# 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 and 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), 1557-1573. doi: 10.1137/070686408. [2] S. R. Arridge and O. Scherzer, Imaging from coupled physics, Inverse Problems, 28 (2012), 080201. doi: 10.1088/0266-5611/28/8/080201. [3] G. Bal, Hybrid inverse problems and systems of partial differential equations, arXiv:1210.0265. [4] ______, Cauchy problem for Ultrasound modulated EIT, Analysis and PDE, 6 (2013), 751-775. doi: 10.2140/apde.2013.6.751. [5] ______, Hybrid Inverse Problems and Internal Functionals, Inside Out, Cambridge University Press, Cambridge, UK, (Editor, G. Uhlmann), 2012. [6] G. Bal, E. Bonnetier, F. Monard and F. Triki, Inverse diffusion from knowledge of power densities, Inverse Problems and Imaging, 7 (2013), 353-375. 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), 265-280. 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), 1629-1652. 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), 1003-1030. doi: 10.1137/080723521. [10] L. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol.19, AMS, 1998. [11] G. B. Folland, Introduction to Partial Differential Equations, Princeton University Press, Princeton New Jersey, 1995. [12] B. Gebauer and O. Scherzer, Impedance-acoustic tomography, SIAM J. Applied Math., 69 (2008), 565-576. 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), 183-208. 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), 055013, 21 pp. doi: 10.1088/0266-5611/27/5/055013. [16] P. Kuchment and D. Steinhauer, Stabilizing inverse problems by internal data, Inverse Problems, 28 (2012), 20 pp. 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, Columbia University, 2012. [18] F. Monard and G. Bal, Inverse anisotropic conductivity from power densities in dimension $n \ge 3$, Comm. PDE, 38 (2013), 1183-1207. [19] ______, Inverse anisotropic diffusion from power density measurements in two dimensions, Inverse Problems, 28 (2012), 084001. [20] ______, Inverse diffusion problems with redundant internal information, Inv. Probl. Imaging, 6 (2012), 289-313. [21] O. Scherzer, Handbook of Mathematical Methods in Imaging, Springer Verlag, New York, 2011. doi: 10.1007/978-0-387-92920-0. [22] P. Stefanov and G. Uhlmann, Multi-Wave Methods by Ultrasounds, Inside out, Cambridge University Press ( Ed. G. Uhlmann), 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), 1557-1573. doi: 10.1137/070686408. [2] S. R. Arridge and O. Scherzer, Imaging from coupled physics, Inverse Problems, 28 (2012), 080201. doi: 10.1088/0266-5611/28/8/080201. [3] G. Bal, Hybrid inverse problems and systems of partial differential equations, arXiv:1210.0265. [4] ______, Cauchy problem for Ultrasound modulated EIT, Analysis and PDE, 6 (2013), 751-775. doi: 10.2140/apde.2013.6.751. [5] ______, Hybrid Inverse Problems and Internal Functionals, Inside Out, Cambridge University Press, Cambridge, UK, (Editor, G. Uhlmann), 2012. [6] G. Bal, E. Bonnetier, F. Monard and F. Triki, Inverse diffusion from knowledge of power densities, Inverse Problems and Imaging, 7 (2013), 353-375. 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), 265-280. 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), 1629-1652. 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), 1003-1030. doi: 10.1137/080723521. [10] L. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol.19, AMS, 1998. [11] G. B. Folland, Introduction to Partial Differential Equations, Princeton University Press, Princeton New Jersey, 1995. [12] B. Gebauer and O. Scherzer, Impedance-acoustic tomography, SIAM J. Applied Math., 69 (2008), 565-576. 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), 183-208. 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), 055013, 21 pp. doi: 10.1088/0266-5611/27/5/055013. [16] P. Kuchment and D. Steinhauer, Stabilizing inverse problems by internal data, Inverse Problems, 28 (2012), 20 pp. 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, Columbia University, 2012. [18] F. Monard and G. Bal, Inverse anisotropic conductivity from power densities in dimension $n \ge 3$, Comm. PDE, 38 (2013), 1183-1207. [19] ______, Inverse anisotropic diffusion from power density measurements in two dimensions, Inverse Problems, 28 (2012), 084001. [20] ______, Inverse diffusion problems with redundant internal information, Inv. Probl. Imaging, 6 (2012), 289-313. [21] O. Scherzer, Handbook of Mathematical Methods in Imaging, Springer Verlag, New York, 2011. doi: 10.1007/978-0-387-92920-0. [22] P. Stefanov and G. Uhlmann, Multi-Wave Methods by Ultrasounds, Inside out, Cambridge University Press ( Ed. G. Uhlmann), 2012.
