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 NEW RULE FOR CHOICE OF THE REGULARIZATION PARAMETER IN (ITERATED) TIKHONOV METHOD

2009 ◽  
Vol 14 (2) ◽  
pp. 187-198 ◽  
Author(s):  
Toomas Raus ◽  
Uno Hämarik
Keyword(s):  
Hilbert Spaces ◽  
Noise Level ◽  
Regularization Parameter ◽  
Optimal Error Estimates ◽  
A Posteriori ◽  
Optimal Error ◽  
Numerical Examples ◽  
Tikhonov Method ◽  
Mild Assumption ◽  
Ill Posed

We propose a new a posteriori rule for choosing the regularization parameter α in (iterated) Tikhonov method for solving linear ill‐posed problems in Hilbert spaces. We assume that data are noisy but noise level δ is given. We prove that (iterated) Tikhonov approximation with proposed choice of α converges to the solution as δ → 0 and has order optimal error estimates. Under certain mild assumption the quasioptimality of proposed rule is also proved. Numerical examples show the advantage of the new rule over the monotone error rule, especially in case of rough δ.

scholarly journals ON NUMERICAL REALIZATION OF QUASIOPTIMAL PARAMETER CHOICES IN (ITERATED) TIKHONOV AND LAVRENTIEV REGULARIZATION

2009 ◽  
Vol 14 (1) ◽  
pp. 99-108 ◽  
Author(s):  
Toomas Raus ◽  
Uno Hämarik
Keyword(s):  
Hilbert Spaces ◽  
Noise Level ◽  
Regularization Parameter ◽  
Numerical Realization ◽  
A Posteriori ◽  
Parameter Choice ◽  
Linear Combinations ◽  
Choice Alternative ◽  
Lavrentiev Regularization ◽  
Ill Posed

We consider linear ill‐posed problems in Hilbert spaces with noisy right hand side and given noise level. For approximation of the solution the Tikhonov method or the iterated variant of this method may be used. In self‐adjoint problems the Lavrentiev method or its iterated variant are used. For a posteriori choice of the regularization parameter often quasioptimal rules are used which require computing of additionally iterated approximations. In this paper we propose for parameter choice alternative numerical schemes, using instead of additional iterations linear combinations of approximations with different parameters.

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).

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).

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.

scholarly journals IDENTIFICATION OF THE SOURCE FOR FULL PARABOLIC EQUATIONS

2021 ◽  
Vol 26 (3) ◽  
pp. 339-357
Author(s):  
Guillermo Federico Umbricht
Keyword(s):  
Parabolic Equations ◽  
Regularization Parameter ◽  
Estimation Error ◽  
Noisy Data ◽  
Stability And Convergence ◽  
Parameter Choice ◽  
Independent Source ◽  
Numerical Examples ◽  
Ill Posed ◽  
The Stability

In this work, we consider the problem of identifying the time independent source for full parabolic equations in Rn from noisy data. This is an ill-posed problem in the sense of Hadamard. To compensate the factor that causes the instability, a family of parametric regularization operators is introduced, where the rule to select the value of the regularization parameter is included. This rule, known as regularization parameter choice rule, depends on the data noise level and the degree of smoothness that it is assumed for the source. The proof for the stability and convergence of the regularization criteria is presented and a Hölder type bound is obtained for the estimation error. Numerical examples are included to illustrate the effectiveness of this regularization approach.

scholarly journals Penalty-based smoothness conditions in convex variational regularization

2019 ◽  
Vol 27 (2) ◽  
pp. 283-300 ◽  
Author(s):  
Bernd Hofmann ◽  
Stefan Kindermann ◽  
Peter Mathé
Keyword(s):  
Banach Space ◽  
Variational Inequalities ◽  
Tikhonov Regularization ◽  
Hilbert Spaces ◽  
Noise Level ◽  
Regularization Parameter ◽  
The Family ◽  
Ill Posed ◽  
General Convex ◽  
Variational Regularization

Abstract The authors study Tikhonov regularization of linear ill-posed problems with a general convex penalty defined on a Banach space. It is well known that the error analysis requires smoothness assumptions. Here such assumptions are given in form of inequalities involving only the family of noise-free minimizers along the regularization parameter and the (unknown) penalty-minimizing solution. These inequalities control, respectively, the defect of the penalty, or likewise, the defect of the whole Tikhonov functional. The main results provide error bounds for a Bregman distance, which split into two summands: the first smoothness-dependent term does not depend on the noise level, whereas the second term includes the noise level. This resembles the situation of standard quadratic Tikhonov regularization in Hilbert spaces. It is shown that variational inequalities, as these were studied recently, imply the validity of the assumptions made here. Several examples highlight the results in specific applications.

scholarly journals EXTRAPOLATION OF TIKHONOV REGULARIZATION METHOD

2010 ◽  
Vol 15 (1) ◽  
pp. 55-68 ◽  
Author(s):  
Uno Hämarik ◽  
Reimo Palm ◽  
Toomas Raus
Keyword(s):  
Approximate Solution ◽  
Tikhonov Regularization ◽  
Noise Level ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Noisy Data ◽  
Tikhonov Regularization Method ◽  
Guaranteed Accuracy ◽  
Numerical Examples ◽  
Ill Posed

We consider regularization of linear ill‐posed problem Au = f with noisy data fδ, ¦fδ - f¦≤ δ . The approximate solution is computed as the extrapolated Tikhonov approximation, which is a linear combination of n ≥ 2 Tikhonov approximations with different parameters. If the solution u* belongs to R((A*A) n ), then the maximal guaranteed accuracy of Tikhonov approximation is O(δ 2/3) versus accuracy O(δ 2n/(2n+1)) of corresponding extrapolated approximation. We propose several rules for choice of the regularization parameter, some of these are also good in case of moderate over‐ and underestimation of the noise level. Numerical examples are given.

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 Convergence Rate of the Modified Landweber Method for Solving Inverse Potential Problems

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 608
Author(s):  
Pornsarp Pornsawad ◽  
Parada Sungcharoen ◽  
Christine Böckmann
Keyword(s):  
Convergence Rate ◽  
Potential Problem ◽  
Regularization Parameter ◽  
Discrepancy Principle ◽  
Source Condition ◽  
Rate Analysis ◽  
Potential Problems ◽  
Ill Posed ◽  
Landweber Method ◽  
Convergence Rate Analysis

In this paper, we present the convergence rate analysis of the modified Landweber method under logarithmic source condition for nonlinear ill-posed problems. The regularization parameter is chosen according to the discrepancy principle. The reconstructions of the shape of an unknown domain for an inverse potential problem by using the modified Landweber method are exhibited.

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