Multiscale potential theory: with applications to geoscience
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boston [u.a.]
Birkhäuser
2004
|
| Series: | Applied and numerical harmonic analysis
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 509 S. Ill., graph. Darst. |
| ISBN: | 081764105X |
Staff View
MARC
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Record in the Search Index
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| adam_text | ERRATA
Contents
Preface ix
List of Symbols xiii
1 Introduction 1
2 Preliminary Tools 5
2.1 Basic Settings 5
2.1.1 Vectors and Tensors 6
2.1.2 Differential Operators 9
2.1.3 Integral Theorems 12
2.2 Spherical Nomenclature 13
2.2.1 Spherical Functions 13
2.2.2 Spherical Harmonics 20
2.2.3 Spherical Radial Basis Functions 32
2.2.4 An Uncertainty Principle 34
2.3 Sphere Oriented Potential Theory 47
2.3.1 Harmonic Functions 47
2.3.2 Inner/Outer Harmonics 48
2.3.3 Abel Poisson, Neumann, and Stokes Kernel 52
2.3.4 Closed and Complete Systems 55
2.4 Exercises 63
I Well-Posed Problems 69
3 Boundary-Value Problems of Potential Theory 71
3.1 Basic Concepts of Potential Theory 71
3.1.1 Regular Surfaces 71
3.1.2 Function Spaces 74
3.1.3 Limit Formulae and Jump Relations 78
3.2 Exterior Dirichlet and Neumann Problem 84
3.2.1 Formulation and Well-posedness 84
3.2.2 Locally Uniform Approximation 88
3.2.3 A Tree Algorithm for Locally Uniform Approximation 106
3.2.4 Globally Uniform Approximation 115
v
vi Contents
3.3 Exterior Oblique Derivative Problem 119
3.3.1 The Problem 119
3.3.2 Existence and Uniqueness 120
3.3.3 L2-Closure 123
3.3.4 A Regularity Theorem 125
3.3.5 L2-Approximation 126
3.4 Runge-Walsh Approximation by Fourier Expansion 127
3.4.1 Motivation 128
3.4.2 Sobolev Spaces 130
3.4.3 Product Kernels in Ho 136
3.4.4 Spectral Signal-to-Noise Thresholding 138
3.4.5 Fourier Representation of Functions on Regular
Surfaces 142
3.4.6 Fourier Modelling of Boundary-Value Problems . . . 154
3.5 Runge-Walsh Approximation by Wavelet Expansion .... 155
3.5.1 Motivation 156
3.5.2 Scaling Functions 163
3.5.3 Windowed Fourier Transform 170
3.5.4 Continuous Wavelet Transform 177
3.5.5 Scale Discrete Wavelet Transform 184
3.5.6 Least-Squares Approximation 197
3.5.7 Examples 198
3.5.8 Multiscale Signal-to-Noise Thresholding 206
3.5.9 Wavelet Representation of Functions on Regular
Surfaces 209
3.5.10 Wavelet Modelling of Boundary-Value Problems . . 216
3.5.11 A Tree Algorithm Using Data on a Sphere 223
3.6 Runge-Walsh Approximation by Spline-Wavelet Expansion 227
3.6.1 Discrete Boundary-Value Problems 228
3.6.2 Harmonic Splines 229
3.6.3 Stability Theorems 239
3.6.4 Some Kernel Representations 244
3.6.5 A Tree Algorithm Based on Spline-Wavelets 245
3.6.6 Multiplicative Schwarz Alternating Algorithm (MSAA):
A Domain Decomposition Method 255
3.7 Exercises 264
4 Boundary-Value Problems of Elasticity 267
4.1 Basic Concepts of Elasticity 268
4.1.1 The Navier Equation 269
4.1.2 Well-Posedness 271
4.1.3 Polynomial Solutions of the Cauchy-Navier Equation 276
4.1.4 Vector Spherical Harmonics as Boundary Values . . 284
4.1.5 Locally Uniform Approximation 290
Contents vii
4.2 Wavelet Approximation Corresponding to a Spherical
Boundary 293
4.2.1 Dirichlet Problem (Scaling Function) 293
4.2.2 Neumann Problem (Scaling Function) 300
4.2.3 Scale Continuous Wavelets 303
4.2.4 Scale Discrete Wavelets 306
4.3 Spline Approximation Corresponding to a Regular Surface . 308
4.3.1 Discrete Boundary-Value Problems 308
4.3.2 Cauchy-Navier Splines 309
4.3.3 Interpolation by Navier Splines 312
4.3.4 A Stability Theorem 317
4.4 A Tree Algorithm by Combined Interpolation and
Integration 321
4.5 Deformation Analysis for the Lake Blasj0 Area 325
4.6 Exercises 327
II Ill-Posed Problems 331
5 Satellite Problems 333
5.1 Formulation of the Problems 336
5.1.1 The SST Problems 337
5.1.2 The SGG Problem 340
5.2 Uniqueness of the Satellite Problems 343
5.2.1 Uniqueness of the SST Problem 343
