Josef Dick

High-dimensional integration: The quasi-Monte Carlo way

This paper is a contemporary review of QMC (‘quasi-Monte Carlo’) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0,1]s, where s may be large, or even infinite. After a general introduction, the paper surveys recent developments in lattice methods, digital nets, and related themes. Among those recent developments are methods of construction of both lattices and digital nets, to yield QMC rules that have a prescribed rate of convergence for sufficiently smooth functions, and ideally also guaranteed slow growth (or no growth) of the worst-case error as s increases. A crucial role is played by parameters called ‘weights’, since a careful use of the weight parameters is needed to ensure that the worst-case errors in an appropriately weighted function space are bounded, or grow only slowly, as the dimension s increases. Important tools for the analysis are weighted function spaces, reproducing kernel Hilbert spaces, and discrepancy, all of which are discussed with an appropriate level of detail.

Walsh Spaces Containing Smooth Functions and Quasi-Monte Carlo Rules of Arbitrary High Order

We define a Walsh space which contains all functions whose partial mixed derivatives up to order

Good Lattice Rules in Weighted Korobov Spaces with General Weights

\delta \ge 1

Multivariate integration in weighted Hilbert spaces based on Walsh functions and weighted Sobolev spaces

exist and have finite variation. In particular, for a suitable choice of parameters, this implies that certain Sobolev spaces are contained in these Walsh spaces. For this Walsh space we then show that quasi-Monte Carlo rules based on digital

Liberating the weights

(t,\alpha,s)

Explicit Constructions of Quasi-Monte Carlo Rules for the Numerical Integration of High-Dimensional Periodic Functions

-sequences achieve the optimal rate of convergence of the worst-case error for numerical integration. This rate of convergence is also optimal for the subspace of smooth functions. Explicit constructions of digital

On the convergence rate of the component-by-component construction of good lattice rules

(t,\alpha,s)

Construction algorithms for polynomial lattice rules for multivariate integration

-sequences are given, hence providing explicit quasi-Monte Carlo rules which achieve the optimal rate of convergence of the integration error for arbitrarily smooth functions.

Discrepancy Theory and Quasi-Monte Carlo Integration

We study the problem of multivariate integration and the construction of good lattice rules in weighted Korobov spaces with general weights. These spaces are not necessarily tensor products of spaces of univariate functions. Sufficient conditions for tractability and strong tractability of multivariate integration in such weighted function spaces are found. These conditions are also necessary if the weights are such that the reproducing kernel of the weighted Korobov space is pointwise non-negative. The existence of a lattice rule which achieves the nearly optimal convergence order is proven. A component-by-component (CBC) algorithm that constructs good lattice rules is presented. The resulting lattice rules achieve tractability or strong tractability error bounds and achieve nearly optimal convergence order for suitably decaying weights. We also study special weights such as finite-order and order-dependent weights. For these special weights, the cost of the CBC algorithm is polynomial. Numerical computations show that the lattice rules constructed by the CBC algorithm give much smaller worst-case errors than the mean worst-case errors over all quasi-Monte Carlo rules or over all lattice rules, and generally smaller worst-case errors than the best Korobov lattice rules in dimensions up to hundreds. Numerical results are provided to illustrate the efficiency of CBC lattice rules and Korobov lattice rules (with suitably chosen weights), in particular for high-dimensional finance problems.

The construction of good extensible rank-1 lattices

We introduce a weighted reproducing kernel Hilbert space which is based on Walsh functions. The worst-case error for integration in this space is studied, especially with regard to (t, m, s)-nets. It is found that there exists a digital (t, m, s)-net, which achieves a strong tractability worst-case error bound under certain condition on the weights.We also investigate the worst-case error of integration in weighted Sobolev spaces. As the main tool we define a digital shift invariant kernel associated to the kernel of the weighted Sobolev space. This allows us to study the mean square worst-case error of randomly digitally shifted digital (t, m, s)- nets. As this digital shift invariant kernel is almost the same as the kernel for the Hilbert space based on Walsh functions, we can derive results for the weighted Sobolev space based on the analysis of the Walsh function space. We show that there exists a (t, m, s)-net which achieves the best possible convergence order for integration in weighted Sobolev spaces and are strongly tractable under the same condition on the weights as for lattice rules.