TY - JOUR

T1 - On the block Lanczos and block Golub-Kahan reduction methods applied to discrete ill-posed problems

AU - Alqahtani, Abdulaziz

AU - Gazzola, Silvia

AU - Reichel, Lothar

AU - Rodriguez, Giuseppe

N1 - 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?, INdAM-GNCS research project 2019-2020; National Science Foundation, DMS-1720259; DMS-1729509; 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 DMS-1720259 and DMS-1729509. The work of GR was partially supported by the Fondazione di Sardegna 2017 research project ?Algorithms for Approximation with Applications [Acube],? the INdAM-GNCS 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.

PY - 2021/10/31

Y1 - 2021/10/31

N2 - 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.

AB - 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.

KW - Golub–Kahan block bidiagonalization

KW - Tikhonov regularization

KW - large-scale discrete ill-posed problem

KW - symmetric Lanczos block tridiagonalization

UR - http://www.scopus.com/inward/record.url?scp=85103192338&partnerID=8YFLogxK

U2 - 10.1002/nla.2376

DO - 10.1002/nla.2376

M3 - Article

VL - 28

JO - Numerical Linear Algebra with Applications

JF - Numerical Linear Algebra with Applications

SN - 1070-5325

IS - 5

M1 - e2376

ER -