Algorithms for Approximation A Iske J Levesley PDF
Preview Algorithms for Approximation A Iske J Levesley
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 Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare liableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springer.com (cid:1)c Springer-VerlagBerlinHeidelberg2007 Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. TypesettingbytheauthorsusingaSpringerLATEXmacropackage Coverdesign:design&productionGmbH,Heidelberg Printedonacid-freepaper SPIN:11733195 46/SPi 543210 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]
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