Uncertainty Quantification (UQ) is the science of accounting for uncertainty in mathematical models. Research in this area has undergone rapid growth in the last few years and is currently considered a 'hot topic'. This growth has been driven by the need for scientists in today's world to provide decision makers with ever more accurate and reliable predictions that are based on results obtained from mathematical models.
Many physical processes such as fluid flows are governed by partial differential equations (PDEs). In practical applications in the real world, it is unlikely that all the inputs (boundary conditions, geometry, coefficients) for the chosen PDE model will be known. One possibility is to model the quantities that we don't know as random variables. Solving these problems is not always hard in theory but solving them efficiently in practice is a massive challenge.
I am interested in working with students who want to develop numerical analysis and numerical methods (e.g. solvers, error estimators) for solving partial differential equations with uncertain inputs (stochastic PDEs). I welcome any enquiries to work in this area. Specific projects could be theoretical or computational, according to the strengths of the student.
Projects on this topic would suit students who have taken undergraduate courses in numerical analysis and applied mathematics who have a keen interest in computational mathematics and developing practical algorithms. Some prior programming experience is essential.
An Introduction to Computational Stochastic PDEs (Cambridge Texts in Applied Mathematics), G. J. Lord, C.E. Powell and T. Shardlow, 2014.