What is local Monte Carlo simulation?

Question

"The traditional local Monte Carlo method is simple, extremely general, and versatile." -Wang Swendsen, 2002

What does 'local' Monte Carlo mean ? Is there anything called 'global' Monte Carlo?

Answer 1

The qutoe is from a 2002 paper by Swensen and Wang in the Journal of Statistical Physics titled "Transition Matrix Monte Carlo Method." A preprint is available from the ArXiv:

http://arxiv.org/pdf/cond-mat/0104418.pdf

Scanning through the paper quickly, it's apparent that by "local Monte Carlo method", the authors mean the conventional approach to Monte Carlo simulation of an Ising model (or similiar statistical mechanics system) in which you consider flipping one spin at a time, and the probability of that flip happening depends on the state of nearby spins. This is the original kind of Markov Chain Monte Carlo simulation.

In practice, it's often the case that the algorithm will consider many potential state changes with small probabilities, so that most steps are not actually accepted. The authors of this paper are proposing an algorithm that speeds things up by estimating a transition matrix for the next state that the system will actually move to.

Answer 2

As Brian said, "local MC methods" consist in proposing local changes in a configuration in order to minimize some objective functional (usually, the energy of the configuation).

Global (or "cluster") methods are intended to perform the same minimization procedure by making changes over huge subdomains of the configuration space at the same time. This method, called Swendsen-Wang algorithm (PRL, 58 (2) 1987, p:86), is based on a transformation found by Fortuin and Kasteleyn (Physica, 57, 1972, p:536) and proves to be extremely efficient in reducing what is called "critical slowing down", a phenomena occuring in local algorithms due to the fact that the correlation time between configurations can be very large near the critical point (i.e. the point where a phase transition occurs).

In their paper, Swendsen and Wang apply their algorithm in the study of the Ising model of statistical mechanics. This cluster method is extremely efficient, widely used, and has allowed numerical simulations of huge systems. One can easily find source codes in several languages.

A similar cluster method was developed by Ulli Wolff (PRL, 62 (4), 1989, p: 361).

Answer 3

It's hard to tell without more context from the quote, but my best guess is that the reference refers to the fact that practical MC methods sample in the vicinity of the previous sample. A global method would find a new sample at a random location of the search domain, independent of the previous sample's location. But this is inefficient and is a method that is not practical.