Convergence Rates For Lavrentiev-Type Regularization In Hilbert Scales

2008 ◽  
Vol 8 (3) ◽  
pp. 279-293 ◽  
Author(s):  
M.T. NAIR ◽  
U. TAUTENHAHN
Keyword(s):  
Error Bounds ◽  
Convergence Rates ◽  
Noisy Data ◽  
Regularization Methods ◽  
Operator Equations ◽  
Optimal Error ◽  
Regularization Scheme ◽  
Data Regularization ◽  
Ill Posed ◽  
Adjoint Operators

AbstractFor solving linear ill-posed problems with noisy data regularization methods are required. We analyze a simplified regularization scheme in Hilbert scales for operator equations with nonnegative self-adjoint operators. By exploiting the op-erator monotonicity of certain functions, order-optimal error bounds are derived that characterize the accuracy of the regularized approximations. These error bounds have been obtained under general smoothness conditions.

M. A. Krasnosel'skii Theorem and Iterative Methods for Solving Ill-Posed Linear Problems with a Self-Adjoint Operator

2015 ◽  
Vol 15 (3) ◽  
pp. 373-389
Author(s):  
Oleg Matysik ◽  
Petr Zabreiko
Keyword(s):  
Hilbert Space ◽  
Iterative Methods ◽  
Critical Case ◽  
Adjoint Operator ◽  
Successive Approximations ◽  
Operator Equations ◽  
Ill Posed ◽  
Linear Problems ◽  
Adjoint Operators ◽  
Linear Operator Equations

AbstractThe paper deals with iterative methods for solving linear operator equations ${x = Bx + f}$ and ${Ax = f}$ with self-adjoint operators in Hilbert space X in the critical case when ${\rho (B) = 1}$ and ${0 \in \operatorname{Sp} A}$. The results obtained are based on a theorem by M. A. Krasnosel'skii on the convergence of the successive approximations, their modifications and refinements.

scholarly journals The Impact of the Discrepancy Principle on the Tikhonov-Regularized Solutions with Oversmoothing Penalties

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 331
Author(s):  
Bernd Hofmann ◽  
Christopher Hofmann
Keyword(s):  
Case Studies ◽  
Convergence Rates ◽  
Regularization Parameter ◽  
A Priori ◽  
Discrepancy Principle ◽  
Parameter Choice ◽  
Operator Equations ◽  
Analytical Results ◽  
Ill Posed ◽  
The Impact

This paper deals with the Tikhonov regularization for nonlinear ill-posed operator equations in Hilbert scales with oversmoothing penalties. One focus is on the application of the discrepancy principle for choosing the regularization parameter and its consequences. Numerical case studies are performed in order to complement analytical results concerning the oversmoothing situation. For example, case studies are presented for exact solutions of Hölder type smoothness with a low Hölder exponent. Moreover, the regularization parameter choice using the discrepancy principle, for which rate results are proven in the oversmoothing case in in reference (Hofmann, B.; Mathé, P. Inverse Probl. 2018, 34, 015007) is compared to Hölder type a priori choices. On the other hand, well-known analytical results on the existence and convergence of regularized solutions are summarized and partially augmented. In particular, a sketch for a novel proof to derive Hölder convergence rates in the case of oversmoothing penalties is given, extending ideas from in reference (Hofmann, B.; Plato, R. ETNA. 2020, 93).

scholarly journals On fractional asymptotical regularization of linear ill-posed problems in hilbert spaces

2019 ◽  
Vol 22 (3) ◽  
pp. 699-721 ◽  
Author(s):  
Ye Zhang ◽  
Bernd Hofmann
Keyword(s):  
Fractional Order ◽  
Hilbert Spaces ◽  
Regularization Method ◽  
Numerical Realization ◽  
Regularization Methods ◽  
Iterative Regularization ◽  
Operator Equations ◽  
Ill Posed ◽  
One Step ◽  
The One

Abstract In this paper, we study a fractional-order variant of the asymptotical regularization method, called Fractional Asymptotical Regularization (FAR), for solving linear ill-posed operator equations in a Hilbert space setting. We assign the method to the general linear regularization schema and prove that under certain smoothness assumptions, FAR with fractional order in the range (1, 2) yields an acceleration with respect to comparable order optimal regularization methods. Based on the one-step Adams-Moulton method, a novel iterative regularization scheme is developed for the numerical realization of FAR. Two numerical examples are given to show the accuracy and the acceleration effect of FAR.

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.

scholarly journals Arcangeli's type discrepancy principles for a class of regularization methods using a modified projection scheme

2001 ◽  
Vol 6 (6) ◽  
pp. 339-355 ◽  
Author(s):  
M. T. Nair ◽  
M. P. Rajan
Keyword(s):  
Error Estimates ◽  
General Class ◽  
Regularization Parameter ◽  
Regularization Methods ◽  
A Posteriori ◽  
Operator Equations ◽  
Projection Scheme ◽  
Ill Posed

Solodkĭ (1998) applied the modified projection scheme of Pereverzev (1995) for obtaining error estimates for a class of regularization methods for solving ill-posed operator equations. But, no a posteriori procedure for choosing the regularization parameter is discussed. In this paper, we consider Arcangeli's type discrepancy principles for such a general class of regularization methods with modified projection scheme.

Simplified Generalized Gauss–Newton Method for Nonlinear Ill-Posed Operator Equations in Hilbert Scales

2018 ◽  
Vol 18 (4) ◽  
pp. 687-702 ◽  
Author(s):  
Pallavi Mahale ◽  
Pradeep Kumar Dadsena
Keyword(s):  
Approximate Solution ◽  
Error Estimate ◽  
Newton Method ◽  
Operator Equation ◽  
Hilbert Spaces ◽  
Nonlinear Operator ◽  
Optimal Error Estimate ◽  
Operator Equations ◽  
Optimal Error ◽  
Ill Posed

AbstractIn this paper, we study the simplified generalized Gauss–Newton method in a Hilbert scale setting to get an approximate solution of the ill-posed operator equation of the form {F(x)=y} where {F:D(F)\subseteq X\to Y} is a nonlinear operator between Hilbert spaces X and Y. Under suitable nonlinearly conditions on F, we obtain an order optimal error estimate under the Morozov type stopping rule.

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