1,214 research outputs found
Deflated Iterative Methods for Linear Equations with Multiple Right-Hand Sides
A new approach is discussed for solving large nonsymmetric systems of linear
equations with multiple right-hand sides. The first system is solved with a
deflated GMRES method that generates eigenvector information at the same time
that the linear equations are solved. Subsequent systems are solved by
combining restarted GMRES with a projection over the previously determined
eigenvectors. This approach offers an alternative to block methods, and it can
also be combined with a block method. It is useful when there are a limited
number of small eigenvalues that slow the convergence. An example is given
showing significant improvement for a problem from quantum chromodynamics. The
second and subsequent right-hand sides are solved much quicker than without the
deflation. This new approach is relatively simple to implement and is very
efficient compared to other deflation methods.Comment: 13 pages, 5 figure
Deflation of Eigenvalues for Iterative Methods in Lattice QCD
Work on generalizing the deflated, restarted GMRES algorithm, useful in
lattice studies using stochastic noise methods, is reported. We first show how
the multi-mass extension of deflated GMRES can be implemented. We then give a
deflated GMRES method that can be used on multiple right-hand sides of
in an efficient manner. We also discuss and give numerical results on the
possibilty of combining deflated GMRES for the first right hand side with a
deflated BiCGStab algorithm for the subsequent right hand sides.Comment: Lattice2003(machine
Reverse engineering small 4-manifolds
We introduce a general procedure called `reverse engineering' that can be
used to construct infinite families of smooth 4-manifolds in a given
homeomorphism type. As one of the applications of this technique, we produce an
infinite family of pairwise nondiffeomorphic 4-manifolds homeomorphic to
CP^2#3(-CP^2).Comment: 13 pages, 2 figures. This is the final version published in AGT,
volume 7 (2007), pp. 2103-2116
On restarting the Arnoldi method for large nonsymmetric eigenvalue problems
Abstract. The Arnoldi method computes eigenvalues of large nonsymmetric matrices. Restarting is generally needed to reduce storage requirements and orthogonalization costs. However, restarting slows down the convergence and makes the choice of the new starting vector difficult if several eigenvalues are desired. We analyze several approaches to restarting and show why Sorensen’s implicit QR approach is generally far superior to the others. Ritz vectors are combined in precisely the right way for an effective new starting vector. Also, a new method for restarting Arnoldi is presented. It is mathematically equivalent to the Sorensen approach but has additional uses. 1
Deflated BiCGStab for linear equations in QCD problems
The large systems of complex linear equations that are generated in QCD
problems often have multiple right-hand sides (for multiple sources) and
multiple shifts (for multiple masses). Deflated GMRES methods have previously
been developed for solving multiple right-hand sides. Eigenvectors are
generated during solution of the first right-hand side and used to speed up
convergence for the other right-hand sides. Here we discuss deflating
non-restarted methods such as BiCGStab. For effective deflation, both left and
right eigenvectors are needed. Fortunately, with the Wilson matrix, left
eigenvectors can be derived from the right eigenvectors. We demonstrate for
difficult problems with kappa near kappa_c that deflating eigenvalues can
significantly improve BiCGStab. We also will look at improving solution of
twisted mass problems with multiple shifts. Projecting over previous solutions
is an easy way to reduce the work needed.Comment: 7 pages, 4 figures, presented at the XXV International Symposium on
Lattice Field Theory, 30 July - 4 August 2007, Regensburg, German
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