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 A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation

2020 ◽  
Vol 18 (1) ◽  
pp. 1685-1697
Author(s):  
Zhenyu Zhao ◽  
Lei You ◽  
Zehong Meng
Keyword(s):  
Cauchy Problem ◽  
Laplace Equation ◽  
Tikhonov Regularization ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Solution Process ◽  
Tikhonov Regularization Method ◽  
Hermite Expansion ◽  
The Cauchy Problem ◽  
Ill Posed

Abstract In this paper, a Cauchy problem for the Laplace equation is considered. We develop a modified Tikhonov regularization method based on Hermite expansion to deal with the ill posed-ness of the problem. The regularization parameter is determined by a discrepancy principle. For various smoothness conditions, the solution process of the method is uniform and the convergence rate can be obtained self-adaptively. Numerical tests are also carried out to verify the effectiveness of the method.

The Regularization Method Based on Complex Collinearity Diagnostics and Metrics

2013 ◽  
Vol 416-417 ◽  
pp. 1393-1398
Author(s):  
Chao Zhong Ma ◽  
Yong Wei Gu ◽  
Ji Fu ◽  
Yuan Lu Du ◽  
Qing Ming Gui
Keyword(s):  
Tikhonov Regularization ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Measurement Data ◽  
Tikhonov Regularization Method ◽  
Parameter Determination ◽  
Selection Methods ◽  
Ill Posed ◽  
Collinearity Diagnostics ◽  
Regularization Matrix

In a large number of measurement data processing, the ill-posed problem is widespread. For such problems, this paper introduces the solution of ill-posed problem of the unity of expression and Tikhonov regularization method, and then to re-collinearity diagnostics and metrics based on proposed based on complex collinearity diagnostics and the metric regularization method is given regularization matrix selection methods and regularization parameter determination formulas. Finally, it uses a simulation example to verify the effectiveness of the method.

Elastoplasticity 2D Problems: Numerical Applications of the Tikhonov Regularization Method

2015 ◽  
Vol 751 ◽  
pp. 109-117
Author(s):  
Thales Augusto Barbosa Pinto Silva ◽  
Hilbeth Parente Azikri de Deus ◽  
Claudio Roberto Ávila da Silva
Keyword(s):  
Tikhonov Regularization ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Numerical Approach ◽  
Theoretical Development ◽  
Tikhonov Regularization Method ◽  
Equilibrium Curve ◽  
Numerical Examples ◽  
Elastoplastic Materials ◽  
Ill Conditioning

The numerical simulation is widely used, in now days, to verify the viability and to optimize structural mechanic designs. The numerical approach of elastoplastic materials can found some problems related to ill-conditioning of matrices (from FEM systems), associated to the critical points from the snap through or snap back shape of the equilibrium curve. Aiming to overcome this misfortune it is proposed a strategy via Tikhonov regularization method in association with L-curve technique to determine the regularization parameter. This strategy can be used in many numerical applications for structural analysis. The theoretical development about these Some numerical examples are presented to attest the efficiency of this proposed approach.

A fast multilevel iteration method for solving linear ill-posed integral equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Hongqi Yang ◽  
Rong Zhang
Keyword(s):  
Approximate Solution ◽  
Linear System ◽  
Integral Equations ◽  
Tikhonov Regularization ◽  
Iteration Method ◽  
Noise Level ◽  
Regularization Parameter ◽  
Ill Posed ◽  
Linear Integral ◽  
Linear Integral Equations

Abstract We propose a new concept of noise level: R ⁢ ( K * ) \mathcal{R}(K^{*}) -noise level for ill-posed linear integral equations in Tikhonov regularization, which extends the range of regularization parameter. This noise level allows us to choose a more suitable regularization parameter. Moreover, we also analyze error estimates of the approximate solution with respect to this noise level. For ill-posed integral equations, finding fast and effective numerical methods is a challenging problem. For this, we formulate a matrix truncated strategy based on multiscale Galerkin method to generate the linear system of Tikhonov regularization for ill-posed linear integral equations, which greatly reduce the computational complexity. To further reduce the computational cost, a fast multilevel iteration method for solving the linear system is established. At the same time, we also prove convergence rates of the approximate solution obtained by this fast method with respect to the R ⁢ ( K * ) \mathcal{R}(K^{*}) -noise level under the balance principle. By numerical results, we show that R ⁢ ( K * ) \mathcal{R}(K^{*}) -noise level is very useful and the proposed method is a fast and effective method, respectively.

