Research in Uncertainty Quantification
In traditional mathematical modelling of physical processes, we solve PDEs with prescribed inputs. That is, material properties, domain geometries, boundary conditions and forcing terms are all assumed to be known exactly. In real-life applications, scientists never have access to all this information and many text-book deterministic PDE models are essentially useless. If the inputs to the systems under consideration are subject to uncertainty, in the sense that a complete deterministic description is unavailable, we require mathematical techniques for propagating this uncertainty to the output quantities of interest and for computing probabilities of events rather than specific solutions.
An important application is fluid flow in porous media, which is often modelled under the assumption that the conductivity coefficients of the porous medium are known at every spatial location in the flow domain. Simulations based on such over-simplifications cannot provide quantification of probabilities of unfavourable events (e.g., the probability that a concentration of a contaminant transported in the flow exceeds a given level.) Realistically, only a handful of conductivity measurements may be available, yet we would like to compute, at the very least, the expected flow field.
In the Manchester Numerical Analysis Group, in addition to solving classical stochastic differential equations of white noise type, we are particularly interested in solving PDEs with correlated random field inputs. In these models, the uncertain input parameters are treated as random variables at each spatial location, with a known correlation structure. The solution variables sought are consequently also random fields.
We have expertise in using traditional Monte Carlo methods to solve such problems, but are currently exploring newer, more efficient technologies such as stochastic finite element methods (SFEMS). These methods provide a framework for incorporating statistical information about spatial variability in material parameters into computer simulations so that probabilistic information about the solution is obtained directly. Some SFEMs, however, give rise to a very large, coupled system of linear equations, rather than the many smaller decoupled problems encountered in Monte Carlo simulations and consequently have suffered from bad press.
Our current research includes developing fast solvers and robust preconditioners for the linear systems that arise from a range of linear and non-linear stochastic Galerkin formulations of Darcy flow problems and recent grant successes in this area include the project: 'Uncertainty quantification in computer simulations of groundwater flow problems with emphasis on contaminant transport' . Together with colleagues overseas, we are also investigating stochastic mixed finite element schemes for systems of stochastic PDEs, a relatively new and unexplored field.