Adaptive multilevel stochastic collocation methods for uncertainty quantification

Computer simulations in science and engineering rely on mathematical models of the underlying phenomena and processes. These mathematical models are typically written in terms of partial differential equations (PDEs) relating rates of changes of physical quantities (e.g., temperature in a solid or velocity of a flowing fluid) in space and time. Realistic models of complex phenomena and processes must account for the ever-present uncertainties resulting e.g. from imprecise or incomplete knowledge of all inputs to a PDE-based model (such as material properties, initial conditions, external forces, etc.). Examples of such phenomena include wave propagation in inhomogeneous media with uncertain wave characteristics and fluid flow through a porous media with permeability not known precisely at every point in the computational domain. In these cases, instead of standard deterministic models, simulations must rely on probabilistic techniques in order to model the underlying uncertainties in the inputs (using random variables or random fields), analyse how the uncertainties propagate to the model outputs, estimate probabilities of undesirable events (e.g., the contamination of groundwater resulting from a leakage from nuclear waste repository), and perform reliable risk assessments. The models are then represented by PDEs with random data, where both inputs and outputs take the form of random fields.

The development of effective approximation techniques and numerical algorithms for solving PDEs with random inputs is an important task in uncertainty quantification, because it opens the door to realistic simulations and ensures reliable and accurate predictions in the presence of uncertainties. Key mathematical challenges in this research area concern (i) the design of approximation methods with guaranteed and reliable error control, and (ii) the development of provably accurate adaptive algorithms that make the best use of available computational resources. This project will address both aforementioned challenges by developing, analysing, implementing and testing a novel methodology for reliable error estimation and adaptive error control in the framework of a powerful approximation technique for PDEs with random inputs known as the multilevel stochastic collocation finite element method. The project is relevant to many applications in engineering and manufacturing (e.g., in nuclear power industry) where improvements in the efficiency and reliability of numerical methods for uncertainty quantification would speed up decision making and have a direct impact on public safety.