyVectorSpace): return NumpyMatrixOperator(np.zeros((self.range.dim, self.source.dim)), solver_options=options, source_id=self.source.id, range_id=self.range.id, name=self.name + '_jacobian') else: return ZeroOperator(self.range, self.source, name=self.name + '_jacobian') elif hasattr(self, 'operator'): return EmpiricalInterpolatedOperator(self.operator.jacobian(U, mu=mu), self.interpolation_dofs, self.collateral_basis, self.triangular, solver_options=options, name=self.name + '_jacobian') else: restricted_source = self.restricted_operator.source U_dofs = restricted_source.make_array(U.dofs(self.source_dofs)) JU = self.restricted_operator.jacobian(U_dofs, mu=mu) \ .apply(restricted_source.make_array(np.eye(len(self.source_dofs)))) try: if self.triangular: interpolation_coefficients = solve_triangular(self.interpolation_matrix, JU.to_numpy().T, lower=True, unit_diagonal=True).T else: interpolation_coefficients = solve(self.interpolation_matrix, JU.to_numpy().T).T except ValueError: # this exception occurs when AU contains NaNs ... interpolation_coefficients = np.empty((len(JU), len(self.collateral_basis))) + np.nan J = self.collateral_basis.lincomb(interpolation_coefficients) if isinstance(J.space, NumpyVectorSpace): J = NumpyMatrixOperator(J.to_numpy().T, range_id=self.range.id) else: J = VectorArrayOperator(J) return Concatenation([J, ComponentProjection(self.source_dofs, self.source)], solver_options=options, name=self.name + '_jacobian')
def __getstate__(self): d = self.__dict__.copy() del d['_operator'] return d
[docs]class ProjectedEmpiciralInterpolatedOperator(OperatorBase): """A projected |EmpiricalInterpolatedOperator|.""" def __init__(self, restricted_operator, interpolation_matrix, source_basis_dofs, projected_collateral_basis, triangular, solver_options=None, name=None): name = name or f'{restricted_operator.name}_projected' self.__auto_init(locals()) self.source = NumpyVectorSpace(len(source_basis_dofs)) self.range = projected_collateral_basis.space self.linear = restricted_operator.linear self.build_parameter_type(restricted_operator)
[docs] def apply(self, U, mu=None): mu = self.parse_parameter(mu) U_dofs = self.source_basis_dofs.lincomb(U.to_numpy()) AU = self.restricted_operator.apply(U_dofs, mu=mu) try: if self.triangular: interpolation_coefficients = solve_triangular(self.interpolation_matrix, AU.to_numpy().T, lower=True, unit_diagonal=True).T else: interpolation_coefficients = solve(self.interpolation_matrix, AU.to_numpy().T).T except ValueError: # this exception occurs when AU contains NaNs ... interpolation_coefficients = np.empty((len(AU), len(self.projected_collateral_basis))) + np.nan return self.projected_collateral_basis.lincomb(interpolation_coefficients)
[docs] def jacobian(self, U, mu=None): assert len(U) == 1 mu = self.parse_parameter(mu) options = self.solver_options.get('jacobian') if self.solver_options else None if self.interpolation_matrix.shape[0] == 0: return NumpyMatrixOperator(np.zeros((self.range.dim, self.source.dim)), solver_options=options, name=self.name + '_jacobian') U_dofs = self.source_basis_dofs.lincomb(U.to_numpy()[0]) J = self.restricted_operator.jacobian(U_dofs, mu=mu).apply(self.source_basis_dofs) try: if self.triangular: interpolation_coefficients = solve_triangular(self.interpolation_matrix, J.to_numpy().T, lower=True, unit_diagonal=True).T else: interpolation_coefficients = solve(self.interpolation_matrix, J.to_numpy().T).T except ValueError: # this exception occurs when J contains NaNs ... interpolation_coefficients = (np.empty((len(self.source_basis_dofs), len(self.projected_collateral_basis))) + np.nan) M = self.projected_collateral_basis.lincomb(interpolation_coefficients) if isinstance(M.space, NumpyVectorSpace): return NumpyMatrixOperator(M.to_numpy().T, solver_options=options) else: assert not options return VectorArrayOperator(M)
def with_cb_dim(self, dim): assert dim <= self.restricted_operator.range.dim interpolation_matrix = self.interpolation_matrix[:dim, :dim] restricted_operator, source_dofs = self.restricted_operator.restricted(np.arange(dim)) old_pcb = self.projected_collateral_basis projected_collateral_basis = NumpyVectorSpace.make_array(old_pcb.to_numpy()[:dim, :]) old_sbd = self.source_basis_dofs source_basis_dofs = NumpyVectorSpace.make_array(old_sbd.to_numpy()[:, source_dofs]) return ProjectedEmpiciralInterpolatedOperator(restricted_operator, interpolation_matrix, source_basis_dofs, projected_collateral_basis, self.triangular, solver_options=self.solver_options, name=self.name)