Wu, Z.-M., Schaback, R. (1993): Local error estimates for radial basis function interpolation of scattered data, I.M.A. J. Numer. Anal. 13, 13--27. E-mail address: ab223@damtp.cam.ac.uk 14  Home/Search   Document Details and Download   Summary   Related Articles   Check

....x3. Radial Basis Functions as the Local Method It is now our goal to use radial basis functions to de ne the local approximation spaces. To this end and for the reader s convenience, we review some details on local error estimates for radial basis function interpolation, which can be found in [11]. To every conditionally positive de nite function and every region , there exists a natural function space N ( 3 the native Hilbert space. This space can be de ned in various ways, and we assume that the reader is familiar with the concept (see [10] In many cases the native space ....

....of H older continuity, but we need only a weak form. Hence we de ne the space C ) to be the space of all functions f 2 C ) such that their derivatives of order k satisfy f(x) O(kxk 2 ) for kxk 2 0. The local version for error estimates on radial basis functions is taken from [11]. It can be formulated as Theorem 3. Suppose 2 C ) is conditionally positive de nite of 2 (0; 2) and h 0 be given. Let satisfy an interior cone condition with angle and radius r = C s h. Suppose further X = f x 1 ; xM g satis es h h. Then there ....

Wu, Z., and R. Schaback, Local error estimates for radial basis function interpolation of scattered data, IMA J. of Num. Analysis 13 (1993), 13-27. Holger Wendland Institut fur Numerische und Angewandte Mathematik

....and p : maxfd=2 d=p; 0g. This result was rst proved by Duchon [5] for the particular choice of associated with surface splines. When taken with the construction of Light and Wayne, this amounts to the case when m = are integers and k = m 1. The case p = 1 has been settled by Wu and Schaback [13], while the case when is an integer, m = 0, and k = 1 has been handled by Wendland [12] We expect that the theorem remains true in case d=2 bd=2c 1, but our techniques are unable to cope with this case. Without the restriction m , our error analysis breaks down. However, the case m ....

Z. Wu and R. Schaback, Local error estimates for radial basis function interpolation of scattered data, IMA J. Numer. Anal. 13 (1993), 13-27.

....the lines of the previous paper [4] where the e ective re nement rules were motivated on the basis of available local error estimates for radial basis function interpolation. For the special case of thin plate spline interpolation, the local error estimate at x is according to Wu and Schaback [18] of the form ju(x) s(x)j C h k N ; x) 17) where C 0 is a constant depending on u, and (for some radius 0) hN ; x) sup ky xk dN (y) is the local ll distance of N around x, with dN (y) min ky k being the Euclidean distance between the point y and the set N . We ....

Wu, Z., and R. Schaback (1993) Local error estimates for radial basis function interpolation of scattered data. IMA J. of Numerical Analysis 13, 13-27.

....also proposed in these articles. However, in many cases, it is not possible to compute the solution at the best # value directly in finite precision arithmetic due to the severe ill conditioning of the RBF interpolation matrix. This is illustrated by the uncertainty principle of Wu and Schaback [30], which says that the attainable error and the condition number of the RBF interpolation matrix cannot both be small at the same time. The condition number grows both with decreasing h and decreasing #. A method which circumvents the ill conditioning and makes it possible to solve the RBF ....

Z. M. Wu and R. Schaback, Local error estimates for radial basis function interpolation of scattered data. IMA J. Numer. Anal. 13 (1) 13--27 (1993).

....surface spline interpolation is at least p : minfm; m d=2 d=pg for 1 p 1. He actually showed that = o(h p ) for all f 2 H whenever the domain has the cone property (see the following section for the details) Duchon s error analysis was eventually generalized by Wu and Schaback [21] and Wendland [20] to apply to a large family of radial basis function interpolation methods. At the same time, there were e orts to understand the special case when and = hZ . Although this special case is quite di erent from the desired setup, it was a tempting case because it falls in ....

Wu, Z. and R. Schaback, Local error estimates for radial basis function interpolation of scattered data, IMA J. Numer. Anal. 13 (1993), 13-27.

