arXiv subjects

Hassan Ballout

Publications and source records attributed to Hassan Ballout.

2 recordsLinked to original sources

arXiv2026

Nonlinear compressive reduced basis approximation : when Taylor meets Kolmogorov

This paper investigates model reduction methods for efficiently approximating the solution of parameter-dependent PDEs with a multi-parameter vector $\vecμ \in \mathbb{R}^p$. In cases where the Kolmogorov $N$-width decays fast enough, it is effective to approximate the solution as a sum of $N$ separable terms, each being the product of a parameter-dependent coefficient and a space-dependent function. This leads to reduced-order models with $N$ degrees of freedom and complexity of order ${\mathcal O}(N^3)$. However, when the $N$-width decays slowly, $N$ must be large to achieve acceptable accuracy, making cubic complexity prohibitive. The linear complexity measure in terms of Kolmogorov width must be replaced by the Gelfand width, with its associated sensing number. Recent nonlinear approaches based on this notion decompose the $N$ coordinates into two groups: $n$ free variables and $\overline{n}$ dependent variables, where the latter are nonlinear functions of the former ($N= n+\overline n$). Several works have focused on cases where these $\overline{n}$ functions are homogeneous quadratic forms of the $n$ variables, with optimization strategies for choosing $n$ given a target accuracy. A rigorous analysis of the local sensing number is carried out, showing that $n = p$ is optimal and appropriate, at least locally, around a reference point. In practical scenarios involving wide parameter ranges, the condition $p\le n \le p + k$ (with $k$ small) is valid and more robust from continuity arguments. Additionally, the assumption of a quadratic mapping, while justified in a local sense, becomes insufficient. More expressive nonlinear mappings-including those using machine learning-become necessary. This work contributes a theoretical foundation for such strategies and highlights the need for further investigations to push back the Kolmogorov Barrier.

math.NA↗

arXiv2024

Nonlinear compressive reduced basis approximation for multi-parameter elliptic problem

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.

math.NA↗