About the Project
Many optimization problems have a large number of local minima, making it difficult to identify the global minimum, or even a representative collection of good candidate solutions. The aim of this project is to develop and analyse new optimization algorithms that use so-called deflation techniques to systematically find multiple local minima of such problems.
The primary application area is spectroscopic fitting problems arising in Chemistry. In these applications, experimental data can often be explained by several different parameter combinations, and identifying multiple candidate solutions can provide valuable information to the scientist. However, the algorithms developed in this project will be applicable much more broadly throughout computational science and engineering.
Deflation techniques date back at least to the work of James Wilkinson in the 1960s on computing multiple roots of polynomial equations. Although the original approaches suffered from numerical instabilities, modern variants were developed by Patrick Farrell and collaborators in 2015 and successfully applied to finding multiple solutions of nonlinear partial differential equations. More recently, the principal supervisor Webb has extended these ideas to optimization problems, and the resulting methods are already being used in chemistry applications
The basic idea is to modify the optimization problem once a local minimum has been found so that subsequent optimization runs do not converge to the same solution, while leaving other minima unchanged. By repeatedly applying this procedure, it is possible to discover multiple solutions of a problem without relying solely on random initial guesses.
A central question is whether deflation can be incorporated into advanced optimization algorithms without destroying their convergence and robustness properties.
The PhD student will investigate new deflation-based optimization algorithms, incorporating deflation into more sophisticated optimization methods such as trust-region algorithms, quasi-Newton methods, and stochastic optimization methods. The project will involve algorithm development, rigorous mathematical analysis, and the development of practical open-source software. Particular topics of interest include convergence theory, computational efficiency, and the application of the resulting methods to challenging scientific optimization problems.
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 in (or international equivalent) Mathematics or a closely related discipline OR any upper-second class (2:1) honours degree and a Master’s degree at merit (or international equivalent) in Mathematics or a closely related discipline.
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.
FSE_SoNS
Funding Notes
Funding covers tuition fees, an annual stipend at the UKRI minimum rate (currently £21,805 per year), and a research training support grant of up to £2,500 per year for the full 3.5-year programme.