Application to Numerical Integration

Monte Carlo techniques are methods to calculate a numerical quantity by casting it into a stochastic framework and applying repeated simulation using pseudorandom numbers. The expectation E[X] of a random variable X is approximated by a sample mean. In practice, the uniform distribution is sampled by pseudo-random sequences which are then transformed to the law of .
Quasi-Monte Carlo techniques replace pseudo-random sequences with low-discrepancy sequences which have a more uniform behaviour. In a certain sense, Quasi-Monte Carlo methods combine the advantages of Monte Carlo and uniform lattice methods. In particular, less quasi-random samples are needed to achieve a similar level of accuracy as obtained by using pseudo-random sequeces.

We consider a test integrand.

Mathematica can symbolically integrate such functions and obtains an exact value:

First we convert the 3-dimensional Niederreiter sequence in base 3 calculated before as rationals to real numbers. This is just to speed up calculation.

Now we build the sample average over the 1000 samples to approximate the integral:

Converted by Mathematica November 18, 1998