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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 and 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.

[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. 

[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.

[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.

[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.

[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.

[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.

[8]

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

[9]

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

[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.

[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.

[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.

[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.

[14]

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

[15]

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

[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. 

[17]

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

[18]

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

[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.

[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.

[21]

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

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.

[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. 

[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.

[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.

[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.

[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.

[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.

[8]

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

[9]

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

[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.

[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.

[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.

[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.

[14]

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

[15]

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

[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. 

[17]

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

[18]

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

[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.

[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.

[21]

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

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