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

TL;DR

This paper presents a novel finite-dimensional approximation space for high-frequency Helmholtz problems using Gaussian coherent states, achieving efficient and uniform accuracy with fewer degrees of freedom than traditional polynomial methods.

Contribution

Introduction of phase-space localized Gaussian coherent states for discretizing high-frequency Helmholtz solutions, reducing degrees of freedom needed for accurate approximation.

Findings

Discretization spaces achieve uniform approximation error for all wavenumbers.

Number of degrees of freedom scales as k^{d-1/2}, better than polynomial-based methods.

Numerical examples confirm theoretical efficiency and accuracy.

Abstract

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.