6.9 Monte Carlo Method of Numerical Integration

Consider an s-dimensional manifold with coordinates {m ¹ , , m ^s }. For a point of the manifold, we use the notation .

Let (m) be an arbitrary scalar function defined over and assume that we need to evaluate the sum

If has finite volume, the simplest method of evaluating I numerically is to define a regular grid of points in , to compute (m) at each point of the grid, and to approximate the integral in equation (6.94) by a discrete sum. But as the number of points in a regular grid is a rapidly increasing function of the dimension of the space ( N proportional to a constant raised to the power s), the method becomes impractical for large-dimensional spaces (say s ?4). The Monte Carlo method of numerical integration consists of replacing the regular grid of points with a pseudorandom grid generated by a computer code based on a pseudorandom-number generator. Although it is not possible to give any general rule for the number of points needed for an accurate evaluation of the sum (because this number is very much dependent on the form of (m)), it turns out in practical applications that, for well-behaved functions, (m) can be smaller, by some orders of magnitude, than the number of points needed in a regular grid.

Let p (m) be an arbitrary normed (