....on a compact subset Omega of R satisfying a uniform interior cone condition. There is a vast of literature studying this kind of approximation problem by introducing the right space, often called native space, and then giving approximation orders depending on h. We cite for example [4, 8, 9, 11, 19]. Here, we follow [19] because it serves our purposes best and it will come out that the native spaces for our functions are norm equivalent to Sobolev spaces (see theorem 2.1) We start with a positive definite and integrable function Phi and define its Fourier transform by b Phi( ....

....of R satisfying a uniform interior cone condition. There is a vast of literature studying this kind of approximation problem by introducing the right space, often called native space, and then giving approximation orders depending on h. We cite for example [4, 8, 9, 11, 19] Here, we follow [19] because it serves our purposes best and it will come out that the native spaces for our functions are norm equivalent to Sobolev spaces (see theorem 2.1) We start with a positive definite and integrable function Phi and define its Fourier transform by b Phi( 2) Phi(x)e dx: ....

[Article contains additional citation context not shown here]

Z. Wu, R. Schaback, Local error estimates for radial basis function interpolation of scattered data, IMA J. of Numerical Analysis 13 (1993), pp 13-27. 16

....scheme yields optimal interpolants in the Beppo Levi space f 2 C(R ; R) D f 2 L 2 (R ) for all j j = 2 being equipped with the semi norm jf j 2 = f 2f f which represents the bending energy of a thin plate of in nite extent. Due to the results in [16], we obtain global error estimates of the form kc(t; sk L1 C f h h hN = sup y2 min kx yk denotes the ll distance of N in the domain (satisfying an interior cone condition) In our particular application, however, we are mainly interested in the local approximation order ....

Z. Wu, and R. Schaback, Local error estimates for radial basis function interpolation of scattered data, IMA J. Numer. Anal. 13, 1993, 13-27.

....the lines of the previous paper [4] where the e ective re nement rules were motivated on the basis of available local error estimates for radial basis function interpolation. For the special case of thin plate spline interpolation, the local error estimate at x 2 is according to Wu and Schaback [18] of the form ju(x) s(x)j C h k N ; x) 17) where C 0 is a constant depending on u, and (for some radius 0) hN ; x) sup ky xk dN (y) is the local ll distance of N around x, with dN (y) min 2N ky k being the Euclidean distance between the point y and the set N . We ....

Wu, Z., and R. Schaback (1993) Local error estimates for radial basis function interpolation of scattered data. IMA J. of Numerical Analysis 13, 13-27.

....in the Beppo Levi space BL 2 = f 2 C(R 2 ; R) D f 2 L 2 (R 2 ) for all j j = 2 being equipped with the semi norm jf j 2 BL 2 = Z R 2 f 2 x1x1 2f 2 x1x2 f 2 x2x2 dx which represents the bending energy of a thin plate of in nite extent. Due to the results in [16], we obtain global error estimates of the form kc(t; sk L1 ( C f h where h h N ; sup y2 min x2N kx yk denotes the ll distance of N in a bounded and open domain satisfying an interior cone condition. In our particular application, however, we are mainly ....

Z. Wu, and R. Schaback, Local error estimates for radial basis function interpolation of scattered data, IMA J. Numer. Anal. 13, 1993, 13-27.

....theory is to study the accuracy to which s h approximates f over Omega when h 0, under various smoothness assumptions on f . This problem and its version for scattered interpolation points have been investigated by Duchon [7] Arcang eli and Rabut [1] Madych and Nelson [14] 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 ....

....const(fl; Omega Gamma ; 8 h h 0 : 3.5) Proof. We use a well known property of the so called Kriging function associated with the grid Omega hZ d . For a fixed parameter fl 0 and for each sufficiently small h 0, the Kriging function P h : R d [0; 1) is given by (cf. Wu and Schaback [27]) P 2 h (x) Z R d j Theta x (t)j 2 ktk Gammafl Gammad dt ; x 2 R d ; 3.6) where Theta x (t) exp(ix T t) Gamma n X j=1 j (x) exp(ihz T j t) t 2 R d : 3.7) In order to show that the above integral is finite for each x 2 R d , we establish the conditions j Theta ....

