On the Lanczos and Golub-Kahan reduction methods applied to discrete ill-posed problems

Silvia Gazzola, Enyinda Onunwor, Lothar Reichel, Giuseppe Rodriguez

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

The symmetric Lanczos method is commonly applied to reduce large-scale symmetric linear discrete ill-posed problems to small ones with a symmetric tridiagonal matrix. We investigate how quickly the nonnegative subdiagonal entries of this matrix decay to zero. Their fast decay to zero suggests that there is little benefit in expressing the solution of the discrete ill-posed problems in terms of the eigenvectors of the matrix compared with using a basis of Lanczos vectors, which are cheaper to compute. Similarly, we show that the solution subspace determined by the LSQR method when applied to the solution of linear discrete ill-posed problems with a nonsymmetric matrix often can be used instead of the solution subspace determined by the singular value decomposition without significant, if any, reduction of the quality of the computed solution.

Original languageEnglish
Pages (from-to)187-204
Number of pages18
JournalNumerical Linear Algebra with Applications
Volume23
Issue number1
Early online date12 Oct 2015
DOIs
Publication statusPublished - 2016

Keywords

  • Discrete ill-posed problems
  • Golub-Kahan bidiagonalization
  • Lanczos decomposition
  • LSQR
  • TSVD

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Applied Mathematics

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