In an attempt to introduce application scientists and graduate students to the exciting topic of positive definite kernels and radial basis functions, this book presents modern theoretical results on kernel-based approximation methods and demonstrates their implementation in various settings. The authors explore the historical context of this fascinating topic and explain recent advances as strategies to address long-standing problems.
Examples are drawn from fields as diverse as function approximation, spatial statistics, boundary value problems, machine learning, surrogate modeling and finance. Researchers from those and other fields can recreate the results within using the documented MATLAB code, also available through the online library. This combination of a strong theoretical foundation and accessible experimentation empowers readers to use positive definite kernels on their own problems of interest.
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Contents:An Introduction to Kernel-Based Approximation Methods and Their Stable Computation:IntroductionPositive Definite Kernels and Reproducing Kernel Hilbert SpacesExamples of KernelsKernels in MATLABThe Connection to KrigingThe Connection to Green's KernelsIterated Brownian Bridge Kernels: A Green's Kernel ExampleGeneralized Sobolev SpacesAccuracy and Optimality of Reproducing Kernel Hilbert Space Methods"Flat" LimitsThe Uncertainty Principle — An Unfortunate MisconceptionAlternate BasesStable Computation via the Hilbert–Schmidt SVDParameter OptimizationAdvanced Examples:Scattered Data FittingComputer Experiments and Surrogate ModelingStatistical Data Fitting via Gaussian ProcessesMachine LearningDerivatives of Interpolants and Hermite InterpolationKernel-Based Methods for PDEsFinanceAppendices:Collection of Positive Definite Kernels and Their Known Mercer SeriesHow to Choose the Data SitesA Few Facts from Analysis and ProbabilityThe GaussQR Repository in MATLAB
Readership: Graduate students and researchers.
Key Features:The examples in this book are drawn from many different fields which will provide all readers with results close to their own interestsNewer ideas such as generalized Sobolev spaces and alternate/stable bases are developed in a historical context, explaining both these significant new results but also their genesis over many decadesBy using the software library which accompanies the book, application scientists and graduate students can easily study how the authors implemented new theoretical concepts and quickly adapt them to their problems



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