doi: 10.3934/mcrf.2019014

On a logarithmic stability estimate for an inverse heat conduction problem

Department of Mathematics, Faculty of Sciences of Bizerte, 7021 Jarzouna Bizerte, Tunisia

* Corresponding author: Aymen Jbalia

Received  December 2017 Revised  April 2018 Published  November 2018

We are concerned with an inverse problem arising in thermal imaging in a bounded domain $Ω\subset \mathbb{R}^n$, $n=2, 3$. This inverse problem consists in the determination of the heat exchange coefficient $q(x)$ appearing in the boundary of a heat equation with Robin boundary condition.

Citation: Aymen Jbalia. On a logarithmic stability estimate for an inverse heat conduction problem. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2019014
References:
[1]

G. AlessandriniL. Del Piero and L. Rondi, Stable determination of corrosion by a single electrostatic boundary measurement, Inverse Probl., 19 (2003), 973-984.

[2]

G. Alessandrini and E. Sincich, Solving elliptic Cauchy problems and the identification of nonlinear corrosion, J. Comput. Appl. Math., 198 (2007), 307-320. doi: 10.1016/j.cam.2005.06.048.

[3]

M. BellassouedJ. Cheng and M. Choulli, Stability estimate for an inverse boundary coefficient problem in thermal imaging, J. Math Anal. Appl., 343 (2008), 328-336. doi: 10.1016/j.jmaa.2008.01.066.

[4]

M. BellassouedM. Choulli and A. Jbalia, Stability of the determination of the surface impedance of an obstacle from the scattering amplitude, Math. Meth. Appl. Sci., 36 (2013), 2429-2448.

[5]

L. Bourgeois, About stability and regularization of ill-posed elliptic Cauchy problems: the case of C1, 1 domains, Math. Model. Numer. Anal., 44 (2010), 715-735. doi: 10.1051/m2an/2010016.

[6]

K. Bryan and Jr. L. F. Caudill, An inverse problem in thermal imaging, SIAM J. Appl. Math., 56 (1996), 715-735. doi: 10.1137/S0036139994277828.

[7]

K. Bryan and Jr. L. F. Caudill, Uniqueness for a boundary identification problem in thermal imaging. in: J. Graef, R. Shivaji, B. Soni, Zhu (Eds. ), Differential Equations and Computational Simulations III, in: Electron. J. Differ. Equ. Conf., 1 (1998), 23-39.

[8]

S. Busenberg and W. Fang, Identification of semiconductor contact resistivity, Quar. J. Appl. Math., 49 (1991), 639-649. doi: 10.1090/qam/1134746.

[9]

S. ChaabaneI. FellahM. Jaoua and J. Leblond, Logarithmic stability estimates for a robin coefficient in two-dimensional Laplace inverse problems, Inverse Probl., 20 (2004), 47-59.

[10]

S. ChaabaneI. Feki and N. Mars, Numerical reconstruction of a piecewise constant Robin parameter in the two- or three-dimensional case, Inverse Probl., 28 (2012), 065016.

[11]

S. Chaabane and M. Jaoua, Identification of Robin coefficient by means of boundary measurements, Inverse Probl., 15 (1999), 1425-1438. doi: 10.1088/0266-5611/15/6/303.

[12]

J. ChengM. Choulli and J. Lin, Stable determination of a boundary coefficient in an elliptic equation, Math Models Methods Appl Sci., 18 (2008), 107-123. doi: 10.1142/S0218202508002620.

[13]

J. ChengM. Choulli and X. Yang, An iterative BEM for the inverse problem of detecting corrosion in a pipe, Numer. Math. J. Chinese Univ., 14 (2005), 252-266.

[14]

M. Choulli and A. Jbalia, The problem of detecting corrosion by electric measurements revisited, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 643-650. doi: 10.3934/dcdss.2016018.

[15]

M. Choulli, An inverse problem in corrosion detection: Stability estimates, J. Inverse Ill-Posed Probl., 12 (2004), 349-367. doi: 10.1515/1569394042248247.

[16]

M. Choulli, Stability estimates for an inverse elliptic problem, J. Inverse Ill-Posed Probl., 10 (2002), 601-610. doi: 10.1515/jiip.2002.10.6.601.

[17]

W. Fang and E. Cumberbatch, Inverse problems for metal oxide semiconductor field-effect transistor contact resistivity, SIAM J. Appl. Math., 52 (1992), 699-709. doi: 10.1137/0152039.

[18]

W. Fang and M. Lu, A fast collocation method for an inverse boundary value problem, Int. J. Numer. Methods Eng., 59 (2004), 1563-1585. doi: 10.1002/nme.928.

[19]

D. Fasino and G. Inglese, Stability of the solutions of an inverse problem for Laplace's equation in a thin strip, Numer. Func. Anal. Opt., 22 (2001), 549-560. doi: 10.1081/NFA-100105307.

