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Error Estimates and Condition Numbers for Radial Basis Function Interpolation
 Adv. Comput. Math
, 1994
"... : For interpolation of scattered multivariate data by radial basis functions, an "uncertainty relation" between the attainable error and the condition of the interpolation matrices is proven. It states that the error and the condition number cannot both be kept small. Bounds on the Lebesgu ..."
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Cited by 81 (20 self)
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: For interpolation of scattered multivariate data by radial basis functions, an "uncertainty relation" between the attainable error and the condition of the interpolation matrices is proven. It states that the error and the condition number cannot both be kept small. Bounds on the Lebesgue constants are obtained as a byproduct. A variation of the NarcowichWard theory of upper bounds on the norm of the inverse of the interpolation matrix is presented in order to handle the whole set of radial basis functions that are currently in use. 1 Introduction Interpolation by "radial" basis functions requires a function \Phi : IR d ! IR, a space IP d m of dvariate polynomials of degree less than m, and interpolates data values y 1 ; . . . ; yN 2 IR at data locations ("centers") x 1 ; . . . ; xN 2 IR d by solving the system N X j=1 ff j \Phi(x j \Gamma x k ) + Q X `=1 fi ` p ` (x k ) = y k ; 1 k N N X j=1 ff j p i (x j ) + 0 = 0; 1 i Q (1:1) for a basis p 1 ; . . . ; pQ...
Multistep scattered data interpolation using compactly supported radial basis functions
 J. Comp. Appl. Math
, 1996
"... Abstract. A hierarchical scheme is presented for smoothly interpolating scattered data with radial basis functions of compact support. A nested sequence of subsets of the data is computed efficiently using successive Delaunay triangulations. The scale of the basis function at each level is determine ..."
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Cited by 65 (12 self)
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Abstract. A hierarchical scheme is presented for smoothly interpolating scattered data with radial basis functions of compact support. A nested sequence of subsets of the data is computed efficiently using successive Delaunay triangulations. The scale of the basis function at each level is determined from the current density of the points using information from the triangulation. The method is rotationally invariant and has good reproduction properties. Moreover the solution can be calculated and evaluated in acceptable computing time. During the last two decades radial basis functions have become a well established tool for multivariate interpolation of both scattered and gridded data; see [2,7,8,22,25] for some surveys. The major part
Creating Surfaces from Scattered Data Using Radial Basis Functions
 in Mathematical Methods for Curves and Surfaces
, 1995
"... . This paper gives an introduction to certain techniques for the construction of geometric objects from scattered data. Special emphasis is put on interpolation methods using compactly supported radial basis functions. x1. Introduction We assume a sample of multivariate scattered data to be given a ..."
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Cited by 56 (11 self)
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. This paper gives an introduction to certain techniques for the construction of geometric objects from scattered data. Special emphasis is put on interpolation methods using compactly supported radial basis functions. x1. Introduction We assume a sample of multivariate scattered data to be given as a set X = fx 1 ; : : : ; xN g of N pairwise distinct points x 1 ; : : : ; xN in IR d , called centers, together with N points y 1 ; : : : ; yN in IR D . An interpolating curve, surface, or solid to these data will be the range of a smooth function s : IR d oe\Omega ! IR D with s(x k ) = y k ; 1 k N: (1) Likewise, an approximating curve, surface, or solid will make the differences s(x j ) \Gamma y j small, for instance in the discrete L 2 sense, i.e. N X k=1 ks(x k ) \Gamma y k k 2 2 should be small. Curves, surfaces, and solids will only differ by their appropriate value of d = 1; 2, or 3. We use the term (geometric) objects to stand for curves, surfaces, or solids. Not...
Error Estimates for Interpolation By Compactly Supported Radial Basis Functions of Minimal Degree
, 1997
"... We consider error estimates for the interpolation by a special class of compactly supported radial basis functions. These functions consist of a univariate polynomial within their support and are of minimal degree depending on space dimension and smoothness. Their associated "native" Hilb ..."
