Convergence of Increasingly Flat Radial Basis Interpolants to Polynomial Interpolants

Lee, Yeon Ju; Yoon, Gangjoon; Yoon, Jungho

doi:10.1137/050642113

Cited by 25 publications

(31 citation statements)

References 12 publications

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“…It is known that interpolation performed with infinitely smooth kernels, equipped with a shape parameter ε, will yield polynomial interpolation as they approach their ε → 0 "flat" limit (see, e.g., [11,23,33]). Because ε can take positive values, these methods have the potential to achieve accuracy superior to polynomials at essentially the same computational cost.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…This change of basis approach was first used in the pioneering work [16]. Once increasingly flat Gaussian interpolants could be stably computed, the polynomial limit, as predicted in [11,22,23,33], could be numerically confirmed in arbitrary dimensions. Therefore, Gaussians can always produce at least the same accuracy as polynomials because the polynomial result can be obtained in the limit.…”

mentioning

confidence: 99%

“…For many very smooth kernels, or kernels with an increasingly flat parametrization, the computational error may prevent realization of the theoretically optimal accuracy. This problem was analyzed in, e.g., [5,11,23,33,39] and techniques to circumvent this in certain circumstances were discussed in, e.g., [15,16,17].…”

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confidence: 99%

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An introduction to the Hilbert-Schmidt SVD using iterated Brownian bridge kernels

2014

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Kernel-based approximation methods-often in the form of radial basis functions-have been used for many years now and usually involve setting up a kernel matrix which may be ill-conditioned when the shape parameter of the kernel takes on extreme values, i.e., makes the kernel "flat". In this paper we present an algorithm we refer to as the Hilbert-Schmidt SVD and use it to emphasize two important points which-while not entirely new-present a paradigm shift under way in the practical application of kernel-based approximation methods: (i) it is not necessary to form the kernel matrix (in fact, it might even be a bad idea to do so), and (ii) it is not necessary to know the kernel in closed form. While the Hilbert-Schmidt SVD and its two implications apply to general positive definite kernels, we introduce in this paper a class of so-called iterated Brownian bridge kernels which allow us to keep the discussion as simple and accessible as possible.

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Section: Numerical Experimentsmentioning

confidence: 99%

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confidence: 99%

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An introduction to the Hilbert-Schmidt SVD using iterated Brownian bridge kernels

2014

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“…In recent years so-called flat radial basis functions (RBFs) have received much attention in the case when the kernels are infinitely smooth (see, e.g., [5,16,21,22,23,33]). We begin by summarizing the essential insight gained in these papers, and then present some recent results from [38] that deal with radial kernels of finite smoothness in the next subsection.…”

Section: Infinitely Smooth Rbfsmentioning

confidence: 99%

Green’s Functions: Taking Another Look at Kernel Approximation, Radial Basis Functions, and Splines

Fasshauer

2011

Springer Proceedings in Mathematics

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The theories for radial basis functions (RBFs) as well as piecewise polynomial splines have reached a stage of relative maturity as is demonstrated by the recent publication of a number of monographs in either field. However, there remain a number of issues that deserve to be investigated further. For instance, it is well known that both splines and radial basis functions yield "optimal" interpolants, which in the case of radial basis functions are discussed within the so-called native space setting. It is also known that the theory of reproducing kernels provides a common framework for the interpretation of both RBFs and splines. However, the associated reproducing kernel Hilbert spaces (or native spaces) are often not that well understood -especially in the case of radial basis functions. By linking (conditionally) positive definite kernels to Green's functions of differential operators we obtain new insights that enable us to better understand the nature of the native space as a generalized Sobolev space. An additional feature that appears when viewing things from this perspective is the notion of scale built into the definition of these function spaces. Furthermore, the eigenfunction expansion of a positive definite kernel via Mercer's theorem provides a tool for making progress on such important questions as stable computation with flat radial basis functions and dimension independent error bounds.

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“…For Gaussian RBFs, the interpolation matrix is always non-singular for distinct node points. In the flat limit, when ε → 0, most smooth RBFs can diverge for non-unisolvent point sets [14,19,27,21]. However, Gaussian RBFs never diverge [27].…”

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confidence: 99%

Stable Computation of Differentiation Matrices and Scattered Node Stencils Based on Gaussian Radial Basis Functions

Larsson¹,

Lehto²,

Heryudono³

et al. 2013

SIAM J. Sci. Comput.

119

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Abstract. Radial basis function (RBF) approximation has the potential to provide spectrally accurate function approximations for data given at scattered node locations. For smooth solutions, the best accuracy for a given number of node points is typically achieved when the basis functions are scaled to be nearly flat. This also results in nearly linearly dependent basis functions and severe ill-conditioning of the interpolation matrices. Fornberg, Larsson, and Flyer recently generalized the RBF-QR method to provide a numerically stable approach to interpolation with flat and nearly flat Gaussian RBFs for arbitrary node sets in up to three dimensions. In this work, we consider how to extend this method to the task of computing differentiation matrices and stencil weights in order to solve partial differential equations. The expressions for first and second order derivative operators as well as hyperviscosity operators are established, numerical issues such as how to deal with non-unisolvency are resolved, and the accuracy and computational efficiency of the method are tested numerically. The results indicate that using the RBF-QR approach for solving PDE problems can be very competitive compared with using the ill-conditioned direct solution approach or using variable precision arithmetic to overcome the conditioning issue.Key words. radial basis function, flat limit, ill-conditioning, differentiation matrix, stencil weight, RBF, RBF-QR, RBF-FD AMS subject classifications. 65D15, 65D251. Introduction. Radial basis function (RBF) approximation [1,31,4] is emerging as an important method class for interpolation, approximation, and solution of partial differential equations (PDEs) for data given at scattered node locations, with non-trivial geometry, or with computational domains in higher dimensions. The main advantages are the spectral convergence rates that can be achieved using infinitely smooth basis functions, the geometrical flexibility, and the ease of implementation. However, in practical cases, convergence has often been hampered by ill-conditioning as the shape of the basis functions become flatter. The best accuracy for smooth and well resolved solutions is often found in this regime [18,13,19]. Therefore, moving to larger shape parameter values (less flat RBFs) is not a desirable solution to the conditioning problem.The first method that allowed stable computations in the flat RBF regime was the Contour-Padé method derived by Fornberg and Wright [13]. The method works in any number of dimensions, but for relatively low numbers of nodes. Except for the approach in [24], limited to Gaussian RBFs on an equispaced grid in one dimension, the next method that was developed was the RBF-QR method, which was first derived for nodes on the surface of the sphere by Fornberg and Piret [12] and then for general node distributions in up to three dimensions [9]. The RBF-QR methods can be employed for approximations over thousands of nodes.In [9], we gave examples of convergence and performance results in the case of i...

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Convergence of Increasingly Flat Radial Basis Interpolants to Polynomial Interpolants

Cited by 25 publications

References 12 publications

An introduction to the Hilbert-Schmidt SVD using iterated Brownian bridge kernels

An introduction to the Hilbert-Schmidt SVD using iterated Brownian bridge kernels

Green’s Functions: Taking Another Look at Kernel Approximation, Radial Basis Functions, and Splines

Stable Computation of Differentiation Matrices and Scattered Node Stencils Based on Gaussian Radial Basis Functions

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