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December  2018, 12(6): 1293-1308. doi: 10.3934/ipi.2018054

Reconstruction of the coefficients of a star graph from observations of its vertices

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

Received  December 2016 Revised  March 2018 Published  October 2018

Consider a three-edge star graph, made up of unknown Sturm-Liouville operators on each edge. By using the heat propagation through the graph and measuring the heat transfer occurring at its vertices, we show that we can extract enough spectral data to reconstruct the three Sturm-Liouville operators by using the Gelfand-Levitan theory. Furthermore this reconstruction is achieved by a single measurement provided we use a special initial condition.

Citation: Amin Boumenir, Vu Kim Tuan. Reconstruction of the coefficients of a star graph from observations of its vertices. Inverse Problems & Imaging, 2018, 12 (6) : 1293-1308. doi: 10.3934/ipi.2018054
References:
[1]

S. Avdonin and J. Bell, Determining a distributed conductance parameter for a neuronal cable model defined on a tree graph, Inverse Probl. Imaging, 9 (2015), 645-659. doi: 10.3934/ipi.2015.9.645. Google Scholar

[2]

S. AvdoninJ. Bell and K. Nurtazina, Determining distributed parameters in a neuronal cable model on a tree graph, Math. Methods Appl. Sci., 40 (2017), 3973-3981. Google Scholar

[3]

S. Avdonin and S. Nicaise, Source identification for the wave equation on graphs, C. R. Math. Acad. Sci. Paris, 352 (2014), 907-912. doi: 10.1016/j.crma.2014.09.008. Google Scholar

[4]

S. AvdoninF. Gesztesy and K. Makarov, Spectral estimation and inverse initial boundary value problems, Inverse Probl. Imaging, 4 (2010), 1-9. doi: 10.3934/ipi.2010.4.1. Google Scholar

[5]

S. 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. Google Scholar

[6]

S. Avdonin and P. Kurasov, Inverse problems for quantum trees, Inverse Probl. Imaging, 2 (2008), 1-21. doi: 10.3934/ipi.2008.2.1. Google Scholar

[7]

S. AvdoninP. Kurasov and M. Nowaczyk, Inverse problems for quantum trees Ⅱ: Recovering matching conditions for star graphs, Inverse Probl. Imaging, 4 (2010), 579-598. doi: 10.3934/ipi.2010.4.579. Google Scholar

[8]

A. Boumenir, Higher approximation of eigenvalues by sampling, BIT, 40 (2000), 215-225. doi: 10.1023/A:1022334806027. Google Scholar

[9]

A. Boumenir, Irregular sampling and the inverse spectral problem, J. Fourier Anal. Appl., 5 (1999), 373-383. doi: 10.1007/BF01259378. Google Scholar

[10]

A. Boumenir and Vu Kim Tuan, An inverse problem for the heat equation, Proc. Amer. Math. Soc., 138 (2010), 3911-3921. doi: 10.1090/S0002-9939-2010-10297-6. Google Scholar

[11]

A. Boumenir and V. Kim Tuan, Recovery of the heat coefficient by two measurements, Inverse Problems and Imaging, 5 (2011), 775-791. doi: 10.3934/ipi.2011.5.775. Google Scholar

[12]

R. Carlson and V. Pivovarchik, Spectral asymptotics for quantum graphs with equal edge lengths, J. Phys. A: Math. Theor., 41 (2008), 145202, 16 pp. doi: 10.1088/1751-8113/41/14/145202. Google Scholar

[13]

I. Kac and V. Pivovarchik, On multiplicity of a quantum graph spectrum, J. Phys. A: Math. Theor., 44 (2011), 105301, 14 pp. doi: 10.1088/1751-8113/44/10/105301. Google Scholar

[14]

H. P. Kramer, A generalized sampling theorem, J. Math. Phys., 38 (1959), 68-72. doi: 10.1002/sapm195938168. Google Scholar

[15]

B. M. Levitan, Inverse Sturm-Liouville Problems, VNU Science Press, Utrecht, 1987. Google Scholar

[16]

B. M. Levitan and M. G. Gasymov, Determination of a differential equation by two of its spectra, Russ. Math. Surveys, 19 (1964), 3-63. Google Scholar

[17]

V. A. Marchenko, Sturm-Liouville Operators and Applications, Birkhäuser, Basel, 1986. doi: 10.1007/978-3-0348-5485-6. Google Scholar

[18]

V. Pivovarchik and H. Woracek, Eigenvalue asymptotics for a star-graph damped vibrations problem, Asymptot. Anal., 73 (2011), 169-185. Google Scholar

[19]

V. Pivovarchik, Inverse problem for the Sturm-Liouville equation on a simple graph, SIAM J. Math. Anal., 32 (2000), 801-819. doi: 10.1137/S0036141000368247. Google Scholar

