Yvon MadayChaire Informatique et sciences numériquesCollège de FranceAnnée 2025-2026Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Albert Cohen : Optimal Linear and Non-Linear Dimensionality ReductionAlbert CohenProfesseur, Laboratoire Jacques-Louis Lions, université Pierre et Marie Curie, ParisRésuméUnderstanding 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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