The basic goal of topological data analysis is to apply topology-based descriptors
to understand and describe the shape of data. In this context, homology is one of
the most relevant topological descriptors, well-appreciated for its discrete nature,
computability and dimension independence. A further development is provided
by persistent homology, which allows to track homological features along a oneparameter
increasing sequence of spaces. Multiparameter persistent homology, also
called multipersistent homology, is an extension of the theory of persistent homology
motivated by the need of analyzing data naturally described by several parameters,
such as vector-valued functions. Multipersistent homology presents several issues in
terms of feasibility of computations over real-sized data and theoretical challenges
in the evaluation of possible descriptors. The focus of this thesis is in the interplay
between persistent homology theory and discrete Morse Theory. Discrete Morse
theory provides methods for reducing the computational cost of homology and persistent
homology by considering the discrete Morse complex generated by the discrete
Morse gradient in place of the original complex. The work of this thesis addresses
the problem of computing multipersistent homology, to make such tool usable in real
application domains. This requires both computational optimizations towards the
applications to real-world data, and theoretical insights for finding and interpreting
suitable descriptors. Our computational contribution consists in proposing a new
Morse-inspired and fully discrete preprocessing algorithm. We show the feasibility
of our preprocessing over real datasets, and evaluate the impact of the proposed
algorithm as a preprocessing for computing multipersistent homology. A theoretical
contribution of this thesis consists in proposing a new notion of optimality for such
a preprocessing in the multiparameter context. We show that the proposed notion
generalizes an already known optimality notion from the one-parameter case. Under
this definition, we show that the algorithm we propose as a preprocessing is optimal
in low dimensional domains. In the last part of the thesis, we consider preliminary
applications of the proposed algorithm in the context of topology-based multivariate
visualization by tracking critical features generated by a discrete gradient field compatible
with the multiple scalar fields under study. We discuss (dis)similarities of such
critical features with the state-of-the-art techniques in topology-based multivariate
data visualization
