ACM PhD projects

Modern AI systems perform trillions of floating point operations. Quite often, these systems use extremely low floating point precision, making it impossible to justify the accuracy of any final result using traditional floating point error analysis. This project will study the sources of floating point errors in typical AI systems and will seek either to (a) justify the use of low precision, or to (b) give examples where low precision leads to significant inaccuracies. Techniques involved will include numerical analysis, applied linear algebra, applied statistics, optimization and high dimensional analysis. Supporting scientific computing experiments in Pytorch, or a related system, will also form a key part of the project.

This studentship will form a key part of the wider endeavour “Numerical Analysis for Stable AI” (NumAStAI) supported by an Advanced Grant from the European Research Council. The overall team will include two post-doctoral research assistants and one other PhD student. The team will benefit from interactions with many visiting experts, and there is scope for conference travel and external visits. The topic of this studentship overlaps with interdisciplinary concerns around ethics, regulation and privacy, and there will be opportunities to interact with University of Edinburgh colleagues in the Centre for Technomoral Futures and the Generative AI Lab

T. Beuzeville, A. Buttari, S. Gratton, T. Mary. Deterministic and probabilistic rounding error analysis of neural networks in floating-point arithmetic. 2025. hal-04663142v2

Large-scale AI models, including transformers, diffusion-based generators, and neural PDE solvers, now define state-of-the-art performance across scientific and generative domains. However, their computational and memory costs are unsustainable, driving a strong need for compact and efficient architectures that retain expressivity while reducing resource demands.

This PhD project aims to develop and analyze neural architectures that exploit compact representations, embedding mathematical structure directly into the data and the model’s operators. Using structured matrices and tensors, such as sparse [0], low-rank [1], Hierarchical Semi-Separable [2,3], Butterfly [4], Monarch [5] operators, the project will investigate how compact layer architectures and compact embedding manifolds can be used to reduce the number of parameters and data dimension in deep neural models, reducing memory and computational costs without sacrificing expressivity.

Prerequisites: A strong background in applied/computational mathematics, linear algebra, and/or machine learning. Familiarity with PyTorch or JAX and enthusiasm for deep learning are highly desirable.

[0] E. Frantar et al., Scaling Laws for Sparsely-Connected Foundation Models, ICLR 2024

[1] S. Schotthöfer et al., Low-Rank Lottery Tickets: Finding Efficient Low-Rank Neural Networks via Matrix Differential Equations. NeurIPS, 2022.

[2] P. Sittoni et al., Neural-HSS: Hierarchical Semi-Separable Neural PDE Solver. https://openreview.net/pdf?id=CYgn0GEtR2

[3] Y. Fan et al., A Multiscale Neural Network Based on Hierarchical Matrices. Multiscale Modeling & Simulation, 2019.

[4] W. Liu et al., Parameter-Efficient Orthogonal Finetuning via Butterfly Factorization. arXiv:2311.06243v2, 2024.

[5] T. Dao et al., Monarch Matrices: Expressive Structured Matrices for Efficient and Accurate Training. ICML, 2022.

Predicting turbulence and transport in magnetically confined plasmas is one of the major challenges in developing fusion energy. High-fidelity simulations based on nonlinear gyrokinetic theory can accurately model turbulent behaviour in tokamaks but are extremely computationally expensive, often requiring hundreds of thousands of CPU-hours for a single run [1,2].

To make large-scale predictive modelling feasible, researchers use simplified, lower-fidelity models such as linear gyrokinetics or gyro-fluid approximations, but these come at the cost of reduced accuracy [3,4].

This PhD project will explore how machine learning can bridge these fidelity levels, combining the accuracy of high-fidelity models with the efficiency of reduced ones. The work will involve comparing and integrating different turbulence models to identify where simplified approaches remain valid, and developing machine learning-based correction models to improve their predictions. The project will also investigate whether machine learning can reconstruct high-resolution plasma behaviour from lower-resolution simulations, guided by physical insight into nonlinear interactions. In addition, we will explore how recent advances in large-scale deep generative modelling, including diffusion-based and autoregressive architectures, can be adapted to accelerate PDE solvers and plasma turbulence simulations [5-12].

Prerequisites: Background in physics, applied mathematics, or computational science; Experience and enthusiasm for using deep learning software such as PyTorch (or equivalent frameworks) are essential.

This project is partially funded by UKAEA, and has an earlier application deadline of January 9th.

This project will be mostly based at UKAEA in Culham, Oxfordshire. The student will spend some time in Edinburgh, Scotland.

[1] P. Rodriguez-Fernandez et al., Nonlinear gyrokinetic predictions of SPARC burning plasma profiles enabled by surrogate modeling 2022 Nucl. Fusion 62 076036

[2] J. Candy et al., Multiscale-optimized plasma turbulence simulation on petascale architectures. Computers \& Fluids 188 (2019): 125-135.

[3] G. Staebler et al., Quasilinear theory and modelling of gyrokinetic turbulent transport in tokamaks. Nuclear Fusion 64.10 (2024): 103001.

[4] C. Bourdelle et al., A new gyrokinetic quasilinear transport model applied to particle transport in tokamak plasmas. Physics of Plasmas 14.11 (2007).

[5] W. A. Hornsby et al., Gaussian process regression models for the properties of micro-tearing modes in spherical tokamaks. Physics of Plasmas 31.1 (2024).

[6] V. Gopakumar et al., Plasma surrogate modelling using Fourier neural operators. Nuclear Fusion 64.5 (2024): 056025.

[7] L. Zanisi et al., Efficient training sets for surrogate models of tokamak turbulence with active deep ensembles. Nuclear Fusion 64.3 (2024): 036022.