5.2.2 Uniqueness of the SGG Problem 345
5.3 Vectorial/Tensorial Basis Systems 347
5.3.1 Holder Closure 349
5.4 Runge-Walsh Approximation by Wavelet Regularization . . 357
5.4.1 Ill-posedness of the SST/SGG Problem 364
5.4.2 Multiscale Regularization 370
5.5 Exercises 399
6 The Gravimetry Problem 401
6.1 Newton s Gravitational Potential 402
6.1.1 The Solution of the Gravimetry Problem 406
6.2 Regularization 418
6.3 Spectral Reconstruction of the Mass Density Distribution . 419
6.3.1 Spectral Reconstruction of the Harmonic Solution . 419
6.3.2 Approximation of the Anharmonic Part 424
6.4 Multiscale Reconstruction of the Mass Density Distribution 427
6.4.1 Harmonic Scaling Functions 427
6.4.2 Scale Continuous Harmonic Wavelets 438
6.4.3 Scale Discrete Harmonic Wavelets 448
viii Contents
6.4.4 Anharmonic Scaling Functions 456
6.4.5 Scale Continuous Anharmonic Wavelets 464
6.4.6 Scale Discrete Anharmonic Wavelets 468
6.5 Exercises 471
7 Conclusion 473
8 Hints for the Solution of the Exercises 477
References 483
Index 501
|
| any_adam_object | 1 |
| author | Freeden, Willi 1948- Michel, Volker |
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| author_facet | Freeden, Willi 1948- Michel, Volker |
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| bvnumber | BV025347331 |
| classification_rvk | SK 950 |
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| dewey-full | 515.96 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.96 |
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| dewey-sort | 3515.96 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| institution | BVB |
| isbn | 081764105X |
| language | English |
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| physical | XVI, 509 S. Ill., graph. Darst. |
| publishDate | 2004 |
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| publishDateSort | 2004 |
| publisher | Birkhäuser |
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| series2 | Applied and numerical harmonic analysis |
| spelling | Freeden, Willi 1948- Verfasser (DE-588)121167259 aut Multiscale potential theory with applications to geoscience Willi Freeden ; Volker Michel Boston [u.a.] Birkhäuser 2004 XVI, 509 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Applied and numerical harmonic analysis Geowissenschaften (DE-588)4020288-4 gnd rswk-swf Mathematik (DE-588)4037944-9 gnd rswk-swf Potenzialtheorie (DE-588)4046939-6 gnd rswk-swf Potenzialtheorie (DE-588)4046939-6 s DE-604 Geowissenschaften (DE-588)4020288-4 s Mathematik (DE-588)4037944-9 s Michel, Volker Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=019975155&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Freeden, Willi 1948- Michel, Volker Multiscale potential theory with applications to geoscience Geowissenschaften (DE-588)4020288-4 gnd Mathematik (DE-588)4037944-9 gnd Potenzialtheorie (DE-588)4046939-6 gnd |
| subject_GND | (DE-588)4020288-4 (DE-588)4037944-9 (DE-588)4046939-6 |
| title | Multiscale potential theory with applications to geoscience |
| title_auth | Multiscale potential theory with applications to geoscience |
| title_exact_search | Multiscale potential theory with applications to geoscience |
| title_full | Multiscale potential theory with applications to geoscience Willi Freeden ; Volker Michel |
| title_fullStr | Multiscale potential theory with applications to geoscience Willi Freeden ; Volker Michel |
| title_full_unstemmed | Multiscale potential theory with applications to geoscience Willi Freeden ; Volker Michel |
| title_short | Multiscale potential theory |
| title_sort | multiscale potential theory with applications to geoscience |
| title_sub | with applications to geoscience |
| topic | Geowissenschaften (DE-588)4020288-4 gnd Mathematik (DE-588)4037944-9 gnd Potenzialtheorie (DE-588)4046939-6 gnd |
| topic_facet | Geowissenschaften Mathematik Potenzialtheorie |
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