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Multilevel Arnoldi-Tikhonov Regularization Methods for Large-Scale Linear Ill-Posed Systems
Abstract:This paper is devoted to the numerical solution of large-scale linear ill-posed systems. A multilevel regularization method is proposed. This method is based on a synthesis of the Arnoldi-Tikhonov regularization technique and the multilevel technique. We show that if the Arnoldi-Tikhonov method is a regularization method, then the multilevel method is also a regularization one. Numerical experiments presented in this paper illustrate the effectiveness of the proposed method.
Digital Object Identifier (DOI): doi.org/10.5281/zenodo.2021701Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 283
 A. S. Al-Fhaid, S. Serra-Capizzano, D. Sesana, and M. Z. Ullah, Singular-value (and eigenvalue) distribution and krylov preconditioning of sequences of sampling matrices approximating integral operators, Numer. Linear Algebra Appl., 21 (2014), pp. 722–743.
 M. L. Baart, The use of auto-correlation for pseudo-rank determination in noisy ill-conditioned least-squares problems, IMA J. Numer. Anal., 2 (1982), pp. 241–247.
 J. Baglama and L. Reichel, Augmented GMRES-type methods, Numer. Linear Algebra Appl., 14 (2007), pp. 337–350.
 A˚ . Bjo¨rck, A bidiagonalization algorithm for solving large and sparse ill-posed systems of linear equations, BIT, 28 (1988), pp. 659–670.
 , Numerical Methods for Least Squares Problems, SIAM, Philadelphia, 1996.
 D. Calvetti, B. Lewis, and L. Reichel, On the choice of subspace for iterative methods for linear discrete ill-posed problems, Int. J. Appl. Math. Comput. Sci., 11 (2001), pp. 1069–1092.
 , On the regularizing properties of the GMRES method, Numer. Math., 123 (2002), pp. 605–625.
 D. Calvetti, S. Morigi, L. Reichel, and F. Sgallari, Tikhonov regularization and the L-curve for large, discrete ill-posed problems, J. Comput. Appl. Math., 123 (2000), pp. 423–446.
 D. Calvetti and L. Reichel, Tikhonov regularization of large linear problems, BIT, 43 (2003), pp. 263–283.
 H. R. Chan and K. Chen, A multilevel algorithm for simultaneously denoising and deblurring images, SIAM J. Sci. Comput., 32 (2010), pp. 1043–1063.
 T. F. Chan and K. R. Jackson, Nonlinearly preconditioned Krylov subspace methods for discrete Newton algorithms, SIAM J. Sci. Statist. Comput., 5 (1984), pp. 533–542.
 Z. Chen, Y. Xu, and H. Yang, A multilevel augmentation method fo solving ill-posed operator equations, SIAM J. Sci. Statist. Comput., 22 (2006), pp. 155–174.
 M. Donatelli and S. Serra-Capizzano, On the regularizing power of multigrid-type algorithms, SIAM J. Sci. Comput., 27 (2006), pp. 2053–2076.
 H. W. Engl, M. Hanke, and A. Neubauer, Regularization of inverse problems, Kluwer, Dordrecht, 1996.
 M. I. Espanol and M. E. Kilmer, Multilevel approach for signal restoration problems with toeplitz matrices, SIAM J. Sci. Comput., 32 (2010), pp. 299–319.
 J. A. Ezquerro and M. A. Hernandez, On a convex acceleration of Newton’s method, J. Optim. Theory Appl., 100 (1999), pp. 311–326.
 W. Gander, On Halley’s iteration method, Amer. Math. Monthly, 92 (1985), pp. 131–134.
 G. H. Golub and C. F. V. Loan, Matrix Computations, John Hopkins University Press, Baltimore, MD, 3rd ed., 1996.
 M. Hanke and C. R. Vogel, Two-level preconditioners for regularized inverse problems I: theory, Numer. Math., 83 (1999), pp. 385–402.
 P. C. Hansen, Rank Deficient and Discrete Ill-posed Problems, SIAM, Philadelphia, PA, 1998.
 M. Jacobsen, P. C. Hansen, and M. A. Saunders, Subspace preconditioned LSQR for discrete ill-posed problems, BIT, 43 (2003), pp. 975–989.
 C. T. Kelley, Solving Nonlinear Equations with Newton’s Method, SIAM, Philadelphia, PA, 2003.
 J. T. King, Multilevel algoritms for ill-posed problems, Numer. Math., 61 (1992), pp. 311–334.
 E. Klann, R. Ramlau, and L. Reichel, Wavelet-based multilevel methods for linear ill-posed problems, BIT, 51 (2011), pp. 669–694.
 D. P. O. Leary and J. A. Simmons, A bidiagonalization-regularization procedure for large-scale discretizations of ill-posed problems, SIAM J. Sci. Statist. Comput., 2 (1981), pp. 474–489.
 B. Lewis and L. Reichel, Arnoldi-Tikhonov regularization methods, J. Comput. Appl. Math., 226 (2009), pp. 92–102.
 Y. Lin, L. Bao, and Y. Cao, Augmented Arnoldi-Tikhonov regularization rethods for solving large-scale linear ill-posed systems, Mathematical Problems in Engineering, 2013 (2013), pp. 1–8.
 S. Morigi, L. Reichel, and F. Sgallari, Cascadic multilevel methods for fast nonsymmetric blur- and noise-removal, Appl. Numer. Math., 60 (2010), pp. 378–396.
 B. N. Parlett, The Symmetric Eigenvalue Problem, SIAM, Philadelphia, PA, 1998.
 D. L. Phillips, A technique for the numerical solution of certain integral equations of the first kind, J. ACM, 9 (1962), pp. 84–97.
 L. Reichel and A. Shyshkov, A new zero-finder for tikhonov regularization, BIT Numer. Math., 48 (2008), pp. 627–643.
 , Cascadic multilevel methods for ill-posed problems, J. Comput. Appl. Math., 233 (2010), pp. 1314–1325.
 L. Reichel and Q. Ye, Breakdown-free GMRES for singular systems, SIAM J. Matrix Anal. Appl., 26 (2005), pp. 1001–1021.
 A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems, Wiley, New York, 1977.
 A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov, and A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems, Kluwe, Dordrecht, 1995.