# zbMATH — the first resource for mathematics

The MFS for numerical boundary identification in two-dimensional harmonic problems. (English) Zbl 1259.65194
Summary: In this study, we briefly review the applications of the method of fundamental solutions to inverse problems over the last decade. Subsequently, we consider the inverse geometric problem of identifying an unknown part of the boundary of a domain in which the Laplace equation is satisfied. Additional Cauchy data are provided on the known part of the boundary. The method of fundamental solutions is employed in conjunction with regularization in order to obtain a stable solution. Numerical results are presented and discussed.

##### MSC:
 65N80 Fundamental solutions, Green’s function methods, etc. for boundary value problems involving PDEs 65N21 Numerical methods for inverse problems for boundary value problems involving PDEs
##### Software:
HYBRJ; minpack; NAG
Full Text:
##### References:
 [1] Alves, C.J.S.; Chen, C.S., A new method of fundamental solutions applied to nonhomogeneous elliptic problems, Advances in computational mathematics, 23, 125-142, (2005) · Zbl 1070.65119 [2] Alves, C.J.S.; Colaço, M.J.; Leitão, V.M.A.; Martins, N.F.M.; Orlande, H.R.B.; Roberty, N.C., Recovering the source term in a linear diffusion problem by the method of fundamental solutions, Inverse problems in science and engineering, 16, 1005-1021, (2005) [3] Alves, C.J.S.; Martins, N.F.M., Reconstruction of inclusions or cavities in potential problems using the MFS, (), 51-73 [4] Alves, C.J.S.; Martins, N.F.M., The direct method of fundamental solutions and the inverse kirsch – kress method for the reconstruction of elastic inclusions or cavities, Journal of integral equations and applications, 21, 153-178, (2009) · Zbl 1166.74008 [5] Alves, C.J.S.; Martins, N.F.M., On the determination of a Robin boundary coefficient in an elastic cavity using the MFS, (), 125-139 · Zbl 1157.74015 [6] Alves, C.J.S.; Martins, N.F.M.; Roberty, N.C., Full identification of acoustic sources with multiple frequencies and boundary measurements, Inverse problems and imaging, 3, 275-294, (2009) · Zbl 1187.65124 [7] Beretta, E.; Vessella, S., Stable determination of boundaries from Cauchy data, SIAM journal on mathematical analysis, 30, 220-232, (1998) · Zbl 0928.35201 [8] Birginie, J.M.; Allard, F.; Kassab, A.J., Application of trigonometric boundary elements to heat and mass transfer problems, (), 65-74 · Zbl 0861.65097 [9] Borman, D.; Ingham, D.B.; Johansson, B.T.; Lesnic, D., The method of fundamental solutions for detection of cavities in EIT, Journal of integral equations and applications, 21, 381-404, (2009) · Zbl 1189.78059 [10] Chen, C.W.; Young, D.L.; Tsai, C.C.; Murugesan, K., The method of fundamental solutions for inverse 2D Stokes problems, Computational mechanics, 37, 2-14, (2005) · Zbl 1158.76392 [11] Cheng, J.; Hon, Y.C.; Yamamoto, M., Conditional stability estimation for an inverse boundary problem with non-smooth boundary in $$\mathbb{R}^3$$, Transactions of the American mathematical society, 353, 4123-4138, (2001) · Zbl 0970.35163 [12] Dong, C.F.; Sun, F.Y.; Meng, B.Q., A method of fundamental solutions for inverse heat conduction problems in an anisotropic medium, Engineering analysis with boundary elements, 31, 75-82, (2007) · Zbl 1195.80036 [13] Fairweather, G.; Karageorghis, A., The method of fundamental solutions for elliptic boundary value problems, Advances in computational mathematics, 9, 69-95, (1998) · Zbl 0922.65074 [14] Fairweather, G.; Karageorghis, A.; Martin, P.A., The method of fundamental solutions for scattering and radiation problems, Engineering analysis with boundary elements, 27, 759-769, (2003) · Zbl 1060.76649 [15] Fasino, D.; Inglese, G., An inverse Robin problem for Laplace’s equation: theoretical results and numerical methods, Inverse problems, 15, 41-48, (1999) · Zbl 0922.35188 [16] Garbow BS, Hillstrom KE, More JJ. MINPACK Project. Argonne National Laboratory; 1980. [17] Golberg, M.A.; Chen, C.S., The method of fundamental solutions for potential Helmholtz and diffusion problems, (), 105-176 · Zbl 0945.65130 [18] Hansen, P.C.; Leary, O’D.P., The use of the L-curve in the regularization of discrete ill-posed problems, SIAM journal on scientific computing, 14, 1487-1503, (1993) · Zbl 0789.65030 [19] Hon, Y.C.; Li, M., A computational method for inverse free boundary determination problem, International journal for numerical methods in engineering, 73, 1291-1309, (2008) · Zbl 1158.80328 [20] Hon, Y.C.; Wei, T., A meshless computational method for solving inverse heat conduction