Chapter 12 - Monte Carlo methods

Abstract

This chapter discusses Monte Carlo solution methods, in which the properties of random numbers are exploited to solve inverse problems. The simplest of these is the Monte Carlo search, in which trial solutions are randomly generated, and the one that most reduces the prediction error (or, equivalently, maximized the likelihood) is considered to be the best estimate of the solution to the inverse problem. The Simulated Annealing method improves upon the Monte Carlo search by modifying the likelihood to include a “temperature” parameter. This parameter is systematically varied during the procedure, resulting in a search process that is initially random to become more and more directed. The relationship between Simulated Annealing and Metropolis-Hastings sampling is identified, and the latter developed into an alternate solution method called the Markov Chain Monte Carlo (MCMC) method. It is used to produce an ensemble of many probable models that can be considered the solution to the inverse problem. Finally, the MCMC method is extended to trans-dimensional models, which evaluate several disparate types of models simultaneously.