Modern Regularization Methods for Inverse Problems
It serves as a broad survey for researchers in inverse problems, image processing, and compressed sensing, but is incremental in nature.
This paper provides a comprehensive overview of modern nonlinear regularization methods for inverse problems, covering variational methods, statistical inverse problems, multiscale decompositions, and learning theory, while highlighting their analysis, applications, and future research directions.
Regularization methods are a key tool in the solution of inverse problems. They are used to introduce prior knowledge and make the approximation of ill-posed (pseudo-)inverses feasible. In the last two decades interest has shifted from linear towards nonlinear regularization methods even for linear inverse problems. The aim of this paper is to provide a reasonably comprehensive overview of this development towards modern nonlinear regularization methods, including their analysis, applications, and issues for future research. In particular we will discuss variational methods and techniques derived from those, since they have attracted particular interest in the last years and link to other fields like image processing and compressed sensing. We further point to developments related to statistical inverse problems, multiscale decompositions, and learning theory.