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29
Kernel Techniques: From Machine Learning to Meshless Methods
, 2006
"... Kernels are valuable tools in various fields of Numerical Analysis, including approximation, interpolation, meshless methods for solving partial differential equations, neural networks, and Machine Learning. This contribution explains why and how kernels are applied in these disciplines. It uncovers ..."
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Cited by 28 (7 self)
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Kernels are valuable tools in various fields of Numerical Analysis, including approximation, interpolation, meshless methods for solving partial differential equations, neural networks, and Machine Learning. This contribution explains why and how kernels are applied in these disciplines. It uncovers the links between them, as far as they are related to kernel techniques. It addresses nonexpert readers and focuses on practical guidelines for using kernels in applications.
Solving Differential Equations with Radial Basis Functions: Multilevel Methods and Smoothing
 Advances in Comp. Math
"... . Some of the meshless radial basis function methods used for the numerical solution of partial differential equations are reviewed. In particular, the differences between globally and locally supported methods are discussed, and for locally supported methods the important role of smoothing within a ..."
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Cited by 26 (7 self)
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. Some of the meshless radial basis function methods used for the numerical solution of partial differential equations are reviewed. In particular, the differences between globally and locally supported methods are discussed, and for locally supported methods the important role of smoothing within a multilevel framework is demonstrated. A possible connection between multigrid finite elements and multilevel radial basis function methods with smoothing is explored. Various numerical examples are also provided throughout the paper. 1. Introduction During the past few years the idea of using socalled meshless methods for the numerical solution of partial differential equations (PDEs) has received much attention throughout the scientific community. As a few representative examples we mention Belytschko and coworker's results [3] using the socalled elementfree Galerkin method, Duarte and Oden's work [11] using hp clouds, Babuska and Melenk 's work [1] on the partition of unity method, ...
Error Bounds for Solving Pseudodifferential Equations on Spheres by Collocation with Zonal Kernels
 J. Approx. Theory
, 1999
"... . The problem of solving pseudodifferential equations on spheres by collocation with zonal kernels is considered and bounds for the approximation error are established. The bounds are given in terms of the maximum separation distance of the collocation points, the order of the pseudodifferential ..."
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Cited by 10 (1 self)
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. The problem of solving pseudodifferential equations on spheres by collocation with zonal kernels is considered and bounds for the approximation error are established. The bounds are given in terms of the maximum separation distance of the collocation points, the order of the pseudodifferential operator, and the smoothness of the employed zonal kernel. A byproduct of the results is an improvement on the previously known convergence order estimates for Lagrange interpolation. 2000 Mathematics Subject Classification. Primary 41A25, 65N15, 65N35; Secondary 41A05, 41A30, 41A63, 58J40, 65D05. Key words and phrases. Pseudodifferential equation, collocation, zonal kernel, interpolation, approximation order, sphere, positive definite function, radial basis function. x1. Introduction Data fitting and solving differential and integral equations on the sphere are areas of growing interest with applications to physical geodesy, potential theory, oceanography, and meteorology [6,10]. As...
Spherical Splines for Data Interpolation and Fitting
 SIAM J. Scientific Computing
, 2005
"... Abstract. We study minimal energy interpolation, discrete and penalized least squares approximation problems on the unit sphere using nonhomogeneous spherical splines. Several numerical experiments are conducted to compare approximating properties of homogeneous and nonhomogeneous splines. Our numer ..."
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Cited by 9 (4 self)
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Abstract. We study minimal energy interpolation, discrete and penalized least squares approximation problems on the unit sphere using nonhomogeneous spherical splines. Several numerical experiments are conducted to compare approximating properties of homogeneous and nonhomogeneous splines. Our numerical experiments show that nonhomogeneous splines have certain advantages over homogeneous splines. Key words. spherical splines, data fitting AMS subject classifications. 65D05, 65D07, 65D17
Multivariate Splines for Data Fitting and Approximation
"... Abstract. Methods for scattered data fitting using multivariate splines will be surveyed in this paper. Existence, uniqueness, and computational algorithms for these methods, as well as their approximation properties will be discussed. Some applications of multivariate splines for data fitting will ..."
