H. Wendland, Error Estimates For Interpolation By Compactly Supported Radial Basis Functions Of Minimal Degree, J. Approx. Theory 93 (1998) 258--272. 15  Home/Search   Document Details and Download   Summary   Related Articles   Check

....yield general approximation bounds. This is also a new quantity that has not been considered in the traditional approximation literature. For example, the approximation property of kernel representation is typically studied using standard analytical techniques such as Fourier analysis (see [7, 11] and references therein) Such results usually depend on many complicated quantities as well as the data dimensionality. Compared with these previous results, our bounds using kp(x)k [n] are much simpler and more general. However, we do not discuss speci c approximation consequences of our ....

Holger Wendland. Error estimates for interpolation by compactly supported radial basis functions of minimal degree. J. Approx. Theory, 93(2):258-272, 1998.

....are explicit. See x5.1 and x5.2) Finally, we mention that in x3.3 we give some new results that allow one to make make use of radial basis functions (RBFs) on R m 1 to obtain strictly positive definite functions on S m Gamma1 . We use this to take a compactly supported RBF in R 5 [16] and produce a locally supported, strictly postive definite function on S m , m 3. 1.2 Sobolev Spaces In this section we briefly describe the notion of a Sobolev space on a C 1 manifold; for proofs and more references, see [1, 8] The Riemannian metric g ij on M m induces the standard ....

....is a positive definite function on S m . Moreover, if neither of the two functions, G(cos( Sigma G( Gamma cos( is a polynomial in cos( then G is strictly positive definite on S m Gamma1 . This corollary allows us to draw upon some of the new, compactly supported RBFs that Wendland [16] has constructed. Although these RBFs have the slight disadvantage of being dimension dependent, their having compact 7 support more than makes up for it. One of these, the RBF below, is positive definite on R 5 and is C 2 . OE 5;1 (r) 1 Gamma r) 5 (5r 1) 3.10) Using the ....

H. Wendland, "Error estimates for interpolation by compactly supported radial basis functions of minimal degree," preprint.

....to a positive factor) be found in the table below, where K denotes the modified Bessel function. A complete list of all common radial basis functions along with their Fourier transforms and corresponding functions F OE can be found in [12] and for compactly supported radial basis functions see [14]. OE OE(s) F 2 OE (h) OE TPS s Gammad Gamma2 h 2 OE MQ GammaK (d 1) 2 (s) Delta s Gamma(d 1) 2 e Gammaff=h , ff 0 OE G e Gammas 2 =4 e Gammaff=h 2 , ff 0 OE IMQ K (d Gamma1) 2 (s) Delta s Gamma(d Gamma1) 2 e Gammaff=h , ff 0 In summary, we ....

....condition c 1 k k Gammad Gammas 1 OE(k k) c 2 k k Gammad Gammas 1 (7) for k k 1, it turns out that F OE (h) h s1=2 . The basis functions OE TPS and OE W are two possible examples. The latter satisfies (7) for d = 3 with s 1 = 3, and consequently F OE W (h) h 3=2 (cf. [14]) By simple conclusion, it is straightforward to carry over the local error estimate (6) for bounding the least squares error EX;Z (f) in terms of the density h ae;X;Y = max y2Y h ae;X (y) of the set X around the points from Y . Corollary 3.2: Let OE and ae be as in Theorem 3.1. Then, there ....

H. Wendland, Error estimates for interpolation by compactly supported radial basis functions of minimal degree, J. Approx. Theory 93 (1998), 258--272.

....CSRBFs are generally expressed as 1 r j j n p( r j j ) with the following conditions 1 r j j n = 1 r j j n ; 0 r j j 1; 0; r j j 1; 3. 46) The formulation of the explicit formulae for the polynomial p( r j j ) can be found at [26] to [28]. Generally not all RBFs in (3.45a d) are globally de ned. The global RBFs require the solving of a system of equations with a full coecient matrix. This may hinder the applicability of the method in dealing with large scale problems. Solving a system of equations with a full matrix can be ....

H. Wendland, \Error estimates for interpolation by compactly supported radial basis functions of minimal degree", J. Approx. theory, Vol. 93, No. 2, pp. 258-272, 1998.

.... 6 PD3 6;4 (r) I 4 6 (r) 1 r) 10 (5 50r 210r 2 450r 3 429r 4 ) C 8 PD3 7;5 (r) I 5 7 (r) 1 r) 12 (9 108r 566r 2 1644r 3 2697r 4 2048r 5 ) C 10 PD3 Error bounds for approximation of f 2 H s (R d ) by CSRBFs are given in [21] and are of the form k f s f k L1 Ch k 1 2 k f k H s (R d ) 3.16) where H s (R d ) is the Sobolev space with s = d 2 k 1 2 and k 1 for d = 1; 2. h = sup x2 min k x x j k for x j 2 R d ; j = 1; 2; d. Recently Fasshauer [11, 22] has given a multilevel ....

H. Wendland, "Error estimates for interpolation by compactly supported radial basis functions of minimal degree", J. Approx. Theory, Vol. 93, No. 2, pp. 258-272, 1998.

