### INVERSE CONDUCTIVITY PROBLEM WITH INTERNAL DATA

Faouzi Triki1, Tao Yin2

1. 1. Laboratoire Jean Kuntzmann, UMR CNRS 5224, Université Grenoble-Alpes, 700 Avenue Centrale, 38401 Saint-Martin- d'Hères, France;
2. Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Science, Beijing 100190, China
• Received:2021-03-26 Revised:2021-07-31 Published:2023-03-14
• Contact: Faouzi Triki, Email:faouzi.triki@univ-grenoble-alpes.fr
• Supported by:
This work was supported in part by the grant ANR-17-CE40-0029 of the French National Research Agency ANR (project MultiOnde).

Faouzi Triki, Tao Yin. INVERSE CONDUCTIVITY PROBLEM WITH INTERNAL DATA[J]. Journal of Computational Mathematics, 2023, 41(3): 483-501.

This paper concerns the reconstruction of a scalar coefficient of a second-order elliptic equation in divergence form posed on a bounded domain from internal data. This problem finds applications in multi-wave imaging, greedy methods to approximate parameterdependent elliptic problems, and image treatment with partial differential equations. We first show that the inverse problem for smooth coefficients can be rewritten as a linear transport equation. Assuming that the coefficient is known near the boundary, we study the well-posedness of associated transport equation as well as its numerical resolution using discontinuous Galerkin method. We propose a regularized transport equation that allow us to derive rigorous convergence rates of the numerical method in terms of the order of the polynomial approximation as well as the regularization parameter. We finally provide numerical examples for the inversion assuming a lower regularity of the coefficient, and using synthetic data.

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 [1] D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, volume 224, springer, 2015.[2] H. Ammari, J. Garnier, H. Kang, L.H. Nguyen and L. Seppecher, Multi-Wave Medical Imaging:Mathematical Modelling & Imaging Reconstruction, World Scientific, London, 2017.[3] M. Choulli and F. Triki, New stability estimates for the inverse medium problem with internal data, SIAM J. Math. Anal, 47:3(2015), 1778-1799.[4] G. Bal and G. Uhlmann, Reconstruction of coefficients in scalar second-order elliptic equations from knowledge of their solutions, Comm. Pure Appl. Math., 66:10(2013), 1629-1652.[5] E. Bonnetier, M. Choulli and F. Triki, Stability for quantitative photoacoustic tomography revisited. Res. Math. Sci., 9:24(2022).[6] G. Bal and K. Ren, Multi-source quantitative photoacoustic tomography in a diffusive regime, Inverse Problems, 27:7(2011), 075003.[7] W. Naetar and O. Scherzer, Quantitative photoacoustic tomography with piecewise constant material parameters, SIAM Journal on Imaging Sciences, 7:3(2014), 1755-1774.[8] M. Briane, Reconstruction of isotropic conductivities from non smooth electric fields, ESAIM:Math. Model. Numer. Anal., 52:3(2018), 1173-1193.[9] M. Briane, G.W. Milton and A. Treibergs, Which electric fields are realizable in conducting materials, ESAIM:Math. Model. Numer. Anal., 48:2(2014), 307-323.[10] M. Choulli and E. Zuazua, Lipschitz dependence of the coefficients on the resolvent and greedy approximation for scalar elliptic problems, C. R. Math. Acad. Sci. Paris, Ser I, 354(2016), 1174-1187.[11] P. Binev, A. Cohen, W. Dahmen, R. DeVore, G. Petrova and P. Wojtaszczyk, Convergence rates for greedy algorithms in reduced basis methods, SIAM J. Math. Anal., 43:3(2011), 1457-1472.[12] P. Pavel and J. Sokolowski, Compressible Navier-Stokes equations:theory and shape optimization, Birkhäuser Basel, 2012.[13] G. Fichera, Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Nat. Sez. I, 5:8(1956), 1-30.[14] J.J. Kohn and L. Nirenberg, Degenerate elliptic-parabolic equations of second order, Comm. Pure Appl. Math., 20(1967), 797-872.[15] O.A. Oleinik and E.V. Radkevic, Second Order Equations with Nonnegative Characteristic Form, Plenum Press, New York, 1973.[16] G.R. Richter, An inverse problem for the steady state diffusion equation, SIAM Journal on Applied Mathematics, 41:2(1981), 210-221.[17] K.O. Friedrichs, Symmetric positive linear differential equations, Comm. Pure Appl. Math., 11(1958), 333-418.[18] A. Ern, J.L. Guermond and G. Caplain, An intrinsic criterion for the bijectivity of hilbert operators related to friedrichs' systems, Communications in Partial Differential Equations, 32(2007), 317- 341.[19] N. Antonić and K. Burazin, Intrinsic boundary conditions for friedrichs systems, Communications in Partial Differential Equations, 35:9(2010), 1690-1715.[20] D.A.D. Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods, Mathématiques & Applications, Springer-Verlag, 2012.[21] G. Alessandrini, An identification problem for an elliptic equation in two variables, Annalidi Matematica Pura ed Applicata, 145(1986), 265-295.[22] R.A. Adams and J.F. Fournier, Sobolev Spaces (Second ed.), Academic Press, 2003.[23] B. Rivière, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations:Theory and Implementation, SIAM, 2008.[24] N. Garofalo and F.H. Lin, Monotonicity properties of variational integrals, ap weights and unique continuation, Indiana Univ. Math. J., 35:2(1986), 245-268.
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