  A persistence module $M$, with coefficients in a field $\mathbb{F}$, is a
finite-dimensional linear representation of an equioriented quiver of type
$A_n$ or, equivalently, a graded module over the ring of polynomials
$\mathbb{F}[x]$. It is well-known that $M$ can be written as the direct sum of
indecomposable representations or as the direct sum of cyclic submodules
generated by homogeneous elements. An interval basis for $M$ is a set of
homogeneous elements of $M$ such that the sum of the cyclic submodules of $M$
generated by them is direct and equal to $M$. We introduce a novel algorithm to
compute an interval basis for $M$. Based on a flag of kernels of the structure
maps, our algorithm is suitable for parallel or distributed computation and
does not rely on a presentation of $M$. This algorithm outperforms the approach
via the presentation matrix and Smith Normal Form. We specialize our parallel
approach to persistent homology modules, and we close by applying the proposed
algorithm to tracking harmonics via Hodge decomposition.
