Local error estimates for adaptive simulation of the Reaction-Diffusion Master Equation via operator splitting

TL;DR

This paper develops a systematic way to estimate and control the local error in operator splitting methods for simulating the reaction-diffusion master equation, enabling adaptive timestep selection for improved efficiency.

Contribution

It introduces a novel approach to estimate local errors and adaptively select timesteps in operator splitting methods for RDME simulation, enhancing accuracy and efficiency.

Findings

Derived local error estimates for operator splitting in RDME

Proposed an adaptive timestep strategy based on error estimates

Extended the DFSP method to include temporal adaptivity

Abstract

The efficiency of exact simulation methods for the reaction-diffusion master equation (RDME) is severely limited by the large number of diffusion events if the mesh is fine or if diffusion constants are large. Furthermore, inherent properties of exact kinetic-Monte Carlo simulation methods limit the efficiency of parallel implementations. Several approximate and hybrid methods have appeared that enable more efficient simulation of the RDME. A common feature to most of them is that they rely on splitting the system into its reaction and diffusion parts and updating them sequentially over a discrete timestep. This use of operator splitting enables more efficient simulation but it comes at the price of a temporal discretization error that depends on the size of the timestep. So far, existing methods have not attempted to estimate or control this error in a systematic manner. This makes the solvers hard to use for practitioners since they must guess an appropriate timestep. It also makes the solvers potentially less efficient than if the timesteps are adapted to control the error. Here, we derive estimates of the local error and propose a strategy to adaptively select the timestep when the RDME is simulated via a first order operator splitting. While the strategy is general and applicable to a wide range of approximate and hybrid methods, we exemplify it here by extending a previously published approximate method, the Diffusive Finite-State Projection (DFSP) method, to incorporate temporal adaptivity.