Recently I began a project on MCMC and I think it may be helpful if some simple cases are given for me to do some simulations and analyses on. Could someone help introduce several simple, classical examples or models for me? Such as Ising model? How can I compare my results with others? To re-implement others' algorithms and simulate again?
Simple models for applying Markov chain Monte Carlo:
Hard spheres, Lennard-Jones, Potts models, etc.
In the literature you can find results to check your codes-
I'd recommend you look in
http://arxiv.org/abs/cond-mat/9612186
Introduction To Monte Carlo Algorithms
Hi,
I think that artificial examples with Gaussian mixtures or banana shaped distributions are pretty nice and quite used in literature.
Regards,
Luca
Maybe you can try diffuse reflectance on the basis of Farrell, Patterson & Wilson: A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo. Medical Physics, 1992, Vol.(19):4, 879-888. This model is generally used for validation of optical Monte Carlo computations.
Look at the book Kurt Binder, Dieter W. Heerman: Monte Carlo Simulation in Statistical Physics. It provides many examples.
Simple 2D ising model is the best testing case for MC. Correct Solutions are known and can be found in any simulation textbook, or stat. physics, K.Binder and/or D.Heermann if you can find them, but any other decent book on MC simulations starts with it. It's very easy to code. I would suggest start focusing on making sure it runs correctly, implement it as close to the formulas as possible, first, and then start worrying about making it faster. This way you will have a reliable baseline to make sure any new improved version of the program runs correctly. HTH, Good Luck.
Actually, if you are looking for something REALLY simple - try random walk of a single particle. First - just in space, to look at the simple diffusion, and if you want something more exciting - add external potential, first with the single potential energy minimum - then two minima, watch how it jumps between the vicinities of the minima - it's fun.
Most physical problems suggest very low depth of Markov chain, like many versions of random walk problems offered here multiple times. For higher-depth Markov chain, you might be interested to check other sciences. Like, say, modeling errors in the theory of information coding.
Hi,
Look at http://www.oecd-nea.org/pt/docs/iem/jeju02/session5/SessionV-08.pdf perhaps, and more generally the NEA benchmark data bank,
JB
I agree with Michael: looking at how far a random walker moves in 1D, 2D, and 3D as a function of time is a nice introduction that balances programming a simple model with relatively straightforward analysis (which is a big part of MCMC!). Calculating magnetization as a function of applied field in the 2D Ising model is the next logical step. I would also recommend programming examples and comparing them against models for which the Markov transition matrix can be written exactly. For example, program a walker that moves between three sites (left, middle, and right), and calculate using a Markov matrix what the fixed point solutions are for different transition probabilities between those three sites. This is an important exercise because it will give you hands-on experience comparing the statistical error in the discrete model vs. exact solutions, and can show you how tiny programming mistakes can violate the detailed balance condition in even the simplest models.
Also, some "simple" models aren't all that simple, btw. The 3D Ising model is one, as are hard disks in 2D. We had to implement a massively parallel method to observe a very subtle phase transition in the hard disk system (http://arxiv.org/abs/1211.1646), and obeying detailed balance in parallelized methods is not easy!
It might helpful if you start with random walk in 1D and 2D
I got an opportunity to do some research on random walk here is an attachment of algorithm made using Scilab language it will help you to got an idea about your simulation
Then follow the idea of Michael Kotelyanskii and Eric Jankowski they are right
Hi,
You might take a look at my attached work on Monte Carlo in petroleum industry. It is a classical work with some spices of novelty.
Luck
Article Estimation of in Place Hydrocarbon Volume in Multilayered Re...
Random walk/diffusion problems are probably easiest. Another good one is a fluid of hard spheres (or disks in 2D), for here the density–pressure relation (“equation of state”) and the radial distribution function are well known.
Well a classical simulation of 2 particles with lennard jones interaction between them is the simplest. It will give you basic knowledge on deterministic simulations. Thats how I started !!!
Hi.
Calculating of Coulomb energy of spheroid with uniform charge distribution could be an example.
Hello,
I think that the Monod's simplest model is a good example. Solving nonlinear differential system is often difficult and generally has no analytic solution.
The steady state of th system is easy to find, and thus confirme the results of the simulation (also, the analytic solution for steady states stability is also easy, confirming the convergence of the simulation.
The Monod's model, that describes the chemostat ("open fermentor") is widely used and can be easily found on the Web (three ordinary differentiel equations and an explicit representation of the microorganisms growth).
Have fun !
Jacques Thierie
You should look at the web page of the Winter Simulation Conference. You will find various high-Quality tutorials.
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