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Manuscript Title: Quasi-Monte Carlo methods for lattice systems: a first look
Authors: K. Jansen, H. Leovey, A. Ammon, A. Griewank, M. Müller-Preussker
Program title: qar-0.1
Catalogue identifier: AERJ_v1_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 185(2014)948
Programming language: C and C++.
Computer: PC.
Operating system: Tested on GNU/Linux, should be portable to other operating systems with minimal efforts.
Has the code been vectorised or parallelized?: No
RAM: The memory usage directly scales with the number of samples and dimensions: Bytes used = "number of samples" x "number of dimensions" x 8 Bytes (double precision).
Keywords: Quasi-Monte Carlo, Quantum field theory, Lattice regularisation, Anharmonic oscillator, Reweighting.
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 regularisation. In this formulation the path integral is discretised 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 behaviour 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 behaviour.

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=216 = 65536 (including 9 replica runs) and d=100 should not take much longer than one minute.