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Using Gaussian Eigenfunctions to Solve Boundary Value Problems

Published online by Cambridge University Press:  03 June 2015

Michael McCourt
Michael McCourt*
Affiliation:
Center for Applied Mathematics, Cornell University, Ithaca, NY 14853, USA
*
Corresponding author. Email: mjm458@cornell.edu
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Abstract

Kernel-based methods are popular in computer graphics, machine learning, and statistics, among other fields; because they do not require meshing of the domain under consideration, higher dimensions and complicated domains can be managed with reasonable effort. Traditionally, the high order of accuracy associated with these methods has been tempered by ill-conditioning, which arises when highly smooth kernels are used to conduct the approximation. Recent advances in representing Gaussians using eigenfunctions have proven successful at avoiding this destabilization in scattered data approximation problems. This paper will extend these techniques to the solution of boundary value problems using collocation. The method of particular solutions will also be considered for elliptic problems, using Gaussian eigenfunctions to stably produce an approximate particular solution.

Keywords

65N3565N80Meshless methodmethod of particular solutionsboundary value problem
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 5 , Special Issue 4 , August 2013 , pp. 569 - 594
Copyright
Copyright © Global-Science Press 2013

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References

[1]Adibi, H. and Es’haghi, J., Numerical solution for biharmonic equation using multilevel radial basis functions and domain decomposition methods, Appl. Math. Comput., 186(1) (2007), pp. 246–255.Google Scholar
[2]Atluri, S. N. and Zhu, T., A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics, Comput. Mech., 22(2) (1998), pp. 117–127.Google Scholar
[3]Bardhan, J. P., Eisenberg, R. S., and Gillespie, D., Discretization of the induced-charge boundary integral equation, Phys. Rev. E, 80 (2009), 011906.Google Scholar
[4]Betcke, T. and Trefethen, L. N., Reviving the method of particular solutions, SIAM Rev., 47(3) (2005), pp. 469–491.Google Scholar
[5]Chen, C. S., Lee, S., and Huang, C.-S., Derivation of particular solutions using Chebyshev polynomial based functions, Int. J. Comput. Meth., 4(1) (2007), pp. 15–32.Google Scholar
[6]Choi, S.-C.T., Paige, C. C., and Saunders, M. A., MINRES-QLP: A Krylov subspace method for indefinite or singular symmetric systems, SIAM J. Sci. Comput., 33(4) (2011), pp. 1810–1836.Google Scholar
[7]Driscoll, T. A. and Fornberg, B., Interpolation in the limit of increasingly flat radial basis functions, Comput. Math. Appl., 43(3–5) (2002), pp. 413–422.Google Scholar
[8]Fairweather, G. and Karageorghis, A., The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math., 9 (1998), pp. 69–95.CrossRefGoogle Scholar
[9]Fasshauer, G. E., Solving partial differential equations by collocation with radial basis functions, In M’haut’e, C.Rabut A. Le and Schumaker, L. L., editors, Surface Fitting and Multiresolution Methods, pages 131–138, University Press, 1997.Google Scholar
[10]Fasshauer, G. E., Meshfree Approximation Methods with MATLAB, World Scientific Publishing Co., Inc., River Edge, NJ, USA, 2007.CrossRefGoogle Scholar
[11]Fasshauer, G. E. and McCourt, M., Stable evaluation of Gaussian RBF interpolants, SIAM J. Sci. Comput., 34(2) (2012), pp. A737–A762.CrossRefGoogle Scholar
[12]Fedoseyev, A. I., Friedman, M. J., and Kansa, E. J., Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary, Comput. Math. Appl., 43(35) (2002), pp. 439–455.Google Scholar
[13]Flyer, N. and Fornberg, B., Radial basis functions: Developments and applications to planetary scale flows, Comput. Fluids, 46(1) (2011), pp. 23–32. 10th ICFD Conference Series on Numerical Methods for Fluid Dynamics (ICFD 2010).Google Scholar
[14]Flyer, N., Lehto, E., Blaise, S., Wright, G. B., and St-Cyr, A., A guide to RBF-generated finite differences for nonlinear transport: Shallow water simulations on a sphere, J. Comput. Phys., 231(11) (2012), pp. 4078–4095.Google Scholar
[15]Fornberg, B., Driscoll, T. A., Wright, G., and Charles, R., Observations on the behavior of radial basis function approximations near boundaries, Comput. Math. Appl., 43(35) (2002), pp. 473–490.Google Scholar
[16]Fornberg, B., Larsson, E., and Flyer, N., Stable computations with Gaussian radial basis functions, SIAM J. Sci. Comput., 33(2) (2011), pp. 869–892.Google Scholar
[17]Fornberg, B. and Piret, C., On choosing a radial basis function and a shape parameter when solving a convective PDE on a sphere, J. Comput. Phys., 227(5) (2008), pp. 2758–2780.Google Scholar
[18]Fornberg, B. and Piret, C., A stable algorithm for flat radial basis functions on a sphere, SIAM J. Sci. Comput., 30(1) (2008), pp. 60–80.CrossRefGoogle Scholar
[19]Fornberg, B. and Wright, G., Stable computation of multiquadric interpolants for all values of the shape parameter, Comput. Math. Appl., 48(56) (2004), pp. 853–867.Google Scholar
[20]Fuselier, E. J. and Wright, G. B., A high-order kernel method for diffusion and reaction-diffusion equations on surfaces, eprint arXiv:1206.0047, May 2012.Google Scholar
