A practical algorithm for determining the optimal pseudo-boundary in the method of fundamental solutions

Advances in Applied Mathematics and Mechanics

Volumn 1, Issue 4, 2009, Pages 510-528

  Karageorghis, A ^a  

^a UNIVERSITY OF CYPRUS   (Cyprus)

Author keywords

Elliptic boundary value problems; function minimization; Method of fundamental solutions

Indexed keywords

EID: 77952010032     PISSN: 20700733     EISSN: 20751354     Source Type: Journal    
DOI: 10.4208/aamm.09-m0916     Document Type: Article

References (32)

1. R. P. Brent, Algorithms for Minimization without Derivatives, Prentice-Hall, Engle-wood Cliffs, New Jersey, 1973.
2. D. Borman, D. B. Ingham, B. T. Johansson and D. Lesnic, The method of fundamental solutions for detection of cavities in EIT, J. Integral Equations Appl., to be appeared.
3. A. P. Cisilino and B. Sensale, Application of a simulated annealing algorithm in the optimal placement of the source points in the method of the fundamental solutions, Comput. Mech., 28 (2002), pp. 129-136.
4. P. J. Davis, Circulant Matrices, John Wiley & Sons, New York, 1979.
5. T. W. Drombosky, A. L. Meyer and L. Ling, Applicability of the method of fundamental solutions, Eng. Anal. Bound. Elem., 27 (2009), pp. 637-643.
6. G. Fairweather and R. L. Johnston, The method of fundamental solutions for problems in potential theory, in: Treatment of Integral Equations by Numerical Methods (Durham, 1982), Academic Press, London, pp. 349-359, 1982.
7. G. Fairweather and A. Karageorghis, The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math., 9 (1998), pp. 69-95.
8. G. Fairweather, A. Karageorghis and P. A. Martin, The method of fundamental solutions for scattering and radiation problems, Eng. Anal. Bound. Elem., 27 (2003), pp. 759-769.
9. G. E. Forsythe, M. A. Malcolm and C. B. Moler, Computer Methods for Mathematical Computations, Prentice-Hall, Englewood Cliffs, New Jersey, 1976.
10. P. E. Gill and W. Murray, Safeguarded steplength algorithms for optimization using descent methods, NPL Report NAC 37, National Physical Laboratory, 1973.
11. M. A. Golberg and C. S. Chen, Discrete Projection Methods for Integral Equations, Computational Mechanics Publications, Southampton, 1997.
12. M. A. Golberg and C. S. Chen, The method of fundamental solutions for potential, Helmholtz and diffusion problems, in: Boundary Integral Methods and Mathematical As-pects, ed. M. A. Golberg, WIT Press/Computational Mechanics Publications, Boston, pp. 103-176, 1999.
13. P. Gorzelańczyk and J. A. Kołodziej, Some remarks concerning the shape of the source contour with application of the method of fundamental solutions to elastic torsion of prismatic rods, Eng. Anal. Bound. Elem., 32 (2008), pp. 64-75.
14. R. L. Johnston and G. Fairweather, The method of fundamental solutions for problems in potential flow, Appl. Math. Model., 8 (1984), pp. 265-270.
15. R. L. Johnston and R. Mathon, The computation of electric dipole fields in conducting media, Int. J. Numer. Meth. Eng., 14 (1979), pp. 1739-1760.
16. D. Kahaner, C. Moler, and S. Nash, Numerical Methods and Software, Prentice Hall, Englewood Cliffs, New Jersey, 1989.
17. A. Karageorghis, The method of fundamental solutions for the solution of steady-state free boundary problems, J. Comput. Phys., 98 (1992), pp. 119-128.
18. A. Karageorghis and D. Lesnic, Detection of cavities using the method of fundamental solutions, Inverse. Probl. Sci. En., to be appeared.
19. A. Karageorghis and Y-S. Smyrlis, Matrix decomposition algorithms related to the MFS for axisymmetric problems, in: Recent Advances in Boundary Element Methods, eds. G. D. Manolis and D. Polyzos, Springer, New York, pp. 223-237, 2009.
20. M. Katsurada, A mathematical study of the charge simulation method II, J. Fac. Sci., Univ. of Tokyo, Sect. 1A, Math., 36 (1989), pp. 135-162,.
21. M. Katsurada and H. Okamoto, A mathematical study of the charge simulation method I, Journal of the Faculty of Science. University of Tokyo. Section IA. Mathematics, 35 (1988), pp. 507-518.
22. L. Marin, A meshless method for solving the Cauchy problem in three-dimensional elastostatics, Comput. Math. Appl., 50 (2005), pp. 73-92.
23. L. Marin and D. Lesnic, The method of fundmental solutions for the Cauchy problem associated with two-dimensional Helmholtz-type equations, Comput. Struct., 83 (2005), pp. 267-278.
24. L. Marin and D. Lesnic, The method of fundamental solutions for inverse boundary value problems associated with the two-dimensional biharmonic equation, Math. Comput. Model., 42 (2005), pp. 261-278.
25. R. Mathon and R. L. Johnston, The approximate solution of elliptic boundary-value problems by fundamental solutions, SIAM J. Numer. Anal., 14 (1977), pp. 638-650.
26. Numerical Algorithms Group Library Mark 21, NAG(UK) Ltd, Wilkinson House, Jordan Hill Road, Oxford, UK (2007).
27. A. Poullikkas, A. Karageorghis, G. Georgiou and J. Ascough, The method of fundamental solutions for Stokes flows with a free surface, Numer. Meth. Part. D. E., 14 (1998), pp. 667-678.
28. W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Fortran version), Cambridge University Press, Cambridge, 1989.
29. R. Schaback, Adaptive numerical solution of MFS systems, in: The Method of Fundamental Solutions-A Meshless Method, eds. C.S. Chen, A Karageorghis and Y.S. Smyrlis, Dynamic Publishers, Inc., pp. 1-27, 2008.
30. Y-S. Smyrlis and A. Karageorghis, Numerical analysis of the MFS for certain harmonic problems, Modélisation Mathématique et Analyse Numérique, 38 (2004), pp. 495-517.
31. R. Tankelevich, G. Fairweather, A. Karageorghis and Y-S. Smyrlis, Potential field based geometric modeling using the method of fundamental solutions, Inter. J. Numer. Meth. Eng., 68 (2006), pp. 1257-1280.
32. Y. Wang and Y. Rudy, Application of the method of fundamental solutions to potential-based inverse electrocardiography, Ann. Biomed. Eng., 34 (2006), pp. 1272-1288.