Sobolev-type Error Estimates for Interpolation by Radial Basis Functions

Sobolev-type Error Estimates for Interpolation by Radial Basis Functions Holger Wendland Abstract.

We generalize techniques dating back to Duchon [4] for error estimates for interpolation by thin plate splines to basis functions with positive and algebraically decaying Fourier transform. We include Lp-estimates for 1 p < 2 that can also be applied to thin plate spline approximation.

x1. Introduction

Radial basis functions are a well-established tool for multivariate approximation problems. A radial basis interpolant to a continuous function f : IRd ! IR on a set X = fx1 ; : : : ; xN g is formed by

sf (x) =

N X j =1

j (x ? xj ):

Here : IRd ! IR is a xed, positive de nite and symmetric function, and the coecients j are determined by the interpolation conditions sf (xj ) = f (xj ), 1 j N . A more general setting adds certain polynomials to sf to form the interpolant and allows to be a more general function. For details we refer the reader to the overview articles [3, 5, 6, 8]. In many cases, the function is radial in the sense (x) = (kxk2), x 2 IRd. In this paper we are mainly interested in basis functions : IRd ! IR that are in L1(IRd) and possess Fourier transforms ^ (!) = (2)?d=2

Z

ixT ! dx ( x ) e d

IR

which satisfy

c1 (1 + k!k2)?d?2k?1 ^ (!) c2 (1 + k!k2)?d?2k?1 Curves and Surfaces in Geometric Design A. Le Mehaute, C. Rabut, and L. L. Schumaker (eds.), pp. 1{8. Copyright c 1997 by Vanderbilt University Press, Nashville, TN. ISBN 1-xxxxx-xxx-x. All rights of reproduction in any form reserved.

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H. Wendland

with constants 0 < c1 c2. For convenience, we give two instances of such basis functions (x) = (r), r = kxk2 : The Sobolev splines (r) = rk+1=2 Kk+1=2 (r) with the modi ed Bessel function K of the third kind and the compactly supported radial basis functions of minimal degree d;k(r). Both functions are positive de nite on IRd (cf. [10, 11]). Every basis function with a Fourier transform satisfying (1) allows to equip W2s(IRd), s = 2d + k + 21 , with an inner product Z f^(!)^g(!) ? d= 2 (f; g) := (2) ^ (!) d! IRd that induces a norm, equivalent to the usual Sobolev norm. In this topology the Sobolev space W2s(IRd ) possesses also a reproducing kernel that can be used to represent f 2 W2s (IRd) in the form f (x) = (f; ( ? x)) : (2) If we now write the 'radial' basis function interpolant to a function f 2 s W2 (IRd ) on a set of pairwise distinct centers X = fx1 ; : : : ; xN g as

sf (x) =

N X j =1

uj (x)f (xj )

with cardinal functions uj (x), which is always possible, we can use (2) to bound the interpolation error by

jf (x) ? sf (x)j = (f; ( ? x) ? kf k PX;u(x) with the so-called power

N X j =1

uj (x)( ? xj ))

function

PX;u(x) P (x) := k( ? x) ?

N X j =1

uj (x)( ? xj )k :

On account of the norm equivalence we derive jf (x) ? sf (x)j C kf kW2s (IRd ) PX;u(x): (3) The features of the power function were thoroughly investigated in a more general setting in [12]. For the next result we need a de nition. We say that a set has the cone property i there exist > 0 and r > 0 such that for all t 2 a unit vector (t) exists such that the cone C (t) := ft + : 2 IRd; kk2 = 1; T (t) cos ; 0 rg is contained in .

