A progressive reduced basis/empirical interpolation method for nonlinear parabolic problems

TL;DR

This paper introduces PREIM, a new progressive method combining Reduced Basis and Empirical Interpolation to efficiently solve parametrized nonlinear parabolic problems, reducing offline costs while maintaining accuracy.

Contribution

The paper develops PREIM, a novel progressive RB-EIM approach that enriches both approximations simultaneously, improving efficiency for nonlinear parabolic problems.

Findings

PREIM reduces offline computational costs.

Maintains high accuracy in online RB approximations.

Effective for nonlinear heat transfer problems.

Abstract

We investigate new developments of the combined Reduced-Basis and Empirical Interpolation Methods (RB-EIM) for parametrized nonlinear parabolic problems. In many situations, the cost of the EIM in the offline stage turns out to be prohibitive since a significant number of nonlinear time-dependent problems need to be solved using the high-fidelity (or full-order) model. In the present work, we develop a new methodology, the Progressive RB-EIM (PREIM) method for nonlinear parabolic problems.The purpose is to reduce the offline cost while maintaining the accuracy of the RB approximation in the online stage. The key idea is a progressive enrichment of both the EIM approximation and the RB space, in contrast to the standard approach where the EIM approximation and the RB space are built separately. PREIM uses high-fidelity computations whenever available and RB computationsotherwise. Another key feature of each PREIM iteration is to select twice the parameter in a greedy fashion, the second selection being made after computing the high-fidelity solution for the firstly selected value of the parameter. Numerical examples are presented on nonlinear heat transfer problems.