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On approximate cardinal preconditioning methods for solving PDEs with radial basis functions. (English) Zbl 1182.65174

Summary: The approximate cardinal basis function (ACBF) preconditioning technique has been used to solve partial differential equations (PDEs) with radial basis functions (RBFs). In [Ling L, Kansa EJ. A least-squares preconditioner for radial basis functions collocation methods. Adv Comput Math; in press], a preconditioning scheme that is based upon constructing the least-squares approximate cardinal basis function from linear combinations of the RBF-PDE matrix elements has shown very attractive numerical results. This preconditioning technique is sufficiently general that it can be easily applied to many differential operators.
In this paper, we review the ACBF preconditioning techniques previously used for interpolation problems and investigate a class of preconditioners based on the one proposed in [Ling L, Kansa EJ. A least-squares preconditioner for radial basis functions collocation methods. Adv Comput Math; in press] when a cardinality condition is enforced on different subsets. We numerically compare the ACBF preconditioners on several numerical examples of Poisson’s, modified Helmholtz and Helmholtz equations, as well as a diffusion equation and discuss their performance.

MSC:

65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
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[1] Baxter, B.J.C., Preconditioned conjugate gradients, radial basis functions, and Toeplitz matrices, Comput math appl, 43, 3-5, 305-318, (2002), Radial basis functions and partial differential equations · Zbl 1002.65018
[2] Beatson, R.; Greengard, L., A short course on fast multipole methods. in: wavelets, multilevel methods and elliptic PDEs (leicester 1996), Numer math sci comput, (1997), Oxford University Press New York, p. 1-37
[3] 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), Radial basis functions and their applications · Zbl 0940.65011
[4] Beatson, R.K.; Cherrie, J.B.; Ragozin, D.L., Fast evaluation of radial basis functions: methods for four-dimensional polyharmonic splines, SIAM J math anal, 32, 6, 1272-1310, (2001), [electronic] · Zbl 0988.65007
[5] Beatson, R.K.; Light, W.A., Fast evaluation of radial basis functions: methods for two-dimensional polyharmonic splines, IMA J numer anal, 17, 3, 343-372, (1997) · Zbl 0929.65004
[6] 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), [electronic] · Zbl 0982.65015
[7] Beatson, R.K.; Newsam, G.N., Fast evaluation of radial basis functions. I, Comput math appl, 24, 12, 7-19, (1992), Advances in the theory and applications of radial basis functions · Zbl 0765.65021
[8] Buhmann, M.D., Multivariate cardinal interpolation with radial-basis functions, Constr approx, 6, 3, 225-255, (1990) · Zbl 0719.41007
[9] Buhmann, M.D., Multivariate interpolation in odd-dimensional Euclidean spaces using multiquadrics, Constr approx, 6, 1, 21-34, (1990) · Zbl 0682.41007
[10] Buhmann, M.D.; Micchelli, C.A., Multiquadric interpolation improved, Comput math appl, 24, 12, 21-25, (1992), Advances in the theory and applications of radial basis functions · Zbl 0764.41001
[11] Carslaw, H.S.; Jaeger, J.C., Condution of heat in solids, (1988), Oxford Science Publications/Clarendon Press/Oxford University Press New York · Zbl 0972.80500
[12] Chen, W., Orthonormal RBF wavelet and ridgelet-like series and transforms for high-dimensional problems, Int J nonlinear sci numer simulat, 2, 2, 155-160, (2001) · Zbl 1062.42028
[13] Chen, W., Symmetric boundary knot method, Eng anal bound elem, 26, 6, 489-494, (2002) · Zbl 1006.65500
[14] Chen, W.; Tanaka, M., A meshless, integration-free, and boundary-only RBF technique, Comput math appl, 43, 3-5, 379-391, (2002) · Zbl 0999.65142
[15] Cheng, A.H.-D.; Golberg, M.A.; Kansa, E.J.; Zammito, G., Exponential convergence and h-c multiquadric collocation method for partial differential equations, Numer meth partial differ equations, 19, 5, 571-594, (2003) · Zbl 1031.65121
[16] Cherrie, J.B.; Beatson, R.K.; Newsam, G.N., Fast evaluation of radial basis functions: methods for generalized multiquadrics in Rn, SIAM J sci comput, 23, 5, 1549-1571, (2002), [electronic] · Zbl 1009.65007
[17] Fornberg B, Wright G. Stable computation of multiquadric interpolants for all values of the shape parameter. Comput Math Appl; 2004;48(5-6): 853-867. · Zbl 1072.41001
