We establish a probe type reconstruction scheme for identifying an inclusion inside a heat conductive medium by nondestructive testing called thermography. For the one space dimension, this has been already achieved by Y. Daido, H. Kang and G. Nakamura. The present paper shows that their result can be generalized to higher space dimension.
@article{ASNSP_2010_5_9_4_725_0,
author = {Isakov, Victor and Kim, Kyoungsun and Nakamura, Gen},
title = {Reconstruction of an unknown inclusion by thermography},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
pages = {725--758},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 9},
number = {4},
year = {2010},
mrnumber = {2789473},
zbl = {1215.35169},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2010_5_9_4_725_0/}
}
TY - JOUR AU - Isakov, Victor AU - Kim, Kyoungsun AU - Nakamura, Gen TI - Reconstruction of an unknown inclusion by thermography JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2010 SP - 725 EP - 758 VL - 9 IS - 4 PB - Scuola Normale Superiore, Pisa UR - http://www.numdam.org/item/ASNSP_2010_5_9_4_725_0/ LA - en ID - ASNSP_2010_5_9_4_725_0 ER -
%0 Journal Article %A Isakov, Victor %A Kim, Kyoungsun %A Nakamura, Gen %T Reconstruction of an unknown inclusion by thermography %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2010 %P 725-758 %V 9 %N 4 %I Scuola Normale Superiore, Pisa %U http://www.numdam.org/item/ASNSP_2010_5_9_4_725_0/ %G en %F ASNSP_2010_5_9_4_725_0
Isakov, Victor; Kim, Kyoungsun; Nakamura, Gen. Reconstruction of an unknown inclusion by thermography. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 4, pp. 725-758. http://www.numdam.org/item/ASNSP_2010_5_9_4_725_0/
[1] and , Stable determination of an inclusion by boundary measurements, SIAM J. Math. Anal. 37 (2005), 200–217. | MR | Zbl
[2] , Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa, Cl. Sci. 22 (1968), 607–694. | EuDML | Numdam | MR | Zbl
[3] and , The significance of damage and defects and their detection in composite materials: A review, J. Strain Anal. Eng. Des. 27 (1992), 29–42.
[4] , and , A probe method for the inverse boundary value problem of non-stationary heat equations, Inverse Problems 23 (2007), 1787–1800. | MR | Zbl
[5] , and , Estimate of the fundamental solution for parabolic operators with discontinuous coefficients, preprint (http://arxiv.org).
[6] and , Stable determination of the discontinuous conductivity coefficient of a parabolic equation, preprint (http://arxiv.org). | MR | Zbl
[7] and , On uniqueness of recovery of the discontinuous conductivity coefficient of a parabolic equation, SIAM J. Math. Anal. 28 (1997), 49–59. | MR | Zbl
[8] , “Partial Differential Equations of Parabolic Type", Prentice-Hall, INC. 1964. | MR | Zbl
[9] , An asymptotic expansion for the heat equation, Arch. Ration. Mech. Anal. 41 (1971), 163–218. | MR | Zbl
[10] , “The Analysis of Linear Partial Differential Operator III", A series of comprehensive study in Mathematics, Springer-Verlag, 1985. | MR
[11] , Reconstruction of inclusion from boundary measurements, J. Inverse Ill-Posed Probl. 10 (2002) 37–65. | MR | Zbl
[12] , “Inverse Problems for Partial Differential Equations", Applied Mathematical Sciences, Vol. 127, Springer 1988.
[13] , “Three-dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity", North-Holland series in Applied Mathematics and Mechanics, Vol. 25, North-Holland Publishing Company, 1979. | MR
[14] , “Second Order Parabolic Differential Equations", World Scientific Publishing Co., Inc., 1996. | MR | Zbl
[15] and , Estimates for elliptic systems from composite material, Comm. Pure Appl. Math., 56 (2003), 892–925. | MR | Zbl
[16] , and , A review of image analysis techniques applied in transient thermographic nondestructive testing, Nondestruct. Test. Eval. 6 (1992) 343–364.
[17] , and , Inverse method using infrared thermography for surface temperature and heat flux measurements, 20th International Congress on Instrumentation in Aerospace Simulation Facilities, 2003, ICIASF ’03, 118–126.
[18] , and , Numerical implementation of dynamical probe method, J. Comput. Math. 28 (2010), 87–104. | MR | Zbl
[19] , “Partial Differential Equations", Cambridge University Press, 1987. | MR | Zbl
