Chapter 8 - Linear Operator Equations

Abstract

Monte Carlo (MC) methods have been applied to the solution of linear operator equations. The methods differ somewhat from MC as quadrature and involve the concept of random walks. Two- and three-dimensional problems are often solved by “walking on circles” and “walking on spheres” algorithms. Eigenvalue problems can also be solved by MC methods. Certain time-dependent problems are approached using reverse-time MC. This chapter gives a fairly thorough treatment of the various MC methods that are used to construct estimates of solutions to linear algebraic, differential, and integral equations. Although MC is often not the method of choice for solving linear operator equations, it is instructive to see how MC random walks can lead to solutions and in some cases, MC is the best approach. Fifteen example problems are given, in order to illustrate the methods used.