Iske / Levesley Algorithms for Approximation

Proceedings of the 5th International Conference, Chester, July 2005
1. Auflage 2006
ISBN: 978-3-540-46551-5
Verlag: Springer Berlin Heidelberg
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 389 Seiten

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.

Autoren/Hrsg.

Iske, Armin

Levesley, Jeremy

Inhaltsverzeichnis

1;Preface;5
2;Contents;7
3;List of Contributors;11
4;Part I Imaging and Data Mining;14
4.1;Ranking as Function Approximation;15
4.1.1;1 Introduction;15
4.1.2;2 Measures of Ranking Quality;17
4.1.3;3 Support Vector Ranking;18
4.1.4;4 Perceptron Ranking;19
4.1.5;5 Neural Network Ranking;20
4.1.6;6 Ranking as Learning Structured Outputs;25
4.1.7;Acknowledgement;29
4.1.8;References;29
4.2;Two Algorithms for Approximation in Highly Complicated Planar Domains;31
4.2.1;1 Introduction;31
4.2.2;2 Some Facts about Polynomial Approximation in Planar Domains;32
4.2.3;3 Distance Defect Ratio as a Measure for Domain Singularity;35
4.2.4;4 Algorithm 1: Geometry-Driven Binary Partition;37
4.2.5;5 Algorithm 2: Dimension-Elevation;39
4.2.6;Acknowledgement;41
4.2.7;References;42
4.3;Computational Intelligence in Clustering Algorithms, With Applications;43
4.3.1;1 Introduction;43
4.3.2;2 Clustering Algorithms;44
4.3.3;3 Neural Networks-Based Clustering;46
4.3.4;4 Kernel-Based Clustering;49
4.3.5;5 Applications;51
4.3.6;6 Conclusions;57
4.3.7;Acknowledgement;58
4.3.8;References;58
4.4;Energy-Based Image Simplification with Nonlocal Data and Smoothness Terms;63
4.4.1;1 Introduction;63
4.4.2;2 The Filtering Framework;64
4.4.3;3 Minimisation Methods;66
4.4.4;4 Numerical Experiments;70
4.4.5;5 Conclusions;71
4.4.6;Acknowledgement;71
4.4.7;References;71
4.5;Multiscale Voice Morphing Using Radial Basis Function Analysis;73
4.5.1;1 Introduction;73
4.5.2;2 Description of the System;74
4.5.3;3 Wavelet Analysis;76
4.5.4;4 Radial Basis Functions and Network Training;76
4.5.5;5 Voice Conversion;78
4.5.6;6 Results and Evaluation;79
4.5.7;7 Conclusion;80
4.5.8;References;80
4.6;Associating Families of Curves Using Feature Extraction and Cluster Analysis;82
4.6.1;1 Introduction;82
4.6.2;2 Feature Extraction;83
4.6.3;3 Normalisation;84
4.6.4;4 Standardisation;85
4.6.5;5 Clustering;87
4.6.6;6 Results;89
4.6.7;7 Conclusions and Further Development;90
4.6.8;Acknowledgement;91
4.6.9;References;91
5;Part II Numerical Simulation;92
5.1;Particle Flow Simulation by Using Polyharmonic Splines;93
5.1.1;1 Introduction;93
5.1.2;2 Hyperbolic Problems;95
5.1.3;3 Basic Lagrangian and Eulerian Particle Methods;96
5.1.4;4 Reconstruction by Polyharmonic Splines;99
5.1.5;5 Tracer Transportation over the Arctic Stratosphere;103
5.1.6;6 Oil Reservoir Simulation: The Five-Spot Problem;105
5.1.7;Acknowledgement;108
5.1.8;References;110
5.2;Enhancing SPH using Moving Least-Squares and Radial Basis Functions;113
5.2.1;1 Introduction;113
5.2.2;2 Variationally-Consistent Hydrodynamic Equations;115
5.2.3;3 Moving Least-Squares and Radial Basis Functions;118
5.2.4;4 Numerical Results;119
5.2.5;Acknowledgements;122
5.2.6;References;122
5.3;Stepwise Calculation of the Basin of Attraction in Dynamical Systems Using Radial Basis Functions;123
5.3.1;1 Radial Basis Functions and a Cauchy Problem;123
5.3.2;2 Application to Dynamical Systems;127
5.3.3;3 Stepwise Calculation of the Basin of Attraction;129
