Stable PDE solution methods for large multiquadric shape parameters

CMES - Computer Modeling in Engineering and Sciences

Volumn 25, Issue 1, 2008, Pages 23-41

  Libre, Nicolas Ali ^a   Emdadi, Arezoo ^a   Kansa, Edward J ^b   Rahimian, Mohammad ^a   Shekarchi, Mohammad ^a  

^a UNIVERSITY OF TEHRAN   (Iran)

^b University of California   (United States)

Author keywords

Asymmetric collocation; Improved truncated singular value decomposition; Meshless radial basis functions; Multiquadric; Partial differential equations

Indexed keywords

ASYMPTOTIC ANALYSIS; NUMERICAL METHODS; PARAMETER ESTIMATION; PARTIAL DIFFERENTIAL EQUATIONS; SINGULAR VALUE DECOMPOSITION;

ASYMMETRIC COLLOCATION; GAUSSIAN ELIMINATION; MULTIQUADRIC SHAPE PARAMETERS;

RADIAL BASIS FUNCTION NETWORKS;

EID: 41649110069     PISSN: 15261492     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (47)

1. Brown, D.; Ling, L.; Kansa, E.J.; Levesley, J. (2005): On approximate cardinal preconditioning methods for radial functions, Eng. Anal. Bound. Elem. Vol. 29, pp. 343-353.
2. Buhmann, M.D.; Micchelli, CA. (1992): Multiquadric interpolation improved, Comput. Math. Appl. Vol. 24 (12), pp.21-25.
3. Buhmann, M.D.; Micchelli, C. A. (1992): Spline prewavelets for nonuniform nodes, Numer. Math., Vol. 61,pp.455-474.
4. Buhmann, M.D. (1995): Multiquadric prewavelets on nonequally spaced knots in one dimension, Math. Comput. Vol. 64, pp.1611-1625.
5. Chen,J.S.; Liu, W.K. (2004): Preface, Comput. Meth Appl. Mech. Eng. Vol. 193, pp. iii-vi.
6. Cheng, A.H.D.; Golberg, M.A; Kansa, E.J.T; Zammito, T. (2003): Exponential convergence and h-c multiquadric collocation method for partial differential equations, Num. Meth. PDEs. Vol. 19, pp.571-594.
7. Chui, C.K.; Stoeckler, J.; Ward, J.D. (1996): Analytic wavelets generated by radial functions, Adv. Comput. Math. Vol. 5, pp. 95-123.
8. de Boor, C.; Kreiss, H.O. (1986): On the condition of linear systems associated with discretized BVPs of ODEs, SIAM J. Numer. Anal. Vol. 23, pp. 936-939.
9. Driscoll, T.A.; Heryudono, A.R.H. (2007): Adaptive residual subsampling for radial basis function interpolation collocations problems, Comput Math Appl. Vol. 53, pp.927-939.
10. Fasshauer, G.E. (1999): Solving differential equations with radial basis functions: multilevel methods and smoothing, Adv. Comput. Math. Vol. 11, pp.139-159.
11. Fasshauer, G.E. (2007): Meshfree approximation methods with MATLAB, World Scientific Publ., Singapore.
12. Fedoseyev, A.I.; Friedman, M.J.; Kansa, E.J. (2002): Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary, Comput. Math. Appl., Vol. 43 (3-5), pp. 491-500.
13. Fornberg, B.; Driscoll, T.A. (2002): Interpolation in the limit of increasingly flat radial basis functions. Comput Math Appl. Vol. 43 (3-5), pp.413-21.
14. Fornberg, B.; Wright, G. (2004): Stable computation of multiquadric interpolants for all values of the shape parameter. Comput Math Appl. Vol. 48 (5-6), pp. 853-867.
15. Fornberg, B.; Wright, G.; Larsson, E. (2004): Some observations regarding interpolants in the limit of flat radial basis functions. Comput Math Appl. Vol. 47, pp.37-55.
16. Fornberg, B.; Zuev, J. (2007): The Runge phenomenon and spatially variable shape parameters in RBF interpolation, Comput Math Appl. Vol. 54, pp. 379-398.
17. Franke, R. (1982): Scattered data interpolation: tests of some methods, Math. Comput. Vol. 38, pp. 181-200.
18. Hansen, P.C. (1997):.Rank-deficient and discrete ill posed problems: Numerical aspects of linear inversion, SIAM Publications, Philadelphia, PA.
19. Hardy, R.L. (1971): Multiquadric equations of topography and other irregular surfaces, J. Geo-Phys. Res. Vol. 176,pp.1905-1915.
20. Hardy, R.L. (1990): Theory and application of the multiquadric-biharmonic method: 20 years of discovery, Comput. Math. Appl. Vol. 19 (6-8), pp. 163-208.
21. Hon, Y.C.; Schaback, R.; Zhou, X. (2003): An adaptive greedy algorithm for solving large RBF collocation problems, Numer. Algor. Vol. 24, pp. 13-25.
22. Huang, C.-S.; Lee, C.-F.; Cheng, A.H.D. (2007): Error estimate, optimal shape parameter, and high precision computation of multiquadric collocation method, Eng. Anal. Bound. Elem. Vol. 31, pp. 615-623.
