February  2011, 5(1): 219-236. doi: 10.3934/ipi.2011.5.219

Structural stability in a minimization problem and applications to conductivity imaging

1. 

Department of Mathematics, University of Central Florida, Orlando, FL 32816, United States

Received  July 2009 Revised  July 2010 Published  February 2011

We consider the problem of minimizing the functional $\int_\Omega a|\nabla u|dx$, with $u$ in some appropriate Banach space and prescribed trace $f$ on the boundary. For $a\in L^2(\Omega)$ and $u$ in the sample space $H^1(\Omega)$, this problem appeared recently in imaging the electrical conductivity of a body when some interior data are available. When $a\in C(\Omega)\cap L^\infty(\Omega)$, the functional has a natural interpretation, which suggests that one should consider the minimization problem in the sample space $BV(\Omega)$. We show the stability of the minimum value with respect to $a$, in a neighborhood of a particular coefficient. In both cases the method of proof provides some convergent minimizing procedures. We also consider the minimization problem for the non-degenerate functional $\int_\Omega a\max\{|\nabla u|,\delta\}dx$, for some $\delta>0$, and prove a stability result. Again, the method of proof constructs a minimizing sequence and we identify sufficient conditions for convergence. We apply the last result to the conductivity problem and show that, under an a posteriori smoothness condition, the method recovers the unknown conductivity.
Citation: M. Zuhair Nashed, Alexandru Tamasan. Structural stability in a minimization problem and applications to conductivity imaging. Inverse Problems and Imaging, 2011, 5 (1) : 219-236. doi: 10.3934/ipi.2011.5.219
References:
[1]

G. Alessandrini, An identification problem for an elliptic equation in two variables, Ann. Mat. Pura Appl., 145 (1986), 265-295. doi: 10.1007/BF01790543.

[2]

H. Ammari, E. Bonnetier, Y. Capdeboscq, M. Tanter and M. Fink, Electrical impedance tomography by elastic deformation, SIAM J. Appl. Math., 68 (2008), 1557-1573. doi: 10.1137/070686408.

[3]

L. C. Evans and M. Gariepy, "Measure Theory and Fine Properties of Functions," CRC Press, Boca Raton, FL, 1992.

[4]

B. Gebauer and O. Scherzer, Impedance-acoustic tomography, SIAM J. Appl. Math., 69 (2008), 565-576. doi: 10.1137/080715123.

[5]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 edition, Springer-Verlag, Berlin, 2001.

[6]

E. Giusti, "Minimal Surfaces and Functions of Bounded Variation," Monographs in Mathematics 80, Birkhäuser Verlag, Basel, 1984.

[7]

M. L. Joy, A. Nachman, K. F. Hasanov, R. S. Yoon and A. W. Ma, A new approach to current density impedance imaging (CDII), Proceedings ISMRM, #2356 (Kyoto, Japan), 2004.

[8]

S. Kim, O. Kwon, J. K. Seo and J. R. Yoon, On a nonlinear partial differential equation arising in magnetic resonance electrical impedance tomography, SIAM J. Math. Anal., 34 (2002), 511-526. doi: 10.1137/S0036141001391354.

[9]

O. Kwon, J. Y. Lee and and J. R. Yoon, Equipotential line method for magnetic resonance electrical impedance tomography, Inverse Problems, 18 (2002), 1089-1100. doi: 10.1088/0266-5611/18/4/310.

[10]

O. Kwon, E. J. Woo, J. R. Yoon and J. K. Seo, Magnetic resonance electric impedance tomography (MREIT): Simulation study of J-substitution algorithm, IEEE Trans. Biomed. Eng., 49 (2002), 160-167. doi: 10.1109/10.979355.

[11]

J. Y. Lee, A reconstruction formula and uniqueness of conductivity in MREIT using two internal current distributions, Inverse Problems, 20 (2004), 847-858. doi: 10.1088/0266-5611/20/3/012.

[12]

X. Li, Y. Xu and B. He, Imaging electrical impedance from acoustic measurements by means of magnetoacoustic tomography with magnetic Induction (MAT-MI), IEEE Transactions on Biomedical Engineering, 54 (2007), 323-330. doi: 10.1109/TBME.2006.883827.

[13]

A. Nachman, A. Tamasan and A. Timonov, Conductivity imaging with a single measurement of boundary and interior data, Inverse Problems, 23 (2007), 2551-2563. doi: 10.1088/0266-5611/23/6/017.

[14]

A. Nachman, A. Tamasan and A. Timonov, Recovering the conductivity from a single measurement of interior data, Inverse Problems, 25 (2009), 035014, 16pp.

[15]

A. Nachman, A. Tamasan and A. Timonov, Reconstruction of planar conductivities in subdomains from incomplete data, SIAM J. Appl. Math., 70 (2010), 3342-3362. doi: 10.1137/10079241X.

[16]

M. Z. Nashed and O. Scherzer, Stable approximation of nondifferentiable optimization problems with variational inequalities, Contemp. Math., 204 (1997), 155-170.

