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A fast algorithm for inverse medium problems with multiple excitations

Friday, November 4, 3PM – 4PM
ACE 6.304

George Biros, ICES

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.

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