Uniform Distribution and Quasi-Monte Carlo Methods: Discrepancy, Integration and Applications

Medientyp: E-Book; Konferenzbericht

Titel: Uniform Distribution and Quasi-Monte Carlo Methods : Discrepancy, Integration and Applications

Enthält: Preface; Contents; Metric number theory, lacunary series and systems of dilated functions; 1 Uniform distribution modulo 1; 2 Metric number theory; 3 Discrepancy; 4 Lacunary series; 5 Almost everywhere convergence; 6 Sums involving greatest common divisors; Strong uniformity; 1 Introduction; 2 Superuniformity and super-duper uniformity; 2.1 Superuniformity of the typical billiard paths; 2.2 Super-duper uniformity of the 2-dimensional ray; 3 Superuniformmotions; 3.1 Billiards in other shapes; 3.2 Superuniformity of the geodesics on an equifacial tetrahedron surface
Discrepancy theory and harmonic analysis1 Introduction; 2 Exponential sums; 3 Fourier analysis methods; 3.1 Rotated rectangles; 3.2 The lower bound for circles; 3.3 Further remarks; 4 Dyadic harmonic analysis: discrepancy function estimates; 4.1 Lp -discrepancy, 1 < p < ∞ ; 4.2 The L∞discrepancy estimates; 4.3 The other endpoint, L1; Explicit constructions of point sets and sequences with low discrepancy; 1 Introduction; 2 Lower bounds; 3 Upper bounds; 4 Digital nets and sequences; 5 Walsh series expansion of the discrepancy function
6 The construction of finite point sets according to Chen and Skriganov7 The construction of infinite sequences according to Dick and Pillichshammer; 8 Extensions to the Lqdiscrepancy; 9 Extensions to Orlicz norms of the discrepancy function; Subsequences of automatic sequences and uniform distribution; 1 Introduction; 2 Automatic sequences; 3 Subsequences along the sequence nc; 4 Polynomial subsequences; 5 Subsequences along the primes; On Atanassov's methods for discrepancy bounds of low-discrepancy sequences; 1 Introduction; 2 Atanassov's methods for Halton sequences
2.1 Review of Halton sequences2.2 Review of previous bounds for the discrepancy of Halton sequences; 2.3 Atanassov's methods applied to Halton sequences; 2.4 Scrambling Halton sequences with matrices; 3 Atanassov's method for(t,s)-sequences ; 3.1 Review of (t,s)-sequences; 3.2 Review of bounds for the discrepancy of (t,s)-sequences; 3.3 Atanassov'smethod applied to (t,s)-sequences; 3.4 The special case of even bases for (t,s)-sequences; 4 Atanassov's methods for generalized Niederreiter sequences and (??, e, ??)- sequences; The hybrid spectral test: a unifying concept; 1 Introduction
2 Adding digit vectors3 Notation; 4 The hybrid spectral test; 5 Examples; 5.1 Example I: Integration lattices; 5.2 Example II: Extreme and star discrepancy; Tractability of multivariate analytic problems; 1 Introduction; 2 Tractability; 3 A weighted Korobov space of analytic functions; 4 Integration in H(Ks,a,b) ; 5 L2-approximation inH(Ks,a,b); 6 Conclusion and outlook; Discrepancy estimates for sequences: new results and open problems; 1 Introduction; 2 Metrical and average type discrepancy estimates for digital point sets and sequences and for good lattice point sets
3 Discrepancy estimates for and applications of hybrid sequences

Beteiligte: Kritzer, Peter [Hrsg.]; Niederreiter, Harald [Sonstige Person, Familie und Körperschaft]; Pillichshammer, Friedrich [Sonstige Person, Familie und Körperschaft]; Winterhof, Arne [Sonstige Person, Familie und Körperschaft]

Erschienen: Berlin [u.a.]: De Gruyter, 2014

Umfang: Online-Ressource

Sprache: Englisch

DOI: 10.1515/9783110317930

ISBN: 9783110317947; 9783110317930

Identifikator:

RVK-Notation: SK 180 : Zahlentheorie
SK 820 : Stochastische Prozesse

Schlagwörter: Gleichmäßige Verteilung > Monte-Carlo-Simulation

Entstehung:

Anmerkungen: Literaturangaben

Beschreibung: Biographical note: Peter Kritzer, Harald Niederreiter, Friedrich Pillichshammer, and Arne Winterhof, JKU Linz, Austria.

This bookissummarizing the results of theworkshop "Uniform Distribution and Quasi-Monte Carlo Methods" of the RICAM Special Semester on "Applications of Algebra and Number Theory" in October 2013. Thesurvey articlesin this bookfocus 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.

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