Variance Reduction with Array-RQMC for Tau-Leaping Simulation of Stochastic Biological and Chemical Reaction Networks

TL;DR

This paper demonstrates that Array-RQMC significantly reduces variance in tau-leaping simulations of stochastic biological and chemical networks, achieving reductions by factors of thousands compared to traditional methods.

Contribution

It introduces the application of Array-RQMC to tau-leaping simulations, highlighting the importance of sorting functions and reaction counts for maximizing variance reduction.

Findings

Variance reductions by factors in the thousands achieved.

Array-RQMC outperforms traditional Monte Carlo in all experiments.

Choice of sorting function influences efficiency but does not worsen performance.

Abstract

We explore the use of Array-RQMC, a randomized quasi-Monte Carlo method designed for the simulation of Markov chains, to reduce the variance when simulating stochastic biological or chemical reaction networks with $\tau$-leaping. The task is to estimate the expectation of a function of molecule copy numbers at a given future time $T$ by the sample average over $n$ sample paths, and the goal is to reduce the variance of this sample-average estimator. We find that when the method is properly applied, variance reductions by factors in the thousands can be obtained. These factors are much larger than those observed previously by other authors who tried RQMC methods for the same examples. Array-RQMC simulates an array of realizations of the Markov chain and requires a sorting function to reorder these chains according to their states, after each step. The choice of sorting function is a key ingredient for the efficiency of the method, although in our experiments, Array-RQMC was never worse than ordinary Monte Carlo, regardless of the sorting method. The expected number of reactions of each type per step also has an impact on the efficiency gain.