- 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.
- 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.
- 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.
- Interior point methods: I have also investigated interior point methods for solving quadratic and nonlinear programming problems, including such problems where the constraints are given by systems of PDEs.
- 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.
- Industrial applications: I have a number of existing industry collaborations related to the optimization of real-world systems, and numerical software development. If this work is of interest, please do get in touch by email or drop by my office.
Current and Recent PhD Students
2022-present: Bernhard Heinzelreiter
2021-present: Andrés Miniguano-Trujillo
2021-present: Karolína Benková
2019-present: Jonna Roden
2019-present: Filippo Zanetti (second supervisor)
2018-present: Mildred Aduamoah
2018-2022: Santolo Leveque (Santolo is now a Research Fellow at Scuola Normale Superiore, Pisa, Italy)
2018-2022: Kresimir Mihic (second supervisor)
2017-2022: Spyros Pougkakiotis (second supervisor; Spyros is now a Research Associate at Yale University, USA)
Recent Conference and Workshop Involvement
Jun 2022: 23rd International Linear Algebra Society (ILAS) Conference, Galway, Ireland
Jun 2022: Householder Meeting XXI, Fasano, Italy (invited by organizing committee)
Jun 2022: Preconditioning Techniques for Large Sparse Matrix Problems, Chemnitz, Germany (plenary speaker)
Apr 2022: Recent Advances in Numerical Linear Algebra for PDEs, Optimization, and Data Assimilation (organiser, with Jemima Tabeart)
Jan 2022: SIAM-UKIE Annual Meeting, Online (plenary speaker)
Jul 2021: INdAM Workshop: Brainstorming on Fractional Differential Equations: Modelling, Discretization, and Numerical Solvers, Rome, Italy (invited speaker)
Feb 2020: Numerical Linear Algebra for PDEs and Optimization, Padova, Italy (co-organiser)
Jul 2019: Advances in Preconditioners and Huge-Scale Optimization, ICMS, Edinburgh (organiser)
Jun 2019: Beyond the Discrete: Iterative Methods from the Continuum Perspective, Trinity College Dublin, Ireland (member of organising committee)
Apr 2019: Young Mathematicians Colloquium, Birmingham (plenary speaker)
Jun 2018: SIAM Conference on Imaging Science, Bologna, Italy
May-Jun 2018: Sixth Scottish Partial Differential Equation Colloquium, Edinburgh (invited speaker)
May 2018: SIAM Conference on Applied Linear Algebra, Hong Kong
Apr 2018: New Directions in Applied Linear Algebra, Numerical Solutions of PDEs, and Applications, ICMS, Edinburgh (organiser)
Sep 2017: Linear Algebra for PDEs and Optimization (organiser)
Jul 2017: 14th International Symposium on Orthogonal Polynomials, Special Functions and Applications, Canterbury (member of local organising committee)
Jun 2017: Householder Symposium on Numerical Linear Algebra XX, Blacksburg, USA (plenary speaker)
Having received a DPhil in Numerical Analysis from the University of Oxford, I moved to the University of Edinburgh in 2013 to take up a Whittaker Research Fellowship at the School of Mathematics. From 2015-2017 I was a Lecturer in Mathematics at the University of Kent, and in 2015 I also received an EPSRC Fellowship. I was awarded an IMA Leslie Fox (2nd) Prize in Numerical Analysis in 2015, and a University of Kent Faculty of Sciences Research Award in 2016. My main research interests arise from the modelling and numerics of optimal control processes, and in particular their application to practical and industrial problems. I have also worked on the numerical solution of PDEs from fluid dynamics and chemical processes, as well as on other areas of computational optimization, and I take a keen interest in writing computer software for real-world mathematical problems.â€‹
Formerly an EPSRC Fellow and a Fellow of The Alan Turing Institute
Fellow of the Higher Education Academy
Member of the EPSRC Peer Review College
Member of the GAMM Activity Groups on Applied and Numerical Linear Algebra & Optimization with PDEs
Member of Numerical Algorithms Group (NAG), Oxford
Postgraduate Certificate in Higher Education University of Kent (2016)
DPhil in Numerical Analysis University of Oxford (2013)
MSc in Mathematical Modelling and Scientific Computing University of Oxford (2009)
BA in Mathematics University of Oxford (2008)