  Recently, some works have suggested methods to combine variational
probabilistic inference with Monte Carlo sampling. One promising approach is
via local optimal transport. In this approach, a gradient steepest descent
method based on local optimal transport principles is formulated to transform
deterministically point samples from an intermediate density to a posterior
density. The local mappings that transform the intermediate densities are
embedded in a reproducing kernel Hilbert space (RKHS). This variational mapping
method requires the evaluation of the log-posterior density gradient and
therefore the adjoint of the observational operator. In this work, we evaluate
nonlinear observational mappings in the variational mapping method using two
approximations that avoid the adjoint, an ensemble based approximation in which
the gradient is approximated by the particle covariances in the state and
observational spaces the so-called ensemble space and an RKHS approximation in
which the observational mapping is embedded in an RKHS and the gradient is
derived there. The approximations are evaluated for highly nonlinear
observational operators and in a low-dimensional chaotic dynamical system. The
RKHS approximation is shown to be highly successful and superior to the
ensemble approximation.
