## Local Error Estimates for Radial Basis Function Interpolation of Scattered Data (1992)

Venue: | IMA J. Numer. Anal |

Citations: | 93 - 20 self |

### BibTeX

@ARTICLE{Wu92localerror,

author = {Zong-min Wu and Robert Schaback},

title = {Local Error Estimates for Radial Basis Function Interpolation of Scattered Data},

journal = {IMA J. Numer. Anal},

year = {1992},

volume = {13},

pages = {13--27}

}

### Years of Citing Articles

### OpenURL

### Abstract

Introducing a suitable variational formulation for the local error of scattered data interpolation by radial basis functions OE(r), the error can be bounded by a term depending on the Fourier transform of the interpolated function f and a certain "Kriging function", which allows a formulation as an integral involving the Fourier transform of OE. The explicit construction of locally well--behaving admissible coefficient vectors makes the Kriging function bounded by some power of the local density h of data points. This leads to error estimates for interpolation of functions f whose Fourier transform f is "dominated" by the nonnegative Fourier transform / of /(x) = OE(kxk) in the sense R j f j 2 / \Gamma1 dt ! 1. Approximation orders are arbitrarily high for interpolation with Hardy multiquadrics, inverse multiquadrics and Gaussian kernels. This was also proven in recent papers by Madych and Nelson, using a reproducing kernel Hilbert space approach and requiring the same h...

### Citations

275 | Interpolation of scattered data: distance matrices and conditionally positive definite functions
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- 1986
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Citation Context ... q, including the case q = Q = 0, the nonsingularity of the (M + Q) \Theta (M + Q) system (1.2), written as ` A P P T 0 ' ` a b ' = ` f 0 ' (1:3) in matrix notation, has been established by Micchelli =-=[8]-=- and Powell [12]. Following [8], we assume F (r) = OE( p r) to be conditionally strictly positive definite of order q, which implies that A is positive definite on the subset of vectors u 2 IR M satis... |

265 |
Radial basis functions for multivariable interpolation: a review
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- 1987
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Citation Context ...he case q = Q = 0, the nonsingularity of the (M + Q) \Theta (M + Q) system (1.2), written as ` A P P T 0 ' ` a b ' = ` f 0 ' (1:3) in matrix notation, has been established by Micchelli [8] and Powell =-=[12]-=-. Following [8], we assume F (r) = OE( p r) to be conditionally strictly positive definite of order q, which implies that A is positive definite on the subset of vectors u 2 IR M satisfying P T u = 0.... |

119 |
Multivariate interpolation and conditionally positive definite functions II
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- 1990
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Citation Context ...centres x j near some x 2\Omega by h ae (x) := max y2K ae (x) min 1jM ky \Gamma x j k for some fixed ae ? 0 , where K ae (x) = fy 2 IR n fi fi kx \Gamma yksaeg, using the Euclidean norm k:k (see also =-=[9]-=-, [10], [11]). Our goal is to prove local error bounds of the following form: Given constants ae 2 IR?0 ; q 2 IN0 , a radial basis function OE and a certain function space F OE to be described below, ... |

54 |
l’erreur d’interpolation des fonctions de plusieurs variables par les Dmsplines
- Sur
- 1978
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Citation Context ... i for �� 2 IN n . Convergence of interpolation on regular grids has been studied extensively by Buhmann (see e.g: [1] for a comprehensive treatment) and others. For the case of scattered data Duc=-=hon [3]-=- treated the thin--plate spline case OE(r) = r 2 log r, while Jackson [6] proved a general, but non--quantitative convergence result. The dissertation of Wu [13] of 1986 contained a rather general Hil... |

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- 1986
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Citation Context ...j ) = 0; 1sjsM for p 2 IP q implies p = 0 (1:4) on q and the positions of data points is required. The numerical problems of (1.3) can be overcome by preconditioning methods (see Dyn, Levin and Rippa =-=[5]-=-, and Dyn [4]), making the radial basis function approach a promising tool for multivariate interpolation. We consider interpolation of values f i = f(x i ) of a smooth function f on a domain\Omega ` ... |

42 |
On the locality of functions
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- 2011
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Citation Context ...the interesting radial basis functions OE, however, we have to use generalized Fourier transforms in (3.1) and (3.2). If (3.1) is interpreted as a generalized Fourier transform (e.g.: in the sense of =-=[7]), t-=-he same interpretation applies to (3.2). Fortunately, we can circumvent these peculiarities, because the function g U (t) := M X j=1 u j e ihx j ;ti \Gamma (it) �� e ihx;ti (3:3) (for U , x j , x,... |

39 |
Interpolation of scattered data by radial functions
- Dyn
- 1987
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Citation Context ...M for p 2 IP q implies p = 0 (1:4) on q and the positions of data points is required. The numerical problems of (1.3) can be overcome by preconditioning methods (see Dyn, Levin and Rippa [5], and Dyn =-=[4]-=-), making the radial basis function approach a promising tool for multivariate interpolation. We consider interpolation of values f i = f(x i ) of a smooth function f on a domain\Omega ` IR n . There ... |

21 |
Multivariate interpolation using radial basis functions
- Buhmann
- 1989
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Citation Context ... Here and in the sequel we use the standard multi--index notation with j��j := P i �� i for �� 2 IN n . Convergence of interpolation on regular grids has been studied extensively by Buhman=-=n (see e.g: [1]-=- for a comprehensive treatment) and others. For the case of scattered data Duchon [3] treated the thin--plate spline case OE(r) = r 2 log r, while Jackson [6] proved a general, but non--quantitative c... |

12 |
An order of convergence for some radial basis functions
- Jackson
- 1989
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Citation Context ... studied extensively by Buhmann (see e.g: [1] for a comprehensive treatment) and others. For the case of scattered data Duchon [3] treated the thin--plate spline case OE(r) = r 2 log r, while Jackson =-=[6]-=- proved a general, but non--quantitative convergence result. The dissertation of Wu [13] of 1986 contained a rather general Hilbert space theory for Kriging and related radial basis function methods i... |

7 |
Error estimates for multiquadric interpolation
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- 1991
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Citation Context ...ur of the Fourier transform of OE near infinity. However, similar convergence orders are obtained, if multiquadric interpolation with a fixed constant c is used on a regular grid (see Buhmann and Dyn =-=[2]). ffl Our-=- methods require the interpolated functions to have Fourier transforms which are "dominated" by b OE in the sense of (4.2). This requirement is not surprising, because a reasonable local con... |

4 | Error Bounds for Multiquadric Interpolation, in: Approximation Theory VI: Vol 2 - Madych, Nelson - 1989 |

4 |
Die Kriging--Methode zur Losung mehrdimensionaler Interpolationsprobleme, Dissertation, Gottingen 1986 Author's addresses: Zong--min Wu
- Wu
- 1986
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Citation Context .... For the case of scattered data Duchon [3] treated the thin--plate spline case OE(r) = r 2 log r, while Jackson [6] proved a general, but non--quantitative convergence result. The dissertation of Wu =-=[13]-=- of 1986 contained a rather general Hilbert space theory for Kriging and related radial basis function methods in the case of a single variable, including error estimates and convergence results. Rece... |