Regularization in Banach spaces — convergence rates by approximative source conditions

2009 ◽  
Vol 17 (1) ◽  
Author(s):  
T. Hein
Keyword(s):  
Banach Spaces ◽  
Convergence Rates

On smoothness conditions and convergence rates in the CLT in Banach spaces

1993 ◽  
Vol 96 (2) ◽  
pp. 137-151
Author(s):  
Vidmantas Bentkus ◽  
Friedrich G�tze
Keyword(s):  
Banach Spaces ◽  
Convergence Rates

scholarly journals The Ivanov regularized Gauss–Newton method in Banach space with an a posteriori choice of the regularization radius

2019 ◽  
Vol 27 (4) ◽  
pp. 539-557
Author(s):  
Barbara Kaltenbacher ◽  
Andrej Klassen ◽  
Mario Luiz Previatti de Souza
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Newton Method ◽  
Noise Level ◽  
Numerical Experiments ◽  
Convergence Rates ◽  
A Priori ◽  
Discrepancy Principle ◽  
A Posteriori ◽  
Source Condition

Abstract In this paper, we consider the iteratively regularized Gauss–Newton method, where regularization is achieved by Ivanov regularization, i.e., by imposing a priori constraints on the solution. We propose an a posteriori choice of the regularization radius, based on an inexact Newton/discrepancy principle approach, prove convergence and convergence rates under a variational source condition as the noise level tends to zero and provide an analysis of the discretization error. Our results are valid in general, possibly nonreflexive Banach spaces, including, e.g., {L^{\infty}} as a preimage space. The theoretical findings are illustrated by numerical experiments.

scholarly journals A converse result for Banach space convergence rates in Tikhonov-type convex regularization of ill-posed linear equations

2018 ◽  
Vol 26 (5) ◽  
pp. 639-646 ◽  
Author(s):  
Jens Flemming
Keyword(s):  
Banach Spaces ◽  
Convergence Rates ◽  
A Priori ◽  
Linear Equations ◽  
Rate Function ◽  
Bregman Distances ◽  
Error Measures ◽  
Ill Posed ◽  
General Convex ◽  
Linear Operator Equations

Abstract We consider Tikhonov-type variational regularization of ill-posed linear operator equations in Banach spaces with general convex penalty functionals. Upper bounds for certain error measures expressing the distance between exact and regularized solutions, especially for Bregman distances, can be obtained from variational source conditions. We prove that such bounds are optimal in case of twisted Bregman distances, a specific a priori parameter choice, and low regularity of the exact solution, that is, the rate function is also an asymptotic lower bound for the error measure. This result extends existing converse results from Hilbert space settings to Banach spaces without adhering to spectral theory.

scholarly journals Quasi-solution of linear inverse problems in non-reflexive Banach spaces

2018 ◽  
Vol 26 (5) ◽  
pp. 689-702 ◽  
Author(s):  
Christian Clason ◽  
Andrej Klassen
Keyword(s):  
Banach Spaces ◽  
Convergence Rates ◽  
Metric Projection ◽  
Discrepancy Principle ◽  
Reflexive Banach Spaces ◽  
Parameter Choice ◽  
Linear Inverse Problems ◽  
Backtracking Line Search ◽  
Ill Posed ◽  
Inverse Source Problems

Abstract We consider the method of quasi-solutions (also referred to as Ivanov regularization) for the regularization of linear ill-posed problems in non-reflexive Banach spaces. Using the equivalence to a metric projection onto the image of the forward operator, it is possible to show regularization properties and to characterize parameter choice rules that lead to a convergent regularization method, which includes the Morozov discrepancy principle. Convergence rates in a suitably chosen Bregman distance can be obtained as well. We also address the numerical computation of quasi-solutions to inverse source problems for partial differential equations in {L^{\infty}(\Omega)} using a semi-smooth Newton method and a backtracking line search for the parameter choice according to the discrepancy principle. Numerical examples illustrate the behavior of quasi-solutions in this setting.

Existence of variational source conditions for nonlinear inverse problems in Banach spaces

2018 ◽  
Vol 26 (2) ◽  
pp. 277-286 ◽  
Author(s):  
Jens Flemming
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Banach Spaces ◽  
Regularization Method ◽  
Multiple Solutions ◽  
Convergence Rates ◽  
Nonlinear Inverse Problems ◽  
Range Type ◽  
Ill Posed ◽  
Tikhonov’S Regularization

AbstractVariational source conditions proved to be useful for deriving convergence rates for Tikhonov’s regularization method and also for other methods. Up to now, such conditions have been verified only for few examples or for situations which can be also handled by classical range-type source conditions. Here we show that for almost every ill-posed inverse problem variational source conditions are satisfied. Whether linear or nonlinear, whether Hilbert or Banach spaces, whether one or multiple solutions, variational source conditions are a universal tool for proving convergence rates.

Convex Tikhonov regularization in Banach spaces: New results on convergence rates

2016 ◽  
Vol 24 (3) ◽  
Author(s):  
Stefan Kindermann
Keyword(s):  
Exact Solution ◽  
Variational Inequality ◽  
Banach Spaces ◽  
Tikhonov Regularization ◽  
Convergence Rates ◽  
Operator Equations ◽  
Ill Posed ◽  
Fidelity Term ◽  
The Difference ◽  
Regularized Solution

AbstractTikhonov regularization in Banach spaces with convex penalty and convex fidelity term for linear ill-posed operator equations is studied. As a main result, convergence rates in terms of the Bregman distance of the regularized solution to the exact solution is proven by imposing a generalization of the established variational inequality conditions on the exact solution. This condition only involves a decay rate of the difference of the penalty functionals in terms of the residual.

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