[21]Golberg, M. A. and Chen, C. S., Discrete Projection Methods for Integral Equations, Computational Mechanics Publications, 1997.Google Scholar
[22]Golberg, M. A. and Chen, C. S., The method of fundamental solutions for potential, Helmholtz and diffusion problems, Computational Engineering Computational Mechanics Publications, WIT Press, 1998.Google Scholar
[23]Hall, W. S., The Boundary Element Method, Solid Mechanics and its Applications, Kluwer Academic Publishers, 1994.Google Scholar
[24]Halton, J. H., On the efficiency of certain quasi-random sequences of points in evaluating multidimensional integrals, Numerische Mathematik, 2 (1960), pp. 84–90.CrossRefGoogle Scholar
[25]Hon, Y. C., A quasi-radial basis functions method for American options pricing, Comput. Math. Appl., 43(35) (2002), pp. 513–524.CrossRefGoogle Scholar
[26]Hon, Y. C. and Schaback, R., On unsymmetric collocation by radial basis functions, Appl. Math. Comput., 119(23) (2001), pp. 177–186.Google Scholar
[27]Ingber, M. S. and Phan-Thien, N., A boundary element approach for parabolic differential equations using a class of particular solutions, Appl. Math. Model., 16(3) (1992), pp. 124–132.Google Scholar
[28]Iserles, A., A first course in the numerical analysis of differential equations, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2009.Google Scholar
[29]Jardin, S. C., Bateman, G., Hammett, G. W., and Ku, L. P., On 1D diffusion problems with a gradient-dependent diffusion coefficient, J. Comput. Phys., 227(20) (2008), pp. 8769–8775.CrossRefGoogle Scholar
[30]Jumarhon, B., Amini, S. and Chen, K., The Hermite collocation method using radial basis functions, Eng. Anal. Bound. Elem., 24(78) (2000), pp. 607–611.Google Scholar
[31]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(89) (1990), pp. 147–161.Google Scholar
[32]Krasny, R. and Wang, L., Fast evaluation of multiquadric RBF sums by a Cartesian treecode, SIAM J. Sci. Comput., 33(5) (2011), pp. 2341–2355.Google Scholar
[33]Larsson, E. and Fornberg, B., Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions, Comput. Math. Appl., 49 (2005), pp. 103–130.Google Scholar
[34]Larsson, E. and Heryudono, A., A partition of unity radial basis function collocation method for partial differential equations, in preparation, 2013.Google Scholar
[35]Lee, Y. J., Yoon, G. J., and Yoon, J., Convergence of increasingly flat radial basis interpolants to polynomial interpolants, SIAM J. Math. Anal., 39(2) (2007), pp. 537–553.Google Scholar
[36]Li, J. and Hon, Y. C., Domain decomposition for radial basis meshless methods, Numer. Methods Partial Differential Equations, 20(3) (2004), pp. 450–462.Google Scholar
[37]Li, X. and Pozrikidis, C., The effect of surfactants on drop deformation and on the rheology of dilute emulsions in Stokes flow, J. Fluid Mech., 341 (1997), pp. 165–194.Google Scholar
[38]Ling, Leevan and Trummer, Manfred R., Adaptive multiquadric collocation for boundary layer problems, J. Comput. Appl. Math., 188(2) (2006), pp. 265–282.Google Scholar
[39]McCourt, M., Building Infrastructure for Multiphysics Simulations. PhD thesis, Cornell University, 2012.Google Scholar
[40]Miele, A. and Iyer, R. R., General technique for solving nonlinear, two-point boundary-value problems via the method of particular solutions, J. Opt. Theory Appl., 5 (1970), pp. 382–399.CrossRefGoogle Scholar
[41]Partridge, P. W., Brebbia, C. A., and Wrobel, L. C., The dual reciprocity boundary element method, International Series on Computational Engineering, Computational Mechanics Publications, 1992.Google Scholar
[42]Pazouki, Maryam and Schaback, Robert, Bases for kernel-based spaces, J. Comput. Appl. Math., 236(4) (2011), pp. 575–588.Google Scholar
[43]Rasmussen, C. E. and Williams, C. K. I., Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning), The MIT Press, 2005.Google Scholar
[44]Rodhe, B., A discontinuous Galerkin method with local radial basis function interpolation, Uptec report f 07 066, School of Engineering, Uppsala University, Uppsala, Sweden, 2007.Google Scholar
[45]Sarra, S. A., A numerical study of the accuracy and stability of symmetric and asymmetric RBF collocation methods for hyperbolic PDEs, Numer. Methods Partial Differential Equations, 24(2) (2008), pp. 670–686.Google Scholar
[46]Schaback, R., Convergence of unsymmetric kernel-based meshless collocation methods, SIAM J. Numer. Anal., 45 (2007), pp. 333–351.Google Scholar
[47]Schaback, R., Unsymmetric meshless methods for operator equations, Numerische Mathematik, 114 (2010), pp. 629–651.Google Scholar
[48]Shokri, A. and Dehghan, M., A Not-a-Knot meshless method using radial basis functions and predictorcorrector scheme to the numerical solution of improved Boussinesq equation, Comput. Phys. Commun., 181(12) (2010), pp. 1990–2000.Google Scholar
[49]Stein, M. L., Interpolation of Spatial Data: Some Theory for Kriging, Springer Series in Statistics, Springer, 1999.Google Scholar
[50]Trefethen, L. N., Spectral Methods in Matlab, Software, Environments, Tools, Society for Industrial and Applied Mathematics, 2000.CrossRefGoogle Scholar
[51]Van Loan, C.>F., The ubiquitous Kronecker product, J. Comput. Appl. Math., 123(12) (2000), pp. 85–100.Google Scholar
[52]Wendland, H., Scattered Data Approximation, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, 2005.Google Scholar