Numerical resolution of moderately high-dimensional nonlinear PDEs remains a huge challenge due to the curse of dimensionality for the classical numerical methods including finite difference, finite element and spectral methods. Starting from the weak formulation of the Lawson-Euler scheme, this paper proposes a stochastic particle method (SPM) by tracking the deterministic motion, random jump, resampling and reweighting of particles. Real-valued weighted particles are adopted by SPM to approximate the high-dimensional solution, which automatically adjusts the point distribution to intimate the relevant feature of the solution. A piecewise constant reconstruction with virtual uniform grid is employed to evaluate the nonlinear terms, which fully exploits the intrinsic adaptive characteristic of SPM. Combining both, SPM can achieve the goal of adaptive sampling in time. Numerical experiments on the 6-D Allen-Cahn equation and the 7- D Hamiltonian-Jacobi-Bellman equation demonstrate the potential of SPM in solving moderately high-dimensional nonlinear PDEs efficiently while maintaining an acceptable accuracy
The ability to segment moving objects from three-dimensional (3D) LiDAR scans is critical to advancing autonomous driving technology, facilitating core tasks like localization, collision avoidance, and path planning. In this paper, we introduce a novel deep neural network designed to enhance the performance of 3D LiDAR point cloud moving object segmentation (MOS) through the integration of image gradient information and the principle of motion consistency. Our method processes sequential range images, employing depth pixel difference convolution (DPDC) to improve the efficacy of dilated convolutions, thus boosting spatial information extraction from range images. Additionally, we incorporate Bayesian filtering to impose posterior constraints on predictions, enhancing the accuracy of motion segmentation. To handle the issue of uneven object scales in range images, we develop a novel edge-aware loss function and use a progressive training strategy to further boost performance. Our method is validated on the SemanticKITTI-based LiDAR MOS benchmark, where it significantly outperforms current state-of-the-art (SOTA) methods, all while working directly on two-dimensional (2D) range images without requiring mapping.