August  2012, 6(3): 423-446. doi: 10.3934/ipi.2012.6.423

Probing for inclusions in heat conductive bodies

1. 

Aix-Marseille University, France

2. 

University of Tsukuba, Japan

3. 

Aix-Marseille Université, France

4. 

University of Helsinki, Finland, Finland

Received  September 2010 Revised  May 2012 Published  September 2012

This work deals with an inverse boundary value problem arising from the equation of heat conduction. Mathematical theory and algorithm is described in dimensions 1--3 for probing the discontinuous part of the conductivity from local temperature and heat flow measurements at the boundary. The approach is based on the use of complex spherical waves, and no knowledge is needed about the initial temperature distribution. In dimension two we show how conformal transformations can be used for probing deeper than is possible with discs. Results from numerical experiments in the one-dimensional case are reported, suggesting that the method is capable of recovering locations of discontinuities approximately from noisy data.
Citation: Patricia Gaitan, Hiroshi Isozaki, Olivier Poisson, Samuli Siltanen, Janne Tamminen. Probing for inclusions in heat conductive bodies. Inverse Problems & Imaging, 2012, 6 (3) : 423-446. doi: 10.3934/ipi.2012.6.423
References:
[1]

K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane,, Ann. of Math, 163 (2006), 265.  doi: 10.4007/annals.2006.163.265.  Google Scholar

[2]

K. Bryan and L. F. Caudill Jr, Stability and reconstruction for an inverse problem for the heat equation,, Inverse Problems, 14 (1998), 1429.   Google Scholar

[3]

R. Chapko, R. Kress and J. R. Yoon, On the numerical solution of an inverse boundary value problem for the heat equation,, Inverse Problems, 14 (1998), 853.  doi: 10.1088/0266-5611/14/4/006.  Google Scholar

[4]

T. P. Fredman, A boundary identification method for an inverse heat conduction problem with an application in ironmaking,, Heat Mass Transfer, 41 (2004), 95.   Google Scholar

[5]

T. Ide, H. Isozaki, S. Nakata, S. Siltanen and G. Uhlmann, Probing for electrical inclusions with complex spherical waves,, Comm. in Pure and Appl. Math., 60 (2007), 1415.  doi: 10.1002/cpa.20194.  Google Scholar

[6]

M. Ikehata, Extracting discontinuity in a heat conductiong body. One-space dimensional case,, Appl. Anal., 86 (2007), 963.  doi: 10.1080/00036810701460834.  Google Scholar

[7]

M. Ikehata and M. Kawashita, An inverse problem for a three-dimensional heat equation in thermal imaging and the enclosure method,, \arXiv{1002.4457}., ().   Google Scholar

[8]

M. Ikehata and M. Kawashita, On the reconstruction of inclusions in a heat conductivity body from dynamical boundary data over a finite interval,, Inverse Problems, 26 (2010).  doi: 10.1088/0266-5611/26/9/095004.  Google Scholar

[9]

H. Kang, J. K. Seo and D. Sheen, The inverse conductivity problem with one measurement: stability and estimation of size,, SIAM J. Math. Anal., 28 (1997), 1389.  doi: 10.1137/S0036141096299375.  Google Scholar

[10]

N. S. Mera, The method of fundamental solutions for the backward heat conduction problem,, Inverse Probl. Sci. Eng., 13 (2005), 65.  doi: 10.1080/10682760410001710141.  Google Scholar

[11]

J. Sylvester and G. Uhlmann, A global uniquness theorem for an inverse boundary value problem,, Ann. of Math., 125 (1987), 153.  doi: 10.2307/1971291.  Google Scholar

[12]

S. Vessella, Quantitative estimates of unique continuation for parabolic equations, determination of time-varying boundaries and optimal stability estimates,, Topical Review, 24 (2008).   Google Scholar

[13]

