Sparse Recovery in Inverse Problems

Within this chapter we present recent results on sparse recovery algorithms for inverse and ill-posed problems, i.e. we focus on those inverse problems in which we can assume that the solution has a sparse series expansion with respect to a preassigned basis or frame. The presented approaches to approximate solutions of inverse problems are limited to iterative strategies that essentially rely on the minimization of Tikhonov-like variational problems, where the sparsity constraint is integrated through `p norms. In addition to algorithmic and computational aspects, we also discuss in greater detail regularization properties that are required for cases in which the operator is ill-posed and no exact data are given. Such scenarios reflect realistic situations and manifest therefore its great suitability for “real-life” applications.