TITLE:
A Novel Modified TSVD Method and Truncated Reorthogonalized Golub-Kahan Bidiagonalization Method for Discrete Ill-Posed Problems
AUTHORS:
Yin Wu, Lei Zhao
KEYWORDS:
Discrete Ill-Posed Problems, Truncated Singular Value Decomposition, Reorthogonalized Golub-Kahan Bidiagonalization, Gram-Schmidt
JOURNAL NAME:
Open Journal of Applied Sciences,
Vol.16 No.2,
February
14,
2026
ABSTRACT: Truncated singular value decomposition (TSVD) and Golub-Kahan diagonalization are two elementary techniques for solving a least squares problem from a linear discrete ill-posed problems. For small to medium sized ill-posed problems, we propose a novel Modified Truncated Singular Value Decomposition (NMTSVD) based on the MTSVD method. This method presents three approaches to select truncation indices, leading to three approximate matrices of the coefficient matrix
A
. And the relationships of these approximate matrices are established in terms of the spectral condition number. For large-scale unstable discrete ill-posed problems, we propose a truncated Reorthogonalized Golub-Kahan Bidiagonalization (RGKB) method by integrating the Golub-Kahan bidiagonalization process, truncation index, and the Gram-Schmidt reorthogonalization procedure. Numerical experiments are presented to illustrate the effectiveness of the methods NMTSVD and RGKB.