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：本文介绍一种新颖算法，该算法收敛于高维Hamilton-Jacobi方程的水平集凸粘性解。该算法适用于广义曲率运动PDE，以及近期发展的用于Tukey深度（一种数据点统计深度度量）的Hamilton-Jacobi方程。本研究的主要贡献是提出了一种新的单调格式来近似梯度方向，该格式允许对沿梯度方向及其正交方向上的纯偏导数进行单调离散化。我们提供了该算法在任意维度的规则笛卡尔网格和非结构点云上的收敛性分析，并通过数值实验展示了该算法在逼近二维仿射流解以及高维数据集（如MNIST和FashionMNIST）的Tukey深度度量方面的有效性。