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Stability of conductivities in an inverse problem in the reaction-diffusion system in electrocardiology
1. | Institut de Mathématiques de Bordeaux, UMR CNRS 5251, Université de Bordeaux, 3 ter Place de la Victoire, 33076 Bordeaux cedex, France, France |
2. | School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000 |
References:
[1] |
M. Bendahmane and F. W. Chaves-Silva, Controllability of a degenerating reaction-diffusion system in electrocardiology,, to appear in SIAM Journal on Control and Optimization, ().
|
[2] |
M. Bendahmane and K. H. Karlsen, Analysis of a class of degenerate reaction-diffusion systems and the bidomain model of cardiac tissue, Netw. Heterog. Media, 1 (2006), 185-218.
doi: 10.3934/nhm.2006.1.185. |
[3] |
M. Bendahmane and K. H. Karlsen, Convergence of a finite volume scheme for the bidomain model of cardiac tissue, Appl. Numer. Math., 59 (2009), 2266-2284.
doi: 10.1016/j.apnum.2008.12.016. |
[4] |
M. Bendahmane, R. Bürger and R. Ruiz Baier, A finite volume scheme for cardiac propagation in media with isotropic conductivities, Math. Comp. Simul., 80 (2010), 1821-1840.
doi: 10.1016/j.matcom.2009.12.010. |
[5] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. |
[6] |
A. L. Bukhgeĭm, Carleman estimates for Volterra operators and uniqueness of inverse problems, in Non-classical Problems of Mathematical Physics, Computing Center of Siberian Branch of Soviet Academy of Sciences, Novosibirsk, 1981, 56-64. |
[7] |
A. L. Bukhgeim, Introduction to the Theory of Inverse Problems, VSP, Utrecht, 2000. |
[8] |
A. L. Bukhgeim and M. V. Klibanov, Uniqueness in the large class of multidimensional inverse problems, (Russian) Dokl. Akad. Nauk SSSR, 260 (1981), 269-272. |
[9] |
K. C. Chang, Methods in Nonlinear Analysis, Springer-Verlag Berlin Heidelberg, Netherlands, 2005. |
[10] |
P. Colli Franzone and G. Savaré, Degenerate evolution systems modeling the cardiac electric field at micro- and macroscopic level, in Evolution Equations, Semigroups and Functional Analysis (Milano, 2000), Progr. Nonlinear Differential Equations Appl., 50, Birkhäuser, Basel, 2002, 49-78. |
[11] |
M. Cristofol, P. Gaitan and H. Ramoul, Inverse problems for a $2 \times 2$ reaction-diffusion system using a Carleman estimate with one observation, Inverse Problems, 22 (2006), 1561-1573.
doi: 10.1088/0266-5611/22/5/003. |
[12] |
A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, 34, RIM Seoul National University, Korea, 1996. |
[13] |
O.Yu. Imanuvilov, M. Yamamoto, Lipschitz stability in inverse problems by Carleman estimates, Inverse Problems, 14 (1998), 1229-1245.
doi: 10.1088/0266-5611/14/5/009. |
[14] |
O. Yu. Imanuvilov and M. Yamamoto, Carleman estimates for the non-stationary Lamé system and the application to an inverse problem, ESAIM, COCV, 11 (2005), 1-56.
doi: 10.1051/cocv:2004030. |
[15] |
V. Isakov, Carleman estimates and applications to inverse problems, Milan J. Math., 72 (2004), 249-271.
doi: 10.1007/s00032-004-0033-6. |
[16] |
M. V. Klibanov, Carleman estimates and inverse problems in the lasrt two decades, in Surveys on Solution Methods for Inverse Problems, Springer, Vienna, 2000, 119-146. |
[17] |
M. V. Klibanov and M. Yamamoto, Lipschitz stability of an inverse problem for an acoustic equation, Appl. Anal., 85 (2006), 515-538.
doi: 10.1080/00036810500474788. |
[18] |
M. A. Krasnoselskii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964. |
[19] |
G. Lebeau and L. Robbiano, Contrôle exact de l'equation de la chaleur, Comm. Partial Differential Equations, 20 (1995), 335-356.
doi: 10.1080/03605309508821097. |
[20] |
J.-P. Puel and M. Yamamoto, On a global estimate in a linear inverse hyperbolic problem, Inverse Problems, 12 (1996), 995-1002.
doi: 10.1088/0266-5611/12/6/013. |
[21] |
K. Sakthivel, N. Baranibalan, J.-H. Kim and K. Balachandran, Stability of diffusion coefficients in an inverse problem for the Lotka-Volterra competition system, Acta Appl. Math., 111 (2010), 129-147.
doi: 10.1007/s10440-009-9455-z. |
[22] |
Z. Q. Wu, J. X. Yin and C. P. Wang, Elliptic and Parabolic Equations, World Scientific Publishing Co. Pte. Ltd, 2003.
doi: 10.1142/6238. |
[23] |
M. Yamamoto, Uniqueness and stability in multidimensional hyperbolic inverse problems, J. Math. Pures Appl., 78 (1999), 65-98.
doi: 10.1016/S0021-7824(99)80010-5. |
[24] |
G. Yuan and M. Yamamoto, Lipshitz stability in the determination of the principal part of a parabolic equation, ESAIM: Control Optim. Calc. Var., 15 (2009), 525-554.
