American Institute of Mathematical Sciences

Advanced
November  2011, 5(4): 775-791. doi: 10.3934/ipi.2011.5.775

Recovery of the heat coefficient by two measurements

Amin Boumenir 1, and  Vu Kim Tuan 2,

1. 

Department of Mathematics, Kuwait University, Safat -13060, Kuwait
2. 

Department of Mathematics, University of West Georgia, Carrollton, GA 30118, United States

Received  March 2010 Revised  August 2011 Published  November 2011

We prove that it takes at most two measurements on the boundary to recover the heat coefficient of a one dimensional heat equation if its lower bound is known. Otherwise a finite number of measurements is needed. We also provide a new constructive algorithm for its recovery. Using asymptotics of eigenfunctions of the associated Sturm-Liouville problem we show that a hot spot initial condition generates all, except maybe a finite number of boundary spectral data. Then a counting argument based on the method of false position helps search for the number of missing boundary spectral data which is then unraveled by a finite number of measurements. Finally, we show how the boundary spectral data is converted into spectral data, and the well known Gelfand-Levitan-Gasymov inverse spectral theory of Sturm-Liouville operators yields the reconstruction of the heat coefficient uniquely.
Keywords: Heat equation, boundary spectral data., inverse problem.
Mathematics Subject Classification: Primary: 35R30; Secondary: 34K2.
Citation: Amin Boumenir, Vu Kim Tuan. Recovery of the heat coefficient by two measurements. Inverse Problems & Imaging, 2011, 5 (4) : 775-791. doi: 10.3934/ipi.2011.5.775
References:
[1]

A. L. Andrew, Computing Sturm-Liouville potentials from two spectra,, Inverse Problems, 22 (2006), 2069.  doi: 10.1088/0266-5611/22/6/010.  Google Scholar

[2]

S. A. Avdonin and S. A. Ivanov, "Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems,", Cambridge University Press, (1995).   Google Scholar

[3]

S. A. Avdonin, M. I. Belishev and Yu. Rozhkov, The BC-method in the inverse problem for the heat equation,, J. Inv. Ill-Posed Probl., 5 (1997), 309.  doi: 10.1515/jiip.1997.5.4.309.  Google Scholar

[4]

S. A. Avdonin and M. I. Belishev, Boundary control and dynamical inverse problem for nonselfadjoint Sturm-Liouville operator (BC-method),, in, 25 (1996), 429.   Google Scholar

[5]

S. A. Avdonin, S. Lenhart and V. Protopopescu, Determining the potential in the Schrödinger equation from the Dirichlet to Neumann map by the boundary control method. Inverse problems: Modeling and simulation,, J. Inv. Ill-Posed Probl., 13 (2005), 317.  doi: 10.1515/156939405775201718.  Google Scholar

[6]

M. I. Belishev, A canonical model of a dynamical system with boundary control in the inverse heat conduction problem, (in Russian),, Algebra i Analiz, 7 (1995), 3.   Google Scholar

[7]

A. Boumenir, The recovery of analytic potentials,, Inverse Problems, 15 (1999), 1405.  doi: 10.1088/0266-5611/15/6/302.  Google Scholar

[8]

A. Boumenir and Vu Kim Tuan, An inverse problem for the heat equation,, Proc. Am. Math. Soc., 138 (2010), 3911.  doi: 10.1090/S0002-9939-2010-10297-6.  Google Scholar

[9]

A. Boumenir and Vu Kim Tuan, Recovery of a heat equation by four measurements at one end,, Numer. Funct. Anal. Optim., 31 (2010), 155.  doi: 10.1080/01630560903574993.  Google Scholar

[10]

R. H. Fabiano, R. Knobel and B. D. Lowe, A finite difference algorithm for an inverse Sturm-Liouville problem,, IMA J. Appl. Math., 15 (1995), 75.   Google Scholar

[11]

V. Isakov, On uniqueness in inverse problems for semilinear parabolic equations,, Arch. Rational Mech. Anal., 124 (1993), 1.  doi: 10.1007/BF00392201.  Google Scholar

