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 $x_1,\dots,x_n\subset Ω$. 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 $(u(x_i))_{i=1,\dots,n}$. 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 $Ω$. Numerical illustrations show that our approach compares favorably with reconstruction methods using other basis functions, and other types of regularization.

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