Far field operator splitting by principal component pursuit

We consider scattering of time-harmonic plane waves by an ensemble of well-separated compactly supported inhomogeneous scatterers. The far field operator, which maps superpositions of plane wave incident fields to the far field patterns of the associated scattered fields, is commonly used as an idealized description of data sets obtained in corresponding remote sensing experiments. Suppose that some a priori information about the approximate position of just one of the scatterers in the ensemble is available. This article is about recovering the far field operator associated to this single scatterer from the far field operator associated to the whole collection of scatterers. Due to multiple scattering effects this is a nonlinear inverse problem. We show that an approximate solution can be obtained by decomposing the far field operator into a sparse component and a low-rank component, and we apply a convex program called principal component pursuit for this purpose. We give necessary and sufficient conditions for unique solvability, establish a stability result and provide numerical examples to illustrate our theoretical findings.