[8] H. Wang et al, Recent Advances on Machine Learning for Computational Fluid Dynamics: A Survey, arxiv:2408.12171

[9] T. Li et al, Synthetic Lagrangian turbulence by generative diffusion models, Nature Machine Intelligence 2024

[10] I. Price et al, Probabilistic weather forecasting with machine learning, Nature 2025

[11] H.A. Majid et al, Test-Time Control Over Accuracy-Cost Trade-Offs in Neural Physics Simulators via Recurrent Depth, NeurIPS 2025

[12] H.A. Majid et al, Solaris: A Foundation Model for the Sun, NeurIPS 2024

Many complex systems, from physics and engineering to biological and social systems, are described by high dimensional parametric PDEs, such as Fokker–Planck or Vlasov–Fokker–Planck equations. These models capture the evolution of probability densities under the combined effects of transport, interaction, and diffusion. Yet their numerical solution remains one of the grand challenges of scientific computing: the state space is high-dimensional, the interactions are nonlinear, and uncertainty affects almost every parameter of interest.

In recent years, a new paradigm has emerged at the interface between numerical analysis and machine learning — that of operator learning. Instead of solving a PDE anew for every choice of input, an operator-learning model aims to approximate the solution operator itself: a map from coefficients or initial conditions directly to the solution. Architectures such as Deep Operator Networks (DeepONets) and Fourier Neural Operators (FNOs) have shown remarkable success for fluid and diffusion equations, yet their potential in more challenging settings, where nonlocality, conservation, and high-dimensional uncertainty coexist, remains largely unexplored.

This PhD project aims to develop mathematically informed operator-learning frameworks for high dimensional PDEs with uncertainty. The candidate will combine tools from kinetic and mean-field theory, numerical analysis, machine learning, and uncertainty quantification to design data-driven surrogates that respect physical principles such as mass conservation, entropy dissipation, and energy balance.
Possible directions include learning effective drift–diffusion operators, coupling neural surrogates with classical solvers, and analyzing generalization properties in the mean-field or stochastic limit.

Beyond its theoretical aspects, the project connects to applications in multiscale plasma systems, and probabilistic modeling of collective dynamics. The candidate will join an active research environment with strong collaborations across Europe through the DataHyking Marie-Curie Doctoral Network (www.datahyking.eu), engaging with groups working on numerical analysis, kinetic equations, and scientific machine learning.

1. Li, Z., Kovachki, N. B., Azizzadenesheli, K., Liu, B., Bhattacharya, K., Stuart, A. M., & Anandkumar, A. (2021). Fourier Neural Operator for Parametric Partial Differential Equations. International Conference on Learning Representations (ICLR). arXiv:2010.08895

2. Lu, L., Jin, P., Pang, G., Zhang, Z., & Karniadakis, G. E. (2021). Learning Nonlinear Operators via DeepONet based on the Universal Approximation Theorem of Operators. Nature Machine Intelligence, 3(3), 218–229. arXiv:1910.03193

3. Wei Chen, Giacomo Dimarco, Lorenzo Pareschi, Structure and asymptotic preserving deep neural surrogates for uncertainty quantification in multiscale kinetic equations, preprint arXiv:2506.10636, 2025.

Inverse problems are central to many imaging applications, including medical imaging, remote sensing, and astronomy. For a few decades, iterative optimisation methods have been state-of-the art for solving such problems. However, they often face limitations in terms of computational and reconstruction efficiency. Recent advances in deep learning offer powerful tools for solving inverse problems, but concerns about the trustworthiness, interpretability, and reliability of these methods hinder their broader adoption in critical applications.

This project aims to develop trustworthy deep learning strategies for inverse problems by leveraging powerful mathematical tools such as optimisation, Bayesian and optimal transport theories. Specifically, we will focus on hybrid novel strategies mixing the power of deep learning and neural networks, with the theoretical guarantees of mathematics. In this context, three main research directions are of great interest: building powerful models for imaging inverse problems; investigating robustness and convergence guarantees of data-driven methods; and exploring frugal learning strategies to tackle computational complexity challenges.
Ultimately, the goal is to bridge the gap between data-driven deep learning approaches and traditional model-based methods, offering a framework that is both practically effective and theoretically grounded for solving complex inverse problems in imaging.
Such a project englobes research directions at the core of multiple communities including inverse problems, computational imaging, optimisation/OR, computational mathematics, machine learning.

Pre-requisites: Background on topics related to optimisation, OR, optimal transport, or foundation of machine learning would be appreciated, as well as knowledge of Python.

Potential collaborators: Julie Delon (ENS Ulm Paris), Nelly Pustelnik (ENS Lyon), Ulugbek Kamilov (University of Wisconsin–Madison).

Many problems in science and engineering involve an unknown complex process, which it is not possible to observe fully and accurately. The goal is then to reconstruct the unknown process, given a small number of direct or indirect observations. Mathematically, this problem can be reformulated as reconstructing a function from limited information available, such as a small number of function evaluations. Statistical approaches, such as interpolation or regression using Gaussian processes, provide us with a best guess of the unknown function, as well as a measure of how confident we are in our reconstruction. There are many open questions related to efficient computations with and convergence properties of these methodologies, including challenges in high dimensional input or output spaces, goal-oriented experimental design, the use of deep Gaussian processes, and improving the methodology by incorporating information about the process such as partial differential equation constraints.

A PhD project in this area will combine ideas from machine learning, numerical analysis and statistics, and could focus on computational or analytical aspects.

Potential co-supervisors: Natalia Bochkina (UoE), Konstantinos Zygalakis (UoE).

