Multilevel Hybrid Chernoff Tau-leap

TL;DR

This paper extends the hybrid Chernoff tau-leap method to multilevel Monte Carlo, enabling efficient and accurate stochastic simulations with nearly optimal computational complexity.

Contribution

It introduces a novel multilevel coupling algorithm and variance estimation technique for the hybrid Chernoff tau-leap method, improving efficiency over previous single-level approaches.

Findings

Achieves computational complexity of order TOL^{-2} with smaller constants.

Demonstrates substantial gains over single-level and SSA methods.

Bounds global error within prescribed tolerance and confidence level.

Abstract

In this work, we extend the hybrid Chernoff tau-leap method to the multilevel Monte Carlo (MLMC) setting. Inspired by the work of Anderson and Higham on the tau-leap MLMC method with uniform time steps, we develop a novel algorithm that is able to couple two hybrid Chernoff tau-leap paths at different levels. Using dual-weighted residual expansion techniques, we also develop a new way to estimate the variance of the difference of two consecutive levels and the bias. This is crucial because the computational work required to stabilize the coefficient of variation of the sample estimators of both quantities is often unaffordable for the deepest levels of the MLMC hierarchy. Our method bounds the global computational error to be below a prescribed tolerance, $TOL$, within a given confidence level. This is achieved with nearly optimal computational work. Indeed, the computational complexity of our method is of order $\mathcal{O}\left(TOL^{-2}\right)$, the same as with an exact method, but with a smaller constant. Our numerical examples show substantial gains with respect to the previous single-level approach and the Stochastic Simulation Algorithm.