A. Bejancu, Jr., Local accuracy for radial basis function interpolation on finite uniform grids. J. Approx. Theory 99 (2) 242--257 (1999).  Home/Search   Document Details and Download   Summary   Related Articles   Check

....the references therein. There are two main groups of radial basis functions, piecewise smooth and infinitely smooth. Some examples of both are given in Table 1. Typically, the piecewise smooth RBFs lead to an algebraic rate of convergence to the desired function as the number of points increase [17, 18], whereas the infinitely smooth RBFs yield a spectral or even faster rate of convergence [19, 20] This is of course assuming that the desired function itself is smooth. # Uppsala University, Department of Information Technology, Scientific Computing, Box 337, SE 751 05 Uppsala, Sweden ....

A. Bejancu, Jr., Local accuracy for radial basis function interpolation on finite uniform grids. J. Approx. Theory 99 (2) 242--257 (1999).

.... error f T f can be written, rst of all, as (f T 00 f) T 00 f T f ) Since , it follows that T f = T T f and hence we obtain f T f = f T 00 f) T 00 (f T f) I II: To express the error in the above form was rst suggested by Mike Powell and can be found in Bejancu [3]. That I decays like O(h ) can be obtained by interpolating between results of Duchon [9] and Matveev [16] For II, we imitate Bejancu s approach, and express II in terms of the Lagrange basis as (1.4) II = T (f T f) L : That the right side converges meaningfully and that equality ....

....modi cations: Regarding f , we assume that f 2 W , and regarding we assume only that regularity property. De nition. For r 0, let r denote the sub domain r : fx 2 : dist(x; rg: Matveev [17] has shown that for a xed r 0 (6. 1) kf T fk ) as h 0 (see also [3]) Our purpose here is to show that (6.1) is still valid when r equals a suciently large constant multiple of h jlog hj. We will thus show that as far as the order of convergence is concerned, the boundary e ects (which degrade the rate of convergence) are con ned to a boundary layer no wider than ....

A. Bejancu, Local accuracy for radial basis function interpolation on nite uniform grids, Journal of Approximation Theory 99 (1999), 242-257.

....[19] It becomes interesting now to see if it is possible to approach L p approximation of order 2m is one changes the rules of the game so as to disabe the boundary effects. One approach is to measure the error not on all of Omega Gamma but rather on a compact subset of Omega Gamma Bejancu [1] has considered the case when Omega is the open unit cube (0 : 1) d and the interpolation points are those points of the grid hZ d which lie in the closed cube [0 : 1] d . He shows that if K is a compact subset of (0 : 1) d and f is sufficiently smooth, then kf Gamma T Xi fk L1 ....

Bejancu A., Local accuracy for radial basis function interpolation on finite uniform grids, manuscript.

....s f;X k L1 ( Omega ) Ch s1=2 kfk Phi (3.7) for f 2 G Omega ; Phi . Inverse Theorems for RBF 5 Actually, in [14] the theorem is stated in a more localized version, but the proof holds true in this situation. There are several other papers giving error bounds of this form, some of them are [1, 3, 5, 12]. Next, we need a stability result on the interpolation process. Therefore, we define the separation distance q X : 1 2 min j 6=k kx j Gamma x k k 2 and cite from [9] Theorem 3.3. Let Phi 2 cpd(m) satisfy the decay condition (2.5) For X = fx 1 ; xN g Omega denote by AX; Phi the ....

....d; Delta Gamma x) x 2 Xg P d m and Theorem 3.2 leads to the error bound kf Gamma s f;X k L1 ( Omega ) ch Gamma d 2 X kfk Phi d; 7.16) For a restricted set of functions f , an improvement in [10] yields kf Gamma s f;X k L1 ( Omega ) c f h 2 Gammad X : 7. 17) In [1] the following improved error estimate is given: Theorem 7.1. Suppose Omega is a cube in IR d and the set of centers X h are given by the grid points hZZ d Omega . If f 2 Lip(2 1; Omega ) then the error can be bounded by kf Gamma s f;Xh k L1 (K) c f h 2 (7.18) for every compact ....

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Bejancu, A., Local accuracy for radial basis function interpolation on finite uniform grids, Preprint Cambridge 1997.

.... Wu and Schaback [27] Powell [22] Matveev [16] Light and Wayne [13] Schaback [24, 25] and Johnson [9] 12] who estimated the dependence on h of the error (or of some of its derivatives) in the uniform or L p norm (1 p 1) over the domain Omega Gamma Further, Matveev [17] and Bejancu [2, 3] proved that the decay of the error as h 0 is significantly faster over a compact subset K of the interior of Omega Gamma Specifically, for any sufficiently differentiable function f , we have max x2K jf(x) Gamma s h (x)j = O(h fl d ) as h 0 ; 1.7) which matches the maximal ....

....which completes the proof of Theorem 1. 2 Remark 1 The exponent fl d in the approximation order (2.6) is maximal, in the sense that there exists a sufficiently smooth bump data function f for which the left hand side of (2. 6) does not tend to zero faster than O(h fl d ) as h 0 (cf. Bejancu [2, 3]) Remark 2 In the case d = 3 and fl = 1, but under different hypotheses on the data function, the maximal convergence order O(h 4 ) for approximation with the corresponding type of surface splines has also been obtained by Hardy and Nelson [8] 3 The Lebesgue Inequality and Kriging Functions ....

Bejancu, A. (1999) Local accuracy for radial basis function interpolation on finite uniform grids. J. Approx. Theory 99, 242--257

....without requiring end conditions for f , even if the order of differentiability of f is increased. Thus the approximation order of the semi cardinal scheme is half of that of the cardinal scheme for general sufficiently smooth data functions. It is now straightforward to modify the arguments of [1] in order to deduce the same order of convergence uniformly on a finite interval, when interpolating with natural splines at equally spaced points, assuming no end conditions for the data function. Acknowledgements. This work was supported by a Research Fellowship of Clare College, Cambridge. ....

Bejancu, A. (1999) Local accuracy for radial basis function interpolation on finite uniform grids. J. Approx. Theory 99, 242--257

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Bej. Bejancu A., Local accuracy for radial basis function interpolation on finite uniform grids, manuscript. B1. Buhmann, M.D. (1990), Multivariate cardinal interpolation with radial basis functions, Constr.

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Bej. Bejancu A. 1998, Local accuracy for radial basis function interpolation on finite uniform grids, manuscript.

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