The structure of iterative methods for symmetric linear discrete ill-posed problems

Abstract

The iterative solution of large linear discrete ill-posed problems with an error contaminated data vector requires the use of specially designed methods in order to avoid severe error propagation. Range restricted minimal residual methods have been found to be well suited for the solution of many such problems. This paper discusses the structure of matrices that arise in a range restricted minimal residual method for the solution of large linear discrete ill-posed problems with a symmetric matrix. The exploitation of the structure results in a method that is competitive with respect to computer storage, number of iterations, and accuracy.

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Notes

  1. 1.

    Algorithm 3.1 refers to the algorithm in Section 3 unless explicitly stated otherwise.

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Acknowledgments

We would like to thank the referees for comments. The work of F. M. was supported by Dirección General de Investigación Científica y Técnica, Ministerio de Economía y Competitividad of Spain under grant MTM2012-36732-C03-01. Work of L. R. was supported by Universidad Carlos III de Madrid in the Department of Mathematics during the academic year 2010-2011 within the framework of the Chair of Excellence Program and by NSF grant DMS-1115385.

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Correspondence to L. Reichel.

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Dedicated to Axel Ruhe on the occasion of his 70th birthday.

Communicated by Peter Benner.

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Dykes, L., Marcellán, F. & Reichel, L. The structure of iterative methods for symmetric linear discrete ill-posed problems. Bit Numer Math 54, 129–145 (2014). https://doi.org/10.1007/s10543-014-0476-2

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Keywords

  • Ill-posed problem
  • Iterative method
  • Truncated iteration

Mathematics Subject Classification (2000)

  • 65F10
  • 65F22