Exercise "Monte Carlo integration"

1. (6 points) Plain Monte Carlo integration

   • Implement plain Monte Carlo multi-dimensional integration, e.g. along the following lines:
     function plainmc(f, a, b, N){ var randomx = function(a,b) [a[i]+Math.random()*(b[i]-a[i]) for (i in a)]; var volume=1; for(var i in a) volume*=b[i]-a[i]; var sum=0,sum2=0; for(var i=0;i<N;i++){var fx=f(randomx(a,b)); sum += fx; sum2 += fx*fx} var mean = sum/N; // <f_i> var sigma = Math.sqrt(sum2/N - mean*mean); // sigma² = <(f_i)²> - <f_i>² var SIGMA = sigma/Math.sqrt(N); // SIGMA² = <Q²> - <Q>² return [mean*volume, SIGMA*volume]; }
   • Calculate some interesting integrals with your Monte-Carlo routine.
   • Calculate

     ∫0π  dx/π ∫0π  dy/π ∫0π  dz/π [1-cos(x)cos(y)cos(z)]-1 = Γ(1/4)4/(4π3) ≈ 
     1.3932039296856768591842462603255
     

     
     This is a difficult singular integral, do not use it for your tests and error estimates, use something less sungular.

2. (3 points)

   • Check that the error of the plain Monte-Carlo method behaves as O(1/√N).

3. (1 point)

   • Implement recursive stratified sampling.