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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 Lebesgue constant ..."
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Cited by 80 (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...
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...
A Radial Basis Function Method for Global Optimization
 JOURNAL OF GLOBAL OPTIMIZATION
, 1999
"... We introduce a method that aims to find the global minimum of a continuous nonconvex function on a compact subset of R^d. It is assumed that function evaluations are expensive and that no additional information is available. Radial basis function interpolation is used to define a utility function. T ..."
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Cited by 49 (1 self)
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We introduce a method that aims to find the global minimum of a continuous nonconvex function on a compact subset of R^d. It is assumed that function evaluations are expensive and that no additional information is available. Radial basis function interpolation is used to define a utility function. The maximizer of this function is the next point where the objective function is evaluated. We show that, for most types of radial basis functions that are considered in this paper, convergence can be achieved without further assumptions on the objective function. Besides, it turns out that our method is closely related to a statistical global optimization method, the Palgorithm. A general framework for both methods is presented. Finally, a few numerical examples show that on the set of DixonSzego test functions our method yields favourable results in comparison to other global optimization methods.
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 35 (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...
Error estimates for scattered data interpolation on spheres
 MATH. COMP
, 1999
"... We study Sobolev type estimates for the approximation order resulting from using strictly positive definite kernels to do interpolation on the nsphere. The interpolation knots are scattered. Our approach partly follows the general theory of Golomb and Weinberger and related estimates. These error e ..."
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Cited by 34 (4 self)
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We study Sobolev type estimates for the approximation order resulting from using strictly positive definite kernels to do interpolation on the nsphere. The interpolation knots are scattered. Our approach partly follows the general theory of Golomb and Weinberger and related estimates. These error estimates are then based on series expansions of smooth functions in terms of spherical harmonics. The Markov inequality for spherical harmonics is essential to our analysis and is used in order to find lower bounds for certain sampling operators on spaces of spherical harmonics.
Approximation by Radial Basis Functions with Finitely Many Centers
, 1996
"... : Interpolation by translates of "radial" basis functions \Phi is optimal in the sense that it minimizes the pointwise error functional among all comparable quasiinterpolants on a certain "native" space of functions F \Phi . Since these spaces are rather small for cases where \Phi is smooth, we st ..."
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Cited by 24 (8 self)
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: Interpolation by translates of "radial" basis functions \Phi is optimal in the sense that it minimizes the pointwise error functional among all comparable quasiinterpolants on a certain "native" space of functions F \Phi . Since these spaces are rather small for cases where \Phi is smooth, we study the behavior of interpolants on larger spaces of the form F \Phi 0 for less smooth functions \Phi 0 . It turns out that interpolation by translates of \Phi to mollifications of functions f from F \Phi 0 yields approximations to f that attain the same asymptotic error bounds as (optimal) interpolation of f by translates of \Phi 0 on F \Phi 0 . AMS Classification: 41A15, 41A25, 41A30, 41A63, 65D10 Keywords: Radial Basis Functions, Multivariate Approximation, Approximation Order 1 Introduction Given a continuous realvalued function \Phi on IR d and a nonnegative integer m, we consider approximations by finitely many translates \Phi(\Delta \Gamma x j ); 1 j N , of \Phi together with...
A stable algorithm for flat radial basis functions
 SIAM J. Sci. Comp
"... Abstract. When radial basis functions (RBFs) are made increasingly flat, the interpolation error typically decreases steadily until some point when Rungetype oscillations either halt or reverse this trend. Because the most obvious method to calculate an RBF interpolant becomes a numerically unstabl ..."
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Cited by 19 (6 self)
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Abstract. When radial basis functions (RBFs) are made increasingly flat, the interpolation error typically decreases steadily until some point when Rungetype oscillations either halt or reverse this trend. Because the most obvious method to calculate an RBF interpolant becomes a numerically unstable algorithm for a stable problem in case of nearflat basis functions, there will typically also be a separate point at which disasterous illconditioning enters. We introduce here a new method, RBFQR, which entirely eliminates such illconditioning, and we apply it in the special case when the data points are distributed over the surface of a sphere. This algorithm works even for thousands of node points, and it allows the RBF shape parameter to be optimized without the limitations imposed by stability concerns. Since interpolation in the flat RBF limit on a sphere is found to coincide with spherical harmonics interpolation, new insights are gained as to why the RBF approach (with nonflat basis functions) often is the more accurate of the two methods. Key words. Radial basis functions, RBF, shape parameter, sphere, spherical harmonics. 1. Introduction. Numerical
Local Accuracy for Radial Basis Function Interpolation on Finite Uniform Grids
 J. Approx. Theory
, 1997
"... We consider interpolation on a finite uniform grid by means of one of the radial basis functions (RBF) OE(r) = r for fl ? 0, fl 62 2N or OE(r) = r ln r for fl 2 2N + . For each positive integer N , set h = N and let fx i : i = 1; 2; : : : ; (N + 1) g be the vertices of the uniform gri ..."
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Cited by 14 (1 self)
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We consider interpolation on a finite uniform grid by means of one of the radial basis functions (RBF) OE(r) = r for fl ? 0, fl 62 2N or OE(r) = r ln r for fl 2 2N + . For each positive integer N , set h = N and let fx i : i = 1; 2; : : : ; (N + 1) g be the vertices of the uniform grid of meshsize h on the unit ddimensional cube [0; 1] . Given f : [0; 1] ! R, let sh be its unique RBF interpolant at the grid vertices: sh (x i ) = f(x i ), i = 1; 2; : : : ; (N + 1) . For h ! 0, we show that the uniform norm of the error f \Gamma sh on a compact subset K of the interior of [0; 1] enjoys the same rate of convergence to zero as the error of RBF interpolation on the infinite uniform grid hZ , provided that f is a data function whose partial derivatives in the interior of [0; 1] up to a certain order can be extended to Lipschitz functions on [0; 1] .
Error Estimates For Approximation By Radial Basis Functions
 in Approximation Theory, Wavelets and Applications
, 1995
"... . This paper considers the problem of error estimates for interpolation by radial basis functions. To this end, a recap of the theory of bounding linear functionals in Hilbert spaces is presented. We begin with a normed linear space X . Let fl 1 ; : : : ; fl m be linear `information functionals' on ..."
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Cited by 10 (2 self)
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. This paper considers the problem of error estimates for interpolation by radial basis functions. To this end, a recap of the theory of bounding linear functionals in Hilbert spaces is presented. We begin with a normed linear space X . Let fl 1 ; : : : ; fl m be linear `information functionals' on X such that, for a given f 2 X , the `information' fl i (f) = ff i ; i = 1; : : : ; m is known. With this data, we wish to compute fl(f) where fl is another linear functional on X . By taking a set of distinct points A = fa 1 ; : : : ; am g and choosing the information functionals to be point evaluations, that is, fl i (f) = f(a i ); i = 1; : : : ; m with fl(f) = f(x) for some fixed point x, we obtain a general interpolation problem. We will, however, concentrate on interpolation by radial basis functions and it will become clear that the analysis which leads to the error estimates can subsequently be used to characterise the interpolant itself. Thus, the theory presented here is a very ...