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

## 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.