Convergence analysis of an optimally accurate frozen multi-level projected steepest descent iteration for solving inverse problems |
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Institution: | Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, 247667, India;School of Mathematical Sciences, Tianjin Normal University, Tianjin, China;School of Mathematical Sciences and LPMC, Nankai University, Tianjin, China;School of Statistics, University of International Business and Economics, Beijing 100029, PR China;Department of Statistics, Stanford University, 390 Jane Stanford Way, Stanford, 94305, CA, USA |
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Abstract: | In this paper, we introduce a novel projected steepest descent iterative method with frozen derivative. The classical projected steepest descent iterative method involves the computation of derivative of the nonlinear operator at each iterate. The method of this paper requires the computation of derivative of the nonlinear operator only at an initial point. We exhibit the convergence analysis of our method by assuming the conditional stability of the inverse problem on a convex and compact set. Further, by assuming the conditional stability on a nested family of convex and compact subsets, we develop a multi-level method. In order to enhance the accuracy of approximation between neighboring levels, we couple it with the growth of stability constants. This along with a suitable discrepancy criterion ensures that the algorithm proceeds from level to level and terminates within finite steps. Finally, we discuss an inverse problem on which our methods are applicable. |
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Keywords: | Nonlinear ill-posed equations Regularization Inverse problems Iterative regularization methods Multi-level approach |
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