[Article contains additional citation context not shown here]

Wu, Z.-M., Schaback, R. (1993) Local error estimates for radial basis function interpolation of scattered data. IMA J. Numer. Anal. 13, 13--27 University of Cambridge Department of Applied Mathematics and Theoretical Physics Silver Street, Cambridge, England CB3 9EW 15

....that of Golomb and Weinberger [10] More recently, Duchon [5, 6] and Madych and Nelson [15, 16] have extensively investigated approximation properties of functions having nonnegative Fourier transforms or, more generally, conditionally positive definite functions of order n on R s . Wu Schaback [30] used kriging methods, which are based on variational techniques, in their investigation of the approximation properties for this class of functions. In this framework, the approximation rates apply to functions in a certain induced space. The object of this paper is to investigate variational ....

Z. Wu and R. Schaback, "Local error estimates for radial basis function interpolation of scattered data," IMA journal of Numerical Analysis 13, (1993), 13-27.

....almost everywhere. By jf j 2 OE = 2) Gammad Z R d j f( j 2 OE( d the space F OE is equipped with a (semi )norm j Delta j OE with nullspace P d m , and F OE =P d m is a Hilbert space. Further details on the structure of the spaces F OE can be found in [11] According to [7, 15], the power function itself can for all of the radial basis function from the introduction be bounded above by PX;OE (x) F OE (h ae;X (x) where for a positive radius ae 0, the quantity h ae;X (x) max y2B ae (x) min 1jN ky Gamma x j k reflects the local density of the points from X ....

Z. Wu and R. Schaback, Local error estimates for radial basis function interpolation of scattered data, IMA J. of Num. Anal. 13 (1993), 13--27.

....d , these functions can pointwise be evaluated by solving the linear system 0 A OE;X PX P T X 0 1 A Delta u(x) v(x) R OE;X (x) Sm (x) 2.4) where v(x) v 1 (x) v Q (x) T 2 R Q and Sm (x) p 1 (x) p Q (x) T . Further details can be found in [27]. For recording the numerical results in Section 5, frequent point evaluations of the power function were required. For reasons of efficiency we preferred to work with an alternative representation of the power function as given by the following Theorem 2.2. For every X 2 X d and OE 2 cpd(m; d) ....

Z. Wu and R. Schaback, Local error estimates for radial basis function interpolation of scattered data, IMA J. of Numerical Analysis 13 (1993), 13--27.

....scattered data interpolation had been presented by Franke [5] in 1982. He showed that MQ have the best performance in terms of accuracy among the methods tested. Micchelli (1986) 6] later gave a proof on the solvability of the RBFs interpolation. In the early 1990 s Madych et al. [7] and Wu et al. [8] had shown that the RBFs interpolant consists of super convergent property. Hardy (1990) provided a comprehensive review of the successful applications of radial basis functions in [9] An intensive study on the theory of the RBFs approximation had been presented by Powell [10] Kansa [11] and ....

Z. Wu and R. Schaback, \Local error estimate for radial basis function interpolation of scattered data", IMA J. Num. Anal., Vol. 13, pp. 13-27, 1993.

....x and the data point x j . The solvability of the resultant system requires that the basis function is positive de nite in the sense that for all choices of points x 1 ; xN and all natural number N , the matrix ( x m x l ) 1 l;m N ) is positive de nite. Refer to Powell [17] and Wu [21, 22] for details on applying RBFs for scattered data interpolation. The use of the univariate function in r in principle saves the computational time for evaluating the approximation of the solution. Furthermore, this representation enjoys the bene t of spatial independence and data structure is ....

Z. Wu & R. Schaback, Local error estimates for radial basis function interpolation of scattered data, IMA J. Numer. Anal., Vol. 13, pp. 13-27, 1993.

....and Duchon s [7] thin plate spline r 2 ln r schemes performed optimally. The idea of the RBFs is to form a basis of function space V = f (kx x j k)g where (r) is a univariate function with r = kx x j k denotes the radial distance between x and the data point x j . Refer to Powell [24] and Wu [31] for details on applying RBFs for scattered data interpolation. The use of the univariate function in r in principle saves the computational time for evaluating the approximation of the solution. Furthermore, this representation enjoys the bene t of spatial independence and data structure is ....

....problems like American option pricing [17] The computations showed the de nite advantages in using this truly mesh free MQ RBFs for solving various initial and boundary values problems. The existence, uniqueness, and convergence proofs in applying the RBFs were given by Micchelli [21] Wu et al. [31] [33] Franke and Schaback [10] Powell [24] Madych and Nelson [20] and Wendland [28] for scattered data interpolation and solving PDEs respectively. Since the radial basis functions are smooth, it can easily be extended to solve high order and high dimensional di erential equations. This ....

