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Mathematics of Computation

On the step-by-step construction of quasi–Monte Carlo integration rules that achieve strong tractability error bounds in weighted Sobolev spaces

Ian H. Sloan, Frances Y. Kuo, Stephen Joe

Published Mar 20, 2002• 91 citations
Publisher

Abstract

We develop and justify an algorithm for the construction of quasi–Monte Carlo (QMC) rules for integration in weighted Sobolev spaces; the rules so constructed are shifted rank-1 lattice rules. The parameters characterising the shifted lattice rule are found “component-by-component”: the ( d + 1 d+1 )-th component of the generator vector and the shift are obtained by successive 1 1 -dimensional searches, with the previous d d components kept unchanged. The rules constructed in this way are shown to achieve a strong tractability error bound in weighted Sobolev spaces. A search for n n -point rules with n n prime and all dimensions 1 to d d requires a total cost of O ( n 3 d 2 ) O(n^3d^2) operations. This may be reduced to O ( n 3 d ) O(n^3d) operations at the expense of O ( n 2 ) O(n^2) storage. Numerical values of parameters and worst-case errors are given for dimensions up to 40 and n n up to a few thousand. The worst-case errors for these rules are found to be much smaller than the theoretical bounds.

DOI
10.1090/s0025-5718-02-01420-5

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We develop and justify an algorithm for the construction of quasi–Monte Carlo (QMC) rules for integration in weighted Sobolev spaces; the rules so constructed are shifted rank-1 lattice rules. The parameters characterising the shifted lattice rule are found “component-by-component”: the ( d + 1 d+1 )-th component of the generator vector and the shift are obtained by successive 1 1 -dimensional searches, with the previous d d components kept unchanged. The rules constructed in this way are shown to achieve a strong tractability error bound in weighted Sobolev spaces. A search for n n -point rules with n n prime and all dimensions 1 to d d requires a total cost of O ( n 3 d 2 ) O(n^3d^2) operations. This may be reduced to O ( n 3 d ) O(n^3d) operations at the expense of O ( n 2 ) O(n^2) storage. Numerical values of parameters and worst-case errors are given for dimensions up to 40 and n n up to a few thousand. The worst-case errors for these rules are found to be much smaller than the theoretical bounds.
Shifted rank-1 lattice rules, a special class of quasi-Monte Carlo methods, have recently been proposed by the present authors for the integration of functions belonging to certain "weighted" Sobolev spaces. The shifts in these rules were generated in a deterministic manner. In contrast, in this paper we generate these shifts randomly. This allows probabilistic estimates for the error in a given integral. It also reduces the number of operations required to find the generating vectors for the underlying lattice rules component-by-component. The rules thus constructed achieve a worst-case strong tractability error bound in an average or probabilistic sense.
Recently, quasi-Monte Carlo methods have been successfully used for approximating multiple integrals in hundreds of dimensions in mathematical finance, and were significantly more efficient than Monte Carlo methods. To understand the apparent success of quasi-Monte Carlo methods for multiple integration, one popular approach is to study worst-case error bounds in weighted function spaces in which the importance of the variables is moderated by some sequences of weights. Ideally, a family of quasi-Monte Carlo methods in some weighted function space should be strongly tractable. Strong tractability means that the minimal number of quadrature points n needed to reduce the initial error by a factor of ε is bounded by a polynomial in ε⁻¹ independently of the dimension d. Several recent publications show the existence of lattice rules that satisfy the strong tractability error bounds in weighted Korobov spaces of periodic integrands and weighted Sobolev spaces of non-periodic integrands. However, those results were non-constructive and thus give no clues as to how to actually construct these lattice rules. In this thesis, we focus on the construction of quasi-Monte Carlo methods that are strongly tractable. We develop and justify algorithms for the construction of lattice rules that achieve strong tractability error bounds in weighted Korobov and Sobolev spaces. The parameters characterizing these lattice rules are found ‘component-by-component’: the (d + 1)-th components are obtained by successive 1-dimensional searches, with the previous d components kept unchanged. The cost of these algorithms vary from O(nd²) to O(n³d²) operations. With currently available technology, they allow construction of rules easily with values of n up to several million and dimensions d up to several hundred.
In this paper, we study quasi-Monte Carlo (QMC) integration in weighted Sobolev spaces. In contrast to many previous results the QMC algorithms considered here are of open type, i.e., they are extensible in the number of sample points without having to discard the samples already used. As the underlying integration nodes we consider randomized Halton sequences in prime bases p=(p1,...,ps)\boldsymbol{p}=(p_1,...,p_s)p=(p1​,...,ps​) for which we study the root mean square (RMS) worst-case error. The randomization method is a p\boldsymbol{p}p-adic shift which is based on p\boldsymbol{p}p-adic arithmetic. The obtained error bounds are optimal in the order of magnitude of the number of sample nodes. Furthermore we obtain conditions on the coordinate weights under which the error bounds are independent of the dimension sss. In terms of the field of Information-Based Complexity this means that the corresponding QMC rule achieves a strong polynomial tractability error bound. Our findings on the RMS worst-case error of randomized Halton sequences can be carried over to the RMS L2L_2L2​-discrepancy. Except for the p\boldsymbol{p}p-adic shift our results are fully constructive and no search algorithms (such as the component-by-component algorithm) are required.
We study quasi-Monte Carlo (QMC) integration of smooth functions defined over the multidimensional unit cube. Inspired by a recent work of Pan and Owen, we study a new construction-free median QMC rule which can exploit the smoothness and the weights of function spaces adaptively. For weighted Korobov spaces, we draw a sample of rrr independent generating vectors of rank-1 lattice rules, compute the integral estimate for each, and approximate the true integral by the median of these rrr estimates. For weighted Sobolev spaces, we use the same approach but with the rank-1 lattice rules replaced by high-order polynomial lattice rules. A major advantage over the existing approaches is that we do not need to construct good generating vectors by a computer search algorithm, while our median QMC rule achieves almost the optimal worst-case error rate for the respective function space with any smoothness and weights, with a probability that converges to 1 exponentially fast as rrr increases. Numerical experiments illustrate and support our theoretical findings.
Polynomial lattice point sets are polynomial versions of classical lattice point sets and among the most widely used classes of node sets for quasi- Monte Carlo integration. In this paper, we study the worst-case integration error of digitally shifted polynomial lattice point sets and give step by step construction algorithms to obtain polynomial lattices that achieve a low worst-case error in certain weighted Sobolev spaces. The construction algorithm is a so-called com- ponent by component algorithm, choosing one component of the relevant point set at a time. Furthermore, under certain conditions on the weights, we achieve that there is only a polynomial or even no dependence of the worst-case error on the dimension of the integration problem.
This short article studies a deterministic quasi-Monte Carlo lattice rule in weighted unanchored Sobolev spaces of smoothness 1 . Building on the error analysis by Kazashi and Sloan, we prove the existence of unshifted rank-1 lattice rules that achieve a worst-case error of O ( n − 1 / 4 ( log n ) 1 / 2 ) , with the implied constant independent of the dimension, under certain summability conditions on the weights. Although this convergence rate is inferior to the one achievable for the shifted-averaged root mean squared worst-case error, the result does not rely on random shifting or transformation and holds unconditionally without any conjecture, in contrast to the result of Kazashi and Sloan.