  The Discrete Morse Theory of Forman appeared to be useful for providing
filtration-preserving reductions of complexes in the study of persistent
homology. So far, the algorithms computing discrete Morse matchings have only
been used for one-dimensional filtrations. This paper is perhaps the first
attempt in the direction of extending such algorithms to multidimensional
filtrations. Initial framework related to Morse matchings for the
multidimensional setting is proposed, and a matching algorithm given by King,
Knudson, and Mramor is extended in this direction. The correctness of the
algorithm is proved, and its complexity analyzed. The algorithm is used for
establishing a reduction of a simplicial complex to a smaller but not
necessarily optimal cellular complex. First experiments with filtrations of
triangular meshes are presented.
