scholarly journals Optimal order yielding discrepancy principle for simplified regularization in Hilbert scales: finite-dimensional realizations

2004 ◽  
Vol 2004 (37) ◽  
pp. 1973-1996 ◽  
Author(s):  
Santhosh George ◽  
M. Thamban Nair
Keyword(s):  
Wide Class ◽  
Noise Level ◽  
A Priori ◽  
Approximate Solutions ◽  
Optimal Order ◽  
Discrepancy Principle ◽  
A Posteriori ◽  
Operator Equations ◽  
Finite Dimensional ◽  
Ill Posed

Simplified regularization using finite-dimensional approximations in the setting of Hilbert scales has been considered for obtaining stable approximate solutions to ill-posed operator equations. The derived error estimates using an a priori and a posteriori choice of parameters in relation to the noise level are shown to be of optimal order with respect to certain natural assumptions on the ill posedness of the equation. The results are shown to be applicable to a wide class of spline approximations in the setting of Sobolev scales.

scholarly journals Morozov’s Discrepancy Principle For The Tikhonov Regularization Of Exponentially Ill-Posed Problems

2008 ◽  
Vol 8 (1) ◽  
pp. 86-98 ◽  
Author(s):  
S.G. SOLODKY ◽  
A. MOSENTSOVA
Keyword(s):  
Tikhonov Regularization ◽  
Optimal Order ◽  
Discrepancy Principle ◽  
Operator Equations ◽  
Order Of Accuracy ◽  
Finite Dimensional ◽  
Dimensional Version ◽  
Ill Posed ◽  
Linear Operator Equations ◽  
Morozov’S Discrepancy Principle

Abstract The problem of approximate solution of severely ill-posed problems given in the form of linear operator equations of the first kind with approximately known right-hand sides was considered. We have studied a strategy for solving this type of problems, which consists in combinating of Morozov’s discrepancy principle and a finite-dimensional version of the Tikhonov regularization. It is shown that this combination provides an optimal order of accuracy on source sets

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 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.

Finite-dimensional iteratively regularized processes with an a posteriori stopping for solving irregular nonlinear operator equations

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mikhail Y. Kokurin ◽  
Alexander I. Kozlov
Keyword(s):  
Nonlinear Operator ◽  
A Priori ◽  
Convex Compact ◽  
A Posteriori ◽  
Operator Equations ◽  
Nonlinear Operator Equations ◽  
Finite Dimensional Approximation ◽  
Finite Dimensional ◽  
Dimensional Approximation ◽  
Priori Information

AbstractWe construct and study a class of numerically implementable iteratively regularized Gauss–Newton type methods for approximate solution of irregular nonlinear operator equations in Hilbert space. The methods include a general finite-dimensional approximation for equations under consideration and cover the projection, collocation and quadrature discretization schemes. Using an a posteriori stopping rule for the iterative processes and the standard source condition on the solution, we establish accuracy estimates for the approximations generated by the methods. We also investigate projected versions of the processes which take into account a priori information about a convex compact containing the solution. An iteratively regularized quadrature process is applied to an inverse 2D problem of gravimetry.

scholarly journals Regularization of backward time-fractional parabolic equations by Sobolev-type equations

2020 ◽  
Vol 28 (5) ◽  
pp. 659-676
Author(s):  
Dinh Nho Hào ◽  
Nguyen Van Duc ◽  
Nguyen Van Thang ◽  
Nguyen Trung Thành
Keyword(s):  
Error Estimates ◽  
Parabolic Equations ◽  
Regularization Parameter ◽  
A Priori ◽  
Sobolev Type ◽  
A Posteriori ◽  
Parameter Choice ◽  
Numerical Tests ◽  
Ill Posed ◽  
Parameter Choice Rules

AbstractThe problem of determining the initial condition from noisy final observations in time-fractional parabolic equations is considered. This problem is well known to be ill-posed, and it is regularized by backward Sobolev-type equations. Error estimates of Hölder type are obtained with a priori and a posteriori regularization parameter choice rules. The proposed regularization method results in a stable noniterative numerical scheme. The theoretical error estimates are confirmed by numerical tests for one- and two-dimensional equations.

scholarly journals COMPLEXITY ESTIMATES FOR SEVERELY ILL-POSED PROBLEMS UNDER A POSTERIORI SELECTION OF REGULARIZATION PARAMETER

2017 ◽  
Vol 22 (3) ◽  
pp. 283-299
Author(s):  
Sergii G. Solodky ◽  
Ganna L. Myleiko ◽  
Evgeniya V. Semenova
Keyword(s):  
Integral Equations ◽  
Regularization Parameter ◽  
Efficient Algorithms ◽  
Optimal Order ◽  
A Posteriori ◽  
Order Of Accuracy ◽  
Tikhonov Method ◽  
Discrete Information ◽  
Ill Posed ◽  
Selection Of

In the article the authors developed two efficient algorithms for solving severely ill-posed problems such as Fredholm’s integral equations. The standard Tikhonov method is applied as a regularization. To select a regularization parameter we employ two different a posteriori rules, namely, discrepancy and balancing principles. It is established that proposed strategies not only achieved optimal order of accuracy on the class of problems under consideration, but also they are economical in the sense of used discrete information.

A derivative-free iterative method for nonlinear ill-posed equations with monotone operators

2017 ◽  
Vol 25 (5) ◽  
pp. 543-551 ◽  
Author(s):  
Santhosh George ◽  
M. Thamban Nair
Keyword(s):  
Iterative Method ◽  
Convergence Analysis ◽  
Monotone Operator ◽  
Regularization Parameter ◽  
A Priori ◽  
Monotone Operators ◽  
Adaptive Procedure ◽  
Operator Equations ◽  
Derivative Free ◽  
Ill Posed

AbstractRecently, Semenova [12] considered a derivative free iterative method for nonlinear ill-posed operator equations with a monotone operator. In this paper, a modified form of Semenova’s method is considered providing simple convergence analysis under more realistic nonlinearity assumptions. The paper also provides a stopping rule for the iteration based on an a priori choice of the regularization parameter and also under the adaptive procedure considered by Pereverzev and Schock [11].

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