Solving differential equations with radial basis functions: Multilevel methods and smoothing

Advances in Computational Mathematics

Volumn 11, Issue 2-3, 1999, Pages 139-159

  Fasshauer, Gregory E ^a  

^a Illinois Institute of Technology   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0041593773     PISSN: 10197168     EISSN: None     Source Type: Journal    
DOI: 10.1023/a:1018919824891     Document Type: Article

References (61)

1. I. Babuška and M. Melenk, The partition of unity method, Internat. J. Numer. Methods Engrg. 40 (1997) 727-758.
2. R.K. Beatson and W.A. Light, Fast evaluation of radial basis functions: Methods for 2-dimensional polyharmonic splines, IMA J. Numer. Anal. 17 (1997) 343-372.
3. T. Belytschko, Y. Krongauz, D. Organ, M. Fleming and P. Krysl, Meshless methods: an overview and recent developments, Comput. Methods Appl. Mech. Engrg. 139 (1996) 3-47.
4. D. Braess, Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics (Cambridge University Press, Cambridge, 1997).
5. S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods (Springer, New York, 1994).
6. C.S. Chen and C.A. Brebbia, The dual reciprocity method for Helmholtz-type operators, in: Boundary Elements, Vol. 20, eds. A. Kassab, C.A. Brebbia and M. Chopra (Computational Mechanics Publications, 1998) pp. 495-504.
7. C.S. Chen, C.A. Brebbia and H. Power, Dual reciprocity method using compactly supported radial basis functions, Comm. Numer. Methods Engrg., to appear.
8. A. Cohen, W. Dahmen and R. DeVore, Adaptive wavelet methods for elliptic operator equations -convergence rates, Preprint (1998).
9. D. Djokovic̀ and R. van Damme, Radial splines and moving grids, Preprint (1996).
10. D. Djokovic̀ and R. van Damme, Moving node methods for PDE's using radial basis functions and B-splines, Preprint (1997).
11. C.A. Duarte and J.T. Oden, H-p clouds - an h-p meshless method, Numer. Methods Partial Differential Equations 4 (1996).
12. M.R. Dubal, Construction of three-dimensional black-hole initial data via multiquadrics, Phys. Rev. D 45 (1992) 1178-1187.
13. M.R. Dubal, Domain decomposition and local refinement for multiquadric approximations. I: Second-order equations in one-dimension, J. Appl. Sci. Comput. 1 (1994) 146-171.
14. M.R. Dubal, S.R. Oliveira and R.A. Matzner, Solution of elliptic equations in numerical relativity using multiquadrics, in: Approaches to Numerical Relativity, ed. R. d'Inverno (Cambridge University Press, Cambridge, 1992) pp. 265-280.
15. J. Duchon, Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces, Rev. Française Automat. Informat. Rech. Oper., Anal. Numer. 10 (1976) 5-12.
16. G.E. Fasshauer, Solving partial differential equations by collocation with radial basis functions, in: Surface Fitting and Multiresolution Methods, eds. A. Le Méhauté, C. Rabut and L.L. Schumaker (Vanderbilt University Press, Nashville, TN, 1997) pp. 131-138.
17. G.E. Fasshauer, On smoothing for multilevel approximation with radial basis functions, in: Approximation Theory, Vol. 9, eds. C.K. Chui and L.L. Schumaker (Vanderbilt University Press, Nashville, TN, 1998), to appear.
18. G.E. Fasshauer and J.W. Jerome, Multistep approximation algorithms: Improved convergence rates through postconditioning with smoothing kernels, Adv. Comput. Math. 10 (1999) 1-27.
19. G.E. Fasshauer and L.L. Schumaker, Scattered data fitting on the sphere, in: Mathematical Methods for Curves and Surfaces II, eds. M. Dæhlen, T. Lyche and L.L. Schumaker (Vanderbilt University Press, Nashville, TN, 1998) pp. 117-166.
20. A.C. Faul and M.J.D. Powell, Proof of convergence of an iterative technique for thin plate spline interpolation in two dimensions, Adv. Comput. Math. 11 (1999), this issue.
21. M.S. Floater and A. Iske, Multistep scattered data interpolation using compactly supported radial basis functions, J. Comput. Appl. Math. 73 (1996) 65-78.
22. C. Franke and R. Schaback, Solving partial differential equations by collocation using radial basis functions, Appl. Math. Comput. 93 (1998) 73-82.
23. C. Franke and R. Schaback, Convergence orders of meshless collocation methods using radial basis functions, Adv. Comput. Math. 8 (1998) 381-399.
24. M. Golberg, Recent developments in the numerical evaluation of particular solutions in the boundary element method, Appl. Math. Comput. 75 (1996) 91-101.
25. M.A. Golberg, C.S. Chen and H. Bowman, Some recent results and proposals for the use of radial basis functions in the BEM, Eng. Anal, with Bound. Elem., to appear.
26. M.A. Golberg, C.S. Chen, H. Bowman and H. Power, Some comments on the use of radial basis functions in the dual reciprocity method, Comput. Mech. 21 (1998) 141-148.
27. M. Golberg, C.S. Chen and S.R. Karur, Improved multiquadric approximation for partial differential equations, Eng. Anal, with Bound. Elem. 18 (1996) 9-17.
28. L.F. Greengard and X. Sun, A new version of the fast Gauss transform, Doc. Math. J. DMV Extra Volume ICM (1998) 575-584.
