Computing the equilibrium of the chemical master equation using tensors
with sliding windows
When studying a system of chemical reactions on the cellular level, it
is often helpful to use the chemical master equation (CME) that results
from modeling the system using a continuous-time Markov chain.
Furthermore, the system’s long-term behavior can be explored by
computing the stationary solution to the CME. However, the number of
states involved grows exponentially with the number of chemical species
tracked. In some cases, the state space may even be countably inﬁnite.
To cope with this issue, a potent strategy is to restrict to a ﬁnite
state projection (FSP) and represent the transition matrix and
probability vector in quantized tensor train (QTT) format. Here, we
employ our adaptive FSP tensor-based solver with sliding windows as well
as the method of using Reaction Rate Equations (RREs) to estimate the
probability mass function when the system is in statistical equilibrium.
Using RREs, we ﬁrst cheaply get an approximation of the steady state,
which is then fed to our adaptive QTT solver to reach the equilibrium
quickly. We refer to this solver as FSP-QTT-SS. We include numerical
experiments to show the eﬃciency of our approach.