Iteratively Regularized Adaptive Finite Element Method for Reconstruction of Coefficients in Maxwell’s System

Abstract

We consider iteratively regularized adaptive finite element method for reconstruction of spatially distributed dielectric permittivity and magnetic permeability functions, " (x) and _ (x) ; x 2 R3 , simultaneously, using time-dependent backscattering data. These functions are unknown coefficients in Maxwell’s system of equations. We formulate our problem as the coefficient inverse problem (CIP) for the full Maxwell’s system. To solve our inverse problem we minimize Tikhonov regularization functional on the locally adaptively refined meshes. In this work, we consider and compare different techniques for choosing regularization parameter in the Tikhonov functional in order to get improved solution of our inverse problem. Our goal is to choose optimized regularization parameters in the solution of our CIP. This means, we choose regularization parameters and parameters in the set up of the program such that we will get best reconstructions of functions " (x) and _ (x) to our backscattering data of the electric field on every iteration of the optimization procedure. Our numerical work consist in the reconstruction of unknown coefficients " (x) and _ (x), on the adaptivity locally refined meshes. Software packages WavES [60] and PETSc [50] are used for computations of reconstructions of these functions. Simulations are done on resources at Chalmers Centre for Computational Science and Engineering (C3SE) provided by the Swedish National Infrastructure for Computing (SNIC).