  Biological and physical systems often exhibit distinct structures at
different spatial/temporal scales. Persistent homology is an algebraic tool
that provides a mathematical framework for analyzing the multi-scale structures
frequently observed in nature. In this paper a theoretical framework for the
algorithmic computation of an arbitrarily good approximation of the persistent
homology is developed. We study the filtrations generated by sub-level sets of
a function $f : X \to \mathbb{R}$, where $X$ is a CW-complex. In the special
case $X = [0,1]^N$, $N \in \mathbb{N}$ we discuss implementation of the
proposed algorithms. We also investigate a priori and a posteriori bounds of
the approximation error introduced by our method.
