  To compute the spatially distributed dielectric constant from the
backscattering data, we study a coefficient inverse problem for a 1D hyperbolic
equation. To solve the inverse problem, we establish a new version of Carleman
estimate and then employ this estimate to construct a cost functional which is
strictly convex on a convex bounded set with an arbitrary diameter in a Hilbert
space. The strict convexity property is rigorously proved. This result is
called the convexification theorem and is considered as the central analytical
result of this paper. Minimizing this convex functional by the gradient descent
method, we obtain the desired numerical solution to the coefficient inverse
problems. We prove that the gradient descent method generates a sequence
converging to the minimizer and we also establish a theorem confirming that the
minimizer converges to the true solution as the noise in the measured data and
the regularization parameter tend to zero. Unlike the methods that are based on
optimization, our convexification method converges globally in the sense that
it delivers a good approximation of the exact solution without requiring any
initial guess. Results of numerical studies of both computationally simulated
and experimental data are presented.
