Sparsity and compressed sensing in inverse problems

Abstract

This chapter is concerned with two important topics in the context of sparse recovery in inverse and ill-posed problems. In first part we elaborate conditions for exact recovery. In particular, we describe how both ℓ1-minimization and matching pursuit methods can be used to regularize ill-posed problems and moreover, state conditions which guarantee exact recovery of the support in the sparse case. The focus of the second part is on the incomplete data scenario. We discuss extensions of compressed sensing for specific infinite dimensional illposed measurement regimes.We are able to establish recovery error estimates when adequately relating the isometry constant of the sensing operator, the ill-posedness of the underlying model operator and the regularization parameter. Finally, we very briefly sketch how projected steepest descent iterations can be applied to retrieve the sparse solution.