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Iterative regularization with minimum-residual methods. (English) Zbl 1113.65037
Summary: We study the regularization properties of iterative minimum-residual methods applied to discrete ill-posed problems. In these methods, the projection onto the underlying Krylov subspace acts as a regularizer, and the emphasis of this work is on the role played by the basis vectors of these Krylov subspaces. We provide a combination of theory and numerical examples, and our analysis confirms the experience that MINRES and MR-II can work as general regularization methods. We also demonstrate theoretically and experimentally that the same is not true, in general, for GMRES and RRGMRES – their success as regularization methods is highly problem dependent.

MSC:
65F22 Ill-posedness and regularization problems in numerical linear algebra
65F10 Iterative numerical methods for linear systems
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[1] P. N. Brown and H. F. Walker, GMRES on (nearly) singular systems, SIAM J. Matrix Anal. Appl., 18 (1997), pp. 37–51. · Zbl 0876.65019 · doi:10.1137/S0895479894262339
[2] D. Calvetti, G. Landi, L. Reichel, and F. Sgallari, Non-negativity and iterative methods for ill-posed problems, Inverse Probl., 20 (2004), pp. 1747–1758. · Zbl 1077.65041 · doi:10.1088/0266-5611/20/6/003
[3] D. Calvetti, B. Lewis, and L. Reichel, GMRES-type methods for inconsistent systems, Linear Algebra Appl., 316 (2000), pp. 157–169. · Zbl 0963.65042 · doi:10.1016/S0024-3795(00)00064-1
[4] D. Calvetti, B. Lewis, and L. Reichel, GMRES, L-curves, and discrete ill-posed problems, BIT, 42 (2002), pp. 44–65. · Zbl 1003.65040 · doi:10.1023/A:1021918118380
[5] D. Calvetti, B. Lewis, and L. Reichel, On the regularizing properties of the GMRES method, Numer. Math., 91 (2002), pp. 605–625. · Zbl 1022.65044 · doi:10.1007/s002110100339
[6] B. Fischer, Polynomial Based Iteration Methods for Symmetric Linear Systems, Wiley Teubner, Stuttgart, 1996. · Zbl 0852.65031
[7] B. Fischer, M. Hanke, and M. Hochbruck, A note on conjugate-gradient type methods for indefinite and/or inconsistent linear systems, Numer. Algorithms, 11 (1996), pp. 181–187. · Zbl 0847.65018 · doi:10.1007/BF02142495
[8] M. Hanke, Conjugate Gradient Type Methods for Ill-Posed Problems, Longman Scientific & Technical, Essex, 1995. · Zbl 0830.65043
[9] M. Hanke, On Lanczos based methods for the regularization of discrete ill-posed problems, BIT, 41 (2001), pp. 1008–1018. · doi:10.1023/A:1021941328858
[10] M. Hanke and J. G. Nagy, Restoration of atmospherically blurred images by symmetric indefinite conjugate gradient techniques, Inverse Probl., 12 (1996), pp. 157–173. · Zbl 0859.65141 · doi:10.1088/0266-5611/12/2/004
[11] P. C. Hansen, Regularization Tools. A Matlab package for analysis and solution of discrete ill-posed problems, Numer. Algorithms, 6 (2004), pp. 1–35. · Zbl 0789.65029 · doi:10.1007/BF02149761
[12] P. C. Hansen and T. K. Jensen, Smoothing-norm preconditioning for regularizing minimum-residual methods, SIAM J. Matrix Anal. Appl., 29 (2006), pp 1–14. · Zbl 1154.65028
[13] P. C. Hansen, J. G. Nagy, and D. P. O’Leary, Deblurring Images: Matrices, Spectra, and Filtering, Fundamentals of Algorithms 3, SIAM, Philadephia, 2006.
[14] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge 1985. · Zbl 0576.15001
[15] M. E. Kilmer, On the regularizing properties of Krylov subspace methods, unpublished; results presented at BIT 40th Anniversary meeting, Lund, Sweden, 2000.
[16] M. E. Kilmer and G. W. Stewart, Iterative regularization and MINRES, SIAM J. Matrix Anal. Appl., 21 (1999), pp. 613–628. · Zbl 0951.65037 · doi:10.1137/S0895479898348623
[17] C. C. Paige and M. A. Saunders, Solution of sparse indefinite systems of linear equations, SIAM J. Numer. Anal., 12 (1975), pp. 617–629. · Zbl 0319.65025 · doi:10.1137/0712047
[18] Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7 (1986), pp. 856–869. · Zbl 0599.65018 · doi:10.1137/0907058
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