Simultaneous state-time approximation of the chemical master equation using tensor product formats

TL;DR

This paper introduces tensor product formats like TT, QTT, and QTT-Tucker to efficiently solve high-dimensional chemical master equations, especially for gene regulation networks, achieving significant reductions in computational complexity.

Contribution

The paper demonstrates the application of novel tensor formats to solve high-dimensional CME problems with proven good separability properties and logarithmic complexity dependence.

Findings

Tensor formats effectively solve high-dimensional CME.

QTT-based methods reduce computational complexity logarithmically.

Approach compares favorably with existing methods.

Abstract

We study the application of the novel tensor formats (TT, QTT, QTT-Tucker) to the solution of $d$-dimensional chemical master equations, applied mostly to gene regulating networks (signaling cascades, toggle switches, phage-$\lambda$). For some important cases, e.g. signaling cascade models, we prove good separability properties of the system operator. The Quantized tensor representations (QTT, QTT-Tucker) are employed in both state space and time, and the global state-time $(d+1)$-dimensional system is solved in the structured form by using the ALS-type iteration. This approach leads to the logarithmic dependence of the computational complexity on the system size. When possible, we compare our approach with the direct CME solution and some previously known approximate schemes, and observe a good potential of the newer tensor methods in simulation of relevant biological systems.