....3.7 applies to the resultant compactly supported function OE = oe j Deltaj (fl Gammad) 2 K (fl Gammad) 2 (j Deltaj) provided oe has a nonnegative Fourier transform. Regarding the applicability of Theorem 3. 7 to Wendland s compactly supported radial functions OE d;k , it is easy to derive from [25] that for d odd, if fl is chosen to satisfy condition (i) then condition (ii) necessarily fails. One expects the same in the case d even, but this has yet to be proven. MICHAEL JOHNSON 11 4. Some Useful Lemmata In this section we gather some technical lemmata which will be used in the ....

Wendland, H., Error estimates for interpolation by compactly supported radial basis functions of minimal degree, J. Approx. Th. 93 (1998), 258--272.

.... Gamma1 ( x Gamma y) is a feasible basis function with respect to L ffi : span f ffi x g x2IR d . The native space F Phi is norm isomorphic to the Sobolev space H oe (IR d ) i.e. F Phi = H oe (IR d ) 16 4. Application to Mixed Linear Problems This is a re formulation of [16], Theorem 2.1, which is based on [20] For example, Wendland s compactly supported radial basis functions satisfy equation (4.2) cf. 16] Theorem 3.6. Now, we examine the Properties of F Psi L 1 . Due to example 3.6, we know that if Phi 0 2 C 2m (IR d ) induces a feasible basis function ....

....isomorphic to the Sobolev space H oe (IR d ) i.e. F Phi = H oe (IR d ) 16 4. Application to Mixed Linear Problems This is a re formulation of [16] Theorem 2.1, which is based on [20] For example, Wendland s compactly supported radial basis functions satisfy equation (4. 2) cf. [16], Theorem 3.6. Now, we examine the Properties of F Psi L 1 . Due to example 3.6, we know that if Phi 0 2 C 2m (IR d ) induces a feasible basis function with respect to L ffi , then Psi L 1 (x; y) L 1 x L 1 y Phi(x; y) Psi L 1 ;0 (x Gamma y) is feasible with respect to ffi x ....

Holger Wendland. Error estimates for interpolation by compactly supported radial basis functions of minimal degree. Gottingen, August 1996. --- to appear in Journal of Approximation Theory, preprint: http: //www.num.math.uni-goettingen.de/wendland/error.ps.gz.

....the two dimensional example we chose the locally supported radial basis functions (r) 1 Gamma r) 4 (4r 1) of Wendland for the approximate inversion T . The reader can easily convince himself that the use of globally supported basis functions in a multilevel framework is pointless. In [15] one can find the following error estimate for these functions: kTX (u)f Gamma fk1 Ch 3 2 kfk d 3 2 ;2 ; which suggests that the loss of derivative fl = 3=2 (see (3.2) The test function for the 1D case is f(x) 15e Gamma1 1 Gamma4(x Gamma1=2) 2 3 4 e Gamma (9x Gamma2) 2 4 3 ....

Wendland, H., Error estimates for interpolation by compactly supported radial basis functions of minimal degree, Universitat Gottingen, 1996. Gregory E. Fasshauer Department of Computer Science and Applied Mathematics

.... using (4) 5) and the fact that all of the functions ;k are supported on [0; 1] k (r) I k ;0 (r) II k Gamma1 ;0 (r) I ;k Gamma1 (r) Z 1 r s ;k Gamma1 (s)ds = Z 1 r s ;k Gamma1 (s)ds: Theorem 1 has a different flavor than the recursion formulae given in [8,9]. It is particularly helpful for our purposes since it sheds some light on the smoothness relations of the functions ;k . On the one hand we see that 6 G. E. Fasshauer the functions in the family (2) which we used for our first set of numerical experiments, are not smoother versions of one ....

Wendland, H., Error estimates for interpolation by compactly supported radial basis functions of minimal degree, J. Approx. Theory, to appear. Gregory E. Fasshauer Department of Computer Science and Applied Mathematics Illinois Institute of Technology Chicago, IL 60616 fass@amadeus.csam.iit.edu

....C 6 , respectively, and strictly positive definite in IR 3 will be referred to later on. At first some researchers hoped that using locally supported basis functions was the solution to the trade off problem, while others were skeptical (and still are) There exist convergence estimates (see [57]) for Wendland s functions which state that kf Gamma sk L1( Omega Gamma Ch 1=2 kfk H oe (IR d ) 2:3) where h denotes the meshsize , i.e. the separation distance of the centers, f 2 H oe (IR d ) generates the data, and H oe is the usual Sobolev space of functions with oe ....

Wendland, H., Error estimates for interpolation by compactly supported radial basis functions of minimal degree, J. Approx. Theory 93 (1998), 258--272.

