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 and 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), 265-299. doi: 10.4007/annals.2006.163.265.

[2]

K. Bryan and L. F. Caudill Jr, Stability and reconstruction for an inverse problem for the heat equation, Inverse Problems, 14 (1998), 1429-1453.

[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), 853-867. doi: 10.1088/0266-5611/14/4/006.

[4]

T. P. Fredman, A boundary identification method for an inverse heat conduction problem with an application in ironmaking, Heat Mass Transfer, 41 (2004), 95-103.

[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), 1415-1442. doi: 10.1002/cpa.20194.

[6]

M. Ikehata, Extracting discontinuity in a heat conductiong body. One-space dimensional case, Appl. Anal., 86 (2007), 963-1005. doi: 10.1080/00036810701460834.

[7]

M. Ikehata and M. Kawashita, An inverse problem for a three-dimensional heat equation in thermal imaging and the enclosure methodarXiv:1002.4457.

[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), 15pp. doi: 10.1088/0266-5611/26/9/095004.

[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), 1389-1405. doi: 10.1137/S0036141096299375.

[10]

N. S. Mera, The method of fundamental solutions for the backward heat conduction problem, Inverse Probl. Sci. Eng., 13 (2005), 65-78. doi: 10.1080/10682760410001710141.

[11]

J. Sylvester and G. Uhlmann, A global uniquness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169. doi: 10.2307/1971291.

[12]

S. Vessella, Quantitative estimates of unique continuation for parabolic equations, determination of time-varying boundaries and optimal stability estimates, Topical Review, Inverse Problems, 24 (2008), 81pp.

[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), 225-232. doi: 10.1016/j.enganabound.2008.04.006.

[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), 551-567. doi: 10.1080/17415970802231610.

[15]

T. Zhou, Reconstructing electromagnetic obstacles by the enclosure method, Inverse Problems and Imaging, 4 (2010), 547-569.

show all references

References:
[1]

K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane, Ann. of Math, 163 (2006), 265-299. doi: 10.4007/annals.2006.163.265.

[2]

K. Bryan and L. F. Caudill Jr, Stability and reconstruction for an inverse problem for the heat equation, Inverse Problems, 14 (1998), 1429-1453.

[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), 853-867. doi: 10.1088/0266-5611/14/4/006.

[4]

T. P. Fredman, A boundary identification method for an inverse heat conduction problem with an application in ironmaking, Heat Mass Transfer, 41 (2004), 95-103.

[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), 1415-1442. doi: 10.1002/cpa.20194.

[6]

M. Ikehata, Extracting discontinuity in a heat conductiong body. One-space dimensional case, Appl. Anal., 86 (2007), 963-1005. doi: 10.1080/00036810701460834.

[7]

M. Ikehata and M. Kawashita, An inverse problem for a three-dimensional heat equation in thermal imaging and the enclosure methodarXiv:1002.4457.

[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), 15pp. doi: 10.1088/0266-5611/26/9/095004.

[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), 1389-1405. doi: 10.1137/S0036141096299375.

[10]

N. S. Mera, The method of fundamental solutions for the backward heat conduction problem, Inverse Probl. Sci. Eng., 13 (2005), 65-78. doi: 10.1080/10682760410001710141.

[11]

J. Sylvester and G. Uhlmann, A global uniquness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169. doi: 10.2307/1971291.

[12]

S. Vessella, Quantitative estimates of unique continuation for parabolic equations, determination of time-varying boundaries and optimal stability estimates, Topical Review, Inverse Problems, 24 (2008), 81pp.

[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), 225-232. doi: 10.1016/j.enganabound.2008.04.006.

[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), 551-567. doi: 10.1080/17415970802231610.

[15]

T. Zhou, Reconstructing electromagnetic obstacles by the enclosure method, Inverse Problems and Imaging, 4 (2010), 547-569.