– E.J. Addy, J. Latz, A.L. Teckentrup. Lengthscale-informed sparse grids for kernel methods in high dimensions. Available as arXiv preprint arXiv:2506.07797.

– J. Latz, A.L. Teckentrup, S. Urbainczyk. Deep Gaussian Process Priors for Bayesian Image Reconstruction. Inverse Problems, 41(6), 065016, 2025 Available as arXiv preprint arXiv:2412.10248.

– C. Osborne, A.L. Teckentrup. Convergence rates of non-stationary and deep Gaussian process regression. Available as arXiv preprint arXiv:2312.07320.

– T. Bai, A.L. Teckentrup, K.C. Zygalakis. Gaussian processes for Bayesian inverse problems associated with linear partial differential equations. Statistics and Computing, 34(4), p.139, 2024. Available as arXiv preprint arXiv:2307.08343.

Inverse problems are concerned with determining causal factors from observed results. Mathematically speaking, we want to reconstruct an unknown object X, given noisy observations of parts of X or another related object Y = f(X). The limited measurements that are available can be combined with expert prior knowledge in a Bayesian statistical framework, resulting in the probability distribution of the unknown X conditioned on the measurements, the so-called posterior distribution.

In practical applications, the goal is often to compute the expected value of a quantity of interest under the posterior distribution. This can be achieved by sampling methods such as Markov chain Monte Carlo (MCMC). However, for real-world, large-scale problems, standard MCMC methods quickly become computationally infeasible, since the number of samples required is large and the cost to produce a single sample is high.

This project will focus on developing an efficient Bayesian inference framework for modern applications. Examples of applications of interest are the inference of physical parameters in mathematical models of root growth in mathematical biology and the inference of cost functions in optimal transport problems with applications in active matter.

The focus of the project can be on theoretical or computational aspects.

Potential co-supervisors: Luke Davis (UoE), Mariya Ptashnyk (HWU).

– T.J. Dodwell, C. Ketelsen, R. Scheichl, A.L. Teckentrup. Multilevel Markov Chain Monte Carlo. SIAM Review, 61(3), 509-545, 2019.

– A.M. Stuart, M.-T. Wolfram. Inverse optimal transport. SIAM Journal on Applied Mathematics 80.1 (2020): 599-619.

Computational imaging often seeks to recover images from indirect, noisy measurements—an inverse problem that modern deep generative models handle increasingly well, but can be slow or opaque. In this project, we will develop fast, modular, and transparent generative methods by turning a well-established statistical sampling procedure (Langevin Markov chain Monte Carlo) into a compact, trainable neural network.

The student working on this project should have experience in stochastic simulation and/or computational statistics. Experience in high-performance computing is desirable but not essential. It is anticipated that this work will have a computational focus, but more theoretical directions are also possible depending on the interests of the student. There is some flexibility in the research directions of the project, depending on the interests of the student.

1. C. K. Mbakam, J. Spence, M. Pereyra, Learning few-step posterior samplers by unfolding and distillation of diffusion models

2. H.A. Abdul-Lateef, M. Pereyra, L. Shaw, K. C. Zygalakis Bayesian computation with generative diffusion models by Multilevel Monte Carlo. Phil. Trans. R. Soc. A.383:20240333, (2025).

3. S. Melidonis, Y. Xi, K. C. Zygalakis, Y. Altmann, and M. Pereyra, Score-Based Denoising Diffusion Models for
Photon-Starved Image Restoration Problems. Transactions on Machine Learning Research, 2835-8856, (2025)

Enhanced sampling is a critical challenge in statistics, artificial intelligence and molecular simulation. In statistics and AI it is used for uncertainty quantification, exploration of Bayesian posteriors, and model selection. In molecular modelling it enables the study of rare events like protein folding by overcoming high free-energy barriers [1,2]. Existing methods work well in low dimension, but suffer from a “curse of dimensionality,” as the bias potential becomes computationally intractable for systems needing many collective variables (CVs) [2].

The aim of this project is to develop and explore novel hybrid numerical algorithms, coupling Langevin path generation with a compact, learnable probabilistic model that replaces the intractable bias grid [3]. Such algorithms may include generative neural networks or samplers running on specialized hardware. The focus is on understanding the scalability and computational efficiency of this hybrid scheme and to leverage novel probabilistic computing approaches, such as quantum algorithms [4] or probabilistic computing, to act as the bias engine for high-dimensional systems.

The student will gain a research foundation in numerical algorithms, the computational modelling of stochastic complex systems, and applying probabilistic (classical and quantum) computing approaches. The connections with physical chemistry open addiitional possibilities for extension, collaboration and dissemination of the research, as well as opportunities for personal scientific development. Collaborations with researchers in Chemistry or Informatics are possibilities.

[1] A. Laio and M. Parrinello (2002). Escaping free-energy minima. \textit{Proceedings of the National Academy of Sciences}, 99(20), 12562–12566. https://doi.org/10.1073/pnas.20242739

[2] A. Barducci, G. Bussi, and M. Parrinello (2008). Well-tempered metadynamics: A smoothly converging and tunable free energy method. \textit{Physical Review Letters}, 100(2), 020603.

[3] P. Wirnsberger, A. J. Ballard, G. Papamakarios, et al. (2020). Targeted free energy estimation via learned mappings. \textit{J. Chem. Phys.}, 153(14), 144112.

[4] C. Zoufal, A. Lucchi, & S. Woerner (2021). Variational Quantum Boltzmann Machines. \textit{Quantum Machine Intelligence}, 3(1), 7.

