论文标题
整个空间上汉密尔顿 - 雅各比方程的Carleman凸式化方法
The Carleman convexification method for Hamilton-Jacobi equations on the whole space
论文作者
论文摘要
我们提出了一种新的全球收敛数值方法，以在$ \ mathbb {r}^d $，$ d \ geq 1 $中求解Hamilton-Jacobi方程。该方法称为卡尔曼凸式化方法。通过Carleman凸式化，我们的意思是我们使用Carleman重量函数来凸出常规的最小二乘不匹配功能。我们将证明一种新版本的凸定理，确保涉及卡尔曼权重函数的不匹配功能严格凸出，因此具有独特的最小化器。此外，我们的凸定理的结果确保了Carleman加权不匹配功能的最小化器是我们要计算的粘度解决方案的近似值。将提出一些1D和2D的数值结果。
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.
