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Regularization Methods for Ill-Posed Problems

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Handbook of Mathematical Methods in Imaging

Abstract

In this chapter are outlined some aspects of the mathematical theory for direct regularization methods aimed at the stable approximate solution of nonlinear ill-posed inverse problems. The focus is on Tikhonov type variational regularization applied to nonlinear ill-posed operator equations formulated in Hilbert and Banach spaces. The chapter begins with the consideration of the classical approach in the Hilbert space setting with quadratic misfit and penalty terms, followed by extensions of the theory to Banach spaces and present assertions on convergence and rates concerning the variational regularization with general convex penalty terms. Recent results refer to the interplay between solution smoothness and nonlinearity conditions expressed by variational inequalities. Six examples of parameter identification problems in integral and differential equations are given in order to show how to apply the theory of this chapter to specific inverse and ill-posed problems.

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  1. School of Mathematical Sciences, Fudan University, Shanghai, China

    Jin Cheng

  2. Faculty of Mathematics, Technische Universität Chemnitz, Chemnitz, Germany

    Bernd Hofmann

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  1. Jin Cheng
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  1. Computational Science Center, University of Vienna, Vienna, Austria

    Otmar Scherzer

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Cheng, J., Hofmann, B. (2014). Regularization Methods for Ill-Posed Problems. In: Scherzer, O. (eds) Handbook of Mathematical Methods in Imaging. Springer, New York, NY. https://doi.org/10.1007/978-3-642-27795-5_3-5

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