The Carleman convexification method for Hamilton-Jacobi equations on the whole space

Abstract: We propose a new globally convergent numerical method to solve Hamilton-Jacobi equations in $\mathbb{R}^d$, $d \geq 1$. This method is named as the Carleman convexification method. By Carleman convexification, we mean that we use a Carleman weight function to convexify the conventional least squares mismatch functional. We will prove a new version of the convexification theorem guaranteeing that the mismatch functional involving the Carleman weight function is strictly convex and, therefore, has a unique minimizer. Moreover, a consequence of our convexification theorem guarantees that the minimizer of the Carleman weighted mismatch functional is an approximation of the viscosity solution we want to compute. Some numerical results in 1D and 2D will be presented.