Faster Evaluation of Multidimensional Integrals

In a recent paper Keister proposed two quadrature rules as alternatives to Monte Carlo for certain multidimensional integrals and reported his test results. In earlier work we had shown that the quasi-Monte Carlo method with generalized Faure points is very effective for a variety of high dimensional integrals occuring in mathematical finance. In this paper we report test results of this method on Keister’s examples of dimension 9 and 25, and also for examples of dimension 60, 80 and 100. For the 25 dimensional integral we achieved accuracy of 10−2 with less than 500 points while the two methods tested by Keister used more than 220, 000 points. In all of our tests, for n sample points we obtained an empirical convergence rate proportional to n−1 rather than the n−1/2 of Monte Carlo.