QMC Integration — Beating Intractability by Weighting the Coordinate Directions

Abstract

How can we understand the apparent success of multiple integration in hundreds or even thousands of dimensions? One way is to assume that the successive coordinate directions are not all equally difficult, and to quantify this by weighting the first coordinate direction by a positive number 71, the second by 72, and so on. We now know that in a classical Sobolev space setting, with mixed first derivatives measured in the L 2 sense and all weights γj equal to 1, the minimum number of points n min(∈, d) needed to reduce the worst case error of a quasi-Monte Carlo (QMC) algorithm from its initial error by a factor of e grows exponentially with the dimension d. In contrast, Sloan and Wozniakowski have shown thatn min(∈, d) is bounded independently of d if (and only if) Σ j γ j < ∞. But there are many other questions, some still open. Can the convergence rate in the case Σ j γ j < ∞ be better than the Monte Carlo rate \( 1/\sqrt n \) (Yes, but so far only if the condition on the weights is strengthened.) Can the results be improved by replacing the equal quadrature weights that characterise the QMC algorithm by other values? (No.) Can QMC point sets with the desired properties actually be constructed? (The original proof of sufficiency of Σ j γ j < ∞ established existence only, and was not constructive.) One construction is available in the form of an algorithm presented recently by Sloan, Kuo and Joe, but with a proven convergence rate that is of order only \( 1/\sqrt n \). Can that order be improved? (Not known.) And do the weighted spaces really have a role in applications such as the Gaussian expectation value integrals that are typical in mathematical finance? This review will trace the story of the QMC algorithm in weighted spaces up to the present day. Only the discussion of Gaussian integrals is new.