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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), 2069-2081. 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, Cambridge, 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), 309-322. 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 "Distributed Parameter Systems: Modelling and Control" (Warsaw, 1995), Control and Cybernetics, 25 (1996), 429-440.  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), 317-330. 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), 3-32; translation in St. Petersburg Math. J., 7 (1996), 869-890.  Google Scholar

[7]

A. Boumenir, The recovery of analytic potentials, Inverse Problems, 15 (1999), 1405-1423. 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), 3911-3921. 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), 155-163. 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), 75-88.  Google Scholar

[11]

V. Isakov, On uniqueness in inverse problems for semilinear parabolic equations, Arch. Rational Mech. Anal., 124 (1993), 1-12. doi: 10.1007/BF00392201.  Google Scholar

[12]

V. Isakov, "Inverse Problems for Partial Differential Equations," Second edition, Applied Mathematical Sciences, 127, Springer, New York, 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, Chapman & Hall/CRC, Boca Raton, FL, 2001.  Google Scholar

[14]

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

[15]

B. M. Levitan and M. G. Gasymov, Determination of a differential equation by two spectra, Uspehi Mat. Nauk, 19 (1964), 3-63.  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, Translations of Mathematical Monographs, Vol. 39, American Mathematical Society, Providence, R.I., 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," Series 2, Vol. 101, Six Papers in Analysis, American Mathematical Society, Providence, R.I., 1973. Google Scholar

[18]

J. R. McLaughlin, Analytical methods for recovering coefficients in differential equations from spectral data, SIAM Rev., 28 (1986), 53-72. doi: 10.1137/1028003.  Google Scholar

[19]

J. R. McLaughlin, Solving inverse problems with spectral data, in "Surveys on Solution Methods for Inverse Problems," 169-194, Springer, Vienna, 2000. 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), 31-50. 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), 482-504. 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, Speech, and Signal Processing, 38 (1990), 814-824. doi: 10.1109/29.56027.  Google Scholar

[23]

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

[24]

T. K. Sarkar, M. C. Wicks, M. Salazar-Palma and R. J. Bonneau, "Smart Antennas," John Wiley & Sons, Hoboken, New Jersey, 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), 2069-2081. 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, Cambridge, 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), 309-322. 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 "Distributed Parameter Systems: Modelling and Control" (Warsaw, 1995), Control and Cybernetics, 25 (1996), 429-440.  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), 317-330. 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), 3-32; translation in St. Petersburg Math. J., 7 (1996), 869-890.  Google Scholar

[7]

A. Boumenir, The recovery of analytic potentials, Inverse Problems, 15 (1999), 1405-1423. 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), 3911-3921. 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), 155-163. 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), 75-88.  Google Scholar

[11]

V. Isakov, On uniqueness in inverse problems for semilinear parabolic equations, Arch. Rational Mech. Anal., 124 (1993), 1-12. doi: 10.1007/BF00392201.  Google Scholar

[12]

V. Isakov, "Inverse Problems for Partial Differential Equations," Second edition, Applied Mathematical Sciences, 127, Springer, New York, 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, Chapman & Hall/CRC, Boca Raton, FL, 2001.  Google Scholar

[14]

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

[15]

B. M. Levitan and M. G. Gasymov, Determination of a differential equation by two spectra, Uspehi Mat. Nauk, 19 (1964), 3-63.  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, Translations of Mathematical Monographs, Vol. 39, American Mathematical Society, Providence, R.I., 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," Series 2, Vol. 101, Six Papers in Analysis, American Mathematical Society, Providence, R.I., 1973. Google Scholar

[18]

J. R. McLaughlin, Analytical methods for recovering coefficients in differential equations from spectral data, SIAM Rev., 28 (1986), 53-72. doi: 10.1137/1028003.  Google Scholar

[19]

J. R. McLaughlin, Solving inverse problems with spectral data, in "Surveys on Solution Methods for Inverse Problems," 169-194, Springer, Vienna, 2000. 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), 31-50. 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), 482-504. 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, Speech, and Signal Processing, 38 (1990), 814-824. doi: 10.1109/29.56027.  Google Scholar

[23]

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

[24]

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

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