Weak error analysis of numerical methods for stochastic models of population processes

TL;DR

This paper develops a framework to analyze the weak error in numerical methods for simulating stochastic population models, highlighting how approximate methods can reduce computational costs despite introducing bias.

Contribution

It introduces a general approach to quantify weak errors in numerical approximations of stochastic population models, considering system scaling and step-size effects.

Findings

Weak error analysis helps understand bias in approximate simulations.

Approximate methods can significantly reduce computational complexity.

The framework applies to various stochastic models of biological populations.

Abstract

The simplest, and most common, stochastic model for population processes, including those from biochemistry and cell biology, are continuous time Markov chains. Simulation of such models is often relatively straightforward as there are easily implementable methods for the generation of exact sample paths. However, when using ensemble averages to approximate expected values, the computational complexity can become prohibitive as the number of computations per path scales linearly with the number of jumps of the process. When such methods become computationally intractable, approximate methods, which introduce a bias, can become advantageous. In this paper, we provide a general framework for understanding the weak error, or bias, induced by different numerical approximation techniques in the current setting. The analysis takes into account both the natural scalings within a given system and the step-size of the numerical method. Examples are provided to demonstrate the main analytical results as well as the reduction in computational complexity achieved by the approximate methods.