A Two-stage Numerical Procedure for an Inverse Scattering Problem
Abstract: In this thesis we study a numerical procedure for the solution of the inverse problem of reconstructing location, shape and material properties (in particular refractive indices) of scatterers located in a known background medium. The data consist of time-resolved backscattered radar signals from a single source position. This relatively small amount of data and the ill-posed nature of the inversion are the main challenges of the problem. Mathematically, the problem is formulated as a coefficient inverse problem for a system of partial differential equations derived from Maxwell's equations. The numerical procedure is divided into two stages. In the first stage, a good initial approximation for the unknown coefficient is computed by an approximately globally convergent algorithm. This initial approximation is refined in the second stage, where an adaptive finite element method is employed to minimize a Tikhonov functional. An important tool for the second stage is a posteriori error estimates -- estimates in terms of known (computed) quantities -- for the difference between the computed coefficient and the true minimizing coefficient. This thesis includes four papers. In the first two, the a posteriori error analysis required for the adaptive finite element method in the second stage is extended from the previously existing indirect error estimators to direct ones. The last two papers concern verification of the two-stage numerical procedure on experimental data. We find that location and material properties of scatterers are obtained already in the first stage, while shapes are significantly improved in the second stage.
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