Sampling and reconstruction of solutions to the Helmholtz equation

Abstract

We consider the inverse problem of reconstructing general solutions to the Helmholtz equation on some domain Ω from their values at scattered points x1;…; xn Ω This problem typically arises when sampling acoustic fields with n microphones for the purpose of reconstructing this field over a region of interest 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(xi))i=1;…;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. Numerical illustrations show that our approach compares favorably with reconstruction methods using other basis functions, and other types of regularization.