Numerical viscosity solutions to Hamilton-Jacobi equations via a Carleman estimate and the convexification method

Abstract

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.

References in corpus (4)

• Homogenization and enhancement for the G-equation
• The gradient descent method for the convexification to solve boundary value problems of quasi-linear PDEs and a coefficient inverse problem
• Convexification for a 3D inverse scattering problem with the moving point source
• Carleman contraction mapping for a 1D inverse scattering problem with experimental time-dependent data

Cited by in corpus (3)

• A Carleman-based numerical method for quasilinear elliptic equations with over-determined boundary data and applications
• The Carleman convexification method for Hamilton-Jacobi equations on the whole space
• The Carleman contraction mapping method for quasilinear elliptic equations with over-determined boundary data