Random and Quasi-Random Point Sets

This volume is a collection of survey papers on recent developments in the fields of quasi-Monte Carlo methods and uniform random number generation.

We will cover a broad spectrum of questions, from advanced metric number theory to pricing financial derivatives.

The Monte Carlo method is one of the most important tools of system modeling.

Deterministic algorithms, so-called uniform random number gen- erators, are used to produce the input for the model systems on computers.

Such generators are assessed by theoretical ("a priori") and by empirical tests.

In the a priori analysis, we study figures of merit that measure the uniformity of certain high-dimensional "random" point sets.

The degree of uniformity is strongly related to the degree of correlations within the random numbers.

The quasi-Monte Carlo approach aims at improving the rate of conver- gence in the Monte Carlo method by number-theoretic techniques.

It yields deterministic bounds for the approximation error.

The main mathematical tool here are so-called low-discrepancy sequences.

These "quasi-random" points are produced by deterministic algorithms and should be as "super"- uniformly distributed as possible.

Hence, both in uniform random number generation and in quasi-Monte Carlo methods, we study the uniformity of deterministically generated point sets in high dimensions.

By a (common) abuse oflanguage, one speaks of random and quasi-random point sets.

The central questions treated in this book are (i) how to generate, (ii) how to analyze, and (iii) how to apply such high-dimensional point sets.