Skip to main content
SpringerLink
Log in

Ambient Approximation on Embedded Submanifolds

  • Published:
Constructive Approximation Aims and scope

Abstract

In this paper, we present a generalization of Lehmann’s approach for solving approximation problems on hypersurfaces to situations with arbitrary codimension. We show that as in the case of hypersurfaces, the method is able to transfer approximation orders from the ambient space to the submanifold. In particular, the resulting approximant is \({\mathrm {C}}^{m-2}\) and the error decays at an optimal \(h^m\) for tensor product B-splines of order m. Additionally, the method is easily implemented and comes with an optimal computational expense when applying quasi interpolation techniques. Applications include, in particular, surfaces embedded into \({{\mathbb {R}}}^4\) but not into \({{\mathbb {R}}}^3\).

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

Access this article

Log in via an institution

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

Similar content being viewed by others

Notes

  1. In fact, we hope that the present result and [31,32,33, 38] will contribute to the application of splines and other types of functions to the solution of PDEs on surfaces with similar approaches.

  2. Further arguments in that respect are presented in Sect. 6.1.

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. Alfeld, P., Neamtu, M., Schumaker, L.L.: Bernstein-Bézier polynomials on spheres and sphere-like surfaces. Comput. Aided Geom. Des. 13(4), 333–349 (1996)

    MATH  Google Scholar 

  5. Alfeld, P., Neamtu, M., Schumaker, L.L.: Fitting scattered data on sphere-like surfaces using spherical splines. J. Comput. Appl. Math. 73(1–2), 5–43 (1996)

    MathSciNet  MATH  Google Scholar 

  6. Atkinson, K., Han, W.: Theoretical Numerical Analysis: A Functional Analysis Framework, vol. 39. Springer, Berlin (2009)

    MATH  Google Scholar 

  7. Atkinson, K., Han, W.: Spherical Harmonics and Approximations on the Unit Sphere: An Introduction. Springer, Berlin (2012)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  9. Baramidze, V., Lai, M.-J., Shum, C.: Spherical splines for data interpolation and fitting. SIAM J. Sci. Comput. 28(1), 241–259 (2006)

    MathSciNet  MATH  Google Scholar 

  10. Behzadan, A., Holst, M.: Sobolev-slobodeckij spaces on compact manifolds, revisited (2017). arXiv preprint arXiv:1704.07930

  11. Berger, M.: A Panoramic View of Riemannian Geometry. Springer, Berlin (2012)

    MATH  Google Scholar 

  12. Bergh, J., Löfström, J.: Interpolation Spaces: An Introduction. Springer, Berlin (1976)

    MATH  Google Scholar 

  13. Davydov, O., Schumaker, L. L.: Scattered data fitting on surfaces using projected Powell-Sabin splines. In: IMA International Conference on Mathematics of Surfaces, pp. 138–153. Springer (2007)

  14. Davydov, O., Schumaker, L.L.: Interpolation and scattered data fitting on manifolds using projected Powell–Sabin splines. IMA J Numer. Anal. 28(4), 785–805 (2008)

    MathSciNet  MATH  Google Scholar 

  15. Deckelnick, K., Dziuk, G., Elliott, C.M., Heine, C.-J.: An h-narrow band finite-element method for elliptic equations on implicit surfaces. IMA J. Numer. Anal. 30(2), 351–376 (2009)

    MathSciNet  MATH  Google Scholar 

  16. Demjanovich, Y.K.: Construction of spaces of local functions on manifolds. Metod. Vychisl 14, 100–109 (1985)

    MathSciNet  Google Scholar 

  17. Dem’yanovich, Y.K.: Local approximations of functions given on manifolds. Transl. Am. Math. Soc. Ser. 2(159), 53–76 (1994)

    MATH  Google Scholar 

  18. Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. des Sci. Math. 136(5), 521–573 (2012)

    MathSciNet  MATH  Google Scholar 

  19. Dziuk, G.: Finite elements for the Beltrami operator on arbitrary surfaces. In: Partial Differential Equations and Calculus of Variations, pp. 142–155. Springer, Berlin (1988)