problem, International series on advances in boundary elements, 13, 135-144, (2002) · Zbl 1011.65065 [21] Hon, Y.C.; Wei, T., A meshless scheme for solving inverse problems of Laplace equation, (), 291-300 · Zbl 1037.65111 [22] Hon, Y.C.; Wei, T., A fundamental solution method for inverse heat conduction problems, Engineering analysis with boundary elements, 28, 489-495, (2004) · Zbl 1073.80002 [23] Hon, Y.C.; Wei, T., The method of fundamental solutions for solving multidimensional heat conduction problems, CMES: computer modeling in engineering & sciences, 7, 119-132, (2005) · Zbl 1114.80004 [24] Hon, Y.C.; Wu, Z., A numerical computation for inverse boundary determination problem, Engineering analysis with boundary elements, 24, 599-606, (2000) · Zbl 0972.65085 [25] Isakov, V., On uniqueness of obstacles and boundary conditions for restricted dynamical scattering data, Inverse problems and imaging, 2, 151-164, (2008) · Zbl 1158.35104 [26] Jin, B.; Marin, L., The method of fundamental solutions for inverse source problems associated with the steady state heat conduction, International journal for numerical methods in engineering, 69, 1572-1589, (2007) · Zbl 1194.80101 [27] Jin, B.; Zheng, Y., A meshless method for some inverse problems associated with the Helmholtz equation, Computer methods in applied mechanics and engineering, 195, 2270-2280, (2006) · Zbl 1123.65111 [28] Jin, B.; Zheng, Y.; Marin, L., The method of fundamental solutions for inverse boundary value problems associated with the steady state heat conduction in anisotropic media, International journal for numerical methods in engineering, 65, 1865-1891, (2006) · Zbl 1124.80400 [29] Karageorghis, A.; Lesnic, D., Detection of cavities using the method of fundamental solutions, Inverse problems in science and engineering, 17, 803-820, (2009) · Zbl 1175.65130 [30] Kaup, P.; Santosa, F., Nondestructive evaluation of corrosion damage using electrostatic boundary measurements, Journal of nondestructive evaluation, 14, 127-136, (1995) [31] Kupradze, V.D.; Aleksidze, M.A., The method of functional equations for the approximate solution of certain boundary value problems, U.S.S.R. computational mathematics and mathematical physics, 4, 82-126, (1964) · Zbl 0154.17604 [32] Lesnic, D.; Berger, J.R.; Martin, P.A., A boundary element regularization method for the boundary determination in potential corrosion damage, Inverse problems in engineering, 10, 163-182, (2002) [33] Marin, L., Numerical solutions of the Cauchy problem for steady-state heat transfer in two-dimensional functionally graded materials, International journal of solids and structures, 42, 4338-4351, (2005) · Zbl 1120.80308 [34] Marin, L., A meshless method for the numerical solution of the Cauchy problem associated with three-dimensional Helmholtz-type equations, Applied mathematics and computation, 165, 355-374, (2005) · Zbl 1070.65115 [35] Marin, L., A meshless method for solving the Cauchy problem in three-dimensional elastostatics, Computers and mathematics with applications, 50, 73-92, (2005) · Zbl 1127.74014 [36] Marin, L., Numerical boundary identification for Helmholtz-type equations, Computational mechanics, 39, 25-40, (2006) · Zbl 1168.80303 [37] Marin, L., The method of fundamental solutions for inverse problems associated with the steady-state heat conduction in the presence of sources, CMES: computer modeling in engineering & sciences, 30, 99-122, (2008) [38] Marin, L., Stable MFS solution to singular direct and inverse problems associated with the Laplace equation subjected to noisy data, CMES: computer modeling in engineering & sciences, 37, 203-242, (2008) [39] Marin, L.; Lesnic, D., BEM first-order regularisation method in linear elasticity for boundary identification, Computer methods in applied mechanics and engineering, 192, 2059-2071, (2003) · Zbl 1041.74075 [40] Marin, L.; Lesnic, D., The method of fundamental solutions for the Cauchy problem in two-dimensional linear elasticity, International journal of solids and structures, 41, 3425-3438, (2004) · Zbl 1071.74055 [41] Marin, L.; Lesnic, D., The method of fundamental solutions for the Cauchy problem associated with two-dimensional Helmholtz-type equations, Computers & structures, 83, 267-278, (2005) · Zbl 1088.35079 [42] Marin, L.; Lesnic, D., The method of fundamental solutions for inverse boundary value problems associated with the two-dimensional biharmonic equation, Mathematical and computer modelling, 42, 261-278, (2005) · Zbl 1088.35079 [43] Martins, N.F.M.; Silvestre, A.L., An iterative MFS approach for the detection of immersed obstacles, Engineering analysis