Uncertainty principles for inverse source and inverse scattering problems

We consider scattering of time-harmonic acoustic waves by an ensemble of compactly supported inhomogeneous objects, called scatterers, in a homogeneous background medium.
The scatterers are illuminated by incident plane waves along all possible illumination directions and, each time, the resulting scattered waves are detected far away along all possible observation directions.
This data can be described by the far field operator, which uniquely determines the scatterers in inverse medium scattering.
In practice, this given data is often noisy and possibly incomplete, which motivates the study of two related inverse problems, that are at the center of our consideration in this thesis.
On the one hand, given the far field operator associated to the ensemble of scatterers, we discuss the nonlinear
inverse problem to recover the far field operators associated to
each of the scatterers individually.
We refer to this problem as \emph{far field operator splitting}, which involves the removal of specific multiple scattering effects.
This problem is closely related to the question whether the components
of the scatterer can be distinguished by means of inverse medium
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scattering in a stable way.
On the other hand, we study the restoration of missing or inaccurate components of the observed far field operator, which we refer to as \emph{far field operator completion}.
Both problems are ill-posed without further assumptions, but we give
sufficient conditions on the diameter of the supports of the
scatterers, the distance between them, and the size of the missing
or corrupted data component to guarantee stable recovery whenever
sufficient a~priori information on the location of the unknown
scatterers is available.
We reformulate both inverse problems as suitable splitting problems and take advantage of the fact that the arising far field operator components can be well approximated by sparse or low rank operators.
This enables us to use techniques from compressed sensing and from low rank matrix completion.
We provide algorithms, error estimates and a stability analysis for two as well as for three dimensions.
Furthermore, we verify our theoretical predictions by numerical examples both on synthetic and on experimental data, and we test to what extent our reconstructions are suitable as an input for an inverse medium scattering problem.