Simultaneous Empirical Interpolation and Reduced Basis method for non-linear problems

TL;DR

This paper introduces the SER algorithm, a simultaneous empirical interpolation and reduced basis method that significantly reduces computational costs for non-linear, non-affinely parametrized PDEs by requiring minimal finite element solves.

Contribution

The paper presents a novel simultaneous EIM and RB algorithm (SER) that improves efficiency by reducing the number of finite element solves needed for non-linear problems.

Findings

SER achieves substantial computational savings.

Requires only N+1 finite element solves for RB dimension N.

Demonstrated effectiveness on benchmark problems.

Abstract

In this paper, we focus on the reduced basis methodology in the context of non-linear non-affinely parametrized partial differential equations in which affine decomposition necessary for the reduced basis methodology are not obtained [4, 3]. To deal with this issue, it is now standard to apply the EIM methodology [8, 9] before deploying the Reduced Basis (RB) methodology. However the computational cost is generally huge as it requires many finite element solves, hence making it inefficient, to build the EIM approximation of the non-linear terms [9, 1]. We propose a simultaneous EIM Reduced basis algorithm, named SER, that provides a huge computational gain and requires as little as N + 1 finite element solves where N is the dimension of the RB approximation. The paper is organized as follows: we first review the EIM and RB methodologies applied to non-linear problems and identify the main issue, then we present SER and some variants and finally illustrates its performances in a benchmark proposed in [9].