Introduction to Monte-Carlo Methods for Transport and Diffusion Equations
The book is devoted to the applications of Monte Carlo methods (i.e., simulation techniques based on random numbers), for solutions of partial differential equations. A probabilistic representation of solutions of these equations is given in the following way: the transport or diffusion type equations are interpreted as the Fokker-Planck equations associated with Markov processes. Transport equations in particle physics, the nonlinear Boltzmann transport equation, and the links between second order partial differential diffusion equations and Brownian motion, are all investigated. For each type of problem, limits of methods are discussed and specific techniques used in practice are described. An essential point is that even if Monte-Carlo methods need not converge, it is always possible to control the reliability of the result using an inexpensive additional calculation. The reader should have a good background in mathematical analysis.