Adaptive numerical algorithms for PDE problems with random i nput data

This project is in the areas of Numerical Analysis, Scientific Computing, and Uncertainty Quantification. It aims at developing and implementing adaptive algorithms underpinned by rigorous mathematical analysis for efficient numerical solution of partial differential equations (PDEs) with uncertainty in input data.

PDEs are key tools in the mathematical modelling of processes in science and engineering. In practical PDE-based models, precise knowledge of inputs (e.g., material properties, initial conditions, external forces) may not be available, or there might be uncertainty about the inputs. In these cases the models are described by PDEs with random data. Such problems arise in many scientific and industrial contexts when it is essential to accurately model complex processes and perform a reliable risk assessment. One of the major challenges in numerical solution of PDEs with random data is the high dimensionality of the resulting discretisations. This motivates the development of robust and effective numerical methods which make best use of available computational resources.

The student will be developing a novel methodology that underpins the design of adaptive algorithms. This will involve:
(i) a posteriori error analysis in the context of stochastic Galerkin and/or stochastic collocation finite element methods;
(ii) convergence analysis of the developed algorithms.
The student will be also implementing the developed algorithms in an open-source software.