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Sobolev-type Estimates

Theorem 1. Let s = d2 + k + 21 and satisfy (1). Let be a bounded

subset of IRd having the cone property. Then there exist constants h0 and C such that for every f 2 W2s (IRd) the interpolant sf on X = fx1; : : : ; xN g

satis es (4) kf ? sf kL1 ( ) C kf kW2s (IRd ) hk+ 21 if h h0 with h de ned as h := sup 1min kx ? xi k2 : (5) iN x2

Thus interpolation with provides at least approximation order k + 1=2. Our main goal is to improve this error bound in two ways. First, we shall use a less restrictive norm on the left hand side in (4). For this, we shall apply techniques dating back to Duchon [4]. After this we shall restrict the function space to a subspace of W2s(IRd) to double the approximation order. Here, we follow the lines of [9]. The nal order will be d + 2k + 1, but only for an Lp norm with 1 p 2 on the left{hand side of (4).

x2. Improvement by Localization

For the rst step we need two results that we cite from the literature. Lemma 2. Let be an open subset of IRd having the cone property. Then there exist M , M1 and h1 > 0 such that to each 0 < h < h1 there corresponds a set Th such that (i) B(t; hS) for all t 2 Th , (ii) P

t2Th B(t; Mh), (iii) t2Th B(t;Mh) M1 . Here A is the characteristic function of the set A and B(t; r) denotes the ball centered at t with radius r. Moreover, if is compact the cardinality of Th is bounded by Ch?d. This is quoted from [4], whereas the next result comes from [2]. Theorem 3. Suppose that has a Lipschitz boundary. Then there is an extension mapping E : Wpm ( ) ! Wpm (IRd) de ned for all non-negative integers m and real numbers p in the range 1 p 1 satisfying Evj = v for all v 2 Wpm ( ) and kEvkWpm(IRd ) C kvkWpm( ); (6) where C is independent of v. If is a ball in IRd the extension Ev 2 Wpm(IRd ) of v 2 Wpm ( ) can even be chosen to satisfy (cf. [1]) jEvjWpk (IRd ) C jvjWpk( ) ; 0 k m; (7) where jvjWpk ( ) denotes the Sobolev semi-norm X p jvjpWpk ( ) := kD vkLp ( ): This leads to

jj=k

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H. Wendland

Lemma 4. For a ball in IRd the constant C in (6) can be chosen to be

independent of the position and the radius of the ball. Proof: The independence of the position is obvious. Let be the unit ball centered at the origin and let h be the ball with radius h. Then we have v 2 Wpm ( ) if and only if vh := v(=h) 2 Wpm (h ) and jvh jWpk (h ) = h(d=p)?kjvjWpk ( ). On account of Evh = (Ev)h we also have jEvhjWpk (IRd ) = h(d=p)?kjEvjWpk (IRd ). Finally (7) yields m X p kEvh kWpm (IRd ) = hd?kpjEvjpWpk (IRd ) k=0 m X p C hd?kpjvjpWpk ( ) k=0 p = C kvh kpWpm (h )

with C depending only on the unit ball .

Now let us set m = dse = d d2 + k + 21 e for the rest of the paper. Here, dse denotes the smallest integer greater than or equal to s. If we take a function f 2 W2m( ) we can extend it to a function f = Ef = E f 2 W2m (IRd) which will again be denoted by f . Finally we introduce the notation f B := EB (f jB ) for any ball B. Thus we have f B jB = f jB and kf B kW2m (IRd ) C kf kW2m(B) (8) with C independent of position and radius of B. Using a modi ed version of (3) for x 2 B in the form jf (x) ? sf (x)j C P (x) k(f ? sf )B kW2m (IRd ) with the power function P , which will be bounded in terms of h later, we get

jf (x) ? sf (x)jp C P p(x) k(f ? sf )B kpW2m (IRd )

and arrive nally at kf ? sf kpLp (B) C vol(B) kP kpL1(B) k(f ? sf )B kpW2m (IRd ):

(9)

Here and in the following C denotes a generic constant. Theorem 5. Let be an open and bounded subset of IRd having the cone property and a Lipschitz boundary. Denote by sf the interpolant to f 2 W2m (IRd) with m = dse. Let s and h be de ned as in Theorem 1. Then we have for suciently small h and p 2

kf ? sf kLp ( ) C h pd +k+ 12 k(f ? sf ) kW2m (IRd ) C h pd +k+ 12 kf ? sf kW2m ( ):

