Sampling and Reconstruction of Solutions to the Helmholtz Equation

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 Ω 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 regular-ization 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 Ω. Numerical illustrations show that our approach compares favorably with reconstruction methods using other basis functions, and other types of regularization.