Review of The Mathematical Basis of Monte Carlo and Quasi Monte Carlo Methods
Shuaiyuan Zhou
In the paper The Mathematical Basis of Monte Carlo and Quasi Monte Carlo Methods,
the author regarded equidistribution properties rather than randomness of the procedure
as a proper justification of both Monte Carlo and Quasi Monte Carlo methods. Two
measurements: extreme discrepancy and mean square discrepancy were presented and the
upper bound of Monte Carlo integration error was discussed. Also, the comparison of
Quasi Monte Carlo methods and Monte Carlo methods was given.
Extreme discrepancy describes the equidistribution properties of the sets of points at
which the integral values are computed, and is noted as D(S), where S is any sequence of
points in the domain of integration – an s-dimensional cube. Another measurement is
mean square discrepancy, which is a measure of the lack of equidistribution of a
sequence of points in a multidimensional cube and is noted as T(S).
Given the definition of discrepancies, the error bound of Monte Carlo integration can be
computed. In one dimension, the error of integration E(f) is bounded as: E(f) ≤ D(S)V(f),
where f denote any integration function and V(f) is the total variation over the interval [0,
1]. And for two dimensional case, the error bound is stated to be: E(f) ≤ V2(f)D(S) +
V(f(x, 1))D(X) + V(f(1, y))D(Y), where X and Y are the projections of the sequence S on
the x-axis and the y-axis.
Comparing extreme and mean square discrepancies, the author stated that T(S) < D(S)
and T(S) can be of a slightly smaller order of magnitude than D(S); so, the usage of mean
square discrepancy T(S) is recommended especially for large values of N. Another
important fact is that T(S) and D(S) are with high probability of the order of magnitude of
N-1/2 if the points are chosen genuinely randomly. Thus, the error bound computed using
T(S) or D(S) is to the accuracy of Monte Carlo methods.
The next part of the paper dealt with Quasi Monte Carlo methods. The idea behind Quasi
Monte Carlo is as following: “because the concept of randomness is vague, it is better to
work with sequences making no pretence of random origin, but can give the best possible
guarantee of accuracy in computing”. According to the author, Quasi Monte Carlo ought
to have the lowest possible extreme or mean square discrepancy. Also, the author in the
paper gave a method for composition of such quasi-random sequence in two dimensions.
Last, the author talked about the difference between Monte Carlo and Quasi Monte Carlo
methods. Monte Carlo is based on sequence of pseudorandom numbers while Quasi
Monte Carlo is based on low-discrepancy sequences. Also, the accuracy of the Quasi
Monte Carlo increases faster than that of Monte Carlo; and the difference is greater if the
integrand is smooth and the number of dimensions of the integral is small.