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Simple stopping criteria for the LSQR method applied to discrete ill-posed problems

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Abstract

The LSQR iterative method is one of the most popular numerical schemes for computing an approximate solution of large linear discrete ill-posed problems with an error-contaminated right-hand side, which represents available data. It is important to terminate the iterations after a suitable number of steps, because too many steps yield an approximate solution that suffers from a large propagated error due to the error in the data, and too few iterations give an approximate solution that may lack many details that can be of interest. When the error in the right-hand side is white Gaussian and a tight bound on its variance is known, the discrepancy principle typically furnishes a suitable termination criterion for the LSQR iterations. However, in many applications in science and engineering that give rise to large linear discrete ill-posed problems, the variance of the error is not known. This has spurred the development of a variety of stopping rules for assessing when to terminate the iterations in this situation. The present paper proposes new simple stopping rules that are based on comparing the residual errors associated with iterates generated by the LSQR and Craig iterative methods.

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References

  1. Baart, M.L.: The use of auto-correlation for pseudorank determination in noisy ill-conditioned linear least-squares problems. IMA J. Numer. Anal. 2, 241–247 (1982)

    Article  MathSciNet  Google Scholar 

  2. Bauer, F., Kindermann, S.: The quasi-optimality criterion for classical inverse problems. Inverse Problems 24, 035002 (2008). (20pp)

    Article  MathSciNet  Google Scholar 

  3. Bauer, F., Lukas, M.A.: Comparing parameter choice methods for regularization of ill-posed problems. Math. Comput. Simulation 81, 1795–1841 (2011)

    Article  MathSciNet  Google Scholar 

  4. Bouhamidi, A., Jbilou, K., Reichel, L., Sadok, H., Wang, Z.: Vector extrapolation applied to truncated singular value decomposition and truncated iteration. J. Eng. Math. 93, 99–112 (2015)

    Article  MathSciNet  Google Scholar 

  5. Brezinski, C., Redivo Zaglia, M., Rodriguez, G., Seatzu, S.: Extrapolation techniques for ill-conditioned linear systems. Numer. Math. 81, 1–29 (1998)

    Article  MathSciNet  Google Scholar 

  6. Brezinski, C., Rodriguez, G., Seatzu, S.: Error estimates for linear systems with applications to regularization. Numer. Algorithms 49, 85–104 (2008)

    Article  MathSciNet  Google Scholar 

  7. Brezinski, C., Rodriguez, G., Seatzu, S.: Error estimates for the regularization of least squares problems. Numer. Algorithms 51, 61–76 (2009)

    Article  MathSciNet  Google Scholar 

  8. Fenu, C., Reichel, L., Rodriguez, G., Sadok, H.: GCV for Tikhonov regularization by partial SVD. BIT Numer. Math. 57, 1019–1039 (2017)

    Article  MathSciNet  Google Scholar 

  9. Fox, L., Goodwin, E.T.: The numerical solution of non-singular linear integral equations. Philos. Trans. Roy. Soc. London. Ser. A. 245, 501–534 (1953)

    Article  MathSciNet  Google Scholar 

  10. Hanke, M.: On Lanczos based methods for the regularization of discrete ill-posed problems. BIT Numer. Math. 41, 1008–1018 (2001)

    Article  MathSciNet  Google Scholar 

  11. Hansen, P.C.: Analysis of discrete ill-posed problems by means of the L-curve. SIAM Rev. 34, 561–580 (1992)

    Article  MathSciNet  Google Scholar 

  12. Hansen, P.C.: Rank-deficient and discrete ill-Posed problems: Numerical aspects of linear inversion. SIAM, Philadelphia (1998)

    Book  Google Scholar 

  13. Hansen, P.C.: Regularization Tools version 4.0 for Matlab 7.3. Numer. Algorithms 46, 189–194 (2007)

    Article  MathSciNet  Google Scholar 

  14. Hansen, P.C.: Discrete inverse problems: insight and algorithms. SIAM, Philadelphia (2010)

    Book  Google Scholar 

  15. Hochstenbach, M.E., Reichel, L., Rodriguez, G.: Regularization parameter determination for discrete ill-posed problems. J. Comput. Appl. Math. 273, 132–149 (2015)

