converges to E(f(U)) almost surely when N tends to infinity. This suggests a very simple algorithm to approximate I: call a random number generator N times and compute the average (??). Observe that the method converges for any integrable function on [0, 1] : f is not necessarily a smooth function. In order to efficiently use the above Monte-Carlo method, we need to know its rate of convergence...

We present an asynchronous algorithm based on Quasi-Monte Carlo methods for the computation of multivari-ate integrals. Randomized Korobov and Richtmyer sequences are applied for problems of moderately high to high dimensions. We propose the use of Sobol rules in those dimensions where the integrand shows particular singular behavior. Timing results on MPI yield good speedup.

We propose nested sequential Monte Carlo (NSMC), a methodology to sample from sequences of probability distributions, even where the random variables are high-dimensional. NSMC generalises the SMC framework by requiring only approximate, properly weighted, samples from the SMC proposal distribution, while still resulting in a correct SMC algorithm. Furthermore, NSMC can in itself be used to pro...

is converging to the expectation Ef [h(X)] when T goes to infinity. Furthermore, the precision of this approximation is exactly of the same kind as the precision of a statistical estimate, in that it usually evolves as O( √ T ). Therefore, once a sample x1, . . . , xT is produced according to a distribution density f , all standard statistical tools, including bootstrap, apply to this sample (w...

Monte Carlo methods refer to any method that uses random numbers to get an approximate answer to a problem. In computer graphics Monte Carlo techniques can be used to perform radiosity calculations and can be used in distribution ray tracing for effects such as soft shadows and motion blur. Although these notes are for a course on radiosity, they have a great deal of material from a previous Si...

These lectures given to graduate students in high energy physics, provide an introduction to Monte Carlo methods. After an overview of classical numerical quadrature rules, Monte Carlo integration together with variance-reducing techniques is introduced. A short description on the generation of pseudo-random numbers and quasi-random numbers is given. Finally, methods to generate samples accordi...