# American Institute of Mathematical Sciences

February  2013, 7(1): 159-182. doi: 10.3934/ipi.2013.7.159

## Inverse problem for a coupled parabolic system with discontinuous conductivities: One-dimensional case

 1 Aix-Marseille Universite, LATP, Technopôle Château-Gombert, 39, rue F. Joliot Curie, 13453 Marseille Cedex 13, France, France 2 Aix-Marseille Universite, CPT, Campus de Luminy, Case 907, 13288 Marseille cedex 9, France 3 Department of Applied Physics, University of Eastern Finland, Kuopio campus, P.O.Box 1627, FIN-70211 Kuopio, Finland

Received  March 2012 Revised  November 2012 Published  February 2013

We study the inverse problem of the simultaneous identification of two discontinuous diffusion coefficients for a one-dimensional coupled parabolic system with the observation of only one component. The stability result for the diffusion coefficients is obtained by a Carleman-type estimate. Results from numerical experiments in the one-dimensional case are reported, suggesting that the method makes possible to recover discontinuous diffusion coefficients.
Citation: Michel Cristofol, Patricia Gaitan, Kati Niinimäki, Olivier Poisson. Inverse problem for a coupled parabolic system with discontinuous conductivities: One-dimensional case. Inverse Problems and Imaging, 2013, 7 (1) : 159-182. doi: 10.3934/ipi.2013.7.159
##### References:
 [1] F. Alvarez, J. Bolte, J. F. Bonnans and F. Silva, Asymptotic expansions for interior penalty solutions of control constrained linear-quadratic problems, Technical Report RR 6863, INRIA, (2009). [2] A. Benabdallah, M. Cristofol, P. Gaitan and M. Yamamoto, Inverse problem for a parabolic system with two components by measurements of one component, Applicable Analysis, 88 (2008), 683-709. doi: 10.1080/00036810802555490. [3] A. Benabdallah, M. Cristofol, P. Gaitan and L. de Teresa, A new Carleman inequality for parabolic systems with a single observation and applications, C. R. Math. Acad. Sci. Paris, 348 (2010), 25-29. doi: 10.1016/j.crma.2009.11.001. [4] A. Benabdallah, Y. Dermenjian and J. Le Rousseau, Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and a inverse problem, Journal of Mathematical Analysis and Applications, 336 (2007), 865-887. doi: 10.1016/j.jmaa.2007.03.024. [5] A. Benabdallah, P. Gaitan and J. Le Rousseau, Stability of discontinuous diffusion coefficients and initial conditions in an inverse problem for the heat equation, SIAM Journal on Control and Optimization, 46 (2007), 1849-1881. doi: 10.1137/050640047. [6] S. Boyd and L. Vandenberghe, "Convex Optimization," Cambridge University Press, Cambridge, 2004. [7] M. Cristofol, P. Gaitan and H. Ramoul, Inverse problems for a two by two reaction-diffusion system using a carleman estimate with one observation, Inverse Problems, 22 (2006), 1561-1573. doi: 10.1088/0266-5611/22/5/003. [8] M. Cristofol, P. Gaitan, H. Ramoul and M. Yamamoto, Identification of two coefficients with data of one component for a nonlinear parabolic system, Applicable Analysis, (2011), 1-9. [9] A. V. Fiacco and G. P. McCormick, "Nonlinear Programming: Sequential Unconstrained Minimization Techniques," John Wiley and Sons, Inc., New York-London-Sydney, 1968. [10] M. Hinze and A. Schiela, Discretization of interior point methods for state constrained elliptic optimal control problems: Optimal error estimates and parameter adjustment, Computational Optimization and Applications, 48 (2010), 581-600. doi: 10.1007/s10589-009-9278-x. [11] V. Kolehmainen, M. Lassas, K. Niinimäki and S. Siltanen, Sparsity-promoting Bayesian inversion, Inverse Problems, 28 (2012), 025005, 28 pp. doi: 10.1088/0266-5611/28/2/025005. [12] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasi-Linear Equations of