Dual Regularized Total Least Squares And Multi-Parameter Regularization

2008 ◽  
Vol 8 (3) ◽  
pp. 253-262 ◽  
Author(s):  
S. LU ◽  
S.V. PEREVERZEV ◽  
U. TAUTENHAHN
Keyword(s):  
Least Squares ◽  
Tikhonov Regularization ◽  
Regularization Method ◽  
Differential Operators ◽  
Total Least Squares ◽  
Right Hand ◽  
Regularization Parameters ◽  
Ill Posed ◽  
Regularized Total Least Squares ◽  
Parameter Regularization

AbstractIn this paper we continue our study of solving ill-posed problems with a noisy right-hand side and a noisy operator. Regularized approximations are obtained by Tikhonov regularization with differential operators and by dual regularized total least squares (dual RTLS) which can be characterized as a special multi-parameter regularization method where one of the two regularization parameters is negative. We report on order optimality results for both regularized approximations, discuss compu-tational aspects, provide special algorithms and show by experiments that dual RTLS is competitive to Tikhonov regularization with differential operators.

scholarly journals Two-Parameter Regularization Method for a Nonlinear Backward Heat Problem with a Conformable Derivative

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Vu Ho ◽  
Donal O’Regan ◽  
Hoa Ngo Van
Keyword(s):  
Integral Equation ◽  
Regularization Method ◽  
Time Variable ◽  
Conformable Derivative ◽  
Heat Problem ◽  
Regularization Parameters ◽  
Ill Posed ◽  
Parameter Regularization ◽  
Two Parameter ◽  
Backward Heat Problem

In this paper, we consider the nonlinear inverse-time heat problem with a conformable derivative concerning the time variable. This problem is severely ill posed. A new method on the modified integral equation based on two regularization parameters is proposed to regularize this problem. Numerical results are presented to illustrate the efficiency of the proposed method.

scholarly journals DISCREPANCY SETS FOR COMBINED LEAST SQUARES PROJECTION AND TIKHONOV REGULARIZATION

2017 ◽  
Vol 22 (2) ◽  
pp. 202-212
Author(s):  
Teresa Reginska
Keyword(s):  
Least Squares ◽  
Tikhonov Regularization ◽  
Discrepancy Principle ◽  
Order Of Accuracy ◽  
Finite Dimensional ◽  
Regularization Parameters ◽  
Ill Posed ◽  
Data Error ◽  
Two Parameter ◽  
Regularized Solution

To solve a linear ill-posed problem, a combination of the finite dimensional least squares projection method and the Tikhonov regularization is considered. The dimension of the projection is treated as the second parameter of regularization. A two-parameter discrepancy principle defines a discrepancy set for any data error bound. The aim of the paper is to describe this set and to indicate its subset such that for regularization parameters from this subset the related regularized solution has the same order of accuracy as the Tikhonov regularization with the standard discrepancy principle but without any discretization.

III-Posedness and Regularization Method for Nonlinear Identification Equations in Time Domain

2012 ◽  
Vol 166-169 ◽  
pp. 3282-3289
Author(s):  
Xian Zhong Xie ◽  
Feng Zhang ◽  
Yang Li ◽  
Yong Tan
Keyword(s):  
Least Squares ◽  
Tikhonov Regularization ◽  
Time Domain ◽  
Regularization Method ◽  
Least Squares Method ◽  
Structural Parameters ◽  
Nonlinear Identification ◽  
Calculating Method ◽  
Ill Posed ◽  
Damped Least Squares

The ill-posedness of nonlinear identification equation in time domain of structural dynamics system is studied and a new calculating method to weaken the influence of ill-posedness is proposed. Damped least squares method is an algorithm of Jacobian matrix positive-definable, which can obtain the solution of ill-posed nonlinear identification equation. But the solution is sensitive to the test noise of response in time domain of the structure. To solve the problem of instability of the solution, a new calculating method is proposed which combines damped least squares method with Tikhonov regularization method. First, the estimate of structural parameters is introduced to Tikhonov regularization function, and a more stable identification equation in time domain can be obtained. Second, the identification equation is solved with damped least squares method, and the iterative result is an approximate solution of the former ill-posed problem. The numerical example shows that the new method in this paper is efficient to solve the ill-posed nonlinear identification equation in time domain.

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.

A model function method in regularized total least squares

2010 ◽  
Vol 89 (11) ◽  
pp. 1693-1703 ◽  
Author(s):  
Shuai Lu ◽  
Sergei V. Pereverzev ◽  
Ulrich Tautenhahn
Keyword(s):  
Least Squares ◽  
Function Method ◽  
Total Least Squares ◽  
Model Function ◽  
Regularized Total Least Squares

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.

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