# American Institute of Mathematical Sciences

February  2014, 8(1): 1-22. doi: 10.3934/ipi.2014.8.1

## Linearized internal functionals for anisotropic conductivities

 1 Department of Applied Physics and Applied Mathematics, Columbia University, 200 S. W. Mudd Building, MC 4701, 500 W. 120th Street, New York, NY 10027 2 Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, United States 3 Department of Mathematics, University of Washington, Seattle WA, 98195, United States

Received  February 2013 Revised  December 2013 Published  March 2014

This paper concerns the reconstruction of an anisotropic conductivity tensor in an elliptic second-order equation from knowledge of the so-called power density functionals. This problem finds applications in several coupled-physics medical imaging modalities such as ultrasound modulated electrical impedance tomography and impedance-acoustic computerized tomography.
We consider the linearization of the nonlinear hybrid inverse problem. We find sufficient conditions for the linearized problem, a system of partial differential equations, to be elliptic and for the system to be injective. Such conditions are found to hold for a lesser number of measurements than those required in recently established explicit reconstruction procedures for the nonlinear problem.
Citation: Guillaume Bal, Chenxi Guo, Francçois Monard. Linearized internal functionals for anisotropic conductivities. Inverse Problems & Imaging, 2014, 8 (1) : 1-22. doi: 10.3934/ipi.2014.8.1
##### 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. [2] S. R. Arridge and O. Scherzer, Imaging from coupled physics,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/8/080201. [3] G. Bal, Hybrid inverse problems and systems of partial differential equations,, , (). [4] ______, Cauchy problem for Ultrasound modulated EIT,, Analysis and PDE, 6 (2013), 751. doi: 10.2140/apde.2013.6.751. [5] ______, Hybrid Inverse Problems and Internal Functionals,, Inside Out, (2012). [6] G. Bal, E. Bonnetier, F. Monard and F. Triki, Inverse diffusion from knowledge of power densities,, Inverse Problems and Imaging, 7 (2013), 353. doi: 10.3934/ipi.2013.7.353. [7] G. Bal, W. Naetar, O. Scherzer and J. Schotland, The levenberg-marquardt iteration for numerical inversion of the power density operator,, J. Ill-posed Inverse Problems, 21 (2013), 265. doi: 10.1515/jip-2012-0091. [8] G. Bal and G. Uhlmann, Reconstruction of coefficients in scalar second-order elliptic equations from knowledge of their solutions,, C.P.A.M., 66 (2013), 1629. doi: 10.1002/cpa.21453. [9] 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. doi: 10.1137/080723521. [10] L. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998). [11] G. B. Folland, Introduction to Partial Differential Equations,, Princeton University Press, (1995). [12] B. Gebauer and O. Scherzer, Impedance-acoustic tomography,, SIAM J. Applied Math., 69 (2008), 565. doi: 10.1137/080715123. [13] A. Grigis and j. Sjöstrand, Microlocal Analysis for Differential Operators: An Introduction,, Cambridge University Press, (1994). [14] P. Kuchment, Mathematics of hybrid imaging. a brief review,, in The Mathematical Legacy of Leon Ehrenpreis. Springer Proceeding in Mathematics, 16 (2012), 183. doi: 10.1007/978-88-470-1947-8_12. [15] P. Kuchment and L. Kunyansky, 2d and 3d reconstructions in acousto-electric tomography,, Inverse Problems, 27 (2011). doi: 10.1088/0266-5611/27/5/055013. [16] P. Kuchment and D. Steinhauer, Stabilizing inverse problems by internal data,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/8/084007. [17] F. Monard, Taming Unstable Inverse Problems. Mathematical Routes Toward High-Resolution Medical Imaging Modalities,, PhD thesis, (2012). [18] F. Monard and G. Bal, Inverse anisotropic conductivity from power densities in dimension $n \ge 3$,, Comm. PDE, 38 (2013), 1183. [19] ______, Inverse anisotropic diffusion from power density measurements in two dimensions,, Inverse Problems, 28 (2012). [20] ______, Inverse diffusion problems with redundant internal information,, Inv. Probl. Imaging, 6 (2012), 289. [21] O. Scherzer, Handbook of Mathematical Methods in Imaging,, Springer Verlag, (2011). doi: 10.1007/978-0-387-92920-0. [22] P. Stefanov and G. Uhlmann, Multi-Wave Methods by Ultrasounds,, Inside out, (2012).

show all references

##### 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. [2] S. R. Arridge and O. Scherzer, Imaging from coupled physics,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/8/080201. [3] G. Bal, Hybrid inverse problems and systems of partial differential equations,, , (). [4] ______, Cauchy problem for Ultrasound modulated EIT,, Analysis and PDE, 6 (2013), 751. doi: 10.2140/apde.2013.6.751. [5] ______, Hybrid Inverse Problems and Internal Functionals,, Inside Out, (2012). [6] G. Bal, E. Bonnetier, F. Monard and F. Triki, Inverse diffusion from knowledge of power densities,, Inverse Problems and Imaging, 7 (2013), 353. doi: 10.3934/ipi.2013.7.353. [7] G. Bal, W. Naetar, O. Scherzer and J. Schotland, The levenberg-marquardt iteration for numerical inversion of the power density operator,, J. Ill-posed Inverse Problems, 21 (2013), 265. doi: 10.1515/jip-2012-0091. [8] G. Bal and G. Uhlmann, Reconstruction of coefficients in scalar second-order elliptic equations from knowledge of their solutions,, C.P.A.M., 66 (2013), 1629. doi: 10.1002/cpa.21453. [9] 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. doi: 10.1137/080723521. [10] L. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998). [11] G. B. Folland, Introduction to Partial Differential Equations,, Princeton University Press, (1995). [12] B. Gebauer and O. Scherzer, Impedance-acoustic tomography,, SIAM J. Applied Math., 69 (2008), 565. doi: 10.1137/080715123. [13] A. Grigis and j. Sjöstrand, Microlocal Analysis for Differential Operators: An Introduction,, Cambridge University Press, (1994). [14] P. Kuchment, Mathematics of hybrid imaging. a brief review,, in The Mathematical Legacy of Leon Ehrenpreis. Springer Proceeding in Mathematics, 16 (2012), 183. doi: 10.1007/978-88-470-1947-8_12. [15] P. Kuchment and L. Kunyansky, 2d and 3d reconstructions in acousto-electric tomography,, Inverse Problems, 27 (2011). doi: 10.1088/0266-5611/27/5/055013. [16] P. Kuchment and D. Steinhauer, Stabilizing inverse problems by internal data,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/8/084007. [17] F. Monard, Taming Unstable Inverse Problems. Mathematical Routes Toward High-Resolution Medical Imaging Modalities,, PhD thesis, (2012). [18] F. Monard and G. Bal, Inverse anisotropic conductivity from power densities in dimension $n \ge 3$,, Comm. PDE, 38 (2013), 1183. [19] ______, Inverse anisotropic diffusion from power density measurements in two dimensions,, Inverse Problems, 28 (2012). [20] ______, Inverse diffusion problems with redundant internal information,, Inv. Probl. Imaging, 6 (2012), 289. [21] O. Scherzer, Handbook of Mathematical Methods in Imaging,, Springer Verlag, (2011). doi: 10.1007/978-0-387-92920-0. [22] P. Stefanov and G. Uhlmann, Multi-Wave Methods by Ultrasounds,, Inside out, (2012).
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