# IAMCS Workshop in Large-Scale Inverse Problems and Uncertainty Quantification

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- Texas A&M University
- College Station, TX
- Stephen W. Hawking Auditorium
- George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy (MIST)

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- George Biros, Georgia Institute of Technology
- A Fast Algorithm for the Inverse Medium Problem with Multiple Sources

### Authors

- George Biros
- Stephanie Chaillat

### Abstract

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 \(N_{\omega}\) 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 parameterization for the scatterer, a dense singular value decomposition for the overall input-output map will have \(\left[\min {\left(N_{s}N_{\omega}N_{d}, N\right)}\right]^{2}\times\max{\left(N_{s}N_{\omega}N_{d}, N\right)}\) cost. We have developed a fast SVD-based preconditioner that brings the cost down to \(O\left(N_{s}N_{\omega}N_{d}N\right)\) thus, providing orders of magnitude improvements over a black-box dense SVD and an unpreconditioned linear iterative solver.