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A discrete adapted hierarchical basis solver for radial basis function interpolation

Abstract

In this paper we develop a discrete Hierarchical Basis (HB) to efficiently solve the Radial Basis Function (RBF) interpolation problem with variable polynomial degree. The HB forms an orthogonal set and is adapted to the kernel seed function and the placement of the interpolation nodes. Moreover, this basis is orthogonal to a set of polynomials up to a given degree defined on the interpolating nodes. We are thus able to decouple the RBF interpolation problem for any degree of the polynomial interpolation and solve it in two steps: (1) The polynomial orthogonal RBF interpolation problem is efficiently solved in the transformed HB basis with a GMRES iteration and a diagonal (or block SSOR) preconditioner. (2) The residual is then projected onto an orthonormal polynomial basis. We apply our approach on several test cases to study its effectiveness.

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References

  1. Alpert, B., Beylkin, G., Coifman, R., Rokhlin, V.: Wavelet-like bases for the fast solution of second-kind integral equations. SIAM J. Sci. Comput. 14, 159–184 (1993)

    MATH  Article  MathSciNet  Google Scholar 

  2. Alpert, B.K.: A class of bases in L 2 for the sparse representation of integral operators. SIAM J. Math. Anal. 24, 246–262 (1993)

    MATH  Article  MathSciNet  Google Scholar 

  3. Amaratunga, K., Castrillon-Candas, J.: Surface wavelets: a multiresolution signal processing tool for 3D computational modeling. Int. J. Numer. Methods Eng. 52, 239–271 (2001)

    MATH  Article  MathSciNet  Google Scholar 

  4. Balay, S., Buschelman, K., Gropp, W.D., Kaushik, D., Knepley, M.G., McInnes, L.C., Smith, B.F., Zhang, H.: http://www.mcs.anl.gov/petsc

  5. Beatson, R., Cherrie, J., Ragozin, D.: Fast evaluation of radial basis functions: methods for four-dimensional polyharmonic splines. SIAM J. Math. Anal. 32(6), 1272–1310 (2001)

    MATH  Article  MathSciNet  Google Scholar 

  6. Beatson, R., Greengard, L.: A short course on fast multipole methods. In: Ainsworth, M., Levesly, J., Light, W., Marietta, M. (eds.) Wavelets, Multilevel Methods, and Elliptic PDE’s. Oxford University Press, Oxford (1997)

    Google Scholar 

  7. Beatson, R.K., Cherrie, J.B., Mouat, C.T.: Fast fitting of radial basis functions: methods based on preconditioned GMRES iteration. Adv. Comput. Math. 11, 253–270 (1999)

    MATH  Article  MathSciNet  Google Scholar 

  8. Beatson, R.K., Light, W.A., Billings, S.: Fast solution of the radial basis function interpolation equations: domain decomposition methods. SIAM J. Sci. Comput. 22(5), 1717–1740 (2000)

    MATH  Article  MathSciNet  Google Scholar 

  9. Beylkin, G., Coifman, R., Rokhlin, V.: Fast wavelet transforms and numerical algorithms I. Commun. Pure Appl. Math. 44, 141–183 (1991)

    MATH  Article  MathSciNet  Google Scholar 

  10. Börm, S., Garcke, J.: Approximating Gaussian processes with h-2 matrices. In: ECML’07: Proceedings of the 18th European Conference on Machine Learning, pp. 42–53. Springer, Berlin (2007)

    Google Scholar 

  11. Borm, S., Grasedyck, L., Hackbusch, W.: Hierarchical matrices. Lecture notes available at www.hmatrix.org/literature.html (2003)

  12. Carnicer, J.W., Dahmen, W., Pena, J.M.: Local decomposition of refinable spaces. Appl. Comput. Harmon. Anal. 3, 127–153 (1996)

    MATH  Article  MathSciNet  Google Scholar 

  13. Carr, J.C., Beatson, R.K., Cherrie, J.B., Mitchell, T.J., Fright, W.R., McCallum, B.C., Evans, T.R.: Reconstruction and representation of 3d objects with radial basis functions. In: SIGGRAPH 2001 Proceedings, pp. 67–76 (2001)

