Algorithms for Approximation

387 Pages·2006·12.1 MB·English

Algorithms for Approximation
A. Iske J. Levesley
Editors
Algorithms
for Approximation
Proceedings of the 5th International
Conference, Chester, July 2005
With 85 Figures and 21 Tables
ABC
Armin Iske Jeremy Levesley
Universität Hamburg University of Leicester
Department Mathematik Department of Mathematics
Bundesstraße55 University Road
20146Hamburg, Germany Leicester LE1 7RH, United Kingdom
E-mail:[email protected] E-mail:[email protected]
The contribution by Alistair Forbes “Algorithms for Structured Gauss-Markov Regression”
is reproduced by permission of the Controller of HMSO, © Crown Copyright 2006
Mathematics Subject Classification (2000): 65Dxx, 65D15, 65D05, 65D07, 65D17
LibraryofCongressControlNumber:2006934297
ISBN-10 3-540-33283-9SpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-33283-1SpringerBerlinHeidelbergNewYork
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Preface
Approximation methods are of vital importance in many challenging applica-
tions from computational science and engineering. This book collects papers
fromworldexpertsinabroadvarietyofrelevantapplicationsofapproximation
theory,includingpatternrecognitionandmachinelearning,multiscalemodel-
ling of fluid flow, metrology, geometric modelling, the solution of differential
equations, and signal and image processing, to mention a few.
The 30 papers in this volume document new trends in approximation
through recent theoretical developments, important computational aspects
andmultidisciplinary applications,whichmakesitaperfecttextforgraduate
studentsandresearchersfromscienceandengineeringwhowishtounderstand
and develop numerical algorithms for solving their specific problems. An im-
portant feature of the book is to bring together modern methods from statis-
tics, mathematical modelling and numerical simulation for solving relevant
problems with a wide range of inherent scales. Industrial mathematicians, in-
cludingrepresentativesfromMicrosoftandSchlumbergermakecontributions,
which fosters the transfer of the latest approximation methods to real-world
applications.
This book grew out of the fifth in the conference series on Algorithms
for Approximation, which took place from 17th to 21st July 2005, in the
beautiful city of Chester in England. The conference was supported by the
NationalPhysicalLaboratoryandtheLondonMathematicalSociety,andhad
around 90 delegates from over 20 different countries.
The book has been arranged in six parts:
Part I. Imaging and Data Mining;
Part II. Numerical Simulation;
Part III.Statistical Approximation Methods;
Part IV. Data Fitting and Modelling;
Part V. Differential and Integral Equations;
Part VI. Special Functions and Approximation on Manifolds.
VI Preface
PartIgrewoutofaworkshopsponsoredbytheLondonMathematicalSo-
ciety on Developments in Pattern Recognition and Data Mining and includes
contributionsfromDonaldWunsch,thePresidentoftheInternationalNeural
NetworksSocietyandChrisBurgesfromMicrosoft.Thenumericalsolutionof
differential equations lies at the heart of practical application of approxima-
tiontheory.Thenexttwopartscontaincontributionsinthisdirection.PartII
demonstrates the growing trend in the transfer of approximation theory tools
to the simulation of physical systems. In particular, radial basis functions are
gaining a foothold in this regard. Part III has papers concerning the solution
ofdifferentialequations,andespeciallydelaydifferentialequations.Thereali-
sationthatstatisticalKrigingmethodsandradialbasisfunctioninterpolation
are two sides of the same coin has led to an increase in interest in statisti-
cal methods in the approximation community. Part IV reflects ongoing work
in this direction. Part V contains recent developments in traditional areas of
approximation theory, in the modelling of data using splines and radial basis
functions. Part VI is concerned with special functions and approximation on
manifolds such as spheres.
Wearegratefultoalltheauthorswhohavesubmittedforthisvolume,es-
pecially for their patience with the editors. The contributions to this volume
haveallbeenrefereed,andthanksgoouttoalltherefereesfortheirtimelyand
considered comments. Finally, we very much appreciate the cordial relation-
ship we have had with Springer-Verlag, Heidelberg, through Martin Peters.