    Article  MathSciNet  Google Scholar 

  16. Kindermann, S.: Convergence analysis of minimization-based noise level-free parameter choice rules for linear ill-posed problems. Electron. Trans. Numer. Anal. 38, 233–257 (2011)

    MathSciNet  MATH  Google Scholar 

  17. Meurant, G.: Computer solution of large linear systems. Elsevier, Amsterdam (1999)

    MATH  Google Scholar 

  18. Meurant, G.: The Lanczos and conjugate gradient algorithms. SIAM, Philadelphia (2006)

    Book  Google Scholar 

  19. Morigi, S., Reichel, L., Sgallari, F., Zama, F.: Iterative methods for ill-posed problems and semiconvergent sequences. J. Comput. Appl. Math. 193, 157–167 (2006)

    Article  MathSciNet  Google Scholar 

  20. Paige, C.C.: Bidiagonalization of matrices and solutions of linear equations. SIAM J. Numer. Anal. 11, 197–209 (1974)

    Article  MathSciNet  Google Scholar 

  21. Paige, C.C., Saunders, M.A.: LSQR: An algorithm for sparse linear equations and sparse least squares. ACM Trans. Math. Soft. 8, 43–71 (1982)

    Article  MathSciNet  Google Scholar 

  22. Park, Y., Reichel, L., Rodriguez, G., Yu, X.: Parameter determination for Tikhonov regularization problems in general form. J. Comput. Appl. Math. 343, 12–25 (2018)

    Article  MathSciNet  Google Scholar 

  23. Phillips, D.L.: A technique for the numerical solution of certain integral equations of the first kind. J. Assoc. Comput. Mach. 9, 84–97 (1962)

    Article  MathSciNet  Google Scholar 

  24. Regińska, T.: A regularization parameter in discrete ill-posed problems. SIAM J. Sci. Comput. 17, 740–749 (1996)

    Article  MathSciNet  Google Scholar 

  25. Reichel, L., Rodriguez, G.: Old and new parameter choice rules for discrete ill-posed problems. Numer. Algorithms 63, 65–87 (2013)

    Article  MathSciNet  Google Scholar 

  26. Reichel, L., Sadok, H.: A new L-curve for ill-posed problems. J. Comput. Appl. Math. 219, 493–508 (2008)

    Article  MathSciNet  Google Scholar 

  27. Saunders, M.A.: Solution of sparse rectangular systems using LSQR and Craig. BIT Numer. Math. 35, 588–604 (1995)

    Article  MathSciNet  Google Scholar 

  28. Shaw, C.B. Jr.: Improvement of the resolution of an instrument by numerical solution of an integral equation. J. Math. Anal. Appl. 37, 83–112 (1972)

    Article  MathSciNet  Google Scholar 

  29. Wilkinson, J.H.: The algebraic eigenvalue problem. Oxford University Press, New York (1965)

    MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the referees for comments that lead to improvements of the presentation. The first author would like to thank the second author for an enjoyable visit to the Université du Littoral, during which this work was initiated. Research by the first author was supported in part by NSF grants DMS-1720259 and DMS-1729509. Research by the third author was supported by the China Scholarship Council (CSC No. 201806180081).

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Authors and Affiliations

  1. Department of Mathematical Sciences, Kent State University, Kent, OH, 44242, USA

    Lothar Reichel & Wei-Hong Zhang

  2. Laboratoire de Mathématiques Pures et Appliquées, Université du Littoral, Centre Universtaire de la Mi-Voix, Batiment H. Poincarré, 50 Rue F. Buisson, BP 699, 62228,, Calais cedex, France

    Hassane Sadok

  3. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, People’s Republic of China

    Wei-Hong Zhang

Authors
  1. Lothar Reichel
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  2. Hassane Sadok
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  3. Wei-Hong Zhang
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Correspondence to Lothar Reichel.

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Dedicated to Gérard Meurant on the occasion of his 70th birthday.

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Reichel, L., Sadok, H. & Zhang, WH. Simple stopping criteria for the LSQR method applied to discrete ill-posed problems. Numer Algor 84, 1381–1395 (2020). https://doi.org/10.1007/s11075-019-00852-1

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