非Lipschitz反应网络有限差分法的方差
Variance of finite difference methods for reaction networks with non-Lipschitz rate functions
摘要：
参数灵敏度分析是物理系统数学模型研究中的一个重要组成部分。由于其简单性，有限差分法被广泛用于这种分析在随机建模的反应网络的研究。已经提出了不同的耦合方法来构建有限差分估计，其中“分裂耦合”，也称为“堆叠耦合”，在绝大多数情况下产生最低的方差。与该耦合相关的分析结果是稀疏的，并且包括在全局Lipschitz强度函数的假设下的耦合过程的方差分析[Anderson，SIAM数值分析，第50卷，2012年]。由于[Anderson，SIAM数值分析，Vol. 50，2012]中使用的全局Lipschitz假设，其主要结果仅适用于文献中发现的一小部分模型，并且证明了类似的结果应该适用于更广泛的一类模型。在本文中，我们证明了这一猜想是真实的，通过证明方差的耦合过程的规模在所需的方式为一大类非Lipschitz模型。我们进一步扩展的分析，以允许在参数的时间依赖性。特别是，二进制系统与或不与时间相关的速率参数，一类模型，占绝大多数的系统在文献中考虑，满足我们的理论假设。
英文摘要：
Parametric sensitivity analysis is a critical component in the study of mathematical models of physical systems. Due to its simplicity, finite difference methods are used extensively for this analysis in the study of stochastically modeled reaction networks. Different coupling methods have been proposed to build finite difference estimators, with the "split coupling," also termed the "stacked coupling," yielding the lowest variance in the vast majority of cases. Analytical results related to this coupling are sparse, and include an analysis of the variance of the coupled processes under the assumption of globally Lipschitz intensity functions [Anderson, SIAM Numerical Analysis, Vol. 50, 2012]. Because of the global Lipschitz assumption utilized in [Anderson, SIAM Numerical Analysis, Vol. 50, 2012], the main result there is only applicable to a small percentage of the models found in the literature, and it was conjectured that similar results should hold for a much wider class of models. In this paper we demonstrate this conjecture to be true by proving the variance of the coupled processes scales in the desired manner for a large class of non-Lipschitz models. We further extend the analysis to allow for time dependence in the parameters. In particular, binary systems with or without time-dependent rate parameters, a class of models that accounts for the vast majority of systems considered in the literature, satisfy the assumptions of our theory.
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