Figure 1.
The neighborhood of $ v_i $
-
Previous Article
Posterior contraction for empirical bayesian approach to inverse problems under non-diagonal assumption
- IPI Home
- This Issue
- Next Article
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 |
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.
Mathematics Subject Classification:
35R30, 35L05, 93B05.
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. |
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. |
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 and 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 and 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 and Related Fields, 2019, 9 (2) : 289-312. doi: 10.3934/mcrf.2019015 |
| [4] |
Xiaoli Feng, Meixia Zhao, Peijun Li, Xu Wang. An inverse source problem for the stochastic wave equation. Inverse Problems and Imaging, 2022, 16 (2) : 397-415. doi: 10.3934/ipi.2021055 |
| [5] |
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 and Control Theory, 2018, 7 (4) : 669-682. doi: 10.3934/eect.2018032 |
| [6] |
Bopeng Rao, Laila Toufayli, Ali Wehbe. Stability and controllability of a wave equation with dynamical boundary control. Mathematical Control and Related Fields, 2015, 5 (2) : 305-320. doi: 10.3934/mcrf.2015.5.305 |
| [7] |
Bastian Gebauer, Nuutti Hyvönen. Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem. Inverse Problems and Imaging, 2008, 2 (3) : 355-372. doi: 10.3934/ipi.2008.2.355 |
| [8] |
Lekbir Afraites. A new coupled complex boundary method (CCBM) for an inverse obstacle problem. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 23-40. doi: 10.3934/dcdss.2021069 |
| [9] |
Laurent Bourgeois, Dmitry Ponomarev, Jérémi Dardé. An inverse obstacle problem for the wave equation in a finite time domain. Inverse Problems and Imaging, 2019, 13 (2) : 377-400. doi: 10.3934/ipi.2019019 |
| [10] |
Lucie Baudouin, Emmanuelle Crépeau, Julie Valein. Global Carleman estimate on a network for the wave equation and application to an inverse problem. Mathematical Control and Related Fields, 2011, 1 (3) : 307-330. doi: 10.3934/mcrf.2011.1.307 |
| [11] |
Xiaoshan Chen, Fahuai Yi. Free boundary problem of Barenblatt equation in stochastic control. Discrete and Continuous Dynamical Systems - B, 2016, 21 (5) : 1421-1434. doi: 10.3934/dcdsb.2016003 |
| [12] |
Alexander Arguchintsev, Vasilisa Poplevko. An optimal control problem by parabolic equation with boundary smooth control and an integral constraint. Numerical Algebra, Control and Optimization, 2018, 8 (2) : 193-202. doi: 10.3934/naco.2018011 |
| [13] |
Muhammad I. Mustafa. On the control of the wave equation by memory-type boundary condition. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 1179-1192. doi: 10.3934/dcds.2015.35.1179 |
| [14] |
Zhiling Guo, Shugen Chai. Exponential stabilization of the problem of transmission of wave equation with linear dynamical feedback control. Evolution Equations and Control Theory, 2022 doi: 10.3934/eect.2022001 |
| [15] |
Hui Yang, Yuzhu Han. Initial boundary value problem for a strongly damped wave equation with a general nonlinearity. Evolution Equations and Control Theory, 2022, 11 (3) : 635-648. doi: 10.3934/eect.2021019 |
| [16] |
Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems and Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271 |
| [17] |
Tony Liimatainen, Lauri Oksanen. Counterexamples to inverse problems for the wave equation. Inverse Problems and Imaging, 2022, 16 (2) : 467-479. doi: 10.3934/ipi.2021058 |
| [18] |
Yu-Feng Sun, Zheng Zeng, Jie Song. Quasilinear iterative method for the boundary value problem of nonlinear fractional differential equation. Numerical Algebra, Control and Optimization, 2020, 10 (2) : 157-164. doi: 10.3934/naco.2019045 |
| [19] |
Peter I. Kogut. On approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2105-2133. doi: 10.3934/dcds.2014.34.2105 |
| [20] |
Jussi Korpela, Matti Lassas, Lauri Oksanen. Discrete regularization and convergence of the inverse problem for 1+1 dimensional wave equation. Inverse Problems and Imaging, 2019, 13 (3) : 575-596. doi: 10.3934/ipi.2019027 |
2020 Impact Factor: 1.639
Tools
Metrics
Other articles
by authors
[Back to Top]