# American Institute of Mathematical Sciences

April  2021, 15(2): 185-199. doi: 10.3934/ipi.2020060

## Leaf Peeling method for the wave equation on metric tree graphs

 1 University of Alaska Fairbanks, Fairbanks, AK 99775-6660, USA 2 Moscow Center for Fundamental and Applied Mathematics, Moscow, 119991, Russia 3 University of Alaska Fairbanks, Fairbanks, AK 99775-6660, USA

* Corresponding author: Yuanyuan Zhao

Received  December 2019 Revised  July 2020 Published  April 2021 Early access  October 2020

Fund Project: The research of the first author was supported in part by the National Science Foundation, grant DMS 1909869 and by the Ministry of Education and Science of Republic of Kazakhstan under the grant No. AP05136197. The research of the second author was supported by the National Science Foundation Graduate Research Fellowship under Grant No. 1242789

We consider the dynamical inverse problem for the wave equation on a metric tree graph and describe the dynamical Leaf Peeling (LP) method. The main step of the method is recalculating the response operator from the original tree to a peeled tree. The LP method allows us to recover the connectivity, potential function on a tree graph and the lengths of its edges from the response operator given on a finite time interval.

Citation: Sergei Avdonin, Yuanyuan Zhao. Leaf Peeling method for the wave equation on metric tree graphs. Inverse Problems and Imaging, 2021, 15 (2) : 185-199. doi: 10.3934/ipi.2020060
##### References:
 [1] F. Ali Mehmeti, Nonlinear Waves in Networks, Mathematical Research, 1994. [2] F. Ali Mehmeti and E. Meister, Regular solutions of transmission and interaction problems for wave equations, Mathematical Methods in the Applied Sciences, 11 (1989), 665-685.  doi: 10.1002/mma.1670110507. [3] S. A. Avdonin, M. I. Belishev and S. A. Ivanov, Boundary control and a matrix inverse problem for the equation, Mathematics of the USSR-Sbornik, 72 (1992), 287-310.  doi: 10.1070/SM1992v072n02ABEH002141. [4] S. Avdonin and J. Bell, Determining a distributed conductance parameter for a neuronal cable model defined on a tree graph, Inverse Problems and Imaging, 9 (2015), 645-659.  doi: 10.3934/ipi.2015.9.645. [5] S. Avdonin, J. Bell, V. Mikhaylov and K. Nurtazina, Source and coefficient identification problems for the wave equation on graphs, Mathematical Methods in the Applied Sciences, 42 (2019), 5029-5039.  doi: 10.1002/mma.5229. [6] S. Avdonin, J. Bell and K. Nurtazina, Determining distributed parameters in a neuronal cable model on a tree graph, Mathematical Methods in the Applied Sciences, 40 (2017), 3973-3981.  doi: 10.1002/mma.4277. [7] S. Avdonin, C. Rivero Abdon, G. Leugering and V. Mikhaylov, On the inverse problem of the two-velocity tree-like graph, Zeit. Angew. Math. Mech., 95 (2015), 1490-1500.  doi: 10.1002/zamm.201400126. [8] S. Avdonin and P. Kurasov, Inverse problems for quantum trees, Inverse Problems and Imaging, 2 (2008), 1-21.  doi: 10.3934/ipi.2008.2.1. [9] S. Avdonin, P. 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. [10] S. Avdonin, G. Leugering and V. Mikhaylov, On an inverse problem for tree-like networks of elastic strings, Zeit. Angew. Math. Mech., 90 (2010), 136-150.  doi: 10.1002/zamm.200900295. [11] S. Avdonin and V. E. Mikhaylov, The boundary control approach to inverse spectral theory, Inverse Problems, 26 (2010), 099801, 2pp. doi: 10.1088/0266-5611/26/9/099801. [12] S. A. Avdonin, V. S. Mikhaylov and K. B. Nurtazina, On inverse dynamical and spectral problems for the wave and schrödinger equations on finite trees. The leaf peeling method, Journal of Mathematical Sciences, 224 (2017), 1-10.  doi: 10.1007/s10958-017-3388-2. [13] S. Avdonin and S. Nicaise, Source identification problems for the wave equation on graphs, Inverse Problems, 31 (2015), 095007, 29pp. doi: 10.1088/0266-5611/31/9/095007. [14] S. Avdonin and Y. Zhao, Exact controllability of the 1-d wave equation on finite metric tree graphs, Applied Mathematics & Optimization, 2019, 1–24. doi: 10.1007/s00245-019-09629-3. [15] M. Belishev, Boundary spectral inverse problem on a class of graphs (trees) by the BC method, Inverse Problems, 20 (2004), 647-672.  doi: 10.1088/0266-5611/20/3/002. [16] G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, 186, American Mathematical Soc., 2013. doi: 10.1090/surv/186. [17] B. M. Brown and R. Weikard, A borg–levinson theorem for trees, in Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, The Royal Society, 461 (2005), 3231–3243. doi: 10.1098/rspa.2005.1513. [18] R. Carlson, Inverse eigenvalue problems on directed graphs, Transactions of the American Mathematical Society, 351 (1999), 4069-4088.  doi: 10.1090/S0002-9947-99-02175-3. [19] R. Dáger and E. Zuazua, Wave Propagation, Observation and Control in 1-D Flexible Multi-Structures, Springer Science & Business Media, 2006. doi: 10.1007/3-540-37726-3. [20] N. I. Gerasimenko and B. S. Pavlov, Scattering problems on noncompact graphs, Theoretical and Mathematical Physics, 74 (1988), 230-240.  doi: 10.1007/BF01016616. [21] J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, On the analysis and control of hyperbolic systems associated with vibrating networks, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 124 (1994), 77-104.  doi: 10.1017/S0308210500029206. [22] V. N. Pivovarchik, An inverse sturm-liouville problem by three spectra, Integral Equations and Operator Theory, 34 (1999), 234-243.  doi: 10.1007/BF01236474. [23] V. Yurko, Inverse spectral problems for sturm–liouville operators on graphs, Inverse Problems, 21 (2005), 1075-1086.  doi: 10.1088/0266-5611/21/3/017.

