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Two and Three Dimensional Partition of Unity Interpolation by Product-Type Functions

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Two and Three Dimensional Partition of Unity Interpolation by Product-Type Functions Appl. Math. Inf. Sci. 9, No. 1, 1-8 (2015) 1 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.12785/amis/090101 Two and Three Dimensional Partition of Unity Interpolation by Product-Type Functions Roberto Cavoretto∗ Department of Mathematics “G. Peano”, University of Torino, Via Carlo Alberto 10, 10123 Torino, Italy Received: 3 Mar. 2014, Revised: 3 Jun. 2014, Accepted: 4 Jun. 2014 Published online: 1 Jan. 2015 Abstract: In this paper we analyze the behavior of product-type radial basis functions (RBFs) and splines, which are used in a partition of unity interpolation scheme as local approximants. In particular, we deal with the case of bivariate and trivariate interpolation on a relatively large number of scattered data points. Thus, we propose the local use of compactly supported RBF and spline interpolants, which take advantage of being expressible in the multivariate setting as a product of univariate functions. Numerical experiments show good accuracy and stability of the partition of unity method combined with these product-type interpolants, comparing it with the one obtained by replacing compactly supported RBFs and splines with Gaussians. Keywords: multivariate approximation, local interpolation schemes, partition of unity methods, radial basis functions, splines, scattered data 1 Introduction can be expressed in the multivariate setting as products of univariate functions. The former are well-known in We consider the problem of interpolating a continuous approximation theory and practice, and are usually used function g : Ω R on a compact domain Ω Rm, as radial functions in the field of multivariate → ⊂ interpolation and approximation (see e.g. [10,19,21,34]). m = 2,3, defined on a finite set XN = xi,i = 1,2,...,N of data points or nodes, which are{ situated in Ω. It} On the other hand, the latter, firstly considered in consists of finding an interpolant I : Ω R such that, probability theory [20,29], have successfully been given the x and the corresponding function→ values g , the proposed for multivariate scattered data interpolation and i i integration in [4,5,6] and for landmark-based image interpolation conditions I (xi) = g(xi), i = 1,2,...,N, are satisfied. registration in [1,3]. We remark that both of these families of univariate functions (depending on a shape In particular, we are interested in considering the parameter) are compactly supported, strictly positive interpolation of large scattered data sets, a problem which definite, and enjoy noteworthy theoretical and has gained much interest in several areas of applied computational properties, such as the spline convergence sciences and scientific computing, where the need of to the Gaussian function. Furthermore, we observe that having accurate, fast and stable algorithms is often the multivariate spline interpolant is noteworthy because essential (see, e.g, [19,22,26,27]). Among the various it is neither a mesh-based formula nor a radial one, but it multivariate approximation techniques, the partition of asymptotically behaves like a Gaussian interpolant (see unity methods such as Shepard’s type interpolants turn [4,9]). Numerical experiments point out that, for out to be particularly effective in meshfree or meshless sufficiently regular basis functions such as the interpolation (see [2,8,12,13,14,16,23,24,28,30,31]). product-type radial basis function and spline of smothness In this paper, we propose the use of product-type C4, the partition of unity scheme combined with RBF and compactly supported radial basis functions (RBFs) and spline interpolants is comparable in accuracy with that splines, which are used in a partition of unity obtained by using Gaussian ones, even if they are usually interpolation scheme as local approximants. Specifically, much better conditioned than Gaussians. here we consider two families of basis functions, known as Wendland functions and Lobachevsky splines, which ∗ Corresponding author e-mail: [email protected] c 2015 NSP Natural Sciences Publishing Cor. 2 R. Cavoretto: Two and Three Dimensional Partition of Unity... The paper is organized as follows. In Section 2 at first where the interpolation matrix we consider the problem of scattered data interpolation by A = ai j = φp j(xi) , i, j = 1,2,...,N, (2) product-type Wendland functions