Speeding up stochastic and deterministic simulation by aggregation: An advanced tutorial

Dynamical models of systems across many branches of science and engineering can be mathematically represented in terms of stochastic processes such as Markov chains, or deterministically through a system of difference or differential equations. Unfortunately, in all but special cases these models do not enjoy analytical solutions, hence one is left with computer-based approaches by means of stochastic simulators and numerical solvers. As a consequence, the computational cost increases with the dimensionality of the model under consideration, hindering our capability of dealing with complex large-scale models arising from accurate mechanistic descriptions of real-world systems. This paper offers an advanced tutorial on an array of recently developed algorithms that seek to tame the complexity of these models by aggregating their constituting systems of equations, leading to lower-dimensional systems that preserve the original dynamics in some appropriate, formal sense.