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  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 & Imaging, doi: 10.3934/ipi.2020060
References:
[1]

F. Ali Mehmeti, Nonlinear Waves in Networks, Mathematical Research, 1994.  Google Scholar

[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.  Google Scholar

[3]

S. A. AvdoninM. 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.  Google Scholar

[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.  Google Scholar

[5]

S. AvdoninJ. BellV. 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.  Google Scholar

[6]

S. AvdoninJ. 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.  Google Scholar

[7]

S. AvdoninC. Rivero AbdonG. 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.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

S. AvdoninG. 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.  Google Scholar

[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.  Google Scholar

[12]

S. A. AvdoninV. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, 186, American Mathematical Soc., 2013. doi: 10.1090/surv/186.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

J. E. LagneseG. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

F. Ali Mehmeti, Nonlinear Waves in Networks, Mathematical Research, 1994.  Google Scholar

[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.  Google Scholar

[3]

S. A. AvdoninM. 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.  Google Scholar

[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.  Google Scholar

[5]

S. AvdoninJ. BellV. 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.  Google Scholar

[6]

S. AvdoninJ. 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.  Google Scholar

[7]

S. AvdoninC. Rivero AbdonG. 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.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

S. AvdoninG. 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.  Google Scholar

[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.  Google Scholar

[12]

S. A. AvdoninV. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, 186, American Mathematical Soc., 2013. doi: 10.1090/surv/186.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

J. E. LagneseG. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  The neighborhood of $ v_i $
Figure 2.  The propagation of $ \delta(t) $ from $ \gamma_i $. Vertex $ v_3 $ may be adjacent to either $ v_1 $ or $ v_2 $
Figure 3.  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
[1]

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

[2]

Lili Chang, Wei Gong, Guiquan Sun, Ningning Yan. PDE-constrained optimal control approach for the approximation of an inverse Cauchy problem. Inverse Problems & Imaging, 2015, 9 (3) : 791-814. doi: 10.3934/ipi.2015.9.791

[3]

Yavar Kian, Morgan Morancey, Lauri Oksanen. Application of the boundary control method to partial data Borg-Levinson inverse spectral problem. Mathematical Control & Related Fields, 2019, 9 (2) : 289-312. doi: 10.3934/mcrf.2019015

[4]

Zhousheng Ruan, Sen Zhang, Sican Xiong. Solving an inverse source problem for a time fractional diffusion equation by a modified quasi-boundary value method. Evolution Equations & Control Theory, 2018, 7 (4) : 669-682. doi: 10.3934/eect.2018032

[5]

Bopeng Rao, Laila Toufayli, Ali Wehbe. Stability and controllability of a wave equation with dynamical boundary control. Mathematical Control & Related Fields, 2015, 5 (2) : 305-320. doi: 10.3934/mcrf.2015.5.305

[6]

Bastian Gebauer, Nuutti Hyvönen. Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem. Inverse Problems & Imaging, 2008, 2 (3) : 355-372. doi: 10.3934/ipi.2008.2.355

[7]

Laurent Bourgeois, Dmitry Ponomarev, Jérémi Dardé. An inverse obstacle problem for the wave equation in a finite time domain. Inverse Problems & Imaging, 2019, 13 (2) : 377-400. doi: 10.3934/ipi.2019019

[8]

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

[9]

Xiaoshan Chen, Fahuai Yi. Free boundary problem of Barenblatt equation in stochastic control. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1421-1434. doi: 10.3934/dcdsb.2016003

[10]

Alexander Arguchintsev, Vasilisa Poplevko. An optimal control problem by parabolic equation with boundary smooth control and an integral constraint. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 193-202. doi: 10.3934/naco.2018011

[11]

Muhammad I. Mustafa. On the control of the wave equation by memory-type boundary condition. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1179-1192. doi: 10.3934/dcds.2015.35.1179

[12]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271

[13]

Yu-Feng Sun, Zheng Zeng, Jie Song. Quasilinear iterative method for the boundary value problem of nonlinear fractional differential equation. Numerical Algebra, Control & Optimization, 2020, 10 (2) : 157-164. doi: 10.3934/naco.2019045

[14]

Jussi Korpela, Matti Lassas, Lauri Oksanen. Discrete regularization and convergence of the inverse problem for 1+1 dimensional wave equation. Inverse Problems & Imaging, 2019, 13 (3) : 575-596. doi: 10.3934/ipi.2019027

[15]

Lauri Oksanen. Solving an inverse problem for the wave equation by using a minimization algorithm and time-reversed measurements. Inverse Problems & Imaging, 2011, 5 (3) : 731-744. doi: 10.3934/ipi.2011.5.731

[16]

Peter I. Kogut. On approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2105-2133. doi: 10.3934/dcds.2014.34.2105

[17]

Masaru Ikehata, Mishio Kawashita. An inverse problem for a three-dimensional heat equation in thermal imaging and the enclosure method. Inverse Problems & Imaging, 2014, 8 (4) : 1073-1116. doi: 10.3934/ipi.2014.8.1073

[18]

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

[19]

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

[20]

Fang Zeng, Pablo Suarez, Jiguang Sun. A decomposition method for an interior inverse scattering problem. Inverse Problems & Imaging, 2013, 7 (1) : 291-303. doi: 10.3934/ipi.2013.7.291

2019 Impact Factor: 1.373

Metrics

  • PDF downloads (19)
  • HTML views (32)
  • Cited by (0)

Other articles
by authors

[Back to Top]