An iterative aggregation/disaggregation procedure for modelling the long-term behaviour of continuous-time evanescent random processes

Abstract

We describe a multilevel aggregation/disaggregation method for finding the quasistationary distribution and decay parameters for continuous-time Markov chains. This complements existing techniques dealing with the discrete-time case, or for finding the stationary distribution. Finding the quasistationary distribution is equivalent to calculating the eigenvector corresponding to the smallest eigenvalue of the q-matrix restricted to the non-absorbing class. The method presented here is similar to an algebraic multigrid, with restriction operators that depend on the current approximation to the solution. The smoothers are short Arnoldi iterations or Gauss-Seidel iterations. Numerical results are presented for a variety of models of differing character, including simple epidemic, bivariate SIS, predator-prey, and the cubic auto-catalator. These indicate that the number of cycles required grows only very slowly with the size of the problem.

Keywords:

• Quasistationary distribution
• numerical algorithm
• V-cycle