Inverse space-dependent force problems for the wave equation

D. Lesnic*, S.O. Hussein, B.T. Johansson

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

The determination of the displacement and the space-dependent force acting on a vibrating structure from measured final or time-average displacement observation is thoroughly investigated. Several aspects related to the existence and uniqueness of a solution of the linear but ill-posed inverse force problems are highlighted. After that, in order to capture the solution a variational formulation is proposed and the gradient of the least-squares functional that is minimized is rigorously and explicitly derived. Numerical results obtained using the Landweber method and the conjugate gradient method are presented and discussed illustrating the convergence of the iterative procedures for exact input data. Furthermore, for noisy data the semi-convergence phenomenon appears, as expected, and stability is restored by stopping the iterations according to the discrepancy principle criterion once the residual becomes close to the amount of noise. The present investigation will be significant to researchers concerned with wave propagation and control of vibrating structures.

Original languageEnglish
Pages (from-to)10-39
Number of pages30
JournalJournal of Computational and Applied Mathematics
Volume306
Early online date13 Apr 2016
DOIs
Publication statusPublished - Nov 2016

Bibliographical note

-© 2016, Elsevier. Licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International http://creativecommons.org/licenses/by-nc-nd/4.0/

Keywords

  • conjugate gradient method
  • finite difference method
  • inverse force problem
  • Landweber method
  • wave equation

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