A low-rank complexity reduction algorithm for the high-dimensional kinetic chemical master equation

TL;DR

This paper introduces a dynamical low-rank algorithm for the chemical master equation that reduces computational complexity by partitioning reaction networks, offering a noise-free alternative to stochastic simulation with improved accuracy and efficiency.

Contribution

The paper presents a novel low-rank approximation method for the CME that partitions reaction networks, reducing dimensionality and computational costs while maintaining accuracy.

Findings

Significant memory and runtime reduction in example cases.

Better accuracy than stochastic simulation algorithm (SSA).

Noise-free computation enabling tail probability resolution.

Abstract

It is increasingly realized that taking stochastic effects into account is important in order to study biological cells. However, the corresponding mathematical formulation, the chemical master equation (CME), suffers from the curse of dimensionality and thus solving it directly is not feasible for most realistic problems. In this paper we propose a dynamical low-rank algorithm for the CME that reduces the dimensionality of the problem by dividing the reaction network into partitions. Only reactions that cross partitions are subject to an approximation error (everything else is computed exactly). This approach, compared to the commonly used stochastic simulation algorithm (SSA, a Monte Carlo method), has the advantage that it is completely noise-free. This is particularly important if one is interested in resolving the tails of the probability distribution. We show that in some cases (e.g. for the lambda phage) the proposed method can drastically reduce memory consumption and run time and provide better accuracy than SSA.