Reconstruction of part of a boundary for the Laplace equation by using a regularized method of fundamental solutions

References

• Hadamard, J, 1923. Lectures on Cauchy's Problem in Linear Partial Differential Equations. London: Oxford University; 1923.
   Google Scholar
• Bukhgeim, AL, Cheng, J, and Yamamoto, M, 1998. "Uniqueness and stability for an inverse problem of determining a part of boundary". In: Inverse Problems in Engineering Mechanics (Nagano, 1998). Oxford: Elsevier; 1998. pp. 327–336.
   Google Scholar
• Cheng, J, Hon, YC, and Yamamoto, M, 2001. Conditional stability estimation for an inverse boundary problem with non-smooth boundary in R3, Trans. Amer. Math. Soc. 353 (10) (2001), pp. 4123–4138.
   Web of Science ®Google Scholar
• Beretta, E, and Vessella, S, 1999. Stable determination of boundaries from cauchy data, SIAM J. Math. Anal. 30 (1) (1999), pp. 220–232.
   Web of Science ®Google Scholar
• Rondi, L, 2000. Uniqueness and stability for the determination of boundary defects by electrostatic measurements, Proc. R. Soc. Edinb. Sect. A 130 (5) (2000), pp. 1119–1151.
   Google Scholar
• McIver, M, 1991. An inverse problem in electromagnetic crack detection, IMA J. Appl. Math. 47 (2) (1991), pp. 127–145.
   Web of Science ®Google Scholar
• Michael, DH, Waechter, RT, and Collins, R, 1982. The measurement of surface cracks in metals by using a.c. electric fields, Proc. R. Soc. Lond. Sect. A 381 (1982), pp. 139–157.
   Web of Science ®Google Scholar
• Charton, M, Reinhardt, HJ, and Hào, DN, 2001. "Numerical solution of a shape optimization problem". In: Proc. Roy. Soc. Lond. Sect. A. Eurotherm Seminar 68. Poitiers, France: Inverse problems and experimental design in thermal and mechanical engineering; 2001.
   Google Scholar
• Nachaoui, A, 2003. An improved implementation of an iterative method in boundary identification problems, Numer. Algorithms 33 (1–4) (2003), pp. 381–398, (International Conference on Numerical Algorithms, Vol. I (Marrakesh, 2001)).
   Web of Science ®Google Scholar
• Lesnic, D, Berger, JR, and Martin, PA, 2002. A boundary element regularization method for the boundary determination in potential corrosion damage, Inv. Probl. Sci. Eng. 10 (2002), pp. 163–182.
   Google Scholar
• Marin, L, and Lesnic, D, 2003. BEM first-order regularisation method in linear elasticity for boundary identification, Comput. Methods Appl. Mech. Eng. 192 (16–18) (2003), pp. 2059–2071.
   Web of Science ®Google Scholar
• Hon, YC, and Wu, Z, 2000. A numerical computation for inverse boundary determination problem, Eng. Anal. Boundary Elements, 24 (7–8) (2000), pp. 599–606.
   Web of Science ®Google Scholar
• Liu, CS, Chang, CW, and Chiang, CY, 2008. A regularized integral equation method for the inverse geometry heat conduction problem, Int. J. Heat. Transf. 51 (21–22) (2008), pp. 5380–5388.
   Web of Science ®Google Scholar
• Powell, MJD, 1987. Radial Basis Functions for Multivariable Interpolation: A Review. Algorithms for Approximation, Clarendon Press; 1987. pp. 143–167.
   Google Scholar
• Wu, Z, 1992. Hermite-birkhoff interpolation of scattered data by radial basis functions, Anal. Theor. Appl. 8 (1992), pp. 1–10.
   Google Scholar
• Golberg, MA, and Chen, CS, 1999. "The method of fundamental solutions for potential, Helmholtz and diffusion problems". In: Boundary Integral Methods: Numerical and Mathematical Aspects. Vol. 1. Boston, MA: WIT Press/Comput. Mech. Publ.; 1999. pp. 103–176.
   Google Scholar
• Fairweather, G, and Karageorghis, A, 1998. The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math. 9 (1–2) (1998), pp. 69–95.
