Source code for pymor.reductors.parabolic

# This file is part of the pyMOR project (http://www.pymor.org).
# Copyright 2013-2018 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.reductors.basic import GenericRBReductor
from pymor.reductors.residual import ResidualReductor, ImplicitEulerResidualReductor
from pymor.operators.constructions import IdentityOperator
from pymor.algorithms.timestepping import ImplicitEulerTimeStepper


[docs]class ParabolicRBReductor(GenericRBReductor): r"""Reduced Basis Reductor for parabolic equations. This reductor uses :class:`~pymor.reductors.basic.GenericRBReductor` for the actual RB-projection. The only addition is the assembly of an error estimator which bounds the discrete l2-in time / energy-in space error similar to [GP05]_, [HO08]_ as follows: .. math:: \left[ C_a^{-1}(\mu)\|e_N(\mu)\|^2 + \sum_{n=1}^{N} \Delta t\|e_n(\mu)\|^2_e \right]^{1/2} \leq \left[ C_a^{-1}(\mu)\Delta t \sum_{n=1}^{N}\|\mathcal{R}^n(u_n(\mu), \mu)\|^2_{e,-1} + C_a^{-1}(\mu)\|e_0\|^2 \right]^{1/2} Here, :math:`\|\cdot\|` denotes the norm induced by the problem's mass matrix (e.g. the L^2-norm) and :math:`\|\cdot\|_e` is an arbitrary energy norm w.r.t. which the space operator :math:`A(\mu)` is coercive, and :math:`C_a(\mu)` is a lower bound for its coercivity constant. Finally, :math:`\mathcal{R}^n` denotes the implicit Euler timestepping residual for the (fixed) time step size :math:`\Delta t`, .. math:: \mathcal{R}^n(u_n(\mu), \mu) := f - M \frac{u_{n}(\mu) - u_{n-1}(\mu)}{\Delta t} - A(u_n(\mu), \mu), where :math:`M` denotes the mass operator and :math:`f` the source term. The dual norm of the residual is computed using the numerically stable projection from [BEOR14]_. Parameters ---------- d The |InstationaryDiscretization| which is to be reduced. RB |VectorArray| containing the reduced basis on which to project. basis_is_orthonormal Indicate whether or not the basis is orthonormal w.r.t. `product`. product The energy inner product |Operator| w.r.t. which the reduction error is estimated and `RB` is orthonormalized. coercivity_estimator `None` or a |Parameterfunctional| returning a lower bound :math:`C_a(\mu)` for the coercivity constant of `d.operator` w.r.t. `product`. """ def __init__(self, d, RB=None, basis_is_orthonormal=None, product=None, coercivity_estimator=None): assert isinstance(d.time_stepper, ImplicitEulerTimeStepper) super().__init__(d, RB, basis_is_orthonormal=basis_is_orthonormal, product=product) self.coercivity_estimator = coercivity_estimator self.residual_reductor = ImplicitEulerResidualReductor( self.RB, d.operator, d.mass, d.T / d.time_stepper.nt, rhs=d.rhs, product=product ) self.initial_residual_reductor = ResidualReductor( self.RB, IdentityOperator(d.solution_space), d.initial_data, product=d.l2_product, riesz_representatives=False ) def _reduce(self): with self.logger.block('RB projection ...'): rd = super()._reduce() with self.logger.block('Assembling error estimator ...'): residual = self.residual_reductor.reduce() initial_residual = self.initial_residual_reductor.reduce() estimator = ParabolicRBEstimator(residual, self.residual_reductor.residual_range_dims, initial_residual, self.initial_residual_reductor.residual_range_dims, self.coercivity_estimator) rd = rd.with_(estimator=estimator) return rd
[docs]class ParabolicRBEstimator(ImmutableInterface): """Instantiated by :class:`ParabolicRBReductor`. Not to be used directly. """ def __init__(self, residual, residual_range_dims, initial_residual, initial_residual_range_dims, coercivity_estimator): self.residual = residual self.residual_range_dims = residual_range_dims self.initial_residual = initial_residual self.initial_residual_range_dims = initial_residual_range_dims self.coercivity_estimator = coercivity_estimator def estimate(self, U, mu, d, return_error_sequence=False): dt = d.T / d.time_stepper.nt C = self.coercivity_estimator(mu) if self.coercivity_estimator else 1. est = np.empty(len(U)) est[0] = (1./C) * self.initial_residual.apply(U[0], mu=mu).l2_norm2()[0] est[1:] = self.residual.apply(U[1:len(U)], U[0:len(U)-1], mu=mu).l2_norm2() est[1:] *= (dt/C**2) est = np.sqrt(np.cumsum(est)) return est if return_error_sequence else est[-1] def restricted_to_subbasis(self, dim, d): if self.residual_range_dims and self.initial_residual_range_dims: residual_range_dims = self.residual_range_dims[:dim + 1] residual = self.residual.projected_to_subbasis(residual_range_dims[-1], dim) initial_residual_range_dims = self.initial_residual_range_dims[:dim + 1] initial_residual = self.initial_residual.projected_to_subbasis(initial_residual_range_dims[-1], dim) return ParabolicRBEstimator(residual, residual_range_dims, initial_residual, initial_residual_range_dims, self.coercivity_estimator) else: self.logger.warning('Cannot efficiently reduce to subbasis') return ParabolicRBEstimator(self.residual.projected_to_subbasis(None, dim), None, self.initial_residual.projected_to_subbasis(None, dim), None, self.coercivity_estimator)