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
This paper addresses the methodological challenges of comparing the iron industries in the Qin-Han and Roman empires by creating "modeling domains" as a pragmatic and utilitarian approach. These domains, built from literary and archaeological evidence, represent generalized rules and frameworks, paired with diachronic, fragmented landscapes that depict the progressive acquisition and integration of lands with established metallurgical traditions. The paper argues that simply reaching this step is not enough, as each domain should be understood as part of a larger aggregative set, with an "external" dimension. The paper further discusses the distancing effect and the need for caution in cross-domain discussions, emphasizing the importance of historical and social specificity. The Roman-Parthian and Han-Nanyue examples are used to illustrate these challenges and opportunities. The paper concludes that the comparative approach should be ever-expanding, leading to a continual dialogue between domains and a deeper understanding of the dynamics of control, trade, and technological exchange in different historical and social contexts.