Parabolic Type," Translations of Mathematical Monographs, 23, AMS, Providence, RI, 1968. [13] J. Le Rousseau and L. Robbiano, Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces, Inventiones Mathematicae, 183 (2011), 245-336. doi: 10.1007/s00222-010-0278-3. [14] J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications," Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968. [15] S. Mehrotra, On the implementation of a primal-dual interior point method, SIAM Journal on Optimization, 2 (1992), 575-601. doi: 10.1137/0802028. [16] I. Neitzel, U. Prüfert and T. Slawig, Strategies for time-dependent PDE control using an integrated modeling and simulation environment. Part one: problems without inequality constraints, Technical Report 408, Matheon, Berlin, (2007). [17] I. Neitzel, U. Prüfert and T. Slawig, Strategies for time-dependent PDE control with inequality constraints using an integrated modeling and simulation environment, Numerical Algorithms, 50 (2008), 241-269. doi: 10.1007/s11075-008-9225-4. [18] J. Nocedal and S. J. Wright, "Numerical Optimization," Second edition, Springer Series in Operations Research and Financial Engineering, Springer, New York, 2006 [19] O. Poisson, Uniqueness and Hölder stability of discontinuous diffusion coefficients in three related inverse problems for the heat equation, Inverse Problems, 24 (2008), 025012, 32 pp. doi: 10.1088/0266-5611/24/2/025012. [20] U. Prüfert and F. Tröltzsch, An interior point method for a parabolic optimal control problem with regularized pointwise state constraints, ZAMM Z. Angew. Math. Mech., 87 (2007), 564-589. doi: 10.1002/zamm.200610337. [21] L. Roques and M. Cristofol, The inverse problem of determining several coefficients in a non linear Lotka-Volterra system, Inverse Problems, 28 (2012), 075007, 12 pp. doi: 10.1088/0266-5611/28/7/075007. [22] K. Sakthivel, N. Branibalan, J.-H. Kim and K. Balachandran, Erratum to: Stability of diffusion coefficients in an inverse problem for the lotka-volterra competition system, Acta Applicandae Mathematicae, 111 (2010), 149-152. doi: 10.1007/s10440-010-9570-x. [23] A. Schiela, Barrier methods for optimal control problems with state constraints, SIAM Journal on Optimization, 20 (2009), 1002-1031. doi: 10.1137/070692789. [24] A. Schiela and A. Günther, An interior point algorithm with inexact step computation in function space for state constrained optimal control, Numerische Mathematik, 119 (2011), 373-407. doi: 10.1007/s00211-011-0381-4. [25] A. Schiela and M. Weiser, Superlinear convergence of the control reduced interior point method for PDE constrained optimization, Computational Optimization and Applications, 39 (2008), 369-393. doi: 10.1007/s10589-007-9057-5. [26] M. Ulbrich and S. Ulbrich, Primal-dual interior point methods for PDE-constrained optimization, Mathematical Programming, 117 (2009), 435-485. doi: 10.1007/s10107-007-0168-7. [27] R. J. Vanderbei and D. F. Shanno, An Interior-point algorith for nonconvex nonlinear programming, Computational Optimization and Applications, 13 (1999), 231-252. doi: 10.1023/A:1008677427361. [28] M. Weiser, T. Gänzler and A. Schiela, A control reduced primal interior point method for a class of control constrained optimal control problems, Computational Optimization and Applications, 41 (2008), 127-145. doi: 10.1007/s10589-007-9088-y. [29] S. J. Wright, "Primal-Dual Interior-Point Methods," SIAM, Philadelphia, PA, 1997. doi: 10.1137/1.9781611971453. [30] W. Wollner, A posteriori error estimates for a finite element discretization of interior point methods for an elliptic optimization problem with state constraints, Computational Optimization and Applications, 47 (2010), 133-159. doi: 10.1007/s10589-008-9209-2.

