# American Institute of Mathematical Sciences

June 2018, 7(2): 197-216. doi: 10.3934/eect.2018010

## The recovery of a parabolic equation from measurements at a single point

 Department of Mathematics, University of West Georgia, Carrollton, GA 30118, USA

* Corresponding author: boumenir@westga.edu

Received  December 2016 Revised  November 2017 Published  May 2018

By measuring the temperature at an arbitrary single point located inside an unknown object or on its boundary, we show how we can uniquely reconstruct all the coefficients appearing in a general parabolic equation which models its cooling. We also reconstruct the shape of the object. The proof hinges on the fact that we can detect infinitely many eigenfunctions whose Wronskian does not vanish. This allows us to evaluate these coefficients by solving a simple linear algebraic system. The geometry of the domain and its boundary are found by reconstructing the first eigenfunction.

Citation: Amin Boumenir, Vu Kim Tuan, Nguyen Hoang. The recovery of a parabolic equation from measurements at a single point. Evolution Equations & Control Theory, 2018, 7 (2) : 197-216. doi: 10.3934/eect.2018010
##### References:
 [1] Sh. A. Alimov, V. A. Il'in and E. M. Nikishin, Questions on the convergence of multiple trigonometric series and spectral expansions Ⅰ, Russian Mathematical Surveys, 31 (1976), 28-83. [2] H. Ammari, An Introduction to Mathematics of Emerging Biomedical Imaging, Mathématiques & Applications 62, Springer, Berlin, 2008. [3] S. A. Avdonin and A. Bulanova, Boundary control approach to the spectral estimation problem: The case of multiple poles, Math. Control Signals Systems, 22 (2011), 245-265. doi: 10.1007/s00498-010-0052-5. [4] S. A. Avdonin, F. Gesztesy and A. Makarov, Spectral estimation and inverse initial boundary value problems, Inverse Probl. and Imaging, 4 (2010), 1-9. doi: 10.3934/ipi.2010.4.1. [5] 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. [6] L. Beilina and M. V. Klibanov, Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems, Springer, New York, 2012. doi: 10.1007/978-1-4419-7805-9. [7] A. Bostan and P. Dumas, Wronskians and linear independence, Amer. Math. Monthly, 117 (2010), 722-727, arXiv: 1301.6598v1. doi: 10.4169/000298910x515785. [8] A. Boumenir and V. K. Tuan, Inverse problems for multidimensional heat equations by measurements at a single point on the boundary, Numer. Funct. Anal. Optim., 30 (2009), 1215-1230. doi: 10.1080/01630560903498979. [9] A. Boumenir and V. K. Tuan, An inverse problem for the wave equation, Journal of Inverse and Ill-posed Problems, 19 (2011), 273-592. doi: 10.1515/JIIP.2011.056. [10] A. S. Demidov and M. Moussaoui, An inverse problem originating from magnetohydrodynamics, Inverse Problems, 20 (2004), 137-154. doi: 10.1088/0266-5611/20/1/008. [11] H. O. Fattorini and D. L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension, Archive for Rational Mechanics and Analysis, 43 (1971), 272-292. doi: 10.1007/BF00250466. [12] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. [13] A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics, Birkhäuser, Basel, 2006. [14] V. A. Il'in and Sh. A. Alimov, Conditions for the convergence of spectral decompositions that correspond to self-adjoint extentions of elliptic operators, Ⅴ, Differential Equations, 10 (1974), 360-377. [15] V. Isakov, Inverse Problems for Partial Differential Equations, 2$^{nd}$ Ed, Applied Mathematical Sciences 127, Springer, New York, 2006. [16] M. V. Klibanov and A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, VSP, Utrecht, The Netherlands, 2004. doi: 10.1515/9783110915549. [17] V. Mikhailov, Equations aux Derivees Partielles, French transl., Mir, Moscow, 1980. [18] K. Wolsson, Linear dependence of a function set of $m$ variables with vanishing generalized Wronskians, Linear Algebra and its Applications, 117 (1989), 73-80. doi: 10.1016/0024-3795(89)90548-X. [19] Z. Wu, J. Yin and C. Wang, Elliptic and Parabolic Equations, World Scientific, Singapore, 2006. doi: 10.1142/6238. [20] L. V. Zhizhiashshvilli, Some problems in the theory of simple and multiple trigonometric series, Russian Mathematical Surveys, 28 (1973), 62-119. [21] E. Zuazua, Controllability and observability of partial differential equations: Some results and open problems, in Handbook of Differential Equations: Evolutionary Equations, Vol. Ⅲ, Elsevier/North-Holland, Amsterdam, 3 (2007), 527-621. doi: 10.1016/S1874-5717(07)80010-7.

