Stochastic Method and Convex Optimization in Inverse Electromagnetic Scattering

The electromagnetic scattering phenomenon is characterized by the electric field integral equation. In inverse problems, this integral equation describes the nonlinear relationship between the unknown complex permittivity distribution in an investigation domain and the observable scattered fields. This thesis tackles the nonlinearity with two approaches. First, the Born iterative method is enhanced by regularization techniques, algebraic reconstruction technique and conjugate gradient; then we use Bayesian inference to estimate the conditional mean of the posterior for the unknown permittivity given scattered field data with Monte Carlo Markov chain (MCMC). The conditional mean estimates not only incorporate prior knowledge from results obtained by BIM, but also avoid the nonlinearity by computing the linear forward model. Second, we consider the inverse scattering problem from a convex optimization perspective, which approximates the inverse scattering as a convex optimization problem and provides an estimation of the internal electric field inside the domain under investigation without a priori knowledge or tuning parameters. Then the estimated internal field is used to reconstruct the unknown contrast permittivity, which is represented by a superposition of level-1 Haar wavelet transform basis functions. Subject to L1-norm constraints of the wavelet coefficients, a LASSO problem that searches for the global minimum of the L2-norm residual is exploited by accounting for the sparsity of the wavelet-based permittivity representation. Numerical results are presented to assess the effectiveness of the proposed formulation against objects with relatively small electric size, and this approach is validated against experimental data.