Federico Poloni: Rigorous approximation of invariant measures using Ulam's method

We focus on the problem of estimating invariant measures for a family of expanding dynamical systems. For this task, we use a technique formally similar to the finite element method (although the underlying problem is not a differential one), known as Ulam's method. Our goal is obtaining in a reasonable time a rigorous estimate that keeps track of every source of error (including inexact linear algebra and IEEE arithmetic error). An important step is bounding rigorously the conditioning of the problem using limited information: (an analogous of) the coercivity constant for the continuous problem is not available. The techniques needed to replace it involve a two-grid approximation strategy and a special inequality for the dynamical system at hand (Lasota-Yorke inequality).