29. R.L. Hardy, Multiquadric equations of topography and other irregular surfaces, J. Geophys. Res. 76 (1971) 1905-1915.
30. Y.C. Hon and X.Z. Mao, An efficient numerical scheme for Burgers' equation, Appl. Math. Comput. 95 (1998) 37-50.
31. Y.-C. Hon and X.-Z. Mao, A radial basis function method for solving options pricing model, Financial Engrg., to appear.
32. A. Iske and T. Sonar, On the structure of function spaces in optimal recovery of point functionals for ENO-schemes by radial basis functions, Numer. Math. 74 (1996) 177-201.
33. J.W. Jerome, An adaptive Newton algorithm based on numerical inversion: regularization as post-conditioner, Numer. Math. 47 (1985) 123-138.
34. B. Jumarhon, S. Amini and K. Chen, Hermite type collocation method using radial basis functions, Preprint (1998).
35. S. Kaczmarz, Approximate solution of systems of linear equations, Internat. J. Control 57 (1993) 1269-1271. (Originally published as: Angenäherte Auflösung von Systemen linearer Gleichungen, Bulletin International de l'Academie Polonaise des Sciences, Lett. A (1937) 355-357.)
36. S. Kaczmarz, Approximate solution of systems of linear equations, Internat. J. Control 57 (1993) 1269-1271. (Originally published as: Angenäherte Auflösung von Systemen linearer Gleichungen, Bulletin International de l'Academie Polonaise des Sciences, Lett. A (1937) 355-357.)
37. E.J. Kansa, Multiquadrics - A scattered data approximation scheme with applications to com-putational fluid-dynamics - II: Solutions to parabolic, hyperbolic and elliptic partial differential equations, Comput. Math. Appl. 19 (1990) 147-161.
38. E.J. Kansa and Y.C. Hon, Circumventing the ill-conditioning problem with multiquadric radial basis functions: Applications to elliptic partial differential equations, Preprint (1998).
39. P. Lancaster and K. Salkauskas, Surfaces generated by moving least squares methods, Math. Comp. 37 (1981) 141-158.
40. W.K. Liu, S. Li and T. Belytschko, Moving least square reproducing kernel methods (I) Methodology and convergence, Comput. Methods Appl. Mech. Engrg. 143 (1997) 113-154.
41. W.R. Madych, Miscellaneous error bounds for multiquadric and related interpolators, Comput. Math. Appl. 24 (1992) 121-138.
42. S.F. McCormick, Multilevel Projection Methods for Partial Differential Equations, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 62 (SIAM, Philadelphia, PA, 1992).
43. G.J. Moridis and E.J. Kansa, The Laplace transform multiquadric method: A highly accurate scheme for the numerical solution of linear partial differential equations, J. Appl. Sci. Comput. 1 (1994) 375-407.
44. J. Moser, A rapidly convergent iteration method and nonlinear partial differential equations I., Ann. Scuola Norm. Pisa XX (1966) 265-315.
45. F.J. Narcowich, R. Schaback and J.D. Ward, Multilevel interpolation and approximation, Appl. Comput. Harmon. Anal., to appear.
46. F.J. Narcowich and J.D. Ward, Generalized Hermite interpolation via matrix-valued conditionally positive definite functions, Math. Comp. 63 (1994) 661-687.
47. M.J.D. Powell, Recent research at Cambridge on radial basis functions, Preprint (1998).
48. C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys. 77 (1988) 439-471.
49. C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, II, J. Comput. Phys. 83 (1989) 32-78.
50. R. Schaback, On the efficiency of interpolation by radial basis functions, in: Surface Fitting and Multiresolution Methods, eds. A. Le Méhauté, C. Rabut and L.L. Schumaker (Vanderbilt University Press, Nashville, TN, 1997) pp. 309-318.
51. R. Schaback, Private communications (1998).
52. K.T. Smith, D.C. Solmon and S.L. Wagner, Practical and mathematical aspects of the problem of reconstructing objects from radiographs, Bull. Amer. Math. Soc. 83 (1977) 1227-1270.
53. T. Sonar, Optimal recovery using thin plate splines in finite volume methods for the numerical solution of hyperbolic conservation laws, IMA J. Numer. Anal. 16 (1996) 549-581.
54. X. Sun, Scattered Hermite interpolation using radial basis functions, Linear Algebra Appl. 207 (1994) 135-146.
55. O.V. Vasilyev and S. Paolucci, A fast adaptive wavelet collocation algorithm for multidimensional PDEs, J. Comput. Phys. 138 (1997) 16-56.
56. O.V. Vasilyev, D.A. Yuen and S. Paolucci, Solving PDEs using wavelets, Comput. Phys. 11 (1997) 429-435.
57. H. Wendland, Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv. Comput. Math. 4 (1995) 389-396.
58. H. Wendland, Error estimates for interpolation by compactly supported radial basis functions of minimal degree, J. Approx. Theory 93 (1998) 258-272.
59. H. Wendland, Meshless Galerkin approximation using radial basis functions, Math. Comp., to appear.
60. H. Wendland, Numerical solution of variational problems by radial basis functions, in: Approximation Theory, Vol. 9, eds. C.K. Chui and L.L. Schumaker (Vanderbilt University Press, Nashville, TN, 1998), to appear.
61. Z. Wu, Hermite-Birkhoff interpolation of scattered data by radial basis functions, Approx. Theory Appl. 8 (1992) 1-10.