A posteriori error estimation and adaptive strategy for PGD model reduction applied to parametrized linear parabolic problems

TL;DR

This paper introduces an a posteriori error estimation and adaptive strategy for PGD model reduction in parametrized linear parabolic problems, ensuring guaranteed accuracy and optimizing the approximation process.

Contribution

It develops a new verification procedure using constitutive relation error for PGD model reduction, including error splitting and adaptive strategies for improved accuracy.

Findings

Effective error bounds for PGD reduced models.

Adaptive strategy improves approximation accuracy.

Validated on multi-parameter mechanical problems.

Abstract

We define an a posteriori verification procedure that enables to control and certify PGD-based model reduction techniques applied to parametrized linear elliptic or parabolic problems. Using the concept of constitutive relation error, it provides guaranteed and fully computable global/goal-oriented error estimates taking both discretization and PGD truncation errors into account. Splitting the error sources, it also leads to a natural greedy adaptive strategy which can be driven in order to optimize the accuracy of PGD approximations. The focus of the paper is on two technical points: (i) construction of equilibrated fields required to compute guaranteed error bounds; (ii) error splitting and adaptive process when performing PGD-based model reduction. Performances of the proposed verification and adaptation tools are shown on several multi-parameter mechanical problems.