Fully discrete computational methods in high dimensions

About the Project

High-dimensional computations are ubiquitous in science and engineering, often arising from models with numerous parameters. For instance, uncertainty quantification (UQ) in fields such as climate modelling, nuclear reactor design, and finance often involves models with thousands of variables, posing significant mathematical challenges.

Our aim is to develop and analyse implementable, fully discrete methods for function approximation, density estimation, and/or time-dependent PDEs or SDEs in high dimensions, with links to UQ and theoretical aspects of machine learning. Applications include improving the efficiency of data assimilation methods and understanding why and how deep learning works.

Recent papers on this topic include:

• Kazashi, Y., Suzuki, Y., & Goda, T. (2025). Optimality of quasi-Monte Carlo methods and suboptimality of the sparse-grid Gauss--Hermite rule in Gaussian Sobolev spaces. arXiv:2509.18712.
• Kaarnioja, V., Kazashi, Y., Kuo, F. Y., Nobile, F., & Sloan, I. H. (2022). Fast approximation by periodic kernel-based lattice-point interpolation with application in uncertainty quantification. Numerische Mathematik, 150(1), 33-77.
• Kazashi, Y., & Nobile, F. (2023). Density estimation in RKHS with application to Korobov spaces in high dimensions. SIAM Journal on Numerical Analysis, 61(2), 1080-1102.
• Kazashi, Y., Nobile, F., & Zoccolan, F. (2025). Dynamical low-rank approximation for stochastic differential equations. Mathematics of Computation, 94(353), 1335-1375.

Eligibility

Applicants should have, or expect to achieve, at least a 2.1 honours degree or a master’s (or international equivalent) in a relevant science or engineering related discipline. Applicants should be familiar with at least one, and preferably two, of:

(a) numerical analysis and real or elementary functional analysis;

(b) probability theory;

(c) implementation of computational methods using, for example, Julia or Python.

Funding

This PhD project is fully funded and home students, and EU students with settled status, are eligible to apply. The successful candidate will receive an annual tax-free stipend set at the UKRI rate (£20,780 for 2025/26) and tuition fees will be paid. We expect the stipend to increase each year.

We recommend that you apply early as the advert may be removed before the deadline.

Before you apply

We strongly recommend that you contact the supervisor(s) for this project before you apply. Please include details of your current level of study, academic background and any relevant experience and include a paragraph about your motivation to study this PhD project.

How to apply

Apply online through our website: https://uom.link/pgr-apply-2425

When applying, you’ll need to specify the full name of this project, the name of your supervisor, if you already having funding or if you wish to be considered for available funding through the university, details of your previous study, and names and contact details of two referees.

Your application will not be processed without all of the required documents submitted at the time of application, and we cannot accept responsibility for late or missed deadlines. Incomplete applications will not be considered.

After you have applied you will be asked to upload the following supporting documents:

• Final Transcript and certificates of all awarded university level qualifications
• Interim Transcript of any university level qualifications in progress
• CV
• Supporting statement: A one or two page statement outlining your motivation to pursue postgraduate research and why you want to undertake postgraduate research at Manchester, any relevant research or work experience, the key findings of your previous research experience, and techniques and skills you’ve developed. (This is mandatory for all applicants and the application will be put on hold without it).
• Contact details for two referees (please make sure that the contact email you provide is an official university/work email address as we may need to verify the reference)
• English Language certificate (if applicable)

If you have any questions about making an application, please contact our admissions team by emailing FSE.doctoralacademy.admissions@manchester.ac.uk.