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Old and new parameter choice rules for discrete ill-posed problems

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Abstract

Linear discrete ill-posed problems are difficult to solve numerically because their solution is very sensitive to perturbations, which may stem from errors in the data and from round-off errors introduced during the solution process. The computation of a meaningful approximate solution requires that the given problem be replaced by a nearby problem that is less sensitive to disturbances. This replacement is known as regularization. A regularization parameter determines how much the regularized problem differs from the original one. The proper choice of this parameter is important for the quality of the computed solution. This paper studies the performance of known and new approaches to choosing a suitable value of the regularization parameter for the truncated singular value decomposition method and for the LSQR iterative Krylov subspace method in the situation when no accurate estimate of the norm of the error in the data is available. The regularization parameter choice rules considered include several L-curve methods, Regińska’s method and a modification thereof, extrapolation methods, the quasi-optimality criterion, rules designed for use with LSQR, as well as hybrid methods.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Kent State University, Kent, OH, 44242, USA

    Lothar Reichel

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

    Giuseppe Rodriguez

Authors
  1. Lothar Reichel
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  2. Giuseppe Rodriguez
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Corresponding author

Correspondence to Giuseppe Rodriguez.

Additional information

Dedicated to Claude Brezinski and Sebastiano Seatzu on the occasion of their 70th birthdays.

Research of L. Reichel was supported in part by NSF grant DMS-1115385 while research of G. Rodriguez was supported in part by MIUR-PRIN grant no. 20083KLJEZ-003.

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Reichel, L., Rodriguez, G. Old and new parameter choice rules for discrete ill-posed problems. Numer Algor 63, 65–87 (2013). https://doi.org/10.1007/s11075-012-9612-8

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  • DOI: https://doi.org/10.1007/s11075-012-9612-8

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