[17]

M. Z. Nashed and O. Scherzer, Stable approximation of a minimal surface problem with variational inequalitites, Abstr. and Appl. Anal., 2 (1997), 137-161. doi: 10.1155/S1085337597000316.

[18]

G. C. Scott, M. L. Joy, R. L. Armstrong and R. M. Henkelman, Measurement of nonuniform current density by magnetic resonance, IEEE Trans. Med. Imag., 10 (1991), 362-374. doi: 10.1109/42.97586.

[19]

N. Zhang, "Electrical Impedance Tomography Based on Current Density Imaging," M. Sc. Thesis, University of Toronto, Canada, 1992.

[20]

E. Zeidler, "Nonlinear Functional Analysis and its Applications. III. Variational Methods and Optimization," Springer-Verlag, New York, 1985.

[21]

W. P. Ziemer, "Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation," Springer-Verlag, New York, 1989.

show all references

References:
[1]

G. Alessandrini, An identification problem for an elliptic equation in two variables, Ann. Mat. Pura Appl., 145 (1986), 265-295. doi: 10.1007/BF01790543.

[2]

H. Ammari, E. Bonnetier, Y. Capdeboscq, M. Tanter and M. Fink, Electrical impedance tomography by elastic deformation, SIAM J. Appl. Math., 68 (2008), 1557-1573. doi: 10.1137/070686408.

[3]

L. C. Evans and M. Gariepy, "Measure Theory and Fine Properties of Functions," CRC Press, Boca Raton, FL, 1992.

[4]

B. Gebauer and O. Scherzer, Impedance-acoustic tomography, SIAM J. Appl. Math., 69 (2008), 565-576. doi: 10.1137/080715123.

[5]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 edition, Springer-Verlag, Berlin, 2001.

[6]

E. Giusti, "Minimal Surfaces and Functions of Bounded Variation," Monographs in Mathematics 80, Birkhäuser Verlag, Basel, 1984.

[7]

M. L. Joy, A. Nachman, K. F. Hasanov, R. S. Yoon and A. W. Ma, A new approach to current density impedance imaging (CDII), Proceedings ISMRM, #2356 (Kyoto, Japan), 2004.

[8]

S. Kim, O. Kwon, J. K. Seo and J. R. Yoon, On a nonlinear partial differential equation arising in magnetic resonance electrical impedance tomography, SIAM J. Math. Anal., 34 (2002), 511-526. doi: 10.1137/S0036141001391354.

[9]

O. Kwon, J. Y. Lee and and J. R. Yoon, Equipotential line method for magnetic resonance electrical impedance tomography, Inverse Problems, 18 (2002), 1089-1100. doi: 10.1088/0266-5611/18/4/310.

[10]

O. Kwon, E. J. Woo, J. R. Yoon and J. K. Seo, Magnetic resonance electric impedance tomography (MREIT): Simulation study of J-substitution algorithm, IEEE Trans. Biomed. Eng., 49 (2002), 160-167. doi: 10.1109/10.979355.

[11]

J. Y. Lee, A reconstruction formula and uniqueness of conductivity in MREIT using two internal current distributions, Inverse Problems, 20 (2004), 847-858. doi: 10.1088/0266-5611/20/3/012.

[12]

X. Li, Y. Xu and B. He, Imaging electrical impedance from acoustic measurements by means of magnetoacoustic tomography with magnetic Induction (MAT-MI), IEEE Transactions on Biomedical Engineering, 54 (2007), 323-330. doi: 10.1109/TBME.2006.883827.

[13]

A. Nachman, A. Tamasan and A. Timonov, Conductivity imaging with a single measurement of boundary and interior data, Inverse Problems, 23 (2007), 2551-2563. doi: 10.1088/0266-5611/23/6/017.

[14]

A. Nachman, A. Tamasan and A. Timonov, Recovering the conductivity from a single measurement of interior data, Inverse Problems, 25 (2009), 035014, 16pp.

[15]

A. Nachman, A. Tamasan and A. Timonov, Reconstruction of planar conductivities in subdomains from incomplete data, SIAM J. Appl. Math., 70 (2010), 3342-3362. doi: 10.1137/10079241X.

[16]

M. Z. Nashed and O. Scherzer, Stable approximation of nondifferentiable optimization problems with variational inequalities, Contemp. Math., 204 (1997), 155-170.

[17]

M. Z. Nashed and O. Scherzer, Stable approximation of a minimal surface problem with variational inequalitites, Abstr. and Appl. Anal., 2 (1997), 137-161. doi: 10.1155/S1085337597000316.

[18]

G. C. Scott, M. L. Joy, R. L. Armstrong and R. M. Henkelman, Measurement of nonuniform current density by magnetic resonance, IEEE Trans. Med. Imag., 10 (1991), 362-374. doi: 10.1109/42.97586.

[19]

N. Zhang, "Electrical Impedance Tomography Based on Current Density Imaging," M. Sc. Thesis, University of Toronto, Canada, 1992.

[20]

E. Zeidler, "Nonlinear Functional Analysis and its Applications. III. Variational Methods and Optimization," Springer-Verlag, New York, 1985.

[21]

W. P. Ziemer, "Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation," Springer-Verlag, New York, 1989.

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