拟蒙特卡罗方法在随机生物系统τ跳跃中的应用
Quasi-Monte Carlo methods applied to tau-leaping in stochastic biological systems
摘要：
拟蒙特卡罗方法已被证明是传统蒙特卡罗方法的有效扩展，其中包括随机微分方程的求积和样本路径模拟问题。通过用低差异数输入流替换模拟过程中的随机数输入流，在金融应用中已经观察到几个数量级的方差减小。充分混合的化学反应网络中的随机效应的分析通常依赖于使用蒙特卡罗方法的样本路径模拟，即使这些方法具有典型的缓慢的$\mathcal{O}(N^{-1/2})$收敛速率作为样本路径$N$的数量的函数。本文研究了一种有效的样本路径模拟程序，即$τ$跳跃的（随机）准蒙特卡罗方法的组合。我们表明，这种组合往往是更有效的比传统的蒙特卡罗模拟的统计误差的衰减。然而，由于化学反应模型的离散性质，所观察到的收敛速率行为是不平凡的。我们解释了这是如何影响性能的准蒙特卡罗方法，在标准正交测试问题。
英文摘要：
Quasi-Monte Carlo methods have proven to be effective extensions of traditional Monte Carlo methods in, amongst others, problems of quadrature and the sample path simulation of stochastic differential equations. By replacing the random number input stream in a simulation procedure by a low-discrepancy number input stream, variance reductions of several orders have been observed in financial applications. Analysis of stochastic effects in well-mixed chemical reaction networks often relies on sample path simulation using Monte Carlo methods, even though these methods suffer from typical slow $\mathcal{O}(N^{-1/2})$ convergence rates as a function of the number of sample paths $N$. This paper investigates the combination of (randomised) quasi-Monte Carlo methods with an efficient sample path simulation procedure, namely $τ$-leaping. We show that this combination is often more effective than traditional Monte Carlo simulation in terms of the decay of statistical errors. The observed convergence rate behaviour is, however, non-trivial due to the discrete nature of the models of chemical reactions. We explain how this affects the performance of quasi-Monte Carlo methods by looking at a test problem in standard quadrature.
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