Research

The majority of my research to date falls within the areas of numerical analysis and scientific computing, with an emphasis on numerical linear algebra and computational optimization. I describe my main research interests below.

Preconditioners for PDE-constrained optimization: The development of fast iterative methods for PDE-constrained optimization, and in particular the construction of effective preconditioners for these problems, is an important and challenging research area. A key component of building these preconditioners is in accurately approximating the Schur complement of the matrix system involved, and we find that with such an approximation we are able to build fast and robust solvers for these problems.

See [1,5,7,8] in Publications.

Time-dependent PDE-constrained optimization and applications: As most real-world problems involve time-dependent (and nonlinear) components, it is desirable for any investigation of PDE-constrained optimization to include these elements. This requires the solution of huge-scale matrix systems, even in comparison to those for equivalent time-independent problems. We find that our methodology can be readily applied to many such problems, which also opens the door to application areas including the modelling of chemical reactions and pattern formation processes within mathematical biology.

See [2,3,6,13,14] in Publications.

Optimal control problems in fluid dynamics: One of the major classes of PDE-constrained optimization formulations arises in the form of flow control problems. I have investigated numerical methods for solving problems involving (time-independent and time-dependent) Stokes and Navier-Stokes equations, with effective preconditioning strategies requiring the precise features of the fluid flow to be taken into account.

See [9,10,11,12] in Publications.

Computation of special functions: Another research area that I am interested in is the development of methods for computing special functions within mathematical physics, in particular hypergeometric functions. It is important to determine effective strategies for carrying out these computations that avoid numerical issues such as roundoff error, cancellation and overflow - the 'best' technique frequently varies depending on the parameter regime being examined.

See [16] in Publications, as well as my MSc thesis.

Radial basis functions/meshless methods: I have also investigated RBF collocation methods for solving PDEs, and have found that the application of these methods is often a viable alternative to finite element methods for solving PDE-constrained optimization problems.

See [4] in Publications.