Journal ArticleDOI
An iterative multi level algorithm for solving nonlinear ill–posed problems
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TLDR
The numerical performance of this multi level algorithm is compared with Landweber's iteration and an advantage of this method is that the convergence analysis can be considered in a family of finite dimensional spaces.Abstract:
The convergence analysis of Landweber's iteration for the solution of nonlinear ill–posed problem has been developed recently by Hanke, Neubauer and Scherzer. In concrete applications, sufficient conditions for convergence of the Landweber iterates developed there (although quite natural) turned out to be complicated to verify analytically. However, in numerical realizations, when discretizations are considered, sufficient conditions for local convergence can usually easily be stated. This paper is motivated by these observations: Initially a discretization is fixed and a discrete Landweber iteration is implemented until an appropriate stopping criterion becomes active. The output is used as an initial guess for a finer discretization. An advantage of this method is that the convergence analysis can be considered in a family of finite dimensional spaces. The numerical performance of this multi level algorithm is compared with Landweber's iteration.read more
Citations
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Journal ArticleDOI
A convergence analysis of iterative methods for the solution of nonlinear ill-posed problems under affinely invariant conditions
TL;DR: In this article, the invariance properties have been used to provide a unified framework for convergence analysis for iterative methods for nonlinear ill-posed problems and prove convergence with rates for the Landweber and the iteratively regularized Gauss-Newton methods.
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Inverse Problems Related to Ion Channel Selectivity
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On convergence rates of inexact Newton regularizations
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Nonlinear Inverse Problems: Theoretical Aspects and Some Industrial Applications
Heinz W. Engl,Philipp Kügler +1 more
TL;DR: Advances in this theory and the development of sophisticated numerical techniques for treating the direct problems allow to address and solve industrial inverse problems on a level of high complexity.
Book ChapterDOI
Convergence Rates Results for Iterative Methods for Solving Nonlinear Ill-Posed Problems
Heinz W. Engl,Otmar Scherzer +1 more
TL;DR: An overview over Tikhonov regularization and iterative regularization techniques is given and the analysis of iterative methods for the solution of well-posed problems is put into perspective.
References
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A practical guide to splines
TL;DR: This book presents those parts of the theory which are especially useful in calculations and stresses the representation of splines as linear combinations of B-splines as well as specific approximation methods, interpolation, smoothing and least-squares approximation, the solution of an ordinary differential equation by collocation, curve fitting, and surface fitting.
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Inverse Acoustic and Electromagnetic Scattering Theory
David Colton,Rainer Kress +1 more
TL;DR: Inverse Medium Problem (IMP) as discussed by the authors is a generalization of the Helmholtz Equation for direct acoustical obstacle scattering in an Inhomogeneous Medium (IMM).
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Regularization of Inverse Problems
TL;DR: Inverse problems have been studied in this article, where Tikhonov regularization of nonlinear problems has been applied to weighted polynomial minimization problems, and the Conjugate Gradient Method has been used for numerical realization.
Journal ArticleDOI
A convergence analysis of the Landweber iteration for nonlinear ill-posed problems
TL;DR: In this paper, the authors proved that the Landweber iteration is a stable method for solving nonlinear ill-posed problems and proposed a stopping rule that yields the convergence rate of O(O( √ Δ ) under appropriate conditions.