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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 |
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.
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. |
show all references
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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