# American Institute of Mathematical Sciences

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

## Inverse diffusion problems with redundant internal information

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

Received  June 2011 Published  May 2012

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.
Citation: François Monard, Guillaume Bal. Inverse diffusion problems with redundant internal information. Inverse Problems & 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.  doi: 10.1137/070686408.  Google Scholar [2] G. Alessandrini and V. Nesi, Univalent $e^\sigma$-harmonic mappings,, Arch. Rat. Mech. Anal., 158 (2001), 155.  doi: 10.1007/PL00004242.  Google Scholar [3] G. Bal, Hybrid inverse problems and internal functionals (review paper),, in, (2012).   Google Scholar [4] _____, Cauchy problem and Ultrasound modulated EIT,, submitted., ().   Google Scholar [5] G. Bal, E. Bonnetier, F. Monard and F. Triki, Inverse diffusion from knowledge of power densities,, Inverse Probl. Imaging, (2012).   Google Scholar [6] G. Bal and K. Ren, Multi-source quantitative photoacoustic tomography,, Inverse Problems, 27 (2011).  doi: 10.1088/0266-5611/27/5/055007.  Google Scholar [7] G. Bal and G. Uhlmann, Inverse diffusion theory for photoacoustics,, Inverse Problems, 26 (2010).  doi: 10.1088/0266-5611/26/8/085010.  Google Scholar [8] A. Calderón, On an inverse boundary value problem,, Seminar on Numerical Analysis and its Applications to Continuum Physics, (1980), 65.   Google Scholar [9] C. Carathéodory, "Calculus of Variations and Partial Differential Equations of the First Order,'', Third edition, (1999).   Google Scholar [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.   Google Scholar [11] L. C. Evans, "Partial Differential Equations,'', Graduate Studies in Mathematics, (1998).   Google Scholar [12] B. Gebauer and O. Scherzer, Impedance-acoustic tomography,, SIAM J. Applied Math., 69 (2009), 565.  doi: 10.1137/080715123.  Google Scholar [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.  doi: 10.1137/S0036141001391354.  Google Scholar [14] P. Kuchment and L. Kunyansky, 2D and 3D reconstructions in acousto-electric tomography,, Inverse Problems, 27 (2011).   Google Scholar [15] J. M. Lee, "Riemannian Manifolds. An Introduction to Curvature,'', Graduate Texts in Mathematics, (1997).   Google Scholar [16] F. Monard, "Taming Unstable Inverse Problems. Mathematical Routes Toward High-Resolution Medical Imaging Modalities,'', Ph.D. thesis, (2012).   Google Scholar [17] A. Nachman, A. Tamasan and A. Timonov, Conductivity imaging with a single measurement of boundary and interior data,, Inverse Problems, 23 (2007), 2551.  doi: 10.1088/0266-5611/23/6/017.  Google Scholar [18] _____, Recovering the conductivity from a single measurement of interior data,, Inverse Problems, 25 (2009).   Google Scholar [19] _____, Current density impedance imaging,, Contemporary Mathematics, (2012).   Google Scholar [20] O. Scherzer, "Handbook of Mathematical Methods in Imaging,'', Springer Verlag, (2011).   Google Scholar [21] M. Spivak, "A Comprehensive Introduction to Differential Geometry, Vol. 2,'', Second edition, (1990).   Google Scholar [22] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem,, Ann. of Math., 125 (1987), 153.  doi: 10.2307/1971291.  Google Scholar [23] M. Taylor, "Partial Differential Equations I, Basic Theory,'', Springer, (1996).   Google Scholar

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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.  doi: 10.1137/070686408.  Google Scholar [2] G. Alessandrini and V. Nesi, Univalent $e^\sigma$-harmonic mappings,, Arch. Rat. Mech. Anal., 158 (2001), 155.  doi: 10.1007/PL00004242.  Google Scholar [3] G. Bal, Hybrid inverse problems and internal functionals (review paper),, in, (2012).   Google Scholar [4] _____, Cauchy problem and Ultrasound modulated EIT,, submitted., ().   Google Scholar [5] G. Bal, E. Bonnetier, F. Monard and F. Triki, Inverse diffusion from knowledge of power densities,, Inverse Probl. Imaging, (2012).   Google Scholar [6] G. Bal and K. Ren, Multi-source quantitative photoacoustic tomography,, Inverse Problems, 27 (2011).  doi: 10.1088/0266-5611/27/5/055007.  Google Scholar [7] G. Bal and G. Uhlmann, Inverse diffusion theory for photoacoustics,, Inverse Problems, 26 (2010).  doi: 10.1088/0266-5611/26/8/085010.  Google Scholar [8] A. Calderón, On an inverse boundary value problem,, Seminar on Numerical Analysis and its Applications to Continuum Physics, (1980), 65.   Google Scholar [9] C. Carathéodory, "Calculus of Variations and Partial Differential Equations of the First Order,'', Third edition, (1999).   Google Scholar [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.   Google Scholar [11] L. C. Evans, "Partial Differential Equations,'', Graduate Studies in Mathematics, (1998).   Google Scholar [12] B. Gebauer and O. Scherzer, Impedance-acoustic tomography,, SIAM J. Applied Math., 69 (2009), 565.  doi: 10.1137/080715123.  Google Scholar [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.  doi: 10.1137/S0036141001391354.  Google Scholar [14] P. Kuchment and L. Kunyansky, 2D and 3D reconstructions in acousto-electric tomography,, Inverse Problems, 27 (2011).   Google Scholar [15] J. M. Lee, "Riemannian Manifolds. An Introduction to Curvature,'', Graduate Texts in Mathematics, (1997).   Google Scholar [16] F. Monard, "Taming Unstable Inverse Problems. Mathematical Routes Toward High-Resolution Medical Imaging Modalities,'', Ph.D. thesis, (2012).   Google Scholar [17] A. Nachman, A. Tamasan and A. Timonov, Conductivity imaging with a single measurement of boundary and interior data,, Inverse Problems, 23 (2007), 2551.  doi: 10.1088/0266-5611/23/6/017.  Google Scholar [18] _____, Recovering the conductivity from a single measurement of interior data,, Inverse Problems, 25 (2009).   Google Scholar [19] _____, Current density impedance imaging,, Contemporary Mathematics, (2012).   Google Scholar [20] O. Scherzer, "Handbook of Mathematical Methods in Imaging,'', Springer Verlag, (2011).   Google Scholar [21] M. Spivak, "A Comprehensive Introduction to Differential Geometry, Vol. 2,'', Second edition, (1990).   Google Scholar [22] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem,, Ann. of Math., 125 (1987), 153.  doi: 10.2307/1971291.  Google Scholar [23] M. Taylor, "Partial Differential Equations I, Basic Theory,'', Springer, (1996).   Google Scholar
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