Quasi-Monte Carlo finite element analysis for wave propagation in heterogeneous random media
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We propose and analyze a quasi-Monte Carlo (QMC) algorithm for efficient simulation of wave propagation modeled by the Helmholtz equation in a bounded region in which the refractive index is random and spatially heterogenous. Our focus is on the case in which the region can contain multiple wavelengths. We bypass the usual sign-indefiniteness of the Helmholtz problem by switching to an alternative sign-definite formulation recently developed by Ganesh and Morgenstern (Numerical Algorithms, 83, 1441-1487, 2020), where the coercivity constant is independent of the wavenumber. The price to pay is that the regularity analysis required for QMC methods becomes much more technical. Nevertheless we obtain a complete analysis with error comprising of stochastic dimension truncation error, finite element error and cubature error, with results comparable to those obtained for the diffusion problem.
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