[12]

V. Isakov, "Inverse Problems for Partial Differential Equations,", Second edition, 127 (2006).   Google Scholar

[13]

A. Katchalov, Y. Kurylev and M. Lassas, "Inverse Boundary Spectral Problems,", Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 123 (2001).   Google Scholar

[14]

B. M. Levitan, "Inverse Sturm-Liouville Problems,", Translated from the Russian by O. Efimov, (1987).   Google Scholar

[15]

B. M. Levitan and M. G. Gasymov, Determination of a differential equation by two spectra,, Uspehi Mat. Nauk, 19 (1964), 3.   Google Scholar

[16]

B. M. Levitan and I. S. Sargsjan, "Introduction to Spectral Theory: Selfadjoint Ordinary Differential Operators,", Translated from the Russian by Amiel Feinstein, (1975).   Google Scholar

[17]

V. A. Marčenko, Some questions in the theory of one-dimensional linear differential operators of the second order (1-104), in, American Mathematical Society Translations, (1973).   Google Scholar

[18]

J. R. McLaughlin, Analytical methods for recovering coefficients in differential equations from spectral data,, SIAM Rev., 28 (1986), 53.  doi: 10.1137/1028003.  Google Scholar

[19]

J. R. McLaughlin, Solving inverse problems with spectral data,, in, (2000), 169.  doi: 10.1007/978-3-7091-6296-5_10.  Google Scholar

[20]

B. D. Lowe and W. Rundell, The determination of a coefficient in a parabolic equation from input sources,, IMA J. Appl. Math., 52 (1994), 31.  doi: 10.1093/imamat/52.1.31.  Google Scholar

[21]

B. D. Lowe, M. Pilant and W. Rundell, The recovery of potentials from finite spectral data,, SIAM J. Math. Anal., 23 (1992), 482.  doi: 10.1137/0523023.  Google Scholar

[22]

Y. Hua and T. K. Sarkar, Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise,, IEEE Transactions of Acoustics, 38 (1990), 814.  doi: 10.1109/29.56027.  Google Scholar

[23]

Y. Hua, A. B. Gershman and Q. Cheng, "High-Resolution and Robust Signal Processing,", Marcel Dekker, (2004).   Google Scholar

[24]

T. K. Sarkar, M. C. Wicks, M. Salazar-Palma and R. J. Bonneau, "Smart Antennas,", John Wiley & Sons, (2003).  doi: 10.1002/0471722839.  Google Scholar

show all references

References:
[1]

A. L. Andrew, Computing Sturm-Liouville potentials from two spectra,, Inverse Problems, 22 (2006), 2069.  doi: 10.1088/0266-5611/22/6/010.  Google Scholar

[2]

S. A. Avdonin and S. A. Ivanov, "Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems,", Cambridge University Press, (1995).   Google Scholar

[3]

S. A. Avdonin, M. I. Belishev and Yu. Rozhkov, The BC-method in the inverse problem for the heat equation,, J. Inv. Ill-Posed Probl., 5 (1997), 309.  doi: 10.1515/jiip.1997.5.4.309.  Google Scholar

[4]

S. A. Avdonin and M. I. Belishev, Boundary control and dynamical inverse problem for nonselfadjoint Sturm-Liouville operator (BC-method),, in, 25 (1996), 429.   Google Scholar

[5]

S. A. Avdonin, S. Lenhart and V. Protopopescu, Determining the potential in the Schrödinger equation from the Dirichlet to Neumann map by the boundary control method. Inverse problems: Modeling and simulation,, J. Inv. Ill-Posed Probl., 13 (2005), 317.  doi: 10.1515/156939405775201718.  Google Scholar

[6]

M. I. Belishev, A canonical model of a dynamical system with boundary control in the inverse heat conduction problem, (in Russian),, Algebra i Analiz, 7 (1995), 3.   Google Scholar

[7]

