Greens Function Monte Carlo Spring 2005

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Greens Function Monte CarloSpring 2005

• By Yaohang Li, Ph.D.
• Department of Computer Science
• North Carolina AT State University
• yaohang_at_ncat.edu

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Review

• Last Class
• Solution of Linear Operator Equations (II)
• Adjoin Method
• Fredholm integral equation
• Dirichlet Problem
• Eigenvalue Problem
• This Class
• PDE
• Greens Function
• Homework Assignment 3
• Next Class
• Random Number Generation

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Greens 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-1LI

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Greens Function

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

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Dirac Delta Function
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Greens Function in Monte Carp

• Greens Function
• G(xx) is a complex expression depending on
• the number of dimensions in the problem
• the distance between x and x
• the boundary condition
• G(xx) is interpreted as a probability of
  walking from x to x
• Each walker at x takes a step sampled from
  G(xx)

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Greens Function for Laplacian

• Laplacian
• Greens Function
• where

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Solution to Laplace Equation using Greens
Function Monte Carlo

• Random Walk on a Mesh
• G is the Greens Function
• The number of times that a walker from the point
  (x,y) lands at the boundary (xb,yb)

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Poissons Equation

• Poissons Equation
• ?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

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Summary

• Greens Function
• Laplaces Equation
• Poissons Equation

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What I want you to do?

• Review Slides
• Read the UNIX handbook if you are not familiar
  with UNIX
• Review basic probability/statistics concepts
• Work on your Assignment 2 and 3