A meshless method for the numerical solution of the Cauchy problem associated with three-dimensional Helmholtz-type equations

Applied Mathematics and Computation

Volumn 165, Issue 2, 2005, Pages 355-374

  Marin, Liviu ^a  

^a UNIVERSITY OF LEEDS   (United Kingdom)

Author keywords

Cauchy problem; Helmholtz type equations; Inverse problem; Meshless method; Method of fundamental solutions; Regularization

Indexed keywords

APPROXIMATION THEORY; BOUNDARY ELEMENT METHOD; CONVERGENCE OF NUMERICAL METHODS; DATA REDUCTION; FINITE DIFFERENCE METHOD; FINITE ELEMENT METHOD; LINEAR ALGEBRA;

CAUCHY PROBLEMS; HELMHOLTZ-TYPE EQUATIONS; INVERSE PROBLEM; MESHLESS METHOD; METHOD OF FUNDAMENTAL SOLUTIONS; REGULARIZATION;

LINEAR EQUATIONS;

EID: 17644417110     PISSN: 00963003     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.amc.2004.04.052     Document Type: Article

References (40)

1. V.D. Kupradze, and M.A. Aleksidze The method of functional equations for the approximate solution of certain boundary value problems USSR. Comp. Math. Math. Phys. 4 1964 82 126
2. R. Mathon, and R.L. Johnston The approximate solution of elliptic boundary value problems by fundamental solutions SIAM J. Numer. Anal. 14 1977 638 650
3. G. Fairweather, and A. Karageorghis The method of fundamentamental solutions for elliptic boundary value problems Adv. Comput. Math. 9 1998 69 95
4. K. Balakrishnan, and P.A. Ramachandran A particular solution Trefftz method for non-linear Poisson problems in heat and mass transfer J. Comput. Phys. 150 1999 239 267
5. M.A. Golberg, and C.S. Chen Discrete projection methods for integral equations 1996 Computational Mechanics Publications Southampton
6. M.A. Golberg, and C.S. Chen The method of fundamental solutions for potential, Helmholtz and diffusion problems M.A. Golberg Boundary Integral Methods: Numerical and Mathematical Aspects 1999 WIT Press and Computational Mechanics Publications Boston 105 176
7. A. Karageorghis, and G. Fairweather The method of fundamental solutions for the numerical solution of the biharmonic equation J. Comput. Phys. 69 1987 434 459
8. A. Poullikkas, A. Karageorghis, and G. Georgiou Methods of fundamental solutions for harmonic and biharmonic boundary value problems Comput. Mech. 21 1998 416 423
9. A. Poullikkas, A. Karageorghis, and G. Georgiou The method of fundamental solutions for inhomogeneous elliptic problems Comput. Mech. 22 1998 100 107
10. A. Poullikkas, A. Karageorghis, and G. Georgiou The numerical solution of three-dimensional Signorini problems with the method of fundamental solutions Eng. Anal. Bound. Elem. 25 2001 221 227
11. A. Karageorghis, and G. Fairweather The method of fundamental solutions for axisymmetric elasticity problems Comput. Mech. 25 2000 524 532
12. J.A. Berger, and A. Karageorghis The method of fundamental solutions for heat conduction in layered materials Int. J. Num. Meth. Eng. 45 1999 1681 1694
13. J.A. Berger, and A. Karageorghis The method of fundamental solutions for layered elastic materials Eng. Anal. Bound. Elem. 25 2001 877 886
14. A. Karageorghis The method of fundamental solutions for the calculation of the eigenvalues of the Helmholtz equation Appl. Math. Lett. 14 2001 837 842
15. A. Poullikkas, A. Karageorghis, and G. Georgiou The numerical solution for three-dimensional elastostatics problems Comput. Struct. 80 2002 365 370
16. P.A. Ramachandran Method of fundamental solutions: Singular value decomposition analysis Commun. J. Numer. Meth. Eng. 18 2002 789 801
