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This book details the development of techniques and ideas from the radial basis function. It begins with a mathematical description of the basic concept of radial function method with chapters progressively delving into the derivation and construction of radial basis functions for large-scale wave propagation problems including singularity problems, high-frequency wave problems and large-scale computation problems. This reference, written by experts in numerical analysis, demonstrates how the functions arise naturally in mathematical analyses of structures responding to external loads. Readers are also equipped with mathematical knowledge about the radial basis function for understanding key algorithms required for practical solutions.

Radial basis functions.

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This book surveys the latest advances in radial basis function (RBF) meshless collocation methods which emphasis on recent novel kernel RBFs and new numerical schemes for solving partial differential equations. The RBF collocation methods are inherently free of integration and mesh, and avoid tedious mesh generation involved in standard finite element and boundary element methods. This book focuses primarily on the numerical algorithms, engineering applications, and highlights a large class of novel boundary-type RBF meshless collocation methods. These methods have shown a clear edge over the traditional numerical techniques especially for problems involving infinite domain, moving boundary, thin-walled structures, and inverse problems. Due to the rapid development in RBF meshless collocation methods, there is a need to summarize all these new materials so that they are available to scientists, engineers, and graduate students who are interest to apply these newly developed methods for solving real world’s problems. This book is intended to meet this need. Prof. Wen Chen and Dr. Zhuo-Jia Fu work at Hohai University. Prof. C.S. Chen works at the University of Southern Mississippi.

Mathematical physics --- Classical mechanics. Field theory --- Applied physical engineering --- Computer science --- theoretische fysica --- toegepaste mechanica --- computers --- informatica --- informaticaonderzoek --- ingenieurswetenschappen --- mechanica --- computerkunde --- Radial basis functions. --- Collocation methods.

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The basin of attraction of an equilibrium of an ordinary differential equation can be determined using a Lyapunov function. A new method to construct such a Lyapunov function using radial basis functions is presented in this volume intended for researchers and advanced students from both dynamical systems and radial basis functions. Besides an introduction to both areas and a detailed description of the method, it contains error estimates and many examples.

Lyapunov functions. --- Radial basis functions. --- Electronic books. -- local. --- Lyapunov functions --- Radial basis functions --- Engineering & Applied Sciences --- Mathematics --- Physical Sciences & Mathematics --- Applied Mathematics --- Calculus --- Mathematical Theory --- Basis functions, Radial --- Functions, Radial basis --- Radial basis function method --- Functions, Liapunov --- Liapunov functions --- Mathematics. --- Approximation theory. --- Dynamics. --- Ergodic theory. --- Differential equations. --- Dynamical Systems and Ergodic Theory. --- Approximations and Expansions. --- Ordinary Differential Equations. --- 517.91 Differential equations --- Differential equations --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Theory of approximation --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems --- Math --- Science --- Approximation theory --- Differentiable dynamical systems. --- Differential Equations. --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Global analysis (Mathematics) --- Topological dynamics

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This book presents the first “How To” guide to the use of radial basis functions (RBF). It provides a clear vision of their potential, an overview of ready-for-use computational tools and precise guidelines to implement new engineering applications of RBF. Radial basis functions (RBF) are a mathematical tool mature enough for useful engineering applications. Their mathematical foundation is well established and the tool has proven to be effective in many fields, as the mathematical framework can be adapted in several ways. A candidate application can be faced considering the features of RBF: multidimensional space (including 2D and 3D), numerous radial functions available, global and compact support, interpolation/regression. This great flexibility makes RBF attractive – and their great potential has only been partially discovered. This is because of the difficulty in taking a first step toward RBF as they are not commonly part of engineers’ cultural background, but also due to the numerical complexity of RBF problems that scales up very quickly with the number of RBF centers. Fast RBF algorithms are available to alleviate this and high-performance computing (HPC) can provide further aid. Nevertheless, a consolidated tradition in using RBF in engineering applications is still missing and the beginner can be confused by the literature, which in many cases is presented with language and symbolisms familiar to mathematicians but which can be cryptic for engineers. The book is divided in two main sections. The first covers the foundations of RBF, the tools available for their quick implementation and guidelines for facing new challenges; the second part is a collection of practical RBF applications in engineering, covering several topics, including response surface interpolation in n-dimensional spaces, mapping of magnetic loads, mapping of pressure loads, up-scaling of flow fields, stress/strain analysis by experimental displacement fields, implicit surfaces, mesh to cad deformation, mesh morphing for crack propagation in 3D, ice and snow accretion using computational fluid dynamics (CFD) data, shape optimization for external aerodynamics, and use of adjoint data for surface sculpting. For each application, the complete path is clearly and consistently exposed using the systematic approach defined in the first section.

Radial basis functions. --- Mathematics. --- Algorithms. --- Computer mathematics. --- Structural mechanics. --- Computational Science and Engineering. --- Algorithm Analysis and Problem Complexity. --- Structural Mechanics. --- Basis functions, Radial --- Functions, Radial basis --- Radial basis function method --- Approximation theory --- Computer science. --- Computer software. --- Mechanics. --- Mechanics, Applied. --- Solid Mechanics. --- Applied mechanics --- Engineering, Mechanical --- Engineering mathematics --- Classical mechanics --- Newtonian mechanics --- Physics --- Dynamics --- Quantum theory --- Software, Computer --- Computer systems --- Informatics --- Science --- Algorism --- Algebra --- Arithmetic --- Computer mathematics --- Electronic data processing --- Mathematics --- Foundations

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“Computational Mathematics, Algorithms, and Data Processing” of MDPI consists of articles on new mathematical tools and numerical methods for computational problems. Topics covered include: numerical stability, interpolation, approximation, complexity, numerical linear algebra, differential equations (ordinary, partial), optimization, integral equations, systems of nonlinear equations, compression or distillation, and active learning.

interpolation --- constraints --- embedded constraints --- generalized multiscale finite element method --- multiscale model reduction --- deep learning --- Deep Neural Nets --- ReLU Networks --- Approximation Theory --- radial basis functions --- native spaces --- truncated function --- approximation --- surface modeling --- second order initial value problems --- linear multistep methods --- Obrechkoff schemes --- trigonometrically fitted --- Darcy-Forchheimer model --- flow in porous media --- nonlinear equation --- heterogeneous media --- finite element method --- multiscale method --- mixed generalized multiscale finite element method --- multiscale basis functions --- two-dimensional domain --- Thiele-like rational interpolation continued fractions with parameters --- unattainable point --- inverse difference --- virtual point --- polynomial chaos --- Szegő polynomials --- directional statistics --- Rogers-Szegő --- state estimation --- clustering