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

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in these cases, the interpretation as a “solution manifold” widely adopted in the model order reduction community is no longer applicable.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