An improved subspace selection algorithm for meshless collocation methods

International Journal for Numerical Methods in Engineering

Volumn 80, Issue 13, 2009, Pages 1623-1639

  Ling, Leevan ^a   Schaback, Robert ^b  

^a HONG KONG BAPTIST UNIVERSITY   (Hong Kong)

^b UNIVERSITY OF GÖTTINGEN   (Germany)

Author keywords

Adaptive greedy algorithm; Collocation; Kansa's method; Radial basis function

Indexed keywords

COLLOCATION METHOD; COMMON PROBLEMS; COMPUTATIONAL PRECISION; DATA POINTS; DATA-DEPENDENT ALGORITHMS; GREEDY ALGORITHMS; ILL-CONDITIONING; KANSA'S METHOD; MESH-LESS METHODS; MESHLESS; METHOD OF FUNDAMENTAL SOLUTIONS; OPTIMAL CENTERS; OPTIMAL LOCATIONS; OPTIMAL PLACEMENTS; RADIAL BASIS FUNCTIONS; RESULTANT MATRICES; SOURCE POINTS; SUBSPACE SELECTION; USER-DEFINED PARAMETERS;

ATTITUDE CONTROL; COMPUTATIONAL FLUID DYNAMICS; IMAGE SEGMENTATION; OPTIMIZATION; PARTIAL DIFFERENTIAL EQUATIONS; RADIAL BASIS FUNCTION NETWORKS;

ADAPTIVE ALGORITHMS;

EID: 64049099076     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.2674     Document Type: Article

References (42)

