High-dimensional computations are ubiquitous in science and engineering, often arising from models with numerous parameters. Uncertainty quantification (UQ) in fields such as climate modelling, nuclear reactor design, and finance, for instance, involves models with thousands of variables, which poses significant challenges for practical computation. Standard methods degrade rapidly as dimension grows.
The project aims to develop and analyse implementable, fully discrete methods for high-dimensional problems, drawing on expertise in numerical analysis, with links to data-driven sciences including UQ and machine learning theory. Specifically, the student will: (i) design algorithms whose cost grows mildly with dimension, exploiting structure such as anisotropy, sparsity, or low effective dimension; (ii) establish rigorous error and complexity bounds for these algorithms; (iii) apply the resulting theory to data assimilation / Bayesian inverse problems, and to the approximation-theoretic questions underlying deep learning.
The work combines approximation theory, numerical analysis of high-dimensional problems, and computational statistics. Techniques will include quasi-Monte Carlo and sparse grid approximations.
This project is expected to start in September 2027.
Before you apply: We strongly recommend that you contact the supervisors for this project before you apply.
How to apply: To be considered for this project you must complete a formal application through our online application portal. If you already have an applicant account this link will directly open an application for PhD School of Natural Sciences Scholarships. If you don’t already have an applicant account, please follow the instructions here.
When applying, please specify the full title and supervisor/s of the project, details of your previous study, and names and contact details of two referees. You must also upload a Supporting Statement describing your motivation to apply to the project, your CV and transcripts of awarded and in-progress university qualifications. Please note late or incomplete applications will not be considered.
Equality, diversity and inclusion are fundamental to the success of The University of Manchester and central to all our activities. A diverse research community strengthens creativity, productivity and quality, while increasing the societal and economic impact of our work. We welcome applicants from all career paths, backgrounds and sections of the community, regardless of age, disability, ethnicity, gender, gender expression, sexual orientation or transgender status.
We welcome applications from candidates returning to study after a career break or experience in other roles. Flexible study arrangements may be available, including part-time study at 50%, 60% or 80%, subject to the requirements of the project and funder.
Eligibility: The standard academic entry requirement for this PhD is an upper second-class (2:1) honours degree (or international equivalent) in relevant science or engineering related discipline PhD OR any upper-second class (2:1) honours degree and a Master’s degree at merit (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.
This project will remain open until filled.
If your application is submitted by 1st November 2026, you can expect a decision by 18th December 2026.
If your application is submitted by 15th January 2027, you can expect a decision by 30th March 2027.
Self or externally funded students can also be considered for this project.
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