doi: 10.1051/cocv:2008043. |
show all references
References:
[1] |
M. Bendahmane and F. W. Chaves-Silva, Controllability of a degenerating reaction-diffusion system in electrocardiology,, to appear in SIAM Journal on Control and Optimization, ().
|
[2] |
M. Bendahmane and K. H. Karlsen, Analysis of a class of degenerate reaction-diffusion systems and the bidomain model of cardiac tissue, Netw. Heterog. Media, 1 (2006), 185-218.
doi: 10.3934/nhm.2006.1.185. |
[3] |
M. Bendahmane and K. H. Karlsen, Convergence of a finite volume scheme for the bidomain model of cardiac tissue, Appl. Numer. Math., 59 (2009), 2266-2284.
doi: 10.1016/j.apnum.2008.12.016. |
[4] |
M. Bendahmane, R. Bürger and R. Ruiz Baier, A finite volume scheme for cardiac propagation in media with isotropic conductivities, Math. Comp. Simul., 80 (2010), 1821-1840.
doi: 10.1016/j.matcom.2009.12.010. |
[5] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. |
[6] |
A. L. Bukhgeĭm, Carleman estimates for Volterra operators and uniqueness of inverse problems, in Non-classical Problems of Mathematical Physics, Computing Center of Siberian Branch of Soviet Academy of Sciences, Novosibirsk, 1981, 56-64. |
[7] |
A. L. Bukhgeim, Introduction to the Theory of Inverse Problems, VSP, Utrecht, 2000. |
[8] |
A. L. Bukhgeim and M. V. Klibanov, Uniqueness in the large class of multidimensional inverse problems, (Russian) Dokl. Akad. Nauk SSSR, 260 (1981), 269-272. |
[9] |
K. C. Chang, Methods in Nonlinear Analysis, Springer-Verlag Berlin Heidelberg, Netherlands, 2005. |
[10] |
P. Colli Franzone and G. Savaré, Degenerate evolution systems modeling the cardiac electric field at micro- and macroscopic level, in Evolution Equations, Semigroups and Functional Analysis (Milano, 2000), Progr. Nonlinear Differential Equations Appl., 50, Birkhäuser, Basel, 2002, 49-78. |
[11] |
M. Cristofol, P. Gaitan and H. Ramoul, Inverse problems for a $2 \times 2$ reaction-diffusion system using a Carleman estimate with one observation, Inverse Problems, 22 (2006), 1561-1573.
doi: 10.1088/0266-5611/22/5/003. |
[12] |
A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, 34, RIM Seoul National University, Korea, 1996. |
[13] |
O.Yu. Imanuvilov, M. Yamamoto, Lipschitz stability in inverse problems by Carleman estimates, Inverse Problems, 14 (1998), 1229-1245.
doi: 10.1088/0266-5611/14/5/009. |
[14] |
O. Yu. Imanuvilov and M. Yamamoto, Carleman estimates for the non-stationary Lamé system and the application to an inverse problem, ESAIM, COCV, 11 (2005), 1-56.
doi: 10.1051/cocv:2004030. |
[15] |
V. Isakov, Carleman estimates and applications to inverse problems, Milan J. Math., 72 (2004), 249-271.
doi: 10.1007/s00032-004-0033-6. |
[16] |
M. V. Klibanov, Carleman estimates and inverse problems in the lasrt two decades, in Surveys on Solution Methods for Inverse Problems, Springer, Vienna, 2000, 119-146. |
[17] |
M. V. Klibanov and M. Yamamoto, Lipschitz stability of an inverse problem for an acoustic equation, Appl. Anal., 85 (2006), 515-538.
doi: 10.1080/00036810500474788. |
[18] |
M. A. Krasnoselskii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964. |
[19] |
G. Lebeau and L. Robbiano, Contrôle exact de l'equation de la chaleur, Comm. Partial Differential Equations, 20 (1995), 335-356.
doi: 10.1080/03605309508821097. |
[20] |
J.-P. Puel and M. Yamamoto, On a global estimate in a linear inverse hyperbolic problem, Inverse Problems, 12 (1996), 995-1002.
doi: 10.1088/0266-5611/12/6/013. |
[21] |
K. Sakthivel, N. Baranibalan, J.-H. Kim and K. Balachandran, Stability of diffusion coefficients in an inverse problem for the Lotka-Volterra competition system, Acta Appl. Math., 111 (2010), 129-147.
doi: 10.1007/s10440-009-9455-z. |
[22] |
Z. Q. Wu, J. X. Yin and C. P. Wang, Elliptic and Parabolic Equations, World Scientific Publishing Co. Pte. Ltd, 2003.
doi: 10.1142/6238. |
[23] |
M. Yamamoto, Uniqueness and stability in multidimensional hyperbolic inverse problems, J. Math. Pures Appl., 78 (1999), 65-98.
doi: 10.1016/S0021-7824(99)80010-5. |
[24] |
G. Yuan and M. Yamamoto, Lipshitz stability in the determination of the principal part of a parabolic equation, ESAIM: Control Optim. Calc. Var., 15 (2009), 525-554.
doi: 10.1051/cocv:2008043. |
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