Skip to main content
Log in

Sparse Data Interpolation and Smoothing on Embedded Submanifolds

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

Energy minimization is one of the properties that make univariate splines so favorable in many problems of approximation and estimation; interpolation in and extrapolation from sparse data sites and smoothing of noisy data in particular. In this paper, we present a novel approach to approximate energy minimization on certain classes of submanifolds that gives rise to new methods for extrapolation and smoothing on submanifolds. To accomplish this, we minimize intrinsic functionals approximately by minimising a suitable extrinsic formulation of the functional augmented by a penalty on the first order normal derivative. The general framework we develop is accompanied by error analysis and exemplified by tensor product B-splines.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Notes

  1. Note that this stands in sharp contrast to the Euclidean setting: In the Euclidean case, the affine functions remain affine polynomials for connected subdomains with a smooth boundary, whereas now cutting away a portion of a smooth compact manifold may introduce affine functions that are not constants!

  2. Just mere polynomials can serve as an example here, as the restrictions of a basis for a polynomial space of fixed degree in \({\mathbb {R}}^d\) is no longer linearly independent in \({\mathbb {R}}^{d-1}\).

References

  1. Adams, R.: Sobolev Spaces. Academic Press, Cambridge (1975)

    MATH  Google Scholar 

  2. Adams, R.A., Fournier, J.J.: Sobolev Spaces. Academic press, Cambridge (2003)

    MATH  Google Scholar 

  3. Agranovich, M.S.: Sobolev Spaces, Their Generalizations and Elliptic Problems in Smooth and Lipschitz Domains. Springer, Berlin (2016)

    Google Scholar 

  4. Aubin, T.: Some Nonlinear Problems in Riemannian Geometry. Springer, Berlin (2013)

    MATH  Google Scholar 

  5. Brezis, H.: Functional Analysis. Sobolev spaces and partial differential equations. Springer, Berlin (2010)

    Google Scholar 

  6. Buhmann, M.D.: Radial Basis Functions: Theory and Implementations. Cambridge University Press, Cambridge (2003)

    MATH  Google Scholar 

  7. Delfour, M., Zolésio, J.-P.: Shapes and Geometries: Analysis, Differential Calculus, and Optimization. Society for Industrial and Applied Mathematics, Philadelphia (2001)

    MATH  Google Scholar 

  8. Dierckx, P.: Curve and Surface Fitting with Splines. Oxford University Press, Oxford (1995)

    MATH  Google Scholar 

  9. Do Carmo, M.P.: Riemannian Geometry. Birkhauser, Basel (1992)

    MATH  Google Scholar 

  10. Dziuk, G., Elliott, C.M.: Finite element methods for surface PDEs. Acta Numer. 22, 289–396 (2013)

    MathSciNet  MATH  Google Scholar 

  11. Fasshauer, G.E.: Meshfree Approximation Methods with Matlab. World Scientific Publishing Company, Singapore (2007)

    MATH  Google Scholar 

  12. Foote, R.L.: Regularity of the distance function. Pro. Am. Math. Soc. 92(1), 153–155 (1984)

    MathSciNet  MATH  Google Scholar 

  13. Franke, R., Nielson, G.: Smooth interpolation of large sets of scattered data. Int. J. Numer. Methods Eng. 15(11), 1691–1704 (1980)

    MathSciNet  MATH  Google Scholar 

  14. Friedman, J., Hastie, T., Tibshirani, R.: The Elements of Statistical Learning. Springer Series in Statistics, 2nd edn. Springer, New York (2009)

    MATH  Google Scholar 

  15. Fuselier, E., Wright, G.B.: Scattered data interpolation on embedded submanifolds with restricted positive definite kernels: Sobolev error estimates. SIAM J. Numer. Anal. 50(3), 1753–1776 (2012)

    MathSciNet  MATH  Google Scholar 

  16. Gallot, S., Hulin, D., Lafontaine, J.: Riemannian Geometry. Springer, Berlin (1990)

    MATH  Google Scholar 

  17. Gordon, W.J., Wixom, J.A.: Shepard’s method of “metric interpolation” to bivariate and multivariate interpolation. Math. Comput. 32(141), 253–264 (1978)

