Randomized residual‐based error estimators for the proper generalized decomposition approximation of parametrized problems

Abstract

This paper introduces a novel error estimator for the Proper Generalized Decomposition (PGD) approximation of parametrized equations. The estimator is intrinsically random: It builds on concentration inequalities of Gaussian maps and an adjoint problem with random right-hand side, which we approximate using the PGD. The effectivity of this randomized error estimator can be arbitrarily close to unity with high probability, allowing the estimation of the error with respect to any user-defined norm as well as the error in some quantity of interest. The performance of the error estimator is demonstrated and compared with some existing error estimators for the PGD for a parametrized time-harmonic elastodynamics problem and the parametrized equations of linear elasticity with a high-dimensional parameter space.in general only as costly as the computation of the PGD approximation or even significantly less expensive, which makes our error estimator strategy attractive from a computational viewpoint. To build this estimator, we first estimate the norm of the error with a Monte Carlo estimator using Gaussian random vectors whose covariance is chosen according to the desired error measure, e.g., user-defined norms or quantity of interest. Then, we introduce a dual problem with random right-hand side the solution of which allows us to rewrite the error estimator in terms of the residual of the original equation. Approximating the random dual problems with the PGD yields a fast-to-evaluate error estimator that has low marginal cost.Next, we relate our error estimator to other stopping criteria that have been proposed for the PGD solution. In [2] the authors suggest letting an error estimator in some linear QoI steer the enrichment of the PGD approximation. In order to estimate the error they employ an adjoint problem with the linear QoI as a right-hand side and approximate this adjoint problem with the PGD as well. It is important to note that, as observed in [2], this type of error estimator seems to require in general a more accurate solution of the dual than the primal problem. In contrast, the framework we present in this paper often allows using a dual PGD approximation with significantly less terms than the primal approximation. An adaptive strategy for the PGD approximation of the dual problem for the error estimator proposed in [2] was suggested in [23] in the context of computational homogenization. In [1] the approach proposed in [2] is extended to problems in nonlinear solids by considering a suitable linearization of the problem.An error estimator for the error between the exact solution of a parabolic PDE and the PGD approximation in a suitable norm is derived in [34] based on the constitutive relation error and the Prager-Synge theorem. The discretization error is thus also assessed, and the computational costs of the error estimator depend on the number of elements in the underlying spatial mesh. Relying on the same techniques an error estimator for the error between the exact solution of a parabolic PDE and a...

Preconditioners for model order reduction by interpolation and random sketching of operators

“…It was first used as a simple sample-based approach for error estimation [9,18]. These ideas then underwent a more significant development in [3,4,25,26]. It has to be noted that the probabilistic error estimator from [26] has a close relation to the multi-purpose preconditioner-based error estimator proposed in the present paper (see Section 1.2 for details).…”

“…A great advantage of certification of the error with Proposition 3.8 and Proposition 3.4 is that such a certification no longer requires B to have a moderate minimal singular value as in (24) and (25). The only requirement is that the preconditioner is such that B is close to R U with the solution space and/or the test space being restricted to the subspace U m .…”

“…The Galerkin methods with an improved stability for ill-conditioned and noncoercive parametric equations were developed in [1,28]. The effective/numerically stable, yet efficient error estimation/certification was addressed in [10,15,27] with deterministic approaches, and in [3,9,14,25,26,29] with statistical learning/randomized approaches.…”

“…The randomized linear algebra recently became a popular approach for the enhancement of MOR methods [3,4,8,9,18,25,26,31]. It was first used as a simple sample-based approach for error estimation [9,18].…”

Special Issue on Credible High‐Fidelity and Low‐Cost Simulations in Computational Engineering

“…The 12 manuscripts in this special issue cover a wide range of techniques related to the above mentioned topics, including: accurate high‐order 1,2 and fast lowest‐order discontinuous Galerkin discretisations; 3,4 mesh, 1‐3,5 degree, 1 and modal basis 6,7 adaptivity; a priori error estimates, 8 a posteriori error indicators, 1,3 error bounds, 5,7,9,10 and error estimates in quantities of interest; 2,6,9,10 a priori 5,10,11 and a posteriori 2,6‐9,12 reduced order models.…”