Workshop on Inverse Problems in Scattering and Imaging

Analysis of Electromagnetic Waves in Random Media Liliana Borcea Department of Mathematics, University of Michigan We study Maxwell’s equations in media with small random fluctuations of the electric permittivity, to obtain a detailed mathematical characterization of the statistics of the electric and magnetic fields at long distances of propagation. We introduce a novel wide-angle wave propagation regime, which is mathematically justified by scaling assumptions. In this regime, we obtain a decomposition of the waves in transverse electric and magnetic modes with random amplitudes. These amplitudes account for the cumulative scattering effects in the medium, and satisfy a system of stochastic differential equations which can be analyzed with the Markov limit theorem. The result is an explicit quantification of the randomization of the waves due to scattering, an understanding of polarization effects, and a mathematical justification of the radiative transport equations with polarization. Babich’s Expansion and the Fast Huygens Sweeping Method for the Helmholtz Wave Equation at High Frequencies Jianliang Qian Department of Mathematics, Michigan State University Starting from Babich’s expansion, we develop a new high-order asymptotic method, which we dub the fast Huygens sweeping method, for solving point-source Helmholtz equations in inhomogeneous media in the high-frequency regime and in the presence of caustics. The first novelty of this method is that we develop a new Eulerian approach to compute the asymptotics, i.e. the traveltime function and amplitude coefficients that arise in Babich’s expansion, yielding a locally valid solution, which is accurate close enough to the source. The second novelty is that we utilize the Huygens–Kirchhoff integral to integrate many locally valid wavefields to construct globally valid wavefields. This automatically treats caustics and yields uniformly accurate solutions both near the source and remote from it. The third novelty is that the butterfly algorithm is adapted to accelerate the Huygens-Kirchhoff summation, achieving nearly optimal complexity O(N logN), where N is the number of mesh points; the complexity prefactor depends on the desired accuracy and is independent of the frequency.