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 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 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.

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].

scholarly journals Tikhonov regularization with ℓ0-term complementing a~convex penalty: ℓ1-convergence under sparsity constraints

2019 ◽  
Vol 27 (4) ◽  
pp. 575-590 ◽  
Author(s):  
Wei Wang ◽  
Shuai Lu ◽  
Bernd Hofmann ◽  
Jin Cheng
Keyword(s):  
Convergence Rates ◽  
Regularization Parameter ◽  
A Priori ◽  
Convex Functional ◽  
Important Special Case ◽  
Bounded Linear ◽  
Sparsity Constraints ◽  
Ill Posed ◽  
Sparse Solutions ◽  
Special Case

Abstract Measuring the error by an {\ell^{1}} -norm, we analyze under sparsity assumptions an {\ell^{0}} -regularization approach, where the penalty in the Tikhonov functional is complemented by a general stabilizing convex functional. In this context, ill-posed operator equations {Ax=y} with an injective and bounded linear operator A mapping between {\ell^{2}} and a Banach space Y are regularized. For sparse solutions, error estimates as well as linear and sublinear convergence rates are derived based on a variational inequality approach, where the regularization parameter can be chosen either a priori in an appropriate way or a posteriori by the sequential discrepancy principle. To further illustrate the balance between the {\ell^{0}} -term and the complementing convex penalty, the important special case of the {\ell^{2}} -norm square penalty is investigated showing explicit dependence between both terms. Finally, some numerical experiments verify and illustrate the sparsity promoting properties of corresponding regularized solutions.

scholarly journals An optimal order yielding discrepancy principle for simplified regularization of ill-posed problems in Hilbert scales

2003 ◽  
Vol 2003 (39) ◽  
pp. 2487-2499 ◽  
Author(s):  
Santhosh George ◽  
M. Thamban Nair
Keyword(s):  
Approximate Solution ◽  
Hilbert Spaces ◽  
Regularization Parameter ◽  
Optimal Order ◽  
Discrepancy Principle ◽  
Parameter Choice ◽  
Bounded Linear ◽  
Choice Strategy ◽  
Ill Posed ◽  
Parameter Choice Strategy

Recently, Tautenhahn and Hämarik (1999) have considered a monotone rule as a parameter choice strategy for choosing the regularization parameter while considering approximate solution of an ill-posed operator equationTx=y, whereTis a bounded linear operator between Hilbert spaces. Motivated by this, we propose a new discrepancy principle for the simplified regularization, in the setting of Hilbert scales, whenTis a positive and selfadjoint operator. When the datayis known only approximately, our method provides optimal order under certain natural assumptions on the ill-posedness of the equation and smoothness of the solution. The result, in fact, improves an earlier work of the authors (1997).

On the Recovery of Continuous Functions from Noisy Fourier Coefficients

2011 ◽  
Vol 11 (1) ◽  
pp. 75-82 ◽  
Author(s):  
Kosnazar Sharipov
Keyword(s):  
Error Bound ◽  
Fourier Coefficients ◽  
Regularization Parameter ◽  
A Priori ◽  
Continuous Functions ◽  
A Posteriori ◽  
Parameter Choice ◽  
Optimal Error ◽  
Ill Posed ◽  
Optimal Error Bound

AbstractWe consider the classical ill-posed problem of the recovery of continuous functions from noisy Fourier coefficients. For the classes of functions given in terms of generalized smoothness, we present a priori and a posteriori regularization parameter choice realizing an order-optimal error bound.