[20]

D. Fujiwara, Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order, Proc.Japan Acad., 43 (1967), 82-86. doi: 10.3792/pja/1195521686.

[21]

P. Germain, Thèse de doctorat: Solutions fortes, solutions faibles d'équations aux dérivées partielles d'évolution, Ecole polytechnique France, 2005.

[22]

L. Hörmander, The Analysis of Partial Differential Operators, 2, 2d ed: Springer-Verlag, Berlin, 1990.

[23]

G. Inglese, An inverse problem in corrosion detection, Inverse Probl., 13 (1977), 977-994. doi: 10.1088/0266-5611/13/4/006.

[24]

M. Jaoua, S. Chaabane, C. Elhechmi, J. Leblond, M. Mahjoub and J. R. Partington, On some robust algorithms for the Robin inverse problem. International conference in honor of Claude Lobry, 2007.

[25]

B. Jin and X. Lu, Numerical identification for a Robin coefficient in parabolic problems, Math. Comp., 81 (2012), 1369-1398. doi: 10.1090/S0025-5718-2012-02559-2.

[26]

B. Jin and J. Zou, Numerical estimation of the Robin coefficient in a stationary diffusion equation, IMA J. Numer. Anal., 30 (2010), 677-701. doi: 10.1093/imanum/drn066.

[27]

B. Jin and J. Zou, Numerical estimation of piecewise constant Robin coefficient, SIAM J. Control Optim., 48 (2009), 1977-2002. doi: 10.1137/070710846.

[28]

P. G. KaupF. Santosa and M. Vogelius, Method for imaging corrosion damage in thin plates from electrostatic data, Inverse Probl., 12 (1996), 279-293.

[29]

F. Lin and W. Fang, A linear integral equation approach to the Robin inverse problem, Inverse Probl., 21 (2005), 1757-1772. doi: 10.1088/0266-5611/21/5/015.

[30]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[31]

M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, Springer-Verlag: New York; 1993.

[32]

Z. SunY. JiaoB. Jin and X. Lu, Numerical identification of a sparse Robin coefficient, Adv. Comput. Math., 41 (2015), 131-148. doi: 10.1007/s10444-014-9352-5.

[33]

F. M. White, Heat and Mass Transfer: Addison-Wesley, Reading, MA, 1988.

[34]

Y. Xu and J. Zou, Analysis of an adaptive finite element method for recovering the Robin coefficient, SIAM J. Control Optimiz., 53 (2015), 622-644. doi: 10.1137/130941742.

[35]

F. YangL. Yan and T. Wei, The identification of a Robin coefficient by a conjugate gradient method, Int. J. Numer. Meth. Engng., 78 (2009), 800-816. doi: 10.1002/nme.2507.

show all references

References:
[1]

G. AlessandriniL. Del Piero and L. Rondi, Stable determination of corrosion by a single electrostatic boundary measurement, Inverse Probl., 19 (2003), 973-984.

[2]

G. Alessandrini and E. Sincich, Solving elliptic Cauchy problems and the identification of nonlinear corrosion, J. Comput. Appl. Math., 198 (2007), 307-320. doi: 10.1016/j.cam.2005.06.048.

[3]

M. BellassouedJ. Cheng and M. Choulli, Stability estimate for an inverse boundary coefficient problem in thermal imaging, J. Math Anal. Appl., 343 (2008), 328-336. doi: 10.1016/j.jmaa.2008.01.066.

[4]

M. BellassouedM. Choulli and A. Jbalia, Stability of the determination of the surface impedance of an obstacle from the scattering amplitude, Math. Meth. Appl. Sci., 36 (2013), 2429-2448.

[5]

L. Bourgeois, About stability and regularization of ill-posed elliptic Cauchy problems: the case of C1, 1 domains, Math. Model. Numer. Anal., 44 (2010), 715-735. doi: 10.1051/m2an/2010016.

[6]

K. Bryan and Jr. L. F. Caudill, An inverse problem in thermal imaging, SIAM J. Appl. Math., 56 (1996), 715-735. doi: 10.1137/S0036139994277828.

[7]

K. Bryan and Jr. L. F. Caudill, Uniqueness for a boundary identification problem in thermal imaging. in: J. Graef, R. Shivaji, B. Soni, Zhu (Eds. ), Differential Equations and Computational Simulations III, in: Electron. J. Differ. Equ. Conf., 1 (1998), 23-39.

[8]

S. Busenberg and W. Fang, Identification of semiconductor contact resistivity, Quar. J. Appl. Math., 49 (1991), 639-649. doi: 10.1090/qam/1134746.

[9]

S. ChaabaneI. FellahM. Jaoua and J. Leblond, Logarithmic stability estimates for a robin coefficient in two-dimensional Laplace inverse problems, Inverse Probl., 20 (2004), 47-59.

