Stochastic simulation is a widely used method for estimating quantities in models of chemical reaction networks where uncertainty plays a crucial role. However, reducing the statistical uncertainty of the corresponding estimators requires the generation of a large number of simulation runs, which is computationally expensive. To reduce the number of necessary runs, we propose a variance reduction technique based on control variates. We exploit constraints on the statistical moments of the stochastic process to reduce the estimators' variances. We develop an algorithm that selects appropriate control variates in an on-line fashion and demonstrate the efficiency of our approach on several case studies.
While the above techniques give a deterministic output, stochastic simulation generates single executions of the stochastic process @cite_16 . This necessitates accumulating large numbers of simulation runs to estimate quantities. This adds a significant computational burden. Consequently, some effort has been directed at lowering this cost. A prominent technique is @math -leaping @cite_4 , which in one step performs multiple instead of only a single reaction. Another approach is to find approximations that are specific to the problem at hand, such as approximations based on time-scale separations @cite_32 @cite_28 .
@cite_28 @cite_16 @cite_4 @cite_32
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Stiffness in chemical reaction systems is a frequently encountered computational problem, arising when different reactions in the system take place at different time-scales. Computational savings can be obtained under time-scale separation. Assuming that the system can be partitioned into slow- and fast- equilibrating subsystems, it is then possible to efficiently simulate the slow subsystem only, provided that the corresponding kinetic laws have been modified so that they reflect their dependency on the fast system. We show that the rate expectation with respect to the fast subsystem’s steady-state is a continuous function of the state of the slow system. We exploit this result to construct an analytic representation of the modified rate functions via statistical modelling, which can be used to simulate the slow system in isolation. The computational savings of our approach are demonstrated in a number of non-trivial examples of stiff systems. The stochastic simulation algorithm (SSA) is an essentially exact procedure for numerically simulating the time evolution of a well-stirred chemically reacting system. Despite recent major improvements in the efficiency of the SSA, its drawback remains the great amount of computer time that is often required to simulate a desired amount of system time. Presented here is the “τ-leap” method, an approximate procedure that in some circumstances can produce significant gains in simulation speed with acceptable losses in accuracy. Some primitive strategies for control parameter selection and error mitigation for the τ-leap method are described, and simulation results for two simple model systems are exhibited. With further refinement, the τ-leap method should provide a viable way of segueing from the exact SSA to the approximate chemical Langevin equation, and thence to the conventional deterministic reaction rate equation, as the system size becomes larger. Reactions in real chemical systems often take place on vastly different time scales, with “fast” reaction channels firing very much more frequently than “slow” ones. These firings will be interdependent if, as is usually the case, the fast and slow reactions involve some of the same species. An exact stochastic simulation of such a system will necessarily spend most of its time simulating the more numerous fast reaction events. This is a frustratingly inefficient allocation of computational effort when dynamical stiffness is present, since in that case a fast reaction event will be of much less importance to the system’s evolution than will a slow reaction event. For such situations, this paper develops a systematic approximate theory that allows one to stochastically advance the system in time by simulating the firings of only the slow reaction events. Developing an effective strategy to implement this theory poses some challenges, but as is illustrated here for two simple systems, when those challenges ...