Inverse diffusion problems with redundant internal information

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May 2012, 6(2): 289-313. doi: 10.3934/ipi.2012.6.289

Inverse diffusion problems with redundant internal information

François Monard ^1, and Guillaume Bal ^1,

1.	Department of Applied Physics and Applied Mathematics, Columbia University, New York NY, 10027

Received June 2011 Published May 2012

Abstract
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This paper concerns the reconstruction of a scalar diffusion coefficient $\sigma(x)$ from redundant functionals of the form $H_i(x)=\sigma^{2\alpha}(x)|\nabla u_i|^2(x)$ where $\alpha\in\mathbb{R}$ and $u_i$ is a solution of the elliptic problem $\nabla\cdot \sigma \nabla u_i=0$ for $1\leq i\leq I$. The case $\alpha=\frac12$ is used to model measurements obtained from modulating a domain of interest by ultrasound and finds applications in ultrasound modulated electrical impedance tomography (UMEIT), ultrasound modulated optical tomography (UMOT) as well as impedance acoustic computerized tomography (ImpACT). The case $\alpha=1$ finds applications in Magnetic Resonance Electrical Impedance Tomography (MREIT).
We present two explicit reconstruction procedures of $\sigma$ for appropriate choices of $I$ and of traces of $u_i$ at the boundary of a domain of interest. The first procedure involves the solution of an over-determined system of ordinary differential equations and generalizes to the multi-dimensional case and to (almost) arbitrary values of $\alpha$ the results obtained in two and three dimensions in [10] and [5], respectively, in the case $\alpha=\frac12$. The second procedure consists of solving a system of linear elliptic equations, which we can prove admits a unique solution in specific situations.

Keywords: hybrid methods, power density measurements, differential geometry., Inverse conductivity, strongly coupled elliptic systems, Calderón's problem.

Mathematics Subject Classification: Primary: 35R30; Secondary: 35J45, 53B2.

Citation: François Monard, Guillaume Bal. Inverse diffusion problems with redundant internal information. Inverse Problems and Imaging, 2012, 6 (2) : 289-313. doi: 10.3934/ipi.2012.6.289

References:

[1]	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.
[2]	G. Alessandrini and V. Nesi, Univalent $e^\sigma$-harmonic mappings, Arch. Rat. Mech. Anal., 158 (2001), 155-171. doi: 10.1007/PL00004242.
[3]	G. Bal, Hybrid inverse problems and internal functionals (review paper), in "Inside Out'' (ed. Gunther Uhlmann), Cambridge University Press, 2012.
[4]	_____, Cauchy problem and Ultrasound modulated EIT, submitted.
[5]	G. Bal, E. Bonnetier, F. Monard and F. Triki, Inverse diffusion from knowledge of power densities, Inverse Probl. Imaging, in press, 2012.
[6]	G. Bal and K. Ren, Multi-source quantitative photoacoustic tomography, Inverse Problems, 27 (2011). doi: 10.1088/0266-5611/27/5/055007.
[7]	G. Bal and G. Uhlmann, Inverse diffusion theory for photoacoustics, Inverse Problems, 26 (2010), 085010. doi: 10.1088/0266-5611/26/8/085010.
[8]	A. Calderón, On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics, Soc. Brasileira de Matematica, Rio de Janeiro, (1980), 65-73.
[9]	C. Carathéodory, "Calculus of Variations and Partial Differential Equations of the First Order,'' Third edition, AMS, Chelsea, 1999.
[10]	Y. Capdeboscq, J. Fehrenbach, F. de Gournay and O. Kavian, Imaging by modification: Numerical reconstruction of local conductivities from corresponding power density measurements, SIAM Journal on Imaging Sciences, 2 (2009), 1003-1030.
[11]	L. C. Evans, "Partial Differential Equations,'' Graduate Studies in Mathematics, Vol. 19, AMS, 1998.
[12]	B. Gebauer and O. Scherzer, Impedance-acoustic tomography, SIAM J. Applied Math., 69 (2009), 565-576. doi: 10.1137/080715123.
[13]	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.
[14]	P. Kuchment and L. Kunyansky, 2D and 3D reconstructions in acousto-electric tomography, Inverse Problems, 27 (2011), 055013.
[15]	J. M. Lee, "Riemannian Manifolds. An Introduction to Curvature,'' Graduate Texts in Mathematics, Vol. 176, Springer, 1997.
[16]	F. Monard, "Taming Unstable Inverse Problems. Mathematical Routes Toward High-Resolution Medical Imaging Modalities,'' Ph.D. thesis, Columbia University, New York, 2012.
[17]	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.
[18]	_____, Recovering the conductivity from a single measurement of interior data, Inverse Problems, 25 (2009), 035014.
[19]	_____, Current density impedance imaging, Contemporary Mathematics, American Mathematical Society, in press, 2012.
[20]	O. Scherzer, "Handbook of Mathematical Methods in Imaging,'' Springer Verlag, New York, 2011.
[21]	M. Spivak, "A Comprehensive Introduction to Differential Geometry, Vol. 2,'' Second edition, Publish or perish, 1990.
[22]	J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169. doi: 10.2307/1971291.
[23]	M. Taylor, "Partial Differential Equations I, Basic Theory,'' Springer, New York, 1996.

