Model order reduction for parametrized nonlinear hyperbolic problems as an application to Uncertainty Quantification

TL;DR

This paper develops reduced order modeling techniques for nonlinear hyperbolic conservation laws, enabling efficient uncertainty quantification through Monte Carlo methods, by addressing challenges posed by moving waves and discontinuities.

Contribution

It introduces a parameter-time framework and a POD-EIM-Greedy algorithm for hyperbolic systems, along with an error indicator and upper bound for model accuracy.

Findings

Effective ROM for hyperbolic laws with discontinuities

Error bounds for reduced solutions

Enhanced UQ efficiency with ROM

Abstract

In this work, we focus on reduced order modeling (ROM) techniques for hyperbolic conservation laws with application in uncertainty quantification (UQ) and in conjunction with the well-known Monte Carlo sampling method. Because we are interested in model order reduction (MOR) techniques for unsteady non-linear hyperbolic systems of conservation laws, which involve moving waves and discontinuities, we explore the parameter-time framework and in the same time we deal with nonlinearities using a POD-EIM-Greedy algorithm \cite{Drohmann2012}. We provide under some hypothesis an error indicator, which is also an error upper bound for the difference between the high fidelity solution and the reduced one.