show all references

##### References:
 [1] Sh. A. Alimov, V. A. Il'in and E. M. Nikishin, Questions on the convergence of multiple trigonometric series and spectral expansions Ⅰ, Russian Mathematical Surveys, 31 (1976), 28-83. [2] H. Ammari, An Introduction to Mathematics of Emerging Biomedical Imaging, Mathématiques & Applications 62, Springer, Berlin, 2008. [3] S. A. Avdonin and A. Bulanova, Boundary control approach to the spectral estimation problem: The case of multiple poles, Math. Control Signals Systems, 22 (2011), 245-265. doi: 10.1007/s00498-010-0052-5. [4] S. A. Avdonin, F. Gesztesy and A. Makarov, Spectral estimation and inverse initial boundary value problems, Inverse Probl. and Imaging, 4 (2010), 1-9. doi: 10.3934/ipi.2010.4.1. [5] 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. [6] L. Beilina and M. V. Klibanov, Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems, Springer, New York, 2012. doi: 10.1007/978-1-4419-7805-9. [7] A. Bostan and P. Dumas, Wronskians and linear independence, Amer. Math. Monthly, 117 (2010), 722-727, arXiv: 1301.6598v1. doi: 10.4169/000298910x515785. [8] A. Boumenir and V. K. Tuan, Inverse problems for multidimensional heat equations by measurements at a single point on the boundary, Numer. Funct. Anal. Optim., 30 (2009), 1215-1230. doi: 10.1080/01630560903498979. [9] A. Boumenir and V. K. Tuan, An inverse problem for the wave equation, Journal of Inverse and Ill-posed Problems, 19 (2011), 273-592. doi: 10.1515/JIIP.2011.056. [10] A. S. Demidov and M. Moussaoui, An inverse problem originating from magnetohydrodynamics, Inverse Problems, 20 (2004), 137-154. doi: 10.1088/0266-5611/20/1/008. [11] H. O. Fattorini and D. L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension, Archive for Rational Mechanics and Analysis, 43 (1971), 272-292. doi: 10.1007/BF00250466. [12] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. [13] A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics, Birkhäuser, Basel, 2006. [14] V. A. Il'in and Sh. A. Alimov, Conditions for the convergence of spectral decompositions that correspond to self-adjoint extentions of elliptic operators, Ⅴ, Differential Equations, 10 (1974), 360-377. [15] V. Isakov, Inverse Problems for Partial Differential Equations, 2$^{nd}$ Ed, Applied Mathematical Sciences 127, Springer, New York, 2006. [16] M. V. Klibanov and A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, VSP, Utrecht, The Netherlands, 2004. doi: 10.1515/9783110915549. [17] V. Mikhailov, Equations aux Derivees Partielles, French transl., Mir, Moscow, 1980. [18] K. Wolsson, Linear dependence of a function set of $m$ variables with vanishing generalized Wronskians, Linear Algebra and its Applications, 117 (1989), 73-80. doi: 10.1016/0024-3795(89)90548-X. [19] Z. Wu, J. Yin and C. Wang, Elliptic and Parabolic Equations, World Scientific, Singapore, 2006. doi: 10.1142/6238. [20] L. V. Zhizhiashshvilli, Some problems in the theory of simple and multiple trigonometric series, Russian Mathematical Surveys, 28 (1973), 62-119. [21] E. Zuazua, Controllability and observability of partial differential equations: Some results and open problems, in Handbook of Differential Equations: Evolutionary Equations, Vol. Ⅲ, Elsevier/North-Holland, Amsterdam, 3 (2007), 527-621. doi: 10.1016/S1874-5717(07)80010-7.
${{\varphi }_{1}}$
$\Omega$
${{\varphi }_{2}}$
${{\varphi }_{12}}$
