  The inverse problem of estimating dielectric constants of explosives using
boundary measurements of one component of the scattered electric field is
addressed. It is formulated as a coefficient inverse problem for a hyperbolic
differential equation. After applying the Laplace transform, a new cost
functional is constructed and a variational problem is formulated. The key
feature of this functional is the presence of the Carleman Weight Function for
the Laplacian. The strict convexity of this functional on a bounded set in a
Hilbert space of an arbitrary size is proven. This allows for establishing the
global convergence of the gradient descent method. Some results of numerical
experiments are presented.
