Faster Evaluation of Multidimensional Integrals

Abstract

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.

Fast convergence of quasi-Monte Carlo for a class of isotropic integrals

“…In [11], the empirical convergence rate of QMC is proportional to n −1 , as if it sees that this is really a one-dimensional problem. In contrast, the empirical convergence rate of MC remains proportional to n −1/2 ; it does not see that the problem is really one dimensional.…”

“…The examples originally considered by Keister [2,5] and later by Papageorgiou and Traub [11], and Novak et al [9] belong to F since f = cos. The example f (r) = (1 + r 2 ) 1/2 in [2, 9] also belongs to F .…”

“…For instance, the discrepancy of the points used by QMC-GF in [11] is small for n ≤ 10 6 and d ≤ 100. This implies that they can be used to efficiently evaluate integrals of functions in the class F .…”

“…In this paper we prove that the worst case speed of convergence of QMC for a class of isotropic functions (which includes the ones tested in [11]) is of order √ log n/n. Thus, QMC has two advantages over MC for this class of integrals:…”

“…Papageorgiou and Traub [11] used QMC-GF, a QMC method using points from the generalized Faure 1 sequence [15], for a model isotropic problem suggested by a physicist B. Keister [5]. Their tests on high dimensional instances of this problem (d ranging from 25 to 100) showed the superiority of QMC-GF over MC and over Keister's proposed quadrature rules.…”