[School of Natural Sciences PhD Scholarships] Non-Intrusive Approximation for Systems of Time-Dependent Parametric PDEs

About the project

About the Project In applied mathematics we frequently encounter physics-based models consisting of partial differential equations (PDEs) with inputs that are uncertain in real-world settings (eg material coefficients, boundary conditions etc). In uncertainty quantification (UQ), we represent uncertain model inputs as functions of random variables, leading to parametric PDEs posed on high dimensional domains. In numerical analysis, the challenge is then to design solution algorithms that deliver approximations of quantities of interest (QoIs) that are provably accurate and computationally feasible to compute. In forward UQ, one aims to understand how uncertainty in model inputs affects uncertainty in QoIs related to model solutions. Naive sampling methods typically require the repeated numerical solution of the original PDE for a huge number of samples of the random inputs, incurring a high computational cost. Over the last three decades, several families of numerical schemes have been developed to approximate solutions of parametric PDEs. However, the vast majority of work done by numerical analysts to date to rigorously analyse the performance of such methods is limited to simple PDE classes. For more complex coupled and time-dependent systems of parametric PDEs, much remains to be done to develop and analyse appropriate numerical solution strategies for forward UQ. In this project, the main goal would be to develop and analyse a non-intrusive UQ method for a problem driven by a real-world application (such as a time-dependent poroelasticity or fluid flow model). There is flexibility for the project to be adapted to the skills and interests of the student. Particular numerical analysis challenges that could be addressed include: investigating regularity of solutions with respect to random input parameters, establishing error bounds for approximations, improving computational aspects of solvers for efficient implementation, and incorporating adaptivity into numerical schemes, balancing discretisation errors associated with time, space and parameter domains Suitable applicants must have a background up to to MSc-level (or equivalent) in numerical analysis, including PDEs, as well as programming experience, for example in MATLAB or Python. 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 Applied Mathematics OR any upper-second class (2:1) honours degree and a Master’s degree at merit (or international equivalent) in Applied Mathematics. A background in numerical analysis of PDEs is preferred. 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. FSESoNS

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