   Web of Science ®Google Scholar
• Chen, W, and Tanaka, M, 2000. New insights in boundary-only and domain-type RBF methods, Int. J. Nonlinear Sci. Numer. Simul. 1 (3) (2000), pp. 145–151.
   Google Scholar
• Chen, W, and Tanaka, M, 2002. A meshless, integration-free, and boundary-only RBF technique, Comput. Math. Appl. 43 (3–5) (2002), pp. 379–391.
   Web of Science ®Google Scholar
• Hon, YC, and Li, M, 2008. A computational method for inverse free boundary determination problem, Int. J. Numer. Methods Eng. 73 (9) (2008), pp. 1291–1309.
   Web of Science ®Google Scholar
• Mera, NS, and Lesnic, D, 2005. A three-dimensional boundary determination problem in potential corrosion damage, Comput. Mech. 36 (2) (2005), pp. 129–138.
   Web of Science ®Google Scholar
• Mera, NS, Elliott, L, and Ingham, DB, 2005. Boundary identification for a 3D laplace equation using a genetic algorithm, Comput. Methods Appl. Mech. Eng. 194 (42–44) (2005), pp. 4411–4424.
   Web of Science ®Google Scholar
• Alessandrini, G, 1993. Stable determination of a crack from boundary measurements, Proc. R. Soc. Edin. Sect. A 123 (3) (1993), pp. 497–516.
   Google Scholar
• Marin, L, 2006. Numerical boundary identification for Helmholtz-type equations, Comput. Mech. 39 (1) (2006), pp. 25–40.
   Web of Science ®Google Scholar
• Marin, L, and Lesnic, D, 2005. The method of fundamental solutions for the cauchy problem associated with two-dimensional Helmholtz-type equations, Comput. Struct. 83 (4–5) (2005), pp. 267–278.
   Web of Science ®Google Scholar
• Jin, B, and Zheng, Y, 2006. A meshless method for some inverse problems associated with the Helmholtz equation, Comput. Methods Appl. Mech. Eng. 195 (19–22) (2006), pp. 2270–2288.
   Web of Science ®Google Scholar
• Wei, T, Hon, YC, and Ling, L, 2007. Method of fundamental solutions with regularization techniques for cauchy problems of elliptic operators, Eng. Anal. Boundary Elements 31 (4) (2007), pp. 373–385.
   Web of Science ®Google Scholar
• Hon, YC, and Wei, T, 2004. A fundamental solution method for inverse heat conduction problem, Eng. Anal. Boundary Elements 28 (5) (2004), pp. 489–495.
   Web of Science ®Google Scholar
• Hon, YC, and Wei, T, 2005. The method of fundamental solution for solving multidimensional inverse heat conduction problems, CMES Comput. Model. Eng. Sci. 7 (2) (2005), pp. 119–132.
   Web of Science ®Google Scholar
• Kythe, PK, 1996. Fundamental Solutions for Differential Operators and Applications. Boston, MA: Birkhäuser; 1996.
   Google Scholar
• Hansen, PC, 1998. "Rank-deficient and discrete ill-posed problems". In: SIAM Monographs on Mathematical Modeling and Computation. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM); 1998.
   Google Scholar
• Engl, HW, Hanke, M, and Neubauer, A, 1996. "Regularization of Inverse Problems". In: Math. Appl. 375. Dordrecht: Kluwer Academic Publishers Group; 1996.
   Google Scholar
• Hansen, PC, 1994. Regularization tools: A Matlab package for analysis and solution of discrete ill-posed problems, Numer. Algorithms 6 (1–2) (1994), pp. 1–35.
   Google Scholar
• Tautenhahn, U, and Hämarik, U, 1999. The use of monotonicity for choosing the regularization parameter in ill-posed problems, Inv. Probl. 15 (6) (1999), pp. 1487–1505.
   Web of Science ®Google Scholar
• Wahba, G, 1977. Practical approximate solutions to linear operator equations when the data are noisy, SIAM J. Numer. Anal. 14 (4) (1977), pp. 651–667.
   Web of Science ®Google Scholar
• Vogel, CR, 2002. "Computational methods for inverse problems". In: Frontiers Appl. Math. 375. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM); 2002.
   Google Scholar