The Institute of Data, Imaging and Communications is seeking a PhD candidate to explore new approaches for numerical simulation using the probabilistic framework of Monte Carlo geometry processing. This research will focus on parabolic models, particularly the advection-diffusion equation that governs gas plume dispersion in congested urban settings.
Monte Carlo geometry processing methods offer a highly flexible, parallelisable, and mesh-free approach, making them appropriate for simulating PDEs in complex or dynamically changing geometries. Unlike traditional methods such as Finite Element or Finite Difference schemes that require domain discretisation, Monte Carlo approaches allow for modelling on the edge, bypassing the need to invert large and potentially ill-conditioned matrices. Originally developed for efficient rendering in computer vision, Monte Carlo methods are now being adapted to solve PDEs across a variety of fields. Leveraging recent advancements like the walk-on-spheres and walk-on-stars algorithms, designed for elliptic models, this project aims to deliver fast-converging variants of such methods for time-dependent parabolic problems.
A key challenge in this is to establish an analogue to the mean-value property in space-time domains. Moreover, whilst Monte Carlo methods offer significant advantages in accommodating dynamic boundary conditions, they also present unique challenges. Achieving fast convergence and managing statistical noise are ongoing areas of research, as these factors are crucial for applications requiring precision and computational efficiency.
This probabilistic framework also supports inverse problem-solving, such as detecting plume sources, which involves inferring release characteristics in geometrically complex environments. Synergistically, Monte Carlo methods allow for inherent uncertainty quantification, and this is particularly useful in situations with sparse measurements or stochastic model behaviours. The topic is also amenable for cross-fertilisation of ideas from randomised numerical linear algebra, exploiting the low-dimensional structure in the kernel of the advection-diffusion model.
Potential applications for this research include environmental monitoring and national security in urban settings. This project is well-suited for candidates with a strong foundation in mathematics, stochastic processes, and Monte Carlo methods.
Further Information:
The University of Edinburgh is committed to equality of opportunity for all its staff and students, and promotes a culture of inclusivity. Please see details here: https://www.ed.ac.uk/equality-diversity
Closing Date:
Principal Supervisor:
Assistant Supervisor:
Eligibility:
A first-class honors degree in Mathematics, Statistics or Computer Science, and preferably an MSc in a closely related topic, e.g. data science or computational and applied mathematics.
A background in stochastic differential equations is necessary.
Further information on English language requirements for EU/Overseas applicants.
Funding:
Applications are welcomed from self-funded students, or Home students who are applying for scholarships from the University of Edinburgh or elsewhere. Further information and other funding options.
Informal Enquiries:
Dr Nick Polydorides (n.polydorides@ed.ac.uk)
