An inverse problem for the heat equation

Authors:
Amin Boumenir and Vu Kim Tuan

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

MSC (2010):
Primary 35R30, 34K29

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

Published electronically:
July 1, 2010

MathSciNet review:
2679613

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

- Alan L. Andrew,
*Computing Sturm-Liouville potentials from two spectra*, Inverse Problems**22**(2006), no. 6, 2069–2081. MR**2277530**, DOI https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
35R30,
34K29

Retrieve articles in all journals with MSC (2010): 35R30, 34K29

Additional Information

**Amin Boumenir**

Affiliation:
Department of Mathematics, University of West Georgia, Carrollton, Georgia 30118

MR Author ID:
288615

Email:
boumenir@westga.edu

**Vu Kim Tuan**

Affiliation:
Department of Mathematics, University of West Georgia, Carrollton, Georgia 30118

Email:
vu@westga.edu

Keywords:
Heat equation,
inverse spectral problem

Received by editor(s):
October 5, 2007

Published electronically:
July 1, 2010

Communicated by:
Peter A. Clarkson

Article copyright:
© Copyright 2010
American Mathematical Society