Solving Partial Differential Equations with Bayesian Inference (Advert ref: RPG24-R/EE/MPEE/REGNIER)

Overview of the project

This project is being offered as part of the Leverhulme Trust grant awarded to Northumbria University entitled “A stochastic approach to solar magnetic field extrapolations”. The project proposes to investigate novel and advanced statistical and stochastic methods to solve partial differential equations and their applications to a particular research field (solar physics) providing challenging observational datasets. 

About the Project

Poisson’s equation is a well-known partial differential equation which appears in a broad scope of research investigations (e.g. electromagnetism, plasma physics, quantum physics). Mathematically, the deterministic solutions are well known as harmonic functions and a variety of analytical and numerical methods exists for different geometry and complex systems. To include the measurement errors as well as the possible fundamental mechanisms, we propose to introduce the stochastic nature of the boundary conditions. The method that you will develop relies on using Bayesian inference to obtain a fast and accurate solution to the Poisson’s equation. As a testbed for applying our mathematical development, you will develop and apply the Bayesian inference method to data provided by observations of the solar magnetic field: these observations provide boundary conditions to solve the differential equations and sparse measurement errors. The mathematical description of the magnetic field as a 3D vector field follows partial differential equations that can be of elliptic (like Poisson’s equation). It can also be extended to hyperbolic or mixed type equations. Especially, the project will focus on the behaviour and stability of solutions in simply and multiply connected domains with complex topology, and the existence of bifurcations.   

You will apply Bayesian inference method to develop robust algorithms to study the existence, uniqueness, and stability of solutions from known theoretical solutions and from solar observations. You will learn various, relevant numerical techniques, and thus will become an expert in this active field of research of stochastic partial differential equations. You will be provided the appropriate training and personal development opportunities to advance the project and your career.

You will join a strong, supportive Mathematics and Statistics group which includes the Solar and Space Physics research group (https://www.northumbria.ac.uk/about-us/academic-departments/mathematics-physics-and-electrical-engineering/research/solar-and-space-physics/ ), the Applied Statistics group (https://www.northumbria.ac.uk/about-us/academic-departments/mathematics-physics-and-electrical-engineering/research/applied-statistics/ ), and the Mathematics of Complex and Nonlinear Phenomena group (https://www.northumbria.ac.uk/about-us/academic-departments/mathematics-physics-and-electrical-engineering/research/mathematics-of-complex-and-nonlinear-phenomena/ ), which are pursuing high-international-priority research. The groups have regular discussion sessions (including with visitors) and run established research seminar series and journal clubs; all of which creates an ideal research environment to support your learning. This PhD project would be suitable for applicants with undergraduate and/or master degrees in Applied Mathematics and Applied Statistics (or related disciplines). Prior experience with astrophysical/solar data is not expected as you will be provided with specific training in all relevant areas. You will conduct research investigations, write publications, disseminate your findings at national and international conferences, and engage in international collaborations.

Academic Enquiries

This project is supervised by Dr Stephane Regnier. For informal queries, please contact [Email Address Removed]. For all other enquiries relating to eligibility or application process please email Admissions ([Email Address Removed]).

Eligibility Requirements:

• Academic excellence i.e. 2:1 (or equivalent GPA from non-UK universities with preference for 1st class honours); or a Masters (preference for Merit or above); or APEL evidence of substantial practitioner achievement.

• Appropriate IELTS score, if required.

• Applicants cannot apply if they are already a PhD holder or if currently engaged in Doctoral study at Northumbria or elsewhere.

To be classed as a Home student, candidates must:

• Be a UK National (meeting residency requirements), or

• have settled status, or

• have pre-settled status (meeting residency requirements), or

• have indefinite leave to remain or enter.

If a candidate does not meet the criteria above, they would be classed as an International student.

For further details on how to apply see https://www.northumbria.ac.uk/research/postgraduate-research-degrees/how-to-apply/

In your application, please include a research proposal of approximately 1,000 words and the advert reference (e.g. RPG24-R/…).

Deadline for applications: 25 July 2024

Start date of course: 1 October 2024 or 1 March 2025

Northumbria University is committed to creating an inclusive culture where we take pride in, and value, the diversity of our postgraduate research students. We encourage and welcome applications from all members of the community. The University holds a bronze Athena Swan award in recognition of our commitment to advancing gender equality, we are a Disability Confident Leader, a member of the Race Equality Charter and are participating in the Stonewall Diversity Champion Programme. We also hold the HR Excellence in Research award for implementing the concordat supporting the career Development of Researchers and are members of the Euraxess initiative to deliver information and support to professional researchers.