Recently quasi-Monte Carlo algorithms have been successfully used for multivariate integration of high dimension d, and were significantly more efficient than Monte Carlo algorithms. The existing theory of the worst case error bounds of quasi-Monte Carlo algorithms does not explain this phenomenon. This paper presents a partial answer to why quasi-Monte Carlo algorithms can work well for arbitrarily large d. It is done by identifying classes of functions for which the effect of the dimension d is negligible. These are weighted classes in which the behavior in the successive dimensions is moderated by a sequence of weights. We prove that the minimal worst case error of quasi-Monte Carlo algorithms does not depend on the dimension d iff the sum of the weights is finite. We also prove that under this assumption the minimal number of function values in the worst case setting needed to reduce the initial error by " is bounded by C " \Gammap , where the exponent p 2 [1; 2], and C depends ...

Abstract. An error bound for multidimensional quadrature is derived that includes the Koksma-Hlawka inequality as a special case. This error bound takes the form of a product of two terms. One term, which depends only on the integrand, is defined as a generalized variation. The other term, which depends only on the quadrature rule, is defined as a generalized discrepancy. The generalized discrepancy is a figure of merit for quadrature rules and includes as special cases the L p-star discrepancy and Pα that arises in the study of lattice rules.

Integration lattices are one of the main types of low discrepancy sets used in quasi-Monte Carlo methods. However, they have the disadvantage of being of fixed size. This article describes the construction of an infinite sequence of points, the first b m of which form a lattice for any non-negative integer m. Thus, if the quadrature error using an initial lattice is too large, the lattice can be extended without discarding the original points. Generating vectors for extensible lattices are found by minimizing a loss function based on some measure of discrepancy or nonuniformity of the lattice. The spectral test used for finding pseudo-random number generators is one important example of such a discrepancy. The performance of the extensible lattices proposed here is compared to that of other methods for some practical quadrature problems.

In recent years, Smolyak quadrature rules (also called quadratures on hyperbolic cross points or sparse grids) have gained interest as a possible competitor to number theoretic quadratures for high dimensional problems. A standard way of comparing the quality of multivariate quadrature formulas consists in computing their L2-discrepancy. Especially for larger dimensions, such computations are a highly complex task. In this paper we develop a fast recursive algorithm for computing the L2-discrepancy (and related quality measures) of general Smolyak quadratures. We carry out numerical comparisons between the discrepancies of certain Smolyak rules, Hammersley and Monte Carlo sequences.

Quasi-Monte Carlo (QMC) methods have begun to displace ordinary Monte Carlo (MC) methods in many practical problems. It is natural and obvious to combine QMC methods with traditional variance reduction techniques used in MC sampling, such as control variates. There can,

This paper analyzes some schemes for reducing the computational burden of digital scrambling. Some such schemes have been shown not to affect the mean squared L² discrepancy. This paper shows that some discrepancy-preserving alternative scrambles can change the variance in scrambled net quadrature. Even the rate of convergence can be adversely affected by alternative scramblings. Finally, some alternatives reduce the computational burden and can also be shown to improve the rate of convergence for the variance, at least in dimension 1.

Summary. We propose and analyze a quasi-Monte Carlo (QMC) method for simulating a discrete-time Markov chain on a discrete state space of dimension s ≥ 1. Several paths of the chain are simulated in parallel and reordered at each step. We provide a convergence result when the number N of simulated paths increases toward infinity. Finally, we present the results of some numerical experiments showing that our QMC algorithm converges faster as a function of N, at least in some situations, than the corresponding Monte Carlo (MC) method.

Abstract. We analyze a quasi-Monte Carlo method to solve the initial-value problem for a system of differential equations y ′ (t) =f(t, y(t)). The function f is smooth in y and we suppose that f and D1 yf are of bounded variation in t and that D2 yf is bounded in a neighborhood of the graph of the solution. The method is akin to the second order Heun method of the Runge-Kutta family. It uses a quasi-Monte Carlo estimate of integrals. The error bound involves the square of the step size as well as the discrepancy of the point set used for quasi-Monte Carlo approximation. Numerical experiments show that the quasi-randomized method outperforms a recently proposed randomized numerical method. 1.

Abstract. This paper analyzes a Monte Carlo algorithm for solving Smoluchowski’s coagulation equation. A finite number of particles approximates the initial mass distribution. Time is discretized and random numbers are used to move the particles according to the coagulation dynamics. Convergence is proved when quasi-random numbers are utilized and if the particles are relabeled according to mass in every time step. The results of some numerical experiments show that the error of the new algorithm is smaller than the error of a standard Monte Carlo algorithm using pseudo-random numbers without reordering the particles.

We study the problem of constructing rank-1 lattice rules which have good bounds on the “weighted star discrepancy”. Here the non-negative weights are general weights rather than the product weights considered in most earlier works. In order to show the existence of such good lattice rules, we use an averaging argument, and a similar argument is used later to prove that these lattice rules may be obtained using a component-by-component (CBC) construction of the generating vector. Under appropriate conditions on the weights, these lattice rules satisfy strong tractability bounds on the weighted star discrepancy. Particular classes of weights known as “order-dependent” and “finite-order ” weights are then considered and we show that the cost of the construction can be very much reduced for these two classes of weights.