Accelerating iterative linear equation solver using modified domain-wall fermion matrix in lattice QCD simulations

Abstract

Lattice simulations of Quantum Chromodynamics (QCD) enable one to calculate the low-energy properties of the strong interaction among quarks and gluons based on the first principle. The most time-consuming part of the numerical simulations of lattice QCD is typically solving a linear equation for the quark matrix. In particular, a discretized quark formulation called the domain-wall fermion operator requires a high numerical cost, while retaining the lattice version of the chiral symmetry to good precision. The domain-wall operator is defined on a five-dimensional (5D) space extending the four-dimensional (4D) spacetime with an extra fifth coordinate. After solving the linear equation in 5D space, the result vector is projected onto the original 4D space. There is a variant of the domain-wall operator that improves the convergence of the 5D linear equation while unchanging the 4D solution vector. In this paper, we examine how this variant of the domain-wall operator accelerates the iterative linear equation solver in practical setups. We also measure the eigenvalues of the operator and compare the condition number with the convergence of the solver. We use a generic lattice QCD code set Bridge++ that is planned to be released including the improved form of the domain-wall operator examined in this work with code for the GPU.

Figures (6)

• Figure 1: Converged number of CG iteration for $\beta=6.0$ on $(L=16)^4$ lattice, $M_0 = 1.8$ (without smearing), $(b,c)=(1.5,0.5)$ (top panels) $L_s=8$ (left) and 16 (right) and $(b,c)=(1.0,1.0)$ (bottom panels).
• Figure 2: Converged number of CG iteration for $\beta=6.0$ on $(L=16)^4$ lattice, $M_0 = 1.0$ (with smearing), $(b,c)=(1.5,0.5)$ (top panels) $L_s=8$ (left) and 16 (right) and $(b,c)=(1.0,1.0)$ (bottom panels).
• Figure 3: Result of the condition number on the $16^4$ lattice at $\beta=6.0$ for the domain-wall operator without link smearing with parameters $M_0=1.8$, $(b,c)=(1.5,0.5)$ (top panels) and $(1.0,1.0)$ (bottom), $m=0.001$. The left panels show the observed condition number on three configurations. The right panels show the condition numbers normalized by the values for $\alpha=1$.
• Figure 4: Result of the condition number on the $16^4$ lattice at $\beta=6.0$ for the domain-wall operator with link smearing with parameters $M_0=1.0$, $(b,c)=(1.5,0.5)$ (top panels) and $(1.0,1.0)$ (bottom), $m=0.001$. The left panels show the observed condition number on three configurations. The right panels show the condition numbers normalized by the values for $\alpha=1$.
• Figure 5: Converged numbers of CG iterations on $32^4$ lattice at $\beta=6.0$. Top panels show $M_0 = 1.8$ (without smearing), $(b,c)=(1.5,0.5)$ (left) and $(b,c)=(1.0,1.0)$ (right). Bottom panels show $M_0=1.0$ (with smearing), $(b,c)=(1.5,0.5)$ (left) and $(b,c)=(1.0,1.0)$ (right). In both cases, $L_s=8$ is used.