ε-SUPERPOSITION AND TRUNCATION DIMENSIONS IN AVERAGE AND PROBABILISTIC SETTINGS FOR ∞-VARIATE LINEAR PROBLEMS

This thesis is a representation of my contribution to the paper of the same name I co-author with Dr. Wasilkowski. It deals with linear problems defined on γ-weighted normed spaces of functions with infinitely many variables. In particular, I describe methods and discuss results for ε-truncation and ε-superposition methods. I show through these results that the ε-truncation and ε-superposition dimensions are small under modest error demand ε. These positive results are derived for product weights and the so-called anchored decomposition

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