An algorithmic comparison of the Hyper-Reduction and the Discrete Empirical Interpolation Method for a nonlinear thermal problem

TL;DR

This paper compares Hyper-Reduction and Discrete Empirical Interpolation Method for nonlinear thermal problems, analyzing their efficiency, accuracy, and differences through algorithmic and numerical assessments.

Contribution

It provides a detailed algorithmic and methodological comparison of Hyper-Reduction and (D)EIM, highlighting their advantages and limitations for nonlinear model reduction.

Findings

Hyper-Reduction and (D)EIM improve computational efficiency over Galerkin RB.

Both methods offer comparable accuracy in nonlinear thermal problems.

Results demonstrate significant speed-ups compared to finite element simulations.

Abstract

A novel algorithmic discussion of the methodological and numerical differences of competing parametric model reduction techniques for nonlinear problems are presented. First, the Galerkin reduced basis (RB) formulation is presented which fails at providing significant gains with respect to the computational efficiency for nonlinear problems. Renown methods for the reduction of the computing time of nonlinear reduced order models are the Hyper-Reduction and the (Discrete) Empirical Interpolation Method (EIM, DEIM). An algorithmic description and a methodological comparison of both methods are provided. The accuracy of the predictions of the hyper-reduced model and the (D)EIM in comparison to the Galerkin RB is investigated. All three approaches are applied to a simple uncertainty quantification of a planar nonlinear thermal conduction problem. The results are compared to computationally intense finite element simulations.