  We consider the problem of developing a method to reconstruct a potential $q$
from the partial data Dirichlet-to-Neumann map for the Schr\"odinger equation
$(-\Delta_g+q)u=0$ on a fixed admissible manifold $(M,g)$. If the part of the
boundary that is inaccessible for measurements satisfies a flatness condition
in one direction, then we reconstruct the local attenuated geodesic ray
transform of the one-dimensional Fourier transform of the potential $q$. This
allows us to reconstruct $q$ locally, if the local (unattenuated) geodesic ray
transform is constructively invertible. We also reconstruct $q$ globally, if
$M$ satisfies certain concavity condition and if the global geodesic ray
transform can be inverted constructively. These are reconstruction procedures
for the corresponding uniqueness results given by Kenig and Salo. Moreover, the
global reconstruction extends and improves the constructive proof of Nachman
and Street in Euclidean setting. We derive a certain boundary integral equation
which involves the given partial data and describes the traces of complex
geometrical optics solutions. For construction of complex geometrical optics
solutions, following Nachman and Street and improving their arguments, we use a
new family of Green's functions for the Laplace-Beltrami operator and the
corresponding single layer potentials. The constructive inversion problem for
local or global geodesic ray transforms is one of the major topics of interest
in integral geometry.
