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Randomized residual‐based error estimators for the proper generalized decomposition approximation of parametrized problems
Author(s) -
Smetana Kathrin,
Zahm Olivier
Publication year - 2020
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.6339
Subject(s) - estimator , mathematics , residual , norm (philosophy) , mathematical optimization , gaussian , algorithm , statistics , physics , quantum mechanics , political science , law
Summary 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.