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{\large George Biros \\
{\bf FaIMS: A fast algorithm for the inverse medium problem in acoustic scattering}}

266 Ferst Dr \\ Georgia Tech \\ Atlanta \\ GA 30332
\\
{\tt gbiros@acm.org}\\
Stephanie Chaillat\end{center}

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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