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Random and Quasi-Random Point Sets

Random and Quasi-Random Point Sets
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Field name Details
Dewey Class 519.5
Title Random and Quasi-Random Point Sets ([EBook] /) / edited by Peter Hellekalek, Gerhard Larcher.
Added Personal Name Hellekalek, Peter editor.
Larcher, Gerhard editor.
Other name(s) SpringerLink (Online service)
Publication New York, NY : : Springer New York : : Imprint: Springer, , 1998.
Physical Details XII, 334 p. 9 illus. : online resource.
Series Lecture Notes in Statistics 0930-0325 ; ; 138
ISBN 9781461217022
Summary Note 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.:
Contents note From Probabilistic Diophantine Approximation to Quadratic Fields -- 1 Part I: Super Irregularity -- 2 Part II: Probabilistic Diophantine Approximation -- 3 Part III: Quadratic Fields and Continued Fractions -- 4 Part IV: Class Number One Problems -- 5 Part V: Cesaro Mean of % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqaaaaaaaaaWdbe % aadaaeqaWdaeaapeWaaeWaa8aabaWdbmaacmaapaqaa8qacaWGUbGa % eqySdegacaGL7bGaayzFaaGaeyOeI0IaaGymaiaac+cacaaIYaaaca % GLOaGaayzkaaaal8aabaWdbiaad6gaaeqaniabggHiLdaaaa!42C9!$$ \sum\nolimits_n {\left( {\left\{ {n\alpha } \right\} - 1/2} \right)} $$ -- 6 References -- On the Assessment of Random and Quasi-Random Point Sets -- 1 Introduction -- 2 Chapter for the Practitioner -- 3 Mathematical Preliminaries -- 4 Uniform Distribution Modulo One -- 5 The Spectral Test -- 6 The Weighted Spectral Test -- 7 Discrepancy -- 8 Summary -- 9 Acknowledgements -- 10 References -- Lattice Rules: How Well Do They Measure Up? -- 1 Introduction -- 2 Some Basic Properties of Lattice Rules -- 3 A General Approach to Worst-Case and Average-Case Error Analysis -- 4 Examples of Other Discrepancies -- 5 Shift-Invariant Kernels and Discrepancies -- 6 Discrepancy Bounds -- 7 Discrepancies of Integration Lattices and Nets -- 8 Tractability of High Dimensional Quadrature -- 9 Discussion and Conclusion -- 10 References -- Digital Point Sets: Analysis and Application -- 1 Introduction -- 2 The Concept and Basic Properties of Digital Point Sets -- 3 Discrepancy Bounds for Digital Point Sets -- 4 Special Classes of Digital Point Sets and Quality Bounds -- 5 Digital Sequences Based on Formal Laurent Series and Non-Archimedean Diophantine Approximation -- 6 Analysis of Pseudo-Random-Number Generators by Digital Nets -- 7 The Digital Lattice Rule -- 8 Outlook and Open Research Topics -- 9 References -- Random Number Generators: Selection Criteria and Testing -- 1 Introduction -- 2 Design Principles and Figures of Merit -- 3 Empirical Statistical Tests -- 4 Examples of Empirical Tests -- 5 Collections of Small RNGs -- 6 Systematic Testing for Small RNGs -- 7 How Do Real-Life Generators Fare in These Tests? -- 8 Acknowledgements -- 9 References -- Nets, (ts)-Sequences, and Algebraic Geometry -- 1 Introduction -- 2 Basic Concepts -- 3 The Digital Method -- 4 Background on Algebraic Curves over Finite Fields -- 5 Construction of (ts)-Sequences -- 6 New Constructions of (tms)-Nets -- 7 New Algebraic Curves with Many Rational Points -- 8 References -- Financial Applications of Monte Carlo and Quasi-Monte Carlo Methods -- 1 Introduction -- 2 Monte Carlo Methods for Finance Applications -- 3 Speeding Up by Quasi-Monte Carlo Methods -- 4 Future Topics -- 5 References.
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