  We compute approximate solutions to inverse problems for determining
parameters in differential equation models with stochastic data on output
quantities. The formulation of the problem and modeling framework define a
solution as a probability measure on the parameter domain for a given
$\sigma-$algebra. In the case where the number of output quantities is less
than the number of parameters, the inverse of the map from parameters to data
defines a type of generalized contour map. The approximate contour maps define
a geometric structure on events in the $\sigma-$algebra for the parameter
domain. We develop and analyze an inherently non-intrusive method of sampling
the parameter domain and events in the given $\sigma-$algebra to approximate
the probability measure. We use results from stochastic geometry for point
processes to prove convergence of a random sample based approximation method.
We define a numerical $\sigma-$algebra on which we compute probabilities and
derive computable estimates for the error in the probability measure. We
present numerical results to illustrate the various sources of error for a
model of fluid flow past a cylinder.
