Advertisement

Journal of Scientific Computing

, Volume 67, Issue 3, pp 1089–1109 | Cite as

Preconditioning for Radial Basis Function Partition of Unity Methods

  • Alfa Heryudono
  • Elisabeth Larsson
  • Alison Ramage
  • Lina von Sydow
Article

Abstract

Meshfree radial basis function (RBF) methods are of interest for solving partial differential equations due to attractive convergence properties, flexibility with respect to geometry, and ease of implementation. For global RBF methods, the computational cost grows rapidly with dimension and problem size, so localised approaches, such as partition of unity or stencil based RBF methods, are currently being developed. An RBF partition of unity method (RBF–PUM) approximates functions through a combination of local RBF approximations. The linear systems that arise are locally unstructured, but with a global structure due to the partitioning of the domain. Due to the sparsity of the matrices, for large scale problems, iterative solution methods are needed both for computational reasons and to reduce memory requirements. In this paper we implement and test different algebraic preconditioning strategies based on the structure of the matrix in combination with incomplete factorisations. We compare their performance for different orderings and problem settings and find that a no-fill incomplete factorisation of the central band of the original discretisation matrix provides a robust and efficient preconditioner.

Keywords

Radial basis function Partition of unity RBF–PUM Iterative method Preconditioning Algebraic preconditioner 