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Proof: The proof technique follows Duchon [4] and Light and Wayne [7]. We have to give the proof in detail, because we want to modify it for 1 p < 2. We use Lemma 2 and (9) to get for h h1

Z

kf ? sf kpLp ( ) = jf ? sf jpdx

Z

X

t2Th B (t;Mh) C kP kpL1( )

S

j(f ? sf )B(t;Mh)jpdx

X

vol(B(t; Mh)) k(f ? sf )B(t;Mh)kpW2m (IRd)

t2Th X p d k(f ? sf )B(t;Mh)kpW2m (IRd ) = C h kP kL1 ( ) t2Th

with = Pt2Th B(t; Mh ). Now, we apply (8) and for 2 p < 1 Jensen's P p 1 =p inequality ( aj ) ( a2j )1=2 to derive kf ? sf kpLp ( ) X kf ? sf kpW2m (B(t;Mh)) C hdkP kpL1 ( ) t2Th

C hd kP kpL1 ( )

X

t2T

kf ? sf k2W2m(B(t;Mh))

!p 2

0 h Z 1p X X = C hdkP kpL1 ( ) @ jD (f ? sf )j2dxA jjm t2Th B (t;Mh) 0 Z 1p X X (f ? sf )j2 dxA = C hdkP kpL1 ( ) @ j D B ( t;Mh ) jjm IRd t2Th 0 Z 1p X M1 jD (f ? sf )j2 dxA ; Chd kP kpL1 ( ) @ 2

2

2

jjm IRd

where we made use of Lemma 2 again. A nal application of (6) yields kf ? sf kLp ( ) C h pd kP kL1 ( )kf ? sf kW2m (IRd ) = C h pd kP kL1 ( )k(f ? sf ) kW2m (IRd) C h pd kP kL1 ( ) kf ? sf kW2m ( ): If we choose h so small that1(M + 1)h h0 with h0 from Theorem 1, we can bound kP kL1 ( ) by Chk+ 2 and the proof is nished for 2 p < 1. The case p = 1 is covered by Theorem 1. We now turn to the case of 1 p < 2, where we have to replace Jensen's inequality in a suitable way.

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Corollary 6. In the setting of the last theorem we have for 1 p < 2 the additional estimate

kf ? sf kLp ( ) Ch 2d +k+ 21 kf ? sf kW2m(IRd ):

Proof:We adopt the notation of the last proof, but replace Jensen's inequal 1 = 2 1 =p P P p for nonnegative aj (cf [2], p ity by M M 1=p?1=2 Mj=1 a2j j =1 aj P 121). Keeping also in mind that t2Th 1 Ch?d, we get kf ? sf kpLp ( ) X kf ? sf kpW2m (B(t;Mh)) C hdkP kpL1( ) t2Th

C hdkP kpL1( ) dp

Ch2

X !1? p X 1

2

t2Th t2Th p p kP kL1( ) kf ? sf kW2m (IRd)

kf ? sf k2W2m (B(t;Mh))

!p 2

If we insert the estimate for the power function as before we arrive at the stated inequality.

x3. Doubling the Approximation Order

Let IRd still be an open and bounded set with Lipschitz boundary having the cone property. To further improve the approximation order, we have to restrict ourselves to a subspace of W2m ( ) that we introduce now by

!_ ^

V2m ( ) := ff 2 W2m ( ) : f 2 W22m (IRd); supp f^ g:

Here, f _ denotes the inverse Fourier transform of f and f the extension of f 2 W2m( ) to W2m(IRd). We shall again use the symbol f for f . Note that in the case of the compactly supported basis functions V2m ( ) consists of those f 2 W2m ( ) that have an extension f 2 W22m (IRd) that can be written as f = hf with an L2-function hf supported in . In general, f has already to be an element of W22m ( ) and has to satisfy certain boundary conditions determined by the de nition of V2m ( ). Theorem 7. Let the assumptions be as in the last theorem. For f 2 V2m ( ), the interpolating process satis es for p 2 and h sucently small

kf ? sf kLp ( ) C h d2 + dp +2k+1 kf kW22m (IRd ):

