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Applied Math Seminar – Khanh Dinh (University of Alabama)
March 30, 2018 @ 11:00 am - 12:00 pm
Title: An adaptive Magnus expansion method for solving the chemical master equation with time-dependent propensities
Abstract: The chemical master equation (CME) is a system of ordinary differential equations (ODEs) to model the chemical interaction of molecular species. The large size of the state space of the system makes solving the CME difficult, and this has motivated reduction strategies such as
the finite state projection (FSP). Moreover, if the reaction rates are functions of the time, the CME becomes an ODE problem with time-dependent coefficients. Solution techniques include Monte Carlo algorithms, such as the stochastic simulation algorithm (SSA) or ODE
solvers, such as Adams-PECE, Runge-Kutta and backward-differentiation formula (BDF). There are also Magnus-based solvers that have however not been thoroughly explored in the CME context. Here we introduce an adaptive time-stepping Magnus-SSA algorithm, in which the
CME is solved using a Magnus expansion with not only a variable time-step but also with a variable state space that changes at each step via the SSA, and several error approximation approaches are attempted to monitor the adaptivity. We perform comparative tests against
the classical Adams-PECE, Runge-Kutta and BDF methods on three biological problems, showing that the proposed adaptive Magnus-based variants can be efficient when the CME with time-dependent rates is stiff.