An iterative and adaptive lie-group method for solving the calderón inverse problem

CMES - Computer Modeling in Engineering and Sciences

Volumn 64, Issue 3, 2010, Pages 299-326

  Liu, Chein Shan ^a   Atluri, Satya N ^b  

^a NATIONAL TAIWAN UNIVERSITY   (Taiwan)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Calder n's inverse problem; Inverse Cauchy problem; Iterative method; Lie group adaptive method; Parameter identification problem

Indexed keywords

ADAPTIVE METHODS; CAUCHY DATA; CAUCHY PROBLEMS; CONDUCTIVITY COEFFICIENT; EXACT SOLUTION; FINITE STRIP METHOD; FINITE STRIPS; ILL-POSEDNESS; IMPEDANCE FUNCTIONS; INVERSE CONDUCTIVITY PROBLEM; ITERATION PROCESS; MATHEMATICAL ALGORITHMS; PARAMETER IDENTIFICATION; PARAMETER IDENTIFICATION PROBLEMS; SELF-ADAPTIVE; SUB-DOMAINS;

INVERSE PROBLEMS; ITERATIVE METHODS; ORDINARY DIFFERENTIAL EQUATIONS; PARAMETER ESTIMATION; SOIL STRUCTURE INTERACTIONS; STRIP METAL;

PROBLEM SOLVING;

EID: 77957563701     PISSN: 15261492     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (33)

