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Compressive sensing principles and iterative sparse recovery for inverse and ill-posed problems

In this paper, we are concerned with compressive sampling strategies and sparse recovery principles for linear inverse and ill-posed problems. As the main result, we provide compressed measurement models for ill-posed problems and recovery accuracy estimates for sparse approximations of the solution of the underlying inverse problem. The main ingredients are variational formulations that allow the treatment of ill-posed operator equations in the context of compressively sampled data. In particular, we rely on Tikhonov variational and constrained optimization formulations. One essential difference to the classical compressed sensing framework is the incorporation of joint sparsity measures allowing the treatment of infinite-dimensional reconstruction spaces. The theoretical results are furnished with a number of numerical experiments.