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Register a new userApplication of block Krylov subspace algorithms to the Wilson-Dirac equation with multiple right-hand sides in lattice QCD
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It is well known that the block Krylov subspace solvers work efficiently for some cases of the solution of differential equations with multiple right-hand sides. In lattice QCD calculation of physical quantities on a given configuration demands us to solve the Dirac equation with multiple sources. We show that a new block Krylov subspace algorithm recently proposed by the authors reduces the computational cost significantly without loosing numerical accuracy for the solution of the O(a)-improved Wilson-Dirac equation.
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There exist two major problems in application of the conventional block BiCGSTAB method to the O(a)-improved Wilson-Dirac equation with multiple right-hand-sides: One is the deviation between the true and the recursive residuals. The other is the convergence failure observed at smaller quark masses for enlarged number of the right-hand-sides. The block BiCGGR algorithm which was recently proposed by the authors succeeds in solving the former problem. In this article we show that a preconditioning technique allows us to improve the convergence behavior for increasing number of the right-hand-sides.
Many Krylov subspace methods for shifted linear systems take advantage of the invariance of the Krylov subspace under a shift of the matrix. However, exploiting this fact in the non-Hermitian case introduces restrictions; e.g., initial residuals must be collinear and this collinearity must be maintained at restart. Thus we cannot simultaneously solve shifted systems with unrelated right-hand sides using this strategy, and all shifted residuals cannot be simultaneously minimized over a Krylov subspace such that collinearity is maintained. It has been shown that this renders them generally incompatible with techniques of subspace recycling [Soodhalter et al. APNUM 14]. This problem, however, can be overcome. By interpreting a family of shifted systems as one Sylvester equation, we can take advantage of the known shift invariance of the Krylov subspace generated by the Sylvester operator. Thus we can simultaneously solve all systems over one block Krylov subspace using FOM or GMRES type methods, even when they have unrelated right-hand sides. Because residual collinearity is no longer a requirement at restart, these methods are fully compatible with subspace recycling techniques. Furthermore, we realize the benefits of block sparse matrix operations which arise in the context of high-performance computing applications. In this paper, we discuss exploiting this Sylvester equation point of view which has yielded methods for shifted systems which are compatible with unrelated right-hand sides. From this, we propose a recycled GMRES method for simultaneous solution of shifted systems.Numerical experiments demonstrate the effectiveness of the methods.
We consider a quadrature-based eigensolver to find eigenpairs of Hermitian matrices arising in lattice quantum chromodynamics. To reduce the computational cost for finding eigenpairs of such Hermitian matrices, we propose a new technique for solving shifted linear systems with complex shifts by means of the shifted CG method. Furthermore using integration paths along horizontal lines corresponding to the real axis of the complex plane, the number of iterations for the shifted CG method is also reduced. Some numerical experiments illustrate the accuracy and efficiency of the proposed method by comparison with a conventional method.
We apply chiral perturbation theory to the pseudoscalar meson mass and decay constant data obtained in the PACS-CS Project toward 2+1 flavor lattice QCD simulations with the O(a)-improved Wilson quarks. We examine the existence of chiral logarithms in the quark mass range from m_{ud}=47 MeV down to 6 MeV on a (2.8 fm)^3 box with the lattice spacing a=0.09 fm. Several low energy constants are determined. We also discuss the magnitude of finite size effects based on chiral perturbation theory.
As computing resources are limited, choosing the parameters for a full Lattice QCD simulation always amounts to a compromise between the competing objectives of a lattice spacing as small, quarks as light, and a volume as large as possible. Aiming to push unquenched simulations with the Wilson action towards the computationally expensive regime of small quark masses we address the question whether one can possibly save computing time by extrapolating results from small lattices to the infinite volume, prior to the usual chiral and continuum extrapolations. In the present work the systematic volume dependence of simulated pion and nucleon masses is investigated and compared with a long-standing analytic formula by Luescher and with results from Chiral Perturbation Theory. We analyze data from Hybrid Monte Carlo simulations with the standard (unimproved) two-flavor Wilson action at two different lattice spacings of a=0.08fm and 0.13fm. The quark masses considered correspond to approximately 85 and 50% (at the smaller a) and 36% (at the larger a) of the strange quark mass. At each quark mass we study at least three different lattices with L/a=10 to 24 sites in the spatial directions (L=0.85-2.08fm).
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