Abstract We suggest in this paper two new random walk based stochastic algorithms for solving high-dimensional PDEs for domains with complicated geometrical structure. The first one, a Random Walk on Spheres (RWS) algorithm is developed for solving systems of coupled drift–diffusion–reaction equations where the random walk is living both on randomly sampled spheres and inside the relevant balls. The second method suggested solves transient anisotropic diffusion equations, where the random walk is carried out on random rectangular parallelepipeds inside the domain. The two methods are mesh-free both in space and time, and are well applied to solve high-dimensional problems with complicated domains. The algorithms are based on tracking the trajectories of the diffusing particles exactly in accordance with the probabilistic distributions derived from the explicit representation of the relevant Green functions for a sphere and a parallelepiped. They can be conveniently used not only for the solutions, but also for a direct calculation of fluxes to any part of the boundary without calculating the whole solution in the domain. Applications to exciton transport in semiconductors and related cathodoluminescence imaging of a set of randomly distributed threading dislocations are presented.

中文翻译：

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漂移-扩散-反应方程系统的无网格随机算法和各向异性扩散通量计算

摘要 我们在本文中提出了两种新的基于随机游走的随机算法，用于求解具有复杂几何结构的域的高维偏微分方程。第一个是球体随机游走 (RWS) 算法，用于求解耦合漂移 - 扩散 - 反应方程的系统，其中随机游走同时存在于随机采样的球体上和相关球内。建议的第二种方法解决瞬态各向异性扩散方程，其中随机游走是在域内的随机长方体上进行的。这两种方法在空间和时间上都是无网格的，很好地应用于解决复杂域的高维问题。该算法基于精确地根据从球体和平行六面体的相关格林函数的显式表示导出的概率分布来跟踪扩散粒子的轨迹。它们不仅可以方便地用于解，而且可以直接计算到边界任何部分的通量，而无需计算域中的整个解。介绍了半导体中激子传输的应用以及一组随机分布的螺纹位错的相关阴极发光成像。也可以直接计算边界任何部分的通量，而无需计算域中的整个解。介绍了半导体中激子传输的应用以及一组随机分布的螺纹位错的相关阴极发光成像。也可以直接计算边界任何部分的通量，而无需计算域中的整个解。介绍了半导体中激子传输的应用以及一组随机分布的螺纹位错的相关阴极发光成像。