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Topic: Metropolis algorithm

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Random walk Monte Carlo

Metropolis light transport

Simulated annealing

VEGAS algorithm

Importance sampling

Genetic algorithm

Stratified sampling

Stochastic tunneling

Monte Carlo method

Nicholas Metropolis

Booker Ervin

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Myrtle Creek, Oregon

Stanislaw Marcin Ulam

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	The Metropolis Algorithm
		The Metropolis algorithm [#!METROPOLIS!#] was invented by Metropolis et al in 1953 for sampling an arbitrary probability distribution.
		In the Metropolis algorithm, new states or configurations of the system are found by systematically moving through all the lattice sites and updating the spin variables.
		The Metropolis algorithm for a spin model is well suited to parallelism, since it is regular--so one can use standard data decomposition of the lattice onto the processors to get good load balance, and local--the update of a site depends only on its nearest neighbors, so only local communication is required.
	www.hpjava.org /theses/shko/thesis_paper/node68.html (388 words)

	Nicholas Metropolis - Wikipedia, the free encyclopedia
		Nicholas Constantine Metropolis (June 11, 1915 – October 17, 1999) was a mathematician and physicist.
		Metropolis contributed several original ideas to mathematics and physics.
		Perhaps the most widely known is the Monte Carlo method.
	en.wikipedia.org /wiki/Nicholas_Metropolis (86 words)

	[No title]
		For this algorithm, the “smart” decision — stay or move — is allowed to depend not only on di, the degree of the vertex where you are, but also on dj, the degree of the vertex where you propose to go.
		To implement the general version of the Metropolis algorithm, you compare all pairs pij and pji, and, whenever the two probabilities are not equal, you force them to be equal by inserting a coin flip to reduce whichever probability is larger.
		The Metropolis algorithm provides an alternative strategy: Each time you choose a string and toss a coin, instead of using the outcome of the toss to decide whether or not to record the length, use it instead to decide which length to record, the current one, or a duplicate of the previous one.
	www.mtholyoke.edu /courses/gcobb/stat344/mcmc/cclich6.doc (6190 words)

	Physics Today September 2000
		Metropolis returned to Chicago at the end the war, but went back to Los Alamos in 1948 to take on the challenge of building a computer that would implement the rapidly developing concepts in digital computation.
		Metropolis also was active in organizing the data and storing the results on nuclear structure, which was a rapidly developing field of physics at the time.
		Metropolis also was greatly concerned about problems with the foundations of mathematics itself that were revealed through computer use.
	www.aip.org /pt/vol-53/iss-10/p100.html (1107 words)

	The American Statistician: A history of the Metropolis-Hastings algorithm.@ HighBeam Research (Site not responding. Last check: 2007-11-05)
		The Metropolis-Hastings algorithm is an extremely popular Markov chain Monte Carlo technique among statisticians.
		We relate reasons for the delay in the acceptance of the algorithm and reasons for its recent popularity.
		The Metropolis-Hastings (M-H) algorithm, a Markov chain Monte Carlo (MCMC) method, is one of the most popular techniques used by statisticians...
	www.highbeam.com /library/doc0.asp?DOCID=1G1:111015346&refid=ip_encyclopedia_hf (207 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		Thus instead of mutating directly in the path space, mutations are realized in the infinite dimensional unit cube of pseudo-random numbers and these points are transformed to the path space according to BRDF sampling, light source sampling and Russian roulette.
		Due to the fact that some samples are generated independently of the previous sample, this method can also be considered as a combination of the Metropolis algorithm with a classical random walk.
		Metropolis light transport is good in rendering bright image areas but poor in rendering dark sections since it allocates samples proportionally to the pixel luminance.
	www.cg.tuwien.ac.at /research/publications/bibtex.php?publ_id=Szirmay-2001-METR (303 words)

