Ebook: 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.

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This book sumarizes recent theoretical and practical developments. The generation and the assessment of pseudo- and quasi-random point sets is one of the basic tasks of applied mathematics and statistics, with implications for Monte Carlo methods, stochastic simulation, and applied statistics. They are also of strong theoretical interest, with applications to algebraic geometry, metric number theory, probability theory, and cryptology.

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This book sumarizes recent theoretical and practical developments. The generation and the assessment of pseudo- and quasi-random point sets is one of the basic tasks of applied mathematics and statistics, with implications for Monte Carlo methods, stochastic simulation, and applied statistics. They are also of strong theoretical interest, with applications to algebraic geometry, metric number theory, probability theory, and cryptology.
Content:
Front Matter....Pages i-xii
From Probabilistic Diophantine Approximation to Quadratic Fields....Pages 1-48
On the Assessment of Random and Quasi-Random Point Sets....Pages 49-108
Lattice Rules: How Well Do They Measure Up?....Pages 109-166
Digital Point Sets: Analysis and Application....Pages 167-222
Random Number Generators: Selection Criteria and Testing....Pages 223-265
Nets, (t, s)-Sequences, and Algebraic Geometry....Pages 267-302
Financial Applications of Monte Carlo and Quasi-Monte Carlo Methods....Pages 303-332
Back Matter....Pages 333-334

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This book sumarizes recent theoretical and practical developments. The generation and the assessment of pseudo- and quasi-random point sets is one of the basic tasks of applied mathematics and statistics, with implications for Monte Carlo methods, stochastic simulation, and applied statistics. They are also of strong theoretical interest, with applications to algebraic geometry, metric number theory, probability theory, and cryptology.
Content:
Front Matter....Pages i-xii
From Probabilistic Diophantine Approximation to Quadratic Fields....Pages 1-48
On the Assessment of Random and Quasi-Random Point Sets....Pages 49-108
Lattice Rules: How Well Do They Measure Up?....Pages 109-166
Digital Point Sets: Analysis and Application....Pages 167-222
Random Number Generators: Selection Criteria and Testing....Pages 223-265
Nets, (t, s)-Sequences, and Algebraic Geometry....Pages 267-302
Financial Applications of Monte Carlo and Quasi-Monte Carlo Methods....Pages 303-332
Back Matter....Pages 333-334
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