Adaptive Numerical Algorithms for Forward UQ in Time-Dependent PDEs

Uncertainty quantification (UQ) is a rapidly-evolving field, incorporating several traditional mathematical disciplines. This project will develop new adaptive numerical algorithms for the forward propagation of uncertainty in time-dependent CFD (computational fluid dynamics) models.

When we use mathematical models to simulate real-world processes (such as fluid flows) we frequently encounter situations where we are uncertain about one or more of the inputs (viscosity, material parameters, initial conditions, geometry etc). In forward UQ, the main aim is to assess the impact of uncertainty in the model inputs on quantities of interest associated with the model's outputs. For this, we require computationally efficient numerical methods that can take in a probability distribution for the model's inputs and deliver accurate approximations of statistical quantities of interest related to the model's outputs. For time-dependent problems, and especially those with non-smooth solutions, the approximation space often needs to be adapted in time to maintain accuracy. How to design adaptive numerical algorithms with guaranteed error control is highly challenging.

This project is a numerical analysis project that will develop new adaptive numerical schemes for forward UQ driven by rigorous error estimation.