Sparsity and Compressed Sensing in Inverse Problems

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 condi-tions for exact recovery. In particular, we describe how both `1-minimization and matching pursuit methods can be used to regularize ill-posed problems and more-over, 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 ex-tensions of compressed sensing for specific infinite dimensional ill-posed measure-ment regimes. We are able to establish recovery error estimates when adequately relating the isometry constant of the sensing operator, the ill-posedness of the un-derlying model operator and the regularization parameter. Finally, we very briefly sketch how projected steepest descent iterations can be applied to retrieve the sparse solution