Green’s Function Monte Carlo Fall 2013 By Yaohang Li, Ph.D.

Review Last Class This Class Next Class Solution of Linear Operator Equations Ulam-von Neumann Algorithm Adjoin Method Fredholm integral equation Dirichlet Problem Eigenvalue Problem This Class PDE Green’s Function Next Class Random Number Generation

Green’s Function (I) Consider a PDE written in a general form L(x)u(x)=f(x) L(x) is a linear differential operator u(x) is unknown f(x) is a known function The solution can be written as u(x)=L-1(x)f(x) L-1L=I

Green’s Function The inverse operator Property of the Green’s Function G(x; x’) is the Green’s Function kernel of the integral two-point function depends on x and x’ Property of the Green’s Function Solution to the PDE

Dirac Delta Function

Green’s Function in Monte Carlo G(x;x’) is a complex expression depending on the number of dimensions in the problem the distance between x and x’ the boundary condition G(x;x’) is interpreted as a probability of “walking” from x’ to x Each walker at x’ takes a step sampled from G(x;x’)

Green’s Function for Laplacian where

Solution to Laplace Equation using Green’s Function Monte Carlo Random Walk on a Mesh G is the Green’s Function The number of times that a walker from the point (x,y) lands at the boundary (xb,yb)

Poisson’s Equation Poisson’s Equation Approximation Random Walk Method u(r)=-4(r) Approximation Random Walk Method n: walkers i: the points visited by the walker The second term is the estimation of the path integral

Summary Green’s Function Laplace’s Equation Poisson’s Equation

What I want you to do? Review Slides Review basic probability/statistics concepts Work on your Assignment 4