39,211 research outputs found
Deflated GMRES for Systems with Multiple Shifts and Multiple Right-Hand Sides
We consider solution of multiply shifted systems of nonsymmetric linear
equations, possibly also with multiple right-hand sides. First, for a single
right-hand side, the matrix is shifted by several multiples of the identity.
Such problems arise in a number of applications, including lattice quantum
chromodynamics where the matrices are complex and non-Hermitian. Some Krylov
iterative methods such as GMRES and BiCGStab have been used to solve multiply
shifted systems for about the cost of solving just one system. Restarted GMRES
can be improved by deflating eigenvalues for matrices that have a few small
eigenvalues. We show that a particular deflated method, GMRES-DR, can be
applied to multiply shifted systems. In quantum chromodynamics, it is common to
have multiple right-hand sides with multiple shifts for each right-hand side.
We develop a method that efficiently solves the multiple right-hand sides by
using a deflated version of GMRES and yet keeps costs for all of the multiply
shifted systems close to those for one shift. An example is given showing this
can be extremely effective with a quantum chromodynamics matrix.Comment: 19 pages, 9 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
A flexible and adaptive Simpler GMRES with deflated restarting for shifted linear systems
In this paper, two efficient iterative algorithms based on the simpler GMRES
method are proposed for solving shifted linear systems. To make full use of the
shifted structure, the proposed algorithms utilizing the deflated restarting
strategy and flexible preconditioning can significantly reduce the number of
matrix-vector products and the elapsed CPU time. Numerical experiments are
reported to illustrate the performance and effectiveness of the proposed
algorithms.Comment: 17 pages. 9 Tables, 1 figure; Newly update: add some new numerical
results and correct some typos and syntax error
Restarted Hessenberg method for solving shifted nonsymmetric linear systems
It is known that the restarted full orthogonalization method (FOM)
outperforms the restarted generalized minimum residual (GMRES) method in
several circumstances for solving shifted linear systems when the shifts are
handled simultaneously. Many variants of them have been proposed to enhance
their performance. We show that another restarted method, the restarted
Hessenberg method [M. Heyouni, M\'ethode de Hessenberg G\'en\'eralis\'ee et
Applications, Ph.D. Thesis, Universit\'e des Sciences et Technologies de Lille,
France, 1996] based on Hessenberg procedure, can effectively be employed, which
can provide accelerating convergence rate with respect to the number of
restarts. Theoretical analysis shows that the new residual of shifted restarted
Hessenberg method is still collinear with each other. In these cases where the
proposed algorithm needs less enough CPU time elapsed to converge than the
earlier established restarted shifted FOM, weighted restarted shifted FOM, and
some other popular shifted iterative solvers based on the short-term vector
recurrence, as shown via extensive numerical experiments involving the recent
popular applications of handling the time fractional differential equations.Comment: 19 pages, 7 tables. Some corrections for updating the reference
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