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Variants of the Krylov-based Finite State Projection algorithm for solving the Chemical Master Equation

From Q-bio

The dynamics of the composition of molecules that regulate the signaling and metabolism within biological cells can be modeled as a continuous time Markov chain, whose underlying probability distribution is governed by the so-called Chemical Master Equation (CME). Solving the CME is a computationally extensive task, since the size of the problem grows exponentially with the number of chemical species involved. Addressing this so-called ‘curse of dimensionality’ was the catalyst of the Finite State Projection (FSP) algorithm of Munsky and Khammash that truncated the state space to a more tractable size. The subsequent Krylov-based FSP provided an early improvement over the basic FSP by using the Krylov subspace method in Sidje’s Expokit to compute the matrix exponential, as well as a built-in step-by-step integration scheme. We will discuss how some parameters of the algorithm, such as the dimension of the Krylov basis, can be made variable to adapt to the problem size. We also discuss a promising approach in selecting the projection space, which is a crucial component in any FSP-based methods. Preliminary numerical results will be reported. Supported by NSF Grant 1320849.