A. Boumenir, The recovery of analytic potentials,, Inverse Problems, 15 (1999), 1405.  doi: 10.1088/0266-5611/15/6/302.  Google Scholar

[8]

A. Boumenir and Vu Kim Tuan, An inverse problem for the heat equation,, Proc. Am. Math. Soc., 138 (2010), 3911.  doi: 10.1090/S0002-9939-2010-10297-6.  Google Scholar

[9]

A. Boumenir and Vu Kim Tuan, Recovery of a heat equation by four measurements at one end,, Numer. Funct. Anal. Optim., 31 (2010), 155.  doi: 10.1080/01630560903574993.  Google Scholar

[10]

R. H. Fabiano, R. Knobel and B. D. Lowe, A finite difference algorithm for an inverse Sturm-Liouville problem,, IMA J. Appl. Math., 15 (1995), 75.   Google Scholar

[11]

V. Isakov, On uniqueness in inverse problems for semilinear parabolic equations,, Arch. Rational Mech. Anal., 124 (1993), 1.  doi: 10.1007/BF00392201.  Google Scholar

[12]

V. Isakov, "Inverse Problems for Partial Differential Equations,", Second edition, 127 (2006).   Google Scholar

[13]

A. Katchalov, Y. Kurylev and M. Lassas, "Inverse Boundary Spectral Problems,", Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 123 (2001).   Google Scholar

[14]

B. M. Levitan, "Inverse Sturm-Liouville Problems,", Translated from the Russian by O. Efimov, (1987).   Google Scholar

[15]

B. M. Levitan and M. G. Gasymov, Determination of a differential equation by two spectra,, Uspehi Mat. Nauk, 19 (1964), 3.   Google Scholar

[16]

B. M. Levitan and I. S. Sargsjan, "Introduction to Spectral Theory: Selfadjoint Ordinary Differential Operators,", Translated from the Russian by Amiel Feinstein, (1975).   Google Scholar

[17]

V. A. Marčenko, Some questions in the theory of one-dimensional linear differential operators of the second order (1-104), in, American Mathematical Society Translations, (1973).   Google Scholar

[18]

J. R. McLaughlin, Analytical methods for recovering coefficients in differential equations from spectral data,, SIAM Rev., 28 (1986), 53.  doi: 10.1137/1028003.  Google Scholar

[19]

J. R. McLaughlin, Solving inverse problems with spectral data,, in, (2000), 169.  doi: 10.1007/978-3-7091-6296-5_10.  Google Scholar

[20]

B. D. Lowe and W. Rundell, The determination of a coefficient in a parabolic equation from input sources,, IMA J. Appl. Math., 52 (1994), 31.  doi: 10.1093/imamat/52.1.31.  Google Scholar

[21]

B. D. Lowe, M. Pilant and W. Rundell, The recovery of potentials from finite spectral data,, SIAM J. Math. Anal., 23 (1992), 482.  doi: 10.1137/0523023.  Google Scholar

[22]

Y. Hua and T. K. Sarkar, Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise,, IEEE Transactions of Acoustics, 38 (1990), 814.  doi: 10.1109/29.56027.  Google Scholar

[23]

Y. Hua, A. B. Gershman and Q. Cheng, "High-Resolution and Robust Signal Processing,", Marcel Dekker, (2004).   Google Scholar

[24]

T. K. Sarkar, M. C. Wicks, M. Salazar-Palma and R. J. Bonneau, "Smart Antennas,", John Wiley & Sons, (2003).  doi: 10.1002/0471722839.  Google Scholar

[1]

Yavar Kian, Morgan Morancey, Lauri Oksanen. Application of the boundary control method to partial data Borg-Levinson inverse spectral problem. Mathematical Control & Related Fields, 2019, 9 (2) : 289-312. doi: 10.3934/mcrf.2019015
[2]

Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems & Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121
[3]

Li Liang. Increasing stability for the inverse problem of the Schrödinger equation with the partial Cauchy data. Inverse Problems & Imaging, 2015, 9 (2) : 469-478. doi: 10.3934/ipi.2015.9.469
[4]

