Source code for pymor.reductors.parabolic

# 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.reductors.basic import InstationaryRBReductor
from pymor.reductors.residual import ResidualReductor, ImplicitEulerResidualReductor
from pymor.operators.constructions import IdentityOperator
from pymor.algorithms.timestepping import ImplicitEulerTimeStepper


[docs]class ParabolicRBReductor(InstationaryRBReductor): r"""Reduced Basis Reductor for parabolic equations. This reductor uses :class:`~pymor.reductors.basic.InstationaryRBReductor` 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^{-2}(\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 ---------- fom The |InstationaryModel| which is to be reduced. RB |VectorArray| containing the reduced basis on which to project. 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 `fom.operator` w.r.t. `product`. """ def __init__(self, fom, RB=None, product=None, coercivity_estimator=None, check_orthonormality=None, check_tol=None): if not isinstance(fom.time_stepper, ImplicitEulerTimeStepper): raise NotImplementedError if fom.mass is not None and fom.mass.parametric and '_t' in fom.mass.parameter_type: raise NotImplementedError super().__init__(fom, RB, product=product, check_orthonormality=check_orthonormality, check_tol=check_tol) self.coercivity_estimator = coercivity_estimator self.residual_reductor = ImplicitEulerResidualReductor( self.bases['RB'], fom.operator, fom.mass, fom.T / fom.time_stepper.nt, rhs=fom.rhs, product=product ) self.initial_residual_reductor = ResidualReductor( self.bases['RB'], IdentityOperator(fom.solution_space), fom.initial_data, product=fom.l2_product, riesz_representatives=False ) def assemble_estimator(self): 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) 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 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.__auto_init(locals()) def estimate(self, U, mu, m, return_error_sequence=False): dt = m.T / m.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] if '_t' in self.residual.parameter_type: t = 0 for n in range(1, m.time_stepper.nt + 1): t += dt mu['_t'] = t est[n] = self.residual.apply(U[n], U[n-1], mu=mu).l2_norm2() else: est[1:] = self.residual.apply(U[1:], 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, m): 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)