  We formulate a novel approach to solve a class of stochastic problems,
referred to as data-consistent inverse (DCI) problems, which involve the
characterization of a probability measure on the parameters of a computational
model whose subsequent push-forward matches an observed probability measure on
specified quantities of interest (QoI) typically associated with the outputs
from the computational model. Whereas prior DCI solution methodologies focused
on either constructing non-parametric estimates of the densities or the
probabilities of events associated with the pre-image of the QoI map, we
develop and analyze a constrained quadratic optimization approach based on
estimating push-forward measures using weighted empirical distribution
functions. The method proposed here is more suitable for low-data regimes or
high-dimensional problems than the density-based method, as well as for
problems where the probability measure does not admit a density. Numerical
examples are included to demonstrate the performance of the method and to
compare with the density-based approach where applicable.