and Lobachevsky { } { } splines, recalling their analytic expressions and some is symmetric and depends on the values of p and δ in (1). properties; then, we describe the partition of unity Since these compactly supported RBFs are strictly method, which makes use of product-type RBF and spline positive definite, the interpolation matrix A in (2) is interpolants as local approximants. In Section 3 we refer positive definite for any set of distinct nodes (see [19]). to the corresponding partition of unity algorithms Furthermore, since Wendland functions are designed for bivariate and trivariate interpolation. Section (univariate) strictly positive definite functions, we can 4 summarizes several numerical results in order to construct multivariate strictly positive definite functions analyze accuracy and stability of the local approximation from univariate ones (see, e.g., [34]). scheme combined with compactly supported radial basis λ λ λ function and spline interpolants, also comparing their Theorem 1.Suppose that 1, 2,..., m are strictly R errors with those of the Gaussian. Finally, Section 5 deals positive definite and integrable functions on , then with conclusions and future work. m Λ(x)= λ1(x1)λ2(x2) λm(xm), x = (x1,x2,...,xm) R , ··· ∈ is a strictly positive definite function on Rm. 2 Partition of unity scheme Let us consider a continuous function g : Ω R on a 2.1.2 Spline functions compact domain Ω Rm, m 1,→ a set ⊂ ≥ XN = xi = (x1i,x2i,...,xmi),i = 1,2,...,N Ω of For even n 2, we construct the product-type spline { } ⊂ ≥ scattered data points, and the set interpolant of g at the nodes xi in the form GN = g(xi),i = 1,2,...,N of the corresponding function{ values. } N F˜n(x)= ∑ c˜jφ˜n j(x), x Ω, j=1 ∈ 2.1 Product-type functions requiring F˜n(xi)= g(xi), i = 1,2,...,N. The interpolant F˜n is a linear combination of products of univariate shifted 2.1.1 Radial basis functions and rescaled functions fn∗, known as Lobachevsky splines [4], i.e. In order to construct an interpolation formula generated by m compactly supported RBFs, for even p 0 we considerthe φ˜ φ˜ α α ≥ n j(x) n j(x; )= ∏ fn∗( (xh xh j)), (3) product-type interpolant of the form ≡ h=1 − N where for j = 1,2,...,N φ Ω Fp(x)= ∑ c j p j(x), x , n 1 n n j=1 ∈ α k fn∗( (xh xh j)) = n ∑ ( 1) − 3 2 (n 1)! k 0 − k r − = requiring Fp(xi)= g(xi), i = 1,2,...,N. The interpolant n 1 n − Fp is a linear combination of products of univariate shifted α(xh xh j) + (n 2k) , ζ × 3 − − and rescaled functions p, called Wendland functions [32], r + i.e. or, equivalently, m φ φ δ ζ δ n 1 n α p j(x) p j(x; )= ∏ p( (xh xh j)), (1) 2 + 2 √ 3 x ≡ h=1 − n 1 k n f ∗(α(xh xh j)) = ∑ ( 1) n − 3 2n(n 1)! − k where, setting r = (xh xh j), for j = 1,2,...,N r − k=0 − n 1 n − ζ δ . δ α 0( r) = (1 r)+ , (xh xh j) + (n 2k) , ζ δ . − δ 3 δ × 3 − − 2( r) = (1 r)+ (3 r + 1), r . − 5 α R+ ζ δ δ δ 2 δ and is a shape parameter. Here the support of fn∗ 4( r) = (1 r)+ 8( r) + 5 r + 1 , ∈ . − 7 3 2 is given by [ √3n/α,√3n/α]. The coefficients c˜ = c˜j ζ6(δr) = (1 δr) 21(δr) + 19(δr) + 7δr + 1 , − + are obtained− by solving the system of linear equations{ } and δ R+ is a shape parameter. Note that the support of ∈ A˜c˜ = g, ζp is [ 1/δ,1/δ]. The coefficients c = c j are computed − { } by solving the linear system whose interpolation matrix Ac = g, A˜ = a˜i j = φ˜n j(xi) , i, j = 1,2,...,N, (4) { } { } c 2015 NSP Natural Sciences Publishing Cor. Appl. Math. Inf. Sci. 9, No. 1, 1-8 (2015) / www.naturalspublishing.com/Journals.asp 3 α Ω Rm Ω d is symmetric and depends on the choice of n and in (3). Definition 1.Let be a bounded set. Let j=1 As also these splines turn out to be strictly positive definite be an open and bounded⊆ covering of Ω. This means{ } that for any even n 2, the interpolation matrix A˜ in (4) is Ω Ω d Ω all j are open and bounded and that j=1 j. Set positive definite≥ for any distinct node set. ⊆ δ = diam(Ω )= sup Ω x y . We call a family of j j x,y j 2 S From the central limit theorem (see [20,29]), we ∈ ||d − || k m ∞ nonnegative functions Wj with Wj C (R ) a remark that the m-variate spline converges for n to { } j=1 ∈ the m-variate Gaussian, i.e. → k-stable partition of unity with respect to the covering d Ω j if m α2 ∑m 2 j=1 1 ( i=1 xi ) { } lim ∏ f ∗(αxi)= exp − . Ω n ∞ n (2π)m/2 2 1)supp(Wj) j; → i=1 d ⊆ 2)∑ Wj(x) 1 on Ω; j=1 ≡ Hence, these product-type functions asymptotically 3)for every β Nm with β k there exists a constant behave like radial functions, though they are not radial in ∈ 0 | |≤ Cβ > 0 such that themselves.
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