Abstract.

Deflation techniques are typically used to shift isolated clusters of small eigenvalues in order to obtain a tighter distribution and a smaller condition number. Such changes induce a positive effect in the convergence behavior of Krylov subspace methods, which are among the most popular iterative solvers for large sparse linear systems. We develop a deflation strategy for symmetric saddle point matrices by taking advantage of their underlying block structure. The vectors used for deflation come from an elliptic singular value decomposition relying on the generalized Golub–Kahan bidiagonalization process. The block targeted by deflation is the off-diagonal one since it features a problematic singular value distribution for certain applications. One example is the Stokes flow in elongated channels, where the off-diagonal block has several small, isolated singular values, depending on the length of the channel. Applying deflation to specific parts of the saddle point system is important when using solvers such as CRAIG, which operates on individual blocks rather than the whole system. The theory is developed by extending the existing framework for deflating square matrices before applying a Krylov subspace method such as MINRES. Numerical experiments confirm the merits of our strategy and lead to interesting questions about using approximate vectors for deflation.

Keywords

  1. Golub–Kahan bidiagonalization
  2. eigenvalue deflation
  3. singular value decomposition
  4. saddle point problems
  5. Stokes equation

MSC codes

  1. 15A18
  2. 35P15
  3. 65F10
  4. 65F15
  5. 65N22

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Information & Authors

Information

Published In

cover image SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Matrix Analysis and Applications
Pages: 203 - 231
ISSN (online): 1095-7162

History

Submitted: 6 December 2022
Accepted: 2 October 2023
Published online: 17 January 2024

Keywords

  1. Golub–Kahan bidiagonalization
  2. eigenvalue deflation
  3. singular value decomposition
  4. saddle point problems
  5. Stokes equation

MSC codes

  1. 15A18
  2. 35P15
  3. 65F10
  4. 65F15
  5. 65N22

Authors

Affiliations

Chair for Computer Science 10 - System Simulation, Friedrich-Alexander University Erlangen-Nuremberg, 91058, Erlangen, Germany.
CERFACS, 31057, Toulouse, France.
Ulrich Rüde
Chair for Computer Science 10 - System Simulation, Friedrich-Alexander University Erlangen-Nuremberg, 91058, Erlangen, Germany.

Funding Information

Bavarian Academic Center for Central, Eastern and Southeastern Europe (BAYHOST)
Funding: This research was supported by the Bavarian Academic Center for Central, Eastern and Southeastern Europe (BAYHOST).

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