Spectral properties of radiation for the Helmholtz equation with a random coefficient

Abstract

Abstract We consider the Helmholtz equation [ Δ + k 0 2 ( 1 + q ( x , ω ) ) ] u = − f in R R 2 and R R 3 , where the coefficient q ( x , ω ) is a Gaussian random field. For any open ball B that includes the support of x ↦ q ( x , ω ) , we approximate and characterize spectrally the source-to-measurement map f ↦ u | ∂ B in the weak scattering/low-frequency setting. To this end, we first analyze the case with a deterministic coefficient q(x), and here we discover and quantify a ‘spectral leakage’ effect caused by the presence of the medium. Furthermore, we compare numerically our first-order asymptotic expansion of the source-to-measurement map with the map achieved by a Finite Element Method implementation of an example radiation problem. Our results are applicable in the analysis of the robustness of the solution of inverse source problems in the presence of deterministic and random media.