Research interests
My research is in numerical analysis and computational mathematics, and focuses on foundational aspects of numerical solutions of differential equations, in particular, approximation of functions and numerical linear algebra. More recently, I have begun to consider numerical algorithms in the context of quantum mechanics and inhomogeneous radiation chemistry.
Computational Mathematics in Quantum Mechanics
- Spectral methods on the whole real line
- High order Zassenhaus splitting methods for the Schrödinger equation and Dirac equation
- Participant in grant of Karolina Kropielnicka, Computational Mathematics in Quantum Mechanics funded by National Science Center (NCN), Poland.
Numerical Methods for Inhomogeneous Radiation Chemistry
- Numerical solution of nonlinear, nonlocal reaction-diffusion-advection equations
- Nonlocal PDEs because radiolised chemical species can be electrically charged
- Coupling of homogeneous and inhomogeneous differential equations and moving boundaries
- Work with Fred Currell (Dept Chemistry & Dalton Cumbria Facility, University of Manchester) and our team.
- PDRA: Mel O'Leary (based at Dalton Cumbria Facility)
- Funded by contract with Sellafield Ltd.
Ill-conditioned function approximation
- Frames are redundant sets of functions (like a basis but without linear independence), which offer great freedom for the design of function approximation space but extreme ill-conditioning.
- Randomised numerical linear algebra techniques for extremely ill-conditioned least squares problems
- Work with Daan Huybrechs (KU Leuven) and Yuji Nakatsukasa (Oxford)
Computational Spectral Theory
- Solvability Complexity Index of computing the spectra of infinite dimensional Jacobi operators (with Sheehan Olver)
- Infinite dimensional QL algorithm (with Sheehan Olver)
- Inverse eigenvalue problems in spectroscopy (with Mike Baker (Dept Chemistry, Manchester))
PhD students
I am always looking out for strong students who want to do a PhD in Numerical Analysis. Contact me if you are interested. Here is a list of my current PhD students:
- Georgia Bradshaw, Spectral methods for inhomogeneous radiation chemistry (Sep 2020-present)
- Alban Bloor Riley, Inverse eigenalue problems in spin spectroscopy (Sep 2021-present)
- Cyrus Wyett, Radiolytic hydrogen production in heterogenous systems with plant relevance (Chemistry, Sep 2021-present)