Linear Operator Equations

Abstract

This chapter is devoted to the use of Monte Carlo to numerically solve equations involving linear operators. Several discrete random walks are used to solve a system of linear algebraic equations. Both forward and adjoint approaches are introduced. Next, MC techniques using continuous random walks are used to solve Fredholm integral equations. Linear differential operators also are discussed. The continuous random walk algorithm known as “walking on spheres” is used to solve boundary value problems defined by important differential equations. In particular, Laplace, Poisson, and Helmholtz equations are studied in 2- and 3-dimensions. The use of MC for eigenvalue problems involving linear operators is discussed. Keywords: linear algebraic equation, MC for linear operators; Fredholm integral equations; linear differential equations; Laplace equation; Poisson equation; Helmholtz equation; walking on spheres.