Abstract
Lattice QCD calculations including the effects of one or more non-degenerate sea quark flavors are conventionally performed using the Rational Hybrid Monte Carlo (RHMC) algorithm, which computes the square root of the determinant of $\mathscr{D}^{\dagger} \mathscr{D}$, where $\mathscr{D}$ is the Dirac operator. The special case of two degenerate quark flavors with the same mass is described directly by the determinant of $\mathscr{D}^{\dagger} \mathscr{D}$ --- in particular, no square root is necessary --- enabling a variety of algorithmic developments, which have driven down the cost of simulating the light (up and down) quarks in the isospin-symmetric limit of equal masses. As a result, the relative cost of single quark flavors --- such as the strange or charm --- computed with RHMC has become more expensive. This problem is even more severe in the context of our measurements of the $\Delta I = 1/2$ $K \rightarrow \pi \pi$ matrix elements on lattice ensembles with $G$-parity boundary conditions, since $G$-parity is associated with a doubling of the number of quark flavors described by $\mathscr{D}$, and thus RHMC is needed for the isospin-symmetric light quarks as well. In this paper we report on our implementation of the exact one flavor algorithm (EOFA) introduced by the TWQCD collaboration for simulations including single flavors of domain wall quarks. We have developed a new preconditioner for the EOFA Dirac equation, which both reduces the cost of solving the Dirac equation and allows us to re-use the bulk of our existing high-performance code. Coupling these improvements with careful tuning of our integrator, the time per accepted trajectory in the production of our 2+1 flavor $G$-parity ensembles with physical pion and kaon masses has been decreased by a factor of 4.2.
Highlights
Lattice QCD simulations are typically performed using variants of the hybrid Monte Carlo (HMC) algorithm, which includes the effects of dynamical sea quarks through the determinant of a fermion matrix evaluated by stochastically sampling a discretized QCD path integral
While rational functions are in many ways a good choice—they are economical in the sense that the inverse square root can usually be well-approximated by a modest number of terms, and the multishift conjugate gradient (CG) algorithm can be used to efficiently invert ðD†D þ βkÞ for all k simultaneously—the additional complexity of evaluating fðD†DÞ and the associated molecular dynamics pseudofermion force makes single flavor rational hybrid Monte Carlo (RHMC) simulations significantly more costly than degenerate two flavor HMC simulations at the same bare quark mass
While exact one flavor algorithm (EOFA) and RHMC are formally equivalent in the sense of Eq (21)—in the limit of infinite statistics they produce the same quark determinant, and the same ensemble averages for physical observables—it is in principle possible that the two algorithms approach this limit at different rates
Summary
Lattice QCD simulations are typically performed using variants of the hybrid Monte Carlo (HMC) algorithm, which includes the effects of dynamical sea quarks through the determinant of a fermion matrix evaluated by stochastically sampling a discretized QCD path integral. In this work we discuss the RBC/UKQCD Collaboration’s implementation and tests of the exact one flavor algorithm, as well as the use of EOFA in generating gauge field configurations for our ongoing first-principles calculation of the ratio of standard model parameters ε0/ε from ΔI 1⁄4 1/2 K → ππ decays with Gparity boundary conditions. We will demonstrate that a significant improvement over the RHMC algorithm in terms of wall clock time is possible with EOFA after introducing a variety of preconditioning and tuning techniques Work in this direction was presented at the 34th International Symposium on Lattice Field Theory [6]; here we will elaborate on the details and discuss our first large-scale EOFA calculation
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