Colloque - Albert Cohen : Optimal Linear and Non-Linear Dimensionality Reduction

Épisode de Informatique et sciences numériques (2025-2026) - Yvon Maday par Collège de France · 23 juin 2026 · 56 min

Yvon Maday

Chaire Informatique et sciences numériques

Collège de France

Année 2025-2026

Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Albert Cohen : Optimal Linear and Non-Linear Dimensionality Reduction

Albert Cohen

Professeur, Laboratoire Jacques-Louis Lions, université Pierre et Marie Curie, Paris

Résumé

Understanding how to optimally approximate general compact sets by finite dimensional spaces is of central interest for designing efficient numerical methods in forward simulation or inverse problems. The concept of n-width, introduced in 1936 by Kolmogorov, is well tailored to linear approximation methods. The interest for n-width has recently been revived by the approximation of parametrized/stochastic PDEs, and the development of reduced basis methods. We briefly survey some now classical results.

We then focus on analogous concepts for nonlinear approximation which are still the object of current research, motivated in particular by the development of neural networks, and possible applications to hyperbolic parametrized PDEs for which linear methods are not effective. We discuss a general framework that allows to embrace various concepts of linear and nonlinear widths, and present some recent results and relevant open problems within this framework.

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