5.3.4;References;131
5.4;Integro-Differential Equation Models and Numerical Methods for Cell Motility and Alignment;133
5.4.1;1 Introduction;133
5.4.2;2 Cell Motility Integro-Differential Equation Models;134
5.4.3;3 Cell Alignment Integro-Differential Models;136
5.4.4;4 Some Computational Results;140
5.4.5;5 Further Work;141
5.4.6;References;142
5.5;Spectral Galerkin Method Applied to Some Problems in Elasticity;145
5.5.1;1 Introduction;145
5.5.2;2 Spectral Galerkin Method;146
5.5.3;3 Linear Elasticity;149
5.5.4;4 Friction Contact;151
5.5.5;5 Conclusions;153
5.5.6;References;154
6;Part III Statistical Approximation Methods;155
6.1;Bayesian Field Theory Applied to Scattered Data Interpolation and Inverse Problems;156
6.1.1;1 Introduction;156
6.1.2;2 Scattered Data Interpolation;157
6.1.3;3 Deterministic Scattered Data Interpolation;158
6.1.4;4 Statistical Scattered Data Interpolation;159
6.1.5;5 Inverse Problems;171
6.1.6;6 Concluding Discussion;173
6.1.7;Acknowledgement;174
6.1.8;References;174
6.2;Algorithms for Structured Gauss-Markov Regression;176
6.2.1;1 Introduction;176
6.2.2;2 Uncertainty Matrix Associated with Data Points;177
6.2.3;3 Fitting Parametric Surfaces to Data;179
6.2.4;4 Generalised Distance Regression;181
6.2.5;5 Solution of the Generalised Footpoint Problem;186
6.2.6;6 Surface Fitting for Structured Uncertainty Matrices;191
6.2.7;7 Concluding Remarks;192
6.2.8;Acknowledgement;193
6.2.9;References;193
6.3;Uncertainty Evaluation in Reservoir Forecasting by Bayes Linear Methodology;195
6.3.1;1 Introduction;195
6.3.2;2 Bayes Linear Methodology;196
6.3.3;3 Construction of the Emulator;197
6.3.4;4 Numerical Results for the PUNQS Test Case;199
6.3.5;5 Conclusion;203
6.3.6;Acknowledgement;204
6.3.7;References;204
7;Part IV Data Fitting and Modelling;205
7.1;Integral Interpolation;206
7.1.1;1 Introduction;206
7.1.2;2 Integral Interpolation and Interpolation with General Functionals;210
7.1.3;3 Computational Issues;213
7.1.4;4 Some Explicit Line Sources;215
7.1.5;5 Some Explicit Ball Sources in R3;219
7.1.6;6 An Application: Approximating Track Data with Line Sources;222
7.1.7;Acknowledgement;224
7.1.8;References;225
7.2;Shape Control in Powell-Sabin Quasi- Interpolation;226
7.2.1;1 Introduction;226
7.2.2;2 Tensioned Powell-Sabin Finite Element;227
7.2.3;3 Tensioned Powell-Sabin Quadratic B-splines;232
7.2.4;4 Discrete Quasi-Interpolants with Tension Properties;236
7.2.5;5 Numerical Examples;241
7.2.6;6 Conclusion;245
7.2.7;References;245
7.3;Approximation with Asymptotic Polynomials;247
7.3.1;1 Introduction;247
7.3.2;2 Asymptotic Polynomials;248
7.3.3;3 Approximation with Asymptotic Polynomials;248
7.3.4;4 Example Applications;251
7.3.5;5 Concluding Remarks;253
7.3.6;Acknowledgement;253
7.3.7;References;253
7.4;Spline Approximation Using Knot Density Functions;255
7.4.1;1 Introduction;255
7.4.2;2 Definition of Flexi-Knot Splines;256
7.4.3;3 Approximation with Flexi-Knot Splines;257
7.4.4;4 Example Applications;260
7.4.5;5 Concluding Remarks;263
7.4.6;Acknowledgement;264
7.4.7;References;264
7.5;Neutral Data Fitting by Lines and Planes;265
7.5.1;1 Introduction;265
7.5.2;2 Neutral Data Fitting in Two Dimensions;266
7.5.3;3 Three Dimensions;271
7.5.4;Acknowledgement;274
7.5.5;References;274
7.6;Approximation on an Infinite Range to Ordinary Differential Equations Solutions by a Function of a Radial Basis Function;275
