Powell, M.J.D. (1994): The uniform convergence of thin plate spline interpolation in two dimensions, Numer. Math. 68, 107--128.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....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 decay of the ....

Powell, M.J.D. (1994) The uniform convergence of thin plate spline interpolation in two dimensions. Numer. Math. 68, 107--128

....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 ....

M.J.D. Powell. The uniform convergence of thin{plate spline interpolation in two dimensions. Numer. Math., 67:107-128, 1994.

....6= o(ffi m 1=p ) where Bn(1 Gamma h)B can be interpreted as the boundary layer within Omega of depth h. Thus it appears that our inability to achieve L p approximation of order 2m is due primarily to boundary effects. This corroborates experimental evidence reported by Powell and Beatson [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 ....

Powell, M.J.D. (1994), The uniform convergence of thin plate spline interpolation in two dimensions, Numer. Math. 68, 107--128.

....Remark 4. In one dimension, Theorem 2 recovers the result of [9] for natural cubic splines where the order 7=2 for pointwise estimates is proven. Remark 5. If one seeks to improve the error bound given by (3) it is not sucient to increase the regularity of the function f: The numerical results in [8] for thin plate spline interpolation in two dimensions (n = m = 2) show that even for analytic functions f the order of accuracy is between one and two. It is boundary e ects that prevent achievement of higher orders. These boundary e ects can be suppressed by imposing boundary conditions on the ....

Powell M.J.D. 1994 The uniform convergence of thin plate spline interpolation in two dimensions. Numerische Mathematik 68, 107-128.

....that his interpolant preserves polynomials of degree k Gamma 1. Since his interpolants are all special cases of the ones just described, the polynomial preservation property of U , that is, Up = p for all p 2 Pi k Gamma1 , is clear. Another approach, taken by Madych and Nelson [7] Powell [10] and Schaback [12] uses a pointwise error estimator. This estimator involves an expression which Schaback calls the power function. Both Powell and Schaback compute this power function in some sense. Their estimates coincide with those of Duchon, although one should note that Schaback is ....

....and = n 1 in Equation (1) then we need to take j to be the smallest integer greater than or equal to (n 3) 2. Let f 2 C (j) Omega Gamma1 Then jf(x) Gamma (U h f) x)j = O(h 3=2 ) as h 0 for all x 2 Omega Gamma We emphasize that in contrast to other authors (Meinguet [9] Powell [10]) we do not make any assumption about the dimension n here. Also the assumption that = n 1 is supposed to convey to the reader that linear polynomials are being used in both these interpolants. 2 Variational Theory In this short section we describe how the salient features of the seminal ....

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Powell, M.J.D. The uniform convergence of thin plate spline interpolation in two dimensions, Numer. Math. 68 (1), 1994, 107--128.

....that the thin plate splines are a member of the class. Our approach is via the work of Golomb and Weinberger [2] together with some interpretations of that work by Meinguet [8] These are fine papers which are still well worth reading. In the course of our work, we will establish results of Powell [10] and Schaback [12] We begin with a normed linear space X . Let fl 1 ; fl m be linear information functionals on X . The intention is that for a given f 2 X , the information fl i (f) ff i , i = 1; m is known. From this information it 9 is desired to compute a value fl(f) ....

....2 X by hf; gi = X jffj=k c ff Z IR n (D ff f) x) D ff g) x)dx: 14 Here the fc ff : jffj = kg are chosen so that the semi norm is rotationally invariant. Explicitly, these parameters are specified by the formal expansion k k 2k 2 = X jffj=k c ff 2ff : The discussion of Powell [10] is confined to the case n = k = 2 when hf; fi = Z Z IR 2 2 f s 2 2 2 2 f s t 2 2 f t 2 2 dsdt: If hf; fi = 0, then it is straightforward to conclude in Powell s case that f is a linear polynomial. In general, the kernel of h Delta; Deltai is the ....

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M.J.D. Powell (1993), "The uniform convergence of thin plate spline interpolation in two dimensions", Report No. DAMTP 1993/NA16, University of Cambridge.

....function f : R 2 R. We say that the local approximation order of the thin plate spline interpolation scheme at x 0 is p, iff jf(x 0 hx) Gamma s h (x 0 hx)j = O(h p ) h 0 holds for every x = P N i=1 ff i x i with P N i=1 ff i = 1. Using an earlier approximation result given in [14] by Powell, we deduce a lower bound for the local approximation order according to Definition 2.1. Powell proves in [14] Lemma 2.3 Proposition 2.2: Let f : R 2 R be a function that has square integrable second derivatives. Let I(f) Z R 2 2 f 2 2 2 2 f j ....

....is p, iff jf(x 0 hx) Gamma s h (x 0 hx)j = O(h p ) h 0 holds for every x = P N i=1 ff i x i with P N i=1 ff i = 1. Using an earlier approximation result given in [14] by Powell, we deduce a lower bound for the local approximation order according to Definition 2.1. Powell proves in [14], Lemma 2.3 Proposition 2.2: Let f : R 2 R be a function that has square integrable second derivatives. Let I(f) Z R 2 2 f 2 2 2 2 f j 2 2 f j 2 2 1 and x = P N i=1 ff i x i where the multipliers ff i sum to 1: Then the error of the ....

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M.J.D. Powell, The uniform convergence of thin plate spline interpolation in two dimensions, Numerische Mathematik 68, 1994, 107-128.

....k k 2l h( d C Z k k d Gamma1 h( d Cd 2m Gamma1 : Thus, from equation (26) we obtain O(h m Gamma 1 2 ) convergence in this case. Similarly, we obtain O(h m ) convergence for h(x) kxk 2m log kxk, m 2 IN 0 , in even space dimension. This agrees with the results of Powell [10], who gives O(h) error estimates for thin plate spline (m=1) interpolation in two dimensions. 20 ....

Powell, M.J.D. (1993) The Uniform Convergence of Thin Plate Spline Interpolation in Two Dimensions, Report no. DAMTP 1993/NA16, University of Cambridge.

....to N , it follows that p = 0, i.e. ffl kj = 0, 1 j d k , 0 k Gamma 1. Hence u is determined uniquely by the given equations. 5 Error bounds In this section we show that our interpolants have good behaviour with regard to error bounds. We will follow the approach of Schaback [12] and Powell [10] rather than the original material of Golomb and Weinburger [6] since this will produce a more immediate development of the results. We maintain the same setup as in previous sections, so that a 1 ; a 2 ; am are the interpolation points and a 1 ; a 2 ; a l are unisolvent with ....

M.J.D. Powell, The uniform convergence of thin plate spline interpolation in two dimensions, Numerische Math. 68 (1), 1994, 107-128.

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Powell, M.J.D. (1994): The uniform convergence of thin plate spline interpolation in two dimensions, Numer. Math. 68, 107--128.

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Powell, M.J.D. (1994): The uniform convergence of thin plate spline interpolation in two dimensions, Numer. Math. 68, 107--128.

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P2. Powell, M.J.D. (1994), The uniform convergence of thin plate spline interpolation in two dimensions, Numer. Math. 68, 107--128.

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P2. Powell M.J.D., The uniform convergence of thin plate spline interpolation in two dimensions, Numer. Math. 68 (1994), 107--128.

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M.J.D. Powell (1993), The uniform convergence of thin plate spline interpolation in two dimensions, Report no. DAMTP 1993/NA16, University of Cambridge.

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Powell, M.J.D. The uniform convergence of thin plate spline interpolation in two dimensions, Numer. Math. 68 (1), 1994, 107--128.

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