[Article contains additional citation context not shown here]

Z. Wu and R. Schaback, Local error estimates for radial basis function interpolation of scattered data, IMA J. Numer. Anal., 13, pp. 13-27, 1993.

....Table 1: PD RBFs certain cases, polynomials up to some small degree m 1 have to be added to ensure a safe reconstruction. More precisely, the functions in question have to be conditionally positive definite (CPD) of some order m, which is zero in the cases of Table 1. Details can be found in [50] or any survey on radial basis functions. We list such functions in Table 2, providing the minimal order m. Note that in these cases the user has to add all polynomials of degree up to m 1 to ensure a safe reconstruction. The situation Linear #(r) r m = 1 Cubic #(r) r 3 m = 2 ....

Z. Wu and R. Schaback. Local error estimates for radial basis function interpolation of scattered data. IMA Journal of Numerical Analysis, 13:13--27, 1993.

.... Fourier transform of the function OE satisfies OE o(1 jwj) GammaK Gammad with spatial dimension d (Here, d = 1 in the discussion of this paper) the approximation order of the interpolation is O(h K 2 ) where h is the density of the points (see Madych Nelson [9] and Wu Schaback [21] for details) The Fourier transform of the radial function OE(x) is said to possess a monotone decay of order K if OE(w) O(1 jwj) GammaK Gamma1 . To handle the time derivative variable, an approximation of the solution in the function space OE T ( which satisfies equation (2.3) on ....

.... error of the function 2 u sol x 2 which is given by f(x; X a j (x) 2 u sol (x j ; x 2 ] Gamma 2 u sol (x; x 2 whose optimal error is of the following order min Z ( X a j (x)e ix j Gamma e ix ) OE( d O(h K=2 ) Using the result of [21], we have (for K 2) kf(x; k k f( k O(h K=2 ) Finally, in the Fourier transformed space, we have u ini ( e Gammaw 2 u ini ( 0) u sol ( e Gammaw 2 [u ini ( 0) Z 0 e 2 t f( t)dt] u sol ( Gamma u ini ( e Gammaw 2 Z 0 e ....

Wu Zongmin and Schaback R., Local Error Estimates for Radial Basis Function Interpolation of Scattered Data, IMA J. Numer. Anal., Vol. 13, pp. 13-27, 1993. 21

....want to focus on the reproduction quality and start with the remark that the classical error bounds for radial basis function interpolation in the nonstationary setting are local. This is not directly stated in the literature, but can be read between the lines of the various proof techniques, e.g. [40, 27]. In principle, if the ll distance h : h(X; of (1) is small enough, and if local reconstruction is to be done at some point x 2 one can con ne the local interpolant to data at points x j with kx x j k 2 ch with a suitable constant c 1. Thus the number of locally required data points can ....

Z. Wu and R. Schaback. Local error estimates for radial basis function interpolation of scattered data. IMA Journal of Numerical Analysis, 13:13{ 27, 1993.

No context found.

Wu, Z.-M., Schaback, R. (1993): Local error estimates for radial basis function interpolation of scattered data, I.M.A. J. Numer. Anal. 13, 13--27. E-mail address: ab223@damtp.cam.ac.uk 14

No context found.

Z. Wu and R. Schaback, Local error estimates for radial basis function interpolation of scattered data, IMA J. Numer. Anal. 13 (1993), 13-27.

No context found.

Wu, Z.-M., Schaback, R. (1993): Local error estimates for radial basis function interpolation of scattered data, I.M.A. J. Numer. Anal. 13, 13--27. E-mail address: ab223@damtp.cam.ac.uk 21

No context found.

Z. Wu and R. Schaback, Local error estimates for radial basis function interpolation of scattered data, IMA J. Numer. Anal. 13 (

No context found.

Wu, Z. and R. Schaback, Local error estimates for radial basis function interpolation of scattered data, IMA J. of Numerical Analysis 13 (1993), 13--27. Holger Wendland Institut fur Numerische und Angewandte Mathematik

No context found.

Wu, Z. and R. Schaback, Local error estimates for radial basis function interpolation of scattered data, IMA J. Numer. Anal., 13, 13-27, (1993).

First 50 documents

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC