  The forcing of particles in turbulent environments influences dynamical
properties pertinent to many fundamental applications involving particle-flow
interactions. Current study explores the determination of forcing for one-way
coupled passive particles, under the assumption that the ambient velocity
fields are known. When measurements regarding particle locations are available
but sparse, direct evaluation of the forcing is intractable. Nevertheless, the
forcing for finite-size particles can be determined using adjoint-based data
assimilation. This inverse problem is formulated with the framework of
optimization, where the cost function is defined as the difference between the
measured and predicted particle locations. The gradient of the cost function,
with respect to the forcing can be calculated from the adjoint dynamics. When
measurements are subject to Gaussian noise, samples within the probability
distribution of the forcing can be drawn using Hamiltonian Monte Carlo. The
algorithm is tested in the Arnold-Beltrami-Childress flow as well as the
homogeneous isotropic turbulence. Results demonstrate that the forcing can only
be determined accurately for particle Reynolds number between 1 and 5, where
the majority of Reynolds number history along the particle trajectory falls in.