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

    MathSciNet  MATH  Google Scholar 

  21. Foley, T.A., Lane, D.A., Nielson, G.M., Franke, R., Hagen, H.: Interpolation of scattered data on closed surfaces. Comput. Aided Geom. Des. 7(1–4), 303–312 (1990)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  23. Freeden, W.: Spherical spline interpolation: basic theory and computational aspects. J. Comput. Appl. Math. 11(3), 367–375 (1984)

    MathSciNet  MATH  Google Scholar 

  24. 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 

  25. Fuselier, E.J., Wright, G.B.: A high-order kernel method for diffusion and reaction–diffusion equations on surfaces. J. Sci. Comput. 56(3), 535–565 (2013)

    MathSciNet  MATH  Google Scholar 

  26. Gu, X., He, Y., Qin, H.: Manifold splines. Graph. Models 68(3), 237–254 (2006)

    MATH  Google Scholar 

  27. Hangelbroek, T., Narcowich, F.J., Sun, X., Ward, J.D.: Kernel approximation on manifolds II: the \(\text{ L }_{\infty } \) norm of the \(\text{ L }_2\) projector. SIAM J. Math. Anal. 43(2), 662–684 (2011)

    MathSciNet  MATH  Google Scholar 

  28. 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 

  29. 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 

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

    MathSciNet  MATH  Google Scholar 

  31. Lehmann, N.: Modeling with Ambient B-Splines. Logos Verlag Berlin GmbH, Berlin (2014)

    MATH  Google Scholar 

  32. Lehmann, N., Maier, L.-B., Odathuparambil, S., Reif, U.: Approximation and modeling with ambient B-splines. In: AIP Conference Proceedings, vol. 1738. AIP Publishing, p. 050002 (2016)

  33. Lehmann, N., Maier, L.-B., Odathuparambil, S., Reif, U.: Ambient approximation on hypersurfaces. Constr. Approx. 49, 175–190 (2019)

    MathSciNet  MATH  Google Scholar 

  34. Lunardi, A.: Interpolation Theory. Scuola Normale Superiore, Pisa (1998)

    MATH  Google Scholar 

  35. Maier, L.-B.: Ambient approximation of functions and functionals on embedded submanifolds. PhD thesis, Technische Universität Darmstadt (2018)

  36. Majeed, M., Cirak, F.: Isogeometric analysis using manifold-based smooth basis functions. Comput. Methods Appl. Mech. Eng. 316, 547–567 (2017)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  38. Odathuparambil, S.: Ambient spline approximation on manifolds. PhD thesis, Technische Universität Darmstadt (2016)

  39. Piret, C.: The orthogonal gradients method: a radial basis functions method for solving partial differential equations on arbitrary surfaces. J. Comput. Phys. 231(14), 4662–4675 (2012)

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  41. 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 

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

    MATH  Google Scholar 

  43. Schumaker, L.L., Traas, C.: Fitting scattered data on spherelike surfaces using tensor products of trigonometric and polynomial splines. Numer. Math. 60(1), 133–144 (1991)

    MathSciNet  MATH  Google Scholar 

  44. Tondeur, P.: Foliations on Riemannian Manifolds. Springer, Berlin (1988)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  47. Wardetzky, M.: Convergence of the Cotangent Formula: An Overview. In Discrete Differential Geometry, pp. 275–286. Springer, Berlin (2008)

    MATH  Google Scholar 

  48. Wendland, H.: Moving Least Squares Approximation on the Sphere. Mathematical Methods for Curves and Surfaces, pp. 517–526. Vanderbilt University, Nashville (2001)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Techn. Univ. Darmstadt, Darmstadt, Germany

    L.-B. Maier

Authors
  1. L.-B. Maier
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L.-B. Maier.

Additional information

Communicated by Larry L. Schumaker.

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. Ambient Approximation on Embedded Submanifolds. Constr Approx 52, 183–211 (2020). https://doi.org/10.1007/s00365-020-09502-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00365-020-09502-5

Keywords

Mathematics Subject Classification

Search

Navigation