A popular method to compute first-passage probabilities in continuous-time Markov chains is by numerically inverting their Laplace transforms. Past decades, the scientific computing community has developed excellent numerical methods for solving problems governed by partial differential equations (PDEs), making the availability of a Laplace transform not necessary here for computational purposes. In this study we demonstrate that numerical PDE problem solvers are suitable for computing first passage times, and can be very efficient for this purpose. By doing extensive computational experiments, we show that modern PDE problem solvers can outperform numerical Laplace transform inversion, even if a transform is available. When the Laplace transform is explicit (e.g. does not require the computation of an eigensystem), numerical transform inversion remains the primary method of choice.

中文翻译：

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使用数值 PDE 问题求解器计算马尔可夫调制流体模型的首次通过时间

在连续时间马尔可夫链中计算首次通过概率的一种流行方法是通过数值反转它们的拉普拉斯变换。在过去的几十年里，科学计算界已经开发出优秀的数值方法来解决偏微分方程 (PDE) 控制的问题，因此这里不需要使用拉普拉斯变换来进行计算。在这项研究中，我们证明了数值 PDE 问题求解器适用于计算首次通过时间，并且可以非常有效地用于此目的。通过进行大量的计算实验，我们表明现代 PDE 问题求解器可以胜过数值拉普拉斯变换反演，即使变换可用。当拉普拉斯变换是显式的（例如不需要计算特征系统），