Numerical analysis using generalised pattern search for a discrete fermionic lattice model of the vacuum [Elektronische Ressource] / vorgelegt von Wätzold Plaum
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Numerical analysis using generalised pattern search for a discrete fermionic lattice model of the vacuum [Elektronische Ressource] / vorgelegt von Wätzold Plaum

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Numerical Analysis UsingGeneralised Pattern Searchfor a Discrete Fermionic LatticeModel of the VacuumDISSERTATIONZUR ERLANGUNG DES DOKTORGRADESDER NATURWISSENSCHAFTEN (DR. RER. NAT.)AN DER NWF I – MATHEMATIK¨DER UNIVERSITAT REGENSBURGvorgelegt vonWa¨tzold Plaum aus Regensburg2009Promotionsgesuch eingereicht am: 9. Juni 2009Die Arbeit wurde angeleitet von: Prof. Dr. Felix FinsterPru¨fungsausschuss: Prof. Dr. Felix FinsterProf. Dr. Georg DolzmannProf. Dr. Bernd AmmannProf. Dr. Gu¨nter Tammefu¨r Laurin und LeonhardIt’s more fun to compute!Kraftwerk – Computerwelt04ContentsAbstract 9Acknowledgments 11Declaration of Symbols 13Introduction 151 A new Model for a discrete Vacuum 171.1 Introducing Remarks to the Theory of the Fermionic Projector . . . . . . 171.2 A Variational Principle in Discrete Space-Time . . . . . . . . . . . . . . 181.3 The Spherically Symmetric Discretization . . . . . . . . . . . . . . . . . 201.4 The Variational Principle on the Lattice . . . . . . . . . . . . . . . . . . 261.5 Definition of the Model and Basic Properties . . . . . . . . . . . . . . . . 301.6 Existence of Minimisers . . . . . . . . . . . . . . . . . . . . . . . . . . 321.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 The Numerical Challenge of Optimisation 352.1 General Introduction into the Problem of Mixed Integer Nonlinear Pro-gramming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.1.

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Published 01 January 2009
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Numerical Analysis Using
Generalised Pattern Search
for a Discrete Fermionic Lattice
Model of the Vacuum
DISSERTATION
ZUR ERLANGUNG DES DOKTORGRADES
DER NATURWISSENSCHAFTEN (DR. RER. NAT.)
AN DER NWF I – MATHEMATIK
¨DER UNIVERSITAT REGENSBURG
vorgelegt von
Wa¨tzold Plaum aus Regensburg
2009Promotionsgesuch eingereicht am: 9. Juni 2009
Die Arbeit wurde angeleitet von: Prof. Dr. Felix Finster
Pru¨fungsausschuss: Prof. Dr. Felix Finster
Prof. Dr. Georg Dolzmann
Prof. Dr. Bernd Ammann
Prof. Dr. Gu¨nter Tammefu¨r Laurin und Leonhard
It’s more fun to compute!
Kraftwerk – Computerwelt0
4Contents
Abstract 9
Acknowledgments 11
Declaration of Symbols 13
Introduction 15
1 A new Model for a discrete Vacuum 17
1.1 Introducing Remarks to the Theory of the Fermionic Projector . . . . . . 17
1.2 A Variational Principle in Discrete Space-Time . . . . . . . . . . . . . . 18
1.3 The Spherically Symmetric Discretization . . . . . . . . . . . . . . . . . 20
1.4 The Variational Principle on the Lattice . . . . . . . . . . . . . . . . . . 26
1.5 Definition of the Model and Basic Properties . . . . . . . . . . . . . . . . 30
1.6 Existence of Minimisers . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2 The Numerical Challenge of Optimisation 35
2.1 General Introduction into the Problem of Mixed Integer Nonlinear Pro-
gramming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.1.1 Relaxation Methods . . . . . . . . . . . . . . . . . . . . . . . . 36
2.1.2 Search Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2 Intermediate Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.3 Generalised Pattern Search Methods . . . . . . . . . . . . . . . . . . . . 42
2.3.1 Positive Spanning Sets . . . . . . . . . . . . . . . . . . . . . . . 42
2.3.2 The basic GPS Algorithm . . . . . . . . . . . . . . . . . . . . . 42
2.3.3 GPS for MVP Problems . . . . . . . . . . . . . . . . . . . . . . 44
2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3 First Numerical Explorations 51
3.1 The General Assumption for the Numerical Analysis . . . . . . . . . . . 51
50 CONTENTS
3.2 Qualitative Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3 General Remarks concerning the Task of Optimisation . . . . . . . . . . 54
3.4 Minima for Small Systems . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.4.1 Systems with one occupied state . . . . . . . . . . . . . . . . . . 55
3.4.2 Systems with two varied states . . . . . . . . . . . . . . . . . . . 56
3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4 Complete Enumerations for more Complex Systems 59
4.1 Spontaneous Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . 59
4.2 Enumerations for n= 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.2.1 Complete enumeration . . . . . . . . . . . . . . . . . . . . . . . 62
