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The triangle method for finding the corner of the L-curve. (English) Zbl 1014.65022
Summary: The conjugate gradient method (CG) has an intrinsic regularization property when applied to systems of linear equations with ill-conditioned matrices. This regularization property is specially useful when either the right-hand side or the coefficient matrix, or both are given with errors. The regularization parameter is the iteration number, and in order to find this parameter, the L-curve is used.
Here we present a novel method to find the corner of the L-curve, that determines the regularizing iteration number. Numerical results on the collection of test problems of P. C. Hansen [SIAM J. Sci. Comput. 16, No. 2, 506-512 (1995; Zbl 0820.65020)] are given to illustrate the potentiality of the method.

MSC:
65F10 Iterative numerical methods for linear systems
65F30 Other matrix algorithms (MSC2010)
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[1] Bjorck, A.; Elfving, T.; Strakos, Z., Stability of conjugate gradient and Lanczos methods for linear least squares problems, SIAM J. matrix anal. appl., 19, 3, 720-736, (1998) · Zbl 0924.65035
[2] Calvetti, D.; Golub, G.H.; Reichel, L., Estimation of the L-curve via Lanczos bidiagonalization, Bit, 39, 603-619, (1999) · Zbl 0945.65044
[3] D. Calvetti, P.C. Hansen, L. Reichel, L-curve curvature bounds via Lanczos bidiagonalization, Report IMM-2001-5, Technical University of Denmark, May 2001 · Zbl 1029.65041
[4] Engl, H.; Hanke, M.; Neubauer, A., Regularization of inverse problems, (1996), Kluwer Dordrecht · Zbl 0859.65054
[5] Golub, G.H.; Van-Loan, C.F., Matrix computations, (1996), The Johns Hopkins Univ. Press Baltimore, MD · Zbl 0865.65009
[6] Guerra, V.; Hernandez, V., Numerical aspects in locating the corner of the L-curve, (), 121-131 · Zbl 0989.65053
[7] Hanke, M., Conjugate gradient type methods for ill-posed problems, Research notes in mathematics series, (1995), CRC Press Boca Raton, FL
[8] Hansen, P.C., Regularization tools: A MATLAB package for analysis and solution of discrete ill-posed problems, Numer. algorithms, 6, 1-35, (1994) · Zbl 0789.65029
[9] Hansen, P.C., Test matrices for regularization methods, SIAM J. sci. comput., 16, 506-512, (1995) · Zbl 0820.65020
[10] Hansen, P.C., Rank-deficient and discrete ill-posed problems, (1998), SIAM Philadelphia, PA
[11] Hansen, P.C.; O’Leary, D., The use of the L-curve in the regularization of discrete ill-posed problems, SIAM J. sci. comput., 14, 6, 1487-1503, (1993) · Zbl 0789.65030
[12] Kaufman, L.; Neumaier, A., PET regularization by envelope guided conjugate gradients, IEEE trans. med. imag., 15, (1996)
[13] Lawson, C.L.; Hanson, R.J., Solving least squares problems, (1974), Prentice-Hall Englewood Cliffs, NJ, reprinted by SIAM, Philadelphia, PA, 1995 · Zbl 0185.40701
[14] Miller, K., Least squares methods for ill-posed problems with a prescribed bound, SIAM J. math. anal., 1, 52-74, (1970) · Zbl 0214.14804
[15] Natterer, F., The mathematics of computerized tomography, (1986), Wiley Chichester · Zbl 0617.92001
[16] Nemirovskii, A.S., The regularization properties of the adjoint gradient method in ill-posed problems, USSR comput. math. phys., 26, 2, 332-347, (1986)
[17] Tikhonov, A.N., Solution of incorrectly formulated problems and the regularization method, Soviet math. dokl., 4, 1035-1038, (1963), English translation of Dokl. Akad. Nauk SSSR 151 (1963) 501-504 · Zbl 0141.11001
[18] Van der Sluis, A.; Van der Vorst, H.A., The rate of convergence of conjugate gradients, Numer. math., 48, 543-560, (1986) · Zbl 0596.65015
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