show all references

##### References:
 [1] F. Alvarez, J. Bolte, J. F. Bonnans and F. Silva, Asymptotic expansions for interior penalty solutions of control constrained linear-quadratic problems, Technical Report RR 6863, INRIA, (2009). [2] A. Benabdallah, M. Cristofol, P. Gaitan and M. Yamamoto, Inverse problem for a parabolic system with two components by measurements of one component, Applicable Analysis, 88 (2008), 683-709. doi: 10.1080/00036810802555490. [3] A. Benabdallah, M. Cristofol, P. Gaitan and L. de Teresa, A new Carleman inequality for parabolic systems with a single observation and applications, C. R. Math. Acad. Sci. Paris, 348 (2010), 25-29. doi: 10.1016/j.crma.2009.11.001. [4] A. Benabdallah, Y. Dermenjian and J. Le Rousseau, Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and a inverse problem, Journal of Mathematical Analysis and Applications, 336 (2007), 865-887. doi: 10.1016/j.jmaa.2007.03.024. [5] A. Benabdallah, P. Gaitan and J. Le Rousseau, Stability of discontinuous diffusion coefficients and initial conditions in an inverse problem for the heat equation, SIAM Journal on Control and Optimization, 46 (2007), 1849-1881. doi: 10.1137/050640047. [6] S. Boyd and L. Vandenberghe, "Convex Optimization," Cambridge University Press, Cambridge, 2004. [7] M. Cristofol, P. Gaitan and H. Ramoul, Inverse problems for a two by two reaction-diffusion system using a carleman estimate with one observation, Inverse Problems, 22 (2006), 1561-1573. doi: 10.1088/0266-5611/22/5/003. [8] M. Cristofol, P. Gaitan, H. Ramoul and M. Yamamoto, Identification of two coefficients with data of one component for a nonlinear parabolic system, Applicable Analysis, (2011), 1-9. [9] A. V. Fiacco and G. P. McCormick, "Nonlinear Programming: Sequential Unconstrained Minimization Techniques," John Wiley and Sons, Inc., New York-London-Sydney, 1968. [10] M. Hinze and A. Schiela, Discretization of interior point methods for state constrained elliptic optimal control problems: Optimal error estimates and parameter adjustment, Computational Optimization and Applications, 48 (2010), 581-600. doi: 10.1007/s10589-009-9278-x. [11] V. Kolehmainen, M. Lassas, K. Niinimäki and S. Siltanen, Sparsity-promoting Bayesian inversion, Inverse Problems, 28 (2012), 025005, 28 pp. doi: 10.1088/0266-5611/28/2/025005. [12] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasi-Linear Equations of Parabolic Type," Translations of Mathematical Monographs, 23, AMS, Providence, RI, 1968. [13] J. Le Rousseau and L. Robbiano, Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces, Inventiones Mathematicae, 183 (2011), 245-336. doi: 10.1007/s00222-010-0278-3. [14] J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications," Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968. [15] S. Mehrotra, On the implementation of a primal-dual interior point method, SIAM Journal on Optimization, 2 (1992), 575-601. doi: 10.1137/0802028. [16] I. Neitzel, U. Prüfert and T. Slawig, Strategies for time-dependent PDE control using an integrated modeling and simulation environment. Part one: problems without inequality constraints, Technical Report 408, Matheon, Berlin, (2007). [17] I. Neitzel, U. Prüfert and T. Slawig, Strategies for time-dependent PDE control with inequality constraints using an integrated modeling and simulation environment, Numerical Algorithms, 50 (2008), 241-269. doi: 10.1007/s11075-008-9225-4. [18] J. Nocedal and S. J. Wright, "Numerical Optimization," Second edition, Springer Series in Operations Research and Financial Engineering, Springer, New York, 2006 [19] O. Poisson, Uniqueness and Hölder stability of discontinuous diffusion coefficients in three related inverse problems for the heat equation, Inverse Problems, 24 (2008), 025012, 32 pp. doi: 10.1088/0266-5611/24/2/025012. [20] U. Prüfert and F. Tröltzsch, An interior point method for a parabolic optimal control problem with regularized pointwise state constraints, ZAMM Z. Angew. Math. Mech., 87 (2007), 564-589. doi: 10.1002/zamm.200610337. [21] L. Roques and M. Cristofol, The inverse problem of determining several coefficients in a non linear Lotka-Volterra system, Inverse Problems, 28 (2012), 075007, 12 pp. doi: 10.1088/0266-5611/28/7/075007. [22] K. Sakthivel, N. Branibalan, J.-H. Kim and K. Balachandran, Erratum to: Stability of diffusion coefficients in an inverse problem for the lotka-volterra competition system, Acta Applicandae Mathematicae, 111 (2010), 149-152. doi: 10.1007/s10440-010-9570-x. [23] A. Schiela, Barrier methods for optimal control problems with state constraints, SIAM Journal on Optimization, 20 (2009), 1002-1031. doi: 10.1137/070692789. [24] A. Schiela and A. Günther, An interior point algorithm with inexact step computation in function space for state constrained optimal control, Numerische Mathematik, 119 (2011), 373-407. doi: 10.1007/s00211-011-0381-4. [25] A. Schiela and M. Weiser, Superlinear convergence of the control reduced interior point method for PDE constrained optimization, Computational Optimization and Applications, 39 (2008), 369-393. doi: 10.1007/s10589-007-9057-5. [26] M. Ulbrich and S. Ulbrich, Primal-dual interior point methods for PDE-constrained optimization, Mathematical Programming, 117 (2009), 435-485. doi: 10.1007/s10107-007-0168-7. [27] R. J. Vanderbei and D. F. Shanno, An Interior-point algorith for nonconvex nonlinear programming, Computational Optimization and Applications, 13 (1999), 231-252. doi: 10.1023/A:1008677427361. [28] M. Weiser, T. Gänzler and A. Schiela, A control reduced primal interior point method for a class of control constrained optimal control problems, Computational Optimization and Applications, 41 (2008), 127-145. doi: 10.1007/s10589-007-9088-y. [29] S. J. Wright, "Primal-Dual Interior-Point Methods," SIAM, Philadelphia, PA, 1997. doi: 10.1137/1.9781611971453. [30] W. Wollner, A posteriori error estimates for a finite element discretization of interior point methods for an elliptic optimization problem with state constraints, Computational Optimization and Applications, 47 (2010), 133-159. doi: 10.1007/s10589-008-9209-2.
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