George Biros
FaIMS: A fast algorithm for the inverse medium problem in acoustic scattering

266 Ferst Dr
Georgia Tech
Atlanta
GA 30332
gbiros@acm.org
Stephanie Chaillat

We consider the inverse medium problem for the time-harmonic wave equation with broadband and multi-point illumination in the low frequency regime. Such a problem finds many applications in geosciences (e.g. ground penetrating radar), non-destructive evaluation (acoustics), and medicine (optical tomography). We use an integral-equation (Lippmann-Schwinger) formulation, which we discretize using a quadrature method. We consider only small perturbations (Born approximation). To solve this inverse problem, we use a least-squares formulation. We present a new fast algorithm for the efficient solution of this particular least-squares problem.

If $N_{\fr}$ is the number of excitation frequencies, $N_{s}$ the number of different source locations for the point illuminations, $N_{d}$ the number of detectors, and $N$ the parametrization for the scatterer, a dense singular value decomposition for the overall input-output map will have $[\min(N_{s} N_{\fr}N_{d}, N)]^{2} \times \max(N_{s} N_{\fr}N_{d}, N)$ cost. We have developed a fast SVD-based preconditioner that brings the cost down to $O( N_{s}N_{\fr} N_{d} N)$ thus, providing orders of magnitude improvements over a black-box dense SVD and an unpreconditioned linear iterative solver.