A Variant of Projection-Regularization Method for Ill-Posed Linear Operator Equations

2020 ◽  
pp. 2150008
Author(s):  
Bechouat Tahar ◽  
Boussetila Nadjib ◽  
Rebbani Faouzia
Keyword(s):  
Approximate Solution ◽  
Linear Operator ◽  
Tikhonov Regularization ◽  
Hilbert Spaces ◽  
Regularization Method ◽  
Tikhonov Regularization Method ◽  
Operator Equations ◽  
Ill Posed ◽  
Rank Approximation ◽  
Linear Operator Equations

In this paper, we report on a strategy for computing the numerical approximate solution for a class of ill-posed operator equations in Hilbert spaces: [Formula: see text]. This approach is a combination of Tikhonov regularization method and the finite rank approximation of [Formula: see text]. Finally, numerical results are given to show the effectiveness of this method.

Comparison of Tikhonov Regularization and Adaptive Regularization for III-Posed Problems

2013 ◽  
Vol 380-384 ◽  
pp. 1193-1196
Author(s):  
Wen Dong ◽  
Tao Sun
Keyword(s):  
Tikhonov Regularization ◽  
Regularization Method ◽  
Information Science ◽  
Tikhonov Regularization Method ◽  
Adaptive Regularization ◽  
Numerical Examples ◽  
Close Relationship ◽  
Ill Posed ◽  
The Stability ◽  
Science Information

nverse problems are important interdisciplinary subject, which receive more and more attention in recent years in the areas of mathematics, computer science, information science and other applied natural sciences. There is close relationship between inverse problems and ill-posedness. Regularization is an important strategy when computing the ill-posed problems to maintain the stability of the computation.This paper compares a new regularization method,which is called Adaptive regularization, with the traditional Tikhonov regularization method. The conclusion that Adaptive regularization method is a stronger regularization method than the traditional Tikhonov regularization method can be made by computing some numerical examples.

scholarly journals The Simplified Tikhonov Regularization Method for Identifying the Unknown Source for the Modified Helmholtz Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
Fan Yang ◽  
HengZhen Guo ◽  
XiaoXiao Li
Keyword(s):  
Exact Solution ◽  
Helmholtz Equation ◽  
Tikhonov Regularization ◽  
Regularization Method ◽  
Numerical Results ◽  
Tikhonov Regularization Method ◽  
Convergence Estimate ◽  
Modified Helmholtz Equation ◽  
Unknown Source ◽  
Ill Posed

This paper discusses the problem of determining an unknown source which depends only on one variable for the modified Helmholtz equation. This problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. The regularization solution is obtained by the simplified Tikhonov regularization method. Convergence estimate is presented between the exact solution and the regularization solution. Moreover, numerical results are presented to illustrate the accuracy and efficiency of this method.

scholarly journals A modified Tikhonov regularization method for a class of inverse parabolic problems

2020 ◽  
Vol 28 (1) ◽  
pp. 181-204
Author(s):  
Nabil Saouli ◽  
Fairouz Zouyed
Keyword(s):  
Boundary Conditions ◽  
Exact Solutions ◽  
Tikhonov Regularization ◽  
Regularization Method ◽  
Parabolic Problem ◽  
Tikhonov Regularization Method ◽  
Parabolic Problems ◽  
Initial Condition ◽  
Ill Posed ◽  
Convergence Estimates

AbstractThis paper deals with the problem of determining an unknown source and an unknown initial condition in a abstract final value parabolic problem. This problem is ill-posed in the sense that the solutions do not depend continuously on the data. To solve the considered problem a modified Tikhonov regularization method is proposed. Using this method regularized solutions are constructed and under boundary conditions assumptions, convergence estimates between the exact solutions and their regularized approximations are obtained. Moreover numerical results are presented to illustrate the accuracy and efficiency of the proposed method.

scholarly journals A Revised Tikhonov Regularization Method for a Cauchy Problem of Two-Dimensional Heat Conduction Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Songshu Liu ◽  
Lixin Feng
Keyword(s):  
Cauchy Problem ◽  
Heat Conduction ◽  
Tikhonov Regularization ◽  
Regularization Method ◽  
Heat Conduction Equation ◽  
Conduction Equation ◽  
Tikhonov Regularization Method ◽  
Two Dimensional ◽  
Convergence Estimate ◽  
Ill Posed

In this paper we investigate a Cauchy problem of two-dimensional (2D) heat conduction equation, which determines the internal surface temperature distribution from measured data at the fixed location. In general, this problem is ill-posed in the sense of Hadamard. We propose a revised Tikhonov regularization method to deal with this ill-posed problem and obtain the convergence estimate between the approximate solution and the exact one by choosing a suitable regularization parameter. A numerical example shows that the proposed method works well.

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