Numerical analysis of adaptive UQ algorithms for PDEs with random inputs

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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.