Editing Global optimization (section)

===Parallel tempering===
{{main|Parallel tempering}}
'''Parallel tempering''', also known as '''replica exchange MCMC sampling''', is a [[simulation]] method aimed at improving the dynamic properties of  [[Monte Carlo method]] simulations of physical systems, and of [[Markov chain Monte Carlo]] (MCMC) sampling methods more generally.  The replica exchange method was originally devised by Swendsen,<ref>Swendsen RH and Wang JS (1986) [https://www.researchgate.net/profile/Robert_Swendsen/publication/13255490_Replica_Monte_Carlo_Simulation_of_Spin-Glasses/links/0046352309b5f54715000000.pdf Replica Monte Carlo simulation of spin glasses] Physical Review Letters 57 : 2607–2609</ref> then extended by Geyer<ref>C. J. Geyer, (1991) in ''Computing Science and Statistics'', Proceedings of the 23rd Symposium on the Interface, American Statistical Association, New York, p. 156.</ref> and later developed, among others, by [[Giorgio Parisi]].,<ref>{{cite journal
 |author = Marco Falcioni and Michael W. Deem
 |year=1999
 |title = A Biased Monte Carlo Scheme for Zeolite Structure Solution
 |journal = J. Chem. Phys.
 |volume = 110 |issue = 3 |pages = 1754–1766
 |doi=10.1063/1.477812
|arxiv = cond-mat/9809085|bibcode = 1999JChPh.110.1754F|s2cid=13963102
 }}</ref><ref>David J. Earl and Michael W. Deem (2005) [http://www.rsc.org/Publishing/Journals/CP/article.asp?doi=b509983h "Parallel tempering: Theory, applications, and new perspectives"],  ''Phys. Chem. Chem. Phys.'',  7, 3910</ref>
Sugita and Okamoto formulated a [[molecular dynamics]] version of parallel tempering:<ref>{{cite journal
 |author = Y. Sugita and Y. Okamoto
 |year=1999
 |title = Replica-exchange molecular dynamics method for protein folding
 |journal = Chemical Physics Letters
 |volume = 314 |issue=1–2
 |pages = 141–151
 |doi=10.1016/S0009-2614(99)01123-9
|bibcode=1999CPL...314..141S}}</ref> this is usually known as replica-exchange molecular dynamics or REMD.

Essentially, one runs ''N'' copies of the system, randomly initialized, at different temperatures. Then, based on the Metropolis criterion  one exchanges configurations at different temperatures. The idea of this method
is to make configurations at high temperatures available to the simulations at low temperatures and vice versa.
This results in a very robust ensemble which is able to sample both low and high energy configurations.
In this way, thermodynamical properties such as the specific heat, which is in general not well computed in the canonical ensemble, can be computed with great precision.