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Regularized FixedPoint Iterations for Nonlinear Inverse Problems
, 2005
"... In this paper we introduce a derivativefree, iterative method for solving nonlinear illposed problems F x = y, where instead of y noisy data yδ with �y − yδ � ≤ δ are given and F: D(F) ⊆ X → Y is a nonlinear operator between Hilbert spaces X and Y. This method is defined by splitting the operator ..."
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In this paper we introduce a derivativefree, iterative method for solving nonlinear illposed problems F x = y, where instead of y noisy data yδ with �y − yδ � ≤ δ are given and F: D(F) ⊆ X → Y is a nonlinear operator between Hilbert spaces X and Y. This method is defined by splitting the operator F into a linear part A and a nonlinear part G, such that F = A + G. Then iterations are organized as Auk+1 = yδ − Guk. In the context of illposed problems we consider the situation when A does not have a bounded inverse, thus each iteration needs to be regularized. Under some conditions on the operators A and G we study the behavior of the iteration error. We obtain its stability with respect to the iteration number k as well as the optimal convergence rate with respect to the noise level δ, provided that the solution satisfies a generalized source condition. As an example, we consider an inverse problem of initial temperature reconstruction for a nonlinear heat equation, where the nonlinearity appears due to radiation effects. The obtained iteration error in the numerical results has the theoretically expected behavior. The theoretical assumptions are illustrated by a computational experiment.
Calibration and Rescaling Principles for Nonlinear Inverse Heat Conduction and Parameter Estimation Problems
, 2015
"... I am submitting herewith a dissertation written by Yinyuan Chen entitled "Calibration and Rescaling Principles for Nonlinear Inverse Heat Conduction and Parameter Estimation Problems. " I have examined the final electronic copy of this dissertation for form and content and recommend that i ..."
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I am submitting herewith a dissertation written by Yinyuan Chen entitled "Calibration and Rescaling Principles for Nonlinear Inverse Heat Conduction and Parameter Estimation Problems. " I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in