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

2. 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

3. 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

4. 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

5. Dirac Delta Function

6. 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’)

7. Green’s Function for Laplacian
where

8. 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)

9. 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

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

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