  We propose a globally convergent numerical method, called the
convexification, to numerically compute the viscosity solution to first-order
Hamilton-Jacobi equations through the vanishing viscosity process where the
viscosity parameter is a fixed small number. By convexification, we mean that
we employ a suitable Carleman weight function to convexify the cost functional
defined directly from the form of the Hamilton-Jacobi equation under
consideration. The strict convexity of this functional is rigorously proved
using a new Carleman estimate. We also prove that the unique minimizer of the
this strictly convex functional can be reached by the gradient descent method.
Moreover, we show that the minimizer well approximates the viscosity solution
of the Hamilton-Jacobi equation as the noise contained in the boundary data
tends to zero. Some interesting numerical illustrations are presented.
