Large-scale Tikhonov regularization of total least squares

For large-scale problems, how to establish an algorithm with high accuracy and stability is particularly important. In this paper, the Householder bidiagonalization total least squares (HBITLS) algorithm and nonlinear iterative partial least squares for total least squares (NIPALS-TLS) algorithm were established, by which the same approximate TLS solutions was obtained. In addition, the propagation of the roundoff error for the process of the HBITLS algorithm was analyzed, and the mixed forward-backward stability of these two algorithms was proved. Furthermore, an upper bound of roundoff error was derived, which presents a more detailed and clearer approximation of the computed solution.

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Novel Iterative Truncated Total Least Squares Algorithm for Image Restoration

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.401-403.1397 ◽

2013 ◽

Vol 401-403 ◽

pp. 1397-1400

Author(s):

Lei Zhang ◽

Yue Yun Cao ◽

Zi Chun Yang

Keyword(s):

Image Restoration ◽

Least Squares ◽

Large Scale ◽

Total Least Squares ◽

System Matrix ◽

Scale Problem ◽

Truncation Parameter ◽

Lanczos Bidiagonalization ◽

Ill Posed ◽

Truncated Total Least Squares

Image restoration is a typical ill-posed inverse problem, which can be solved by a successful total least squares (TLS) method when not only the observation but the system matrix is also contaminated by addition noise. Considering the image restoration is a large-scale problem in general, project the TLS problem onto a subspace defined by a Lanczos bidiagonalization algorithm, and then the Truncated TLS method is applied on the subspace. Therefore, a novel iterative TTLS method, involving appropriate the choice of truncation parameter, is proposed. Finally, an Image reconstruction example is given to illustrate the effectiveness and robustness of proposed algorithm.

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Dual Regularized Total Least Squares And Multi-Parameter Regularization

Computational Methods in Applied Mathematics ◽

10.2478/cmam-2008-0018 ◽

2008 ◽

Vol 8 (3) ◽

pp. 253-262 ◽

Cited By ~ 12

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.

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Methods for Large Scale Total Least Squares Problems

SIAM Journal on Matrix Analysis and Applications ◽

10.1137/s0895479899355414 ◽

2000 ◽

Vol 22 (2) ◽

pp. 413-429 ◽

Cited By ~ 22

Author(s):

Åke Björck ◽

P. Heggernes ◽

P. Matstoms

Keyword(s):

Least Squares ◽

Large Scale ◽

Total Least Squares ◽

Least Squares Problems

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