with boundary elements, 32, 517-524, (2008) · Zbl 1244.76082 [44] Mathon, R.; Johnston, R.L., The approximate solution of elliptic boundary value problems by fundamental solutions, SIAM journal on numerical analysis, 14, 638-650, (1977) · Zbl 0368.65058 [45] McIver, M., An inverse problem in electromagnetic crack detection, IMA journal of applied mathematics, 47, 127-145, (1991) · Zbl 0727.73063 [46] Mera, N.S., The method of fundamental solutions for the backward heat conduction problem, Inverse problems in science and engineering, 13, 79-98, (2005) · Zbl 1194.80107 [47] Mera, N.S.; Lesnic, D., A three-dimensional boundary determination problem in potential corrosion damage, Computational mechanics, 36, 129-138, (2005) · Zbl 1100.78022 [48] Numerical Algorithms Group Library Mark 21. NAG(UK) Ltd, Wilkinson House, Jordan Hill Road, Oxford, UK, 2007. [49] Nili Ahmadabadi, M.; Arab, M.; Maalek Ghaini, F.M., The method of fundamental solutions for the inverse space-dependent heat source problem, Engineering analysis with boundary elements, 33, 1231-1235, (2009) · Zbl 1180.80054 [50] Ohe, T.; Ohnaka, K., Uniqueness and convergence of numerical solution of the Cauchy problem for the Laplace equation by a charge simulation method, Japanese journal of industrial and applied mathematics, 21, 339-359, (2004) · Zbl 1078.65081 [51] Shigeta, T.; Young, D.L., Method of fundamental solutions with optimal regularization techniques for the Cauchy problem of the Laplace equation with singular points, Journal of computational physics, 228, 1903-1915, (2009) · Zbl 1161.65353 [52] Tikhonov, A.N.; Leonov, A.S.; Yagola, A.G., Nonlinear ill-posed problems, (1998), Chapman & Hall London · Zbl 0920.65038 [53] Valle, M.F.; Colaço, M.J.; Neto, F.S., Estimation of the heat transfer coefficient by means of the method of fundamental solutions, Inverse problems in science and engineering, 16, 777-795, (2008) · Zbl 1154.65076 [54] Vessella, S., Quantitative estimates of unique continuation for parabolic equations, determination of unknown time-varying boundaries and optimal stability estimates, Inverse problems, 24, 023001, (2008), (81p.) · Zbl 1154.35095 [55] Vogelius, M.; Xu, J., A nonlinear elliptic boundary value problem related to corrosion modeling, Quarterly of applied mathematics, 56, 479-505, (1998) · Zbl 0954.35067 [56] Wang, Y.; Rudy, Y., Application of the method of fundamental solutions to potential-based inverse electrocardiography, Annals of biomedical engineering, 34, 1272-1288, (2006) [57] Wei, T.; Li, Y.S., An inverse boundary problem for one-dimensional heat equation with a multilayer domain, Engineering analysis with boundary elements, 33, 225-232, (2009) · Zbl 1244.80005 [58] Wei T, Zhou DY. Convergence analysis for the Cauchy problem of Laplace’s equation by a regularized method of fundamental solutions. Advances in Computational Mathematics 2010;33:491-510. · Zbl 1213.65136 [59] Wei, T.; Hon, Y.C.; Ling, L., Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators, Engineering analysis with boundary elements, 31, 373-385, (2007) · Zbl 1195.65206 [60] Yan, L.; Fu, C.L.; Yang, F.L., The method of fundamental solutions for the inverse heat source problem, Engineering analysis with boundary elements, 32, 216-222, (2008) · Zbl 1244.80026 [61] Yan, L.; Fu, C.L.; Yang, F.L., A meshless method for solving an inverse spacewise-dependent heat source problem, Journal of computational physics, 228, 123-136, (2009) · Zbl 1157.65444 [62] Yang, F.L.; Yan, L.; Wei, T., Reconstruction of the corrosion boundary for the Laplace equation by using a boundary collocation method, Mathematics and computers in simulation, 79, 2148-2156, (2009) · Zbl 1161.65363 [63] Yang, F.L.; Yan, L.; Wei, T., Reconstruction of part of a boundary for the Laplace equation by using a regularized method of fundamental solutions, Inverse problems in science and engineering, 17, 1113-1128, (2009) · Zbl 1178.65134 [64] Young, D.L.; Tsai, C.C.; Chen, C.W.; Fan, C.M., The method of fundamental solutions and condition number analysis for inverse problems of Laplace equation, Computers and mathematics with applications, 55, 1189-1200, (2008) · Zbl 1143.65087 [65] Zeb, A.; Ingham, D.B.; Lesnic, D., The method of fundamental solutions for a biharmonic boundary determination, Computational mechanics, 42, 371-379, (2008) · Zbl 1163.65078 [66] Zhou, D.; Wei, T., The method of fundamental solutions for solving a Cauchy problem of Laplace’s equation in a multi-connected domain, Inverse problems in science and engineering, 16, 389-411, (2008) · Zbl 1258.65102
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.