Leicester, June 2006 Armin Iske
Jeremy Levesley
Contents
Part I Imaging and Data Mining
Ranking as Function Approximation
Christopher J.C. Burges .......................................... 3
Two Algorithms for Approximation in Highly Complicated
Planar Domains
Nira Dyn, Roman Kazinnik ....................................... 19
Computational Intelligence in Clustering Algorithms, With
Applications
Rui Xu, Donald Wunsch II........................................ 31
Energy-Based Image Simplification with Nonlocal Data and
Smoothness Terms
Stephan Didas, Pavel Mr´azek, Joachim Weickert..................... 51
Multiscale Voice Morphing Using Radial Basis Function
Analysis
Christina Orphanidou, Irene M. Moroz, Stephen J. Roberts............ 61
Associating Families of Curves Using Feature Extraction and
Cluster Analysis
Jane L. Terry, Andrew Crampton, Chris J. Talbot ................... 71
Part II Numerical Simulation
Particle Flow Simulation by Using Polyharmonic Splines
Armin Iske...................................................... 83
VIII Contents
Enhancing SPH using Moving Least-Squares and Radial Basis
Functions
Robert Brownlee, Paul Houston, Jeremy Levesley, Stephan Rosswog.....103
Stepwise Calculation of the Basin of Attraction in Dynamical
Systems Using Radial Basis Functions
Peter Giesl......................................................113
Integro-Differential Equation Models and Numerical Methods
for Cell Motility and Alignment
Athena Makroglou................................................123
Spectral Galerkin Method Applied to Some Problems in
Elasticity
Chris J. Talbot ..................................................135
Part III Statistical Approximation Methods
Bayesian Field Theory Applied to Scattered Data
Interpolation and Inverse Problems
Chris L. Farmer .................................................147
Algorithms for Structured Gauss-Markov Regression
Alistair B. Forbes ................................................167
Uncertainty Evaluation in Reservoir Forecasting by Bayes
Linear Methodology
Daniel Busby, Chris L. Farmer, Armin Iske .........................187
Part IV Data Fitting and Modelling
Integral Interpolation
Rick K. Beatson, Michael K. Langton ..............................199
Shape Control in Powell-Sabin Quasi-Interpolation
Carla Manni ....................................................219
Approximation with Asymptotic Polynomials
Philip Cooper, Alistair B. Forbes, John C. Mason ....................241
Spline Approximation Using Knot Density Functions
Andrew Crampton, Alistair B. Forbes...............................249
Neutral Data Fitting by Lines and Planes
Tim Goodman, Chris Tofallis......................................259
Contents IX
Approximation on an Infinite Range to Ordinary Differential
Equations Solutions by a Function of a Radial Basis Function
Damian P. Jenkinson, John C. Mason..............................269
Weighted Integrals of Polynomial Splines
Mladen Rogina ..................................................279
Part V Differential and Integral Equations
On Sequential Estimators for Affine Stochastic Delay
Differential Equations
Uwe Ku¨chler, Vyacheslav Vasiliev..................................287
Scalar Periodic Complex Delay Differential Equations: Small
Solutions and their Detection
Neville J. Ford, Patricia M. Lumb .................................297
Using Approximations to Lyapunov Exponents to Predict
Changes in Dynamical Behaviour in Numerical Solutions to
Stochastic Delay Differential Equations
Neville J. Ford, Stewart J. Norton .................................309
Superconvergence of Quadratic Spline Collocation for
Volterra Integral Equations
Darja Saveljeva..................................................319
Part VI Special Functions and Approximation on Manifolds
Asymptotic Approximations to Truncation Errors of Series
Representations for Special Functions
Ernst Joachim Weniger...........................................331
Strictly Positive Definite Functions on Generalized Motion
Groups
Wolfgang zu Castell, Frank Filbir ..................................349
Energy Estimates and the Weyl Criterion on Compact
Homogeneous Manifolds
Steven B. Damelin, Jeremy Levesley, Xingping Sun ..................359
MinimalDiscreteEnergyProblemsandNumericalIntegration
on Compact Sets in Euclidean Spaces
Steven B. Damelin, Viktor Maymeskul..............................369
X Contents
Numerical Quadrature of Highly Oscillatory Integrals Using
Derivatives
Sheehan Olver ...................................................379
Index..........................................................387
List of Contributors
Rick K. Beatson Andrew Crampton
University of Canterbury University of Huddersfield
Dept of Mathematics and Statistics School of Computing and Engineering
Christchurch 8020, New Zealand Huddersfield HD1 3DH, UK
[email protected] [email protected]
Christopher J.C. Burges
Microsoft Research
Steven B. Damelin
One Microsoft Way
University of Minnesota
Redmond, WA 98052-6399, U.S.A.
Institute Mathematics & Applications
[email protected]
Minneapolis, MN 55455, U.S.A.
Daniel Busby [email protected]
Schlumberger
Abingdon Technology Center
Abingdon OX14 1UJ, UK Stephan Didas
[email protected] Saarland University
Mathematics and Computer Science
Robert Brownlee D-66041 Saarbru¨cken, Germany
University of Leicester [email protected]
Department of Mathematics
Leicester LE1 7RH, UK
[email protected] Nira Dyn
Tel-Aviv University
Wolfgang zu Castell
School of Mathematical Sciences
GSF - National Research Center for
Tel-Aviv 69978, Israel
Environment and Health
[email protected]
D-85764 Neuherberg, Germany
[email protected]
Philip Cooper Chris L. Farmer
University of Huddersfield Schlumberger
School of Computing and Engineering Abingdon Technology Center
Huddersfield HD1 3DH, UK Abingdon OX14 1UJ, UK
[email protected] [email protected]