An adaptive solution to the chemical master equation using tensors.

Area of Science:

• Computational Chemistry
• Applied Mathematics
• Chemical Kinetics

Background:

• The chemical master equation (CME) describes the dynamics of chemical systems but is computationally intractable for high-dimensional problems due to the curse of dimensionality.
• Existing numerical methods struggle to scale efficiently with the number of species or reaction channels.

Purpose of the Study:

• To develop a scalable numerical method for solving the CME in high-dimensional systems.
• To leverage the quantized tensor train (QTT) format for efficient representation and computation of the CME solution.

Main Methods:

• A numerical scheme based on the quantized tensor train (QTT) format was developed to represent the CME solution in a compressed form.
• The finite state projection was adapted within the QTT framework, allowing adaptive expansion based on error criteria.
• A QTT-formatted matrix exponential was evaluated using inexact uniformization and the alternating minimal energy algorithm.

Main Results:

• The proposed QTT-based method achieves a linear scaling with system dimension, overcoming the curse of dimensionality.
• An efficient test was developed to detect the attainment of equilibrium distribution by exploiting the QTT structure.
• Numerical tests on high-dimensional problems, previously out of reach, were successfully performed.

Conclusions:

• The QTT-based numerical scheme provides a powerful and scalable approach for solving high-dimensional chemical master equations.
• This method significantly expands the scope of problems addressable in computational chemistry and systems biology.
• The technique offers a computationally efficient alternative to classical methods for complex chemical kinetics analysis.