We show how stiffness manifests itself in the simulation of chemical reactions at both the continuous-deterministic level and the discrete-stochastic level. Existing discrete stochastic simulation methods, such as the stochastic simulation algorithm and the ͑explicit͒ tau-leaping method, are both exceedingly slow for such systems. We propose an implicit tau-leaping method that can take much larger time steps for many of these problems.

Abstract. We investigate some basic properties of the proper orthogonal decomposition (POD) method as it is applied to data compression and model reduction of finite dimensional nonlinear systems. First we provide an analysis of the errors involved in solving a nonlinear ODE initial value problem using a POD reduced order model. Then we study the effects of small perturbations in the ensemble of data from which the POD reduced order model is constructed on the reduced order model. We explain why in some applications this sensitivity is a concern while in others it is not. We also provide an analysis of computational complexity of solving an ODE initial value problem and study the computational savings obtained by using a POD reduced order model. We provide several examples to illustrate our theoretical results.

Parametric sensitivity of biochemical networks is an indispensable tool for studying system robustness properties, estimating network parameters, and identifying targets for drug therapy. For discrete stochastic representations of biochemical networks where Monte Carlo methods are commonly used, sensitivity analysis can be particularly challenging, as accurate finite difference computations of sensitivity require a large number of simulations for both nominal and perturbed values of the parameters. In this paper we introduce the common random number ͑CRN͒ method in conjunction with Gillespie's stochastic simulation algorithm, which exploits positive correlations obtained by using CRNs for nominal and perturbed parameters. We also propose a new method called the common reaction path ͑CRP͒ method, which uses CRNs together with the random time change representation of discrete state Markov processes due to Kurtz to estimate the sensitivity via a finite difference approximation applied to coupled reaction paths that emerge naturally in this representation. While both methods reduce the variance of the estimator significantly compared to independent random number finite difference implementations, numerical evidence suggests that the CRP method achieves a greater variance reduction. We also provide some theoretical basis for the superior performance of CRP. The improved accuracy of these methods allows for much more efficient sensitivity estimation. In two example systems reported in this work, speedup factors greater than 300 and 10 000 are demonstrated.

Abstract. We develop a theory of local errors for the explicit and implicit tau-leaping methods for simulating stochastic chemical systems, and we prove that these methods are first-order consistent. Our theory provides local error formulae that could serve as the basis for future stepsize control techniques. We prove that, for the special case of systems with linear propensity functions, both tau-leaping methods are first-order convergent in all moments. We provide a stiff stability analysis of the mean of both leaping methods, and we confirm that the implicit method is unconditionally stable in the mean for stable systems. Finally, we give some theoretical and numerical examples to illustrate these results.

Tau-leaping methods have recently been proposed for the acceleration of discrete stochastic simulation of chemically reacting systems. This paper considers the numerical stability of these methods. The concept of stochastic absolute stability is defined, discussed, and applied to the following leaping methods: the explicit tau, implicit tau, and trapezoidal tau.