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Cited by 43 (6 self)
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We consider error estimates for the interpolation by a special class of compactly supported radial basis functions. These functions consist of a univariate polynomial within their support and are of minimal degree depending on space dimension and smoothness. Their associated "native" Hilbert spaces are shown to be normequivalent to Sobolev spaces. Thus we can derive approximation orders for functions from Sobolev spaces which are comparable to those of thinplatespline interpolation. Finally, we investigate the numerical stability of the interpolation process.
Multivariate Interpolation and Approximation by Translates of a Basis Function
, 1995
"... . This contribution will touch the following topics: ffl Short introduction into the theory of multivariate interpolation and approximation by finitely many (irregular) translates of a (not necessarily radial) basis function, motivated by optimal recovery of functions from discrete samples. ffl Na ..."
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Cited by 36 (7 self)
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. This contribution will touch the following topics: ffl Short introduction into the theory of multivariate interpolation and approximation by finitely many (irregular) translates of a (not necessarily radial) basis function, motivated by optimal recovery of functions from discrete samples. ffl Native spaces of functions associated to conditionally positive definite functions, and relations between such spaces. ffl Error bounds and condition numbers for interpolation of functions from native spaces. ffl Uncertainty Relation: Why are good error bounds always tied to bad condition numbers? ffl Shift and Scale: How to cope with the Uncertainty Relation? x1. Introduction and Overview This contribution contains the author's view of a certain area of multivariate interpolation and approximation. It is not intended to be a complete survey of a larger area of research, and it will not account for the history of the theory it deals with. Related surveys are [15, 21, 22, 27, 30, 47, 48, 58...
On approximate approximations using Gaussian kernels
 IMA Journal of Numerical Analysis
, 1996
"... This paper discusses quasiinterpolation and interpolation with Gaussians from a new point of view concerning accuracy in numerical computations. Estimates are obtained showing a high order approximation up to some saturation error negligible in numerical applications. The construction of local hig ..."
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Cited by 33 (5 self)
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This paper discusses quasiinterpolation and interpolation with Gaussians from a new point of view concerning accuracy in numerical computations. Estimates are obtained showing a high order approximation up to some saturation error negligible in numerical applications. The construction of local high order quasiinterpolation formulas is given.
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 29 (8 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.
Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions
, 2003
"... ..."
Comparison of Radial Basis Function Interpolants
 In Multivariate Approximation. From CAGD to Wavelets
, 1995
"... This paper compares radial basis function interpolants on different spaces. The spaces are generated by other radial basis functions, and comparison is done via an explicit representation of the norm of the error functional. The results pose some new questions for further research. x1. Introduction ..."
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Cited by 26 (7 self)
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This paper compares radial basis function interpolants on different spaces. The spaces are generated by other radial basis functions, and comparison is done via an explicit representation of the norm of the error functional. The results pose some new questions for further research. x1. Introduction We consider interpolation of realvalued functions f defined on a set \Omega ` IR d ; d 1. These functions are evaluated on a set X := fx 1 ; : : : ; xNX g of NX 1 pairwise distinct points x 1 ; : : : ; xNX in \Omega\Gamma If N 2; d 2 and\Omega ` IR d are given with\Omega containing at least an interior point, it is well known that there is no Ndimensional space of continuous functions on\Omega that contains a unique interpolant for every f and every set X = fx 1 ; : : : ; xNX g ae\Omega ` IR d consisting of N = NX data points. Thus the family of interpolants must necessarily depend on X. This can easily be achieved by using translates \Phi(x \Gamma x j ) of a single continu...
Improved error bounds for scattered data interpolation by radial basis functions
 Math. Comp
, 1999
"... Abstract. If additional smoothness requirements and boundary conditions are met, the well–known approximation orders of scattered data interpolants by radial functions can roughly be doubled. 1. ..."
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Cited by 25 (6 self)
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Abstract. If additional smoothness requirements and boundary conditions are met, the well–known approximation orders of scattered data interpolants by radial functions can roughly be doubled. 1.