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If p 2, we get

kf ? sf kLp ( ) C hd+2k+1 kf kW22m (IRd ):

Proof: Following [9], we rst establish the inequality kf ? sf k2W2m (IRd ) C kf ? sf kL2( ) kf kW22m (IRd )

by making use of the orthogonal relation satis ed by sf in the (; ) inner product. Applications of Parseval's equality and the Cauchy-Schwarz inequality under consideration of the de nition of V2m( ) yields kf ? sf k2W2m (IRd ) C (f ? sf ; f ? sf ) = C (f ? sf ; f ) !_ Z (f d Z ^ ^ ? s ) f f f d! = C =C d ^ ^ (f ? sf ) d! IR

C =C

Z _ 2 ^^

f=

d!

Z 2 ^^ f= d!

IRd C kf kW22m (IRd)kf

1 2

1 2

kf ? sf kL2 ( )

kf ? sf kL2 ( )

? sf kL2 ( ):

The last inequality is valid, because from (1) it follows that

Z 2 1=2 ^^ IRd

f= d!

C kf kW22s(IRd ) C kf kW22m(IRd ):

After this we use Theorem 5 with p = 2 to get kf ? sf k2W2m(IRd ) C h 2d +k+ 12 kf ? sf kW2m (IRd )kf kW22m (IRd ): A simple reduction leads to kf ? sf kW2m (IRd ) C h d2 +k+ 21 kf kW22m (IRd ) and a last application of Theorem 5 to kf ? sf kLp ( ) C h 2d + dp +2k+1kf kW22m (IRd ) for p 2 or to kf ? sf kLp ( ) C hd+2k+1kf kW22m (IRd ) for 1 p < 2. Note that in case of even space dimension d = 2n, the condition on f 2 V2m ( ) can be weakened by only demanding f 2 W22s (IRd). For thin plate spline interpolation with (r) = r , or for (r) = r log r, the techniques of Corollary 6 and Theorem 7 yield Lp-approximation orders d + for 1 p 2.

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H. Wendland

References 1. Adams, R. A., Sobolev Spaces, Academic Press, New York, 1975. 2. Brenner, S. C. and L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, 1994. 3. Buhmann, M. D., New developments in the theory of radial basis function interpolation, in Multivariate Approximations: From CAGD to Wavelets, K. Jetter and F. I. Utreras (eds.), World Scienti c, Singapore, 1993, 35{ 75. 4. Duchon, J., Sur l'erreur d'interpolation des fonctions de plusieurs variables par les Dm -splines, R.A.I.R.O. Analyse numerique 12 no. 4 (1978), 325{334. 5. Dyn, N., Interpolation of scattered data by radial functions, in Topics in Multivariate Approximation, C. K. Chui, L. L. Schumaker and F. I. Utreras (eds.), Academic Press, New York, 1987, 47{61. 6. Powell, M. J. D., The theory of radial basis function approximation in 1990, in Advances in Numerical Analysis II: Wavelets, Subdivision Algorithms and Radial Basis Functions, W. Light (ed.), Clarendon Press, Oxford, 1992, 105{210. 7. Light, W. and H. Wayne, Some remarks on power functions and error estimates for radial basis function interpolation, Preprint, Leicester, 1996. 8. Schaback, R., Creating surfaces from scattered data using radial basis functions, in Mathematical Methods for Curves and Surfaces, M. Dhlen, T. Lyche and L. Schumaker (eds.), Vanderbilt University Press, Nashville, 1995, 477{496. 9. Schaback, R., Improved error bounds for scattered data interpolation by radial basis functions, Preprint, Gottingen, 1996. 10. Wendland, H., Piecewise polynomial, positive de nite and compactly supported radial functions of minimal degree, Advances in Computational Mathematics 4 (1995), 389{396. 11. Wendland, H., Error estimates for interpolation by compactly supported radial basis functions of minimal degree, Preprint, Gottingen, 1996. 12. 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 Universitat Gottingen 37083 Gottingen Germany [email protected]