    Google Scholar 

  14. Carr, J.C., Beatson, R.K., McCallum, B.C., Fright, W.R., McLennan, T.J., Mitchell, T.J.: Smooth surface reconstruction from noisy range data. In: Proceedings of the 1st International Conference on Computer Graphics and Interactive Techniques in Australasia and South East Asia, pp. 119–126 (2003)

    Chapter  Google Scholar 

  15. Casciola, G., Lazzaro, D., Montefusco, L., Morigi, S.: Shape preserving surface reconstruction using locally anisotropic RBF interpolants. Comput. Math. Appl. 51(8), 1185–1198 (2006)

    MATH  Article  MathSciNet  Google Scholar 

  16. Casciola, G., Montefusco, L., Morigi, S.: The regularizing properties of anisotropic radial basis functions. Appl. Math. Comput. 190(2), 1050–1062 (2007)

    MATH  Article  MathSciNet  Google Scholar 

  17. Castrillon-Candas, J., Amaratunga, K.: Spatially adapted multiwavelets and sparse representation of integral operators on general geometries. SIAM J. Sci. Comput. 24(5), 1530–1566 (2003)

    MATH  Article  MathSciNet  Google Scholar 

  18. Castrillon-Candas, J.E., Li, J.: Fast solver for radial basis function interpolation with anisotropic spatially varying kernels. In preparation

  19. Cherrie, J., Beatson, R., Newsam, G.: Fast evaluation of radial basis functions: methods for generalized multiquadrics in R n. SIAM J. Sci. Comput. 23(5), 1549–1571 (2002)

    MATH  Article  MathSciNet  Google Scholar 

  20. D’Heedene, S., Amaratunga, K.S., Castrillón-Candás, J.E.: Generalized hierarchical bases: a Wavelet-Ritz-Galerkin framework for Lagrangian FEM. Eng. Comput. 22(1), 15–37 (2005)

    MATH  Article  Google Scholar 

  21. Duan, Y.: A note on the meshless method using radial basis functions. Comput. Math. Appl. 55(1), 66–75 (2008)

    MATH  Article  MathSciNet  Google Scholar 

  22. Duan, Y., Tan, Y.J.: A meshless Galerkin method for Dirichlet problems using radial basis functions. J. Comput. Appl. Math. 196(2), 394–401 (2006)

    MATH  Article  MathSciNet  Google Scholar 

  23. Duchon, J.: Splines minimizing rotation invariant semi-norms in Sobolev spaces. In: Schempp, W., Zeller, K. (eds.) Constructive Theory of Functions of Several Variables. Lecture Notes in Math., vol. 571, pp. 85–100. Springer, Berlin (1977)

    Chapter  Google Scholar 

  24. Franke, R.: Scattered data interpolation: tests of some methods. Math. Comput. 38(157), 181–201 (1982)

    MATH  MathSciNet  Google Scholar 

  25. Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996)

    MATH  Google Scholar 

  26. Greengard, L., Rokhlin, V.: New version of the fast multipole method for the Laplace equation in three dimensions. Acta Numer. 6, 229–269 (1997)

    Article  MathSciNet  Google Scholar 

  27. Gumerov, N.A., Duraiswami, R.: Fast radial basis function interpolation via preconditioned Krylov iteration. SIAM J. Sci. Comput. 29(5), 1876–1899 (2007)

    MATH  Article  MathSciNet  Google Scholar 

  28. Hestenes, M.R., Stiefel, E.: Methods of conjugate gradients for solving linear systems. J. Res. Natl. Bur. Stand. 49, 409–436 (1952)

    MATH  Article  MathSciNet  Google Scholar 

  29. Jourdan, A.: How to repair a second-order surface for computer experiments by kriging. Laboratoire de Mathématiques et de leurs Applications de Pau, LMA-PAU, CNRS: UMR5142, Université de Pau et des Pays de l’Adour (2007). 18 pages

