  We consider the problem of efficiently computing a discrete Morse complex on
simplicial complexes of arbitrary dimension and very large size. Based on a
common graph-based formalism, we analyze existing data structures for
simplicial complexes, and we define an efficient encoding for the discrete
Morse gradient on the most compact of such representations. We theoretically
compare methods based on reductions and coreductions for computing a discrete
Morse gradient, proving that the combination of reductions and coreductions
produces new mutually equivalent approaches. We design and implement a new
algorithm for computing a discrete Morse complex on simplicial complexes. We
show that our approach scales very well with the size and the dimension of the
simplicial complex also through comparisons with the only existing
public-domain algorithm for discrete Morse complex computation. We discuss
applications to the computation of multi-parameter persistent homology and of
extrema graphs for visualization of time-varying 3D scalar fields.