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References:

[1]	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.
[2]	G. Alessandrini and V. Nesi, Univalent $e^\sigma$-harmonic mappings, Arch. Rat. Mech. Anal., 158 (2001), 155-171. doi: 10.1007/PL00004242.
[3]	G. Bal, Hybrid inverse problems and internal functionals (review paper), in "Inside Out'' (ed. Gunther Uhlmann), Cambridge University Press, 2012.
[4]	_____, Cauchy problem and Ultrasound modulated EIT, submitted.
[5]	G. Bal, E. Bonnetier, F. Monard and F. Triki, Inverse diffusion from knowledge of power densities, Inverse Probl. Imaging, in press, 2012.
[6]	G. Bal and K. Ren, Multi-source quantitative photoacoustic tomography, Inverse Problems, 27 (2011). doi: 10.1088/0266-5611/27/5/055007.
[7]	G. Bal and G. Uhlmann, Inverse diffusion theory for photoacoustics, Inverse Problems, 26 (2010), 085010. doi: 10.1088/0266-5611/26/8/085010.
[8]	A. Calderón, On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics, Soc. Brasileira de Matematica, Rio de Janeiro, (1980), 65-73.
[9]	C. Carathéodory, "Calculus of Variations and Partial Differential Equations of the First Order,'' Third edition, AMS, Chelsea, 1999.
[10]	Y. Capdeboscq, J. Fehrenbach, F. de Gournay and O. Kavian, Imaging by modification: Numerical reconstruction of local conductivities from corresponding power density measurements, SIAM Journal on Imaging Sciences, 2 (2009), 1003-1030.
[11]	L. C. Evans, "Partial Differential Equations,'' Graduate Studies in Mathematics, Vol. 19, AMS, 1998.
[12]	B. Gebauer and O. Scherzer, Impedance-acoustic tomography, SIAM J. Applied Math., 69 (2009), 565-576. doi: 10.1137/080715123.
[13]	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.
[14]	P. Kuchment and L. Kunyansky, 2D and 3D reconstructions in acousto-electric tomography, Inverse Problems, 27 (2011), 055013.
[15]	J. M. Lee, "Riemannian Manifolds. An Introduction to Curvature,'' Graduate Texts in Mathematics, Vol. 176, Springer, 1997.
[16]	F. Monard, "Taming Unstable Inverse Problems. Mathematical Routes Toward High-Resolution Medical Imaging Modalities,'' Ph.D. thesis, Columbia University, New York, 2012.
[17]	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.
[18]	_____, Recovering the conductivity from a single measurement of interior data, Inverse Problems, 25 (2009), 035014.
[19]	_____, Current density impedance imaging, Contemporary Mathematics, American Mathematical Society, in press, 2012.
[20]	O. Scherzer, "Handbook of Mathematical Methods in Imaging,'' Springer Verlag, New York, 2011.
[21]	M. Spivak, "A Comprehensive Introduction to Differential Geometry, Vol. 2,'' Second edition, Publish or perish, 1990.
[22]	J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169. doi: 10.2307/1971291.
[23]	M. Taylor, "Partial Differential Equations I, Basic Theory,'' Springer, New York, 1996.

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