Multigrid preconditioning for the overlap operator in lattice QCD

Abstract

The overlap operator is a lattice discretization of the Dirac operator of quantum chromodynamics, the fundamental physical theory of the strong interaction between the quarks. As opposed to other discretizations it preserves the important physical property of chiral symmetry, at the expense of requiring much more effort when solving systems with this operator. We present a preconditioning technique based on another lattice discretization, the Wilson-Dirac operator. The mathematical analysis precisely describes the effect of this preconditioning in the case that the Wilson-Dirac operator is normal. Although this is not exactly the case in realistic settings, we show that current smearing techniques indeed drive the Wilson-Dirac operator towards normality, thus providing a motivation why our preconditioner works well in computational practice. Results of numerical experiments in physically relevant settings show that our preconditioning yields accelerations of up to one order of magnitude.

Scaling properties of multiscale equilibration

“…Multiscale methods have been applied successfully in a variety of ways to facilitate Markov chain Monte Carlo (MCMC) simulations in lattice quantum chromodynamics (QCD). These applications range from Dirac operator inversion [1][2][3][4] to evaluation of correlation functions and other observables [5][6][7][8], and have resulted in significant increases in computational efficiency, and reductions in uncertainties of stochastically estimated observables. Implementation of a multiscale algorithm for gauge field updating in lattice QCD, however, remains an open challenge despite some early progress for simpler theories [9][10][11][12][13][14].…”

Adaptive aggregation-based domain decomposition multigrid for twisted mass fermions

“…[7,8,9,10], referred to as MG-GCR (Multi Grid -Generalized Conjugate Residual) which is part of the USQCD package QOPQDP [11] and; iii) an adaptive aggregation-based domain decomposition multigrid approach [1], referred to as DD-αAMG, recently made publicly available in the DDalphaAMG library [12]. Although these solvers have been developed for clover Wilson fermions they can be extended to other fermion discretization schemes as has been done for example for the overlap operator using DD-αAMG [13] or for Domain-Wall fermions using MG-GCR [14].…”

First Numerical Experiences with Overlap Fermions based on the Brillouin Kernel

“…With the Brillouin kernel even a single iteration of f 11 seems to establish an operator with good chiral properties (at least if starting from D ker = D B with sufficient link smearing). Moreover, since the low-lying physical eigenvalues hardly change, it seems like a self-suggesting idea to use D B or D W to precondition the f 11 approximant [11], and the latter action to precondition a higher-order approximant, e.g. KL44 given by f 44 = f 11 ( f 11 ) [9].…”