  Information about the behavior of dynamical systems can often be obtained by
analyzing the eigenvalues and corresponding eigenfunctions of linear operators
associated with a dynamical system. Examples of such operators are the
Perron-Frobenius and the Koopman operator. In this paper, we will review
different methods that have been developed over the last decades to compute
finite-dimensional approximations of these infinite-dimensional operators -
e.g. Ulam's method and Extended Dynamic Mode Decomposition (EDMD) - and
highlight the similarities and differences between these approaches. The
results will be illustrated using simple stochastic differential equations and
molecular dynamics examples.
