Leaf Peeling method for the wave equation on metric tree graphs

Sergei Avdonin; Yuanyuan Zhao

doi:10.3934/ipi.2020060

Article Contents

2021, Volume 15, Issue 2: 185-199. Doi: 10.3934/ipi.2020060

This issue Previous Article Next Article Posterior contraction for empirical bayesian approach to inverse problems under non-diagonal assumption

Leaf Peeling method for the wave equation on metric tree graphs

Sergei Avdonin^1,2, and
Yuanyuan Zhao^3, ,

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
^* Corresponding author: Yuanyuan Zhao

Received: November 30, 2019

Revised: June 30, 2020

Early access: October 2020

Published: April 2021

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

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Abstract

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.

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Mathematics Subject Classification: 35R30, 35L05, 93B05.

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Figure 1. The neighborhood of $ v_i $

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

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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

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