Recovery of the heat coefficient by two measurements

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[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.
[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.
[7]	A. Boumenir, The recovery of analytic potentials, Inverse Problems, 15 (1999), 1405-1423. doi: 10.1088/0266-5611/15/6/302.
[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.
[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.
[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.
[11]	V. Isakov, On uniqueness in inverse problems for semilinear parabolic equations, Arch. Rational Mech. Anal., 124 (1993), 1-12. doi: 10.1007/BF00392201.
[12]	V. Isakov, "Inverse Problems for Partial Differential Equations," Second edition, Applied Mathematical Sciences, 127, Springer, New York, 2006.
[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.
[14]	B. M. Levitan, "Inverse Sturm-Liouville Problems," Translated from the Russian by O. Efimov, VSP, Zeist, 1987.
[15]	B. M. Levitan and M. G. Gasymov, Determination of a differential equation by two spectra, Uspehi Mat. Nauk, 19 (1964), 3-63.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[23]	Y. Hua, A. B. Gershman and Q. Cheng, "High-Resolution and Robust Signal Processing," Marcel Dekker, New York-Basel, 2004.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[7]	A. Boumenir, The recovery of analytic potentials, Inverse Problems, 15 (1999), 1405-1423. doi: 10.1088/0266-5611/15/6/302.
[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.
[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.
[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.
[11]	V. Isakov, On uniqueness in inverse problems for semilinear parabolic equations, Arch. Rational Mech. Anal., 124 (1993), 1-12. doi: 10.1007/BF00392201.
[12]	V. Isakov, "Inverse Problems for Partial Differential Equations," Second edition, Applied Mathematical Sciences, 127, Springer, New York, 2006.
[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.
[14]	B. M. Levitan, "Inverse Sturm-Liouville Problems," Translated from the Russian by O. Efimov, VSP, Zeist, 1987.
[15]	B. M. Levitan and M. G. Gasymov, Determination of a differential equation by two spectra, Uspehi Mat. Nauk, 19 (1964), 3-63.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[23]	Y. Hua, A. B. Gershman and Q. Cheng, "High-Resolution and Robust Signal Processing," Marcel Dekker, New York-Basel, 2004.
[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.

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