# zbMATH — the first resource for mathematics

Iterative methods for ill-posed problems and semiconvergent sequences. (English) Zbl 1092.65025
Summary: Iterative schemes, such as LSQR and RRGMRES, are among the most efficient methods for the solution of large-scale ill-posed problems. The iterates generated by these methods form semiconvergent sequences. A meaningful approximation of the desired solution of an ill-posed problem often can be obtained by choosing a suitable member of this sequence. However, it is not always a simple matter to decide which member to choose. Semiconvergent sequences also arise when approximating integrals by asymptotic expansions, and considerable experience and analysis of how to choose a suitable member of a semiconvergent sequence in this context are available. The present note explores how the guidelines developed within the context of asymptotic expansions can be applied to iterative methods for ill-posed problems.

##### MSC:
 65F10 Iterative numerical methods for linear systems 65F22 Ill-posedness and regularization problems in numerical linear algebra
##### Software:
Regularization tools
Full Text:
##### References:
 [1] Björck, Å., Numerical methods for least squares problems, (1996), SIAM Philadelphia · Zbl 0734.65031 [2] Calvetti, D.; Lewis, B.; Reichel, L., On the choice of subspace for iterative methods for linear discrete ill-posed problems, Internat. J. appl. math. comput. sci., 11, 1069-1092, (2001) · Zbl 0994.65043 [3] Calvetti, D.; Reichel, L.; Zhang, Q., Conjugate gradient algorithms for symmetric inconsistent linear systems, (), 267-272 [4] Castellanos, J.L.; Gómez, S.; Guerra, V., The triangle method for finding the corner of the L-curve, Appl. numer. math., 43, 359-373, (2002) · Zbl 1014.65022 [5] Dahlquist, G.; Björck, Å., Numerical methods, (2003), Dover Mineola [6] Ferreira, C.; López, J.L.; Temme, N.M., Asymptotic approximations of integrals: an introduction, with recent developments and applications to orthogonal polynomials, Elec. trans. numer. anal., 19, 58-83, (2005), Available at $$<$$> · Zbl 1122.41018 [7] Hanke, M., Conjugate gradient type methods for ill-posed problem, (1995), Longman Essex · Zbl 0830.65043 [8] Hanke, M.; Hansen, P.C., Regularization methods for large-scale problems, Surv. math. ind., 3, 253-315, (1993) · Zbl 0805.65058 [9] Hansen, P.C., Regularization tools: a Matlab package for analysis and solution of discrete ill-posed problems, Numer. algorithms, 6, 1-35, (1994), Software is available in Netlib at $$<$$> · Zbl 0789.65029 [10] Hansen, P.C., Rank-deficient and discrete ill-posed problems, (1998), SIAM Philadelphia [11] Kaufman, L.; Neumaier, A., Regularization of ill-posed problems by envelope guided conjugate gradients, J. comput. graph. statist., 6, 451-463, (1997) [12] Phillips, D.L., A technique for the numerical solution of certain integral equations of the first kind, J. ACM, 9, 84-97, (1962) · Zbl 0108.29902
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.