# American Institute of Mathematical Sciences

May  2015, 9(2): 371-393. doi: 10.3934/ipi.2015.9.371

## Some geometric inverse problems for the linear wave equation

 1 University of Sevilla, Dpto. E.D.A.N., Aptdo. 1160, 41080, Sevilla, Spain, Spain

Received  June 2014 Revised  November 2014 Published  March 2015

In this paper we consider some geometric inverse problems for the linear wave equation. We prove uniqueness results, we present some reconstruction algorithms and we perform numerical experiments in dimensions one and two.
Citation: Anna Doubova, Enrique Fernández-Cara. Some geometric inverse problems for the linear wave equation. Inverse Problems & Imaging, 2015, 9 (2) : 371-393. doi: 10.3934/ipi.2015.9.371
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##### References:
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