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

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

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 $Ax=b$ 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

Some criteria for determining recognizability of a set

Let an be the number of strings of length n in a set A ⊆ ∑*, where ∑ is a finite alphabet. Several criteria for determining that a set is not recognizable by a finite automaton are given, based solely on the sequence {an}. The sequence {an} is also used to define a finitely addititive probability measure on all recognizable sets