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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 L1 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 )) xy 4( x, y ) 4 4 •Random Walk Method E (u ( x, y )) 1 1 f ( x , y ) ( xi , , yi , ) n xy 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