Identification problem for the wave equation with Neumann data input and Dirichlet data observations

Xiaobing Feng; Suzanne Lenhart; Vladimir Protopopescu; Lizabeth Rachele; Brian Sutton

doi:10.1016/S0362-546X(02)00295-X

Identification problem for the wave equation with Neumann data input and Dirichlet data observations

Xiaobing Feng
, Suzanne Lenhart
, Vladimir Protopopescu
, Lizabeth Rachele
, Brian Sutton

Research output: Contribution to journal › Article › peer-review

9 Scopus citations

Abstract

We seek to identify the dispersive coefficient in a wave equation with Neumann boundary conditions in a bounded space-time domain from imprecise observations of the solution on the boundary of the spatial domain (Dirichlet data). The problem is regularized and solved by casting it into an optimal control setting. By letting the "cost of the control" tend to zero, we obtain the limit of the corresponding control sequence, which we identify with the sought dispersive coefficient. The corresponding solution of the wave equation is interpreted as the possibly nonunique projection of the observation vector onto the range of the Neumann-to-Dirichlet maps corresponding to a single input Neumann data, as the dispersive coefficient is varied. Several numerical examples illustrate the merits and limitations of the procedure.

Original language	English
Pages (from-to)	1777-1795
Number of pages	19
Journal	Nonlinear Analysis, Theory, Methods and Applications
Volume	52
Issue number	7
DOIs	https://doi.org/10.1016/S0362-546X(02)00295-X
State	Published - Mar 2003

Funding

Research supported by the US Department of Energy, Office of Basic Energy Sciences, under Contract No. AC05-00 OR 22725 with UT-Battelle, LCC. “This submitted manuscript has been authored by a contractor of the US Government under Contract No. AC 05-00 OR 22725. Accordingly, the US Government retains a nonexclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for US Government purpose”.

Keywords

Coefficient identification
Dirichlet-Neumann map
Optimal control
Wave equation

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10.1016/S0362-546X(02)00295-X

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title = "Identification problem for the wave equation with Neumann data input and Dirichlet data observations",

abstract = "We seek to identify the dispersive coefficient in a wave equation with Neumann boundary conditions in a bounded space-time domain from imprecise observations of the solution on the boundary of the spatial domain (Dirichlet data). The problem is regularized and solved by casting it into an optimal control setting. By letting the {"}cost of the control{"} tend to zero, we obtain the limit of the corresponding control sequence, which we identify with the sought dispersive coefficient. The corresponding solution of the wave equation is interpreted as the possibly nonunique projection of the observation vector onto the range of the Neumann-to-Dirichlet maps corresponding to a single input Neumann data, as the dispersive coefficient is varied. Several numerical examples illustrate the merits and limitations of the procedure.",

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T1 - Identification problem for the wave equation with Neumann data input and Dirichlet data observations

AU - Feng, Xiaobing

AU - Lenhart, Suzanne

AU - Protopopescu, Vladimir

AU - Rachele, Lizabeth

AU - Sutton, Brian

PY - 2003/3

Y1 - 2003/3

N2 - We seek to identify the dispersive coefficient in a wave equation with Neumann boundary conditions in a bounded space-time domain from imprecise observations of the solution on the boundary of the spatial domain (Dirichlet data). The problem is regularized and solved by casting it into an optimal control setting. By letting the "cost of the control" tend to zero, we obtain the limit of the corresponding control sequence, which we identify with the sought dispersive coefficient. The corresponding solution of the wave equation is interpreted as the possibly nonunique projection of the observation vector onto the range of the Neumann-to-Dirichlet maps corresponding to a single input Neumann data, as the dispersive coefficient is varied. Several numerical examples illustrate the merits and limitations of the procedure.

AB - We seek to identify the dispersive coefficient in a wave equation with Neumann boundary conditions in a bounded space-time domain from imprecise observations of the solution on the boundary of the spatial domain (Dirichlet data). The problem is regularized and solved by casting it into an optimal control setting. By letting the "cost of the control" tend to zero, we obtain the limit of the corresponding control sequence, which we identify with the sought dispersive coefficient. The corresponding solution of the wave equation is interpreted as the possibly nonunique projection of the observation vector onto the range of the Neumann-to-Dirichlet maps corresponding to a single input Neumann data, as the dispersive coefficient is varied. Several numerical examples illustrate the merits and limitations of the procedure.

KW - Coefficient identification

KW - Dirichlet-Neumann map

KW - Optimal control

KW - Wave equation

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DO - 10.1016/S0362-546X(02)00295-X

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AN - SCOPUS:0037333655

SN - 0362-546X

VL - 52

SP - 1777

EP - 1795

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

IS - 7

ER -

Identification problem for the wave equation with Neumann data input and Dirichlet data observations

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