Representing model-form uncertainty from missing microstructural information

Abstract

A common challenge in modeling multiscale phenomena lies in representing the dependence of macroscopic quantities on microscale dynamics. Incomplete information or limitations in computational resources make it impossible to resolve the microscale dynamics and their effect on those at the macroscale. To obtain a model of the phenomenon that can be used to make predictions, approximations must be made. For instance, it is commonly assumed that microscale effects on the macroscale can be represented with macroscopic quantities, effectively removing any dependence on the microstate. Such approximations introduce uncertainty in the model. When the approximations are invalid, the uncertainty is significant and must be quantified to assess the reliability of the model. This work focuses on the formulation of a model-form uncertainty representation to account for such missing dependencies. The process by which a model-form uncertainty representation is formulated is an open area of research, so particular attention is paid to determining the feasibility and inherent challenges of its development.

The representation is developed in the context of a simplified testbed problem, accounting for uncertainty in a model of mean contaminant transport through a heterogeneous porous medium. In heterogeneous media, the evolution of the mean depends on small-scale fluctuations of the flow velocity from its mean and their induced fluctuations on the detailed concentration field. However, these fluctuations can neither be observed nor resolved. In this work, model-form uncertainty caused by the unresolved dependence on the small-scale fluctuations is represented as an infinite-dimensional stochastic operator acting on the mean concentration. Physical constraints are enforced through its eigendecomposition, and uncertainty is encoded in its eigenvalues by casting them as random variables. The feasibility of inferring their mean values using observations of the mean concentration is explored, and a novel method of extracting samples from their distribution using direct numerical simulation is discussed. These findings are used to develop a stochastic model for the probability distribution of the operator's eigenvalues, and its validity is assessed using forward propagation of uncertainty to the mean concentration.