Source code for pymor.reductors.coercive

# This file is part of the pyMOR project (http://www.pymor.org).
# Copyright 2013-2017 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 GenericRBReductor
from pymor.reductors.residual import ResidualReductor
from pymor.vectorarrays.numpy import NumpyVectorSpace


[docs]class CoerciveRBReductor(GenericRBReductor): """Reduced Basis reductor for |StationaryDiscretizations| with coercive linear operator. The only addition to :class:`~pymor.reductors.basic.GenericRBReductor` 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]_. .. [BEOR14] A. Buhr, C. Engwer, M. Ohlberger, S. Rave, A Numerically Stable A Posteriori Error Estimator for Reduced Basis Approximations of Elliptic Equations, Proceedings of the 11th World Congress on Computational Mechanics, 2014. Parameters ---------- d The |Discretization| which is to be reduced. RB |VectorArray| containing the reduced basis on which to project. orthogonal_projection List of keys in `d.operators` for which the corresponding |Operator| should be orthogonally projected (i.e. operators which map to vectors in contrast to bilinear forms which map to functionals). product Inner product for the projection of the |Operators| given by `orthogonal_projection` 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, d, RB=None, orthogonal_projection=('initial_data',), product=None, coercivity_estimator=None): super().__init__(d, RB, orthogonal_projection=orthogonal_projection, product=product) self.coercivity_estimator = coercivity_estimator self.residual_reductor = ResidualReductor(self.RB, self.d.operator, self.d.rhs, product=product) def _reduce(self): with self.logger.block('RB projection ...'): rd = super()._reduce() with self.logger.block('Assembling error estimator ...'): residual = self.residual_reductor.reduce() estimator = CoerciveRBEstimator(residual, tuple(self.residual_reductor.residual_range_dims), self.coercivity_estimator) rd = rd.with_(estimator=estimator) return rd
[docs]class CoerciveRBEstimator(ImmutableInterface): """Instantiated by :class:`CoerciveRBReductor`. Not to be used directly. """ def __init__(self, residual, residual_range_dims, coercivity_estimator): self.residual = residual self.residual_range_dims = residual_range_dims self.coercivity_estimator = coercivity_estimator def estimate(self, U, mu, d): 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, d): 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(GenericRBReductor): """Reductor for linear |StationaryDiscretizations| with affinely decomposed operator and rhs. .. note:: The reductor :class:`CoerciveRBReductor` can be used for arbitrary coercive |StationaryDiscretizations| and offers an improved error estimator with better numerical stability. The only addition is to :class:`~pymor.reductors.basic.GenericRBReductor` is an error estimator, which evaluates the norm of the residual with respect to a given inner product. Parameters ---------- d The |Discretization| which is to be reduced. RB |VectorArray| containing the reduced basis on which to project. orthogonal_projection List of keys in `d.operators` for which the corresponding |Operator| should be orthogonally projected (i.e. operators which map to vectors in contrast to bilinear forms which map to functionals). product Inner product for the projection of the |Operators| given by `orthogonal_projection` 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, d, RB=None, orthogonal_projection=('initial_data',), product=None, coercivity_estimator=None): assert d.linear assert isinstance(d.operator, LincombOperator) assert all(not op.parametric for op in d.operator.operators) if d.rhs.parametric: assert isinstance(d.rhs, LincombOperator) assert all(not op.parametric for op in d.rhs.operators) super().__init__(d, RB, orthogonal_projection=orthogonal_projection, product=product) self.coercivity_estimator = coercivity_estimator self.residual_reductor = ResidualReductor(self.RB, self.d.operator, self.d.rhs, product=product) self.extends = None def _reduce(self): d, RB, extends = self.d, self.RB, self.extends rd = super()._reduce() if extends: old_RB_size = extends[0] old_data = extends[1] else: old_RB_size = 0 # compute data for estimator space = d.operator.source # compute the Riesz representative of (U, .)_L2 with respect to product def riesz_representative(U): if self.product is None: return U.copy() else: return self.product.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 d.rhs.parametric: R_R = space.empty(reserve=1) RR_R = space.empty(reserve=1) append_vector(d.rhs.as_source_array(), R_R, RR_R) else: R_R = space.empty(reserve=len(d.rhs.operators)) RR_R = space.empty(reserve=len(d.rhs.operators)) for op in d.rhs.operators: append_vector(op.as_source_array(), R_R, RR_R) if len(RB) == 0: R_Os = [space.empty()] RR_Os = [space.empty()] elif not d.operator.parametric: R_Os = [space.empty(reserve=len(RB))] RR_Os = [space.empty(reserve=len(RB))] for i in range(len(RB)): append_vector(-d.operator.apply(RB[i]), R_Os[0], RR_Os[0]) else: R_Os = [space.empty(reserve=len(RB)) for _ in range(len(d.operator.operators))] RR_Os = [space.empty(reserve=len(RB)) for _ in range(len(d.operator.operators))] if old_RB_size > 0: for op, R_O, RR_O, old_R_O, old_RR_O in zip(d.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(d.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) rd = rd.with_(estimator=estimator) self.extends = (len(RB), dict(R_R=R_R, RR_R=RR_R, R_Os=R_Os, RR_Os=RR_Os)) return rd
[docs]class SimpleCoerciveRBEstimator(ImmutableInterface): """Instantiated by :class:`SimpleCoerciveRBReductor`. Not to be used directly. """ def __init__(self, estimator_matrix, coercivity_estimator): self.estimator_matrix = estimator_matrix self.coercivity_estimator = coercivity_estimator self.norm = induced_norm(estimator_matrix) def estimate(self, U, mu, d): if len(U) > 1: raise NotImplementedError if not d.rhs.parametric: CR = np.ones(1) else: CR = np.array(d.rhs.evaluate_coefficients(mu)) if not d.operator.parametric: CO = np.ones(1) else: CO = np.array(d.operator.evaluate_coefficients(mu)) C = np.hstack((CR, np.dot(CO[..., np.newaxis], U.data).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, d): cr = 1 if not d.rhs.parametric else len(d.rhs.operators) co = 1 if not d.operator.parametric else len(d.operator.operators) old_dim = d.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)