Monte Carlo

Rating:
Summary: About G.S.Fishman's book
Review: My work concerns numerical simulations and in particular it deals with monte carlo simulations. I have found this book a good comprehensive reference to explore new numerical methods in a handy way, thanks also for the sketch of many algorithms. However, I wish to prevent anybody who thinks of breaking through these topics starting from this book, since they could be overwhelmed by the huge amount of info.

Rating:
Summary: HIGHLY recommended!!
Review: This book gives a rigorous introduction to the main ideas behind the Monte Carlo technique and also gives many concrete examples to illustrate the important concepts. The applications of Monte Carlo are immense, and cover a wide range of fields, including finance, physics, chemistry, computational biology, geology, computational radiology, and network engineering.

The author begins, naturally, with a discussion of how to compute the volume of a high-dimensional body using standard (deterministic) methods and then shows the advantages of using Monte Carlo to find the volume. He does a fine job of assessing the errors in the volume calculation, being careful to distinguish between convergence with probability one and convergence in probability. He also explains the need for specifying confidence levels, and not just the epsilon error term, to determine the smallest sample size that guarantees an error no larger than the specified epsilon. He also gives an interesting application to network reliability in this chapter. The probability that two network nodes are connected is reduced to a volume computation which is then estimated using Monte Carlo sampling. This is an excellent example of how Monte Carlo can be used to arrive at an accurate estimate of an intractable problem. The author gives another example of this later in the chapter wherein he uses Monte Carlo methods in combinatorial probability. Also included in the chapter are some useful hands-on exercises for the reader.

The sometimes tricky procedure of generating samples from a variety of distributions is the subject of the next chapter. The author is careful throughout the chapter to distinguish between results that are exact from a formal standpoint and those that are implementable in practice. In addition, a thorough discussion of the error introduced by using pseudorandom numbers in place of sequences of uniform deviates is given. This is the only book I know of that discusses this issue with the clarity it does. The author treats the inverse transform, composition, acceptance-rejection, ratio-of-uniforms, and exact-approximation in great detail. Here again a very useful set of hands-on exercises is given at the end of the chapter.

Increasing the efficiency of Monte Carlo sampling is the subject of the next chapter, wherein importance sampling, control variates, stratified sampling, correlated sampling, and conditional Monte Carlo are discussed in detail. These methods are usually called variance reduction techniques, but the author gives an interesting argument about why this characterization is not really accurate.

The author brings in conditional sampling, or Markov chain sampling, in the next chapter. It is this approach that has made Monte Carlo such a widely used techniques in science, finance, and network queueing problems. He gives a rather quick overview of the necessary background in Markov chains, and then moves on to discuss neutron transport. It was the intractable nature of the Boltzmann transport equation that gave Monte Carlo its first real application back in the 1940's. The much-used Metropolis

method is discussed later, with close attention paid to the details. This is a section that should be read by anyone interested in Monte Carlo techniques. This is followed by a detailed discussion of Markov random fields, Gibbs sampling, and simulated annealing, all of these being heavily used in applications. And also, no book on Monte Carlo could be complete without a discussion of the three-dimensional Ising model, which is in here. The next chapter concentrates on sampling design and statistical inference, wherein the author discusses how choices of the initial (nonequilibrium) distribution, the number of steps, the number of replications, and the simulation time affect the computational and statistical efficiency of the Monte Carlo simulation. He explains very effectively these issues and also the difficulties involved with path-dependence. Network modelers will appreciate his example of routing algorithm performance. The last chapter treats in great detail procedures for generating pseudorandom numbers. The standard methods for doing this are covered, along with spectral tests and performance issues. More exotic pseudorandom generators using nonlinear recursion are also discussed, but the proofs for these are omittted. This excellent book belongs on the shelves of anyone interested in Monte Carlo techniques. The price is reasonable considering how much time it would take to collect all the results in the book from the literature. It deserves a highest recommendation.