AI 中文总结
该研究提出hp型时间步进谱蒙特卡洛方法，结合残差迭代与谱重构策略，可处理半线性抛物型方程的长时间模拟与初始奇异性，无需线性方程组求解，支持并行计算，经实验验证了其精度与效率。
AI 中文摘要
本文提出一种hp型时间步进谱蒙特卡洛方法用于求解半线性抛物型方程。核心创新在于构建了指数精度的随机算法，该算法在时间和空间方向上结合高斯型节点上的残差迭代格式与基于谱方法的重构策略。针对传统随机算法（如球上行走法）常面临的长时间模拟和初始奇异性难题，我们进一步开发了hp型时间步进框架，该框架采用多个时间步长，并分别使用几何时间分区和线性递增的多项式阶数来处理这些困难。值得注意的是，该算法无需求解传统谱方法所需的线性方程组，且显著支持时间和空间网格点上的并行计算。我们严格证明了多步方法在有限次迭代内的指数收敛速率。大量数值实验被开展，以验证所提方法在长时间模拟、带初始奇异性的问题以及五维问题中的谱精度和计算效率，从而验证了理论结果。
英文摘要
In this paper, we present an $hp$-version time-stepping spectral Monte Carlo method for solving semi-linear parabolic equations. The key innovation lies in constructing an exponentially accurate stochastic algorithm that integrates a residual iteration scheme on Gauss-type nodes in both temporal and spatial directions with a reconstruction strategy rooted in spectral methods. To address the long-time simulations and initial singularities that are often challenging for traditional stochastic algorithms (e.g., walk-on-spheres method), we further develop an $hp$-version time-stepping framework that employs multiple time steps and, respectively, geometric time partitions with linearly increasing polynomial degrees to handle these difficulties. Notably, the proposed algorithm bypasses the need to solve linear systems required by traditional spectral methods and remarkably supports parallel computation at both temporal and spatial grid points. We rigorously establish exponential convergence rates for the multistep method within a finite number of iterations. Extensive numerical experiments are conducted to demonstrate the spectral accuracy and computational efficiency of the proposed method in long-time simulations, problems with initial singularities, and a five-dimensional problem, thereby validating the theoretical results.
Journal refSIAM Journal on Numerical Analysis, 2026
