Analysis of Numerical Methods for Partial Differential Equations with Random Data

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.

In the short term, the major beneficiaries of the proposed research will be academics (see 'Academic Beneficiaries' section for details). We do not anticipate that there will be immediate benefits for the commercial sector or for private businesses. It is our intention to make theoretical and practical advances in approximating solutions to engineering problems that are subject to uncertainty and then to disseminate our findings to mathematicians and engineers in academia and industry. The motivation for doing this is not to produce a marketable product for financial gain but to equip mathematical modellers, in the first instance, with the theoretical and practical tools necessary to tackle safety-critical engineering problems. Uncertainty quantification is a relatively young science and more and more industries are realising that it has an important role to play in assessing risks to public safety. In the long term, new ideas and algorithms for performing uncertainty quantification in mathematical modelling and the ability to reliably and efficiently predict probabilities of undesirable events, will have an impact for policy makers and project managers working in industries which affect public safety. This includes, most significantly, the nuclear power industry, where the storage of nuclear waste underground and the potential contamination of groundwater has been long debated and also the aerospace and construction industries, where the effects of uncertain forcing on structural components have a crucial impact on design. Vehicle manufacturers also carry out a large amount of their product development and testing via simulations of impact tests using finite element analysis. This is a safety-critical application where good uncertainty estimates are vital to the correct interpretation of results. The successful application of stochastic finite element methods to problems such as these would be hugely beneficial to the UK as a whole. With the UK's rapidly increasing population, a growing call for nuclear power and improved health and safety standards, real-time, efficient uncertainty quantification needs to become the norm in engineering. We aim to take a step towards this by drawing the attention of UK mathematicians and engineers to a wealth of open problems. If we can equip scientists in industry with better predictive tools then design improves, expectations are raised, standards improve and as a direct consequence, public safety improves. To achieve this requires also philosophical change and this will not be achieved in a few short years. Mathematical modellers need to be trained first. At the end of the proposed project, the PDRA will have acquired valuable skills in this context, that are highly attractive to above-named industries. To maximize the potential impact of the research outside of academia, we intend to organise a two-day workshop with colleagues from the National Physical Laboratory. This will take place at the end of the first year of the project and the audience and invited speakers will be composed of UK mathematicians and engineers.