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Regularization Methods for Ill-Posed Problems

In: Handbook of Mathematical Methods in Imaging

Author

Listed:
  • Jin Cheng

    (Fudan University, School of Mathematical Sciences)

  • Bernd Hofmann

    (Technische Universität Chemnitz, Faculty of Mathematics)

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.

Suggested Citation

  • Jin Cheng & Bernd Hofmann, 2015. "Regularization Methods for Ill-Posed Problems," Springer Books, in: Otmar Scherzer (ed.), Handbook of Mathematical Methods in Imaging, edition 2, pages 91-123, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4939-0790-8_3
    DOI: 10.1007/978-1-4939-0790-8_3
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