	Parallel implementation of Metropolis algorithm
		The Metropolis algorithm as discussed in Section 3.7 is very suitable to run on a parallel computer.
		A trivial way to parallelize the Metropolis algorithm is to run several simulations with different initial conditions etc simultaneously to improve the statistics in thermodynamic averages.
		The most time consuming part of the Metropolis algorithm is the sweep through all the spins in the lattice.
	www.fysik.uu.se /cmt/berg/node31.html (479 words)

	Metropolis, Nicholas Constantine (1915-1999)
		Metropolis returned to the University of Chicago in 1957 as professor of physics, founded and directed the university's Institute for Computer Research, but came back to Los Alamos in 1965.
		The Metropolis algorithm, first described in a 1953 paper by Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and Edward Teller, was cited in Computing in Science and Engineering as being among the top 10 algorithms having the "greatest influence on the development and practice of science and engineering in the 20th century."
		Metropolis also coined the terms "Monte Carlo" and "MANIAC." The latter was proposed in hope of putting at end to an emerging fad for naming computers with acronyms but, as Metropolis often lamented, it instead had the opposite effect.
	www.mlahanas.de /Greeks/new/Metropolis.htm (1067 words)

	[No title]
		The algorithm simulates the cooling process by gradually lowering the temperature of the system until it converges to a steady, frozen state.
		One of the parameters to the algorithm is the schedule.
		In practise, it is not necessary to let the temperature reach zero because as it approaches zero the chances of accepting a worse move are almost the same as the temperature being equal to zero.
	www.cs.nott.ac.uk /~gxk/aim/notes/simulatedannealing.doc (1398 words)

	math lessons - Metropolis-Hastings algorithm
		In mathematics and physics, the Metropolis-Hastings algorithm is an algorithm used to generate a sequence of samples from the joint distribution of two or more variables.
		This algorithm is an example of a Markov chain Monte Carlo algorithm.
		The Gibbs sampling algorithm is a special case of the Metropolis-Hastings algorithm.
	www.mathdaily.com /lessons/Metropolis-Hastings_algorithm (483 words)

	A catalytic Metropolis-Hastings algorithm (Site not responding. Last check: 2007-11-05)
		The Metropolis-Hastings algorithm produces a sequence of samples from a probability distribution.
		One problems with this algorithm is choosing good random steps to take.
		If the algorithm generates steps in uniformly random direction, it will not be able to move quickly along ridge shaped probability density functions.
	www.logarithmic.net /pfh/blog/01099980282 (172 words)

	The Monte-Carlo method
		Second, you see growth of dendrite structures in the left-bottom part of the pictures, because all spins are flipped by lines from this corner (to avoid this feature one could choose flipped spins at random).
		In the thermostat algorithm we get a single spin (all the rest are fixed) into contact with big thermostat at temperature T.
		In the Metropolis method "freezing" is faster, but sometimes we get two clusters with a big metastable domain wall.
	www.ibiblio.org /e-notes/Perc/monte.htm (468 words)

	Monte Carlo Techniques
		The method calls for a random walk, or a guided random walk, in phase space during which the integrand is evaluated at each step and averaged over.
		The Metropolis algorithm generates a random walk guided by the weight function w(x).
		A concern about the algorithm is that the random numbers generated by the Metropolis algorithm are correlated.
	einstein.drexel.edu /courses/PHYS405/Monte_Carlo (1885 words)

	3.2 Monte Carlo methods
		The Metropolis algorithm generates a random walk of points distributed according to a required probability distribution.
		From an initial ``position'' in phase or configuration space, a proposed ``move'' is generated and the move either accepted or rejected according to the Metropolis algorithm.
		By taking a sufficient number of trial steps all of phase space is explored and the Metropolis algorithm ensures that the points are distributed according to the required probability distribution.
	www.physics.uc.edu /~pkent/thesis/pkthnode19.html (546 words)

	[No title]
		In practice, it is not necessary to let the temperature reach zero because as it approaches zero, the chances of accepting a worse move are almost the same as the temperature being equal to zero.
		However, the cost function is often a bottleneck and it may sometimes be necessary to use Delta evaluation: the difference between the current solution and the neighborhood solution is evaluated.
		The number of computations of the algorithm is the order of nlimit * ntemps (number of iterations at each temperature * number of temperatures), while searching through all possible combination for the best path is in the order of n!.
	courses.washington.edu /inde510/510/SA_Algorithms.doc (1273 words)

