An alternative approach to the oblique derivative problem in potential theory [Elektronische Ressource] / Frank Bauer

Sujets

• Mathematics
• Astronomie

Informations

• Publié par: technische_universitat_kaiserslautern
• Publié le: 01 janvier 2004
• Nombre de lectures: 20
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

An Alternative Approach to the
Oblique Derivative Problem
in Potential Theory
Frank Bauer
Vom Fachbereich Mathematik
der Universität Kaiserslautern
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften
(Doctor rerum naturalium, Dr. rer. nat.)
genehmigte Dissertation
1. Gutachter: Prof. Dr. W. Freeden
2. Gutachter: Prof. Dr. S. Pereverzev
Vollzug der Promotion: 8. September 2004
D 386This thesis has evolved with the advice, feedback and help of many people. In particular
I would like to express my gratitude to Prof. Dr. W. Freeden for giving me the opportu-
nity to work on this subject and supporting me with advice and suggestions throughout
my time. Furthermore I would like to thank HDoz. Dr. V. Michel for valuable discus-
sions and my cooperation partner Dr. P. Schwintzer for answering detailed questions
concerning forthcoming satellite missions.
A part of my work is concerning the symbolical solution of partial differential equations.
At this point I would like to express my thanks to Dr. habil. W. Seiler, M. Hausdorf,
HDoz. Dr. E. Zerz, Dr. A. Quadrat and in particular to V. Levandovskyy for their
valuable ideas and directing me to important literature.
Furthermore I would like to especially thank Prof. Dr. S. Pereverzev for giving a very
interesting lecture course on inverse problems. This lecture constitutes the basis of my
treatment of severely ill-posed problems and gave rise to intense and fertile discussions
on this topic. As I am not an expert in stochastics I gratefully acknowledge the patience
of Dr. J.-P. Stockis who helped me whenever I had a problem at the borderline between
functional analysis and stochastics.
At last I want to thank everybody whom I have not mentioned before including the
members of the Geomathematics Group and my family.
The financial support of the Graduiertenkolleg “Mathematik und Praxis” financing my
two and a half years of research is gratefully acknowledged.Contents
Preface 1
Introduction and Outline 3
1.1 Oblique Derivative Problem . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.2 Oblique Boundary Value Problem . . . . . . . . . . . . . . . . . . 4
1.1.3 Oblique Satellite Problem . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 Operator Split Approach for Δ . . . . . . . . . . . . . . . . . . . 6
1.2.2 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.3 Geoscientifical Problems . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.4 Noise and Regularization . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.5 Unified Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.6 Aspects of Scientific Computing . . . . . . . . . . . . . . . . . . . 8
1.2.7 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Operator Split Approach for Δ 11
2.1 General Problem Setup and Solution Strategy . . . . . . . . . . . . . . . 12
2.1.1 Bidirectional Split Operator . . . . . . . . . . . . . . . . . . . . . 12
2.1.2 Relaxations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.3 Composition and Linearity . . . . . . . . . . . . . . . . . . . . . . 15
2.1.4 Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.5 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Split Operators with Respect to a First Order Operator Condition . . . . 17
2.2.1 First Order Split Operators . . . . . . . . . . . . . . . . . . . . . 19
2.2.2 Second Order Split Operators . . . . . . . . . . . . . . . . . . . . 22
2.2.3 First Computations . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.4 Second Order Terms . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.5 Restriction to Δ = Δ . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.6 Further Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.7 Solving the System . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.8 Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2.9 Composition of Solutions for Pure Second Order Operators . . . . 35
2.3 Split Operators with Respect to a Purely Second Order Operator Condition 36
2.3.1 First Computations . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3.2 Second order terms . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.3.3 Further equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 39ii Contents
2.3.4 Number of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3 Integration 47
3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.1.1 Kelvin Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.1.2 Homogeneous Harmonic Polynomials . . . . . . . . . . . . . . . . 49
3.1.3 Kelvin Transform and Derivatives . . . . . . . . . . . . . . . . . . 50
3.2 Homogeneous Harmonic Polynomials . . . . . . . . . . . . . . . . . . . . 51
3.2.1 Differential Operator d . . . . . . . . . . . . . . . . . . . . . . . 51id
3.2.2 Differential Operator d . . . . . . . . . . . . . . . . . . . . . . . 51xi
3.2.3 Differential Operator d . . . . . . . . . . . . . . . . . . . . . . 56¬xi
3.2.4 Differential Operator d . . . . . . . . . . . . . . . . . . . . . . . 58r
3.3 Kernel Spaces and other Remarks . . . . . . . . . . . . . . . . . . . . . . 59
3.4 Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.4.1 Basis Change Matrices . . . . . . . . . . . . . . . . . . . . . . . . 61
3.4.2 Direct Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.5 Other Basis Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4 Geoscientifical Problems 69
4.1 Data Situation and Open Problems . . . . . . . . . . . . . . . . . . . . . 69
4.2 Mathematical Description . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2.1 The Gravitational Field . . . . . . . . . . . . . . . . . . . . . . . 70
4.2.2 Spectral Representation . . . . . . . . . . . . . . . . . . . . . . . 70
4.2.3 From Data to a Solution . . . . . . . . . . . . . . . . . . . . . . . 70
4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5 Noise and Regularization 73
5.1 Data Error for the Satellite Problem . . . . . . . . . . . . . . . . . . . . 73
5.1.1 Integration Grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.1.2 Stochastical Preliminaries . . . . . . . . . . . . . . . . . . . . . . 74
5.1.3 Uncorrelated Noise Case . . . . . . . . . . . . . . . . . . . . . . . 74
5.1.4 Correlated Noise Case . . . . . . . . . . . . . . . . . . . . . . . . 76
5.1.5 Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.1.6 Noise Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.2 Auto-Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.2.1 Functional Analysis Preliminaries . . . . . . . . . . . . . . . . . . 79
5.2.2 The Situation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.2.3 Regularization with Known Smoothness . . . . . . . . . . . . . . 87
5.2.4 Regularization with Unknown Smoothness . . . . . . . . . . . . . 88
5.2.5 Estimations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.3 Noise Estimation out of Two Start Values . . . . . . . . . . . . . . . . . 102
5.3.1 Ordinary Ill-Posed Case . . . . . . . . . . . . . . . . . . . . . . . 109
5.3.2 Severely Ill-Posed Case . . . . . . . . . . . . . . . . . . . . . . . . 112
5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116Contents iii
6 Combining Data in a Unified Setup 117
6.1 Order of the Solution Scheme . . . . . . . . . . . . . . . . . . . . . . . . 117
6.1.1 Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.1.2 Downward-Continuation . . . . . . . . . . . . . . . . . . . . . . . 118
6.1.3 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.2 The Combination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.3 Error and Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.4 Final Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.5 Conclusion and Demands . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7 Aspects of Scientific Computing 125
7.1 Restrictions and Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.1.1 Satellite Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.1.2 Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.1.3 Data Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.1.4 Data Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.1.5 Regularization Method . . . . . . . . . . . . . . . . . . . . . . . . 127
7.1.6 Auto - Regularization . . . . . . . . . . . . . . . . . . .