We introduce a novel algorithm that converges to level-set convex viscosity solutions of high-dimensional Hamilton-Jacobi equations. The algorithm is applicable to a broad class of curvature motion PDEs, as well as a recently developed Hamilton-Jacobi equation for the Tukey depth, which is a statistical depth measure of data points. A main contribution of our work is a new monotone scheme for approximating the direction of the gradient, which allows for monotone discretizations of pure partial derivatives in the direction of, and orthogonal to, the gradient. We provide a convergence analysis of the algorithm on both regular Cartesian grids and unstructured point clouds in any dimension and present numerical experiments that demonstrate the effectiveness of the algorithm in approximating solutions of the affine flow in two dimensions and the Tukey depth measure of high-dimensional datasets such as MNIST and FashionMNIST.

翻译：我们提出了一种新颖算法，该算法可收敛至高维Hamilton-Jacobi方程的水平集凸黏性解。该算法适用于广泛曲率运动偏微分方程，以及近期针对Tukey深度（一种数据点统计深度度量）开发的Hamilton-Jacobi方程。本研究的主要贡献在于提出了一种用于近似梯度方向的新型正则格式，该格式能够实现在梯度方向及其正交方向上的纯偏导数的正则离散化。我们提供了该算法在任意维度的规则笛卡尔网格与非结构化点云上的收敛性分析，并通过数值实验验证了该算法在近似二维仿射流解及MNIST、FashionMNIST等高维数据集Tukey深度度量方面的有效性。