Lectures on Monte Carlo
Methods
Neal Madras
Monte Carlo methods form an
experimental branch of mathematics that employs simulations driven by random number
generators. These methods are often used when others fail, since they are much less
sensitive to the "curse of dimensionality", which plagues deterministic methods
in problems with a large number of variables. Monte Carlo methods are used in many fields:
mathematics, statistics, physics, chemistry, finance, computer science, and biology, for
instance.
This book is an introduction
to Monte Carlo methods for anyone who would like to use these methods to study various
kinds of mathematical models that arise in diverse areas of application. The book is based
on lectures in a graduate course given by the author. It examines theoretical properties
of Monte Carlo methods as well as practical issues concerning their computer
implementation and statistical analysis. The only formal prerequisite is an undergraduate
course in probability.
The book is intended to be
accessible to students from a wide range of scientific backgrounds. Rather than being a
detailed treatise, it covers the key topics of Monte Carlo methods to the depth necessary
for a researcher to design, implement, and analyze a full Monte Carlo study of a
mathematical or scientific problem. The ideas are illustrated with diverse running
examples. There are exercises sprinkled throughout the text. The topics covered include
computer generation of random variables, techniques and examples for variance reduction of
Monte Carlo estimates, Markov chain Monte Carlo, and statistical analysis of Monte Carlo
output.
100 pages