Mathematics Subject Classification

65F08 65M70 

References

  1. 1.
    Axelsson, O., Neytcheva, M., Ahmad, B.: A comparison of iterative methods to solve complex valued linear algebraic systems. Numer. Algorithms 66(4), 811–841 (2014). doi: 10.1007/s11075-013-9764-1 MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Babuška, I., Melenk, J.M.: The partition of unity method. Int. J. Numer. Methods Eng. 40(4), 727–758 (1997). doi: 10.1002/(SICI)1097-0207(19970228)40:4<727::AID-NME86>3.0.CO;2-N
  3. 3.
    Baxter, B.J.C.: Preconditioned conjugate gradients, radial basis functions, and Toeplitz matrices. Comput. Math. Appl. 43(3–5), 305–318 (2002). doi: 10.1016/S0898-1221(01)00288-7 MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    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(2–3), 253–270 (1999). doi: 10.1023/A:1018932227617 MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    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). doi: 10.1137/S1064827599361771 MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Benzi, M.: Preconditioning techniques for large linear systems: a survey. J. Comp. Phys. 182, 418–477 (2002). doi: 10.1006/jcph.2002.7176 MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Brown, D., Ling, L., Kansa, E., Levesley, J.: On approximate cardinal preconditioning methods for solving PDEs with radial basis functions. Eng. Anal. Bound. Elem. 29(4), 343–353 (2005). doi: 10.1016/j.enganabound.2004.05.006 CrossRefMATHGoogle Scholar
  8. 8.
    Cavoretto, R., De Rossi, A.: Spherical interpolation using the partition of unity method: an efficient and flexible algorithm. Appl. Math. Lett. 25(10), 1251–1256 (2012). doi: 10.1016/j.aml.2011.11.006 MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Cavoretto, R., De Rossi, A.: A meshless interpolation algorithm using a cell-based searching procedure. Comput. Math. Appl. 67(5), 1024–1038 (2014). doi: 10.1016/j.camwa.2014.01.007 MathSciNetCrossRefGoogle Scholar
  10. 10.
    Cavoretto, R., De Rossi, A.: A trivariate interpolation algorithm using a cube-partition searching procedure. arXiv:1409.5423 [math.NA] (2014)
  11. 11.
    Cavoretto, R., De Rossi, A., Donatelli, M., Serra-Capizzano, S.: Spectral analysis and preconditioning techniques for radial basis function collocation matrices. Numer. Linear Algebra Appl. 19(1), 31–52 (2012). doi: 10.1002/nla.774 MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    De Marchi, S., Santin, G.: Fast computation of orthonormal basis for RBF spaces through Krylov space methods. BIT, pp. 1–18 (2014). doi: 10.1007/s10543-014-0537-6
  13. 13.
    Deng, Q., Driscoll, T.A.: A fast treecode for multiquadric interpolation with varying shape parameters. SIAM J. Sci. Comput. 34(2), A1126–A1140 (2012). doi: 10.1137/110836225 MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Driscoll, T.A., Fornberg, B.: Interpolation in the limit of increasingly flat radial basis functions. Comput. Math. Appl. 43(3–5), 413–422 (2002). doi: 10.1016/S0898-1221(01)00295-4 MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Driscoll, T.A., Toh, K.C., Trefethen, L.N.: From potential theory to matrix iterations in six steps. SIAM Rev. 40, 547–578 (1998). doi: 10.1137/S0036144596305582 MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Embree, M.: How descriptive are GMRES convergence bounds? Tech. Rep. 99/08, Oxford University Computing Laboratory Numerical Analysis (1999)Google Scholar
  17. 17.
    Farrell, P., Pestana, J.: Block preconditioners for linear systems arising from multiscale collocation with compactly supported RBFs. Numer. Linear Algebra Appl. 22(4), 731–747 (2015). doi: 10.1002/nla.1984 MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Fasshauer, G.E.: Meshfree approximation methods with MATLAB. Interdisciplinary Mathematical Sciences, vol. 6. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2007). doi: 10.1142/6437
  19. 19.
    Fasshauer, G.E., McCourt, M.J.: Stable evaluation of Gaussian radial basis function interpolants. SIAM J. Sci. Comput. 34(2), A737–A762 (2012). doi: 10.1137/110824784 MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Faul, A.C., Goodsell, G., Powell, M.J.D.: A Krylov subspace algorithm for multiquadric interpolation in many dimensions. IMA J. Numer. Anal. 25(1), 1–24 (2005). doi: 10.1093/imanum/drh021 MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Fornberg, B., Larsson, E., Flyer, N.: Stable computations with Gaussian radial basis functions. SIAM J. Sci. Comput. 33(2), 869–892 (2011). doi: 10.1137/09076756X MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Fornberg, B., Lehto, E., Powell, C.: Stable calculation of Gaussian-based RBF-FD stencils. Comput. Math. Appl. 65(4), 627–637 (2013). doi: 10.1016/j.camwa.2012.11.006 MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Fornberg, B., Piret, C.: A stable algorithm for flat radial basis functions on a sphere. SIAM J. Sci. Comput. 30(1), 60–80 (2007). doi: 10.1137/060671991 MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Fornberg, B., Wright, G.: Stable computation of multiquadric interpolants for all values of the shape parameter. Comput. Math. Appl. 48(5–6), 853–867 (2004). doi: 10.1016/j.camwa.2003.08.010 MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Fornberg, B., Wright, G., Larsson, E.: Some observations regarding interpolants in the limit of flat radial basis functions. Comput. Math. Appl. 47(1), 37–55 (2004). doi: 10.1016/S0898-1221(04)90004-1 MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Fornberg, B., Zuev, J.: The Runge phenomenon and spatially variable shape parameters in RBF interpolation. Comput. Math. Appl. 54(3), 379–398 (2007). doi: 10.1016/j.camwa.2007.01.028 MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Fuselier, E., Hangelbroek, T., Narcowich, F.J., Ward, J.D., Wright, G.B.: