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Error estimates for large-scale ill-posed problems

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Abstract

The computation of an approximate solution of linear discrete ill-posed problems with contaminated data is delicate due to the possibility of severe error propagation. Tikhonov regularization seeks to reduce the sensitivity of the computed solution to errors in the data by replacing the given ill-posed problem by a nearby problem, whose solution is less sensitive to perturbation. This regularization method requires that a suitable value of the regularization parameter be chosen. Recently, Brezinski et al. (Numer Algorithms 49, 2008) described new approaches to estimate the error in approximate solutions of linear systems of equations and applied these estimates to determine a suitable value of the regularization parameter in Tikhonov regularization when the approximate solution is computed with the aid of the singular value decomposition. This paper discusses applications of these and related error estimates to the solution of large-scale ill-posed problems when approximate solutions are computed by Tikhonov regularization based on partial Lanczos bidiagonalization of the matrix. The connection between partial Lanczos bidiagonalization and Gauss quadrature is utilized to determine inexpensive bounds for a family of error estimates.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Kent State University, Kent, 9244, OH, USA

    L. Reichel

  2. Dipartimento di Matematica e Informatica, Università di Cagliari, Viale Merello 92, 09123, Cagliari, Italy

    G. Rodriguez & S. Seatzu

Authors
  1. L. Reichel
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  2. G. Rodriguez
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  3. S. Seatzu
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Corresponding author

Correspondence to L. Reichel.

Additional information

In memory of Gene H. Golub.

This work was supported by MIUR under the PRIN grant no. 2006017542-003 and by the University of Cagliari.

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Reichel, L., Rodriguez, G. & Seatzu, S. Error estimates for large-scale ill-posed problems. Numer Algor 51, 341–361 (2009). https://doi.org/10.1007/s11075-008-9244-1

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