计算松散连接部件大系统中MTTA的张量方法
Tensor methods for the computation of MTTA in large systems of loosely interconnected components
摘要：
我们关心的是计算的平均时间吸收（MTTA）的一个大系统的松散互连组件，建模为连续时间马尔可夫链。特别是，我们表明，分裂的本地和同步转换的较小的子系统，可以制定一个算法的MTTA的计算被证明是线性收敛的。然后，我们展示了如何修改的方法，使其二次收敛，从而克服了困难的问题，收敛速度接近$1$。此外，它表明，这种解耦的本地和同步转换允许轻松地表示所有的矩阵和向量的张量列车（TT）格式的方法中涉及的-我们提供的数值证据表明，这允许处理大型问题高达数十亿的状态-这将是不可行的。
英文摘要：
We are concerned with the computation of the mean-time-to-absorption (MTTA) for a large system of loosely interconnected components, modeled as continuous time Markov chains. In particular, we show that splitting the local and synchronization transitions of the smaller subsystems allows to formulate an algorithm for the computation of the MTTA which is proven to be linearly convergent. Then, we show how to modify the method to make it quadratically convergent, thus overcoming the difficulties for problems with convergent rate close to $1$. In addition, it is shown that this decoupling of local and synchronization transitions allows to easily represent all the matrices and vectors involved in the method in the tensor-train (TT) format - and we provide numerical evidence showing that this allows to treat large problems with up to billions of states - which would otherwise be unfeasible.
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