11 research outputs found
A nested Krylov subspace method to compute the sign function of large complex matrices
We present an acceleration of the well-established Krylov-Ritz methods to
compute the sign function of large complex matrices, as needed in lattice QCD
simulations involving the overlap Dirac operator at both zero and nonzero
baryon density. Krylov-Ritz methods approximate the sign function using a
projection on a Krylov subspace. To achieve a high accuracy this subspace must
be taken quite large, which makes the method too costly. The new idea is to
make a further projection on an even smaller, nested Krylov subspace. If
additionally an intermediate preconditioning step is applied, this projection
can be performed without affecting the accuracy of the approximation, and a
substantial gain in efficiency is achieved for both Hermitian and non-Hermitian
matrices. The numerical efficiency of the method is demonstrated on lattice
configurations of sizes ranging from 4^4 to 10^4, and the new results are
compared with those obtained with rational approximation methods.Comment: 17 pages, 12 figures, minor corrections, extended analysis of the
preconditioning ste
Short-recurrence Krylov subspace methods for the overlap Dirac operator at nonzero chemical potential
The overlap operator in lattice QCD requires the computation of the sign
function of a matrix, which is non-Hermitian in the presence of a quark
chemical potential. In previous work we introduced an Arnoldi-based Krylov
subspace approximation, which uses long recurrences. Even after the deflation
of critical eigenvalues, the low efficiency of the method restricts its
application to small lattices. Here we propose new short-recurrence methods
which strongly enhance the efficiency of the computational method. Using
rational approximations to the sign function we introduce two variants, based
on the restarted Arnoldi process and on the two-sided Lanczos method,
respectively, which become very efficient when combined with multishift
solvers. Alternatively, in the variant based on the two-sided Lanczos method
the sign function can be evaluated directly. We present numerical results which
compare the efficiencies of a restarted Arnoldi-based method and the direct
two-sided Lanczos approximation for various lattice sizes. We also show that
our new methods gain substantially when combined with deflation.Comment: 14 pages, 4 figures; as published in Comput. Phys. Commun., modified
data in Figs. 2,3 and 4 for improved implementation of FOM algorithm,
extended discussion of the algorithmic cos
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