Monte Carlo methods

Rather than cataloguing techniques one by one, this text looks for the common principles that make Monte Carlo approaches work well in practice. Two threads run through it. The first is the random walk, both as a model of stochastic processes found in nature and as a bridge to integral and differential equations. The second is variance reduction, with particular emphasis on importance sampling as a way of getting more accurate results from the same computational effort.

The book begins with a simple case: radiation passing through matter, described by a probabilistic sketch of the particles' interactions, which leads directly to a random walk model of transport. From there it shows how such walks correspond to integral equations. It then extends the same reasoning to the Schrödinger equation, presenting its solution by random walks in an accessible way.

On the variance reduction side, the author covers Monte Carlo estimation of integrals in finitely many dimensions. Importance sampling receives extended treatment, both for its value in numerical quadrature and for its usefulness in solving integral equations. A notable feature is the separate treatment of the Metropolis algorithm as a sampling method, kept clearly distinct from importance sampling.

The book is intended for physicists, chemists, statisticians, mathematicians and computer scientists who want a thorough and engaging entry point into the subject.