Journal
JOURNAL OF COMPUTATIONAL PHYSICS
Volume 451, Issue -, Pages -Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jcp.2021.110828
Keywords
Numerical methods; Convexification method; Gradient descent method; Viscosity solutions; Hamilton-Jacobi equations; Boundary value problems
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This article proposes a globally convergent numerical method, called the convexification, to compute the viscosity solution to first-order Hamilton-Jacobi equations through the vanishing viscosity process. The method employs a suitable Carleman weight function to convexify the cost functional defined directly from the form of the Hamilton-Jacobi equation and utilizes the gradient descent method to find the unique minimizer of this convex functional.
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 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. (C) 2021 Elsevier Inc. All rights reserved.
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