American Institute of Mathematical Sciences

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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

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 & 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.  Google Scholar

[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.  Google Scholar

[3]

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

[4]

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

[5]

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

[6]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

[21]

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

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.  Google Scholar

[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.  Google Scholar

[3]

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

[4]

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

[5]

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

[6]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

[21]

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

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