August  2008, 2(3): 397-409. doi: 10.3934/ipi.2008.2.397

On the regularization of the inverse conductivity problem with discontinuous conductivities

1. 

Dipartimento di Matematica e Informatica, Università degli Studi di Trieste, via Valerio, 12/1, 34127 Trieste, Italy

Received  May 2008 Revised  June 2008 Published  July 2008

We consider the regularization of the inverse conductivity problem with discontinuous conductivities, like for example the so-called inclusion problem. We theoretically validate the use of some of the most widely adopted regularization operators, like for instance total variation and the Mumford-Shah functional, by proving a convergence result for the solutions to the regularized minimum problems.
Citation: Luca Rondi. On the regularization of the inverse conductivity problem with discontinuous conductivities. Inverse Problems and Imaging, 2008, 2 (3) : 397-409. doi: 10.3934/ipi.2008.2.397
References:
[1]

R. Acar and C. R. Vogel, Analysis of bounded variation penalty methods for ill-posed problems, Inverse Problems, 10 (1994), 1217-1229. doi: 10.1088/0266-5611/10/6/003.

[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. doi: 10.1515/jiip.2007.025.

[4]

G. Alessandrini and M. Di Cristo, Stable determination of an inclusion by boundary measurements, SIAM J. Math. Anal., 37 (2005), 200-217. doi: 10.1137/S003614100444191X.

[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. doi: 10.4007/annals.2006.163.265.

[7]

K. Astala, L. Päivärinta and M. Lassas, Calderón's inverse problem for anisotropic conductivity in the plane, Comm. Partial Differential Equations, 30 (2005), 207-224. doi: 10.1081/PDE-200044485.

[8]

H. Attouch, "Variational Convergence for Functions and Operators,'' Pitman Publishing, Boston London Melbourne, 1984.

[9]

A. Braides, "$\Gamma$-convergence for Beginners,'' Oxford University Press, Oxford, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001.

[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. doi: 10.1016/j.jcp.2003.08.003.

[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. doi: 10.1051/cocv:1997113.

[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. doi: 10.1016/j.jcp.2004.11.022.

[13]

G. Dal Maso, "An Introduction to $\Gamma$-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. doi: 10.1088/0266-5611/19/3/313.

[15]

D. C. Dobson and F. Santosa, An image-enhancement technique for electrical impedance tomography, Inverse Problems, 10 (1994), 317-334. doi: 10.1088/0266-5611/10/2/008.

[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. doi: 10.1002/cpa.3160410702.

[19]

R. V. Kohn and M. Vogelius, Determining conductivity by boundary measurements II. Interior results, Comm. Pure Appl. Math., 38 (1985), 643-667. doi: 10.1002/cpa.3160380513.

[20]

A. Lechleiter and A. Rieder, Newton regularizations for impedance tomography: a numerical study, Inverse Problems, 22 (2006), 1967-1987. doi: 10.1088/0266-5611/22/6/004.

[21]

N. G. Meyers, An $L^p$-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 associated variational problems, Comm. Pure Appl. Math., 42 (1989), 577-685. doi: 10.1002/cpa.3160420503.

[23]

L. Rondi, A variational approach to the reconstruction of cracks by boundary measurements, J. Math. Pures Appl. (9), 87 (2007), 324-342. doi: 10.1016/j.matpur.2007.01.007.

[24]

L. Rondi and F. Santosa, Enhanced electrical impedance tomography via the Mumford-Shah functional, ESAIM Control Optim. Calc. Var., 6 (2001), 517-538. doi: 10.1051/cocv:2001121.

[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. doi: 10.1137/0152060.

[26]

J. Sylvester, An anisotropic inverse boundary value problem, Comm. Pure Appl. Math., 43 (1990), 201-232. doi: 10.1002/cpa.3160430203.

[27]

V. V. Vasin, Some tendencies in the Tikhonov regularization of ill-posed problems, J. Inverse Ill-Posed Probl., 14 (2006), 813-840. doi: 10.1515/156939406779768328.

[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. doi: 10.1515/jiip.2007.035.

show all references

References:
[1]

R. Acar and C. R. Vogel, Analysis of bounded variation penalty methods for ill-posed problems, Inverse Problems, 10 (1994), 1217-1229. doi: 10.1088/0266-5611/10/6/003.

[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. doi: 10.1515/jiip.2007.025.

[4]

G. Alessandrini and M. Di Cristo, Stable determination of an inclusion by boundary measurements, SIAM J. Math. Anal., 37 (2005), 200-217. doi: 10.1137/S003614100444191X.

[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. doi: 10.4007/annals.2006.163.265.

[7]

K. Astala, L. Päivärinta and M. Lassas, Calderón's inverse problem for anisotropic conductivity in the plane, Comm. Partial Differential Equations, 30 (2005), 207-224. doi: 10.1081/PDE-200044485.

[8]

H. Attouch, "Variational Convergence for Functions and Operators,'' Pitman Publishing, Boston London Melbourne, 1984.

[9]

A. Braides, "$\Gamma$-convergence for Beginners,'' Oxford University Press, Oxford, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001.

[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. doi: 10.1016/j.jcp.2003.08.003.

[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. doi: 10.1051/cocv:1997113.

[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. doi: 10.1016/j.jcp.2004.11.022.

[13]

G. Dal Maso, "An Introduction to $\Gamma$-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. doi: 10.1088/0266-5611/19/3/313.

[15]

D. C. Dobson and F. Santosa, An image-enhancement technique for electrical impedance tomography, Inverse Problems, 10 (1994), 317-334. doi: 10.1088/0266-5611/10/2/008.

[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. doi: 10.1002/cpa.3160410702.

[19]

R. V. Kohn and M. Vogelius, Determining conductivity by boundary measurements II. Interior results, Comm. Pure Appl. Math., 38 (1985), 643-667. doi: 10.1002/cpa.3160380513.

[20]

A. Lechleiter and A. Rieder, Newton regularizations for impedance tomography: a numerical study, Inverse Problems, 22 (2006), 1967-1987. doi: 10.1088/0266-5611/22/6/004.

[21]

N. G. Meyers, An $L^p$-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 associated variational problems, Comm. Pure Appl. Math., 42 (1989), 577-685. doi: 10.1002/cpa.3160420503.

[23]

L. Rondi, A variational approach to the reconstruction of cracks by boundary measurements, J. Math. Pures Appl. (9), 87 (2007), 324-342. doi: 10.1016/j.matpur.2007.01.007.

[24]

L. Rondi and F. Santosa, Enhanced electrical impedance tomography via the Mumford-Shah functional, ESAIM Control Optim. Calc. Var., 6 (2001), 517-538. doi: 10.1051/cocv:2001121.

[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. doi: 10.1137/0152060.

[26]

J. Sylvester, An anisotropic inverse boundary value problem, Comm. Pure Appl. Math., 43 (1990), 201-232. doi: 10.1002/cpa.3160430203.

[27]

V. V. Vasin, Some tendencies in the Tikhonov regularization of ill-posed problems, J. Inverse Ill-Posed Probl., 14 (2006), 813-840. doi: 10.1515/156939406779768328.

[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. doi: 10.1515/jiip.2007.035.

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