Green’s Function Monte Carlo Fall 2013

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

2. 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

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 • 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 • 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’)

7. Green’s Function for Laplacian • Laplacian • Green’s Function • 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 • 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