Source code for pymor.reductors.coercive

# 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 numpy as np

from pymor.core.interfaces import ImmutableInterface
from pymor.operators.constructions import LincombOperator, induced_norm
from pymor.operators.numpy import NumpyMatrixOperator
from pymor.reductors.basic import StationaryRBReductor
from pymor.reductors.residual import ResidualReductor
from pymor.vectorarrays.numpy import NumpyVectorSpace


[docs]class CoerciveRBReductor(StationaryRBReductor): """Reduced Basis reductor for |StationaryModels| with coercive linear operator. The only addition to :class:`~pymor.reductors.basic.StationaryRBReductor` is an error estimator which evaluates the dual norm of the residual with respect to a given inner product. For the reduction of the residual we use :class:`~pymor.reductors.residual.ResidualReductor` for improved numerical stability [BEOR14]_. Parameters ---------- fom The |Model| which is to be reduced. RB |VectorArray| containing the reduced basis on which to project. product Inner product for the orthonormalization of `RB`, the projection of the |Operators| given by `vector_ranged_operators` and for the computation of Riesz representatives of the residual. If `None`, the Euclidean product is used. coercivity_estimator `None` or a |Parameterfunctional| returning a lower bound for the coercivity constant of the given problem. Note that the computed error estimate is only guaranteed to be an upper bound for the error when an appropriate coercivity estimate is specified. """ def __init__(self, fom, RB=None, product=None, coercivity_estimator=None, check_orthonormality=None, check_tol=None): super().__init__(fom, RB, product=product, check_orthonormality=check_orthonormality, check_tol=check_tol) self.coercivity_estimator = coercivity_estimator self.residual_reductor = ResidualReductor(self.bases['RB'], self.fom.operator, self.fom.rhs, product=product, riesz_representatives=True) def assemble_estimator(self): residual = self.residual_reductor.reduce() estimator = CoerciveRBEstimator(residual, tuple(self.residual_reductor.residual_range_dims), self.coercivity_estimator) return estimator def assemble_estimator_for_subbasis(self, dims): return self._last_rom.estimator.restricted_to_subbasis(dims['RB'], m=self._last_rom)
[docs]class CoerciveRBEstimator(ImmutableInterface): """Instantiated by :class:`CoerciveRBReductor`. Not to be used directly. """ def __init__(self, residual, residual_range_dims, coercivity_estimator): self.__auto_init(locals()) def estimate(self, U, mu, m): est = self.residual.apply(U, mu=mu).l2_norm() if self.coercivity_estimator: est /= self.coercivity_estimator(mu) return est def restricted_to_subbasis(self, dim, m): if self.residual_range_dims: residual_range_dims = self.residual_range_dims[:dim + 1] residual = self.residual.projected_to_subbasis(residual_range_dims[-1], dim) return CoerciveRBEstimator(residual, residual_range_dims, self.coercivity_estimator) else: self.logger.warning('Cannot efficiently reduce to subbasis') return CoerciveRBEstimator(self.residual.projected_to_subbasis(None, dim), None, self.coercivity_estimator)
[docs]class SimpleCoerciveRBReductor(StationaryRBReductor): """Reductor for linear |StationaryModels| with affinely decomposed operator and rhs. .. note:: The reductor :class:`CoerciveRBReductor` can be used for arbitrary coercive |StationaryModels| and offers an improved error estimator with better numerical stability. The only addition is to :class:`~pymor.reductors.basic.StationaryRBReductor` is an error estimator, which evaluates the norm of the residual with respect to a given inner product. Parameters ---------- fom The |Model| which is to be reduced. RB |VectorArray| containing the reduced basis on which to project. product Inner product for the orthonormalization of `RB`, the projection of the |Operators| given by `vector_ranged_operators` and for the computation of Riesz representatives of the residual. If `None`, the Euclidean product is used. coercivity_estimator `None` or a |Parameterfunctional| returning a lower bound for the coercivity constant of the given problem. Note that the computed error estimate is only guaranteed to be an upper bound for the error when an appropriate coercivity estimate is specified. """ def __init__(self, fom, RB=None, product=None, coercivity_estimator=None, check_orthonormality=None, check_tol=None): assert fom.operator.linear and fom.rhs.linear assert isinstance(fom.operator, LincombOperator) assert all(not op.parametric for op in fom.operator.operators) if fom.rhs.parametric: assert isinstance(fom.rhs, LincombOperator) assert all(not op.parametric for op in fom.rhs.operators) super().