Material profile reconstruction using plane electromagnetic waves in PML-truncated heterogeneous domains

摘要

This paper discusses a full-waveform inversion method for reconstructing the spatial distribution of permittivity in heterogeneous infinite domains, using measured electric field intensities at sparse sensor locations. In solving the electromagnetic wave problems numerically, perfectly matched layer (PML) absorbing boundaries are used to truncate the originally infinite extent to a finite computational domain of interest without introducing significant reflections. The full-waveform inversion is based on an optimization scheme where Maxwell's equations endowed with the PML for plane electromagnetic waves are imposed as constraints. The approach seeks the optimal solution of permittivity profile to minimize the objective functional comprising the L~2-norm of a misfit between calculated and measured electric fields. For casting the problem to an unconstrained optimization problem, a Lagrangian is constructed augmenting the objective functional with the PML-endowed Maxwell's equations via Lagrange multipliers. Enforcing the stationarity of the Lagrangian yields time-dependent state, adjoint, and time-invariant control problems, which constitute Karush-Kuhn-Tucker (KKT) conditions for optimal solutions. The permittivity profile of the PML-truncated domain is iteratively updated by solving the KKT conditions in the reduced space of the control variable. A conjugate gradient method with inexact line search is used to update the permittivity profile in each inversion iteration. Tikhonov and total variation regularization schemes are explored to relieve the ill-posedness of the inverse problem. Through a set of numerical results, it is shown that both smooth and sharply-varying permittivity profiles can be recovered successfully using the proposed inversion method.