1. Adler, A.; Guardo, R. (1994): A neural network image reconstruction technique for electrical impedance tomography. IEEE Trans. Med. Imag., vol.13, pp. 594-600.
2. Borcea, L. (2002): Electrical impedance tomography. Inverse Problems, vol.18, pp. R99-R136.
3. Borcea, L.; Gray, G. A.; Zhang, Y. (2003): Variationally constrained numerical solution of electrical impedance tomography. Inverse Problems, vol.19, pp. 1159-1184.
4. Brown, R. M.; Uhlmann, G. A. (1997): Uniqueness in the inverse conductivity problem for non-smooth conductivities in two dimensions. Comm. Part. Diff. Eq., vol.22, pp. 1009-1027.
5. Calderón, A. P. (1980): On an inverse boundary value problem, in Seminar on Numerical Analysis and its Application to Continuum Physics, Rio de Janeiro, Brazil: Soc. Brasileira de Mathematica, 1980, pp. 65-73.
6. Calderón, A. P. (2006): On an inverse boundary value problem. Comp. Appl. Math., vol.25, pp. 133-138.
7. Francini, E. (2000): Recoverning a complex coefficient in a planar domain from the Dirichlet-to-Neumann map. Inverse Problems, vol.16, pp. 107-119.
8. Kaup, P. G.; Santosa, F.; Vogelius, M. (1996): Method for imaging corrosion damage in thin plates from electrostatic data. Inverse Problems, vol.12, pp. 279-293.
9. Knowles, I. (1998): A variational algorithm for electrical impedance tomography. Inverse Problems, vol.14, pp. 1513-1526.
10. Kohn, R. V.; McKenney, A. (1990): Numerical implementation of a variational method for electrical impedance tomography. Inverse Problems, vol.6, pp. 389-414.
11. Levy, S.; Adam, D.; Bresler, Y. (2002): Electromagnetic impedance tomography (EMIT): a new method for impedance imaging. IEEE Trans. Med. Imag., vol.21 pp. 676-687.
12. Liu, C.-S. (2001): Cone of non-linear dynamical system and group preserving schemes. Int. J. Non-Linear Mech., vol.36, pp. 1047-1068.
13. Liu, C.-S. (2006): The Lie-group shooting method for nonlinear two-point boundary value problems exhibiting multiple solutions. CMES: Computer Modeling in Engineering & Sciences, vol.13, pp. 149-163.
14. Liu, C.-S. (2008a): An LGSM to identify nonhomogeneous heat conductivity functions by an extra measurement of temperature. Int. J. Heat Mass Transfer, vol.51, pp. 2603-2613.
15. Liu, C.-S. (2008b): An LGEM to identify time-dependent heat conductivity function by an extra measurement of temperature gradient. CMC: Computers, Materials & Continua, vol.7, pp. 81-95.
16. Liu, C.-S. (2009a): Solving the inverse problems of Laplace equation to determine the Robin coefficient/cracks' position inside a disk. CMES: Computer Modeling in Engineering & Sciences, vol.40, pp. 1-28.
17. Liu, C.-S. (2009b): A two-stage LGSM to identify time-dependent heat source through an internal measurement of temperature. Int. J. Heat Mass Transfer, vol.52, pp. 1635-1642.
18. Liu, C.-S. (2010a): A two-stage Lie-group shooting method (TSLGSM) to identify time-dependent thermal diffusivity. Int. J. Heat Mass Transfer, vol.53, pp. 4876- 4884.
19. Liu, C.-S. (2010b): A highly accurate LGSM for severely ill-posed BHCP under a large noise on the final time data. Int. J. Heat Mass Transfer, vol.53, pp. 4132- 4140.
20. Meeson, S.; Killingback, A. L. T.; Blott, B. H. (1995): The dependence of EIT images on the assumed initial conductivity distributionl: a study of pelvic imaging. Phys. Med. Biol., vol.40, pp. 643-657.
21. Murai, T.; Kagawa, Y. (1985): Electrical impedance computed tomography based on a finite element model. IEEE Trans. Biomed. Eng., vol.32, pp. 177-184.
22. Nachman, A. I. (1988): Reconstructions from boundary measurements. Ann. Math., vol.128, pp. 531-576.
23. Nachman, A. I. (1996): Global uniqueness for a two-dimensional inverse boundary problem. Ann. Math., vol.143, pp. 71-96.
24. Santosa, F.; Vogelius, M.; Xu, J. M. (1999): A non linear elliptic boundary value problem related to corrosion modeling. Quart. Appl. Math., vol.56, pp. 479-505.
25. Siltanen, S.; Mueller, J.; Isaacson, D. (2000): An implementation of the reconstruction algorithm of A Nachman for the 2D inverse conductivity problem. Inverse Problems, vol.16, pp. 681-699.
26. Somersalo, E.; Cheney, M.; Isaacson, D., Isaacson, E. L. (1991): Layer stripping: a direct numerical method for impedance imaging. Inverse Problems, vol.7, pp. 899-926.
27. Stasiak, M.; Sikora, J.; Filipowicz, S. F.; Nita, K. (2007): Principal component analysis and artificial neural network approach to electrical impedance tomography problems approximated by multi-region boundary element method. Eng. Anal. Bound. Elem., vol.31, pp. 713-720.
28. Sylvester, J. (1992): A convergent layer stripping algorithm for radially symmetric impedance tomography problem. Comm. Partial Diff. Eq., vol.17, pp. 1955-1994.
29. Sylvester, J.; Uhlmann, G. (1987): A global uniqueness theorem for an inverse boundary value problem. Ann. Math., vol.125, pp. 153-169.
30. Wexler, A.; Fry, B.; Neuman, M. R. (1985): Impedance-computed tomography algorithm and system. Appl. Optics, vol.24, pp. 3985-3992.
31. Yorkey, T.J.;Webster, J. G.; Tompkins,W. J. (1987): Comparing reconstruction algorithms for electrical impedance tomography. IEEE Trans. Biomed. Eng., vol. 34, pp. 843-852.
32. Zadehkoochak, M.; Hames, T. K.; Blott, B. H.; George, R. F. (1990): A transputer implemented algorithm for electrical impedance tomography. Clin. Phys. Physiol. Measu., vol. 11, pp. 223-230.
33. Zlochiver, S.; Rosenfeld, M.; Abboud, S. (2003): Induced-current electrical impedance tomography: a 2-D theoretical simulation. IEEE Trans. Med. Imag., vol. 22, pp. 1550-1560.