128 research outputs found
Augmented Implicitly Restarted Lanczos Bidiagonalization Methods
New restarted Lanczos bidiagonalization methods for the computation of a few of the largest or smallest singular values of a large matrix are presented. Restarting is carried out by augmentation of Krylov subspaces that arise naturally in the standard Lanczos bidiagonalization method. The augmenting vectors are associated with certain Ritz or harmonic Ritz vectors. Computed examples show the new methods to be competitive with available schemes
Deflation for the off-diagonal block in symmetric saddle point systems
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 like MINRES. Numerical experiments confirm the merits of our strategy
and lead to interesting questions about using approximate vectors for
deflation.Comment: 26 pages, 12 figure
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