  The nuclear fission process is a dramatic example of the large-amplitude
collective motion in which the nucleus undergoes a series of shape changes
before splitting into distinct fragments. This motion can be represented by a
pathway in the many-dimensional space of collective coordinates. The collective
action along the fission pathway determines the spontaneous fission half-lives
as well as mass and charge distributions of fission fragments.
  We study the performance and precision of various methods to determine the
minimum action and minimum-energy fission trajectories in the collective space.
  We apply the nudged elastic band method (NEB), grid-based methods, and Euler
Lagrange approach to the collective action minimization in two and three
dimensional collective spaces.
  The performance of various approaches to the fission pathway problem is
assessed by studying the collective motion along both analytic energy surfaces
and realistic potential energy surfaces obtained with the
Hartree-Fock-Bogoliubov theory. The uniqueness and stability of the solutions
is studied. The NEB method is capable of efficient determination of the exit
points on the outer turning surface that characterize the most probable fission
pathway and constitute the key input for fission studies. This method can also
be used to accurately compute the critical points (i.e., local minima and
saddle points) on the potential energy surface of the fissioning nucleus that
determine the static fission path.
  The NEB method is the tool of choice for finding the least-action and minimum
energy fission trajectories. It will be particularly useful in large-scale
fission calculation of superheavy nuclei and neutron-rich fissioning nuclei
contributing to the astrophysical r-process recycling.
