Scaling properties of multiscale equilibration

W Detmold,M G Endres

doi:10.1103/physrevd.97.074507

Abstract

We investigate the lattice spacing dependence of the equilibration time for a recently proposed multiscale thermalization algorithm for Markov chain Monte Carlo simulations. The algorithm uses a renormalization-group matched coarse lattice action and prolongation operation to rapidly thermalize decorrelated initial configurations for evolution using a corresponding target lattice action defined at a finer scale. Focusing on nontopological long-distance observables in pure $SU(3)$ gauge theory, we provide quantitative evidence that the slow modes of the Markov process, which provide the dominant contribution to the rethermalization time, have a suppressed contribution toward the continuum limit, despite their associated timescales increasing. Based on these numerical investigations, we conjecture that the prolongation operation used herein will produce ensembles that are indistinguishable from the target fine-action distribution for a sufficiently fine coupling at a given level of statistical precision, thereby eliminating the cost of rethermalization.

Highlights

Multiscale methods have been applied successfully in a variety of ways to facilitate Markov chain Monte Carlo (MCMC) simulations in lattice QCD
The benefits of using this algorithm in a QCD context are severalfold. It enables the generation of ensembles with well-distributed topological charge in parameter regimes where topological freezing is problematic
The main focus of the present study is to quantitatively investigate the scaling properties of multiscale thermalization as a function of the lattice spacing, under the assumption that all coarse and fine lattice pairs have been properly matched using renormalization-group matching conditions

Summary

INTRODUCTION

Multiscale methods have been applied successfully in a variety of ways to facilitate Markov chain Monte Carlo (MCMC) simulations in lattice QCD. If the lattice spacing dependence of the rethermalization time scales better than that for autocorrelation times in a conventional approach [e.g., τR 1⁄4 Oð1=azRÞ with a rethermalization exponent zR < 2, for nontopological observables] or the overlap of prolongated fine distributions onto slow modes decreases sufficiently fast with a, multiscale thermalization could offer a new strategy for addressing the problem of critical slowing down. In general not much can be said theoretically about the scaling properties of either with lattice spacing (though our expectation is that the slow modes are generally diffusive for local updating schemes, and have quadratic scaling with inverse lattice spacing), heuristic and perturbative arguments suggest that the overlap factors arising from multiscale thermalization will diminish with lattice spacing for gauge theories in the continuum limit, since configurations become locally smooth, and the interpolation of coarse gauge fields performed prior to the rethermalization step becomes increasingly accurate [15]. The numerical results suggest that there exists a lattice spacing beyond which the unthermalized bias associated with excited modes of the Markov process becomes negligible for a given desired level of statistics

METHODS AND RESULTS

Nmeas þ 1

DISCUSSION