[10]

S. ChaabaneI. Feki and N. Mars, Numerical reconstruction of a piecewise constant Robin parameter in the two- or three-dimensional case, Inverse Probl., 28 (2012), 065016.

[11]

S. Chaabane and M. Jaoua, Identification of Robin coefficient by means of boundary measurements, Inverse Probl., 15 (1999), 1425-1438. doi: 10.1088/0266-5611/15/6/303.

[12]

J. ChengM. Choulli and J. Lin, Stable determination of a boundary coefficient in an elliptic equation, Math Models Methods Appl Sci., 18 (2008), 107-123. doi: 10.1142/S0218202508002620.

[13]

J. ChengM. Choulli and X. Yang, An iterative BEM for the inverse problem of detecting corrosion in a pipe, Numer. Math. J. Chinese Univ., 14 (2005), 252-266.

[14]

M. Choulli and A. Jbalia, The problem of detecting corrosion by electric measurements revisited, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 643-650. doi: 10.3934/dcdss.2016018.

[15]

M. Choulli, An inverse problem in corrosion detection: Stability estimates, J. Inverse Ill-Posed Probl., 12 (2004), 349-367. doi: 10.1515/1569394042248247.

[16]

M. Choulli, Stability estimates for an inverse elliptic problem, J. Inverse Ill-Posed Probl., 10 (2002), 601-610. doi: 10.1515/jiip.2002.10.6.601.

[17]

W. Fang and E. Cumberbatch, Inverse problems for metal oxide semiconductor field-effect transistor contact resistivity, SIAM J. Appl. Math., 52 (1992), 699-709. doi: 10.1137/0152039.

[18]

W. Fang and M. Lu, A fast collocation method for an inverse boundary value problem, Int. J. Numer. Methods Eng., 59 (2004), 1563-1585. doi: 10.1002/nme.928.

[19]

D. Fasino and G. Inglese, Stability of the solutions of an inverse problem for Laplace's equation in a thin strip, Numer. Func. Anal. Opt., 22 (2001), 549-560. doi: 10.1081/NFA-100105307.

[20]

D. Fujiwara, Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order, Proc.Japan Acad., 43 (1967), 82-86. doi: 10.3792/pja/1195521686.

[21]

P. Germain, Thèse de doctorat: Solutions fortes, solutions faibles d'équations aux dérivées partielles d'évolution, Ecole polytechnique France, 2005.

[22]

L. Hörmander, The Analysis of Partial Differential Operators, 2, 2d ed: Springer-Verlag, Berlin, 1990.

[23]

G. Inglese, An inverse problem in corrosion detection, Inverse Probl., 13 (1977), 977-994. doi: 10.1088/0266-5611/13/4/006.

[24]

M. Jaoua, S. Chaabane, C. Elhechmi, J. Leblond, M. Mahjoub and J. R. Partington, On some robust algorithms for the Robin inverse problem. International conference in honor of Claude Lobry, 2007.

[25]

B. Jin and X. Lu, Numerical identification for a Robin coefficient in parabolic problems, Math. Comp., 81 (2012), 1369-1398. doi: 10.1090/S0025-5718-2012-02559-2.

[26]

B. Jin and J. Zou, Numerical estimation of the Robin coefficient in a stationary diffusion equation, IMA J. Numer. Anal., 30 (2010), 677-701. doi: 10.1093/imanum/drn066.

[27]

B. Jin and J. Zou, Numerical estimation of piecewise constant Robin coefficient, SIAM J. Control Optim., 48 (2009), 1977-2002. doi: 10.1137/070710846.

[28]

P. G. KaupF. Santosa and M. Vogelius, Method for imaging corrosion damage in thin plates from electrostatic data, Inverse Probl., 12 (1996), 279-293.

[29]

F. Lin and W. Fang, A linear integral equation approach to the Robin inverse problem, Inverse Probl., 21 (2005), 1757-1772. doi: 10.1088/0266-5611/21/5/015.

[30]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[31]

M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, Springer-Verlag: New York; 1993.

[32]

Z. SunY. JiaoB. Jin and X. Lu, Numerical identification of a sparse Robin coefficient, Adv. Comput. Math., 41 (2015), 131-148. doi: 10.1007/s10444-014-9352-5.

[33]

F. M. White, Heat and Mass Transfer: Addison-Wesley, Reading, MA, 1988.

[34]

Y. Xu and J. Zou, Analysis of an adaptive finite element method for recovering the Robin coefficient, SIAM J. Control Optimiz., 53 (2015), 622-644. doi: 10.1137/130941742.

[35]

F. YangL. Yan and T. Wei, The identification of a Robin coefficient by a conjugate gradient method, Int. J. Numer. Meth. Engng., 78 (2009), 800-816. doi: 10.1002/nme.2507.

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