FaIMS: A fast algorithm for the inverse medium problem with multiple frequencies and multiple sources for the scalar Helmholtz equation

We propose an algorithm to compute an approximate singular value decomposition (SVD) of least-squares operators related to linearized inverse medium problems with multiple events. Such factorizations can be used to accelerate matrix-vector multiplications and to precondition iterative solvers. We describe the algorithm in the context of an inverse scattering problem for the low-frequency time-harmonic wave equation with broadband and multi-point illumination. This model finds many applications in science and engineering (e.g., seismic imaging, subsurface imaging, impedance tomography, non-destructive evaluation, and diffuse optical tomography). We consider small perturbations of the background medium and, by invoking the Born approximation, we obtain a linear least-squares problem. The scheme we describe in this paper constructs an approximate SVD of the Born operator (the operator in the linearized least-squares problem). The main feature of the method is that it can accelerate the application of the Born operator to a vector. If Nω is the number of illumination frequencies, Ns the number of illumination locations, Nd the number of detectors, and N the discretization size of the medium perturbation, a dense singular value decomposition of the Born operator requires O(min(NsNωNd,N)]2×max(NsNωNd,N)) operations. The application of the Born operator to a vector requires O(NωNsμ(N)) work, where μ(N) is the cost of solving a forward scattering problem. We propose an approximate SVD method that, under certain conditions, reduces these work estimates significantly. For example, the asymptotic cost of factorizing and applying the Born operator becomes O(μ(N)Nω). We provide numerical results that demonstrate the scalability of the method.