Suman Kumar Sahoo, Manmohan Vashisth. A partial data inverse problem for the convection-diffusion equation. Inverse Problems & Imaging, 2020, 14 (1) : 53-75. doi: 10.3934/ipi.2019063
[5]

Sergei A. Avdonin, Sergei A. Ivanov, Jun-Min Wang. Inverse problems for the heat equation with memory. Inverse Problems & Imaging, 2019, 13 (1) : 31-38. doi: 10.3934/ipi.2019002
[6]

Umberto Biccari. Boundary controllability for a one-dimensional heat equation with a singular inverse-square potential. Mathematical Control & Related Fields, 2019, 9 (1) : 191-219. doi: 10.3934/mcrf.2019011
[7]

Sergei Avdonin, Fritz Gesztesy, Konstantin A. Makarov. Spectral estimation and inverse initial boundary value problems. Inverse Problems & Imaging, 2010, 4 (1) : 1-9. doi: 10.3934/ipi.2010.4.1
[8]

Masaru Ikehata, Mishio Kawashita. An inverse problem for a three-dimensional heat equation in thermal imaging and the enclosure method. Inverse Problems & Imaging, 2014, 8 (4) : 1073-1116. doi: 10.3934/ipi.2014.8.1073
[9]

Sombuddha Bhattacharyya. An inverse problem for the magnetic Schrödinger operator on Riemannian manifolds from partial boundary data. Inverse Problems & Imaging, 2018, 12 (3) : 801-830. doi: 10.3934/ipi.2018034
[10]

Aymen Jbalia. On a logarithmic stability estimate for an inverse heat conduction problem. Mathematical Control & Related Fields, 2019, 9 (2) : 277-287. doi: 10.3934/mcrf.2019014
[11]

Pedro Caro. On an inverse problem in electromagnetism with local data: stability and uniqueness. Inverse Problems & Imaging, 2011, 5 (2) : 297-322. doi: 10.3934/ipi.2011.5.297
[12]

Victor Isakov. On uniqueness in the inverse conductivity problem with local data. Inverse Problems & Imaging, 2007, 1 (1) : 95-105. doi: 10.3934/ipi.2007.1.95
[13]

Luciano Pandolfi. Riesz systems, spectral controllability and a source identification problem for heat equations with memory. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 745-759. doi: 10.3934/dcdss.2011.4.745
[14]

Alexander Gladkov. Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2053-2068. doi: 10.3934/cpaa.2017101
[15]

Alexei Rybkin. On the boundary control approach to inverse spectral and scattering theory for Schrödinger operators. Inverse Problems & Imaging, 2009, 3 (1) : 139-149. doi: 10.3934/ipi.2009.3.139
[16]

Nguyen Huy Tuan, Mokhtar Kirane, Long Dinh Le, Van Thinh Nguyen. On an inverse problem for fractional evolution equation. Evolution Equations & Control Theory, 2017, 6 (1) : 111-134. doi: 10.3934/eect.2017007
[17]

Zhousheng Ruan, Sen Zhang, Sican Xiong. Solving an inverse source problem for a time fractional diffusion equation by a modified quasi-boundary value method. Evolution Equations & Control Theory, 2018, 7 (4) : 669-682. doi: 10.3934/eect.2018032
[18]

Kazuhiro Ishige, Asato Mukai. Large time behavior of solutions of the heat equation with inverse square potential. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4041-4069. doi: 10.3934/dcds.2018176
[19]

Guangsheng Wei, Hong-Kun Xu. On the missing bound state data of inverse spectral-scattering problems on the half-line. Inverse Problems & Imaging, 2015, 9 (1) : 239-255. doi: 10.3934/ipi.2015.9.239
[20]

Mi-Ho Giga, Yoshikazu Giga, Takeshi Ohtsuka, Noriaki Umeda. On behavior of signs for the heat equation and a diffusion method for data separation. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2277-2296. doi: 10.3934/cpaa.2013.12.2277

2018 Impact Factor: 1.469

Tools

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences