Performance bounds for Reduced Order Models with Application to Parametric Transport

TL;DR

This paper establishes performance bounds for reduced order models applied to parametric transport problems, highlighting limitations of linear methods and proposing benchmarks for nonlinear techniques.

Contribution

It introduces non-trivial benchmarks for reduced order models in transport problems and proves lower bounds for their performance.

Findings

Linear reduced basis methods have slow convergence for transport problems.

Nonlinear model reduction benchmarks can be trivial if not properly structured.

Lower bounds for the performance of reduced models in transport equations are established.

Abstract

The Kolmogorov $n$-width is an established benchmark to judge the performance of reduced basis and similar methods that produce linear reduced spaces. Although immensely successful in the elliptic regime, this width, shows unsatisfactory slow convergence rates for transport dominated problems. While this has triggered a large amount of work on nonlinear model reduction techniques, we are lacking a benchmark to evaluate their optimal performance. Nonlinear benchmarks like manifold/stable/Lipschitz width applied to the solution manifold are often trivial if the degrees of freedom exceed the parameter dimension and ignore desirable structure as offline/online decompositions. In this paper, we show that the same benchmarks applied to the full reduced order model pipeline from PDE to parametric quantity of interest provide non-trivial benchmarks and we prove lower bounds for transport equations.