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This book is summarizing the results of the workshop "Uniform
Distribution and Quasi-Monte Carlo Methods" of the RICAM Special
Semester on "Applications of Algebra and Number Theory" in October
2013. The survey articles in this book focus on number theoretic
point constructions, uniform distribution theory, and quasi-Monte
Carlo methods. As deterministic versions of the Monte Carlo method,
quasi-Monte Carlo rules enjoy increasing popularity, with many
fruitful applications in mathematical practice, as for example in
finance, computer graphics, and biology. The goal of this book is
to give an overview of recent developments in uniform distribution
theory, quasi-Monte Carlo methods, and their applications,
presented by leading experts in these vivid fields of research.
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."
Monte Carlo methods are numerical methods based on random sampling
and quasi-Monte Carlo methods are their deterministic versions.
This volume contains the refereed proceedings of the Second
International Conference on Monte Carlo and Quasi-Monte Carlo
Methods in Scientific Computing which was held at the University of
Salzburg (Austria) from July 9--12, 1996. The conference was a
forum for recent progress in the theory and the applications of
these methods. The topics covered in this volume range from
theoretical issues in Monte Carlo and simulation methods,
low-discrepancy point sets and sequences, lattice rules, and
pseudorandom number generation to applications such as numerical
integration, numerical linear algebra, integral equations, binary
search, global optimization, computational physics, mathematical
finance, and computer graphics. These proceedings will be of
interest to graduate students and researchers in Monte Carlo and
quasi-Monte Carlo methods, to numerical analysts, and to
practitioners of simulation methods.
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