[18] Franke, C.; Schaback, R., Convergence order estimates of meshless collocation methods using radial basis functions, Adv comput math, 8, 4, 381-399, (1998) · Zbl 0909.65088
[19] Franke, C.; Schaback, R., Solving partial differential equations by collocation using radial basis functions, Appl math comput, 93, 1, 73-82, (1998) · Zbl 0943.65133
[20] Greengard, L.; Rokhlin, V., A fast algorithm for particle simulations, J comput phys, 73, 325-348, (1987) · Zbl 0629.65005
[21] Hon, Y.C., A quasi-radial basis functions method for American options pricing, Comput math appl, 43, 3-5, 513-524, (2002), Radial basis functions and partial differential equations · Zbl 1073.91588
[22] Hon, Y.C.; Mao, X.Z., A multiquadric interpolation method for solving initial value problems, J sci comput, 12, 1, 51-55, (1997) · Zbl 0907.65062
[23] Hon, Y.C.; Mao, X.Z., An efficient numerical scheme for Burgers’ equation, Appl math comput, 95, 1, 37-50, (1998) · Zbl 0943.65101
[24] Hon, Y.C.; Mao, X.Z., A radial basis function method for solving options pricing models, Financial eng, 8, 1, 31-49, (1999)
[25] Kansa, E.J., Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics. I. surface approximations and partial derivative estimates, Comput math appl, 19, 8/9, 127-145, (1990) · Zbl 0692.76003
[26] Kansa, E.J., Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics. II. solutions to parabolic, hyperbolic and elliptic partial differential equations, Comput math appl, 19, 8/9, 147-161, (1990) · Zbl 0850.76048
[27] Kansa, E.J.; Hon, Y.C., Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to elliptic partial differential equations, Comput math appl, 39, 7/8, 123-137, (2000) · Zbl 0955.65086
[28] Larsson E, Fornberg B. A numerical study of radial basis functions based solution methods for elliptic pdes. Comput Math Appl; 2003; 46(5-6): 891-902. · Zbl 1049.65136
[29] Li, J.; Hon, Y.C., Domain decomposition for radial basis meshless methods, Numer meth pdes, 20, 3, 450-462, (2004) · Zbl 1048.65124
[30] Li X, Chen CS. A mesh free method using hyperinterpolation and fast Fourier transform for solving differential equations. Eng Anal Bound Elem, 2004;28(10): 1253-1260. · Zbl 1081.65110
[31] Ling L, Kansa EJ. A least-squares preconditioner for radial basis functions collocation methods. Adv Comput Math; 2005; 23(1-2): 31-54. · Zbl 1067.65136
[32] Ling L, Kansa EJ. Preconditioning for radial basis functions with domain decomposition methods. Math Comput Model; in press. · Zbl 1077.41008
[33] Madych, W.R., Miscellaneous error bounds for multiquadric and related interpolators, Comput math appl, 24, 12, 121-128, (1992), Advances in the theory and applications of radial basis functions · Zbl 0766.41003
[34] Madych, W.R.; Nelson, S.A., Multivariate interpolation and conditionally positive definite functions, Approx theory appl, 4, 4, 77-89, (1988) · Zbl 0703.41008
[35] Mouat CT. Fast algorithms and preconditioning techniques for fitting radial basis functions. PhD Thesis. Mathematics Department, University of Canterbury, Christchurch, New Zealand; 2001.
[36] Powell, M.J.D., The theory of radial basis function approximation in 1990, In: advances in numerical analysis, vol. II (lancaster, 1990). Oxford sci publ, (1992), Oxford University Press New York, p. 105-210 · Zbl 0787.65005
[37] Schaback, R., Error estimates and condition numbers for radial basis function interpolation, Adv comput math, 3, 3, 251-264, (1995) · Zbl 0861.65007
[38] Schaback, R., Multivariate interpolation and approximation by translates of a basis function, In: approximation theory VIII, vol. 1 (College Station, TX, 1995). ser approx decompos, vol. 6, (1995), World Scientific River Edge, NJ · Zbl 1139.41301
[39] Schaback, R., On the efficiency of interpolation by radial basis functions, In: surface Fitting and multiresolution methods (chamonix-mont-blanc, 1996), (1997), Vanderbilt University Press Nashville, TN, p. 309-318 · Zbl 0937.65013
[40] Wu, Z.M.; Schaback, R., Local error estimates for radial basis function interpolation of scattered data, IMA J numer anal, 13, 1, 13-27, (1993) · Zbl 0762.41006
[41] Yoon, J., Spectral approximation orders of radial basis function interpolation on the Sobolev space, SIAM J math anal, 33, 4, 946-958, (2001), [electronic] · Zbl 0996.41002
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