×

Sparse data interpolation and smoothing on embedded submanifolds. (English) Zbl 1443.41004

Quasi-interpolation especially, and also interpolation, are popular tools in the approximation of functions and data in one or more dimensions for the purpose of smoothing (for data with noise) or extra- or interpolation.
Several very good methods use approximations from spaces of piecewise polynomials, splines, or the related radial basis functions. Both of those have certain minimising properties especially when interpolation at data points in multivariable real spaces, spheres or manifolds is carried out (sometimes called kriging). A particular advantage of most of these algorithms is that they can be used when data are arbitrarily distributed (“scattered”).
They minimise certain (semi-)norms (“energy norms”) directly related to the form of the splines (or otherwise) that span the linear approximation spaces. In many instances, these semi-norms can be identified via the generalised Fourier transforms of the kernels (B-splines, radial basis functions, thin-plate splines, multiquadrics etc). They are often Euclidean norms of certain partial derivatives of the approximants, so the approximations take place in Sobolev space. Especially when smoothing is wished for, these minimisation features render the described methods so attractive.
In this article, these concepts are generalised to manifolds on which the approximations and approximants are defined. The simplest, quite well-studied cases are those of spheres, where the distances used in the arguments of the radial basis functions, are geodesics, but this paper is more general. The mentioned minimising properties are now guaranteed by minimising functions with constraints by penalty functions. The objective functions being minimised are the aforementioned energies defined by intrinsic differential operators on the manifolds, the constraints are the interpolation conditions.
Some numerical examples using tensor-product splines as bases are presented to demonstrate the efficiency of the algorithms.

MSC:

41A15 Spline approximation
41A05 Interpolation in approximation theory
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Software:

