Low-rank tensor methods for Markov chains with applications to tumor progression models

Abstract: Continuous-time Markov chains describing interacting processes exhibit a state space that grows exponentially in the number of processes. This state-space explosion renders the computation or storage of the time-marginal distribution, which is defined as the solution of a certain linear system, infeasible using classical methods. We consider Markov chains whose transition rates are separable functions, which allows for an efficient low-rank tensor representation of the operator of this linear system. Typically, the right-hand side also has low-rank structure, and thus we can reduce the cost for computation and storage from exponential to linear. Previously known iterative methods also allow for low-rank approximations of the solution but are unable to guarantee that its entries sum up to one as required for a probability distribution. We derive a convergent iterative method using low-rank formats satisfying this condition. We also perform numerical experiments illustrating that the marginal distribution is well approximated with low rank.