The main focus of this project will be the data-driven analysis of complex dynamical systems exhibiting multiple time scales. Based on simulation or measurement data only, it is possible to extract dominant spatio-temporal patterns, which can then, for instance, be used for dimensionality reduction, the detection of metastable or coherent sets, system identification, or control. Due to the sheer size of the data sets, kernel-based approaches or deep learning techniques might be required to mitigate the curse of dimensionality. The successful candidate will develop, optimize, and implement novel methods to analyze high-dimensional time-series data in order to gain insights into the characteristic properties of the underlying dynamical system. Of particular interest are molecular dynamics problems (analysis of protein folding processes), fluid dynamics problems (detection of coherent sets, ocean dynamics), quantum physics problems (stochastic formulations of quantum mechanics, quantum computing), and graphs and graphons (random walks, transfer operators, spectral clustering).

Potential collaborators: Feliks Nüske, Max Planck Institute for Dynamics of Complex Technical Systems.

https://www.aimsciences.org/article/doi/10.3934/jcd.2016003

https://www.sciencedirect.com/science/article/pii/S0167278919306086

https://www.sciencedirect.com/science/article/pii/S0167278925003161

Discontinuous Galerkin (dG) methods, a class of finite element methods, have received astounding popularity over the last 20 years as a framework of numerical approximation to PDE problems, especially in the contexts of solid mechanics and fluid dynamics. Within the last decade, dG methods have been successfully generalised to be able to admit computational meshes of arbitrary shapes, allowing for unprecedented potential in computational complexity reduction [dG1,dG2].

To harvest this potential, a key broad challenge is the development of efficient solution algorithms for the, typically vast in size, linear algebraic systems resulting from the computer implementation of dG methods, including preconditioned iterative methods for such systems [Pr1,Pr2]. This PhD project aims to address exactly this challenge for a number of industrially/practically relevant stationary and time-dependent problems, and so bridge a key gap to even wider applicability of dG schemes for these classes of problems.

The PhD project is relatively flexible and open in terms of the profile and background of the selected PhD student and can range from theoretical analysis of dG methods, numerical linear algebra for the resulting systems of equations, to computer implementation, or even code development in parallel computing architectures. It is anticipated that there will be opportunity to work with an industry partner, should this be of interest to the student.

[dG1] Cangiani, A., Dong, Z., Georgoulis, E. H., Houston, P. hp-version discontinuous Galerkin methods on polygonal and polyhedral meshes. SpringerBriefs Math. Springer, Cham, 2017. viii+131 pp.

[dG2] Cangiani, A., Dong, Z., Georgoulis, E. H. hp-version discontinuous Galerkin methods on essentially arbitrarily-shaped elements. Math. Comp. 91(2021), no.333, pp.1–35.

[Pr1] Pearson, J. W., Pestana, J. Preconditioners for Krylov subspace methods: an overview.
GAMM-Mitt.43(2020), no.4, e202000015, 35 pp.

In recent years computational mathematics, including numerical linear algebra, has been subject to a “randomized revolution”, enabling classical methods for solving huge-scale problems to be made significantly more efficient. A central technique is randomized sketching, where large matrices are projected onto lower-dimensional subspaces using random transformations. Rigorous results from random matrix theory and compressive sensing lead to guarantees that randomized computations are accurate with overwhelmingly high probability, while the intermediate dimension reduction substantially reduces the computational complexity.

Many developments, including those led by the supervisor, have involved accelerating the solution of linear systems, however the impact of randomized methods is now also evident across a wide variety of additional problem structures. These include eigenvalue computations, low-rank matrix approximations, model reduction, uncertainty quantification, inverse problems, and data-driven PDE models, among others. The goal of this project is to explore and develop state-of-the-art randomized techniques to enhance computational methods arising in discretized PDEs, optimization, and related scientific computing problems. This broad agenda encompasses both theoretical and practical challenges, such as:

– Developing randomized algorithms for accelerating iterative solvers, computation of eigendecompositions, and singular value decompositions.

– Applying randomization to enable efficient model reduction and dimension reduction for high-dimensional PDE-constrained systems.

– Exploring randomized strategies for data assimilation, inverse problems, or optimal experimental design, where large data or parameter spaces are involved.

– Understanding and leveraging the trade-offs between computational cost, accuracy, and probabilistic guarantees inherent in randomized methods.

The student working on this project should have experience in numerical mathematics for scientific applications, including numerical linear algebra, as well as solving PDEs and/or optimization problems. There is some flexibility in the research directions of the project, depending on the interests of the student.

The subject matter of this PhD project is related to the EPSRC grant on “Randomized Numerical Linear Algebra for Optimization and Control of PDEs”, awarded to the supervisor. This means that the student will have an opportunity to work within a team, including Stefan Güttel (University of Manchester), the Co-Investigator of the grant, and the postdoctoral researchers recruited through the grant. There is also the potential to work with the industry partners involved with this grant, if this is of interest to the student.

Some relevant recent work on randomized numerical methods, in particular by the project team, as well as some ideas which link with the topics of the proposed research, is provided in the references below.

Potential collaborator: Stefan Güttel (University of Manchester).

S. Güttel, I. Simunec. A sketch-and select Arnoldi process, SIAM Journal on Scientific Computing 46(4), A2774-A2797, 2024.

L. Burke, S. Güttel. Krylov subspace recycling with randomized sketching for matrix functions, SIAM Journal on Matrix Analysis and Applications 45(4), 2243-2362, 2024.

S. Güttel, J.W. Pearson, Stabilizing randomized GMRES through flexible GMRES, arXiv preprint arXiv:2506.18408, 2025.

P.-G. Martinsson, J. A. Tropp. Randomized numerical linear algebra: Foundations and algorithms, Acta Numerica 29, 403-572, 2020.

