On iterative methods for solving ill-posed problems modeled by partial differential equations

  • 1 Fachbereich Mathematik, JohannWolfgang Goethe Universität, Robert–Mayer–Str. 6–10, 60054 Frankfurt am Main, Germany
  • 2 Department of Mathematics, Federal University of Santa Catarina, P.O. Box 476, 88010-970 Florianópolis, Brazil

Abstract

- We investigate the iterative methods proposed by Maz’ya and Kozlov (see [6, 7]) for solving ill-posed inverse problems modeled by partial differential equations. We consider linear evolutionary problems of elliptic, hyperbolic and parabolic types. Each iteration of the analyzed methods consists in the solution of a well posed problem (boundary value problem or initial value problem respectively). The iterations are described as powers of affine operators, as in [7]. We give alternative convergence proofs for the algorithms by using spectral theory and the fact that the linear parts of these affine operators are non-expansive with additional functional analytical properties (see [9, 10]). Also problems with noisy data are considered and estimates for the convergence rate are obtained under a priori regularity assumptions on the problem data.

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This journal presents original articles on the theory, numerics and applications of inverse and ill-posed problems. These inverse and ill-posed problems arise in mathematical physics and mathematical analysis, geophysics, acoustics, electrodynamics, tomography, medicine, ecology, financial mathematics etc. Articles on the construction and justification of new numerical algorithms of inverse problem solutions are also published.

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