Uncertainty Quantification and the Numerical Solution of Random Differential Equations

Uncertainty Quantification (UQ) is a rapidly developing field within applied mathematics and computational science concerned with identifying, modeling and quantifying the myriad uncertainties arising in computational models, specifically those involving differential equations. Besides discretization error and rounding error, which by now are well understood and can be managed, uncertainties in problem specifications such as coefficient functions, source and boundary data or domain geometry can often dominate the previously mentioned uncertainty and error sources.

In this talk we discuss the prevalent model problem in UQ, stationary diffusion with an uncertain diffusion coefficient which is modeled as a lognormal random field. We emphasize uncertainty modeling aspects in connection with a groundwater contamination problem leading to such a random diffusion equation and present numerical approaches for approximating its solution. In particular, the numerical representation of rough lognormal fields by Karhunen-Loève expansion raises interesting questions of stochastic homogenization. Continuing with the groundwater application, we also indicate how observational data can be incorporated to reduce uncertainty by large-scale Bayesian inference.