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

Kathrin Smetana*, Olivier Zahm

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

2 Citations (Scopus)

Abstract

This article 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.
Original languageEnglish
Pages (from-to)5153-5177
Number of pages25
JournalInternational journal for numerical methods in engineering
Volume121
Issue number23
Early online date27 Feb 2020
DOIs
Publication statusPublished - 15 Dec 2020

Keywords

  • UT-Hybrid-D
  • a posteriori error estimation
  • concentration phenomenon
  • goal-oriented error estimation
  • Monte-Carlo estimator
  • proper generalized decomposition
  • parametrized equations

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