George Biros, Georgia Institute of Technology

IAMCS Workshop in Large-Scale Inverse Problems and Uncertainty Quantification

February 24-25, 2011

Stephen W. Hawking Auditorium

George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy

Texas A&M University

College Station, Texas

A Fast Algorithm for the Inverse Medium Problem with Multiple Sources 

Authors: George Biros and 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_ω 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 [min(N_sN_ωN_d, N)]² × max(N_sN_ωN_d, N) cost. We have developed a fast SVD-based preconditioner that brings the cost down to Ο(N_sN_ωN_dN) thus, providing orders of magnitude improvements over a black-box dense SVD and an unpreconditioned linear iterative solver.