Iterative methods for ill-posed problems and semiconvergent sequences.

*(English)*Zbl 1092.65025Summary: 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 |

##### Keywords:

iterative method; stopping criterion; L-curve; large-scale ill-posed problems; semiconvergent sequences; asymptotic expansions##### Software:

Regularization tools
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\textit{S. Morigi} et al., J. Comput. Appl. Math. 193, No. 1, 157--167 (2006; Zbl 1092.65025)

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##### 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 |

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