  Giant resonances (GRs) are a striking manifestation of collective motions in
mesoscopic systems such as atomic nuclei. Until recently, theoretical
investigations have essentially relied on the (quasiparticle) random phase
approximation ((Q)RPA), and extensions of it, based on phenomenological energy
density functionals (EDFs). As part of a current effort to describe GRs within
an ab initio theoretical scheme, the present work promotes the use of the
projected generator coordinate method (PGCM). This method, which can handle
anharmonic effects while satisfying symmetries of the nuclear Hamiltonian,
displays a favorable (i.e. mean-field-like) scaling with system's size.
Presently focusing on the isoscalar giant monopole resonance (GMR) of light-
and medium-mass nuclei, PGCM's potential to deliver wide-range ab initio
studies of GRs in closed- and open-shell nuclei encompassing pairing,
deformation, and shape coexistence effects is demonstrated. The comparison with
consistent QRPA calculations highlights PGCM's unique attributes and sheds
light on the intricate interplay of nuclear collective excitations. The present
paper is the first in a series of four and focuses on technical aspects and
uncertainty quantification of ab initio PGCM calculations of GMR using the
doubly open-shell $^{46}$Ti as an illustrative example. The second paper
displays results for a set of nuclei of physical interest and proceeds to the
comparison with consistent (deformed) ab initio QRPA calculations. While the
third paper analyzes useful moments of the monopolar strength function and
different ways to access them within PGCM calculations, the fourth paper
focuses on the effect of the symmetry restoration on the monopole strength
function.
