Algorithms for Approximation (eBook)

Proceedings of the 5th International Conference, Chester, July 2005

Armin Iske, Jeremy Levesley (Herausgeber)

eBook Download: PDF

2006 | 2007
XIII, 389 Seiten
Springer Berlin (Verlag)
978-3-540-46551-5 (ISBN)

Infos

Approximation methods are vital in many challenging applications of computational science and engineering.

This is a collection of papers from world experts in a broad variety of relevant applications, including pattern recognition, machine learning, multiscale modelling of fluid flow, metrology, geometric modelling, tomography, signal and image processing.

It documents recent theoretical developments which have lead to new trends in approximation, it gives important computational aspects and multidisciplinary applications, thus making it a perfect fit for graduate students and researchers in science and engineering who wish to understand and develop numerical algorithms for the solution of their specific problems.

An important feature of the book is that it brings together modern methods from statistics, mathematical modelling and numerical simulation for the solution of relevant problems, with a wide range of inherent scales.

Contributions of industrial mathematicians, including representatives from Microsoft and Schlumberger, foster the transfer of the latest approximation methods to real-world applications.

Inhalt

Preface 5
Contents 7
List of Contributors 11
Part I Imaging and Data Mining 14
Ranking as Function Approximation 15
1 Introduction 15
2 Measures of Ranking Quality 17
3 Support Vector Ranking 18
4 Perceptron Ranking 19
5 Neural Network Ranking 20
6 Ranking as Learning Structured Outputs 25
Acknowledgement 29
References 29
Two Algorithms for Approximation in Highly Complicated Planar Domains 31
1 Introduction 31
2 Some Facts about Polynomial Approximation in Planar Domains 32
3 Distance Defect Ratio as a Measure for Domain Singularity 35
4 Algorithm 1: Geometry-Driven Binary Partition 37
5 Algorithm 2: Dimension-Elevation 39
Acknowledgement 41
References 42
Computational Intelligence in Clustering Algorithms, With Applications 43
1 Introduction 43
2 Clustering Algorithms 44
3 Neural Networks-Based Clustering 46
4 Kernel-Based Clustering 49
5 Applications 51
6 Conclusions 57
Acknowledgement 58
References 58
Energy-Based Image Simplification with Nonlocal Data and Smoothness Terms 63
1 Introduction 63
2 The Filtering Framework 64
3 Minimisation Methods 66
4 Numerical Experiments 70
5 Conclusions 71
Acknowledgement 71
References 71
Multiscale Voice Morphing Using Radial Basis Function Analysis 73
1 Introduction 73
2 Description of the System 74
3 Wavelet Analysis 76
4 Radial Basis Functions and Network Training 76
5 Voice Conversion 78
6 Results and Evaluation 79
7 Conclusion 80
References 80
Associating Families of Curves Using Feature Extraction and Cluster Analysis 82
1 Introduction 82
2 Feature Extraction 83
3 Normalisation 84
4 Standardisation 85
5 Clustering 87
6 Results 89
7 Conclusions and Further Development 90
Acknowledgement 91
References 91
Part II Numerical Simulation 92
Particle Flow Simulation by Using Polyharmonic Splines 93
1 Introduction 93
2 Hyperbolic Problems 95
3 Basic Lagrangian and Eulerian Particle Methods 96
4 Reconstruction by Polyharmonic Splines 99
5 Tracer Transportation over the Arctic Stratosphere 103
6 Oil Reservoir Simulation: The Five-Spot Problem 105
Acknowledgement 108
References 110
Enhancing SPH using Moving Least-Squares and Radial Basis Functions 113
1 Introduction 113
2 Variationally-Consistent Hydrodynamic Equations 115
3 Moving Least-Squares and Radial Basis Functions 118
4 Numerical Results 119
Acknowledgements 122
References 122
Stepwise Calculation of the Basin of Attraction in Dynamical Systems Using Radial Basis Functions 123
1 Radial Basis Functions and a Cauchy Problem 123
2 Application to Dynamical Systems 127
3 Stepwise Calculation of the Basin of Attraction 129
References 131
Integro-Differential Equation Models and Numerical Methods for Cell Motility and Alignment 133
1 Introduction 133
2 Cell Motility Integro-Differential Equation Models 134
3 Cell Alignment Integro-Differential Models 136
4 Some Computational Results 140
5 Further Work 141
References 142
Spectral Galerkin Method Applied to Some Problems in Elasticity 145
1 Introduction 145
2 Spectral Galerkin Method 146
3 Linear Elasticity 149
4 Friction Contact 151
5 Conclusions 153
References 154
Part III Statistical Approximation Methods 155
Bayesian Field Theory Applied to Scattered Data Interpolation and Inverse Problems 156
1 Introduction 156
2 Scattered Data Interpolation 157
3 Deterministic Scattered Data Interpolation 158
4 Statistical Scattered Data Interpolation 159
5 Inverse Problems 171
6 Concluding Discussion 173
Acknowledgement 174
References 174
Algorithms for Structured Gauss-Markov Regression 176
1 Introduction 176
2 Uncertainty Matrix Associated with Data Points 177
3 Fitting Parametric Surfaces to Data 179
4 Generalised Distance Regression 181
5 Solution of the Generalised Footpoint Problem 186
6 Surface Fitting for Structured Uncertainty Matrices 191
7 Concluding Remarks 192
Acknowledgement 193
References 193
Uncertainty Evaluation in Reservoir Forecasting by Bayes Linear Methodology 195
