Numerical analysis of adaptive UQ algorithms for PDEs with random inputs

Partial differential equations (PDEs) are key tools in the mathematical modelling of physical processes in science and engineering. Traditional deterministic PDE-based models assume precise knowledge of all inputs (material properties, initial conditions, external forces, etc.). There exist an abundance of numerical methods that can be used to compute a solution to such models to any required accuracy. In practical applications, however, a complete characterisation of all the inputs to a PDE model may not be available. Examples include the modulus of elasticity of a stressed body (in linear elasticity models)and wave characteristics of inhomogeneous media (in wave propagation models). In these cases, simulations based on deterministic models are unable to estimate probabilities of undesirable events (e.g., the fracture of a stressed plate) and, hence, to perform a reliable risk assessment. The emergent area of uncertainty quantification (UQ) deals with mathematical modelling at a different level. It involves the use of probabilistic techniques in order to
(i) determine and quantify uncertainties in the inputs to PDE-based models, and
(ii) analyse how these uncertainties propagate to the outputs
(either the solution to the PDE, or a quantity of interest derived from the solution). The models are then described by PDEs with random data, where both inputs and outputs take the form of random fields. Numerical solution of such a PDE model is significantly more challenging than the solution of the deterministic analogues. The development of robust, accurate, and practical numerical methods for solving associated parameter-dependent PDE models is the central focus of the project.

Numerical methods based on a parametric reformulation of such PDE problems emerged in the engineering literature in the 1990s as more efficient and rapidly convergent alternatives to Monte-Carlo sampling in cases where the dimension of the stochastic space is moderate (of the order of 10 random parameters). Recent research into these methods suggests that their advantageous approximation properties can best be achieved by using an adaptive refinement strategy, when spatial and stochastic components of the approximate solution are judiciously chosen in the course of numerical computation. The design of optimal adaptive algorithms remains an open question however. The proposed research programme aims at the design, theoretical analysis and efficient implementation of the state-of-the-art adaptive algorithms applicable to a range of PDE problems with random inputs. By improving the efficiency and reliability of numerical methods for uncertainty quantification, the research project is directly relevant to the UK societal challenge of managing nuclear waste and minimising the risks of contamination of groundwater.

The goal of the proposed research is to make theoretical and practical advances in approximating solutions to partial differential equations with inputs that are subject to epistemic uncertainty. There are many applications in engineering and manufacturing where improvements in the efficiency and reliability of uncertainty quantification (UQ) would have a direct impact on public safety by better informing policy makers who set health and safety standards. This includes the nuclear power industry, where the storage of nuclear waste underground and the potential contamination of groundwater have been long debated, the aerospace and construction industries, where the effects of uncertain forcing on structural components have a crucial impact on design, and vehicle manufacturing, where simulations of impact tests are used to develop products. The proposed research also has a potential economic impact for such industries who have traditionally relied on inefficient Monte Carlo sampling for performing UQ. More efficient adaptive UQ algorithms could lead to significant reductions in computation times and hence speedier conclusions for product design and decision making.

In the long term, the potential beneficiaries of the proposed research are the whole of the UK public and UK industries who perform UQ to inform the design of their products. In the short term, this requires engagement with scientists and industrialists who can filter research results to the upper levels of management and government. It is the mathematicians and engineers working in industry who we must engage with first and equip with the necessary tools (analysis, algorithms, codes) to perform efficient UQ. We will do this by
- organising an industrial engagement workshop,
- maintaining strong links with data analysis and uncertainty evaluation research group at the National Physical Laboratory,
- contributing to the steering group of the KTN special interest group on UQ & management in high-value manufacturing,
- exploring Knowledge Transfer Partnerships with engineering companies,
- giving public lectures on mathematical modelling and uncertainty.