Abstract
Monte Carlo simulations of quantum field theories on a lattice become increasingly expensive as the continuum limit is approached since the cost per independent sample grows with a high power of the inverse lattice spacing. Simulations on fine lattices suffer from critical slowdown, the rapid growth of autocorrelations in the Markov chain with decreasing lattice spacing a. This causes a strong increase in the number of lattice configurations that have to be generated to obtain statistically significant results. In this paper, hierarchical sampling methods to tame this growth in autocorrelations are discussed. Combined with multilevel variance reduction techniques, this significantly reduces the computational cost of simulations for given tolerances ε_{disc }on the discretisation error and ε_{stat }on the statistical error. For an observable with lattice errors of order α and an integrated autocorrelation time that grows like τ_{int }∼ a^{z}, multilevel Monte Carlo can reduce the cost from O(ε_{stat}^{2 }ε_{disc}^{(1+z)/α}) to O(ε_{stat}^{2 }log(ε_{disc})^{2}+ε_{disc}^{1}) or O(ε_{stat}^{2}+ε_{disc}^{1}). Even higher performance gains are expected for nonperturbative simulations of quantum field theories in D dimensions. The efficiency of the approach is demonstrated on two nontrivial model systems in quantum mechanics, including a topological oscillator that is badly affected by critical slowdown due to freezing of the topological charge. On fine lattices, the new methods are several orders of magnitude faster than standard, single level sampling based on Hybrid Monte Carlo. For high resolutions, multilevel Monte Carlo can be used to accelerate even the cluster algorithm for the topological oscillator. Performance is further improved through perturbative matching. This guarantees efficient coupling of theories on the multilevel lattice hierarchy, which have a natural interpretation in terms of effective theories obtained by renormalisation group transformations.
Original language  English 

Article number  114512 
Number of pages  24 
Journal  Physical Review D 
Volume  102 
DOIs  
Publication status  Published  18 Dec 2020 
Keywords
 Multilevel Monte Carlo
 Path Integral
 Hierarchical Methods
 Numerical Algorithm
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