Simulation and the Monte Carlo Method

This third edition of a well-known textbook on Monte Carlo methods follows a second edition from 2008 and adds two new chapters, new exercises and revised versions of three chapters. The second edition was reviewed here.

The book was designed for use by undergraduates in the sciences and engineering. The authors suggest that readers need a basic mathematical background that includes an elementary course in probability and statistics. They say of the new edition that it “gives a fully updated and comprehensive account of the major topics in Monte Carlo simulation.”

The idea of Monte Carlo simulations is usually attributed to Ulam, von Neumann and Metropolis, who developed the approach in the mid to late 1940s while working on nuclear weapons at Los Alamos. The method is computationally intensive, so it became practical as the computer age began and its use has expanded as computational power and resources have grown. Monte Carlo methods continue to be used extensively in physics but now are employed very broadly in engineering, science, business and finance.

This new edition offers an opportunity to take another look at the book and consider more carefully what it offers and to whom. The two new chapters introduce new advanced methods and thus continue the trend of the second edition by pushing to include more state-of-the-art techniques.

The first four chapters introduce the basic ideas of discrete event simulation using Monte Carlo techniques. The authors quickly review basic probability, provide methods for generating random numbers with several prescribed distributions, introduce the notion of discrete event simulation, and then describe statistical analysis of simulation results. This is the core of the book and the most useful for those new to the field. The authors move quickly through the topics. While there are examples, they generally do not have much elaboration.

The next three chapters explore important related topics, including minimizing the variance of the simulation output and doing sensitivity analyses. One of the main approaches to variance reduction involves importance sampling, and the authors consider several approaches to this. The final three chapters take up more advanced topics. The two new chapters consider splitting methods and stochastic enumeration. Splitting methods use sequential sampling to decompose a hard estimation problem into several easier small ones. Stochastic enumeration studies tree search and tree counting.

My main concern with this book is that it is not well matched to students and readers new to the subject. The introductory parts handle some subjects very nicely, but the examples are too abbreviated and much of the needed “how to” is missing. The book would benefit from at least one extended example of a Monte Carlo simulation carried through from beginning to end with subsequent analysis. Students using the book are also going to need more than the elementary probability and statistics that the authors recommend. The Fischer information matrix and entropy, for example, are usually not part of an elementary course.

The second major part of the book tends more toward a survey of more advanced techniques. While more expert readers will appreciate it, the value for novices is not so clear.

I have had occasion to develop and use Monte Carlo simulations several times in many different contexts. Most often the goal was to characterize the performance of a large system, often with a hundred or more variables. Sometimes the analysis was intended to evaluate a design before it was implemented, sometimes to identify critical parameters for existing operational systems, and sometimes to estimate the cost of manufacturing a product. If I (when new to the subject) had tried to use the current book to learn what I needed to know, I would have been rather frustrated.

The first part of this book might profitably be used in an introductory course provided the instructor took time to fill in details and flesh out the examples. As a whole the text seems best suited for more experienced readers looking to review the subject or experts looking for new techniques.

Readers might consider Monte Carlo Methods by Kalos and Whitlock as an alternative textbook. Although it tends to focus more on physics applications, its pace is more deliberate, its examples more elaborate, and it takes up some of the important “how to” questions that the current book does not.

Bill Satzer (bsatzer@gmail.com) was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.