The Essence of Monte Carlo

11 June 2018

Eric Veach wrote his thesis on “Robust Monte Carlo Methods for Light Transport Simulation”. This is a famous piece of work, considered a groundbreaking work in rendering and ubiquitous among graphics researchers and professionals. On page 12, there is a sentence which reads

The principle of Monte Carlo methods is not that the samples are truly random, but that random samples could be used in their place.

It took me some time to understand what he means, and I am still not sure if I have interpret it well, but pay attention to the following piece of code, using two slightly different functions to compute the value of $\pi$ and see if you see my point.

import numpy as np                                                              
def mc1(N):                                                                     
    c = 0                                                                       
    N = int(N)                                                                  
    for i in range(N):                                                          
        x = np.random.random()                                                  
        y = np.random.random()                                                  
        if (x-0.5)**2 + (y-0.5)**2 < 0.25:                                      
            c = c+1                                                             
    return 4.0*float(c)/N                                                       
def mc2(N):                                                                     
    c = 0                                                                       
    _N = int(np.sqrt(N))                                                        
    for i in range(_N):                                                         
        for j in range(_N):                                                     
            x = float(i)/_N                                                     
            y = float(j)/_N                                                     
            if (x-0.5)**2 + (y-0.5)**2 < 0.25:                                  
                c = c+1                                                         
    return 4.0*float(c)/(_N*_N)                                                 
print mc1(1e1), mc2(1e1)                                                        
print mc1(1e2), mc2(1e2)                                                        
print mc1(1e3), mc2(1e3)                                                        
print mc1(1e4), mc2(1e4)                                                        
print mc1(1e5), mc2(1e5)                                                        
print mc1(1e6), mc2(1e6)                                                        
print mc1(1e7), mc2(1e7)  

which generates the output

3.2 1.77777777778
3.16 2.76
3.148 3.08012486993
3.132 3.13
3.1404 3.14088287133
3.14302 3.1413
3.1420544 3.14153285317

The true value of $\pi$ upto 5 decimal places is $3.14159$.

My two cents!