zbMATH — the first resource for mathematics

The approximation power of moving least-squares. (English) Zbl 0911.41016
The author investigates a general method to approximate linear functionals on spaces of multivariate functions, namely the so-called moving least-squares method. Given a data set $$\{L_i(f) \}_{i=1}^l$$ (for example, $$f(x_i)$$ for sample points $$x_i\in \mathbb{R}^d$$), an unknown functional $$L(f)$$ (for example, $$f(x)$$ for another point $$x$$) is approximated by $$\sum_{i=1}^l a_iL_i(f)$$, where the coefficients are obtained by minimizing the quadratic form $$\sum_{i=1}^l w(L,L_i)a_i^2$$ under the linear constraints $$\sum_{i=1}^l a_iL_i(p_j)$$, $$j=1,\dots,J$$. This means that the elements $$p_j$$ (typically polynomials of low degree) are reproduced, while the nonnegative weight $$w(L,L_i)$$ is a separation measure between the functionals $$L_i$$ and $$L$$ (such as a radial function of the distance of the sample point $$x_i$$ from $$x$$). It is shown how this constrained minimization can be solved in a matrix formulation. For the multivariate interpolation problem and for specific choices of the weights, $$C^\infty$$-smoothness in $$x$$ and the approximation order on suitable data sets are established. The paper concludes with several numerical examples, including some in a bi- and trivariate setting as well as with data-dependent separation measures.
Reviewer: E.Quak (Oslo)

MSC:
 41A45 Approximation by arbitrary linear expressions 41A25 Rate of convergence, degree of approximation 65D15 Algorithms for approximation of functions
Full Text:
References:
 [1] F. Abramovici, 1984 The Shepard interpolation as the best average of a set of data, Technical Report, Tel-Aviv University. [2] M. D. Buhmann, N. Dyn, and D. Levin, On quasi-interpolation by radial basis functions with scattered centres, Constr. Approx. 11 (1995), no. 2, 239 – 254. · Zbl 0824.41010 [3] G. Backus and F. Gilbert, 1967 Numerical applications of a formalism for geophysical inverse problems, Geophys. J.R. Astr. Soc. 13 247-276. [4] G. Backus and F. Gilbert, 1968 The resolving power of gross Earth data, Geophys. J.R. Astr. Soc. 16 169-205. · Zbl 0177.54102 [5] G. Backus and F. Gilbert, Uniqueness in the inversion of inaccurate gross Earth data, Philos. Trans. Roy. Soc. London Ser. A 266 (1970), no. 1173, 123 – 192. [6] L. P. Bos and K. Šalkauskas, Moving least-squares are Backus-Gilbert optimal, J. Approx. Theory 59 (1989), no. 3, 267 – 275. · Zbl 0729.41008 [7] Nira Dyn, David Levin, and Samuel Rippa, Data dependent triangulations for piecewise linear interpolation, IMA J. Numer. Anal. 10 (1990), no. 1, 137 – 154. · Zbl 0699.65004 [8] Reinhard Farwig, Rate of convergence of Shepard’s global interpolation formula, Math. Comp. 46 (1986), no. 174, 577 – 590. · Zbl 0607.41005 [9] Reinhard Farwig, Multivariate interpolation of arbitrarily spaced data by moving least squares methods, J. Comput. Appl. Math. 16 (1986), no. 1, 79 – 93. · Zbl 0636.65007 [10] Richard Franke, Scattered data interpolation: tests of some methods, Math. Comp. 38 (1982), no. 157, 181 – 200. · Zbl 0476.65005 [11] Richard Franke and Greg Nielson, Smooth interpolation of large sets of scattered data, Internat. J. Numer. Methods Engrg. 15 (1980), no. 11, 1691 – 1704. · Zbl 0444.65011 [12] P. Lancaster and K. Salkauskas, Surfaces generated by moving least squares methods, Math. Comp. 37 (1981), no. 155, 141 – 158. · Zbl 0469.41005 [13] D. H. McLain, 1974 Drawing contours from arbitrary data points, Comput. J. 17 318-324. [14] D. H. McLain, Two dimensional interpolation from random data, Comput. J. 19 (1976), no. 2, 178 – 181. · Zbl 0321.65009 [15] D. Shepard, 1968 A two dimensional interpolation function for irregularly spaced data, Proc. 23th Nat. Conf. ACM, 517-523.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.