T. Wagner, J.W. Pearson, M. Stoll. A preconditioned interior point method for support vector machines using an ANOVA-decomposition and NFFT-based matrix-vector products, arXiv preprint arXiv:2312.00538, 2024.

S. Güttel, J.W. Pearson. A spectral-in-time Newton-Krylov method for nonlinear PDE-constrained optimization, IMA Journal of Numerical Analysis 42(2), 1478-1499, 2022.

J.W. Pearson, A. Potschka. On symmetric positive definite preconditioners for multiple saddle-point systems, IMA Journal of Numerical Analysis 44(3), 1731-1750, 2024.

S. Leveque, J.W. Pearson. Parameter-robust preconditioning for Oseen iteration applied to stationary and instationary Navier–Stokes control, SIAM Journal on Scientific Computing 44(3), B694-B722, 2022.

Optimization problems constrained by partial differential equations (PDEs) form a highly relevant class of challenges in science and engineering, with applications ranging from fluid flow control and medical image processing to modeling in chemical, biological, and industrial systems. In particular, practical examples where such PDE-constrained optimization problems arise include model predictive control in transport and thermodynamic processes, as well as optimal sensor placement in mechanical systems.

A central focus of this project is the discretization of PDEs in both space and time, which is essential for transforming continuous mathematical models into large-scale algebraic systems suitable for numerical optimization. The choice and implementation of space–time discretization strategies are pivotal, as they directly affect the size, structure, and solvability of the resulting systems. These discretizations must not only be accurate but also tailored to enable scalable and efficient computation.

The project addresses the key challenge of developing fast and robust numerical solvers for the large, structured systems that emerge from these space–time discretizations. Of particular interest is the design and analysis of iterative solvers equipped with advanced preconditioning techniques, enabling the solution of problems at scales far beyond the reach of traditional methods. Some particular challenges, which could be incorporated into the PhD project if of interest to the student, include:

– Exploring iterative solvers aligned with specific time discretization schemes, particularly those that facilitate parallel computation across time steps. This includes methods such as multiple shooting, domain decomposition, and space–time multigrid, all of which benefit from an integrated space–time perspective.

– Incorporating advanced linear algebra techniques within solvers tailored to space–time Galerkin discretizations, including discontinuous Galerkin methods. These high-order discretizations offer both flexibility and accuracy in time-dependent settings, but require specialized solvers for practical use.

– Extending the multiple saddle point framework—originally developed for steady-state PDE-constrained optimization—to time-dependent problems. This direction provides a novel opportunity to merge space–time discretizations with powerful theoretical tools for unsteady systems.

Some examples of work previously undertaken on effective discretizations and numerical methods for PDEs and PDE-constrained optimization, with PhD students at Edinburgh, are provided in the references below.

The student working on this project should have experience in numerical mathematics for scientific applications, including numerical methods for PDEs and/or optimization problems, as well as numerical linear algebra. Interest in high-performance computing is desirable but not essential. There is some flexibility in the research directions of the project, depending on the interests of the student.

S. Pougkakiotis, J.W. Pearson, S. Leveque, J. Gondzio. Fast solution methods for convex quadratic optimization of fractional differential equations, SIAM Journal on Matrix Analysis and Applications 41(3), 1443-1476, 2020.

S. Leveque, J.W. Pearson. Parameter-robust preconditioning for Oseen iteration applied to stationary and instationary Navier–Stokes control, SIAM Journal on Scientific Computing 44(3), B694-B722, 2022.

M. Aduamoah, B.D. Goddard, J.W. Pearson, J.C. Roden. Pseudospectral methods and iterative solvers for optimization problems from multiscale particle dynamics, BIT Numerical Mathematics 62(4), 1703-1743, 2022.

J. Gondzio, S. Pougkakiotis, J.W. Pearson. General-purpose preconditioning for regularized interior point methods, Computational Optimization and Applications 83(3), 727-757, 2022.

A. Miniguano-Trujillo, J.W. Pearson, B.D. Goddard. Efficient nonlocal linear image denoising: Bilevel optimization with Nonequispaced Fast Fourier Transform and matrix-free preconditioning, SIAM Journal on Imaging Sciences 18(3), 1857-1903, 2025.

B. Heinzelreiter, J.W. Pearson, Diagonalization-based parallel-in-time preconditioners for instationary fluid flow control problems, arXiv preprint arXiv:2405.18964, 2025.

(1) N. Joshi and A. Olde Daalhuis, Exponentially-improved asymptotics for q-difference equations: 2ϕ0 and qPi, Indagationes Mathematicae, 36 (2025), pp. 1555–1571.
( https://www.sciencedirect.com/science/article/pii/S0019357725000084 )

(2) https://arxiv.org/abs/2510.24485

The Discrete Element Method (DEM) is a computational method, widely used in both industry and academia, primarily to model the dynamics of dry systems containing many small particles [1]. Typical examples include grains, minerals, powders, and pills. There is increasing industrial interest in extending the applicability of DEM to systems where the particles are suspended in a ‘liquid bath’. In particular, systems in which the particle density is high typically form pastes or slurries, which have a wide range of applications in healthcare, electronics, and other formulated products.

The two principle challenges that need to be overcome before DEM can be used to accurately model such systems are:

1. The presence of a liquid bath results in additional forces between the particles. In the systems considered here, the dominant terms arise from ‘lubrication’: for example, if one tries to push two particles together then (on a highly simplified level) one must do more work to also remove the intervening liquid. In many-particle systems, lubrication forces can qualitatively and quantitatively change the dynamics. [2] We have made some recent progress on this, but there are still open questions, particularly in terms of non-spherical particles and systematic comparisons to other approaches.

