Monte-Carlo Course Materials

These are materials for the course "Monte-Carlo Methods" taught by Jonathan Goodman in the Spring term of 1997 at the Courant Institute at NYU. The course meets in room 713 in the main building. from 5pm. until 7pm Wednesday evenings.

Course outline:

This course will cover the basics of Monte-Carlo and discuss several active research areas where surprising and beautiful discoveries have recently been made. The prerequisites are an understanding of basic probability, linear algebra, Fourier series and the diffusion equation. The course is suitable for graduate students in Mathematics, Computer Science, Physics, Chemistry, and other departments. For those who actually regester for the course, the grade will be based on a small project, which may be theoretical (read, develop and report on one or more research papers) or computational. Background material on Monte-Carlo methods will come from basic texts such as Hammersley and Handscomb, Kalos and Whitlock, Alan Sokal's notes, Bratley, Fox, and Schrage, and Binder and Heermann. The advanced topics will come from research and survey papers in the literature. I will cover advanced topics as time and interest allows. Possible topics are:
(i) Fast methods for sampling lattice fields, the work of Adler, Parisi, Goodman, Sokal, Swendsen, Wang, and Wolff.
(ii) Effective methods for accurate estimation of very small probabilities: the "epsilon-delta" methods of Karp and Luby and others.
(iii) Umbrella sampling and simulated tempering: the work of Valleau, Swensen, Ferrenberg, Parisi, Madras, and others.
(iv) "Rapidly mixing" Markov chains for solving counting and sampling problems in computer science and statists.
(v) Accurate and efficient methods for sampling path integrals: the work of Ceperley and others.
(vi) Diffusion Monte-Carlo for quantum mechanical problems having fermionic symmetry: the work of Kalos, Anderson, Akao, Goodman.
(vii) Accurate time stepping methods for stochastic differential equations with absorbing or reflecting boundaries.

Course notes written by Goodman:


These are preliminary versions of course notes that I hope will improve with revisions over time. If you have comments or have found mistakes, please let me know. My email and home page URL are at the bottom of this page.

Introducion to Monte Carlo and pseudo random number generators, in postscript format, or LaTeX format.
Direct sampling methods, mapping and rejection, in postscript format, or LaTeX format.

Other interesting sources:


Introducion to Monte Carlo, by Alan Sokal, in postscript format only.
Exact sampling of the invariant invariant density for a Markov chain, papers by James Propp and David Wilson.

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