|
Abstract |
The Markov chain Monte Carlo (MCMC) method is a versatile tool in statistical
physics to evaluate multi-dimensional integrals numerically. For the method to
work effectively, one must consider the following key issues: representation of
problems in the MCMC framework, choice of ensemble, selection of candidate
states, optimization of transition kernel, algorithm for choosing a
configuration according to the transition probabilities, and
parallelization/optimization for target computer architecture [1-3]. We show
that the unconventional approaches based on the geometric allocation of
probabilities or weights can improve the dynamics and scaling of the Monte Carlo
simulation in several aspects [4]. Particularly, the approach using the
irreversible kernel can reduce or sometimes completely eliminate the rejection
of trial move in the Markov chain [4,5]. We also discuss how the space-time
interchange trick can reduce the computational time especially for the case
where the number of candidates is large, such as models with long-range
interactions [4,6] and random-bit generation problems [7].
References:
[1] S. Todo, in Strongly Correlated Systems: Numerical Methods (Springer Series
in Solid-State Sciences), ed. A. Avella, F. Mancini, p. 153 (Springer-Verlag,
Berlin, 2013).
[2] S. Todo, H. Matsuo, H. Shitara, Comp. Phys. Comm. 239, 84 (2019).
[3] B. Bauer, et al, J. Stat. Mech. P05001 (2011).
[4] S. Todo and H. Suwa, J. Phys.: Conf. Ser. 473, 012013 (2013).
[5] H. Suwa and S. Todo, Phys. Rev. Lett. 105, 120603 (2010).
[6] K. Fukui and S. Todo, J. Comp. Phys. 228, 2629 (2009).
[7] H. Watanabe, S. Morita, S. Todo, and N. Kawashima, J. Phys. Soc. Jpn. 88,
024004 (2019).
|
