GMRES Computation of High Frequency Electrical Field Propagation in Land Mine Detection

The ultimate goal of the authors is to apply inverse problem techniques to image land mines using an electromagnetic signal originated by ground penetrating radar. Specifically, the intention is to use the recently developed elliptic systems method, which has been successfully applied by these authors to the problem of imaging biological tissues using lasers. As the first step it is necessary to develop a fast and accurate numerical method for the solution of the forward problem to simulate data for the inverse problem. The main challenge is the requirement of solving a Helmholtz-like equation for high frequencies which is excessively time consuming using standard direct solution techniques. A novel accurate and rapid numerical procedure for the solution of this equation is described in this paper. The kernel of this algorithm is a combination of GMRES and a carefully chosen preconditioner, the solution of which is found using a fast transform method. The extended problem of solving Helmholtz-like equations for many frequencies is also considered. Numerical results for realistic ranges of parameters in soil and mine-like targets, the investigation of the impact of the size of the truncation region on accuracy, and the sensitivity of detector readings to changes in the media are presented.