Kritzer / Niederreiter / Pillichshammer Uniform Distribution and Quasi-Monte Carlo Methods

Discrepancy, Integration and Applications
1. Auflage 2014
ISBN: 978-3-11-031793-0
Verlag: De Gruyter
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, Band 15, 269 Seiten

Reihe: Radon Series on Computational and Applied MathematicsISSN

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.

Zielgruppe

Researchers in Mathematics, Finance, Biology, and Computer Science; Academic libraries

Autoren/Hrsg.

Kritzer, Peter

Niederreiter, Harald

Pillichshammer, Friedrich

Winterhof, Arne

Weitere Mitwirkende

Aistleitner, Christoph

Beck, Jozsef

Bilyk, Dmitriy

Dick, Josef

Drmota, Michael

Faure, Henri

Hellekalek, Peter

Larcher, Gerhard

Leobacher, Gunther

Nuyens, Dirk

Wozniakowski, Henryk

Fachgebiete

• Mathematik | Informatik Mathematik Stochastik Wahrscheinlichkeitsrechnung
• Mathematik | Informatik Mathematik Algebra Algebraische Strukturen, Gruppentheorie
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle

Inhaltsverzeichnis

1;Preface;5
2;Contents;7
3;Metric number theory, lacunary series and systems of dilated functions;11
3.1;1 Uniform distribution modulo 1;12
3.2;2 Metric number theory;14
3.3;3 Discrepancy;16
3.4;4 Lacunary series;17
3.5;5 Almost everywhere convergence;20
3.6;6 Sums involving greatest common divisors;22
4;Strong uniformity;27
4.1;1 Introduction;27
4.2;2 Superuniformity and super-duper uniformity;36
4.2.1;2.1 Superuniformity of the typical billiard paths;36
4.2.2;2.2 Super-duper uniformity of the 2-dimensional ray;47
4.3;3 Superuniformmotions;51
4.3.1;3.1 Billiards in other shapes;51
4.3.2;3.2 Superuniformity of the geodesics on an equifacial tetrahedron surface;52
5;Discrepancy theory and harmonic analysis;55
5.1;1 Introduction;55
5.2;2 Exponential sums;56
5.3;3 Fourier analysis methods;59
5.3.1;3.1 Rotated rectangles;59
5.3.2;3.2 The lower bound for circles;61
5.3.3;3.3 Further remarks;63
5.4;4 Dyadic harmonic analysis: discrepancy function estimates;64
5.4.1;4.1 Lp -discrepancy, 1 < p <

8 ;65
5.4.2;4.2 The L8
discrepancy estimates;66
5.4.3;4.3 The other endpoint, L1
;68
6;Explicit constructions of point sets and sequences with low discrepancy;73
6.1;1 Introduction;73
6.2;2 Lower bounds;75
6.3;3 Upper bounds;77
6.4;4 Digital nets and sequences;79
6.5;5 Walsh series expansion of the discrepancy function;81
6.6;6 The construction of finite point sets according to Chen and Skriganov;87
6.7;7 The construction of infinite sequences according to Dick and Pillichshammer;89
6.8;8 Extensions to the Lq
discrepancy;92
6.9;9 Extensions to Orlicz norms of the discrepancy function;93
7;Subsequences of automatic sequences and uniform distribution;97
7.1;1 Introduction;97
7.2;2 Automatic sequences;100
7.3;3 Subsequences along the sequence nc
;103
7.4;4 Polynomial subsequences;105
7.5;5 Subsequences along the primes;108
8;On Atanassov’s methods for discrepancy bounds of low-discrepancy sequences;115
8.1;1 Introduction;115
8.2;2 Atanassov’s methods for Halton sequences;117
8.2.1;2.1 Review of Halton sequences;117
8.2.2;2.2 Review of previous bounds for the discrepancy of Halton sequences;118
8.2.3;2.3 Atanassov’s methods applied to Halton sequences;118
8.2.4;2.4 Scrambling Halton sequences with matrices;123
8.3;3 Atanassov’s method for
(t,s)-sequences ;128
8.3.1;3.1 Review of (t,s)-sequences
;128
8.3.2;3.2 Review of bounds for the discrepancy of (t,s)-sequences
;129
8.3.3;3.3 Atanassov’smethod applied to (t,s)-
sequences;129
8.3.4;3.4 The special case of even bases for (t,s)-sequences;131
8.4;4 Atanassov’s methods for generalized Niederreiter sequences and (??, e, ??)- sequences;134
9;The hybrid spectral test: a unifying concept;137
9.1;1 Introduction;137
9.2;2 Adding digit vectors;139
9.3;3 Notation;142
9.4;4 The hybrid spectral test;144
9.5;5 Examples;147
9.5.1;5.1 Example I: Integration lattices;147
9.5.2;5.2 Example II: Extreme and star discrepancy;150
10;Tractability of multivariate analytic problems;157
10.1;1 Introduction;157
10.2;2 Tractability;159
10.3;3 A weighted Korobov space of analytic functions;164
10.4;4 Integration in H(Ks,a,b) ;166
10.5;5 L2-approximation in
H(Ks,a,b);172
10.6;6 Conclusion and outlook;179
11;Discrepancy estimates for sequences: new results and open problems;181
11.1;1 Introduction;181
11.2;2 Metrical and average type discrepancy estimates for digital point sets and sequences and for good lattice point sets;184
11.3;3 Discrepancy estimates for and applications of hybrid sequences;191
11.4;4 Miscellaneous problems;195
12;A short introduction to quasi-Monte Carlo option pricing;201
12.1;1 Overview;201
12.2;2 Foundations of financial mathematics;202
12.2.1;2.1 Bonds, stocks and derivatives;202
12.2.2;2.2 Arbitrage and the no-arbitrage principle;204
12.2.3;2.3 The Black–Scholesmodel;206
12.2.4;2.4 SDE models;207
12.2.5;2.5 Lévy models;209
12.2.6;2.6 Examples;210
12.3;3 MC and QMC simulation;211
12.3.1;3.1 Nonuniform random number generation;211
12.3.2;3.2 Generation of Brownian paths;218
12.3.3;3.3 Generation of Lévy paths;224
12.3.4;3.4 Multilevel (quasi-)Monte Carlo;226
12.3.5;3.5 Examples;228
13;The construction of good lattice rules and polynomial lattice rules;233
13.1;1 Lattice rules and polynomial lattice rules;233
13.1.1;1.1 Lattice rules;234
13.1.2;1.2 Polynomial lattice rules;235
13.2;2 The worst-case error;237
13.2.1;2.1 Koksma–Hlawka error bound;237
13.2.2;2.2 Lattice rules;239
13.2.3;2.3 Polynomial lattice rules;242
13.3;3 Weighted worst-case errors;246
13.4;4 Some standard spaces;248
13.4.1;4.1 Lattice rules and Fourier spaces;248
13.4.2;4.2 Randomly-shifted lattice rules and the unanchored Sobolev space;249
13.4.3;4.3 Tent-transformed lattice rules and the cosine space;251
13.4.4;4.4 Polynomial lattice rules and Walsh spaces;253
13.5;5 Component-by-component constructions;255
13.5.1;5.1 Component-by-component construction;255
13.5.2;5.2 Fast component-by-component construction;259
13.6;6 Conclusion;262
14;Index;267

Über Autor(innen)

Peter Kritzer, Harald Niederreiter, Friedrich Pillichshammer, and Arne Winterhof, JKU Linz, Austria.