- Titel
- Discrepancy, Quasi-Monte Carlo and Energy
- Untertitel
- Optimal Point Sets in the Cube and Torus
- Zitierfähige Url:
- https://nbn-resolving.org/urn:nbn:de:bsz:ch1-qucosa2-1026269
- Datum der Einreichung
- 16.10.2025
- Datum der Verteidigung
- 26.01.2026
- ISBN
- 978-3-96100-317-4
- E-ISBN
- 978-3-96100-318-1
- DOI
- https://doi.org/10.51382/978-3-96100-318-1
- Abstract (EN)
- Real world data is frequently given via a continuous function relating two or more quantities of interest with each other. However, for various reasons we might only have a finite number of samples to work with at any time. This leads to the question of how one can extract as much information as possible from a given number of sample nodes. In this thesis we look at such problems from a geometric, harmonic analytic and approximation theoretic point of view. We start by giving a comprehensive overview on uniform point sets (in the sense of low discrepancy, quasi-Monte Carlo worst case errors and energy minimizers) as well as an extensive account of techniques and known results in this area. Throughout the thesis we focus on point sets in the cube and torus, first showing that certain combinatorially defined point sets connect the periodic structure of the torus with the nonperiodic one of the cube. We then look at the problem of energy minimizers in the torus in a product sense, in particular analyzing the optimality of so called Fibonacci lattices for this purpose. To conclude we turn to the question of whether one can give simple constructions of optimal point sets for the quasi-Monte Carlo method in the cube in any dimension, particularly arguing for the use of tent transformed order 2 nets in this context.
- Freie Schlagwörter (EN)
- Energy minimization, Quasi-Monte Carlo method, Worst-case error bounds, Discrepancy theory
- Klassifikation (DDC)
- 515
- 516
- 518
- Normschlagwörter (GND)
- Harmonische Analyse, Globale Optimierung, Numerische Integration
- GutachterIn
- Prof. Dr. Tino Ullrich
- Prof. Dr. Dmitriy Bilyk
- A.Univ.-Prof. Dr. Friedrich Pillichshammer
- BetreuerIn Hochschule / Universität
- Prof. Dr. Tino Ullrich
- Verlag
- Universitätsverlag Chemnitz, Chemnitz
- Den akademischen Grad verleihende / prüfende Institution
- Technische Universität Chemnitz, Chemnitz
- Förder- / Projektangaben
- Sächsische Aufbaubank (SAB)
ESF Plus
Sensorintegration und effiziente Datenanalyse für die Regelung von Wasserstoffbrennstoffzellen
(ReSIDA-H2)
ID: 10064931 - Version / Begutachtungsstatus
- publizierte Version / Verlagsversion
- URN Qucosa
- urn:nbn:de:bsz:ch1-qucosa2-1026269
- Veröffentlichungsdatum Qucosa
- 02.06.2026
- Dokumenttyp
- Dissertation
- Sprache des Dokumentes
- Englisch
- Lizenz / Rechtehinweis
CC BY 4.0- Inhaltsverzeichnis
1 Introduction 2 Preliminaries I: Kernels and linear programming 3 Preliminaries II: Discrepancy and related notions on the cube and torus 4 Precise relations among discrepancies 5 Linear programming methods for tensor product energies 6 Order 2 nets, the tent transform and quasi-Monte Carlo over nonperiodic Sobolev spaces