Computational complexity analysis for Monte Carlo approximations of classically scaled population processes

TL;DR

This paper compares the computational complexity of various Monte Carlo simulation strategies for population processes, including biochemical systems, under different bias and accuracy conditions, highlighting when multilevel Monte Carlo methods are most efficient.

Contribution

It provides a systematic analysis of the efficiency of Monte Carlo methods, including multilevel variants, for population process simulations with a novel asymptotic complexity framework.

Findings

Multilevel Monte Carlo for diffusion approximation is most efficient if bias is small.

Tau-leaping methods are preferable if diffusion bias exceeds error tolerance.

The analysis introduces a new asymptotic regime based on model scaling parameters.

Abstract

We analyze and compare the computational complexity of different simulation strategies for Monte Carlo in the setting of classically scaled population processes. This allows a range of widely used competing strategies to be judged systematically. Our setting includes stochastically modeled biochemical systems. We consider the task of approximating the expected value of some path functional of the state of the system at a fixed time point. We study the use of standard Monte Carlo when samples are produced by exact simulation and by approximation with tau-leaping or an Euler-Maruyama discretization of a diffusion approximation. Appropriate modifications of recently proposed multilevel Monte Carlo algorithms are also studied for the tau-leaping and Euler-Maruyama approaches. In order to quantify computational complexity in a tractable yet meaningful manner, we consider a parameterization that, in the mass action chemical kinetics setting, corresponds to the classical system size scaling. We base the analysis on a novel asymptotic regime where the required accuracy is a function of the model scaling parameter. Our new analysis shows that, under the specific assumptions made in the manuscript, if the bias inherent in the diffusion approximation is smaller than the required accuracy, then multilevel Monte Carlo for the diffusion approximation is most efficient, besting multilevel Monte Carlo with tau-leaping by a factor of a logarithm of the scaling parameter. However, if the bias of the diffusion model is greater than the error tolerance, or if the bias can not be bounded analytically, multilevel versions of tau-leaping are often the optimal choice.