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

TL;DR

This paper introduces a new framework using Sectional Kolmogorov N-widths for parametrized variational problems with parameter-dependent function spaces, providing exponential approximation rates especially for Friedrichs' systems.

Contribution

It develops a fiber bundle-based approach and extends Kolmogorov N-width theory to parameter-dependent spaces, addressing limitations of the solution manifold concept.

Findings

Exponential approximation rates for the N-width under certain conditions

A sufficient criterion for Friedrichs' systems to achieve these rates

Application of the framework to various parametrized PDE problems

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in these cases the interpretation as a "solution manifold" widely adopted in the model order reduction community is no longer applicable.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.