Random Inverse Problems with Structural and Probabilistic Ambiguities

Abstract

In this paper, we investigate a computational class of random inverse problems that incorporates model uncertainties through random variable parameters nonlinearly in the forward model as well as additive observational uncertainty. Random inverse problems with nonlinear parameter dependencies may arise in engineering, geophysics, image processing or uncertainty quantification. We study a perspective on structural ambiguities due to the non-injectivity of the forward model together with probabilistic ambiguities by assigning mixture model densities with separate components to the parameters, which leads to an in general complex forward model, observation model and posterior. As a result, the mixture-model parameters in the forward model can be viewed as simultaneously describing aspects of both the nonlinear ambiguity and the uncertainty of the inverse problem. The presented solution algorithms are based on Bayesian inversion and Monte Carlo computation for three observation scenarios leading to posterior densities for the input for given output samples or an observed output density. We apply the derived algorithms to analytic 1D and 2D quadratic models, to an epidemiological inverse parameter estimation and to a heat equation for inverse source localization. We numerically demonstrate in which observation scenarios the proposed algorithm can resolve probabilistic ambiguities in the solution and in which they cannot and show the interaction patterns between these two types of ambiguities. The results suggest that the proposed perspective is useful in making residual structural ambiguities visible in highly ambiguous inverse problems, including cases with a finite and an infinite number of solutions.