A parameter choice strategy for the regularized approximation of Fredholm integral equations of the first kind

2010 ◽  
Vol 87 (11) ◽  
pp. 2612-2622 ◽  
Author(s):  
M. P. Rajan
Keyword(s):  
Integral Equations ◽  
Fredholm Integral Equations ◽  
Parameter Choice ◽  
Choice Strategy ◽  
Parameter Choice Strategy

Multiscale Galerkin methods for the nonstationary iterated Tikhonov method with a modified posteriori parameter selection

2018 ◽  
Vol 26 (1) ◽  
pp. 109-120
Author(s):  
Xingjun Luo ◽  
Zhaofu Ouyang ◽  
Chunmei Zeng ◽  
Fanchun Li
Keyword(s):  
Convergence Rates ◽  
Regularization Parameter ◽  
Fredholm Integral Equations ◽  
Galerkin Methods ◽  
A Posteriori ◽  
Parameter Choice ◽  
Compression Technique ◽  
Choice Strategy ◽  
Optimal Convergence Rates ◽  
Parameter Choice Strategy

AbstractIn this paper, we consider a fast multiscale Galerkin method with compression technique for solving Fredholm integral equations of the first kind via the nonstationary iterated Tikhonov regularization. A modified a posteriori regularization parameter choice strategy is established, which leads to optimal convergence rates.

On the Parameter Choice in the Multilevel Augmentation Method

2020 ◽  
Vol 20 (3) ◽  
pp. 555-571
Author(s):  
Suhua Yang ◽  
Xingjun Luo ◽  
Chunmei Zeng ◽  
Zhihai Xu ◽  
Wenyu Hu
Keyword(s):  
Tikhonov Regularization ◽  
Regularization Method ◽  
Numerical Experiments ◽  
Convergence Rates ◽  
Tikhonov Regularization Method ◽  
Fredholm Integral Equations ◽  
Parameter Choice ◽  
Choice Strategy ◽  
Ill Posed ◽  
Parameter Choice Strategy

AbstractIn this paper, we apply the multilevel augmentation method for solving ill-posed Fredholm integral equations of the first kind via iterated Tikhonov regularization method. The method leads to fast solutions of the discrete regularization methods for the equations. The convergence rates of iterated Tikhonov regularization are achieved by using a modified parameter choice strategy. Finally, numerical experiments are given to illustrate the efficiency of the method.

A parameter choice strategy for a multilevel augmentation method in iterated Lavrentiev regularization

2018 ◽  
Vol 26 (2) ◽  
pp. 153-170 ◽  
Author(s):  
Chunmei Zeng ◽  
Xingjun Luo ◽  
Suhua Yang ◽  
Fanchun Li
Keyword(s):  
Integral Equation ◽  
Numerical Experiments ◽  
Convergence Rates ◽  
Parameter Choice ◽  
Lavrentiev Regularization ◽  
Choice Strategy ◽  
Ill Posed ◽  
Parameter Choice Strategy

AbstractIn this paper we apply the multilevel augmentation method to solve an ill-posed integral equation via the iterated Lavrentiev regularization. This method leads to fast solutions of discrete iterated Lavrentiev regularization. The convergence rates of the iterated Lavrentiev regularization are achieved by using a certain parameter choice strategy. Finally, numerical experiments are given to illustrate the efficiency of the method.

A STOPPING RULE FOR THE CONJUGATE GRADIENT REGULARIZATION METHOD APPLIED TO INVERSE PROBLEMS IN ACOUSTICS

2006 ◽  
Vol 14 (04) ◽  
pp. 397-414 ◽  
Author(s):  
THOMAS DELILLO ◽  
TOMASZ HRYCAK
Keyword(s):  
Conjugate Gradient ◽  
A Priori ◽  
A Priori Information ◽  
Parameter Choice ◽  
Regularization Algorithm ◽  
Lanczos Process ◽  
Choice Strategy ◽  
Priori Information ◽  
The One ◽  
Parameter Choice Strategy

We present a novel parameter choice strategy for the conjugate gradient regularization algorithm which does not assume a priori information about the magnitude of the measurement error. Our approach is to regularize within the Krylov subspaces associated with the normal equations. We implement conjugate gradient via the Lanczos bidiagonalization process with reorthogonalization, and then we construct regularized solutions using the SVD of a bidiagonal projection constructed by the Lanczos process. We compare our method with the one proposed by Hanke and Raus and illustrate its performance with numerical experiments, including detection of acoustic boundary vibrations.

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).

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