On The Analog Of The Monotone Error Rule For Parameter Choice In The (Iterated) Lavrentiev Regularization

2008 ◽  
Vol 8 (3) ◽  
pp. 237-252 ◽  
Author(s):  
U HAMARIK ◽  
R. PALM ◽  
T. RAUS
Keyword(s):  
Hilbert Spaces ◽  
Noise Level ◽  
Regularization Parameter ◽  
Optimal Error Estimates ◽  
Discrepancy Principle ◽  
Parameter Choice ◽  
Optimal Error ◽  
Numerical Examples ◽  
Lavrentiev Regularization ◽  
Ill Posed

AbstractWe consider linear ill-posed problems in Hilbert spaces with a noisy right hand side and a given noise level. To solve non-self-adjoint problems by the (it-erated) Tikhonov method, one effective rule for choosing the regularization parameter is the monotone error rule (Tautenhahn and Hamarik, Inverse Problems, 1999, 15, 1487– 1505). In this paper we consider the solution of self-adjoint problems by the (iterated) Lavrentiev method and propose for parameter choice an analog of the monotone error rule. We prove under certain mild assumptions the quasi-optimality of the proposed rule guaranteeing convergence and order optimal error estimates. Numerical examples show for the proposed rule and its modifications much better performance than for the modified discrepancy principle.

scholarly journals Method of iteration of solving invariant equations with an approximate operator in the case of an arbitrary choice of the regularization parameter

2019 ◽  
Vol 54 (4) ◽  
pp. 408-416
Author(s):  
O. V. Matysik ◽  
V. F. Savchuk
Keyword(s):  
Error Estimate ◽  
Error Estimates ◽  
Regularization Parameter ◽  
A Priori ◽  
Operator Equations ◽  
Mathematical Economics ◽  
Prove Theorem ◽  
A Value ◽  
Ill Posed ◽  
The Right

In the introduction, the object of investigation is indicated – incorrect problems described by first-kind operator equations. The subject of the study is an explicit iterative method for solving first-kind equations. The aim of the paper is to prove the convergence of the proposed method of simple iterations with an alternating step alternately and to obtain error estimates in the original norm of a Hilbert space for the cases of self-conjugated and non self-conjugated problems. The a priori choice of the regularization parameter is studied for a source-like representable solution under the assumption that the operator and the right-hand side of the equation are given approximately. In the main part of the work, the achievement of the stated goal is expressed in four reduced and proved theorems. In Section 1, the first-kind equation is written down and a new explicit method of simple iteration with alternating steps is proposed to solve it. In Section 2, we consider the case of the selfconjugated problem and prove Theorem 1 on the convergence of the method and Theorem 2, in which an error estimate is obtained. To obtain an error estimate, an additional condition is required – the requirement of the source representability of the exact solution. In Section 3, the non-self-conjugated problem is solved, the convergence of the proposed method is proved, which in this case is written differently, and its error estimate is obtained in the case of an a priori choice of the regularization parameter. In sections 2 and 3, the error estimates obtained are optimized, that is, a value is found – the step number of the iteration, in which the error estimate is minimal. Since incorrect problems constantly arise in numerous applications of mathematics, the problem of studying them and constructing methods for their solution is topical. The obtained results can be used in theoretical studies of solution of first-kind operator equations, as well as applied ill-posed problems encountered in dynamics and kinetics, mathematical economics, geophysics, spectroscopy, systems for complete automatic processing and interpretation of experiments, plasma diagnostics, seismic and medicine.

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.

scholarly journals Two fractional regularization methods for identifying the radiogenic source of the Helium production-diffusion equation

2021 ◽  
Vol 6 (10) ◽  
pp. 11425-11448
Author(s):  
Xuemin Xue ◽  
◽  
Xiangtuan Xiong ◽  
Yuanxiang Zhang ◽  
Keyword(s):  
Diffusion Equation ◽  
Source Term ◽  
Regularization Parameter ◽  
A Priori ◽  
Regularization Methods ◽  
A Posteriori ◽  
Parameter Choice ◽  
Ill Posed ◽  
Parameter Choice Rules ◽  
Regularized Solution

<abstract><p>The predication of the helium diffusion concentration as a function of a source term in diffusion equation is an ill-posed problem. This is called inverse radiogenic source problem. Although some classical regularization methods have been considered for this problem, we propose two new fractional regularization methods for the purpose of reducing the over-smoothing of the classical regularized solution. The corresponding error estimates are proved under the a-priori and the a-posteriori regularization parameter choice rules. Some numerical examples are shown to display the necessarity of the methods.</p></abstract>

Sign in / Sign up

Export Citation Format

Share Document