Monte Carlo and Quasi-Monte Carlo Sampling (Springer Series in Statistics)

This is a standard "cookbook" of numerical procedures, and the authors have made the various books, Numerical Recipes in C , Numerical Recipes in Fortran 77 and Numerical Recipes in Fortran 90 available online. The books are divided up into numerous PDF files, one per book section where each chapter has many sections. A good link for these is here. These notes are very rare, and there may not be a hard bound copy of these notes anywhere outside the US.

Many of the topics that appear in the Lezioni Fermiane, and other of Mark Kac's more famous work appears here. This is, for example, the first place that Kac's stochastic version telegrapher's equation is written up. A PDF file with these notes can be found here. This book has a very nice introduction to functional integration and its relationship to partial differential equations via the Feynman-Kac formula.

In addition, it provides introductory material on probability and measure theory as they relate to the theory of Brownian motion. The book is no longer in print, but a PDF version of the book can be downloaded here. This is a review paper that provides an interesting perspective from This paper can be found here.

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This paper is a very comprehensive review of Monte Carlo methods in Of particular interest is the thorough job done on linear algebra. This paper presents the Ulam and von Neumann method for solving a linear system, which was never published in the open literature by the originators of the method.

This paper gives a good overview of Monte Carlo methods for iterating linear operators. These methods can be used for various fundamental problems in numerical linear algebra as well as the solution of problems amenable to solution via Picard iteration. Meyer, editor, Symposium on Monte Carlo Methods , pp. Wiley, New York, This paper is a comprehensive comparison of deterministic to Monte Carlo methods for solving linear equations.

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These results are dated, but the paper is strong both theoretically and experimentally, and begs for a modern revision. This paper presents Wasow's variance reduction enhancement to the Ulam and von Neumann method for solving a linear system. This paper proves many fundamental properties of the solutions of elliptic, parabolic, and hyperbolic partial differential equations by rigorously analyzing their difference equations. Of particular importance to students of Monte Carlo methods is that they discuss the solution of the difference approximations for elliptic and parabolic equations through probabilistic methods.

It should be a great source of pride to students of Monte Carlo methods that the seminal paper on the numerical analysis of finite difference equations for partial differential equations presents Monte Carlo solutions of the same! Kac, "Can one hear the shape of a drum? This is a classic exposition on the question of whether complete knowledge of the spectrum of the Laplacian for a domain in two-dimensions completely determines its shape.

The answer is no, and was not solved by this paper. However, this paper uses many techniques to answer it, partially.

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For our purposes, it uses stochastic techniques related to probabilistic representations of solutions to elliptic PDEs. The Chauvenet Prize is awarded to the author of an outstanding expository article on a mathematical topic by a member of the MAA. The paper can be found here.

Monte Carlo and Quasi-Monte Carlo Sampling

This paper gives the derivation of a stochastic process which leads to the telegrapher's equation. The paper starts with the wave equation, and adds a stochastic component to it which converts the equation from purely hyperbolic to the weakly but unmistakably parabolic telegrapher's equation. This paper presents the branching Brownian motion representation to the solution of the KPP equation. This theory forms the basis for more general Monte Carlo approaches to nonlinear problems.

Mixture model

This paper is by Mark Kac of the Feynman-Kac formula. It deals with the relationship with Brownian motion, which is a continuous stochastic process, and random walk on discrete grids. The connections are made using exact mathematical treatment of the discrete random walks and using rather simple mathematical ideas to show how deep these connections are. Mark Kac is one of the best mathematical expositors, and this early paper of his is certainly noteworthy in this regard.

Markov chain Monte Carlo

It can be found here. This paper gives a nice introduction to quasi-Monte Carlo methods. In addition, it provides a very simple, and hence easy to follow proof of the Koksma-Hlawka inequality. Lecture notes for the class can be downloaded at the following page. A tentative outline of the course can be found here. Rechnen 9 —] on the transformation of QMC point sets into low discrepancy point sets with respect to non uniform distributions. As a corollary of the latter, we note that we can slightly weaken the assumptions to prove the consistency of SQMC.

Source Bernoulli , Volume 23, Number 4B , February First available in Project Euclid: Permanent link to this document https: Zentralblatt MATH identifier Keywords hidden Markov models low discrepancy particle filtering quasi-Monte Carlo sequential quasi-Monte Carlo smoothing state—space models. Gerber, Mathieu; Chopin, Nicolas. Convergence of sequential quasi-Monte Carlo smoothing algorithms.

Monte Carlo and Quasi-Monte Carlo Sampling - CERN Document Server

Bernoulli 23 , no. More by Mathieu Gerber Search this author in: Google Scholar Project Euclid. More by Nicolas Chopin Search this author in: Read more about accessing full-text Buy article. Download Email Please enter a valid email address.