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##### References:
 [1] F. Ali Mehmeti, Nonlinear Waves in Networks, Mathematical Research, 1994. [2] F. Ali Mehmeti and E. Meister, Regular solutions of transmission and interaction problems for wave equations, Mathematical Methods in the Applied Sciences, 11 (1989), 665-685.  doi: 10.1002/mma.1670110507. [3] S. A. Avdonin, M. I. Belishev and S. A. Ivanov, Boundary control and a matrix inverse problem for the equation, Mathematics of the USSR-Sbornik, 72 (1992), 287-310.  doi: 10.1070/SM1992v072n02ABEH002141. [4] S. Avdonin and J. Bell, Determining a distributed conductance parameter for a neuronal cable model defined on a tree graph, Inverse Problems and Imaging, 9 (2015), 645-659.  doi: 10.3934/ipi.2015.9.645. [5] S. Avdonin, J. Bell, V. Mikhaylov and K. Nurtazina, Source and coefficient identification problems for the wave equation on graphs, Mathematical Methods in the Applied Sciences, 42 (2019), 5029-5039.  doi: 10.1002/mma.5229. [6] S. Avdonin, J. Bell and K. Nurtazina, Determining distributed parameters in a neuronal cable model on a tree graph, Mathematical Methods in the Applied Sciences, 40 (2017), 3973-3981.  doi: 10.1002/mma.4277. [7] S. Avdonin, C. Rivero Abdon, G. Leugering and V. Mikhaylov, On the inverse problem of the two-velocity tree-like graph, Zeit. Angew. Math. Mech., 95 (2015), 1490-1500.  doi: 10.1002/zamm.201400126. [8] S. Avdonin and P. Kurasov, Inverse problems for quantum trees, Inverse Problems and Imaging, 2 (2008), 1-21.  doi: 10.3934/ipi.2008.2.1. [9] S. Avdonin, P. 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. [10] S. Avdonin, G. Leugering and V. Mikhaylov, On an inverse problem for tree-like networks of elastic strings, Zeit. Angew. Math. Mech., 90 (2010), 136-150.  doi: 10.1002/zamm.200900295. [11] S. Avdonin and V. E. Mikhaylov, The boundary control approach to inverse spectral theory, Inverse Problems, 26 (2010), 099801, 2pp. doi: 10.1088/0266-5611/26/9/099801. [12] S. A. Avdonin, V. S. Mikhaylov and K. B. Nurtazina, On inverse dynamical and spectral problems for the wave and schrödinger equations on finite trees. The leaf peeling method, Journal of Mathematical Sciences, 224 (2017), 1-10.  doi: 10.1007/s10958-017-3388-2. [13] S. Avdonin and S. Nicaise, Source identification problems for the wave equation on graphs, Inverse Problems, 31 (2015), 095007, 29pp. doi: 10.1088/0266-5611/31/9/095007. [14] S. Avdonin and Y. Zhao, Exact controllability of the 1-d wave equation on finite metric tree graphs, Applied Mathematics & Optimization, 2019, 1–24. doi: 10.1007/s00245-019-09629-3. [15] M. Belishev, Boundary spectral inverse problem on a class of graphs (trees) by the BC method, Inverse Problems, 20 (2004), 647-672.  doi: 10.1088/0266-5611/20/3/002. [16] G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, 186, American Mathematical Soc., 2013. doi: 10.1090/surv/186. [17] B. M. Brown and R. Weikard, A borg–levinson theorem for trees, in Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, The Royal Society, 461 (2005), 3231–3243. doi: 10.1098/rspa.2005.1513. [18] R. Carlson, Inverse eigenvalue problems on directed graphs, Transactions of the American Mathematical Society, 351 (1999), 4069-4088.  doi: 10.1090/S0002-9947-99-02175-3. [19] R. Dáger and E. Zuazua, Wave Propagation, Observation and Control in 1-D Flexible Multi-Structures, Springer Science & Business Media, 2006. doi: 10.1007/3-540-37726-3. [20] N. I. Gerasimenko and B. S. Pavlov, Scattering problems on noncompact graphs, Theoretical and Mathematical Physics, 74 (1988), 230-240.  doi: 10.1007/BF01016616. [21] J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, On the analysis and control of hyperbolic systems associated with vibrating networks, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 124 (1994), 77-104.  doi: 10.1017/S0308210500029206. [22] V. N. Pivovarchik, An inverse sturm-liouville problem by three spectra, Integral Equations and Operator Theory, 34 (1999), 234-243.  doi: 10.1007/BF01236474. [23] V. Yurko, Inverse spectral problems for sturm–liouville operators on graphs, Inverse Problems, 21 (2005), 1075-1086.  doi: 10.1088/0266-5611/21/3/017.
The neighborhood of $v_i$
The propagation of $\delta(t)$ from $\gamma_i$. Vertex $v_3$ may be adjacent to either $v_1$ or $v_2$
A sheaf on a tree graph rooted at $\gamma_m$ (the sheaf is in solid lines), in which $v_0$ is the abscission vertex and $e_0$ is the stem edge
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