### 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 SchrÃ¶dinger 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)