Identification problem for the wave equation with Neumann data input and Dirichlet data observations
type: Item
instance of: scholarly article
title: Identification problem for the wave equation with Neumann data input and Dirichlet data observations
zbMATH Open document ID: 1031.35148
DOI: 10.1016/S0362-546X(02)00295-X
author: Xiaobing Feng
author: Suzanne Lenhart
author: V. Protopopescu
author: Lizabeth V. Rachele
author: Brian D. Sutton
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2003-03-11T00:00:00Z
review text: The identification of the dispersive coefficient \(h(x)\in L^\infty\) in the wave equation in a bounded domain \(\Omega\) with \(C^2\) boundary \[ \begin{gathered} u_{tt}-\Delta u= h(x)u+ f(x,t),\quad (x,t)\in \Omega\times (0,T),\quad f\in L^2,\\ u(x,0)= u_0\in H^1(\Omega),\quad u_t(x,0)= u_1\in L^2(\Omega),\quad x\in\Omega,\\ {\partial u\over\partial n}= g\in H^{1/2}(\partial\Omega\times (0,T))\end{gathered} \] is obtained by minimizing the Tikhonov functional \[ J_\beta(h):= {1\over 2} \Biggl(\int_{\partial\Omega\times (t_1,t_2)} (u(h)- z)^2 ds dt+ \beta \int_\Omega h^2 dx\Biggr), \] over \(h\in L^\infty(\Omega)\), where \(z\in L^2(\partial\Omega\times (t_1,t_2))\) with \(0\leq t_1< t_2\leq T\), is a given data for \(u|_{\partial\Omega\times (t_1,t_2)}\). However, no criterion for choosing the regularization parameter \(\beta> 0\) is given. Furthermore, some of the numerically obtained results for \(h(x)\) are 50\% out of the corresponding analytical solution, showing that a more accurate numerical method for solving the nonlinear control problem is needed in any future work.
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 35L05
Mathematics Subject Classification ID: 65M32
zbMATH DE Number: 1880125
zbMATH Keywords: coefficient identification
zbMATH Keywords: wave equation
zbMATH Keywords: optimal control
zbMATH Keywords: Dirichlet-Neumann map
zbMATH Keywords: Tikhonov's regularization
reviewed by: Daniel Lesnic
MaRDI profile type: Publication
Wikidata QID: Q60231258
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full work available at URL: s0362-546x(02)00295-x
OpenAlex ID: W2093191832
description: scientific article; zbMATH DE number 1880125