A Statistically Based Preconditioner for Two-Dimensional Microwave Tomography

This paper presents an estimation approach to preconditioning for gradient based inverse scattering algorithms. In particular, a two-dimensional inverse problem is considered where the permittivity and conductivity profiles are unknown and the input data consists of the scattered field over a certain bandwidth. A time-domain least-squares formulation is employed and the inversion algorithm is based on a conjugate gradient algorithm together with an FDTD-electromagnetic solver. A Fisher information analysis is used to estimate the Hessian of the error functional. A robust preconditioner is then obtained by choosing a parameter scaling such that the scaled Fisher information has a unit diagonal, cf., the Jacobi preconditioner in numerical analysis. Numerical examples of image reconstruction are included to illustrate the efficiency of the proposed technique.