[1]

Sergei A. Avdonin, Sergei A. Ivanov, Jun-Min Wang. Inverse problems for the heat equation with memory. Inverse Problems and Imaging, 2019, 13 (1) : 31-38. doi: 10.3934/ipi.2019002

[2]

Shuli Chen, Zewen Wang, Guolin Chen. Cauchy problem of non-homogenous stochastic heat equation and application to inverse random source problem. Inverse Problems and Imaging, 2021, 15 (4) : 619-639. doi: 10.3934/ipi.2021008

[3]

Masaru Ikehata, Mishio Kawashita. An inverse problem for a three-dimensional heat equation in thermal imaging and the enclosure method. Inverse Problems and Imaging, 2014, 8 (4) : 1073-1116. doi: 10.3934/ipi.2014.8.1073

[4]

Aymen Jbalia. On a logarithmic stability estimate for an inverse heat conduction problem. Mathematical Control and Related Fields, 2019, 9 (2) : 277-287. doi: 10.3934/mcrf.2019014

[5]

Nguyen Huy Tuan, Mokhtar Kirane, Long Dinh Le, Van Thinh Nguyen. On an inverse problem for fractional evolution equation. Evolution Equations and Control Theory, 2017, 6 (1) : 111-134. doi: 10.3934/eect.2017007

[6]

Kazuhiro Ishige, Asato Mukai. Large time behavior of solutions of the heat equation with inverse square potential. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 4041-4069. doi: 10.3934/dcds.2018176

[7]

Arturo de Pablo, Guillermo Reyes, Ariel Sánchez. The Cauchy problem for a nonhomogeneous heat equation with reaction. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 643-662. doi: 10.3934/dcds.2013.33.643

[8]

Xiangfeng Yang, Yaodong Ni. Extreme values problem of uncertain heat equation. Journal of Industrial and Management Optimization, 2019, 15 (4) : 1995-2008. doi: 10.3934/jimo.2018133

[9]

Tianyu Yang, Yang Yang. A stable non-iterative reconstruction algorithm for the acoustic inverse boundary value problem. Inverse Problems and Imaging, 2022, 16 (1) : 1-18. doi: 10.3934/ipi.2021038

[10]

Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems and Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121

[11]

Xiaoli Feng, Meixia Zhao, Peijun Li, Xu Wang. An inverse source problem for the stochastic wave equation. Inverse Problems and Imaging, 2022, 16 (2) : 397-415. doi: 10.3934/ipi.2021055

[12]

Ahmad Z. Fino, Mokhtar Kirane. The Cauchy problem for heat equation with fractional Laplacian and exponential nonlinearity. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3625-3650. doi: 10.3934/cpaa.2020160

[13]

Umberto Biccari. Boundary controllability for a one-dimensional heat equation with a singular inverse-square potential. Mathematical Control and Related Fields, 2019, 9 (1) : 191-219. doi: 10.3934/mcrf.2019011

[14]

Lacramioara Grecu, Constantin Popa. Constrained SART algorithm for inverse problems in image reconstruction. Inverse Problems and Imaging, 2013, 7 (1) : 199-216. doi: 10.3934/ipi.2013.7.199

[15]

Tiexiang Li, Tsung-Ming Huang, Wen-Wei Lin, Jenn-Nan Wang. On the transmission eigenvalue problem for the acoustic equation with a negative index of refraction and a practical numerical reconstruction method. Inverse Problems and Imaging, 2018, 12 (4) : 1033-1054. doi: 10.3934/ipi.2018043

[16]

Michael V. Klibanov. A phaseless inverse scattering problem for the 3-D Helmholtz equation. Inverse Problems and Imaging, 2017, 11 (2) : 263-276. doi: 10.3934/ipi.2017013

[17]

Kenichi Sakamoto, Masahiro Yamamoto. Inverse source problem with a final overdetermination for a fractional diffusion equation. Mathematical Control and Related Fields, 2011, 1 (4) : 509-518. doi: 10.3934/mcrf.2011.1.509

[18]

Suman Kumar Sahoo, Manmohan Vashisth. A partial data inverse problem for the convection-diffusion equation. Inverse Problems and Imaging, 2020, 14 (1) : 53-75. doi: 10.3934/ipi.2019063

[19]

Daniel G. Alfaro Vigo, Amaury C. Álvarez, Grigori Chapiro, Galina C. García, Carlos G. Moreira. Solving the inverse problem for an ordinary differential equation using conjugation. Journal of Computational Dynamics, 2020, 7 (2) : 183-208. doi: 10.3934/jcd.2020008

[20]

John C. Schotland, Vadim A. Markel. Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation. Inverse Problems and Imaging, 2007, 1 (1) : 181-188. doi: 10.3934/ipi.2007.1.181

2021 Impact Factor: 1.483

Metrics

  • PDF downloads (74)
  • HTML views (0)
  • Cited by (10)

[Back to Top]