We suggest in this paper a new stochastic simulation algorithm for solving narrow escape problems governed by drift-diffusion-reaction equations of high dimension. The developed method drastically improves the efficiency of the diffusion trajectory tracking algorithm by introducing an artificial drift directed to the target position. The method is especially appropriate to solve narrow escape problems for domains of very long extension in one direction which is the case in many practical problems. We present simulation results for a diffusion transport problem of calculation of cathodoluminescence intensity, a diffusion flux of excitons to a threading dislocation, and the electron beam induced current in a semiconductor. The diffusion tracking algorithm is based on the random walk on spheres process. The method is meshless both in space and time, and is well applied to solve high-dimensional problems in 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. 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.

我们建议在本文中提出一种新的随机仿真算法，用于解决由高维漂移-扩散-反应方程控制的狭窄逃生问题。通过引入针对目标位置的人工漂移，该开发方法极大地提高了扩散轨迹跟踪算法的效率。该方法特别适合解决在一个方向上非常长的扩展域的狭窄逃生问题，这在许多实际问题中都是如此。我们提出了模拟结果，用于计算阴极发光强度，激子向穿线位错的扩散通量以及电子束在半导体中感应的电流的扩散传输问题。扩散跟踪算法基于球面上的随机游走过程。该方法在时间和空间上都是无网格的，并很好地解决了复杂领域中的高维问题。该算法基于完全根据从相关格林函数的明确表示中得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，而且可以直接计算通向边界任何部分的通量，而无需计算域中的整个解。