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Non-Gaussian statistical inverse problems. Part II: Posterior convergence for approximated unknowns
Inverse diffusion problems with redundant internal information
1. | Department of Applied Physics and Applied Mathematics, Columbia University, New York NY, 10027 |
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.
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] |
G. Alessandrini and V. Nesi, Univalent $e^\sigma$-harmonic mappings, Arch. Rat. Mech. Anal., 158 (2001), 155-171.
doi: 10.1007/PL00004242. |
[3] |
G. Bal, Hybrid inverse problems and internal functionals (review paper), in "Inside Out'' (ed. Gunther Uhlmann), Cambridge University Press, 2012. |
[4] |
_____, Cauchy problem and Ultrasound modulated EIT, submitted. |
[5] |
G. Bal, E. Bonnetier, F. Monard and F. Triki, Inverse diffusion from knowledge of power densities, Inverse Probl. Imaging, in press, 2012. |
[6] |
G. Bal and K. Ren, Multi-source quantitative photoacoustic tomography, Inverse Problems, 27 (2011).
doi: 10.1088/0266-5611/27/5/055007. |
[7] |
G. Bal and G. Uhlmann, Inverse diffusion theory for photoacoustics, Inverse Problems, 26 (2010), 085010.
doi: 10.1088/0266-5611/26/8/085010. |
[8] |
A. Calderón, On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics, Soc. Brasileira de Matematica, Rio de Janeiro, (1980), 65-73. |
[9] |
C. Carathéodory, "Calculus of Variations and Partial Differential Equations of the First Order,'' Third edition, AMS, Chelsea, 1999. |
[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), 1003-1030. |
[11] |
L. C. Evans, "Partial Differential Equations,'' Graduate Studies in Mathematics, Vol. 19, AMS, 1998. |
[12] |
B. Gebauer and O. Scherzer, Impedance-acoustic tomography, SIAM J. Applied Math., 69 (2009), 565-576.
doi: 10.1137/080715123. |
[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), 511-526.
doi: 10.1137/S0036141001391354. |
[14] |
P. Kuchment and L. Kunyansky, 2D and 3D reconstructions in acousto-electric tomography, Inverse Problems, 27 (2011), 055013. |
[15] |
J. M. Lee, "Riemannian Manifolds. An Introduction to Curvature,'' Graduate Texts in Mathematics, Vol. 176, Springer, 1997. |
[16] |
F. Monard, "Taming Unstable Inverse Problems. Mathematical Routes Toward High-Resolution Medical Imaging Modalities,'' Ph.D. thesis, Columbia University, New York, 2012. |
[17] |
A. Nachman, A. Tamasan and A. Timonov, Conductivity imaging with a single measurement of boundary and interior data, Inverse Problems, 23 (2007), 2551-2563.
doi: 10.1088/0266-5611/23/6/017. |
[18] |
_____, Recovering the conductivity from a single measurement of interior data, Inverse Problems, 25 (2009), 035014. |
[19] |
_____, Current density impedance imaging, Contemporary Mathematics, American Mathematical Society, in press, 2012. |
[20] |
O. Scherzer, "Handbook of Mathematical Methods in Imaging,'' Springer Verlag, New York, 2011. |
[21] |
M. Spivak, "A Comprehensive Introduction to Differential Geometry, Vol. 2,'' Second edition, Publish or perish, 1990. |
[22] |
J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169.
doi: 10.2307/1971291. |
[23] |
M. Taylor, "Partial Differential Equations I, Basic Theory,'' Springer, New York, 1996. |
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] |
G. Alessandrini and V. Nesi, Univalent $e^\sigma$-harmonic mappings, Arch. Rat. Mech. Anal., 158 (2001), 155-171.
doi: 10.1007/PL00004242. |
[3] |
G. Bal, Hybrid inverse problems and internal functionals (review paper), in "Inside Out'' (ed. Gunther Uhlmann), Cambridge University Press, 2012. |
[4] |
_____, Cauchy problem and Ultrasound modulated EIT, submitted. |
[5] |
G. Bal, E. Bonnetier, F. Monard and F. Triki, Inverse diffusion from knowledge of power densities, Inverse Probl. Imaging, in press, 2012. |
[6] |
G. Bal and K. Ren, Multi-source quantitative photoacoustic tomography, Inverse Problems, 27 (2011).
doi: 10.1088/0266-5611/27/5/055007. |
[7] |
G. Bal and G. Uhlmann, Inverse diffusion theory for photoacoustics, Inverse Problems, 26 (2010), 085010.
doi: 10.1088/0266-5611/26/8/085010. |
[8] |
A. Calderón, On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics, Soc. Brasileira de Matematica, Rio de Janeiro, (1980), 65-73. |
[9] |
C. Carathéodory, "Calculus of Variations and Partial Differential Equations of the First Order,'' Third edition, AMS, Chelsea, 1999. |
[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), 1003-1030. |
[11] |
L. C. Evans, "Partial Differential Equations,'' Graduate Studies in Mathematics, Vol. 19, AMS, 1998. |
[12] |
B. Gebauer and O. Scherzer, Impedance-acoustic tomography, SIAM J. Applied Math., 69 (2009), 565-576.
doi: 10.1137/080715123. |
[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), 511-526.
doi: 10.1137/S0036141001391354. |
[14] |
P. Kuchment and L. Kunyansky, 2D and 3D reconstructions in acousto-electric tomography, Inverse Problems, 27 (2011), 055013. |
[15] |
J. M. Lee, "Riemannian Manifolds. An Introduction to Curvature,'' Graduate Texts in Mathematics, Vol. 176, Springer, 1997. |
[16] |
F. Monard, "Taming Unstable Inverse Problems. Mathematical Routes Toward High-Resolution Medical Imaging Modalities,'' Ph.D. thesis, Columbia University, New York, 2012. |
[17] |
A. Nachman, A. Tamasan and A. Timonov, Conductivity imaging with a single measurement of boundary and interior data, Inverse Problems, 23 (2007), 2551-2563.
doi: 10.1088/0266-5611/23/6/017. |
[18] |
_____, Recovering the conductivity from a single measurement of interior data, Inverse Problems, 25 (2009), 035014. |
[19] |
_____, Current density impedance imaging, Contemporary Mathematics, American Mathematical Society, in press, 2012. |
[20] |
O. Scherzer, "Handbook of Mathematical Methods in Imaging,'' Springer Verlag, New York, 2011. |
[21] |
M. Spivak, "A Comprehensive Introduction to Differential Geometry, Vol. 2,'' Second edition, Publish or perish, 1990. |
[22] |
J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169.
doi: 10.2307/1971291. |
[23] |
M. Taylor, "Partial Differential Equations I, Basic Theory,'' Springer, New York, 1996. |
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