Sectional Kolmogorov N-widths for parameter-dependent function spaces: A general framework with application to parametrized Friedrichs' systems

Engwer, Christian; Ohlberger, Mario; Rebelt, Lukas

Research article in digital collection | Preprint

Abstract

We investigate parametrized variational problems where for each parameter the solution may originate from a different parameter-dependent function space. Our main motivation is the theory of Friedrichs' systems, a large abstract class of linear PDE-problems whose solutions are sought in operator- (and thus parameter-)dependent graph spaces. Other applications include function spaces on parametrized domains or discretizations involving data-dependent stabilizers. Concerning the set of all parameter-dependent solutions, we argue that in these cases the interpretation as a "solution manifold" widely adopted in the model order reduction community is no longer applicable. Instead, we propose a novel framework based on the theory of fiber bundles and explain how established concepts such as approximability generalize by introducing a Sectional Kolmogorov N-width. Further, we prove exponential approximation rates of this N-width if a norm equivalence criterion is fulfilled. Applying this result to problems with Friedrichs' structure then gives a sufficient criterion that can be easily verified.

Details about the publication

Name of the repositoryarXiv
Article number2507.00678
Statussubmitted / under review
Release year2025 (01/07/2025)
Language in which the publication is writtenEnglish
DOI10.48550/arXiv.2507.00678
Link to the full texthttps://doi.org/10.48550/arXiv.2507.00678
KeywordsApproximation theory, Model order reduction, Friedrichs’ systems, Parametrized function spaces

Authors from the University of Münster

Engwer, Christian
Professorship for Applications of Partial Differential Equations
Center for Nonlinear Science
Ohlberger, Mario
Professorship of Applied Mathematics, especially Numerics (Prof. Ohlberger)
Center for Nonlinear Science
Center for Multiscale Theory and Computation
Renelt, Lukas
Professorship of Applied Mathematics, especially Numerics (Prof. Ohlberger)