1. Li J, Chen CS. Some observations on unsymmetric radial basis function collocation methods for convection- diffusion problems. International Journal for Numerical Methods in Engineering 2003; 57(8):1085-1094.
2. Chen CS, Ganesh M, Golberg MA, Cheng AH-D. Multilevel compact radial functions based computational schemes for some elliptic problems. Computers and Mathematics with Applications 2002; 43(3-5):359-378.
3. Chen CS, Kuhn G, Li J, Mishuris G. Radial basis functions for solving near singular Poisson problems. Communications in Numerical Methods in Engineering 2003; 19(5):333-347.
4. Chen CS, Brebbia CA, Power H. Dual reciprocity method using compactly supported radial basis functions. Communications in Numerical Methods in Engineering 1999; 15(2):137-150.
5. Golberg MA, Chen CS, Bowman H, Power H. Some comments on the use of radial basis functions in the dual reciprocity method. Computational Mechanics 1998; 21(2):141-148.
6. Karageorghis A, Fairweather G. The method of fundamental solutions for the numerical solution of the biharmonic equation. Journal of Computational Physics 1987; 69:433-459.
7. Young DL, Jane SJ, Fan CM, Murugesan K, Tsai CC. The method of fundamental solutions for 2D and 3D Stokes problems. Journal of Computational Physics 2006; 211(1):1-8.
8. Young DL, Chen KH, Lee CW. Novel meshless method for solving the potential problems with arbitrary domain. Journal of Computational Physics 2005; 209(1):290-321.
9. Chen CS. The method of fundamental solutions for non-linear thermal explosions. Communications in Numerical Methods in Engineering 1995; 11(8):675-681.
10. Young DL, Tsai CC, Lin YC, Chen CS. The method of fundamental solutions for eigenfrequencies of plate vibrations. CMC Computers, Materials and Continua 2006; 4(1):1-10.
11. Hon YC, Mao XZ. A radial basis function method for solving options pricing models. Financial Engineering 1999; 8(1):31-49.
12. Marcozzi MD, Choi S, Chen CS. On the use of boundary conditions for variational formulations arising in financial mathematics. Applied Mathematics and Computation 2001; 124(2):197-214.
13. Chen JS, Hu W, Hu HY. Reproducing kernel enhanced local radial basis collocation method. International Journal for Numerical Methods in Engineering 2008; 75(5):600-627.
14. Cheng AH-D, Cabral JJSP. Direct solution of ill-posed boundary value problems by radial basis function collocation method. International Journal for Numerical Methods in Engineering 2005; 64(1):45-64.
15. Hon YC, Ling L, Liew KM. Numerical analysis of parameters in a laminated beam model by radial basis functions. CMC Computers, Materials and Continua 2005; 2(1):39-50.
16. Hu HY, Chen JS, Hu W. Weighted radial basis collocation method for boundary value problems. International Journal for Numerical Methods in Engineering 2007; 69(13):2736-2757.
17. Ingber MS, Chen CS, Tanski JA. A mesh free approach using radial basis functions and parallel domain decomposition for solving three-dimensional diffusion equations. International Journal for Numerical Methods in Engineering 2004; 60(13):2183-2201.
18. Jin B, Zheng Y. Boundary knot method for some inverse problems associated with the Helmholtz equation. International Journal for Numerical Methods in Engineering 2005; 62(12):1636-1651.
19. Rendall TCS, Allen CB. Unified fluid-structure interpolation and mesh motion using radial basis functions. International Journal for Numerical Methods in Engineering 2008; 74(10):1519-1559.
20. Drombosky TW, Meyer AL, Ling L. Applicability of the method of fundamental solutions. Engineering Analysis with Boundary Elements 2009; 33(5):637-643.
21. Alves CJS, Antunes PRS. The method of fundamental solutions applied to the calculation of eigenfrequencies and eigenmodes of 2D simply connected shapes. CMC Computers, Materials and Continua 2005; 2(4):251-266.
22. Schaback R. Error estimates and condition numbers for radial basis function interpolation. Advances in Computational Mathematics 1995; 3(3):251-264.
23. Schaback R. Multivariate interpolation and approximation by translates of a basis function. Approximation Theory VIII, vol. 1, College Station, TX, 1995. Series in Approximations and Decompositions, vol. 6. World Scientific Publishing: River Edge, NJ, 1995; 491-514.
24. Schaback R. On the efficiency of interpolation by radial basis functions. Surface Fitting and Multiresolution Methods, Chamonix-Mont-Blanc, 1996. Vanderbilt University Press: Nashville, TN, 1997; 309-318.
25. Wertz J, Kansa EJ, Ling L. The role of the multiquadric shape parameters in solving elliptic partial differential equations. Computers and Mathematics with Applications 2006; 51(8):1335-1348.
26. Rippa S. An algorithm for selecting a good value for the parameter c in radial basis function interpolation. Advances in Computational Mathematics 1999; 11(2-3):193-210.
27. Huang C-S, Lee C-F, Cheng AH-D. Error estimate, optimal shape factor, and high precision computation of multiquadric collocation method. Engineering Analysis with Boundary Elements 2007; 31(7):614-623.
28. Ling L, Opfer R, Schaback R. Results on meshless collocation techniques. Engineering Analysis with Boundary Elements 2006; 30(4):247-253.
29. Kansa EJ. Multiquadrics-a scattered data approximation scheme with applications to computational fluiddynamics. I. Surface approximations and partial derivative estimates. Computers and Mathematics with Applications 1990; 19(8-9):127-145.
30. Kansa EJ. Multiquadrics-a scattered data approximation scheme with applications to computational fluiddynamics. II. Solutions to parabolic, hyperbolic and elliptic partial differential equations. Computers and Mathematics with Applications 1990; 19(8-9):147-161.
31. Schaback R, Wendland H. Adaptive greedy techniques for approximate solution of large RBF systems. Numerical Algorithms 2000; 24(3):239-254.
32. Hon YC, Schaback R, Zhou X. An adaptive greedy algorithm for solving large RBF collocation problems. Numerical Algorithms 2003; 32(1):13-25.
33. Lee C-F, Ling L, Schaback R. On convergent numerical algorithms for unsymmetric collocation. Advances in Computational Mathematics 2009; 30(4):339-354.
34. Schaback R. Convergence of unsymmetric kernel-based meshless collocation methods. SIAM Journal on Numerical Analysis 2007; 45(1):333-351.
35. Appleby G, Smolarski DC. A linear acceleration row action method for projecting onto subspaces. Electronic Transactions on Numerical Analysis 2005; 20:253-275.
36. Ling L, Schaback R. Stable and convergent unsymmetric meshless collocation methods. SIAM Journal on Numerical Analysis 2008; 46(3):1097-1115.
37. Cheng AH-D, Golberg MA, Kansa EJ, Zammito G. Exponential convergence and h-c multiquadric collocation method for partial differential equations. Numerical Methods for Partial Differential Equations 2003; 19(5): 571-594.
38. Brown D, Ling L, Kansa EJ, Levesley J. On approximate cardinal preconditioning methods for solving PDEs with radial basis functions. Engineering Analysis with Boundary Elements 2005; 29(4):343-353.
39. Ling L, Kansa EJ. A least-squares preconditioner for radial basis functions collocation methods. Advances in Computational Mathematics 2005; 23(1-2):31-54.
40. Ling L, Kansa EJ. Preconditioning for radial basis functions with domain decomposition methods. Mathematical and Computer Modelling 2004; 40(13):1413-1427.
41. Kwok TO, Ling L. On convergence of a least-squares Kansa's method for the modified Helmholtz equations. Advances in Applied Mathematics and Mechanics 2009; 1(3):367-382.
42. Kansa EJ, Aldredge RC, Ling L. Numerical simulation of two-dimensional combustion using mesh-free methods. Engineering Analysis with Boundary Elements 2009; 33(7):940-950.