À propos de certains problèmes inverses hybrides
Séminaire Laurent Schwartz — EDP et applications (2013-2014), Talk no. 2, 9 p.

Dans cet exposé, nous présentons quelques résultats récents concernant certains problèmes d’identification de paramètres de type hybride, aussi appelés multi-physiques, pour lesquels le modèles physique sous-jacent est une équation aux dérivées partielles elliptique.

@article{SLSEDP_2013-2014____A2_0,
     author = {Alberti, Giovanni S. and Capdeboscq, Yves},
     title = {\`A propos de certains probl\`emes inverses hybrides},
     journal = {S\'eminaire Laurent Schwartz --- EDP et applications},
     publisher = {Institut des hautes \'etudes scientifiques \& Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2013-2014},
     note = {talk:2},
     doi = {10.5802/slsedp.50},
     language = {fr},
     url = {http://www.numdam.org/item/SLSEDP_2013-2014____A2_0}
}
Alberti, Giovanni S.; Capdeboscq, Yves. À propos de certains problèmes inverses hybrides. Séminaire Laurent Schwartz — EDP et applications (2013-2014), Talk no. 2, 9 p. doi : 10.5802/slsedp.50. http://www.numdam.org/item/SLSEDP_2013-2014____A2_0/

[1] G. S. Alberti. On multiple frequency power density measurements. Inverse Problems, 29(11) :115007, 25, 2013. | MR 3116343 | Zbl pre06277373

[2] G. S. Alberti. On multiple frequency power density measurements II. The full Maxwell’s equations. submitted, 2013. | MR 3116343

[3] G. S. Alberti. Enforcing local non-zero-constraints in pde and applications to hybrid imaging problems. sub, page 23, 2014.

[4] G. Alessandrini. Determining conductivity by boundary measurements, the stability issue. In Renato Spigler, editor, Applied and Industrial Mathematics, volume 56 of Mathematics and Its Applications, pages 317–324. Springer Netherlands, 1991. | MR 1147209 | Zbl 0723.35082

[5] G. Alessandrini and V. Nesi. Univalent σ-harmonic mappings. Arch. Rat. Mech. Anal., 158 :155–171, 2001. | MR 1838656 | Zbl 0977.31006

[6] Giovanni Alessandrini, Antonino Morassi, Edi Rosset, and Sergio Vessella. On doubling inequalities for elliptic systems. J. Math. Anal. Appl., 357(2) :349–355, 2009. | MR 2557649 | Zbl 1167.35541

[7] H. Ammari. An introduction to mathematics of emerging biomedical imaging, volume 62 of Mathématiques & Applications (Berlin). Springer, Berlin, 2008. | MR 2440857 | Zbl 1181.92052

[8] H. Ammari, E. Bonnetier, Y. Capdeboscq, M. Tanter, and M. Fink. Electrical impedance tomography by elastic deformation. SIAM J. Appl. Math., 68(6) :1557–1573, 2008. | MR 2424952 | Zbl 1156.35101

[9] H. Ammari, E. Bossy, V. Jugnon, and H. Kang. Mathematical modeling in photoacoustic imaging of small absorbers. SIAM Rev., 52(4) :677–695, 2010. | MR 2736968 | Zbl 1257.74091

[10] H. Ammari, E. Bretin, J. Garnier, and V. Jugnon. Coherent interferometry algorithms for photoacoustic imaging. SIAM J. Numer. Anal., 50(5) :2259–2280, 2012. | MR 3022218 | Zbl 1262.65204

[11] H. Ammari, Y. Capdeboscq, F. de Gournay, A. Rozanova-Pierrat, and F. Triki. Microwave imaging by elastic deformation. SIAM J. Appl. Math., 71(6) :2112–2130, 2011. | MR 2873260 | Zbl 1235.31006

[12] Habib Ammari, Laure Giovangigli, Loc Hoang Nguyen, and Jin-Keun Seo. Admittivity imaging from multi-frequency micro-electrical impedance tomography. arXiv :1403.5708, 2014.

[13] K. Astala and L. Päivärinta. Calderón’s inverse conductivity problem in the plane. Ann. of Math. (2), 163(1) :265–299, 2006. | MR 2195135 | Zbl 1111.35004

[14] G. Bal, E. Bonnetier, F. Monard, and F. Triki. Inverse diffusion from knowledge of power densities. Inverse Probl. Imaging, 7(2) :353–375, 2013. | MR 3063538 | Zbl 1267.35249

[15] Patricia Bauman, Antonella Marini, and Vincenzo Nesi. Univalent solutions of an elliptic system of partial differential equations arising in homogenization. Indiana Univ. Math. J., 50(2) :747–757, 2001. | MR 1871388 | Zbl pre01780879

[16] M. Briane. Isotropic realizability of electric fields around critical points. Discrete and Continuous Dynamical Systems - Series B, 19 :353–372, 2014. | MR 3170189 | Zbl pre06266120

[17] M. Briane, G. W. Milton, and V. Nesi. Change of sign of the corrector’s determinant for homogenization in three-dimensional conductivity. Arch. Ration. Mech. Anal., 173(1) :133–150, 2004. | MR 2073507 | Zbl 1118.78009

[18] M. Briane, G. W. Milton, and A. Treibergs. Which electric fields are realizable in conducting materials ? ESAIM : Mathematical Modelling and Numerical Analysis, 48 :307–323, 3 2014. | Numdam | MR 3177847

[19] A.-P. Calderón. On an inverse boundary value problem. In Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), pages 65–73. Soc. Brasil. Mat., Rio de Janeiro, 1980. | MR 590275

[20] Y. Capdeboscq, J. Fehrenbach, F. de Gournay, and O. Kavian. Imaging by modification : numerical reconstruction of local conductivities from corresponding power density measurements. SIAM J. Imaging Sci., 2(4) :1003–1030, 2009. | MR 2559157 | Zbl 1180.35549

[21] P. Kuchment and L. Kunyansky. Mathematics of photoacoustic and thermoacoustic tomography. In Otmar Scherzer, editor, Handbook of Mathematical Methods in Imaging, pages 817–865. Springer New York, 2011. | MR 2885203 | Zbl 1259.92065

[22] R. S. Laugesen. Injectivity can fail for higher-dimensional harmonic extensions. Complex Variables Theory Appl., 28(4) :357–369, 1996. | MR 1700199 | Zbl 0871.54020

[23] N. Mandache. Exponential instability in an inverse problem for the Schrödinger equation. Inverse Problems, 17(5) :1435, 2001. | MR 1862200 | Zbl 0985.35110

[24] Antonios D. Melas. An example of a harmonic map between Euclidean balls. Proc. Amer. Math. Soc., 117(3) :857–859, 1993. | MR 1112497 | Zbl 0836.54007

[25] Siegfried Momm. Lower bounds for the modulus of analytic functions. Bull. London Math. Soc., 22(3) :239–244, 1990. | MR 1041137 | Zbl 0668.30001

[26] G. Uhlmann. Electrical impedance tomography and Calderón’s problem. Inverse Problems, 25(12) :123011, 2009. | Zbl 1181.35339

[27] K. Wang and M. A. Anastasio. Photoacoustic and thermoacoustic tomography : Image formation principles. In O. Scherzer, editor, Handbook of Mathematical Methods in Imaging, pages 781–815. Springer New York, 2011. | Zbl 1259.78035