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.