 [1] Andrew Homan. Multi-wave imaging in attenuating media. Inverse Problems and Imaging, 2013, 7 (4) : 1235-1250. doi: 10.3934/ipi.2013.7.1235 [2] Liang Huang, Jiao Chen. The boundedness of multi-linear and multi-parameter pseudo-differential operators. Communications on Pure and Applied Analysis, 2021, 20 (2) : 801-815. doi: 10.3934/cpaa.2020291 [3] JIAO CHEN, WEI DAI, GUOZHEN LU. $L^p$ boundedness for maximal functions associated with multi-linear pseudo-differential operators. Communications on Pure and Applied Analysis, 2017, 16 (3) : 883-898. doi: 10.3934/cpaa.2017042 [4] Lanzhe Liu. Mean oscillation and boundedness of Toeplitz Type operators associated to pseudo-differential operators. Communications on Pure and Applied Analysis, 2015, 14 (2) : 627-636. doi: 10.3934/cpaa.2015.14.627 [5] Thomas Kappeler, Riccardo Montalto. Normal form coordinates for the Benjamin-Ono equation having expansions in terms of pseudo-differential operators. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022048 [6] Ildoo Kim. An $L_p$-Lipschitz theory for parabolic equations with time measurable pseudo-differential operators. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2751-2771. doi: 10.3934/cpaa.2018130 [7] Dinh Nguyen Duy Hai. Hölder-Logarithmic type approximation for nonlinear backward parabolic equations connected with a pseudo-differential operator. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1715-1734. doi: 10.3934/cpaa.2022043 [8] Michael Herty, Giuseppe Visconti. Kinetic methods for inverse problems. Kinetic and Related Models, 2019, 12 (5) : 1109-1130. doi: 10.3934/krm.2019042 [9] Agnese Di Castro, Mayte Pérez-Llanos, José Miguel Urbano. Limits of anisotropic and degenerate elliptic problems. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1217-1229. doi: 10.3934/cpaa.2012.11.1217 [10] Hisashi Morioka. Inverse boundary value problems for discrete Schrödinger operators on the multi-dimensional square lattice. Inverse Problems and Imaging, 2011, 5 (3) : 715-730. doi: 10.3934/ipi.2011.5.715 [11] Victor Isakov, Shuai Lu. Inverse source problems without (pseudo) convexity assumptions. Inverse Problems and Imaging, 2018, 12 (4) : 955-970. doi: 10.3934/ipi.2018040 [12] Frank Natterer. Incomplete data problems in wave equation imaging. Inverse Problems and Imaging, 2010, 4 (4) : 685-691. doi: 10.3934/ipi.2010.4.685 [13] Md. Ibrahim Kholil, Ziqi Sun. A uniqueness theorem for inverse problems in quasilinear anisotropic media. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022008 [14] Michel Chipot, Senoussi Guesmia. On the asymptotic behavior of elliptic, anisotropic singular perturbations problems. Communications on Pure and Applied Analysis, 2009, 8 (1) : 179-193. doi: 10.3934/cpaa.2009.8.179 [15] Tony Liimatainen, Lauri Oksanen. Counterexamples to inverse problems for the wave equation. Inverse Problems and Imaging, 2022, 16 (2) : 467-479. doi: 10.3934/ipi.2021058 [16] Deyue Zhang, Yue Wu, Yinglin Wang, Yukun Guo. A direct imaging method for the exterior and interior inverse scattering problems. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022025 [17] Rafael del Rio, Mikhail Kudryavtsev, Luis O. Silva. Inverse problems for Jacobi operators III: Mass-spring perturbations of semi-infinite systems. Inverse Problems and Imaging, 2012, 6 (4) : 599-621. doi: 10.3934/ipi.2012.6.599 [18] Martin Hanke, William Rundell. On rational approximation methods for inverse source problems. Inverse Problems and Imaging, 2011, 5 (1) : 185-202. doi: 10.3934/ipi.2011.5.185 [19] Daijun Jiang, Hui Feng, Jun Zou. Overlapping domain decomposition methods for linear inverse problems. Inverse Problems and Imaging, 2015, 9 (1) : 163-188. doi: 10.3934/ipi.2015.9.163 [20] Dušan D. Repovš. Infinitely many symmetric solutions for anisotropic problems driven by nonhomogeneous operators. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 401-411. doi: 10.3934/dcdss.2019026

2021 Impact Factor: 1.483