	The Metropolis Algorithm
		The Metropolis algorithm provides a prescription for choosing which moves in configuration space to accept or reject.
		To ensure that points are sampled correctly from the probability distribution, the random walk has to be allowed to proceed from some arbitrary initial starting point until the average over an ensemble of moves represents the distribution to be sampled.
		It should be noted that the Metropolis algorithm is just one of a large number of such algorithms, but we have not found any reason to choose a different algorithm.
	www.tcm.phy.cam.ac.uk /~ajw29/thesis/node18.html (555 words)

	Heat Bath: Z_2 symmetry (+-1 Ising spin) (Site not responding. Last check: 2007-11-05)
		The fastest Algorithm is the Restricted Metropolis Algorithm for the Ising ±1 spin.
		The restricted Metropolis Algorithm is the fastest at the critical temperature.
		The Ising ±1 case is the only one where the Metropolis is better than the Heat Bath.
	www.physik.fu-berlin.de /~loison/fast_algorithms/Ising (184 words)

	12.6.2 Cluster Algorithms
		The aim of the cluster update algorithms is to find a suitable collection of spins which can be flipped with relatively little cost in energy.
		The first such algorithm was proposed by Swendsen and Wang [Swendsen:87a], and was based on an equivalence between a Potts spin model [Potts:52a], [Wu:82a] and percolation models [Stauffer:78a], [Essam:80a], for which cluster properties play a fundamental role.
		A variant of this algorithm, for which only a single cluster is constructed and updated at each sweep, has been proposed by Wolff [Wolff:89a].
	www.netlib.org /utk/lsi/pcwLSI/text/node292.html (693 words)

	Importance Sampling and Metropolis Algorithm
		Therefore, a method is needed to restrict the sampling only to the interesting volume in phase space.
		This was solved by Metropolis who proposed a method to generate a sequence of states
		Note that this is the only place where the temperature enters in the algorithm.
	www.fysik.uu.se /cmt/berg/node14.html (228 words)

	Laird Breyer's Markov Chain Monte Carlo (MCMC) Pages
		Metropolis-Hastings algorithms are a class of Markov chains which are commonly used to perform large scale calculations and simulations in Physics and Statistics.
		This was suggested by Gareth Roberts to be a "mode hopping" algorithm.
		This algorithm is an example of the use of auxiliary variables, and operates a little differently from the other algorithms above.
	www.lbreyer.com /classic.html (1579 words)

	Osnat Stramer - Talks (Site not responding. Last check: 2007-11-05)
		"Self-targeting candidates for MCMC algorithm", a memorial session for Richard Tweedie, the JSM meeting, New-York City, New-York, August 2002.
		"On inference for nonlinear diffusion models using the Hastings-Metropolis algorithms", at the workshop on Bayesian inference and stochastic processes in Madrid, June 1998.
		Colloquium "On inference for nonlinear diffusion models using the Hastings-Metropolis algorithms", Department of Statistics & Actuarial Science, 1998.
	www.cs.uiowa.edu /~stramer/talks.html (450 words)

	Path Integration using the Metropolis Algorithm
		The purpose of these problems is to implement the Metropolis algorithm on a 2D lattice to determine the difference between the ground and first excited state energies for both harmonic oscillator and quartic oscillator 1D potentials.
		To accomplish this, we start with an initial path with a set pathlength and then “thermalize” the path by running it through the Metropolis loop 10*N_cor times.
		//implementation of this algorithm, which may be found in his lecture notes.
	webusers.physics.uiuc.edu /~zetienne/probs2-3/probs2-3_writeup.htm (837 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		This paper presents a new mutation strategy for the Metropolis light transport algorithm, which works in the unit cube of pseudo-random numbers instead of mutating in the path space.
		Due to the fact that samples are generated independently in large steps, this method can also be considered as a combination of the Metropolis algorithm with a classical random walk.
		If we use multiple importance sampling for this combination, the combined method will be as good at bright regions as the Metropolis algorithm and at dark regions as random walks.
	www.iit.bme.hu /~szirmay/metroimp4_link.htm (239 words)

