An inverse problem for the heat equation

Boumenir, Amin; Tuan, Vu

doi:10.1090/S0002-9939-2010-10297-6

An inverse problem for the heat equation
HTML articles powered by AMS MathViewer

by Amin Boumenir and Vu Kim Tuan

Proc. Amer. Math. Soc. 138 (2010), 3911-3921

DOI: https://doi.org/10.1090/S0002-9939-2010-10297-6

Published electronically: July 1, 2010

PDF | Request permission

Abstract:

We prove that we can uniquely recover the coefficient of a one dimensional heat equation from a finite set of measurements and provide a constructive procedure for its recovery. The algorithm is based on the well known Gelfand-Levitan-Gasymov inverse spectral theory of Sturm-Liouville operators. By using a hot spot, as a first initial condition, we determine nearly all except maybe a finite number of spectral data. A counting procedure helps detect the number of missing data which is then unraveled by a finite number of measurements.

References

Alan L. Andrew, Computing Sturm-Liouville potentials from two spectra, Inverse Problems 22 (2006), no. 6, 2069–2081. MR 2277530, DOI 10.1088/0266-5611/22/6/010
Alan L. Andrew, Numerov’s method for inverse Sturm-Liouville problems, Inverse Problems 21 (2005), no. 1, 223–238. MR 2146173, DOI 10.1088/0266-5611/21/1/014
S. A. Avdonin and S. A. Ivanov, Riesz bases of exponentials and divided differences, Algebra i Analiz 13 (2001), no. 3, 1–17 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 13 (2002), no. 3, 339–351. MR 1850184
Sergei A. Avdonin and Sergei A. Ivanov, Families of exponentials, Cambridge University Press, Cambridge, 1995. The method of moments in controllability problems for distributed parameter systems; Translated from the Russian and revised by the authors. MR 1366650
S. A. Avdonin, M. I. Belishev, and Yu. S. Rozhkov, The BC-method in the inverse problem for the heat equation, J. Inverse Ill-Posed Probl. 5 (1997), no. 4, 309–322. MR 1473633, DOI 10.1515/jiip.1997.5.4.309
Sergei Avdonin and Mikhail Belishev, Boundary control and dynamical inverse problem for nonselfadjoint Sturm-Liouville operator (BC-method), Control Cybernet. 25 (1996), no. 3, 429–440. Distributed parameter systems: modelling and control (Warsaw, 1995). MR 1408711
S. Avdonin, S. Lenhart, and V. Protopopescu, Determining the potential in the Schrödinger equation from the Dirichlet to Neumann map by the boundary control method, J. Inverse Ill-Posed Probl. 13 (2005), no. 3-6, 317–330. Inverse problems: modeling and simulation. MR 2188615, DOI 10.1163/156939405775201718
M. I. Belishev, A canonical model of a dynamical system with boundary control in the inverse heat conduction problem, Algebra i Analiz 7 (1995), no. 6, 3–32 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 7 (1996), no. 6, 869–890. MR 1381977
Amin Boumenir, The recovery of analytic potentials, Inverse Problems 15 (1999), no. 6, 1405–1423. MR 1733208, DOI 10.1088/0266-5611/15/6/302
Richard H. Fabiano, Roger Knobel, and Bruce D. Lowe, A finite-difference algorithm for an inverse Sturm-Liouville problem, IMA J. Numer. Anal. 15 (1995), no. 1, 75–88. MR 1311338, DOI 10.1093/imanum/15.1.75
G. Freiling and V. Yurko, Inverse Sturm-Liouville problems and their applications, Nova Science Publishers, Inc., Huntington, NY, 2001. MR 2094651
V. Isakov, On uniqueness in inverse problems for semilinear parabolic equations, Arch. Rational Mech. Anal. 124 (1993), no. 1, 1–12. MR 1233645, DOI 10.1007/BF00392201
Victor Isakov, Inverse problems for partial differential equations, 2nd ed., Applied Mathematical Sciences, vol. 127, Springer, New York, 2006. MR 2193218
B. M. Levitan, Inverse Sturm-Liouville problems, VSP, Zeist, 1987. Translated from the Russian by O. Efimov. MR 933088
B. M. Levitan and M. G. Gasymov, Determination of a differential equation by two spectra, Uspehi Mat. Nauk 19 (1964), no. 2 (116), 3–63 (Russian). MR 0162996
Joyce R. McLaughlin, Analytical methods for recovering coefficients in differential equations from spectral data, SIAM Rev. 28 (1986), no. 1, 53–72. MR 828436, DOI 10.1137/1028003
Joyce R. McLaughlin, Solving inverse problems with spectral data, Surveys on solution methods for inverse problems, Springer, Vienna, 2000, pp. 169–194. MR 1766744
Bruce D. Lowe and William Rundell, The determination of a coefficient in a parabolic equation from input sources, IMA J. Appl. Math. 52 (1994), no. 1, 31–50. MR 1270801, DOI 10.1093/imamat/52.1.31
Bruce D. Lowe, Michael Pilant, and William Rundell, The recovery of potentials from finite spectral data, SIAM J. Math. Anal. 23 (1992), no. 2, 482–504. MR 1147873, DOI 10.1137/0523023
Yingbo Hua and Tapan K. Sarkar, Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise, IEEE Trans. Acoust. Speech Signal Process. 38 (1990), no. 5, 814–824. MR 1051029, DOI 10.1109/29.56027
Y. Hua, A.B. Gershman, and Q. Cheng, High-Resolution and Robust Signal Processing. Marcel Dekker, New York–Basel, 2004.
T.K. Sarkar, M.C. Wicks, M. Salazar-Palma, and R.J. Bonneau, Smart Antennas. John Wiley & Sons, Hoboken, New Jersey, 2003.