多级蒙特卡洛τ跃迁的复杂性
Complexity of Multilevel Monte Carlo Tau-Leaping
AI总结：
本文研究多级蒙特卡洛τ跃迁的计算复杂性，通过直接分析差值方差而非二阶矩，得出更精确的理论结果，适用于大规模跳变系统。
摘要：
τ跃迁是一种生成连续时间离散空间马尔可夫链近似路径的流行离散化方法，尤其适用于生物化学反应系统。为了计算此类上下文中的期望值，适当多级蒙特卡洛形式的τ跃迁已被证明能显著提高效率。本文推导了多级蒙特卡洛τ跃迁计算复杂性的新解析结果，这些结果比之前的更为精确。我们避免取渐近极限，专注于一个实际场景，其中系统规模足够大，使得路径上许多事件发生，因此精确模拟路径成本高昂，使得τ跃迁成为有吸引力的选择。我们使用了一种通用系统组件的缩放方式，允许反应速率常数和物种丰度在多个数量级上变化，并利用Kurtz开发的随机时间变换表示。分析的关键特征是当比较相关过程对时，直接分析其差值的方差而非通过二阶矩进行上界估计。使用二阶矩在扩散方程的设置中是自然的，多级方法最初是在此背景下开发的，其中数值方法的强收敛性结果 readily available，但在本文考虑的泊松驱动跳变系统中并不最优。我们还展示了新的分析的计算结果。
英文摘要：
Tau-leaping is a popular discretization method for generating approximate paths of continuous time, discrete space, Markov chains, notably for biochemical reaction systems. To compute expected values in this context, an appropriate multilevel Monte Carlo form of tau-leaping has been shown to improve efficiency dramatically. In this work we derive new analytic results concerning the computational complexity of multilevel Monte Carlo tau-leaping that are significantly sharper than previous ones. We avoid taking asymptotic limits, and focus on a practical setting where the system size is large enough for many events to take place along a path, so that exact simulation of paths is expensive, making tau-leaping an attractive option. We use a general scaling of the system components that allows for the reaction rate constants and the abundances of species to vary over several orders of magnitude, and we exploit the random time change representation developed by Kurtz. The key feature of the analysis that allows for the sharper bounds is that when comparing relevant pairs of processes we analyze the variance of their difference directly rather than bounding via the second moment. Use of the second moment is natural in the setting of a diffusion equation, where multilevel was first developed and where strong convergence results for numerical methods are readily available, but is not optimal for the Poisson-driven jump systems that we consider here. We also present computational results that illustrate the new analysis.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