....not be equally spaced in general. In fact, 3.8) suggests that the mesh size be chosen adaptively. In both the one dimensional and the two dimensional example we chose the locally supported radial basis functions (r) 1 Gamma r) 4 (4r 1) of Wendland for the approximate inversion T . In [13] one can find the following error estimate for these functions: kTX (u)f Gamma fk1 Ch 3 2 kfk d 3 2 ;2 ; which suggests that the loss of derivative fl = 3=2. The test function for the 1D case is f(x) 15e Gamma1 1 Gamma4(x Gamma1=2) 2 3 4 e Gamma (9x Gamma2) 2 4 3 4 e ....

Wendland, H., Error estimates for interpolation by compactly supported radial basis functions of minimal degree, Universitat Gottingen, 1996. Gregory E. Fasshauer and Joseph W. Jerome Department of Mathematics Northwestern University Evanston, IL 60208 fass@math.nwu.edu, jwj@math.nwu.edu

....computation of the values of If on a specified set having N points, e.g. a uniform grid, requires O(nN) additional operations. Methods to reduce this complexity include Powell s development [39] 40] of fast algorithms to compute thin plate spline interpolants and Wendland s construction [53] [54] of positive definite compactly supported radial basis functions. Lack of spatial varying resolution is intrinsic to radial basis function interpolation methods because the basis functions r(jx Gamma x k j) are translates of a single function r(jxj) having effective width w: The accuracy of the ....

H. Wendland (1997), Error estimates for interpolation by compactly supported radial basis functions of minimal degree, Technical Report, Institute fĻur Numerische und Angewendte Mathematik, Universitat GĻottingen.

No context found.

H. Wendland, Error estimates for interpolation by compactly supported radial basis functions of minimal degree, J. Approx. Theory 93 (1998), 258--272.

No context found.

Wendland, H., Error estimates for interpolation by compactly supported radial basis functions of minimal degree, Preprint, Gottingen, 1996.

.... 0 e Gammaffi=h ffi 0 [37] e Gammafir 2 ; fi 0 Gaussians 0 e Gammaffi=h 2 ffi 0 [37] Gamma 2 d=2 = Gamma(k) Delta K k Gammad=2 (r) r=2) k Gammad=2 2k d Sobolev splines 0 h k Gammad=2 as in [65] 1 Gamma r) 4 (1 4r) d 3 Wendland function 0 h 3=2 [58] 3 Generalized interpolation The range of applications of radial basis functions is extremely large, and we conne ourselves here to a still rather general setting where the application wants to solve a linear operator equation Lu = f; L : U F; u 2 U; where f 2 F is given and u 2 U has to be ....

H. Wendland, Error estimates for interpolation by compactly supported radial basis functions of minimal degree, Journal of Approx. Theory, 93, pp. 258-272, 1998.

....Sobolev splines, and compactly supported functions satisfies b Phi 1 ( Ck k Gammad Gamma 2 , b Phi 1 ( 1 k k 2 2 ) Gamma and b Phi 1 ( 1 k k 2 ) Gammad Gamma2 Gamma1 respectively. This is well known for thin plate splines and Sobolev splines and can be found in [15] for the compactly supported functions of minimal degree. Thus fi equals (d ) 2, d 1) 2, respectively. The condition fi k gives the conditions on the parameters. 6. Conclusion We have shown that our approach using radial basis functions leads to the same error bounds in the energy ....

H. Wendland, Error estimates for interpolation by compactly supported radial basis functions of minimal degree, to appear in Journal of Approx. Theory.

No context found.

Wendland, H., Error estimates for interpolation by compactly supported radial basis functions of minimal degree, J. of Approx. Theory 93 (1998), 258-272.

....1 2 (2.5) for k k 1. The upper bound is important for the direct theorems, while the lower bound is necessary for the inverse theorems. This decay condition is, for instance, covered by the thin plate splines and the compactly supported radial basis functions of minimal degree (cf. [13]) But we shall state the inverse theorems also in case of exponentially decaying Fourier transforms which covers Gaussian and (inverse) multiquadrics. 3. Direct theorems There are several papers dealing with direct theorems, but only few have tried to establish inverse theorems. We will briefly ....

Wendland, H., Error estimates for interpolation by compactly supported radial basis functions of minimal degree, J. Approx. Theory 93 (1998), 258-272.

.... of such basis functions Phi(x) OE(r) r = kxk 2 : The Sobolev splines OE(r) r k 1=2 K k 1=2 (r) with the modified Bessel function K of the third kind and the compactly supported radial basis functions of minimal degree OE d;k (r) Both functions are positive definite on IR d (cf. [10, 11]) Every basis function Phi with a Fourier transform satisfying (1) allows to equip W s 2 (IR d ) s = d 2 k 1 2 , with an inner product (f; g) Phi : 2 ) Gammad=2 Z IR d f ( g( Phi( d that induces a norm, equivalent to the usual Sobolev norm. In this topology the ....

Wendland, H., Error estimates for interpolation by compactly supported radial basis functions of minimal degree, Preprint, Gottingen, 1996.

No context found.

H. Wendland, Error Estimates For Interpolation By Compactly Supported Radial Basis Functions Of Minimal Degree, J. Approx. Theory 93 (1998) 258--272. 15

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