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

On Polynomial-Time Property for a Class of Randomized Quadratures

... this paper. In particular, for periodic functions, to enjoy the polynomialtime and/or strongly polynomial-time properties, these methods require less restrictive assumptions on the spaces than the assumptions required by the classical Monte Carlo methods. Recall that polynomial-time property means, roughly, that the errors are bounded from above by a polynomial in s in 1=n. Here n is the number of function values used. The strong polynomial-time property means that the error is bounded by a polynomial in 1=n independently of s

Can any stationary iteration using linear information be globally convergent?

Computer Science Departmen

Weighted Tensor Product Algorithms for Linear Multivariate Problems

We study the "-approximation of linear multivariate problems defined over weighted tensor product Hilbert spaces of functions f of d variables. A class of weighted tensor product (WTP) algorithms is defined which depends on a number of parameters. Two classes of permissible information are studied. all consists of all linear functionals while std consists of evaluations of f or its derivatives. We show that these multivariate problems are sometimes tractable even with a worst-case assurance. We study problem tractability by investigating when a WTP algorithm is a polynomial-time algorithm, that is, when the minimal number of information evaluations is a polynomial in 1=" and d. For all we construct an optimal WTP algorithm and provide a necessary and sufficient condition for tractability in terms of the sequence of weights and the sequence of singular values for d = 1. For std we obtain a weaker result by constructing a WTP algorithm which is optimal only for some weight se..