Abstract:
Complex interacting networks are prevalent across many fields of science and engineering, ranging from chemical kinetics and pharmacology to social sciences. In biochemistry, such networks arise through the reactions between the cell's components such as DNA and RNA molecules. Since these key biomolecules often appear in low copy numbers, the intrinsic randomness of their interactions becomes significant. This calls for a stochastic framework that treats the chemical populations as a continuous-time, discrete-state Markov process. The time-dependent probability distribution of this process is the solution of the chemical master equation (CME). Despite many potential benefits, the CME is notoriously difficult to solve due to the curse of dimensionality. This dissertation is about numerical methods that seek to alleviate this curse. We introduce three tools for this challenging task. The first tool aims to exploit sparsity in the solution of the CME by only keeping states that have a significant probability mass at each time step. It uses the stochastic simulation algorithm to quickly scout these important states, and advance the integration using Krylov subspace approximation techniques. The second tool applies a grid-based aggregation method that is well expressed in terms of the tensor product. It employs a residual-based error control to automatically adapt the aggregation scheme to the region that requires fine details. The third tool leverages novel results in the tensor train format. It combines the finite state projection, tensor-based linear system solvers, and the inexact uniformization method. Numerical results are conducted to show the efficiency of the proposed methods.
