Monte Carlo simulation

1. Monte Carlo Integration:


Monte Carlo integration is numerical integration using random numbers  for the approximate evaluation of definite integrals.

For multidimensional definite integration:

Suppose that the sample has size N and denote the points in the sample by . Then the estimate for the integral is given by

,

where denotes the sample mean of the integrand and V is volume.
The variance of the function can be estimated using

• Matlab code for Monte Carlo  3D integration:

%e.g. MC3D(inline('x.+2*y.*z'), [0 1 0 1 0 1], 5000)
function MC3D(fun, range, N)
     B = range;
     R = rand(3, N);
     %Set the random samplings to the correct intervals
     R(1, :) = (B(2)-B(1))*R(1, :)+ B(1);
     R(2, :) = R(2, :)*(B(4) - B(3)) + B(3);
     R(3, :) = R(3, :)*(B(6) - B(5)) + B(5);
     Volume = (B(2)-B(1))*(B(4)-B(3))*(B(6)-B(5));
     s = feval(fun, R(1,:), R(2,:), R(3,:));
     total = sum(s);
     avgF = total/N;
     Int_result = avgF*Volume;
     fprintf('Approximation: %f',Int_result );

• 
  
  
  
  

  C++ code for MC integration   4/(1.+x*x) from [0 1]

#include <cmath>
#include <iostream>
using namespace std;

//     Here we define various functions called by the main program
//     this function defines the function to integrate
double func(double x);
//     Main function begins here
int main()
{
     int i, n;
     long idum;
     double crude_mc, sum_sigma, fx, variance, exact;
     printf("Read in the number of Monte-Carlo samples\n");
     scanf("%d", &n);
     crude_mc = sum_sigma=0. ; idum=-1 ;  exact=acos(-1.);
//   evaluate the integral with a crude Monte-Carlo method
     for ( i = 1;  i <= n; i++){
           double x=rand()/(RAND_MAX+1.0);
           fx=func(x);
           crude_mc += fx;
           sum_sigma += fx*fx;
      }
      crude_mc = crude_mc/((double) n );
      sum_sigma = sum_sigma/((double) n );
      variance=sum_sigma-crude_mc*crude_mc;
//    final output

 printf("%d var= %12.5E Inum= %12.5E pi= %12.5E\n", n, variance, crude_mc, exact);
      return 0;
}  // end of main program

// this function defines the function to integrate

double func(double x)
{
  double value;
  value = 4/(1.+x*x);
  return value;
} // end of function to evaluate 

2. variational Monte Carlo (VMC) 

Variation Monte Carlo is a quantum Monte Carlo method that applies

the variational method to approximate the ground state of the system.

The expectation value necessary can be written in the x representation as

Following the Monte Carlo method for evaluating integrals, we can interpret

QWalk is a program developed to perform quantum Monte Carlo calculations of electronic structure in molecules and solids. It has been written from the ground up in C++ incorporating a modular approach that makes it extremely extensible as a research vehicle. It has also been written specifically with large-scale parallel processing in mind, making it usable on (and portable to) the fastest computers in the world.

QWalk is freely available for download

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