    MathSciNet  MATH  Google Scholar 

  18. Hangelbroek, T.: Polyharmonic approximation on the sphere. Constr. Approx. 33(1), 77–92 (2011)

    MathSciNet  MATH  Google Scholar 

  19. Hangelbroek, T., Narcowich, F.J., Sun, X., Ward, J.D.: Kernel approximation on manifolds II: the \(L_\infty \) norm of the \(L_2\) projector. SIAM J. Math. Anal. 43(2), 662–684 (2011)

    MathSciNet  MATH  Google Scholar 

  20. Hangelbroek, T., Narcowich, F.J., Ward, J.D.: Kernel approximation on manifolds I: bounding the Lebesgue constant. SIAM J. Math. Anal. 42(4), 1732–1760 (2010)

    MathSciNet  MATH  Google Scholar 

  21. Hangelbroek, T., Narcowich, F.J., Ward, J.D.: Polyharmonic and related kernels on manifolds: interpolation and approximation. Found. Comput. Math. 12(5), 625–670 (2012)

    MathSciNet  MATH  Google Scholar 

  22. Hangelbroek, T., Schmid, D.: Surface spline approximation on SO(3). Appl. Comput. Harmon. Anal. 31(2), 169–184 (2011)

    MathSciNet  MATH  Google Scholar 

  23. Jia, R.-Q.: Approximation by quasi-projection operators in Besov spaces. J. Approx. Theory 162(1), 186–200 (2010)

    MathSciNet  MATH  Google Scholar 

  24. Lancaster, P., Salkauskas, K.: Surfaces generated by moving least squares methods. Math. Comput. 37(155), 141–158 (1981)

    MathSciNet  MATH  Google Scholar 

  25. Laugwitz, D.: Differential and Riemannian Geometry. Academic Press, Cambridge (1965)

    MATH  Google Scholar 

  26. Lee, J.M.: Introduction to Smooth Manifolds. Springer, Berlin (2003)

    Google Scholar 

  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)

  29. Maier, L.-B.: Ambient approximation on embedded submanifolds. Constr. Approx., pp. 1–29 (2020)

  30. Moskowitz, M.A., Paliogiannis, F.: Functions of Several Real Variables. World Scientific, Singapore (2011)

    MATH  Google Scholar 

  31. Petersen, P., Axler, S., Ribet, K.: Riemannian Geometry. Springer, Berlin (2006)

    Google Scholar 

  32. Robeson, S.M.: Spherical methods for spatial interpolation: review and evaluation. Cartogr. Geogr. Inf. Syst. 24(1), 3–20 (1997)

    Google Scholar 

  33. Rychkov, V.S.: 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)

    MathSciNet  MATH  Google Scholar 

  34. Schumaker, L.: Spline Functions: Basic Theory. Cambridge University Press, Cambridge (1981)

    MATH  Google Scholar 

  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, M.L.: Interpolation of Spatial Data: Some Theory for Kriging. Springer, Berlin (2012)

    Google Scholar 

  37. Triebel, H.: Theory of Function Spaces. Birkhäuser Verlag, Basel (1983)

    MATH  Google Scholar 

  38. Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. Johann Ambrosius Barth, Leipzig (1995)

    MATH  Google Scholar 

  39. Triebel, H.: Theory of Function Spaces III. Birkhäuser Verlag, Basel (2006)

    MATH  Google Scholar 

  40. Wahba, G.: Spline interpolation and smoothing on the sphere. SIAM J. Sci. Stat. Comput. 2(1), 5–16 (1981)

    MathSciNet  MATH  Google Scholar 

  41. Wang, R.-H., Wang, J.-X.: Quasi-interpolations with interpolation property. J. Comput. Appl. Math. 163(1), 253–257 (2004)

    MathSciNet  MATH  Google Scholar 

  42. Wendland, H.: Moving least squares approximation on the sphere. In: Mathematical Methods for Curves and Surfaces. Vanderbilt University, pp. 517–526 (2001)

  43. Wendland, H.: Scattered Data Approximation, vol. 17. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to L.-B. Maier.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Maier, LB. Sparse Data Interpolation and Smoothing on Embedded Submanifolds. J Sci Comput 84, 19 (2020). https://doi.org/10.1007/s10915-020-01268-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-020-01268-z

Keywords

Navigation