Adaptive Multidimensional Integration Based on Rank-1 Lattices

Quasi-Monte Carlo methods are used for numerically integrating multivariate functions. However, the error bounds for these methods typically rely on a priori knowledge of some semi-norm of the integrand, not on the sampled function values. In this article, we propose an error bound based on the discrete Fourier coefficients of the integrand. If these Fourier coefficients decay more quickly, the integrand has less fine scale structure, and the accuracy is higher. We focus on rank-1 lattices because they are a commonly used quasi-Monte Carlo design and because their algebraic structure facilitates an error analysis based on a Fourier decomposition of the integrand. This leads to a guaranteed adaptive cubature algorithm with computational cost $O(mb^m)$, where $b$ is some fixed prime number and $b^m$ is the number of data points

Hot new directions for quasi-Monte Carlo research in step with applications

This article provides an overview of some interfaces between the theory of quasi-Monte Carlo (QMC) methods and applications. We summarize three QMC theoretical settings: first order QMC methods in the unit cube $[0,1]^s$ and in $\mathbb{R}^s$, and higher order QMC methods in the unit cube. One important feature is that their error bounds can be independent of the dimension $s$ under appropriate conditions on the function spaces. Another important feature is that good parameters for these QMC methods can be obtained by fast efficient algorithms even when $s$ is large. We outline three different applications and explain how they can tap into the different QMC theory. We also discuss three cost saving strategies that can be combined with QMC in these applications. Many of these recent QMC theory and methods are developed not in isolation, but in close connection with applications

Sub-space approximations for MDO problems with disparate disciplinary variable dependence

The research leading to these results have been funded by the European Union Seventh Framework Programme FP7-PEOPLE-2012-ITN under grant agreement 316394, Aerospace Multidisciplinarity Enabling DEsign Optimization (AMEDEO) Marie Curie Initial Training Network

The Relationship Between Discrepancies Defined on a Domain and on its Subset

Discrepancy, Factorial design, Reproducing kernel, Uniform design, 62K15, 62K99,

On Bounds for Diffusion, Discrepancy and Fill Distance Metrics

Criteria for optimally discretizing measurable sets in Euclidean space is a difficult and old problem which relates directly to the problem of good numerical integration rules or finding points of low discrepancy. On the other hand, learning meaningful descriptions of a finite number of given points in a measure space is an exploding area of research with applications as diverse as dimension reduction, data analysis, computer vision, critical infrastructure, complex networks, clustering, imaging neural and sensor networks, wireless communications, financial marketing and dynamic programming. The purpose of this paper is to show that a general notion of extremal energy as defined and studied recently by Damelin, Hickernell and Zeng on measurable sets X in Euclidean space, defines a diffusion metric on X which is equivalent to a discrepancy on X and at the same time bounds the fill distance on X for suitable measures with discrete support. The diffusion metric is used to learn via normalized graph Laplacian dimension reduction and the discepancy is used to discretize. Dedicated to Alexander Gorban, Andrei Zinovyev and their co-organisers for the wonderful international workshop on large data sets held at the University of Leicester in August 2006

A note on the connection between uniformity and generalized minimum aberration

Primary 62K15, Secondary 62K05, Discrete discrepancy, factorial design, generalized minimum aberration, minimum projection variance, uniformity,