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

Review
• Last 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 -1 L=I

Green’s Function
•
The inverse operator
L

1 f
 
G ( x ; x ' ) f ( x ' ) dx '
– 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
• Green’s Function
– 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
• Laplacian
• Green’s Function
– 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 (x b
,y b
) u ( x , y )

1 n
 b
G ( x , y , x b
, y b
) u ( x b
, y b
)

Poisson’s Equation
• Poisson’s Equation
–  u( r )=-4

( r )
•
Approximation u ( x , y )

1
4
[ u ( x
  x , y )
 u ( x
  x , y )
 u ( x , y
  y )
 u ( x , y
  y ))

1
4
 x
 y 4

( x , y )
• Random Walk Method
E ( u ( x , y ))

1 n

 f ( x

, y

)

1 n
 x
 y
 i

,


( x i ,

, y i ,

)
– n: walkers
– i: the points visited by the walker
– The second term is the estimation of the path integral

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

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