  30. Kolotilina, L.Y., Yeremin, A.Y.: Block SSOR preconditionings for high order 3D FE systems. BIT 29(4), 805–823 (1989)

    MATH  Article  MathSciNet  Google Scholar 

  31. Lophaven, S.N., Nielsen, H.B., Sondergaard, J.: DACE: A Matlab kriging toolbox. Tech. Rep. IMM-TR-2002-12, IMM, Informatics and Mathematical Modeling. Technical University of Denmark (2002)

  32. Lophaven, S.N., Nielsen, H.B., Sondergaard, J.: Aspects of the Matlab toolbox Dace. Tech. Rep. IMM-TR-2002-13, IMM, Informatics and Mathematical Modeling. Technical University of Denmark (2002)

  33. Martin, J.D., Simpson, T.W.: A study on the use of kriging models to approximate deterministic computer models. In: Proceedings of DETC’04 ASME 2004 Design Engineering Technical Conferences and Computers and Information in Engineering Conference (2004)

    Google Scholar 

  34. Martin, J.D., Simpson, T.W.: Use of kriging models to approximate deterministic computer models. AIAA J. 43(4), 853–863 (2005)

    Article  Google Scholar 

  35. Micchelli, C.: Interpolation of scattered data: distance matrices and conditionally positive definite functions. Constr. Approx. 2, 11–22 (1986)

    MATH  Article  MathSciNet  Google Scholar 

  36. Narcowich, F.J., Ward, J.D.: Scattered-data interpolation on R n: error estimates for radial basis and band-limited functions. SIAM J. Math. Anal. 36(1), 284–300 (2004)

    MATH  Article  MathSciNet  Google Scholar 

  37. Nielsen, H.B.: Surrogate models: kriging, radial basis functions, etc. In: Working Group on Matrix Computations and Statistics. Sixth Workshop, Copenhagen, Denmark, April 1–3, 2005. ERCIM: European Research Consortium on Informatics and Mathematics (2005)

  38. Noh, J., Fidaleo, D., Neumann, U.: Animated deformations with radial basis functions. In: VRST’00: Proceedings of the ACM Symposium on Virtual Reality Software and Technology, pp. 166–174. ACM, New York (2000). doi:10.1145/502390.502422

    Chapter  Google Scholar 

  39. Pasciak, J., Bramble, J., Xu, J.: Parallel multilevel preconditioners. Math. Comput. 55, 1–22 (1990)

    MATH  Article  MathSciNet  Google Scholar 

  40. Petersdorff, T.V., Schwab, C.: Wavelet approximation for first kind integral equations on polygons. Numer. Math. 74, 479–516 (1996)

    MATH  Article  MathSciNet  Google Scholar 

  41. Petersdorff, T.v., Schwab, C.: Fully discrete multiscale Galerkin BEM. In: Dahmen, W., Kurdila, A., Oswald, P. (eds.) Multiscale Methods for PDEs, vol. 74, pp. 287–346. Academic Press, San Diego (1997)

    Google Scholar 

  42. Potts, D., Steidl, G., Nieslony, A.: Fast convolution with radial kernels at nonequispaced knots. Numer. Math. 98, 329–351 (2004)

    MATH  Article  MathSciNet  Google Scholar 

  43. Romero, V.J., Swiler, L.P., Giunta, A.A.: Application of finite-element, global polynomial, and kriging response surfaces in progressive lattice sampling designs. In: 8th ASCE Specialty Conference on Probabilistic Mechanics and Structural Reliability (2000)

    Google Scholar 

  44. Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3), 856–869 (1986)

    MATH  Article  MathSciNet  Google Scholar 

  45. Sacks, J., Welch, W.J., Mitchell, T., Wynn, H.: Design and analysis of computer experiments. Stat. Sci. 4(4), 409–435 (1989)

