An analysis of a multi-level projected steepest descent iteration for nonlinear inverse problems in Banach spaces subject to stability constraints
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We consider nonlinear inverse problems described by operator equations in Banach spaces. Assuming conditional stability of the inverse problem, that is, assuming that stability holds on a compact, convex subset of the domain of the operator, we introduce a novel nonlinear projected steepest descent iteration and analyze its convergence to an approximate solution given limited accuracy data. We proceed with developing a multi-level algorithm based on a nested family of compact, convex subsets on which stability holds and the stability constants are ordered. Growth of the stability constants is coupled to the increase in accuracy of approximation between neighboring levels to ensure that the algorithm can continue from level to level until the iterate satisfies a desired discrepancy criterion, after a finite number of steps.
Mathematics Subject Classification35R30 65J22 47J25
The research was initiated at the Isaac Newton Institute for Mathematical Sciences (Cambridge, England) during a programme on Inverse Problems in Fall 2011. The authors would like to thank the anonymous referees for their valuable comments and suggestions to improve the quality of the paper.
- 4.Ammari, H., Bahouri, H., Dos Santos Ferreira, D., Gallagher, I.: Stability estimates for an inverse scattering problem at high frequencies. ArXiv e-prints (2012)Google Scholar
- 10.Cioranescu, I.: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Mathematics and Its Applications, vol. 62. Kluwer Academic Publishers Group, Dordrecht (1990)Google Scholar
- 14.Gilyazov, S.F.: Iterative solution methods for inconsistent linear equations with nonself-adjoint operator. Moscow Univ. Comput. Math. Cybernet. 13, 8–13 (1977)Google Scholar
- 18.Kaltenbacher, B., Neubauer, A., Scherzer, O.: Iterative Regularization Methods for Nonlinear Ill-Posed Problems. Radon Series on Computational and Applied Mathematics, vol. 6. Walter de Gruyter GmbH & Co. KG, Berlin (2008)Google Scholar
- 23.Schuster, T., Kaltenbacher, B., Hofmann, B., Kazimierski, K.S.: Regularization Methods in Banach Spaces. Radon Series on Computational and Applied Mathematics, vol. 10. Walter de Gruyter GmbH & Co. KG, Berlin (2012)Google Scholar
- 24.Teschke, G., Borries, C.: Accelerated projected steepest descent method for nonlinear inverse problems with sparsity constraints. Inverse Probl. 26(2), 025007 (23 pp) (2010)Google Scholar