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Abstract
The reduction of a large‐scale symmetric linear discrete ill‐posed problem with multiple right‐hand sides to a smaller problem with a symmetric block tridiagonal matrix can easily be carried out by the application of a small number of steps of the symmetric block Lanczos method. We show that the subdiagonal blocks of the reduced problem converge to zero fairly rapidly with increasing block number. This quick convergence indicates that there is little advantage in expressing the solutions of discrete ill‐posed problems in terms of eigenvectors of the coefficient matrix when compared with using a basis of block Lanczos vectors, which are simpler and cheaper to compute. Similarly, for nonsymmetric linear discrete ill‐posed problems with multiple right‐hand sides, we show that the solution subspace defined by a few steps of the block Golub–Kahan bidiagonalization method usually can be applied instead of the solution subspace determined by the singular value decomposition of the coefficient matrix without significant, if any, reduction of the quality of the computed solution.
Original language  English 

Article number  e2376 
Journal  Numerical Linear Algebra with Applications 
Volume  28 
Issue number  5 
Early online date  27 Mar 2021 
DOIs  
Publication status  Published  31 Oct 2021 
Bibliographical note
Funding Information:information Engineering and Physical Sciences Research Council, EP/T001593/1; Fondazione di Sardegna, Algorithms for Approximation with Applications; Istituto Nazionale di Alta Matematica ?Francesco Severi?, INdAMGNCS research project 20192020; National Science Foundation, DMS1720259; DMS1729509; Regione Autonoma della Sardegna, RASSR57257 [AMIS]The authors would like to thank the two anonymous referees for their insightful comments that lead to improvements of the presentation. The work of SG was partially supported by EPSRC, under grant EP/T001593/1. Work by LR was supported in part by NSF grants DMS1720259 and DMS1729509. The work of GR was partially supported by the Fondazione di Sardegna 2017 research project ?Algorithms for Approximation with Applications [Acube],? the INdAMGNCS research project ?Tecniche numeriche per l'analisi delle reti complesse e lo studio dei problemi inversi,? and the Regione Autonoma della Sardegna research project ?Algorithms and Models for Imaging Science [AMIS]? (RASSR57257, intervento finanziato con risorse FSC 2014?2020  Patto per lo Sviluppo della Regione Sardegna). This study does not have any conflicts to disclose.
Publisher Copyright:
© 2021 The Authors. Numerical Linear Algebra with Applications published by John Wiley & Sons Ltd.
Keywords
 Golub–Kahan block bidiagonalization
 Tikhonov regularization
 largescale discrete illposed problem
 symmetric Lanczos block tridiagonalization
ASJC Scopus subject areas
 Algebra and Number Theory
 Applied Mathematics
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 1 Finished

Fast and Flexible Solvers for Inverse Problems
Engineering and Physical Sciences Research Council
15/09/19 → 14/09/22
Project: Research council