[20]

V. Kim Tuan and N. Thanh Hong, Interpolation in the Hardy space, Integral Transforms Spec. Funct., 24 (2013), 664-671. doi: 10.1080/10652469.2012.749874. Google Scholar

[21]

A. Zayed, Advances in Shannon's Sampling Theory, CRC Press, 1993. Google Scholar

show all references

References:
[1]

S. Avdonin and J. Bell, Determining a distributed conductance parameter for a neuronal cable model defined on a tree graph, Inverse Probl. Imaging, 9 (2015), 645-659. doi: 10.3934/ipi.2015.9.645. Google Scholar

[2]

S. AvdoninJ. Bell and K. Nurtazina, Determining distributed parameters in a neuronal cable model on a tree graph, Math. Methods Appl. Sci., 40 (2017), 3973-3981. Google Scholar

[3]

S. Avdonin and S. Nicaise, Source identification for the wave equation on graphs, C. R. Math. Acad. Sci. Paris, 352 (2014), 907-912. doi: 10.1016/j.crma.2014.09.008. Google Scholar

[4]

S. AvdoninF. Gesztesy and K. Makarov, Spectral estimation and inverse initial boundary value problems, Inverse Probl. Imaging, 4 (2010), 1-9. doi: 10.3934/ipi.2010.4.1. Google Scholar

[5]

S. 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. Google Scholar

[6]

S. Avdonin and P. Kurasov, Inverse problems for quantum trees, Inverse Probl. Imaging, 2 (2008), 1-21. doi: 10.3934/ipi.2008.2.1. Google Scholar

[7]

S. AvdoninP. Kurasov and M. Nowaczyk, Inverse problems for quantum trees Ⅱ: Recovering matching conditions for star graphs, Inverse Probl. Imaging, 4 (2010), 579-598. doi: 10.3934/ipi.2010.4.579. Google Scholar

[8]

A. Boumenir, Higher approximation of eigenvalues by sampling, BIT, 40 (2000), 215-225. doi: 10.1023/A:1022334806027. Google Scholar

[9]

A. Boumenir, Irregular sampling and the inverse spectral problem, J. Fourier Anal. Appl., 5 (1999), 373-383. doi: 10.1007/BF01259378. Google Scholar

[10]

A. Boumenir and Vu Kim Tuan, An inverse problem for the heat equation, Proc. Amer. Math. Soc., 138 (2010), 3911-3921. doi: 10.1090/S0002-9939-2010-10297-6. Google Scholar

[11]

A. Boumenir and V. Kim Tuan, Recovery of the heat coefficient by two measurements, Inverse Problems and Imaging, 5 (2011), 775-791. doi: 10.3934/ipi.2011.5.775. Google Scholar

[12]

R. Carlson and V. Pivovarchik, Spectral asymptotics for quantum graphs with equal edge lengths, J. Phys. A: Math. Theor., 41 (2008), 145202, 16 pp. doi: 10.1088/1751-8113/41/14/145202. Google Scholar

[13]

I. Kac and V. Pivovarchik, On multiplicity of a quantum graph spectrum, J. Phys. A: Math. Theor., 44 (2011), 105301, 14 pp. doi: 10.1088/1751-8113/44/10/105301. Google Scholar

[14]

H. P. Kramer, A generalized sampling theorem, J. Math. Phys., 38 (1959), 68-72. doi: 10.1002/sapm195938168. Google Scholar

[15]

B. M. Levitan, Inverse Sturm-Liouville Problems, VNU Science Press, Utrecht, 1987. Google Scholar

[16]

B. M. Levitan and M. G. Gasymov, Determination of a differential equation by two of its spectra, Russ. Math. Surveys, 19 (1964), 3-63. Google Scholar

[17]

V. A. Marchenko, Sturm-Liouville Operators and Applications, Birkhäuser, Basel, 1986. doi: 10.1007/978-3-0348-5485-6. Google Scholar

[18]

V. Pivovarchik and H. Woracek, Eigenvalue asymptotics for a star-graph damped vibrations problem, Asymptot. Anal., 73 (2011), 169-185. Google Scholar

[19]

V. Pivovarchik, Inverse problem for the Sturm-Liouville equation on a simple graph, SIAM J. Math. Anal., 32 (2000), 801-819. doi: 10.1137/S0036141000368247. Google Scholar

[20]

V. Kim Tuan and N. Thanh Hong, Interpolation in the Hardy space, Integral Transforms Spec. Funct., 24 (2013), 664-671. doi: 10.1080/10652469.2012.749874. Google Scholar

[21]

A. Zayed, Advances in Shannon's Sampling Theory, CRC Press, 1993. Google Scholar

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