Quasi-Monte Carlo algorithms for unbounded, weighted integration problems

Abstract

In this article we investigate quasi-Monte Carlo (QMC) methods for multidimensional improper integrals with respect to a measure other than the uniform distribution. Additionally, the integrand is allowed to be unbounded at the lower boundary of the integration domain. We establish convergence of the QMC estimator to the value of the improper integral under conditions involving both the integrand and the sequence used. Furthermore, we suggest a modification of an approach proposed by Hlawka and Mu¨ck for the creation of low-discrepancy sequences with regard to a given density, which are suited for singular integrands. r This paper is devoted to quasi-Monte Carlo (QMC) techniques for weighted integration problems of the formwhere H denotes an s-dimensional probability distribution with support K ¼ ½a; b ¼ Q s i¼1 ½a ðiÞ ; b ðiÞ DR s and continuous density hðxÞ: Furthermore, f is a function with singularities on the left boundary of K:Numerical integration problems of this form frequently occur in practice, e.g. in the field of computational finance. A typical example is the estimation of the mean of ARTICLE IN PRESS

Backreaction and continuum limit in a closed universe filled with black holes

“…Unfortunately the Koksma-Hlawka inequality does not work for unbounded functions like (4), whose Hardy-Krause variation is infinite. While many papers have been written on the bounding E if f has a singularity [30,35,36,37], none of them seems to offer methods applicable in our case, where the domain of integration is spherical, the volume form is non-standard and the weights in the average may be more complicated than simply 1 N . We have thus developed our own tools for estimating σ Φ and σ M , based on an improved version of the Koksma-Hlawka inequality, which we will present below.…”

Functions of bounded variation, signed measures, and a general Koksma–Hlawka inequality

“…Chelson's generalized Koksma-Hlawka inequality is mentioned several times in the literature; it is explicitly stated, in differing forms, in [16,17,35,39,40,41,46,47]. In some of these instances it is stated without the transformation procedure and without using the incorrect identity (47) (which means that in those cases it is stated roughly in the same form as our Theorem 1 or Corollary 1).…”

“…To calculate the HK-variation of f , we have to calculate its variation in the sense of Vitali on faces of the form (17). However, the situation becomes much easier if we separately calculate the variations in the sense of Vitali of f + and f − instead.…”

Calculation of improper integrals using uniformly distributed sequences

“…(Results of this kind can be applied to the calculation of such integrals; see, for example, [19], [10] and [7].) For a more complete survey of the existing literature we refer the reader to the monographs by L. Kuipers and H. Niederreiter [12] and M. Drmota and R. F. Tichy [4].…”