Structural stability in a minimization problem and applications to conductivity imaging

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

Structural stability in a minimization problem and applications to conductivity imaging

M. Zuhair Nashed ^1, and Alexandru Tamasan ^1,

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

Received July 2009 Revised July 2010 Published February 2011

Abstract
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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.

Keywords: Non-smooth optimization; bounded variation; degenerate elliptic equations; regularization; conductivity imaging..

Mathematics Subject Classification: Primary 49A45; Secondary 35R3.

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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