23. Ingber, M.S.; Chen, C.S.; Tanski, J.A (2004): A mesh free approach using radial basis functions and parallel domain decomposition for solving three dimensional diffusion equations, Int. J. Num. Meths. Eng. Vol. 60, pp. 2183-2201.
24. Kansa, E.J. (1990a): Multiquadrics-A scattered data approximation scheme with applications to computational fluid dynamics: I. Surface approximations and partial derivative estimates, Comput. Math. Appl. Vol. 19 (6-8), pp. 127-145.
25. Kansa, E.J. (1990b): 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. Vol. 19 (6-8), pp. 147-161.
26. Kansa, E.J.; Hon, Y.C. (2000): Circumventing the ill-conditioning problem with multiquadric radial basis functions: Applications to elliptic partial differential equations, Comput. Math. Applic. Vol. 39 (7/8), pp. 123-137.
27. Ling, L.; Kansa, E.J. (2004): Preconditioning for radial basis functions with domain decomposition, Math. & Comput. Model. Vol. 40, pp. 1413-1427.
28. Ling, L.; Kansa, E.J. (2005): A Least Squares Preconditioner for Radial Basis Functions Collocation Methods, Adv. Comput. Math. Vol. 23, pp. 31-54.
29. Ling, L.; Hon, Y.C. (2005): Improved numerical solver for Kansa's method based on affine space decomposition, Eng. Anal. Bound. Elem. Vol. 29, pp.1077-1085.
30. Ling, L.; Opfer, R.; Schaback, R. (2006): Resuits on meshless collocation techniques, Eng. Anal. Bound. Elem. Vol. 30, pp.147-253.
31. Madych, W.R ; Nelson, S.A. (1990): Multivariate interpolation and conditionally positive definite functions II, Math. Comput. Vol. 54, pp.211-230.
32. Madych, W.R. (1992): Miscellaneous error bounds for multiquadric and related interpolators, Comput. Math. Applic. Vol. 24 (12), pp. 121-138.
33. Mai-Duy, N; Tran-Cong, T. (2001): Numerical solution of Navier-Stokes equations using multiquadric radial basis function networks, Int. J. Num. Meth. Fluids Vol 37, pp. 65-86.
34. Mai-Duy, N; Tran-Cong, T. (2003): Approximation of function and its derivatives using radial basis function networks, Appl. Math. Modelling Vol. 27, pp. 197-220.
35. Mai-Duy, N; Tran-Cong, T. (2007): Solving Partial Differential Equations with Point Collocation And One-Dimensional Integrated Interpolation Schemes, ICCES, vol.3, no.3, pp.127-132.
36. Mai-Duy, N; Khennane, A.; Tran-Cong, T. (2006): Computation of Laminated Composite Plates using Integrated Radial Basis Function Networks, CMC: Computers, Materials, and Continua Vol. 5, no. 1, pp. 63-78.
37. Sarler, B. (2005): A Radial Basis Function Collocation Approach in Computational Fluid Dynamics, CMES: Computer Modeling in Engineering & Sciences Vol. 7, no. 2, pp. 185-194.
38. Schaback, R. (1995): Multivariate interpolation and approximation by translates of a basis function, In: Approximation theory VIII Vol. 1 College Station, TX.
39. Schaback, R. (1997): On the efficiency of interpolation by radial basis functions. In: Surface fitting and multiresolution methods (Chamonix-Mont-Blanc, 1996). Nashville, TN: Vanderbilt University Press, pp. 309-318.
40. Timoshenko, S.P.; Goodier, J.N. (1970): Theory of Elasticity, third ed., McGraw-Hill, New York.
41. Tolstykh, A.I.; Shirobokov, D.A. (2005): Using radial basis functions in a "finite difference mode, CMES: Computer Modeling in Engineer ing & Sciences Vol. 7, no. 2, pp. 207-222.
42. Volokh, K.Y.; Vilnay, O. (2000): Pin-pointing solution of ill-conditioned square systems of linear equations, Appl. Math. Lett. Vol. 13, pp. 119-124.
43. Wen, P.H. ; Hon, Y.C. (2007): Geometrically Nonlinear Analysis of Reissner-Mindlin Plate by Meshless Computation, CMES: Computer Modeling in Engineering & Sciences Vol. 21, no. 3, pp. 177-192.
44. Wertz, J.; Kansa, E.J.; Ling, L. (2006): The role of the Multiquadric Shape Parameters in solving Elliptic Partial Differential Equations, Comput. Math. Applic. Vol. 51 (8), pp. 1335-1348.
45. Yoon, J. (2001): Spectral Approximation Orders of Radial Basis Function Interpolation on the Sobolov Space, SIAM J. Math. Anal.,Vol. 33, pp. 946-958.
46. Young, D.L.; Chen, C.S.; Wong, T.K. (2005): Solution of Maxwell's Equations Using the MQ Method, CMC: Computers, Materials and Continua Vol. 2, no. 4, pp. 267-276.
47. Zienkiewicz, O.C.; Taylor, R.L. (2000): The finite element method, Vol. 1, 5th ed., Butterworth-Heinemann, Oxford.