Randomized residual-based error estimators for the Proper Generalized Decomposition approximation of parametrized problems

TL;DR

This paper presents a new randomized error estimator for the PGD approximation of parametrized problems, leveraging concentration inequalities and adjoint problems, with high-probability effectivity close to one.

Contribution

It introduces a novel probabilistic error estimator for PGD that can accurately estimate errors in various norms and quantities of interest.

Findings

Effectivity close to unity with high probability

Effective for high-dimensional parametrized problems

Demonstrated on elastodynamics and elasticity 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.