 [1] Michael V. Klibanov, Dinh-Liem Nguyen, Loc H. Nguyen, Hui Liu. A globally convergent numerical method for a 3D coefficient inverse problem with a single measurement of multi-frequency data. Inverse Problems & Imaging, 2018, 12 (2) : 493-523. doi: 10.3934/ipi.2018021 [2] Michel Cristofol, Shumin Li, Eric Soccorsi. Determining the waveguide conductivity in a hyperbolic equation from a single measurement on the lateral boundary. Mathematical Control & Related Fields, 2016, 6 (3) : 407-427. doi: 10.3934/mcrf.2016009 [3] Shitao Liu. Recovery of the sound speed and initial displacement for the wave equation by means of a single Dirichlet boundary measurement. Evolution Equations & Control Theory, 2013, 2 (2) : 355-364. doi: 10.3934/eect.2013.2.355 [4] Eva Sincich, Mourad Sini. Local stability for soft obstacles by a single measurement. Inverse Problems & Imaging, 2008, 2 (2) : 301-315. doi: 10.3934/ipi.2008.2.301 [5] Mohsen Tadi. A computational method for an inverse problem in a parabolic system. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 205-218. doi: 10.3934/dcdsb.2009.12.205 [6] 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 [7] Sebastian Acosta. Recovery of the absorption coefficient in radiative transport from a single measurement. Inverse Problems & Imaging, 2015, 9 (2) : 289-300. doi: 10.3934/ipi.2015.9.289 [8] 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 [9] Mourad Choulli, Aymen Jbalia. The problem of detecting corrosion by an electric measurement revisited. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 643-650. doi: 10.3934/dcdss.2016018 [10] Jun Lai, Ming Li, Peijun Li, Wei Li. A fast direct imaging method for the inverse obstacle scattering problem with nonlinear point scatterers. Inverse Problems & Imaging, 2018, 12 (3) : 635-665. doi: 10.3934/ipi.2018027 [11] Abdumajid Begmatov, Akhtam Dzhalilov, Dieter Mayer. Renormalizations of circle hoemomorphisms with a single break point. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4487-4513. doi: 10.3934/dcds.2014.34.4487 [12] Kenichi Sakamoto, Masahiro Yamamoto. Inverse source problem with a final overdetermination for a fractional diffusion equation. Mathematical Control & Related Fields, 2011, 1 (4) : 509-518. doi: 10.3934/mcrf.2011.1.509 [13] John C. Schotland, Vadim A. Markel. Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation. Inverse Problems & Imaging, 2007, 1 (1) : 181-188. doi: 10.3934/ipi.2007.1.181 [14] Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367 [15] 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 [16] Lucie Baudouin, Emmanuelle Crépeau, Julie Valein. Global Carleman estimate on a network for the wave equation and application to an inverse problem. Mathematical Control & Related Fields, 2011, 1 (3) : 307-330. doi: 10.3934/mcrf.2011.1.307 [17] Michael V. Klibanov. A phaseless inverse scattering problem for the 3-D Helmholtz equation. Inverse Problems & Imaging, 2017, 11 (2) : 263-276. doi: 10.3934/ipi.2017013 [18] Jaan Janno, Kairi Kasemets. Uniqueness for an inverse problem for a semilinear time-fractional diffusion equation. Inverse Problems & Imaging, 2017, 11 (1) : 125-149. doi: 10.3934/ipi.2017007 [19] Michel Cristofol, Patricia Gaitan, Kati Niinimäki, Olivier Poisson. Inverse problem for a coupled parabolic system with discontinuous conductivities: One-dimensional case. Inverse Problems & Imaging, 2013, 7 (1) : 159-182. doi: 10.3934/ipi.2013.7.159 [20] J. F. Padial. Existence and estimate of the location of the free-boundary for a non local inverse elliptic-parabolic problem arising in nuclear fusion. Conference Publications, 2011, 2011 (Special) : 1176-1185. doi: 10.3934/proc.2011.2011.1176

2017 Impact Factor: 1.049

## Metrics

• HTML views (153)
• Cited by (0)

• on AIMS