4.2.2 Combined Complete Enumeration and GPS-Search . . . . . . . . 66
4.3 Enumerations for n= 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5 Causal Structure 73
5.1 Varying the System Size . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.2 Varying the Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6 Multiple GPS Search 83
6.1 n= 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.2 n= 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7 Performance and Quality Comparison between discretised and relaxed Search 93
7.1 Performance Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7.2 Quality Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
7.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
8 Local Search with slowly increasing n 97
8.1 Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
8.1.1 “Adding a Particle” . . . . . . . . . . . . . . . . . . . . . . . . . 97
8.1.2 The Method of Scattering . . . . . . . . . . . . . . . . . . . . . 98
8.1.3 Tuning the Scattering Factors α andα . . . . . . . . . . . . . . 100ω τ
8.2 The Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
8.2.1 n= 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
8.2.2 n= 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
8.2.3 n= 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
8.2.4 n= 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
8.3 Considering the Runtime . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6CONTENTS 0
8.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
9 Local Search with fast increasing n 119
9.1 Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
9.2 The calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
9.2.1 Extrapolated Start Values . . . . . . . . . . . . . . . . . . . . . . 120
9.2.2 Dirac Sea like Start Values . . . . . . . . . . . . . . . . . . . . . 124
9.3 Interim Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
9.4 Using Advanced Search Steps . . . . . . . . . . . . . . . . . . . . . . . 126
9.4.1 Global Search for n= 12 . . . . . . . . . . . . . . . . . . . . . 126
9.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
10 Discussion and Conclusion 131
10.1 Discussion of the main Assumption of this Thesis . . . . . . . . . . . . . 131
10.2 Further Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
10.2.1 Ideas for future Research Programs . . . . . . . . . . . . . . . . 132
10.2.2 Technical issues . . . . . . . . . . . . . . . . . . . . . . . . . . 135
A Calculating the Formal Gradient in the τ-subspace 137
B Analysis Data 141
B.1 Data belonging to Chapter 4.2.2 . . . . . . . . . . . . . . . . . . . . . . 141
B.1.1 n= 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
B.2 n= 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
B.3 Data belonging to Subsection 8.1.3 . . . . . . . . . . . . . . . . . . . . 149
B.4 Example for Discrete Symmetries . . . . . . . . . . . . . . . . . . . . . 150
B.5 Increasing n, n= 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
B.5.1 Best 20 Solutions, Global Search . . . . . . . . . . . . . . . . . 152
B.6 Data belonging to Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . 153
B.6.1 Dirac Sea like Starting Values . . . . . . . . . . . . . . . . . . . 154
C Standard Settings of the NOMADm interface 157
70 CONTENTS
8Abstract
This thesis deals with the first numerical analysis of the variation principle concerning the
theory of the Fermionic Projector.
A model for describing discrete fermionic systems is developed, whereas the case of
vacuum is discussed. In the continuous case, vacuum systems can be described by the
Fermionic Projector of the Dirac Sea. The discretisation of this concept allows the de-
scription of physical systems by the introduction of an action principle. In this thesis
systems capable of configuring discretisations of continuous systems with one Dirac Sea
are numerically analysed. For the purpose of a more easy numerical analysis spherical
symmetry in momentum space is introduced.
The numerical problem of MINLP emerging from this setting is treated by the methods
of complete enumeration and the MGPS algorithm, an extension of the method of Gener-
alised Pattern Search.
The general hypothesis of this thesis is that there exist Dirac Sea like minimisers. This
thesis could be confirmed by the numerical results. The model had to undergo some subtle
modifications – which has to be considered technical in nature – to deliver the expected
results. Finally several research programs for further research are addressed, which aim
to bring forward the numerical treatment of this problem from a prototypical state to a
state of high performance parallel computing.
90 CONTENTS
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