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• W78867331 abstract "We consider preconditioned regularized Newton methods tailored to the efficient solution of nonlinear large-scale exponentially ill-posed problems.In the first part of this thesis we investigate convergence and convergence rates of the iteratively regularized Gauss-Newton method under general source conditions both for an a-priori stopping rule and the discrepancy principle. The source condition determines the smoothness of the true solution of the given problem in an abstract setting. Dealing with large-scale ill-posed problems it is in general not realistic to assume that the regularized Newton equations can be solved exactly in each Newton step. Therefore, our convergence analysis includes the practically relevant case that the regularized Newton equations are solved only approximately and the Newton updates are computed by using these approximations.In a second part of this thesis we analyze the complexity of the iteratively regularized Gauss-Newton method assuming that the regularized Newton equations are solved by the conjugate gradient method. This analysis includes both mildly and severely ill-posed problems. As a measure of the complexity we count the number of operator evaluations of the Frechet derivative and its adjoint at some given vectors. Following a common practice for linear ill-posed problems, we express the total complexity of the iteratively regularized Gauss-Newton method in terms of the noise level of the given data.To reduce the total complexity of these regularized Newton methods we consider spectral preconditioners to accelerate the convergence speed of the inner conjugate gradient iterations. We extend our complexity analysis to these preconditioned regularized Newton methods. This investigation gives us the possibility to compare the total complexity of non preconditioned regularized Newton methods and preconditioned ones. In particular we show the superiority of the latter ones in the case of exponentially ill-posed problems.Finally, in a third part we discuss the implementation of preconditioned iteratively regularized Gauss-Newton methods exploiting the close connection of the conjugate gradient method and Lanczos method as well as the fast decay of the eigenvalues corresponding to the linearized operators in the regularized Newton equations. More precisely, we determine by Lanczos method approximations to some of the extremal eigenvalues. These are used to construct spectral preconditioners for the following Newton steps. Developing updating techniques to keep the preconditioner efficient while performing Newtons method the total complexity can be significantly reduced compared to the non preconditioned iteratively regularized Gauss-Newton method. Finally, we illustrate in numerical examples from inverse scattering theory the efficiency of the preconditioned regularized Newton methods compared to other regularized Newton methods." @default.
• W78867331 created "2016-06-24" @default.
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• W78867331 date "2007-09-05" @default.
• W78867331 modified "2023-10-16" @default.
• W78867331 title "Preconditioned Newton methods for ill-posed problems" @default.
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