1 Introduction 195
2 Bayes Linear Methodology 196
3 Construction of the Emulator 197
4 Numerical Results for the PUNQS Test Case 199
5 Conclusion 203
Acknowledgement 204
References 204
Part IV Data Fitting and Modelling 205
Integral Interpolation 206
1 Introduction 206
2 Integral Interpolation and Interpolation with General Functionals 210
3 Computational Issues 213
4 Some Explicit Line Sources 215
5 Some Explicit Ball Sources in R3 219
6 An Application: Approximating Track Data with Line Sources 222
Acknowledgement 224
References 225
Shape Control in Powell-Sabin Quasi- Interpolation 226
1 Introduction 226
2 Tensioned Powell-Sabin Finite Element 227
3 Tensioned Powell-Sabin Quadratic B-splines 232
4 Discrete Quasi-Interpolants with Tension Properties 236
5 Numerical Examples 241
6 Conclusion 245
References 245
Approximation with Asymptotic Polynomials 247
1 Introduction 247
2 Asymptotic Polynomials 248
3 Approximation with Asymptotic Polynomials 248
4 Example Applications 251
5 Concluding Remarks 253
Acknowledgement 253
References 253
Spline Approximation Using Knot Density Functions 255
1 Introduction 255
2 Definition of Flexi-Knot Splines 256
3 Approximation with Flexi-Knot Splines 257
4 Example Applications 260
5 Concluding Remarks 263
Acknowledgement 264
References 264
Neutral Data Fitting by Lines and Planes 265
1 Introduction 265
2 Neutral Data Fitting in Two Dimensions 266
3 Three Dimensions 271
Acknowledgement 274
References 274
Approximation on an Infinite Range to Ordinary Differential Equations Solutions by a Function of a Radial Basis Function 275
1 Introduction 275
2 Special End Point Behaviour 277
3 Summary of Previous Contributions 278
4 Nonlinear ODEs With Known Solutions and Behaviour 278
5 Numerical Examples 281
6 Conclusions 283
References 284
7 Appendix (Blasius Equation) 284
Weighted Integrals of Polynomial Splines 285
1 Introduction and Motivation 285
2 Recurrence for Integrals of Polynomial B-Splines 286
3 Conclusion 289
Acknowledgement 289
References 289
Part V Differential and Integral Equations 291
On Sequential Estimators for Affine Stochastic Delay Differential Equations 292
1 Preliminaries 292
2 Sequential Estimation Procedure 294
Acknowledgement 300
References 301
Scalar Periodic Complex Delay Differential Equations: Small Solutions and their Detection 302
1 Introduction and Background 302
2 Known Analytical Results for the Complex Case 303
3 A Summary of our Methodology 303
4 Numerical Results and their Interpretation 304
References 311
Using Approximations to Lyapunov Exponents to Predict Changes in Dynamical Behaviour in Numerical Solutions to Stochastic Delay Differential Equations 313
1 Introduction 313
2 Dynamical Approach 315
3 Experimental Results 317
References 321
Superconvergence of Quadratic Spline Collocation for Volterra Integral Equations 323
1 Introduction 323
2 Description of the Method and Convergence Theorem 323
3 Superconvergence in the Case c = 1/2 325
4 Superconvergence in the Case c = 1 327
5 Numerical Tests 329
Acknowledgement 330
References 331
Part VI Special Functions and Approximation on Manifolds 332
Asymptotic Approximations to Truncation Errors of Series Representations for Special Functions 333
1 Introduction 333
2 The Euler-Maclaurin Formula 335
3 Asymptotic Approximations to Truncation Errors 337
4 Numerical Analytic Continuation 339
5 The Dirichlet Series for the Riemann Zeta Function 341
6 The Gaussian Hypergeometric Series 343
7 The Asymptotic Series for the Exponential Integral 346
8 Conclusions and Outlook 348
References 349
Strictly Positive Definite Functions on Generalized Motion Groups 351
1 Introduction 351
2 Interpolation of Scattered Data 352
3 Strictly Positive Definite Functions on Semi-Direct Products 355
4 Reflection Invariant Functions 357
References 359
Energy Estimates and the Weyl Criterion on Compact Homogeneous Manifolds 360
1 Introduction 360
2 Weyl’s Criterion 364
3 Energy on Manifolds 366
References 368
Minimal Discrete Energy Problems and Numerical Integration on Compact Sets in Euclidean Spaces 369
1 Introduction 369
2 Point Energies, Separation, and Mesh Norm for Optimal Riesz Points on d- Rectifiable Sets 372
3 Discrepancy and Errors of Numerical Integration on Spheres 375
Acknowledgement 377
References 377
Numerical Quadrature of Highly Oscillatory Integrals Using Derivatives 378
1 Introduction 378
2 Univariate Asymptotic Expansion and Filon-type Methods 379
3 Univariate Levin-type Method 380
4 Multivariate Levin-type Method 382
References 385
Index 386

Produkt-Details

• Erscheint lt. Verlag: 13.12.2006
• Zusatzinfo: XIII, 389 p.
• Verlagsort: Berlin
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Informatik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik
• Schlagworte: algorithms • Alignment • Approximation Theory • computational mathematics • Data Mining • Image Processing • Markov • Mathematical Methods in Computational Science and Engineering • Mathematical Statistics • Metrology • Numerical Algorithms • Numerical analysis • Numerical Integration • Numerical Methods • numerical quadrature • Regression • Scientific Computing • Signal • Simulation • Statistica • Statistics
• ISBN-10: 3-540-46551-0 / 3540465510
• ISBN-13: 978-3-540-46551-5 / 9783540465515