Bayesian inference of an uncertain generalized diffusion operator

This paper defines a novel Bayesian inverse problem to infer an infinite-dimensional uncertain operator appearing in a differential equation, whose action on an observable state variable affects its dynamics. The operator is parametrized using its eigendecomposition, which enables prior information to be incorporated into its formulation. The Bayesian inverse problem is defined in terms of an uncertain, generalized diffusion operator appearing in an evolution equation for a contaminant's transport through a heterogeneous porous medium. Limited observations of the state variable's evolution are used as data for inference, and the dependence on the solution of the inverse problem is studied as a function of the frequency of observations, as well as on whether or not the data is collected as a spatial or time series.