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 & 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, (2009).   Google Scholar

[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.  doi: 10.1080/00036810802555490.  Google Scholar

[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.  doi: 10.1016/j.crma.2009.11.001.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2007.03.024.  Google Scholar

[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.  doi: 10.1137/050640047.  Google Scholar

[6]

S. Boyd and L. Vandenberghe, "Convex Optimization,", Cambridge University Press, (2004).   Google Scholar

[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.  doi: 10.1088/0266-5611/22/5/003.  Google Scholar

[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.   Google Scholar

[9]

A. V. Fiacco and G. P. McCormick, "Nonlinear Programming: Sequential Unconstrained Minimization Techniques,", John Wiley and Sons, (1968).   Google Scholar

[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.  doi: 10.1007/s10589-009-9278-x.  Google Scholar

[11]

V. Kolehmainen, M. Lassas, K. Niinimäki and S. Siltanen, Sparsity-promoting Bayesian inversion,, Inverse Problems, 28 (2012).  doi: 10.1088/0266-5611/28/2/025005.  Google Scholar

[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 (1968).   Google Scholar

[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.  doi: 10.1007/s00222-010-0278-3.  Google Scholar

[14]

J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications,", Vol. 1, (1968).   Google Scholar

[15]

S. Mehrotra, On the implementation of a primal-dual interior point method,, SIAM Journal on Optimization, 2 (1992), 575.  doi: 10.1137/0802028.  Google Scholar

[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, (2007).   Google Scholar

[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.  doi: 10.1007/s11075-008-9225-4.  Google Scholar

[18]

J. Nocedal and S. J. Wright, "Numerical Optimization,", Second edition, (2006).   Google Scholar

[19]

O. Poisson, Uniqueness and Hölder stability of discontinuous diffusion coefficients in three related inverse problems for the heat equation,, Inverse Problems, 24 (2008).  doi: 10.1088/0266-5611/24/2/025012.  Google Scholar

[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.  doi: 10.1002/zamm.200610337.  Google Scholar

[21]

L. Roques and M. Cristofol, The inverse problem of determining several coefficients in a non linear Lotka-Volterra system,, Inverse Problems, 28 (2012).  doi: 10.1088/0266-5611/28/7/075007.  Google Scholar

[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.  doi: 10.1007/s10440-010-9570-x.  Google Scholar

[23]

A. Schiela, Barrier methods for optimal control problems with state constraints,, SIAM Journal on Optimization, 20 (2009), 1002.  doi: 10.1137/070692789.  Google Scholar

[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.  doi: 10.1007/s00211-011-0381-4.  Google Scholar

[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.  doi: 10.1007/s10589-007-9057-5.  Google Scholar

[26]

M. Ulbrich and S. Ulbrich, Primal-dual interior point methods for PDE-constrained optimization,, Mathematical Programming, 117 (2009), 435.  doi: 10.1007/s10107-007-0168-7.  Google Scholar

[27]

R. J. Vanderbei and D. F. Shanno, An Interior-point algorith for nonconvex nonlinear programming,, Computational Optimization and Applications, 13 (1999), 231.  doi: 10.1023/A:1008677427361.  Google Scholar

[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.  doi: 10.1007/s10589-007-9088-y.  Google Scholar

[29]

S. J. Wright, "Primal-Dual Interior-Point Methods,", SIAM, (1997).  doi: 10.1137/1.9781611971453.  Google Scholar

[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.  doi: 10.1007/s10589-008-9209-2.  Google Scholar

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, (2009).   Google Scholar

[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.  doi: 10.1080/00036810802555490.  Google Scholar

[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.  doi: 10.1016/j.crma.2009.11.001.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2007.03.024.  Google Scholar

[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.  doi: 10.1137/050640047.  Google Scholar

[6]

S. Boyd and L. Vandenberghe, "Convex Optimization,", Cambridge University Press, (2004).   Google Scholar

[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.  doi: 10.1088/0266-5611/22/5/003.  Google Scholar

[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.   Google Scholar

[9]

A. V. Fiacco and G. P. McCormick, "Nonlinear Programming: Sequential Unconstrained Minimization Techniques,", John Wiley and Sons, (1968).   Google Scholar

[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.  doi: 10.1007/s10589-009-9278-x.  Google Scholar

[11]

V. Kolehmainen, M. Lassas, K. Niinimäki and S. Siltanen, Sparsity-promoting Bayesian inversion,, Inverse Problems, 28 (2012).  doi: 10.1088/0266-5611/28/2/025005.  Google Scholar

[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 (1968).   Google Scholar

[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.  doi: 10.1007/s00222-010-0278-3.  Google Scholar

[14]

J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications,", Vol. 1, (1968).   Google Scholar

[15]

S. Mehrotra, On the implementation of a primal-dual interior point method,, SIAM Journal on Optimization, 2 (1992), 575.  doi: 10.1137/0802028.  Google Scholar

[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, (2007).   Google Scholar

[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.  doi: 10.1007/s11075-008-9225-4.  Google Scholar

[18]

J. Nocedal and S. J. Wright, "Numerical Optimization,", Second edition, (2006).   Google Scholar

[19]

O. Poisson, Uniqueness and Hölder stability of discontinuous diffusion coefficients in three related inverse problems for the heat equation,, Inverse Problems, 24 (2008).  doi: 10.1088/0266-5611/24/2/025012.  Google Scholar

[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.  doi: 10.1002/zamm.200610337.  Google Scholar

[21]

L. Roques and M. Cristofol, The inverse problem of determining several coefficients in a non linear Lotka-Volterra system,, Inverse Problems, 28 (2012).  doi: 10.1088/0266-5611/28/7/075007.  Google Scholar

[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.  doi: 10.1007/s10440-010-9570-x.  Google Scholar

[23]

A. Schiela, Barrier methods for optimal control problems with state constraints,, SIAM Journal on Optimization, 20 (2009), 1002.  doi: 10.1137/070692789.  Google Scholar

[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.  doi: 10.1007/s00211-011-0381-4.  Google Scholar

[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.  doi: 10.1007/s10589-007-9057-5.  Google Scholar

[26]

M. Ulbrich and S. Ulbrich, Primal-dual interior point methods for PDE-constrained optimization,, Mathematical Programming, 117 (2009), 435.  doi: 10.1007/s10107-007-0168-7.  Google Scholar

[27]

R. J. Vanderbei and D. F. Shanno, An Interior-point algorith for nonconvex nonlinear programming,, Computational Optimization and Applications, 13 (1999), 231.  doi: 10.1023/A:1008677427361.  Google Scholar

[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.  doi: 10.1007/s10589-007-9088-y.  Google Scholar

[29]

S. J. Wright, "Primal-Dual Interior-Point Methods,", SIAM, (1997).  doi: 10.1137/1.9781611971453.  Google Scholar

[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.  doi: 10.1007/s10589-008-9209-2.  Google Scholar

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