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Register a new userApplication of preconditioned block BiCGGR to the Wilson-Dirac equation with multiple right-hand sides in lattice QCD
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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.
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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.
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 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).
We present our result for the $Ktopipi$ decay amplitudes for both the $Delta I=1/2$ and $3/2$ processes with the improved Wilson fermion action. Expanding on the earlier works by Bernard {it et al.} and by Donini {it et al.}, we show that mixings with four-fermion operators with wrong chirality are absent even for the Wilson fermion action for the parity odd process in both channels due to CPS symmetry. Therefore, after subtraction of an effect from the lower dimensional operator, a calculation of the decay amplitudes is possible without complications from operators with wrong chirality, as for the case with chirally symmetric lattice actions. As a first step to verify the possibility of calculations with the Wilson fermion action, we consider the decay amplitudes at an unphysical quark mass $m_K sim 2 m_pi$. Our calculations are carried out with $N_f=2+1$ gauge configurations generated with the Iwasaki gauge action and nonperturbatively $O(a)$-improved Wilson fermion action at $a=0.091,{rm fm}$, $m_pi=280,{rm MeV}$, and $m_K=580,{rm MeV}$ on a $32^3times 64$ ($La=2.9,{rm fm}$) lattice. For the quark loops in the penguin and disconnected contributions in the $I=0$ channel, the combined hopping parameter expansion and truncated solver method work very well for variance reduction. We obtain, for the first time with a Wilson-type fermion action, that ${rm Re}A_0 = 60(36) times10^{ -8},{rm GeV}$ and ${rm Im}A_0 =-67(56) times10^{-12},{rm GeV}$ for a matching scale $q^* =1/a$. The dependence on the matching scale $q^*$ for these values is weak.
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