TIME-DOMAIN INVERSE SCATTERING USING LAGRANGE MULTIPLIERS: AN ITERATIVE FDTD-BASED OPTIMIZATION TECHNIQUE

Abstract

In this paper, we present an electromagnetic inverse scattering technique in the time domain, which is based on the minimization of an augmented cost functional. This functional is composed of an error term representing the discrepancy between the measured and the estimated values of the electromagnetic field, a regularization term related to the first-order Tikhonov's regularization scheme and, finally, an equality-constraints term associated with the fulfillment of the Maxwell's curl equations. These equality constraints are introduced by applying Lagrange multipliers. The minimization of the functional is carried out by applying the Polak-Ribière nonlinear conjugate-gradient algorithm. The required Fr´echet derivatives of the augmented cost functional with respect to the functions that describe the scatterer properties are derived by an analysis based on the calculus of variations. Actually, it is proven that the Lagrange multipliers are waves satisfying the Maxwell's curl equations. Consequently, during each iteration of the minimization procedure, we apply the finite-difference time-domain method and the perfectly-matched-layer absorber to calculate both the electromagnetic fields and the Lagrange multipliers. The presented technique is successfully applied to the reconstruction of multiple two-dimensional scatterers, which can be dielectric, lossy, and magnetic.