Variational methods for uncertainty quantification

Abstract

A very common problem in science and engineering is the determination of the effects of uncertainty or variation in parameters and data on the output of a deterministic nonlinear operator. For example, such variations may describe the effect of experimental error or may arise as part of a sensitivity analysis of the model. In this dissertation, we present an approach for ascertaining the effects of variations and uncertainty in parameters in a differential equation that is based on techniques borrowed from a posteriori error analysis for finite element methods. The generalized Green's function is used to describe how variation propagates into the solution around localized points in the parameter space. This information can be used either to create a higher order method or produce an error estimate for information computed from a given representation. In the latter case, this provides the basis for adaptive sampling. Both the higher order method and the adaptive sampling methods provide a powerful alternative to traditional Monte-Carlo methods, and provide a more detailed picture of how various regions in parameter space affect the output of the nonlinear operator.