17. K. Balakrishnan, R. Sureshkumar, and P.A. Ramachandran An operator splitting-radial basis function method for the solution of transient nonlinear Poisson problems Comput. Math. Appl. 43 2001 289 304
18. D.E. Beskos Boundary element method in dynamic analysis: Part II (1986-1996) ASME Appl. Mech. Rev. 50 1997 149 197
19. J.T. Chen, and F.C. Wong Dual formulation of multiple reciprocity method for the acoustic mode of a cavity with a thin partition J. Sound Vib. 217 1998 75 95
20. I. Harari, P.E. Barbone, M. Slavutin, and R. Shalom Boundary infinite elements for the Helmholtz equation in exterior domains Int. J. Numer. Meth. Eng. 41 1998 1105 1131
21. W.S. Hall, and X.Q. Mao A boundary element investigation of irregular frequencies in electromagnetic scattering Eng. Anal. Bound. Elem. 16 1995 245 252
22. Y. Niwa, S. Kobayashi, and M. Kitahara Determination of eigenvalue by boundary element method P.K. Banerjee R. Shaw Development in Boundary Element Methods 1982 Applied Science Publisher New York Chapter 7
23. A.J. Nowak, and C.A. Brebbia Solving Helmholtz equation by boundary elements using multiple reciprocity method C.M. Calomagno C.A. Brebbia Computer Experiment in Fluid Flow 1989 CMP/Springer Verlag Berlin 265 270
24. J.P. Agnantiaris, D. Polyzer, and D. Beskos Three-dimensional structural vibration analysis by the dual reciprocity BEM Comput. Mech. 21 1998 372 381
25. J. Hadamard Lectures on Cauchy problem in linear partial differential equations 1923 Oxford University Press London
26. M.R. Bai Application of BEM-based acoustic holography to radiation analysis of sound sources with arbitrarily shaped geometries J. Acoust. Soc. Am. 92 1992 533 549
27. B.K. Kim, and J.G. Ih On the reconstruction of the vibro-acoustic field over the surface enclosing an interior space using the boundary element method J. Acoust. Soc. Am. 100 1996 3003 3016
28. Z. Wang, and S.R. Wu Helmholtz equation-least-squares method for reconstructing the acoustic pressure field J. Acoust. Soc. Am. 102 1997 2020 2032
29. S.R. Wu, and J. Yu Application of BEM-based acoustic holography to radiation analysis of sound sources with arbitrarily shaped geometries J. Acoust. Soc. Am. 104 1998 2054 2060
30. T. DeLillo, V. Isakov, N. Valdivia, and L. Wang The detection of the source of acoustical noise in two dimensions SIAM J. Appl. Math. 61 2001 2104 2121
31. T. DeLillo, V. Isakov, N. Valdivia, and L. Wang The detection of surface vibrations from interior acoustical pressure Inverse Probl. 19 2003 507 524
32. L. Marin, L. Elliott, P.J. Heggs, D.B. Ingham, D. Lesnic, and X. Wen An alternating iterative algorithm for the Cauchy problem associated to the Helmholtz equation Comput. Meth. Appl. Mech. Eng. 192 2003 709 722
33. L. Marin, L. Elliott, P.J. Heggs, D.B. Ingham, D. Lesnic, and X. Wen Conjugate gradient-boundary element solution to the Cauchy problem for Helmholtz-type equations Comput. Mech. 31 2003 367 377
34. A.N. Tikhonov, and V.Y. Arsenin Methods for Solving Ill-Posed Problems 1986 Nauka Moscow
35. P.C. Hansen The L-curve and its use in the numerical treatment of inverse problems P. Johnston Computational Inverse Problems in Electrocardiology 2001 WIT Press Southampton 119 142
36. G. Chen, and J. Zhou Boundary Element Methods 1992 Academic Press London
37. P.C. Hansen Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion 1998 SIAM Philadelphia
38. A.N. Tikhonov, A.S. Leonov, and A.G. Yagola Nonlinear Ill-Posed Problems 1998 Chapman & Hall London
39. M. Hanke Limitations of the L-curve method in ill-posed problems BIT 36 1996 287 301
40. C.R. Vogel Non-convergence of the L-curve regularization parameter selection method Inverse Probl. 12 1996 535 547