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Lattice rules are a powerful and popular form of quasi-Monte Carlo
rules based on multidimensional integration lattices. This book
provides a comprehensive treatment of the subject with detailed
explanations of the basic concepts and the current methods used in
research. This comprises, for example, error analysis in
reproducing kernel Hilbert spaces, fast component-by-component
constructions, the curse of dimensionality and tractability,
weighted integration and approximation problems, and applications
of lattice rules.
The contributions in this book focus on a variety of topics related
to discrepancy theory, comprising Fourier techniques to analyze
discrepancy, low discrepancy point sets for quasi-Monte Carlo
integration, probabilistic discrepancy bounds, dispersion of point
sets, pair correlation of sequences, integer points in convex
bodies, discrepancy with respect to geometric shapes other than
rectangular boxes, and also open problems in discrepany theory.
This textbook introduces readers to the basic concepts of
quasi-Monte Carlo methods for numerical integration and to the
theory behind them. The comprehensive treatment of the subject with
detailed explanations comprises, for example, lattice rules,
digital nets and sequences and discrepancy theory. It also presents
methods currently used in research and discusses practical
applications with an emphasis on finance-related problems. Each
chapter closes with suggestions for further reading and with
exercises which help students to arrive at a deeper understanding
of the material presented. The book is based on a one-semester,
two-hour undergraduate course and is well-suited for readers with a
basic grasp of algebra, calculus, linear algebra and basic
probability theory. It provides an accessible introduction for
undergraduate students in mathematics or computer science.
Lattice rules are a powerful and popular form of quasi-Monte Carlo
rules based on multidimensional integration lattices. This book
provides a comprehensive treatment of the subject with detailed
explanations of the basic concepts and the current methods used in
research. This comprises, for example, error analysis in
reproducing kernel Hilbert spaces, fast component-by-component
constructions, the curse of dimensionality and tractability,
weighted integration and approximation problems, and applications
of lattice rules.
Harald Niederreiter's pioneering research in the field of applied
algebra and number theory has led to important and substantial
breakthroughs in many areas. This collection of survey articles has
been authored by close colleagues and leading experts to mark the
occasion of his 70th birthday. The book provides a modern overview
of different research areas, covering uniform distribution and
quasi-Monte Carlo methods as well as finite fields and their
applications, in particular, cryptography and pseudorandom number
generation. Many results are published here for the first time. The
book serves as a useful starting point for graduate students new to
these areas or as a refresher for researchers wanting to follow
recent trends.
Indispensable for students, invaluable for researchers, this
comprehensive treatment of contemporary quasi-Monte Carlo methods,
digital nets and sequences, and discrepancy theory starts from
scratch with detailed explanations of the basic concepts and then
advances to current methods used in research. As deterministic
versions of the Monte Carlo method, quasi-Monte Carlo rules have
increased in popularity, with many fruitful applications in
mathematical practice. These rules require nodes with good uniform
distribution properties, and digital nets and sequences in the
sense of Niederreiter are known to be excellent candidates. Besides
the classical theory, the book contains chapters on reproducing
kernel Hilbert spaces and weighted integration, duality theory for
digital nets, polynomial lattice rules, the newest constructions by
Niederreiter and Xing and many more. The authors present an
accessible introduction to the subject based mainly on material
taught in undergraduate courses with numerous examples, exercises
and illustrations.
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