School of Mathematical & Statistical Sciences Faculty Publications
Document Type
Article
Publication Date
9-30-2026
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.
Recommended Citation
Rim, Donsub, and Gerrit Welper. "Performance bounds for reduced order models with application to parametric transport." Constructive Approximation (2026): 1-35. https://doi.org/10.1007/s00365-026-09784-1
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.
Publication Title
Constructive Approximation
DOI
10.1007/s00365-026-09784-1
Comments
This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.