__init__(fom, RB, product=product, check_orthonormality=check_orthonormality, check_tol=check_tol) self.coercivity_estimator = coercivity_estimator self.residual_reductor = ResidualReductor(self.bases['RB'], self.fom.operator, self.fom.rhs, product=product) self.extends = None def assemble_estimator(self): fom, RB, extends = self.fom, self.bases['RB'], self.extends if extends: old_RB_size = extends[0] old_data = extends[1] else: old_RB_size = 0 # compute data for estimator space = fom.operator.source # compute the Riesz representative of (U, .)_L2 with respect to product def riesz_representative(U): if self.products['RB'] is None: return U.copy() else: return self.products['RB'].apply_inverse(U) def append_vector(U, R, RR): RR.append(riesz_representative(U), remove_from_other=True) R.append(U, remove_from_other=True) # compute all components of the residual if extends: R_R, RR_R = old_data['R_R'], old_data['RR_R'] elif not fom.rhs.parametric: R_R = space.empty(reserve=1) RR_R = space.empty(reserve=1) append_vector(fom.rhs.as_range_array(), R_R, RR_R) else: R_R = space.empty(reserve=len(fom.rhs.operators)) RR_R = space.empty(reserve=len(fom.rhs.operators)) for op in fom.rhs.operators: append_vector(op.as_range_array(), R_R, RR_R) if len(RB) == 0: R_Os = [space.empty()] RR_Os = [space.empty()] elif not fom.operator.parametric: R_Os = [space.empty(reserve=len(RB))] RR_Os = [space.empty(reserve=len(RB))] for i in range(len(RB)): append_vector(-fom.operator.apply(RB[i]), R_Os[0], RR_Os[0]) else: R_Os = [space.empty(reserve=len(RB)) for _ in range(len(fom.operator.operators))] RR_Os = [space.empty(reserve=len(RB)) for _ in range(len(fom.operator.operators))] if old_RB_size > 0: for op, R_O, RR_O, old_R_O, old_RR_O in zip(fom.operator.operators, R_Os, RR_Os, old_data['R_Os'], old_data['RR_Os']): R_O.append(old_R_O) RR_O.append(old_RR_O) for op, R_O, RR_O in zip(fom.operator.operators, R_Os, RR_Os): for i in range(old_RB_size, len(RB)): append_vector(-op.apply(RB[i]), R_O, RR_O) # compute Gram matrix of the residuals R_RR = RR_R.dot(R_R) R_RO = np.hstack([RR_R.dot(R_O) for R_O in R_Os]) R_OO = np.vstack([np.hstack([RR_O.dot(R_O) for R_O in R_Os]) for RR_O in RR_Os]) estimator_matrix = np.empty((len(R_RR) + len(R_OO),) * 2) estimator_matrix[:len(R_RR), :len(R_RR)] = R_RR estimator_matrix[len(R_RR):, len(R_RR):] = R_OO estimator_matrix[:len(R_RR), len(R_RR):] = R_RO estimator_matrix[len(R_RR):, :len(R_RR)] = R_RO.T estimator_matrix = NumpyMatrixOperator(estimator_matrix) estimator = SimpleCoerciveRBEstimator(estimator_matrix, self.coercivity_estimator) self.extends = (len(RB), dict(R_R=R_R, RR_R=RR_R, R_Os=R_Os, RR_Os=RR_Os)) return estimator def assemble_estimator_for_subbasis(self, dims): return self._last_rom.estimator.restricted_to_subbasis(dims['RB'], m=self._last_rom)
[docs]class SimpleCoerciveRBEstimator(ImmutableInterface): """Instantiated by :class:`SimpleCoerciveRBReductor`. Not to be used directly. """ def __init__(self, estimator_matrix, coercivity_estimator): self.__auto_init(locals()) self.norm = induced_norm(estimator_matrix) def estimate(self, U, mu, m): if len(U) > 1: raise NotImplementedError if not m.rhs.parametric: CR = np.ones(1) else: CR = np.array(m.rhs.evaluate_coefficients(mu)) if not m.operator.parametric: CO = np.ones(1) else: CO = np.array(m.operator.evaluate_coefficients(mu)) C = np.hstack((CR, np.dot(CO[..., np.newaxis], U.to_numpy()).ravel())) est = self.norm(NumpyVectorSpace.make_array(C)) if self.coercivity_estimator: est /= self.coercivity_estimator(mu) return est def restricted_to_subbasis(self, dim, m): cr = 1 if not m.rhs.parametric else len(m.rhs.operators) co = 1 if not m.operator.parametric else len(m.operator.operators) old_dim = m.operator.source.dim indices = np.concatenate((np.arange(cr), ((np.arange(co)*old_dim)[..., np.newaxis] + np.arange(dim)).ravel() + cr)) matrix = self.estimator_matrix.matrix[indices, :][:, indices] return SimpleCoerciveRBEstimator(NumpyMatrixOperator(matrix), self.coercivity_estimator)