Sampling and reconstruction of solutions to the Helmholtz equation

TL;DR

This paper presents a method for reconstructing solutions to the Helmholtz equation from scattered point samples, analyzing convergence and regularization, with a focus on Fourier-Bessel functions and plane waves, applicable in acoustic field sampling.

Contribution

It introduces a convergence analysis for least-squares reconstruction of Helmholtz solutions using Fourier-Bessel functions and plane waves, highlighting the benefits of non-uniform sampling distributions.

Findings

Non-uniform sampling with more boundary points improves reconstruction accuracy.

The method achieves favorable results compared to other basis functions.

Regularization is crucial for ensuring convergence of the estimates.

Abstract

We consider the inverse problem of reconstructing general solutions to the Helmholtz equation on some domain $\Omega$ from their values at scattered points $x_1,\dots,x_n\subset \Omega$. This problem typically arises when sampling acoustic fields with $n$ microphones for the purpose of reconstructing this field over a region of interest $\Omega$ contained in a larger domain $D$ in which the acoustic field propagates. In many applied settings, the shape of $D$ and the boundary conditions on its border are unknown. Our reconstruction method is based on the approximation of a general solution $u$ by linear combinations of Fourier-Bessel functions or plane waves. We analyze the convergence of the least-squares estimates to $u$ using these families of functions based on the samples $(u(x_i))_{i=1,\dots,n}$. Our analysis describes the amount of regularization needed to guarantee the convergence of the least squares estimate towards $u$, in terms of a condition that depends on the dimension of the approximation subspace, the sample size $n$ and the distribution of the samples. It reveals the advantage of using non-uniform distributions that have more points on the boundary of $\Omega$. Numerical illustrations show that our approach compares favorably with reconstruction methods using other basis functions, and other types of regularization.