Monte Carlo Methods Fall Semester , 2007

This is a set of lecture notes for a graduate class on Monte Carlo methods given at the Courant Institute of Mathematical Sciences at NYU in the fall of 2005 and again in 2007. During these years important new techniques and applications emerged. At the same time, most basic introductions to Monte Carlo are specialized for one of the application disciplines. These notes attempt to present Monte Carlo principles and methods in a way that will be useful to people from many disciplines. I hope to express ideas from the various Monte Carlo communities in a common mathematical language so that people from each discipline can use tricks invented for other applications. Given the variety of people using Monte Carlo and the range of application areas, it is surprising how far one can go in this way. Roughly speaking, Monte Carlo means computing using random numbers. It is helpful to refine this definition and distinguish between true Monte Carlo and simulation. We take simulation to mean generating individual random objects faithfully according to some model. For example, we might want to see what shape clouds come from a specific model of cloud formation that involves randomness. The point of simulation might not be gather detailed statistics, but just to see what a few random objects look like. By contrast, Monte Carlo uses random numbers as a means to evaluate quantities that themselves are not random. For example, suppose f(x) is the probability density for a one dimensional random variable, X. One way to evaluate A = E[X] is to generate many samples (random variables Xk with probability density f) and average them. More generally, we could generate a number of random objects (e.g. clouds) and collect interesting statistics about them. The difference is that the expected value of X or of some more complex statistic is a property of a random variable but is not itself random. Therefore, it may be possible to evaluate A without generating random samples with probability density f(x). For example, we could estimate A = ∫ xf(x)dx by numerical quadrature. Estimating A by averaging many Xk would be called plain simulation. Practitioners often find clever methods that are better (faster or more accurate). Simulation is mostly programming, but Monte Carlo is all about devising and understanding new computational strategies. Each piece of creativity will improve your results.