In this paper we present a general, axiomatical framework for the rigorous
approximation of invariant densities and other important statistical features
of dynamics. We approximate the system trough a finite element reduction, by
composing the associated transfer operator with a suitable finite dimensional
projection (a discretization scheme) as in the well-known Ulam method.
We introduce a general framework based on a list of properties (of the system
and of the projection) that need to be verified so that we can take advantage
of a so-called ``coarse-fine'' strategy. This strategy is a novel method in
which we exploit information coming from a coarser approximation of the system
to get useful information on a finer approximation, speeding up the
computation. This coarse-fine strategy allows a precise estimation of invariant
densities and also allows to estimate rigorously the speed of mixing of the
system by the speed of mixing of a coarse approximation of it, which can easily
be estimated by the computer.
The estimates obtained here are rigourous, i.e., they come with exact error
bounds that are guaranteed to hold and take into account both the
discretiazation and the approximations induced by finite-precision arithmetic.
We apply this framework to several discretization schemes and examples of
invariant density computation from previous works, obtaining a remarkable
reduction in computation time.
We have implemented the numerical methods described here in the Julia
programming language, and released our implementation publicly as a Julia
package