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Regularization of nonlinear ill-posed problemsby exponentialintegrators

Published online by Cambridge University Press:  08 July 2009

Marlis Hochbruck ,
Michael Hönig  and
Alexander Ostermann
Marlis Hochbruck
Affiliation:
Mathematisches Institut, Heinrich-Heine Universität Düsseldorf, Universitätsstraße 1, 40225 Düsseldorf, Germany. marlis@am.uni-duesseldorf.de; hoenig@am.uni-duesseldorf.de
Michael Hönig
Affiliation:
Mathematisches Institut, Heinrich-Heine Universität Düsseldorf, Universitätsstraße 1, 40225 Düsseldorf, Germany. marlis@am.uni-duesseldorf.de; hoenig@am.uni-duesseldorf.de
Alexander Ostermann
Affiliation:
Institut für Mathematik, Universität Innsbruck, Technikerstraße 13, 6020 Innsbruck, Austria. alexander.ostermann@uibk.ac.at
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Abstract

The numerical solution of ill-posed problems requires suitableregularization techniques. One possible option is to consider timeintegration methods to solve the Showalter differential equationnumerically. The stopping time of the numerical integrator correspondsto the regularization parameter. A number of well-knownregularization methods such as the Landweber iteration or theLevenberg-Marquardt method can be interpreted as variants of theEuler method for solving the Showalter differential equation. Motivated by an analysis of the regularization properties of theexact solution of this equation presented by [U. Tautenhahn, Inverse Problems10 (1994) 1405–1418], we consider a variant of the exponential Euler methodfor solving the Showalter ordinary differential equation. We discuss asuitable discrepancy principle for selecting the step sizes withinthe numerical method and we review the convergence properties of [U. Tautenhahn, Inverse Problems10 (1994) 1405–1418], and of our discrete version [M. Hochbruck et al., Technical Report (2008)].Finally, we present numerical experiments which show that thismethod can be efficiently implemented by using Krylov subspacemethods to approximate the product of a matrix function with avector.

Keywords

Nonlinear ill-posed problemsasymptotic regularizationexponential integratorsvariable step sizesconvergenceoptimal convergence rates.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 4: Special issue on Numerical ODEs today , July 2009 , pp. 709 - 720
Copyright
© EDP Sciences, SMAI, 2009

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