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Computing bivariate splines in scattered data fitting and the finite-element method

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Abstract

A number of useful bivariate spline methods are global in nature, i.e., all of the coefficients of an approximating spline must be computed at one time. Typically this involves solving a system of linear equations. Examples include several well-known methods for fitting scattered data, such as the minimal energy, least-squares, and penalized least-squares methods. Finite-element methods for solving boundary-value problems are also of this type. It is shown here that these types of globally-defined splines can be efficiently computed, provided we work with spline spaces with stable local minimal determining sets.

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References

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  3. Alfeld, P., Neamtu, M., Schumaker, L.L.: Dimension and local bases of homogeneous spline spaces. SIAM J. Math. Anal. 27, 1482–1501 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  4. Awanou, G., Lai, M.J., Wenston, P.: The multivariate spline method for scattered data fitting and numerical solution of partial differential equations. In: Chen, G., Lai, M.-J. (eds.) Wavelets and Splines: Athens 2005, pp. 24–74. Nashboro Press, Brentwood (2006)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  6. Dunavant, D.: High degree efficient symmetrical gaussian quadrature rules for the triangle. Int. J. Numer. Methods Eng. 21, 1129-1148 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  7. Fasshauer, G., Schumaker, L.L.: Minimal energy surfaces using parametric splines. Comput. Aided Geom. Des. 13, 45–79 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  8. Gockenbach, M.S.: Understanding and Implementing the Finite Element Method. SIAM, Philadelphia (2006)

    MATH  Google Scholar 

  9. Golitschek, M., Schumaker, L.L.: Bounds on projections onto bivariate polynomial spline spaces with stable bases. Constr. Approx. 18, 241–254 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  10. Golitschek, M., Schumaker, L.L.: Penalized least squares fitting. Serdica Math. J. 28, 329–348 (2002)

    MATH  MathSciNet  Google Scholar 

  11. von Golitschek, M., Lai, M.-J., Schumaker, L.L.: Error bounds for minimal energy bivariate polynomial splines. Numer. Math. 93, 315–331 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  12. Lai, M.-J.: Multivariate splines for data fitting and approximation. In: Neamtu, M., Schumaker, L.L. (eds.) Approximation Theory XII: San Antonio 2007. Nashboro Press, Brentwood (2008, in press)

    Google Scholar 

  13. Lai, M.J., Schumaker, L.L.: On the approximation power of bivariate splines. Adv. Comp. Math. 9, 251–279 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  14. Lai, M.J., Schumaker, L.L.: Spline Functions on Triangulations. Cambridge University Press, Cambridge (2007)

    MATH  Google Scholar 

  15. Quak, E., Schumaker, L.L.: Calculation of the energy of a piecewise polynomial surface. In: Mason, J.C., Cox, M.G. (eds.) Algorithms for Approximation II, pp. 134–143. Chapman & Hall, London (1990)

    Google Scholar 

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  1. Department of Mathematics, Vanderbilt University, Nashville, TN, 37240, USA

    Larry L. Schumaker

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  1. Larry L. Schumaker
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Correspondence to Larry L. Schumaker.

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Schumaker, L.L. Computing bivariate splines in scattered data fitting and the finite-element method. Numer Algor 48, 237–260 (2008). https://doi.org/10.1007/s11075-008-9175-x

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  • DOI: https://doi.org/10.1007/s11075-008-9175-x

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