Solving Stochastic Inverse Problems using Sigma-Algebras on Contour Maps

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.