On the regularization of the inverse conductivity problem with discontinuous conductivities

Inverse Problems and Imaging

Volumn 2, Issue 3, 2008, Pages 397-409

  Rondi, Luca ^a  

^a UNIVERSITY OF TRIESTE   (Italy)

Author keywords

BV functions; Electrical impedance tomography; Inclusion; Regularization; convergence

Indexed keywords

EID: 74349086162     PISSN: 19308337     EISSN: 19308345     Source Type: Journal    
DOI: 10.3934/ipi.2008.2.397     Document Type: Article

References (28)

1. R. Acar and C. R. Vogel, Analysis of bounded variation penalty methods for ill-posed problems, Inverse Problems, 10 (1994), 1217–1229.
2. R. A. Adams, “Sobolev Spaces,” Academic Press, New York, 1975.
3. G. Alessandrini, Open issues of stability for the inverse conductivity problem, J. Inverse Ill- Posed Probl., 15 (2007), 451–460.
4. G. Alessandrini and M. Di Cristo, Stable determination of an inclusion by boundary mea- surements, SIAM J. Math. Anal., 37 (2005), 200–217.
5. L. Ambrosio, N. Fusco and D. Pallara, “Functions of Bounded Variation and Free Discontinuity Problems,” Clarendon Press, Oxford, 2000.
6. K. Astala and L. Päivärinta, Calderón’s inverse conductivity problem in the plane, Ann. of Math. (2), 163 (2006), 265–299.
7. K. Astala, L. Päivärinta and M. Lassas, Calderón’s inverse problem for anisotropic conduc- tivity in the plane, Comm. Partial Differential Equations, 30 (2005), 207–224.
8. H. Attouch, “Variational Convergence for Functions and Operators,” Pitman Publishing, Boston London Melbourne, 1984.
9. A. Braides, “Γ-convergence for Beginners,” Oxford University Press, Oxford, 2002.
10. T. F. Chan and X.-C. Tai, Level set and total variation regularization for elliptic inverse problems with discontinuous coefficients, J. Comput. Phys., 193 (2004), 40–66.
11. G. Chavent and K. Kunisch, Regularization of linear least squares problems by total bounded variation, ESAIM Control Optim. Calc. Var., 2 (1997), 359–376.
12. E. T. Chung, T. F. Chan and X.-C. Tai, Electrical impedance tomography using level set representation and total variational regularization, J. Comput. Phys., 205 (2005), 357–372.
13. G. Dal Maso, “An Introduction to Γ-convergence,” Birkhäuser, Boston Basel Berlin, 1993.
14. M. Di Cristo and L. Rondi, Examples of exponential instability for inverse inclusion and scattering problems, Inverse Problems, 19 (2003), 685–701.
15. D. C. Dobson and F. Santosa, An image-enhancement technique for electrical impedance tomography, Inverse Problems, 10 (1994), 317–334.
16. H. W. Engl, M. Hanke and A. Neubauer, “Regularization of Inverse Problems,” Kluwer Academic Publishers, Dordrecht Boston London, 1996.
17. T. Gallouet and A. Monier, On the regularity of solutions to elliptic equations, Rend. Mat. Appl. (7), 19 (1999), 471–488.
18. V. Isakov, On uniqueness of recovery of a discontinuous conductivity coefficient, Comm. Pure Appl. Math., 41 (1988), 865–877.
19. R. V. Kohn and M. Vogelius, Determining conductivity by boundary measurements II. Interior results, Comm. Pure Appl. Math., 38 (1985), 643–667.
20. A. Lechleiter and A. Rieder, Newton regularizations for impedance tomography: a numerical study, Inverse Problems, 22 (2006), 1967–1987.
21. N. G. Meyers, An Lp-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa (3), 17 (1963), 189–206.
22. D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and asso- ciated variational problems, Comm. Pure Appl. Math., 42 (1989), 577–685.
23. L. Rondi, A variational approach to the reconstruction of cracks by boundary measurements, J. Math. Pures Appl. (9), 87 (2007), 324–342.
24. L. Rondi and F. Santosa, Enhanced electrical impedance tomography via the Mumford-Shah functional, ESAIM Control Optim. Calc. Var., 6 (2001), 517–538.
25. E. Somersalo, M. Cheney and D. Isaacson, Existence and uniqueness for electrode models for electric current computed tomography, SIAM J. Appl. Math., 52 (1992), 1023–1040.
26. J. Sylvester, An anisotropic inverse boundary value problem, Comm. Pure Appl. Math., 43 (1990), 201–232.
27. V. V. Vasin, Some tendencies in the Tikhonov regularization of ill-posed problems, J. Inverse Ill-Posed Probl., 14 (2006), 813–840.
28. V. V. Vasin, Some approaches to reconstruction of nonsmooth solutions of linear ill-posed problems, J. Inverse Ill-Posed Probl., 15 (2007), 625–640.