Performance Bounds for Reduced Order Models with Application to Parametric Transport

Abstract

The Kolmogorov n -width is an established benchmark to judge the approximation properties of the reduced linear spaces arising from reduced order models (ROMs). 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. Simply replacing the Kolmogorov n -width of the solution manifold with nonlinear benchmarks, like manifold/stable/Lipschitz width, does generally not provide satisfactory results: The performance bounds tend to be trivial if the degrees of freedom exceed the parameter dimension. Furthermore, without linearity, nonlinear width lack structure to implement corresponding online/offline decompositions. This paper introduces a different perspective, where the nonlinear widths are applied to the full reduced order model pipeline from PDE to parametric quantity of interest. We prove that this alternative view provides informative nonlinear benchmarks for transport equations.