2. Current models which couple DEM to a fluid bath through a Computational Fluid Dynamics solver are prohibitively computationally expensive for many applications. They are also generally restricted to the ‘Newtonian’ case where viscous stress is related linearly to strain, whereas slurries exhibit more complex and demanding ‘non-Newtonian’ behaviour, which is crucial to correctly understand their dynamics. [3]

Main Aims: This project can focus on one or both of these related areas. There is also the possibility to collaborate with an industrial partner, who are one of the world-leaders in DEM software development and production. The key applications are in green and sustainable areas, with a strong emphasis on increasing efficiency and decreasing resources, both for DEM itself, and in the application areas.

Useful Background: Interested students should have a strong background in one or more of: mathematical modelling; computation; fluid dynamics; statistical mechanics.

Potential collaborators: Cathal Cummins (HWU), Chris Ness (Engineering, UoE).

[1] Y. Guo, Yu, and J. S. Curtis, Discrete element method simulations for complex granular flows. Ann. Rev. Fluid Mech., 47(1), 21-46, 2015.

[2] B. D. Goddard, R. D. Mills-Williams, and J. Sun, The singular hydrodynamic interactions between two spheres in Stokes flow, Phys. Fluids, 32, 062001, 2020

[3] R. P. Chhabra, Non-Newtonian fluids: an introduction, In: J. Krishnan, A. Deshpande, P. Kumar (eds), Rheology of Complex Fluids, Springer, New York, NY, 2010

This project focuses on modelling and simulation for mixtures of liquids. Such mixtures not only allow us to probe fundamental questions in physics, but they also have a wide range of industrial applications. Particular mixtures of interest here are those of water, alcohol, and oil, which exhibit a range of behaviours depending on the precise proportions of the constituents. If the student is interested then there is an opportunity to perform simple experiments to inform and validate the modelling approaches, as has been done for other systems [1,2,3].

An everyday example of such mixtures can be seen while drinking common Mediterranean spirits, including ouzo and sambuca. The spirit sold in bottles is a clear, single-phase liquid consisting of around 60% water, 40% ethanol (alcohol), and a small amount of anise oil, which gives the drinks their distinctive taste. Water and oil are completely immiscible, but both are fully soluble in alcohol; alcohol has a strong preference for water. In pure Ouzo, there is sufficient alcohol to solubilise the oil, but when even a small amount of water is added, the alcohol partitions with the water, reducing the solubility of the oil and causing the drink to become cloudy. Recent work (combining modelling and experiment) has elucidated the mechanisms behind this behaviour [2,3], but there are many open questions around such mixtures.

The mathematical modelling is based on non-equilibrium statistical mechanics, which has its foundations in statistics and probability, and often uses stochastic and/or partial differential equations (SDEs/PDEs) to describe the dynamics of particles, liquids, and other systems. Here we focus on a well-established approach called Dynamic Density Functional Theory (DDFT), which has a wide range of applications ranging from colloid particles, to liquids (as studied here), cancer modelling, and even how people form and change their opinions [4].

This project will focus on two DDFT approaches, which are very closely related: one comes from lattice models and describes the dynamics of mixtures on liquids over a set of discrete sites. The second is a continuum description through (integro-)PDEs. The former is numerically more tractable, but has limitations such as a fixed length-scale, and poor scaling with the size of the domain. The second has challenges when trying to describe steep interfaces, which are common in the systems studied here, but scales much better with domain size. Part of this project will be to develop numerical schemes; we already have an extensive, efficient, and robust code base for each approach on which this project can build.

Main Aims: To understand, predict, and possibly control, the equilibria and dynamics of ouzo-like systems through a combination of modelling, numerics, and (if there is interest) experiments.

Useful Background: Interested students should have a strong background in one or more of: mathematical modelling; computation (preferably SDEs and/or PDEs); statistical mechanics. No experimental background is required.

[1] Changing the flow profile and resulting drying pattern of dispersion droplets via contact angle modification, C. Morcillo Perez, M. Rey, B. D. Goddard, and J. H. J. Thijssen, https://arxiv.org/abs/2111.00464

[2] Experimental and theoretical bulk phase diagram and interfacial tension of ouzo, A. J. Archer, B. D. Goddard, D. N. Sibley, J. T. Rawlings, R. Broadhurst, F. F. Ouali, and D .J. Fairhurst, Soft Matt., 2024

[3] Coexisting multiphase and interfacial behavior of ouzo, D. N. Sibley, B. D. Goddard, F. F. Ouali, D. J. Fairhurst, and A. J. Archer, Phys. Fluids, 37, 042118, 2025

[4] Classical dynamical density functional theory: from fundamentals to applications, M. te Vrugt, H. Löwen, and R. Wittkowski, Adv. Phys., 69(2), 121-247, 2020

Achieving sustainable human-wildlife coexistence in well-functioning ecosystems is a vitally important and major challenge under global change. In response, novel approaches to Ecosystem Restoration (sometimes generally referred to as Rewilding) have emerged in the past 20 years, and are now starting to establish in the ecological community. Such approaches have shifted the attention from e.g. recovery of single species to ecosystem health as a whole. Efforts in this direction need the support of mathematical and statistical models to help monitor and forecast not just the behaviour of single species but of e.g. ecosystem functions and processes. This project will deal with modelling aspects related to monitoring, modelling and forecasting for Ecosystem Restoration projects and will be in connection with relevant non-academic stakeholders, such as Forest Research, the Center for Ecology and Hydrology, or the High Weald National Landscape.

