arXiv subjects

M. Ingremeau

Publications and source records attributed to M. Ingremeau.

3 recordsLinked to original sources

arXiv2026

An efficient Galerkin method for high-frequency scattering problems using Wilson bases

We propose a new Galerkin discretization scheme for wave scattering problems that is based on microlocalised basis functions. We show that the proposed method can be made uniformly accurate for large wavenumbers $k$ with a number of degrees of freedom only scaling as $k^{d-1/2}$, while leading to an essentially sparse linear system. In contrast, finite element methods are known to require a number of degrees of freedom scaling at least as $k^d$ to achieve the same property. A similar method based on a Gabor frame was previously introduced by two of the authors, but it was suffering from severe conditioning issues. In the present work, by replacing the Gabor frame by a Wilson basis, we completely alleviate this problem. We rigorously establish error estimates and condition number bounds for the proposed method, and we provide one-dimensional numerical examples illustrating our theoretical findings.

math.NA↗

arXiv2023

Decay of coefficients and approximation rates in Gabor Gaussian frames

The aim of this note is to present a self-contained proof of the fact that a function can be approximated using a linear combination of Gaussian coherent states, with a number of terms controlled in terms of the smoothness and of the decay at infinity of the function. This result can easily be obtained using advanced results on modulation spaces, but the proof presented here is completely elementary and self-contained.

math.NA↗

arXiv2022

Efficient approximation of high-frequency Helmholtz solutions by Gaussian coherent states

We introduce new finite-dimensional spaces specifically designed to approximate the solutions to high-frequency Helmholtz problems with smooth variable coefficients in dimension $d$. These discretization spaces are spanned by Gaussian coherent states, that have the key property to be localised in phase space. We carefully select the Gaussian coherent states spanning the approximation space by exploiting the (known) micro-localisation properties of the solution. For a large class of source terms (including plane-wave scattering problems), this choice leads to discrete spaces that provide a uniform approximation error for all wavenumber $k$ with a number of degrees of freedom scaling as $k^{d-1/2}$, which we rigorously establish. In comparison, for discretization spaces based on (piecewise) polynomials, the number of degrees of freedom has to scale at least as $k^d$ to achieve the same property. These theoretical results are illustrated by one-dimensional numerical examples, where the proposed discretization spaces are coupled with a least-squares variational formulation.

math.NA↗