  Let $K$ be a finite simplicial, cubical, delta or CW complex. The persistence
map $\mathrm{PH}$ takes a filter $f:K \rightarrow \mathbb{R}$ as input and
returns the barcodes $\mathrm{PH}(f)$ of the associated sublevel set persistent
homology modules. We address the inverse problem: given a target barcode $D$,
computing the fiber $\mathrm{PH}^{-1}(D)$. For this, we use the fact that
$\mathrm{PH}^{-1}(D)$ decomposes as complex of polyhedra when $K$ is a
simplicial complex, and we generalise this result to arbitrary based chain
complexes. We then design and implement a depth first search algorithm that
recovers the polyhedra forming the fiber $\mathrm{PH}^{-1}(D)$. As an
application, we solve a corpus of 120 sample problems, providing a first
insight into the statistical structure of these fibers, for general CW
complexes.
