An adaptive solution to the chemical master equation using tensors

Solving the chemical master equation directly is difficult due to the curse of dimensionality. We tackle that challenge by a numerical scheme based on the quantized tensor train (QTT) format, which enables us to represent the solution in a compressed form that scales linearly with the dimension. We recast the finite state projection in this QTT framework and allow it to expand adaptively based on proven error criteria. The end result is a QTT-formatted matrix exponential that we evaluate through a combination of the inexact uniformization technique and the alternating minimal energy algorithm. Our method can detect when the equilibrium distribution is reached with an inexpensive test that exploits the structure of the tensor format. We successfully perform numerical tests on high-dimensional problems that had been out of reach for classical approaches.