  This article develops the numerical and theoretical study of a reconstruction
algorithm of a potential in a wave equation from boundary measurements, using a
cost functional built on weighted energy terms coming from a Carleman estimate.
More precisely, this inverse problem for the wave equation consists in the
determination of an unknown time-independent potential from a single
measurement of the Neumann derivative of the solution on a part of the
boundary. While its uniqueness and stability properties are already well known
and studied, a constructive and globally convergent algorithm based on Carleman
estimates for the wave operator was recently proposed in [L. Baudouin, M. de
Buhan and S. Ervedoza, Global carleman estimates for waves and applications,
Comm. Partial Differential Equations 38 (2013), no. 5]. However, the numerical
implementation of this strategy still presents several challenges, that we
propose to address here.
