Think of your favourite PDE. The heat equation, the convection-diffusion equation or, perhaps, the wave equation. In traditional mathematical modelling we solve PDEs for given sets of input data. That is, we assume the material properties, domain geometries, boundary conditions and forcing terms are given to us, before we start solving the equations. 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 uncertain, we require mathematical techniques that propagate this uncertainty to the output quantities of interest and methods 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 permeability coefficients of the porous medium are known at every location in the flow domain. Simulations based on such over-simplifications cannot provide 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 permeability 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, or in other words, random fields. The figure shows two realisations of a random field with a particular statistical distribution, on the unit square. These could represent two possible sets of permeabilities values in a slice of rock, across which we would like to study a fluid flow. Here, red indicates a high value and blue a low value.
We have expertise in using traditional Monte Carlo methods but are currently exploring newer, more efficient stochastic finite element methods (SFEMS). These methods provide a framework for incorporating statistical information about input parameters into computer simulations so that probabilistic information about the solution is obtained directly. Some SFEMs, despite having very desirable accuracy properties, give rise to massive coupled systems of equations. Solving them is extremely challenging and an active research area.
Our current research includes developing fast solvers for the linear systems that arise when we apply SFEMs to a range of linear and non-linear flow problems. We are also investigating stochastic mixed finite element schemes for systems of stochastic PDEs, a relatively new and unexplored field.