I am more than happy to be contacted for informal enquiries!

G.B. Bischetti et al. (2021) Design and temporal issues in Soil Bioengineering structures for the stabilisation of shallow soil movements. Ecolog. Engin.

A. Bhattacharya, B. Hosseini, N.B. Kovachki, A.M. Stuart (2021) Model reduction and neural networks for parametric PDEs. SMAI J Comput Math, 7, 121-157

T.J. Dodwell, C. Ketelsen, R. Scheichl, A.L. Teckentrup (2019) Multilevel Markov chain Monte Carlo. SIAM Review, 61, 509-545

A. Quarteroni, A. Manzoni, F. Negri, Reduced Basis Methods for PDEs: An Introduction. Springer 2016

A.M. Stuart (2010) Inverse problems: A Bayesian perspective. Acta Num., 19, 451-559

Tipping points represent critical thresholds in environmental systems where small perturbations can lead to abrupt and potentially irreversible transitions to alternative states; examples include the collapse of the Atlantic Meridional Overturning Circulation (AMOC) and the runaway melting of ice sheets. These phenomena are inherently multiscale, involving interactions across spatial and temporal scales—from fast atmospheric processes to slow oceanic dynamics.

This project aims to develop multiscale analytical and statistical frameworks for the study of tipping dynamics in climate modelling. Techniques from dynamical systems will be applied in the analysis of couplings between fast and slow processes, whereas the Bayesian approach will be used for the quantification of uncertainties in the evolution of the dynamics. Multiscale analysis and uncertainty quantification in climate modelling will help us identify indicators of critical transitions and analyse multiscale interactions that influence the onset and propagation of tipping events.

C. Budd, C. Griffith, R. Kuske, Dynamic tipping in the non-smooth Stommel-box model, with fast oscillatory forcing. Physica D: Nonlinear Phenomena, 432, 132948, 2022

F. Ragone, J. Wouters, F. Bouchet, Computation of extreme heat waves in climate models using a large deviation algorithm. PNAS, 115, 2018

N. Wunderling et al., Climate tipping point interactions and cascades: a review. Earth Syst. Dynam., 15, 41-74, 2024

The DNA in our cells mutates throughout life. Thousands of such mutations are seen in individual cancers. What causes the mutations? This question is both fundamental to human biology and to clinical decision making. Large-scale mutation datasets are now readily available, making it an exciting time to tackle this question.

In this project we’ll use stochastic modelling to understand how DNA repair processes influence the mutations seen in cancer [1]. We’ll develop new models which we’ll study using analytic techniques and stochastic simulation. Using our models, we’ll use various statistical inference methods (likelihood and Bayesian approaches) to analyse a range of mutation datasets, coming from both human cancers and experimental systems [2]. This project will benefit from co-supervision between the School of Mathematics and the MRC Human Genetics Unit, which will ensure that the mathematical modelling is highly biologically relevant and will provide the student with multidisciplinary training.

The ideal candidate for this project will be enthusiastic about: probability theory and stochastic modelling (e.g. Markov processes); statistical inference with large datasets; and developing their computational skills. No biological knowledge is initially required, but the candidate should be motivated to learn and to answer biologically relevant research questions.

Potential collaborator: Tibor Antal (UoE).

1. Nicholson et al, PNAS, 2024 (https://doi.org/10.1073/pnas.2403871121)

2. Anderson et al, Nature, 2024 (https://doi.org/10.1038/s41586-024-07490-1)

This project will focus on establishing and studying stochastic models of growing cell populations. The main motivation is cancer, but applications to bacterial and viral evolutionary processes will also be considered. The project will initially focus on models of drug resistance evolution, which have applications across all these biological systems. The question is how to administer multiple drugs optimally. Although many attempts have been made to address this question (see e.g. [64] on my webpage), a model which accounts for stochasticity, drug dosage and sensitivity, while also connects to experimental data is missing.

Later, the project can be extended to model initiation, progression and metastasis formation models. Here the student will benefit from existing collaborations with Edinburgh and Oxford Universities and Harvard Medical School. The project involves coding simulations, analysing data, modelling stochastic processes, or using asymptotic methods.

The project can take different directions based on student’s interest. Take a look at my publication list to see some recent works of my students.

Potential collaborators: Michael Nicholson (UoE).

I. Bozic et al:
Evolutionary dynamics of cancer in response to targeted combination therapy
eLife 2013; 2:e00747
https://elifesciences.org/articles/00747#commentC. Nyhoegen et al, The many dimensions of combination therapy: How to combine antibiotics to limit resistance evolution
Evolutionary Applications. 2024;17:e13764.
https://onlinelibrary.wiley.com/doi/epdf/10.1111/eva.13764

A.E. Pomeroy and A.C. Palmer
A Model of Intratumor and Interpatient Heterogeneity Explains Clinical Trials of Curative Combination Therapy for Lymphoma
Blood Cancer Discov 2025;6:254–69
https://pmc.ncbi.nlm.nih.gov/articles/PMC12050944/pdf/bcd-24-0230.pdf

Cells constantly sense and respond to their environment, but how much information can they actually process? To answer this question, scientists often use simplified stochastic models to estimate the mutual information (MI) rate, a key quantity that captures how efficiently signals are transmitted inside cells [1,2]. However, it remains unclear whether these simplified models truly reflect the complexity of real biological systems.

In this project, you will develop analytical and computational techniques to estimate the MI rate and apply them to stochastic biochemical models of varying complexity. By comparing how different levels of model simplification (coarse-graining) affect information flow, you will gain insight into the balance between biological detail and computational tractability.