T. Wei and Y. S. Li, An inverse boundary problem for one-dimensional heat equation with a multilayer domain,, Engineering Analysis with Boundary Elements, 33 (2009), 225.  doi: 10.1016/j.enganabound.2008.04.006.  Google Scholar

[14]

T. Wei and M. Yamamoto, Reconstruction of a moving boundary from Cauchy data in one-dimensional heat equation,, Inverse Problems in Science and Engineering, 17 (2009), 551.  doi: 10.1080/17415970802231610.  Google Scholar

[15]

T. Zhou, Reconstructing electromagnetic obstacles by the enclosure method,, Inverse Problems and Imaging, 4 (2010), 547.   Google Scholar

show all references

References:
[1]

K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane,, Ann. of Math, 163 (2006), 265.  doi: 10.4007/annals.2006.163.265.  Google Scholar

[2]

K. Bryan and L. F. Caudill Jr, Stability and reconstruction for an inverse problem for the heat equation,, Inverse Problems, 14 (1998), 1429.   Google Scholar

[3]

R. Chapko, R. Kress and J. R. Yoon, On the numerical solution of an inverse boundary value problem for the heat equation,, Inverse Problems, 14 (1998), 853.  doi: 10.1088/0266-5611/14/4/006.  Google Scholar

[4]

T. P. Fredman, A boundary identification method for an inverse heat conduction problem with an application in ironmaking,, Heat Mass Transfer, 41 (2004), 95.   Google Scholar

[5]

T. Ide, H. Isozaki, S. Nakata, S. Siltanen and G. Uhlmann, Probing for electrical inclusions with complex spherical waves,, Comm. in Pure and Appl. Math., 60 (2007), 1415.  doi: 10.1002/cpa.20194.  Google Scholar

[6]

M. Ikehata, Extracting discontinuity in a heat conductiong body. One-space dimensional case,, Appl. Anal., 86 (2007), 963.  doi: 10.1080/00036810701460834.  Google Scholar

[7]

M. Ikehata and M. Kawashita, An inverse problem for a three-dimensional heat equation in thermal imaging and the enclosure method,, \arXiv{1002.4457}., ().   Google Scholar

[8]

M. Ikehata and M. Kawashita, On the reconstruction of inclusions in a heat conductivity body from dynamical boundary data over a finite interval,, Inverse Problems, 26 (2010).  doi: 10.1088/0266-5611/26/9/095004.  Google Scholar

[9]

H. Kang, J. K. Seo and D. Sheen, The inverse conductivity problem with one measurement: stability and estimation of size,, SIAM J. Math. Anal., 28 (1997), 1389.  doi: 10.1137/S0036141096299375.  Google Scholar

[10]

N. S. Mera, The method of fundamental solutions for the backward heat conduction problem,, Inverse Probl. Sci. Eng., 13 (2005), 65.  doi: 10.1080/10682760410001710141.  Google Scholar

[11]

J. Sylvester and G. Uhlmann, A global uniquness theorem for an inverse boundary value problem,, Ann. of Math., 125 (1987), 153.  doi: 10.2307/1971291.  Google Scholar

[12]

S. Vessella, Quantitative estimates of unique continuation for parabolic equations, determination of time-varying boundaries and optimal stability estimates,, Topical Review, 24 (2008).   Google Scholar

[13]

T. Wei and Y. S. Li, An inverse boundary problem for one-dimensional heat equation with a multilayer domain,, Engineering Analysis with Boundary Elements, 33 (2009), 225.  doi: 10.1016/j.enganabound.2008.04.006.  Google Scholar

[14]

T. Wei and M. Yamamoto, Reconstruction of a moving boundary from Cauchy data in one-dimensional heat equation,, Inverse Problems in Science and Engineering, 17 (2009), 551.  doi: 10.1080/17415970802231610.  Google Scholar

[15]

T. Zhou, Reconstructing electromagnetic obstacles by the enclosure method,, Inverse Problems and Imaging, 4 (2010), 547.   Google Scholar

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