Computational Mathematics and Computer Modeling with Applications (CMCMA)

Regularization properties of range restricted LSQR method for solving large-scale linear discrete ill-posed problems

https://doi.org/10.48308/CMCMA.3.1.21
Volume 3, Issue 1 - Serial Number 1
December 2024
Pages 21-37
Computational Mathematics and Computer Modeling with Applications (CMCMA)

Document Type : Invited paper

Authors

Hui Zhang 1
Hua Dai 2

1 Department of Basic Courses, Jiangsu Police Institute, Nanjing 210031, P.R. China

2 School of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, P.R. China

Abstract
The LSQR iterative method is one of the most popular methods for solving large-scale linear discrete ill-posed problem $Ax=b$ with an error-contaminated right-hand side.
In this paper, we consider the regularization properties of range restricted LSQR (RRLSQR) method. The iteration number $k$ always acts as the regularization parameter because of the semi-convergence. In order to verify whether or not the RRLSQR method finds a 2-norm filtering best regularization solution for severely, moderately and mildly ill-posed problems, we present the $sin \Theta$ theorems for the 2-norm distances between the $k$ dimensional left and right Krylov subspaces generated by Lanczos bidiagonalization and the $k$ dimensional dominant left and right singular subspaces of $A$, and estimate the distances for the three kinds problems assuming that the singular values are simple, and develop a regularized RRLSQR method for solving linear discrete ill-posed problems. Numerical experiments confirm our theoretical results and show the efficiency of the proposed method. 

Keywords

Linear discrete ill-posed problem
semi-convergence
range restricted LSQR method
regularization property
  • Receive Date 12 August 2024
  • Revise Date 09 September 2024
  • Accept Date 15 September 2024

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