Modern graphics hardware is designed for highly parallel numerical tasks and promises significant cost and performance benefits for many scientific applications. One such application is lattice quantum chromodyamics (lattice QCD), where the main comp
utational challenge is to efficiently solve the discretized Dirac equation in the presence of an SU(3) gauge field. Using NVIDIAs CUDA platform we have implemented a Wilson-Dirac sparse matrix-vector product that performs at up to 40 Gflops, 135 Gflops and 212 Gflops for double, single and half precision respectively on NVIDIAs GeForce GTX 280 GPU. We have developed a new mixed precision approach for Krylov solvers using reliable updates which allows for full double precision accuracy while using only single or half precision arithmetic for the bulk of the computation. The resulting BiCGstab and CG solvers run in excess of 100 Gflops and, in terms of iterations until convergence, perform better than the usual defect-correction approach for mixed precision.
We discuss how dynamical fermion computations may be made yet cheaper by using symplectic integrators that conserve energy much more accurately without decreasing the integration step size. We first explain why symplectic integrators exactly conserve
a ``shadow Hamiltonian close to the desired one, and how this Hamiltonian may be computed in terms of Poisson brackets. We then discuss how classical mechanics may be implemented on Lie groups and derive the form of the Poisson brackets and force terms for some interesting integrators such as those making use of second derivatives of the action (Hessian or force gradient integrators). We hope that these will be seen to greatly improve energy conservation for only a small additional cost and that their use will significantly reduce the cost of dynamical fermion computations.
We introduce a simple general method for finding the equilibrium distribution for a class of widely used inexact Markov Chain Monte Carlo algorithms. The explicit error due to the non-commutivity of the updating operators when numerically integrating
Hamiltons equations can be derived using the Baker-Campbell-Hausdorff formula. This error is manifest in the conservation of a ``shadow Hamiltonian that lies close to the desired Hamiltonian. The fixed point distribution of inexact Hybrid algorithms may then be derived taking into account that the fixed point of the momentum heatbath and that of the molecular dynamics do not coincide exactly. We perform this derivation for various inexact algorithms used for lattice QCD calculations.
