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Modern regularization methods for inverse problems

Published online by Cambridge University Press:  04 May 2018

Martin Benning  and
Martin Burger
Martin Benning
Affiliation:
DAMTP, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK E-mail: [email protected]
Martin Burger
Affiliation:
Institute for Computational and Applied Mathematics, University of Münster, Einsteinstrasse 62, D-48149 Münster, Germany E-mail: [email protected]
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Abstract

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.

Type
Research Article
Information
Acta Numerica , Volume 27 , 01 May 2018 , pp. 1 - 111
Copyright
© Cambridge University Press, 2018 

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