Quasi-Monte Carlo methods for lattice systems: A first look

Abstract

a b s t r a c tWe investigate the applicability of quasi-Monte Carlo methods to Euclidean lattice systems for quantum mechanics in order to improve the asymptotic error behavior of observables for such theories. In most cases the error of an observable calculated by averaging over random observations generated from an ordinary Markov chain Monte Carlo simulation behaves like N −1/2 , where N is the number of observations. By means of quasi-Monte Carlo methods it is possible to improve this behavior for certain problems to N −1 , or even further if the problems are regular enough. We adapted and applied this approach to simple systems like the quantum harmonic and anharmonic oscillator and verified an improved error scaling. Programming language: C and C++. Program summary Computer: PC.Operating system: Tested on GNU/Linux, should be portable to other operating systems with minimal efforts. Has the code been vectorized or parallelized?: No RAM:The memory usage directly scales with the number of samples and dimensions: Bytes used = ''number of samples'' × ''number of dimensions'' × 8 Bytes (double precision).Classification: 4.13, 11.5, 23.External routines: FFTW 3 library (http://www.fftw.org) Nature of problem:Certain physical models formulated as a quantum field theory through the Feynman path integral, such as quantum chromodynamics, require a non-perturbative treatment of the path integral. The only known approach that achieves this is the lattice regularization. In this formulation the path integral is discretized to a finite, but very high dimensional integral. So far only Monte Carlo, and especially Markov chain-Monte Carlo methods like the Metropolis or the hybrid Monte Carlo algorithm have been used to calculate approximate solutions of the path integral. These algorithms often lead to the undesired effect of autocorrelation in the samples of observables and suffer in any case from the slow asymptotic error behavior proportional to N −1/2 , if N is the number of samples. Solution method:This program applies the quasi-Monte Carlo approach and the reweighting technique (respectively the weighted uniform sampling method) to generate uncorrelated samples of observables of the anharmonic oscillator with an improved asymptotic error behavior. Unusual features:The application of the quasi-Monte Carlo approach is quite revolutionary in the field of lattice field theories. Running time:The running time depends directly on the number of samples N and dimensions d. On modern computers a run with up to N = 2 16 = 65536 (including 9 replica runs) and d = 100 should not take much longer than one minute.

On the efficient numerical solution of lattice systems with low-order couplings

“…When the extent of the time lattice is kept short, T 1.5, both, QMC and recursive numerical integration show a comparable performance with an improved error scaling which is faster than 1/ √ N . However, as noted in [7] QMC becomes inefficient when the time extent T is made larger than T = 1.5, independent of the value of the lattice spacing a.…”

“…Adding a non-Gaussian term in case of the anharmonic oscillator, we found α ≈ 0.75 which is not optimal but significantly better than the error scaling of MCMC methods. However, it needs to be mentioned that for certain system sizes, that is, large Euclidean times, the QMC method did not work particularly well for the anharmonic oscillator and no error scaling improvement could be established; cf., [7] for details.…”

Applicability of Quasi-Monte Carlo for lattice systems

“…A mathematical review of the QMC approach can be found in [1]. The major part of this contribution is based on our paper [2] (cf. also [3]).…”

“…The QMC approach provides the potential to circumvent the aforementioned problems, as it exhibits certain favorable properties. Most importantly, it is able to improve the error scaling to N −1 , given that certain conditions are met (see [2]). It is further encouraging to realize that the QMC technique has already been applied successfully in other fields like financial mathematics [4] for example.…”

Overcoming the sign problem in one-dimensional QCD by new integration rules with polynomial exactness

“…Therefore, alternative approaches to MC-MC methods need to be developed and in [5,6] we have proposed and tested Quasi Monte Carlo and iterated numerical integration techniques. These methods can improve the convergence of the involved integrations and also have the potential to deal with the sign problem.…”