Proof Techniques in Quasi-Monte Carlo Theory

TL;DR

This survey reviews key mathematical tools and methods used in quasi-Monte Carlo theory, illustrating their application through examples across various mathematical disciplines.

Contribution

It organizes and discusses a selection of analytical tools relevant to QMC, highlighting their roles and applications in the field.

Findings

Illustrates the use of harmonic analysis in QMC

Shows how algebraic methods aid in discrepancy estimates

Demonstrates probabilistic inequalities in QMC analysis

Abstract

In this survey paper we discuss some tools and methods which are of use in quasi-Monte Carlo (QMC) theory. We group them in chapters on Numerical Analysis, Harmonic Analysis, Algebra and Number Theory, and Probability Theory. We do not provide a comprehensive survey of all tools, but focus on a few of them, including reproducing and covariance kernels, Littlewood-Paley theory, Riesz products, Minkowski's fundamental theorem, exponential sums, diophantine approximation, Hoeffding's inequality and empirical processes, as well as other tools. We illustrate the use of these methods in QMC using examples.