Modern regularization methods for inverse problems. (English) Zbl 1431.65080
Summary: Regularization methods are a key tool in the solution of inverse problems. They are used to introduce prior knowledge and allow a robust approximation of ill-posed (pseudo-) inverses. In the last two decades interest has shifted from linear to nonlinear regularization methods, even for linear inverse problems. The aim of this paper is to provide a reasonably comprehensive overview of this shift 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 them, since they have attracted much recent interest and link to other fields, such as image processing and compressed sensing. We further point to developments related to statistical inverse problems, multiscale decompositions and learning theory.
In particular we will discuss variational methods and techniques derived from them, since they have attracted much recent interest and link to other fields, such as image processing and compressed sensing. We further point to developments related to statistical inverse problems, multiscale decompositions and learning theory.
MSC:
| 65J20 | Numerical solutions of ill-posed problems in abstract spaces; regularization |
| 65J22 | Numerical solution to inverse problems in abstract spaces |
| 65K10 | Numerical optimization and variational techniques |
| 65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |
| 94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |
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