Locating radiating sources for Maxwell's equations using the approximate inverse

We present a new approach to solve inverse source problems for the three-dimensional time-harmonic Maxwell's equations using boundary measurements of the radiated fields. The modelling is based on the formulation as a system of integro-differential equations for the electric field. We introduce a method to recast the intertwined vector equations of Maxwell into decoupled scalar problems. The method of the approximate inverse is used both for regularization and the development of fast algorithms. We make the analysis of the method when data are collected on a spherical setting around the object. Based on the singular value decomposition, we study the smoothing properties for the underlying operator and derive an error estimate for the regularized solution in a Sobolev-space framework. Numerical simulations illustrate the efficiency and practical usefulness of the developed method.