Approximation of solutions to the Helmholtz equation from scattered data

submitted for publication to Applied and Computational Harmonic AnalysisWe consider the problem of reconstructing general solutions to the Helmholtz equation ∆u+λ2u = 0, for some fixed λ > 0, on some domain Ω ⊂ R2 from the data of these functions 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 that is contained in a larger domain D over which the acoustic field is generated. In many applied settings, the boundary conditions satisfied by the acoustic field on ∂D are unknown as well as the exact shape of D. Our reconstruction method is based on the approximation of a general solution u by linear combinations of Fourier-Bessel functions or plane waves ek(x) := eik*x with |k| = λ. We study two different ways of discretizing the infinite dimensional space Vλ of solutions to the Helmholtz equation, leading to two different types of finite dimensional approximation subspaces, and we analyze the convergence of the least squares estimates to u in these subspaces based on the samples (u(xi))i=1,...,n. Our analysis describes the amount of regularization that is 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 and the sample size n. This condition also involves the distribution of the samples and reveals the advantage of using non-uniform distributions that have more points near or 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