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.

翻译：我们提出一种全球趋同数字法,称为“凝结”法,通过粘结参数是一个固定小数的消失粘结过程,从数字上计算汉密尔顿-贾科比等式的第一顺序粘结溶液。通过粘结参数是一个固定小数的消失粘结过程,我们提出一个适当的卡莱曼加权函数,将所考虑的汉密尔顿-贾科比等式直接界定的成本功能混结起来。使用新的卡莱曼估计法,严格证明这一功能的严格凝结。我们还证明,这种严格粘结功能的独特最小化器可以通过梯度下降法达到。此外,我们表明,最小化极接近汉密尔顿-贾科比等式的粘结溶液,因为边界数据中所含的噪音趋向为零。我们提出了一些有趣的数字说明。