	Monte Carlo methods and the Metropolis algorithm (Site not responding. Last check: 2007-11-05)
		Whenever the conformation resulting from the attempted move is refused for any of the three possible reasons, then the new conformation of the chain is the same current conformation.
		Note that the ratio between the acceptance probabilities of a move is related to the acceptance probability for the inverse move by the same relation presented in Eq.
		This is the reason why local moves that are not compatible with the chain conformation or violate the excluded volume condition must be considered during the choice of conformation to be attempted in the simulation in conformational space, even if they will always be refused.
	www.unb.br /ib/cel/chico/artigos/thesis/node6.html (1319 words)

	An adaptive Metropolis algorithm, Heikki Haario, Eero Saksman, Johanna Tamminen
		An adaptive Metropolis algorithm, Heikki Haario, Eero Saksman, Johanna Tamminen
		A proper choice of a proposal distribution for Markov chain Monte Carlo methods, for example for the Metropolis-Hastings algorithm, is well known to be a crucial factor for the convergence of the algorithm.
		Due to the adaptive nature of the process, the AM algorithm is non-Markovian, but we establish here that it has the correct ergodic properties.
	projecteuclid.org /Dienst/UI/1.0/Display/euclid.bj/1080222083 (635 words)

	Metropolis-Hastings Algorithm
		We will now show that the Gibbs sampler discussed in Section 3.3.2 is a special case of a class of MCMC algorithms that are of the so-called Metropolis-Hastings type.
		The classic Metropolis algorithm is obtained if we consider equality in (50), i.e.
		In estimating the approximation error for MCMC simulation algorithms it is useful
	www.mathematik.uni-ulm.de /stochastik/lehre/ss06/markov/skript_engl/node36.html (252 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		The algorithm was originally developed by Jasper A. Vrugt as part of his doctoral dissertation work at the Institute for Biodiversity and Ecosystem Dynamics (IBED), Department of Physical Geography and Soil Science, The University of Amsterdam.
		The algorithm, entitled the Shuffled Complex Evolution Metropolis (SCEM-UA) is developed in collaboration between the University of Arizona and the University of Amsterdam and is presented below.
		The algorithm tentatively assumes that the number of sequences is identical to the number of complexes.
	www.science.uva.nl /ibed/cbpg/software/doc/Manual_SCEM-UA.doc (2990 words)

	Monte Carlos And Metropolis
		The algorithm of choice for Monte Carlo analysis of the Ising Model involves repeated, probabilistic updates of individual spins.
		That is, the frequency with which the algorithm generates any particular type of events must be the same as that specified by the Boltzmann distribution:
		If we imagine repeating this decision process a large number of times from some fixed "Old State", then the fraction of times in which the algorithm chooses the upper branch in decision process will be close to the true probability, P. That is, this very simple procedure satisfies the consistency requirement in Equation 6.
	oscar.cacr.caltech.edu /Hrothgar/Ising/monte.html (2059 words)

	Laboratory conference marks birth of the Metropolis algorithm \| The Newsbulletin \| June 2, 2003
		Since then, the algorithm found widespread use for other areas of research such as finance, statistics, political science and computer science.
		The computational method that Metropolis and others used eventually became known as "The Metropolis algorithm." The Metropolis algorithm made the Monte Carlo method a general purpose fixture for studying the properties of physical systems and sparked the development of other Monte Carlo algorithms.
		On June 9, the Laboratory will mark the publication of the Metropolis algorithm with a three-day conference devoted to a review the use of the Metropolis algorithm and Monte Carlo method in the physical sciences.
	www.lanl.gov /orgs/pa/newsbulletin/2003/06/02/text01.shtml (301 words)

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