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Cited by 9 (7 self)
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Abstract. Methods for scattered data fitting using multivariate splines will be surveyed in this paper. Existence, uniqueness, and computational algorithms for these methods, as well as their approximation properties will be discussed. Some applications of multivariate splines for data fitting will be briefly explained. Some new research initiatives of scattered data fitting will be outlined. Given a set of scattered data, e.g., {(xi, yi, zi), i = 1, · · · , N}, we need to find a smooth function or surface S such that S(xi, yi) = zi, i = 1, · · · , N, if zi are very accurate measurements or
M.J.: Stable evaluation of Gaussian RBF interpolants
 In preparation
"... Abstract. We provide a new way to compute and evaluate Gaussian radial basis function interpolants in a stable way also for small values of the shape parameter, i.e., for “flat ” kernels. This work is motivated by the fundamental ideas proposed earlier by Bengt Fornberg and his coworkers. However, ..."
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Cited by 5 (3 self)
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Abstract. We provide a new way to compute and evaluate Gaussian radial basis function interpolants in a stable way also for small values of the shape parameter, i.e., for “flat ” kernels. This work is motivated by the fundamental ideas proposed earlier by Bengt Fornberg and his coworkers. However, following Mercer’s theorem, an L2(Rd,ρ)orthonormal expansion of the Gaussian kernel allows us to come up with an algorithm that is simpler than the one proposed by Fornberg, Larsson and Flyer and that is applicable in arbitrary space dimensions d. In addition to obtaining an accurate approximation of the RBF interpolant (using many terms in the series expansion of the kernel) we also propose and investigate a highly accurate leastsquares approximation based on early truncation of the kernel expansion. Key words. Radial basis functions, Gaussian kernel, stable evaluation, QR decomposition.
APPROXIMATION ON THE SPHERE USING RADIAL BASIS FUNCTIONS PLUS POLYNOMIALS
"... Abstract. In this paper we analyse a hybrid approximation of functions on the sphere S 2 ⊂ R 3 by radial basis functions combined with polynomials, with the radial basis functions assumed to be generated by a (strictly) positive definite kernel. The approximation is determined by interpolation at sc ..."
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Cited by 4 (1 self)
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Abstract. In this paper we analyse a hybrid approximation of functions on the sphere S 2 ⊂ R 3 by radial basis functions combined with polynomials, with the radial basis functions assumed to be generated by a (strictly) positive definite kernel. The approximation is determined by interpolation at scattered data points, supplemented by side conditions on the coefficients to ensure a square linear system. The analysis is first carried out in the native space associated with the kernel (with no explicit polynomial component, and no side conditions). A more refined error estimate is obtained for functions in a still smaller space. Numerical calculations support the utility of this hybrid approximation. Key words. Scattered data, radial basis functions, spherical harmonics, error estimate. AMS subject classifications. 41A30, 65D30 1. Introduction. In
Galerkin Approximation for Elliptic PDEs on Spheres
 Journal of Approximation Theory
, 2004
"... We discuss a Galerkin approximation scheme for the elliptic partial differential equation −∆u + ω 2 u = f on S n ⊂ R n+1. Here ∆ is the LaplaceBeltrami operator on S n, ω is a nonzero constant and f belongs to C 2k−2 (S n), where k ≥ n/4 + 1, k is an integer. The shifts of a spherical basis functi ..."
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Cited by 3 (0 self)
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We discuss a Galerkin approximation scheme for the elliptic partial differential equation −∆u + ω 2 u = f on S n ⊂ R n+1. Here ∆ is the LaplaceBeltrami operator on S n, ω is a nonzero constant and f belongs to C 2k−2 (S n), where k ≥ n/4 + 1, k is an integer. The shifts of a spherical basis function φ with φ ∈ H τ (S n) and τ> 2k ≥ n/2 + 2 are used to construct an approximate solution. An H 1 (S n)error estimate is derived under the assumption that the exact solution u belongs to C 2k (S n). Key words: spherical basis function, Galerkin method
Interpolating Functions on Lines in 3Space
 Curve and Surface Fitting: Saint Malo 1999
"... Given straight lines L i , i = 1; : : : ; N , in Euclidean 3space with associated function values f i , we study the interpolation problem of constructing a smooth real valued function F which interpolates values f i at given data lines L i . The function F shall be defined on the entire set of l ..."
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Cited by 3 (1 self)
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Given straight lines L i , i = 1; : : : ; N , in Euclidean 3space with associated function values f i , we study the interpolation problem of constructing a smooth real valued function F which interpolates values f i at given data lines L i . The function F shall be defined on the entire set of lines or at least on lines contained in a domain of interest in 3space. x1.