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A joint bidiagonalization based iterative algorithm for large scale general-form Tikhonov regularization. (English) Zbl 1452.65076
Summary: Based on the joint bidiagonalization (JBD) process of the matrix pair \(\{A, L\}\), an iterative regularization algorithm, called JBDQR, is proposed and developed for large scale linear discrete ill-posed problems in general-form Tikhonov regularization. It is proved that the JBDQR iterates take the form of attractive filtered generalized singular value decomposition (GSVD) expansions, where the filters are given explicitly and insightful. This result and a detailed analysis on it show that JBDQR must have the desired semi-convergence property, where the iteration number \(k\) plays the role of the regularization parameter. Embedded with the L-curve criterion or the discrepancy principle that is used to estimate the optimal \(k^\ast\) at which the semi-convergence occurs, JBDQR can compute a satisfying good regularized solution. JBDQR is theoretically solid and effective, and it is simple to implement. Numerical experiments confirm our results and the robustness of JBDQR.

MSC:
65F22 Ill-posedness and regularization problems in numerical linear algebra
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
15A18 Eigenvalues, singular values, and eigenvectors
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