Download Monte Carlo for Partial Differential Equations

Green’s Function Monte Carlo
Fall 2012
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-1L=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 (xb,yb)
u ( x, y) 
1
G( x, y, xb , yb )u ( xb , yb )

n b
Poisson’s Equation
•Poisson’s Equation
– u(r)=-4(r)
•Approximation
1
1
u ( x, y )  [u ( x  x, y )  u ( x  x, y )  u ( x, y  y )  u ( x, y  y ))  xy 4( x, y )
4
4
•Random Walk Method
E (u ( x, y )) 
1
1
f
(
x
,
y
)

 ( xi , , yi , )
   n xy 
n 
i ,
– 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