Localized bases for kernel spaces on the unit sphere. SIAM J. Numer. Anal. 51(5), 2538–2562 (2013). doi: 10.1137/120876940 MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Halton, J.H.: On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Numer. Math. 2, 84–90 (1960). doi: 10.1007/BF01386213 MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Larsson, E., Fornberg, B.: A numerical study of some radial basis function based solution methods for elliptic PDEs. Comput. Math. Appl. 46(5–6), 891–902 (2003). doi: 10.1016/S0898-1221(03)90151-9 MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Larsson, E., Fornberg, B.: Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions. Comput. Math. Appl. 49(1), 103–130 (2005). doi: 10.1016/j.camwa.2005.01.010 MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Larsson, E., Heryudono, A.: A partition of unity radial basis function collocation method for partial differential equations (2015) (in press)Google Scholar
  32. 32.
    Larsson, E., Lehto, E., Heryudono, A., Fornberg, B.: Stable computation of differentiation matrices and scattered node stencils based on Gaussian radial basis functions. SIAM J. Sci. Comput. 35(4), A2096–A2119 (2013). doi: 10.1137/120899108 MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Liesen, J., Strakos̆, Z.: Krylov subspace methods: principles and analysis. In: Stuart, A.M., Süli, E. (eds.) Numerical Mathematics and Scientific Computation, vol. 25. Oxford University Press, Oxford (2013)Google Scholar
  34. 34.
    Ling, L., Kansa, E.J.: A least-squares preconditioner for radial basis functions collocation methods. Adv. Comput. Math. 23(1–2), 31–54 (2005). doi: 10.1007/s10444-004-1809-5 MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    MATLAB: version 8.3.0.532 (R2014a). The MathWorks Inc., Natick, Massachusetts (2014)Google Scholar
  36. 36.
    Meijerink, J.A., van der Vorst, H.A.: An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix. Math. Comp. 31, 148–162 (1977). doi: 10.1090/S0025-5718-1977-0438681-4 MathSciNetMATHGoogle Scholar
  37. 37.
    Müller, S., Schaback, R.: A Newton basis for kernel spaces. J. Approx. Theory 161(2), 645–655 (2009). doi: 10.1016/j.jat.2008.10.014 MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Pazouki, M., Schaback, R.: Bases for kernel-based spaces. J. Comput. Appl. Math. 236(4), 575–588 (2011). doi: 10.1016/j.cam.2011.05.021 MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Rieger, C., Zwicknagl, B.: Sampling inequalities for infinitely smooth functions, with applications to interpolation and machine learning. Adv. Comput. Math. 32(1), 103–129 (2010). doi: 10.1007/s10444-008-9089-0 MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Rieger, C., Zwicknagl, B.: Improved exponential convergence rates by oversampling near the boundary. Constr. Approx. 39(2), 323–341 (2014). doi: 10.1007/s00365-013-9211-5 MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Saad, Y.: Iterative Methods for Sparse Linear Systems. Society for Industrial and Applied Mathematics (2003). doi: 10.1137/1.9780898718003
  42. 42.
    Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869 (1986). doi: 10.1137/0907058 MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Safdari-Vaighani, A., Heryudono, A., Larsson, E.: A radial basis function partition of unity collocation method for convection–diffusion equations arising in financial applications. J. Sci. Comp. 1–27 (2014). doi: 10.1007/s10915-014-9935-9
  44. 44.
    Schoenberg, I.J.: Metric spaces and completely monotone functions. Ann. of Math. (2) 39(4), 811–841 (1938). doi: 10.2307/1968466 MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Shcherbakov, V., Larsson, E.: Radial basis function partition of unity methods for pricing vanilla basket options. Tech. Rep. 2015–001, Department of Information Technology, Uppsala University (2015)Google Scholar
  46. 46.
    Shepard, D.: A two-dimensional interpolation function for irregularly-spaced data. In: Proceedings of the 1968 23rd ACM national conference, ACM ’68, pp. 517–524. ACM, New York, NY, USA (1968). doi: 10.1145/800186.810616
  47. 47.
    Sonneveld, P., van Gijzen, M.B.: IDR(s): a family of simple and fast algorithms for solving large nonsymmetric linear systems. SIAM J. Sci. Comput. 31, 1035–1062 (2008). doi: 10.1137/070685804 MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    von Sydow, L., Höök, L.J., Larsson, E., Lindström, E., Milovanović, S., Persson, J., Shcherbakov, V., Shpolyanskiy, Y., Sirén, S., Toivanen, J., Waldén, J., Wiktorsson, M., Giles, M.B., Levesley, J., Li, J., Oosterlee, C.W., Ruijter, M.J., Toropov, A., Zhao, Y.: BENCHOP–The BENCHmarking project in option pricing. Int. J. Comput. Math. (2015). doi: 10.1080/00207160.2015.1072172
  49. 49.
    van der Vorst, H.A.: Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 13, 631–644 (1992). doi: 10.1137/0913035 MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    Wendland, H.: Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv. Comput. Math. 4(4), 389–396 (1995). doi: 10.1007/BF02123482 MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Wendland, H.: Fast evaluation of radial basis functions: methods based on partition of unity. In: Approximation theory, X (St. Louis, MO, 2001), Innov. Appl. Math., pp. 473–483. Vanderbilt Univ. Press, Nashville, TN (2002)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Alfa Heryudono
    • 1
  • Elisabeth Larsson
    • 2
  • Alison Ramage
    • 3
  • Lina von Sydow
    • 2
  1. 1.Department of MathematicsUniversity of MassachusettsDartmouthUSA
  2. 2.Department of Information TechnologyUppsala UniversityUppsalaSweden
  3. 3.Department of Mathematics and StatisticsUniversity of StrathclydeGlasgowScotland

Personalised recommendations