To cope with an extremely large or even infinite state space when solving the chemical master equation in biological problems, a potent strategy is to restrict to a finite state projection (FSP) and represent the transition matrix and probability vector in quantized tensor train (QTT) format, leading to savings in storage while retaining accuracy. In an earlier adaptive FSP–QTT algorithm, the multidimensional state space was downsized and kept in the form of a hyper rectangle that was updated when needed by selectively doubling some of its side dimensions. However, this could result in a much larger state space than necessary, with the effect of hampering both the execution time and stepping scheme. In this work, we improve the algorithm by enabling sliding windows that can dynamically slide, shrink or expand, with updates driven by a number of stochastic simulation algorithm trajectories. The ensuing state space is a considerably reduced hyper rectangle containing only the most probable states at each time step. Three numerical experiments of varying difficulty are performed to compare our approach with the original adaptive FSP–QTT algorithm.

在求解生物问题中的化学主方程时，为了应对极大甚至无限的状态空间，一种有效的策略是限制为有限状态投影（FSP）并在量化张量序列（QTT）中表示转移矩阵和概率向量格式，从而在保持准确性的同时节省存储空间。在早期的自适应 FSP-QTT 算法中，多维状态空间被缩小并保持为超矩形的形式，在需要时通过选择性地加倍其一些边维度来更新该超矩形。然而，这可能会导致比必要的状态空间大得多，从而阻碍执行时间和步进方案。在这项工作中，我们通过启用可以动态滑动、收缩或扩展的滑动窗口来改进算法，由许多随机模拟算法轨迹驱动的更新。随后的状态空间是一个大大减少的超矩形，仅包含每个时间步最可能的状态。进行了三个不同难度的数值实验，以将我们的方法与原始自适应 FSP-QTT 算法进行比较。