    MATH  Article  MathSciNet  Google Scholar 

  46. Schaback, R.: Error estimates and condition numbers for radial basis function interpolation. Adv. Comput. Math. 3, 251–264 (1995)

    MATH  Article  MathSciNet  Google Scholar 

  47. Schaback, R.: Improved error bounds for scattered data interpolation by radial basis functions. Math. Comput. 68(225), 201–216 (1999)

    MATH  Article  MathSciNet  Google Scholar 

  48. Schroder, P., Sweldens, W.: Rendering Techniques: Spherical Wavelets: Texture Processing. Springer, New York (1995)

    Google Scholar 

  49. Sibson, R., Stone, G.: Computation of thin-plate splines. SIAM J. Sci. Stat. Comput. 12(6), 1304–1313 (1991)

    MATH  Article  MathSciNet  Google Scholar 

  50. Simpson, T.W., Mauery, T.M., Korte, J.J., Branch, M.O., Mistree, F.: Comparison of response surface and kriging models for multidisciplinary design optimization. In: 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, pp. 98–4755 (1998)

    Google Scholar 

  51. Tausch, J., White, J.: Multiscale bases for the sparse representation of boundary integral operators on complex geometry. SIAM J. Sci. Comput. 25(5), 1610–1629 (2003)

    Article  MathSciNet  Google Scholar 

  52. Wu, Z., Schaback, R.: Local error estimates for radial basis function interpolation of scattered data. IMA J. Numer. Anal. 13, 13–27 (1993)

    MATH  Article  MathSciNet  Google Scholar 

  53. Yalavarthy, P.K.: A generalized least squares minimization method for near infrared diffuse optical tomography. PhD thesis, Dartmouth College (2007)

  54. Yalavarthy, P.K., Pogue, B.W., Dehghani, H., Paulsen, K.D.: Weight-matrix structured regularization provides optimal generalized least-squares estimate in diffuse optical tomography. Med. Phys. 34, 2085–2098 (2007)

    Article  Google Scholar 

  55. Yee, P., Haykin, S.: Regularized Radial Basis Function Networks: Theory and Applications. Wiley, New York (2001)

    Google Scholar 

  56. Ying, L., Biros, G., Zorin, D.: A kernel-independent adaptive fast multipole method in two and three dimensions. J. Comput. Phys. 196(2), 591–626 (2004)

    MATH  Article  MathSciNet  Google Scholar 

  57. Zhu, Z., White, J.: FastSies: a fast stochastic integral equation solver for modeling the rough surface effect. In: International Conference on Computer-Aided Design (ICCAD’05), pp. 675–682 (2005)

    Google Scholar 

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Acknowledgements

We are grateful to Lexing Ying for providing a single processor version of the KIFMM3d code. We also appreciate the discussions, assistance and feedback from Raul Tempone, Robert Van De Gein, Vinay Siddavanahalli and the members of the Computational Visualization Center (Institute for Computational Engineering and Sciences) at the University of Texas at Austin. In addition, we appreciate the invaluable feedback from the reviewers of this paper.

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Correspondence to Julio E. Castrillón-Candás.

Additional information

Communicated by Michiel Hochstenbach.

J.E. Castrillón-Candás was supported in part by the Institute for Computational Engineering and Sciences Postdoctoral Fellowship at the University of Texas at Austin.

J. Li was supported in part by grant # NIH R01 GM074258.

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Castrillón-Candás, J.E., Li, J. & Eijkhout, V. A discrete adapted hierarchical basis solver for radial basis function interpolation. Bit Numer Math 53, 57–86 (2013). https://doi.org/10.1007/s10543-012-0397-x

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  • DOI: https://doi.org/10.1007/s10543-012-0397-x

Keywords

  • Radial basis function
  • Interpolation
  • Hierarchical basis
  • Integral equations
  • Fast summation methods
  • Stable completion
  • Lifting
  • Generalized least squares
  • Best linear unbiased estimator

Mathematics Subject Classification (2010)

  • 65D05
  • 65D07
  • 65F25
  • 65F10
  • 62J05
  • 41A15