FITPACK; Matlab
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adams, R., Sobolev Spaces (1975), Cambridge: Academic Press, Cambridge
[2] Adams, RA; Fournier, JJ, Sobolev Spaces (2003), Cambridge: Academic press, Cambridge
[3] Agranovich, MS, Sobolev Spaces, Their Generalizations and Elliptic Problems in Smooth and Lipschitz Domains (2016), Berlin: Springer, Berlin
[4] Aubin, T., Some Nonlinear Problems in Riemannian Geometry (2013), Berlin: Springer, Berlin
[5] Brezis, H., Functional Analysis. Sobolev spaces and partial differential equations (2010), Berlin: Springer, Berlin
[6] Buhmann, MD, Radial Basis Functions: Theory and Implementations (2003), Cambridge: Cambridge University Press, Cambridge · Zbl 1038.41001
[7] Delfour, M.; Zolésio, J-P, Shapes and Geometries: Analysis, Differential Calculus, and Optimization (2001), Philadelphia: Society for Industrial and Applied Mathematics, Philadelphia · Zbl 1002.49029
[8] Dierckx, P., Curve and Surface Fitting with Splines (1995), Oxford: Oxford University Press, Oxford · Zbl 0932.41010
[9] Do Carmo, MP, Riemannian Geometry (1992), Basel: Birkhauser, Basel
[10] Dziuk, G.; Elliott, CM, Finite element methods for surface PDEs, Acta Numer., 22, 289-396 (2013) · Zbl 1296.65156
[11] Fasshauer, GE, Meshfree Approximation Methods with Matlab (2007), Singapore: World Scientific Publishing Company, Singapore · Zbl 1123.65001
[12] Foote, RL, Regularity of the distance function, Pro. Am. Math. Soc., 92, 1, 153-155 (1984) · Zbl 0528.53005
[13] Franke, R.; Nielson, G., Smooth interpolation of large sets of scattered data, Int. J. Numer. Methods Eng., 15, 11, 1691-1704 (1980) · Zbl 0444.65011
[14] Friedman, J.; Hastie, T.; Tibshirani, R., The Elements of Statistical Learning. Springer Series in Statistics (2009), New York: Springer, New York
[15] Fuselier, E.; Wright, GB, Scattered data interpolation on embedded submanifolds with restricted positive definite kernels: Sobolev error estimates, SIAM J. Numer. Anal., 50, 3, 1753-1776 (2012) · Zbl 1251.41004
[16] Gallot, S.; Hulin, D.; Lafontaine, J., Riemannian Geometry (1990), Berlin: Springer, Berlin
[17] Gordon, WJ; Wixom, JA, Shepard’s method of “metric interpolation” to bivariate and multivariate interpolation, Math. Comput., 32, 141, 253-264 (1978) · Zbl 0383.41003
[18] Hangelbroek, T., Polyharmonic approximation on the sphere, Constr. Approx., 33, 1, 77-92 (2011) · Zbl 1213.41009
[19] Hangelbroek, T.; Narcowich, FJ; Sun, X.; Ward, JD, Kernel approximation on manifolds II: the \(L_\infty\) norm of the \(L_2\) projector, SIAM J. Math. Anal., 43, 2, 662-684 (2011) · Zbl 1232.41002
[20] Hangelbroek, T.; Narcowich, FJ; Ward, JD, Kernel approximation on manifolds I: bounding the Lebesgue constant, SIAM J. Math. Anal., 42, 4, 1732-1760 (2010) · Zbl 1219.41003
[21] Hangelbroek, T.; Narcowich, FJ; Ward, JD, Polyharmonic and related kernels on manifolds: interpolation and approximation, Found. Comput. Math., 12, 5, 625-670 (2012) · Zbl 1259.41005
[22] Hangelbroek, T.; Schmid, D., Surface spline approximation on SO(3), Appl. Comput. Harmon. Anal., 31, 2, 169-184 (2011) · Zbl 1223.41006
[23] Jia, R-Q, Approximation by quasi-projection operators in Besov spaces, J. Approx. Theory, 162, 1, 186-200 (2010) · Zbl 1242.41022
[24] Lancaster, P.; Salkauskas, K., Surfaces generated by moving least squares methods, Math. Comput., 37, 155, 141-158 (1981) · Zbl 0469.41005
[25] Laugwitz, D., Differential and Riemannian Geometry (1965), Cambridge: Academic Press, Cambridge
[26] Lee, JM, Introduction to Smooth Manifolds (2003), Berlin: Springer, Berlin
[27] Maier, L.-B.: Ambient residual penalty approximation of partial differential equations on embedded submanifolds. Accepted for publication by Advances in Computational Mathematics
[28] Maier, L.-B.: Ambient Approximation of Functions and Functionals on Embedded Submanifolds. Ph.D. thesis, Technische Universität Darmstadt (2018) · Zbl 1426.41001
[29] Maier, L.-B.: Ambient approximation on embedded submanifolds. Constr. Approx., pp. 1-29 (2020)
[30] Moskowitz, MA; Paliogiannis, F., Functions of Several Real Variables (2011), Singapore: World Scientific, Singapore · Zbl 1233.26001
[31] Petersen, P.; Axler, S.; Ribet, K., Riemannian Geometry (2006), Berlin: Springer, Berlin
[32] Robeson, SM, Spherical methods for spatial interpolation: review and evaluation, Cartogr. Geogr. Inf. Syst., 24, 1, 3-20 (1997)
[33] Rychkov, VS, On restrictions and extensions of the Besov and Triebel-Lizorkin spaces with respect to lipschitz domains, J. Lond. Math. Soc., 60, 1, 237-257 (1999) · Zbl 0940.46017
[34] Schumaker, L., Spline Functions: Basic Theory (1981), Cambridge: Cambridge University Press, Cambridge · Zbl 0449.41004
[35] Shepard, D.: A two-dimensional interpolation function for irregularly-spaced data. In: Proceedings of the 1968 23rd ACM National Conference, pp. 517-524. ACM (1968)
[36] Stein, ML, Interpolation of Spatial Data: Some Theory for Kriging (2012), Berlin: Springer, Berlin
[37] Triebel, H., Theory of Function Spaces (1983), Basel: Birkhäuser Verlag, Basel · Zbl 0546.46028
[38] Triebel, H., Interpolation Theory, Function Spaces, Differential Operators (1995), Leipzig: Johann Ambrosius Barth, Leipzig · Zbl 0830.46028
[39] Triebel, H., Theory of Function Spaces III (2006), Basel: Birkhäuser Verlag, Basel · Zbl 1104.46001
[40] Wahba, G., Spline interpolation and smoothing on the sphere, SIAM J. Sci. Stat. Comput., 2, 1, 5-16 (1981) · Zbl 0537.65008
[41] Wang, R-H; Wang, J-X, Quasi-interpolations with interpolation property, J. Comput. Appl. Math., 163, 1, 253-257 (2004) · Zbl 1048.65012
[42] Wendland, H.: Moving least squares approximation on the sphere. In: Mathematical Methods for Curves and Surfaces. Vanderbilt University, pp. 517-526 (2001) · Zbl 0989.65043
[43] Wendland, H., Scattered Data Approximation (2004), Cambridge: Cambridge University Press, Cambridge
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.