Nonlinear compressive reduced basis approximation for multi-parameter elliptic problem

TL;DR

This paper introduces a nonlinear compressive reduced basis method for multi-parameter elliptic PDEs, demonstrating that the approximation complexity depends on the number of parameters rather than traditional linear width measures.

Contribution

It proposes a new nonlinear approach that better captures solution manifolds, reducing complexity in multi-parameter PDE approximation compared to classical linear methods.

Findings

Complexity relates to the number of parameters, not Kolmogorov N-width.

Nonlinear methods outperform linear approaches in multi-parameter settings.

Rigorous analysis confirms the efficiency of the proposed nonlinear reduced basis.

Abstract

Reduced basis methods for approximating the solutions of parameter-dependant partial differential equations (PDEs) are based on learning the structure of the set of solutions - seen as a manifold ${\mathcal S}$ in some functional space - when the parameters vary. This involves investigating the manifold and, in particular, understanding whether it is close to a low-dimensional affine space. This leads to the notion of Kolmogorov $N$-width that consists of evaluating to which extent the best choice of a vectorial space of dimension $N$ approximates ${\mathcal S}$ well enough. If a good approximation of elements in ${\mathcal S}$ can be done with some well-chosen vectorial space of dimension $N$ -- provided $N$ is not too large -- then a ``reduced'' basis can be proposed that leads to a Galerkin type method for the approximation of any element in ${\mathcal S}$. In many cases, however, the Kolmogorov $N$-width is not so small, even if the parameter set lies in a space of small dimension yielding a manifold with small dimension. In terms of complexity reduction, this gap between the small dimension of the manifold and the large Kolmogorov $N$-width can be explained by the fact that the Kolmogorov $N$-width is linear while, in contrast, the dependency in the parameter is, most often, non-linear. There have been many contributions aiming at reconciling these two statements, either based on deterministic or AI approaches. We investigate here further a new paradigm that, in some sense, merges these two aspects: the nonlinear compressive reduced basisapproximation. We focus on a simple multiparameter problem and illustrate rigorously that the complexity associated with the approximation of the solution to the parameter dependant PDE is directly related to the number of parameters rather than the Kolmogorov $N$-width.