7.6.1;1 Introduction;275
7.6.2;2 Special End Point Behaviour;277
7.6.3;3 Summary of Previous Contributions;278
7.6.4;4 Nonlinear ODEs With Known Solutions and Behaviour;278
7.6.5;5 Numerical Examples;281
7.6.6;6 Conclusions;283
7.6.7;References;284
7.6.8;7 Appendix (Blasius Equation);284
7.7;Weighted Integrals of Polynomial Splines;285
7.7.1;1 Introduction and Motivation;285
7.7.2;2 Recurrence for Integrals of Polynomial B-Splines;286
7.7.3;3 Conclusion;289
7.7.4;Acknowledgement;289
7.7.5;References;289
8;Part V Differential and Integral Equations;291
8.1;On Sequential Estimators for Affine Stochastic Delay Differential Equations;292
8.1.1;1 Preliminaries;292
8.1.2;2 Sequential Estimation Procedure;294
8.1.3;Acknowledgement;300
8.1.4;References;301
8.2;Scalar Periodic Complex Delay Differential Equations: Small Solutions and their Detection;302
8.2.1;1 Introduction and Background;302
8.2.2;2 Known Analytical Results for the Complex Case;303
8.2.3;3 A Summary of our Methodology;303
8.2.4;4 Numerical Results and their Interpretation;304
8.2.5;References;311
8.3;Using Approximations to Lyapunov Exponents to Predict Changes in Dynamical Behaviour in Numerical Solutions to Stochastic Delay Differential Equations;313
8.3.1;1 Introduction;313
8.3.2;2 Dynamical Approach;315
8.3.3;3 Experimental Results;317
8.3.4;References;321
8.4;Superconvergence of Quadratic Spline Collocation for Volterra Integral Equations;323
8.4.1;1 Introduction;323
8.4.2;2 Description of the Method and Convergence Theorem;323
8.4.3;3 Superconvergence in the Case c = 1/2;325
8.4.4;4 Superconvergence in the Case c = 1;327
8.4.5;5 Numerical Tests;329
8.4.6;Acknowledgement;330
8.4.7;References;331
9;Part VI Special Functions and Approximation on Manifolds;332
9.1;Asymptotic Approximations to Truncation Errors of Series Representations for Special Functions;333
9.1.1;1 Introduction;333
9.1.2;2 The Euler-Maclaurin Formula;335
9.1.3;3 Asymptotic Approximations to Truncation Errors;337
9.1.4;4 Numerical Analytic Continuation;339
9.1.5;5 The Dirichlet Series for the Riemann Zeta Function;341
9.1.6;6 The Gaussian Hypergeometric Series;343
9.1.7;7 The Asymptotic Series for the Exponential Integral;346
9.1.8;8 Conclusions and Outlook;348
9.1.9;References;349
9.2;Strictly Positive Definite Functions on Generalized Motion Groups;351
9.2.1;1 Introduction;351
9.2.2;2 Interpolation of Scattered Data;352
9.2.3;3 Strictly Positive Definite Functions on Semi-Direct Products;355
9.2.4;4 Reflection Invariant Functions;357
9.2.5;References;359
9.3;Energy Estimates and the Weyl Criterion on Compact Homogeneous Manifolds;360
9.3.1;1 Introduction;360
9.3.2;2 Weyl’s Criterion;364
9.3.3;3 Energy on Manifolds;366
9.3.4;References;368
9.4;Minimal Discrete Energy Problems and Numerical Integration on Compact Sets in Euclidean Spaces;369
9.4.1;1 Introduction;369
9.4.2;2 Point Energies, Separation, and Mesh Norm for Optimal Riesz Points on d- Rectifiable Sets;372
9.4.3;3 Discrepancy and Errors of Numerical Integration on Spheres;375
9.4.4;Acknowledgement;377
9.4.5;References;377
9.5;Numerical Quadrature of Highly Oscillatory Integrals Using Derivatives;378
9.5.1;1 Introduction;378
9.5.2;2 Univariate Asymptotic Expansion and Filon-type Methods;379
9.5.3;3 Univariate Levin-type Method;380
9.5.4;4 Multivariate Levin-type Method;382
9.5.5;References;385
10;Index;386