Sectional Kolmogorov N-widths for parameter-dependent function spaces: A general framework with application to parametrized Friedrichs' systems
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| Format: | Preprint |
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2025
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| _version_ | 1866909669968052224 |
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| author | Engwer, Christian Ohlberger, Mario Renelt, Lukas |
| author_facet | Engwer, Christian Ohlberger, Mario Renelt, Lukas |
| contents | 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in these cases the interpretation as a "solution manifold" widely adopted in the model order reduction community is no longer applicable.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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_00678 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sectional Kolmogorov N-widths for parameter-dependent function spaces: A general framework with application to parametrized Friedrichs' systems Engwer, Christian Ohlberger, Mario Renelt, Lukas Numerical Analysis 65M22, 41A46, 65N30, 65J05 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. |
| title | Sectional Kolmogorov N-widths for parameter-dependent function spaces: A general framework with application to parametrized Friedrichs' systems |
| topic | Numerical Analysis 65M22, 41A46, 65N30, 65J05 |
| url | https://arxiv.org/abs/2507.00678 |