Analysis of Numerical Methods for Partial Differential Equations with Random Data

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Physical processes such as fluid flow in porous rocks or the deformation of a metal plate subject to an in-plane stress are modelled mathematically using partial differential equations (PDEs). In traditional deterministic modelling, input data for PDEs (e.g. material parameters, boundary data and source terms etc.) are assumed to be known at every point in the computational domain. Highly accurate numerical approximations of solutions that correspond to particular realizations of the inputs are then sought. For example, a typical groundwater flow model requires, as inputs, the geometry of the domain of interest, the permeability coefficients of the porous medium through which the fluid is moving, pressure head measurements at the boundary of the flow domain and the locations of any sources or sinks in the flow. If we assume we know what these quantities are, everywhere, then we can proceed to find the single solution (fluid velocity and corresponding pressure), if it exists, that corresponds to those particular inputs. However, we as human beings have limited resources and the permeability coefficients, in particular, can never be recorded exactly at every point in the flow domain. Indeed, in most engineering problems, one or more inputs to the governing PDEs will not be known to the modeller, save perhaps at a few isolated locations. If we are honest about our imperfect knowledge ('epistemic uncertainty') in inputs to PDEs then we need to approach our modelling in a different way and pose PDE problems in a probabilistic framework. We need numerical approximation schemes that can accommodate uncertain inputs and then allow us to quantify the resulting uncertainty in the output variables. For instance, we need to estimate probabilities of undesirable events that are critical to public safety such as a chemical being transported to a particular location in groundwater, or the fracture of a stressed metal plate. In short, it is imperative to incorporate uncertainty into mathematical models of physical processes so that risk assessments can be performed.We can easily represent the unknown inputs in PDEs as random quantities; we are then faced with solving so-called stochastic PDEs. For material parameters such as permeability coefficients or the modulus of elasticity of an elastic body, it is usually the case that their values, at two distinct spatial locations, are associated and so it is appropriate to talk about correlated random data (rather than white noise). In the past, solving PDEs with correlated random data has been avoided due to limitations in computing resources or else hampered by very primitive approximation schemes. Simply averaging multiple solutions that correspond to particular realisations of the inputs can result in a huge amount of wasted computation time. Recently, more sophisticated numerical methods for approximating solutions to PDEs with correlated random data have been proposed. Unfortunately, this work has been restricted to scalar, elliptic PDEs and the question of efficient linear algebra for the resulting linear systems of equations has been largely overlooked. So-called stochastic Galerkin methods, in particular, have attractive approximation properties but have been somewhat ignored due to a lack of robust solvers. More simplistic schemes which require less user know-how but ultimately more computing time to implement, have been popularised. The aim of this project is to investigate approximation schemes for quantifying uncertainty in more complex engineering problems modelled by systems of PDEs with two output variables (e.g. groundwater flow in a random porous medium). We will extend and test the efficiency of approximation schemes introduced for scalar PDEs with random data, paying significant attention to the development of efficient linear algebra techniques for solving the resulting linear systems of equations.