This interdisciplinary project is ideal for students in Mathematics, Physics, or related disciplines who want to apply their technical and problem-solving skills to biological questions. No prior knowledge of biology is required. The project will be jointly supervised by Dr. Nikola Popovic (School of Mathematics; Nikola.Popovic@ed.ac.uk) and Prof. Ramon Grima (School of Biological Sciences; Ramon.Grima@ed.ac.uk).

[1] Tostevin, Filipe, and Pieter Rein Ten Wolde. “Mutual information between input and output trajectories of biochemical networks.” Physical Review Letters 102.21 (2009): 218101.

[2] Moor, Anne-Lena, and Christoph Zechner. “Dynamic information transfer in stochastic biochemical networks.” Physical Review Research 5.1 (2023): 013032.

Mixed-mode oscillations (MMOs)—recurrent patterns that alternate between small- and large-amplitude oscillations—are well understood in singularly perturbed systems of ordinary differential equations, where geometric singular perturbation theory (GSPT) provides a mature framework for their analysis. In contrast, our understanding of MMOs in spatially extended systems remains largely non-existent.

This project will address that gap by identifying mechanisms that induce spatiotemporal mixed-mode patterns in (systems of) reaction-diffusion equations; specific aims include a description of their bifurcations in parameter space and the characterisation of their structure and robustness. Methodologically, the project will rely on dynamical systems techniques for partial differential equations: we will apply generalisations of GSPT that exploit an underlying slow–fast structure, in combination with the desingularisation technique known as “blow-up”. Analytical findings will be validated by numerical simulation and continuation. The project is motivated by impactful applications in the life sciences where spatial coupling and slow-fast kinetics coexist; examples include intracellular and intercellular calcium dynamics [2]; early afterdepolarisation in cardiac tissue [3], and the Belousov–Zhabotinsky reaction in chemically reacting media [4].

This project is suitable for students with a solid grounding in dynamical systems and asymptotic analysis, and an interest in mathematical biology; experience with numerical simulation or continuation is beneficial, but not essential. Depending on the student’s interests, there is scope to tailor the balance between rigorous analysis, mathematical modelling, biological application, and numerical computation. The project will be jointly supervised by Dr. Nikola Popovic (School of Mathematics; Nikola.Popovic@ed.ac.uk) and Prof. Andrew Goryachev (School of Biological Sciences; Andrew.Goryachev@ed.ac.uk).

[1] Kuehn, C. (2015). Multiple Time Scale Dynamics. Applied Mathematical Sciences 191. Springer Cham.

[2] Goldbeter, A., Dupont, G., and Berridge, M. J. (1990). Minimal model for signal-induced Ca2+ oscillations and waves. Proceedings of the National Academy of Sciences USA 87(4), 1461–1465.

[3] Echebarria, B. and Karma, A. (2002). Instability and spatiotemporal dynamics of alternans in paced cardiac tissue. Physical Review Letters 88(20), 208101.

[4] Vanag, V. K. and Epstein, I. R. (2001). Pattern formation in a tunable medium: the Belousov–Zhabotinsky reaction in a microemulsion. Physical Review Letters 87(22), 228301.

Active matter is a rapidly growing area of nonequilibrium statistical mechanics where constituent particles turnover energy to sustain persistent dynamics [1]. Examples include the flocking of birds, bacteria, and reactive colloids. Due to the lack of a well-defined global stationary distribution and the presence of complex interactions, the modelling of active matter via computer simulations is challenging. Another mathematical and computational challenge is how best to quantitatively describe and predict the geometry of these systems [2].

The key aim of this project is to develop and explore numerical simulation algorithms, such as molecular dynamics (e.g., Langevin) and Monte Carlo (e.g., kinetic Monte Carlo) [3] for a broad class of active matter toy models. Such models include driven hard spheres, flying spins (or arrows), and possibly active polymers. A key challenge is to interrogate the efficiency of the algorithms and how to leverage machine learning to probe the nonequilibrium nature of the system [4].

The student will gain a research foundation in numerical algorithms, the computational modelling of stochastic complex systems, and applying machine learning approaches.

[1] Symmetry, Thermodynamics, and Topology in Active Matter, M. J. Bowick, N. Fakhri, M. C. Marchetti, S. Ramaswamy. Physical Review X (2022)

[2] Insertion space in repulsive active matter, L. Davis & K. Proesmans. arXiv 2509.08131(2025)

[3] Efficient numerical algorithms for the generalized Langevin equation, B. Leimkuhler & Matthias Sachs. SIAM Journal of Scientific Computing A365-A388 (2022)

[4] Machine learning for active matter, F. Cichos, K. Gustavsson, B. Mehlig & G. Volpe. Nature Machine Intelligence 2, 94-103 (2020)

Active matter is a rapidly emerging field in theoretical and statistical physics which concerns itself with systems, such as living systems, that exist far from thermal equilibrium. Indeed, due to their rich behavior, there is a great potential in controlling the emergent collective states of active matter, living or artificial, to design and control optimized physical systems whose functions surpass passive — equilibrium — technology [1,2]. Such functions could include the transportation of material against chemical gradients and the sustaining of cyclic states that perform useful work. However, the non-equilibrium and noisy nature of active matter presents significant challenges to the framework of equilibrium statistical mechanics, and thus makes exploring design and control strategies difficult. To overcome these challenges, this project aims to leverage recent theoretical results [3], machine learning approaches in active matter [4], and automatic differentiation techniques, that have seen success in active [5] and quantum optimal control problems [6], to optimize particle-based molecular dynamics simulations of active matter.

This project has the scope to cover in-depth theoretical/analytical and computational aspects, and the student can expect to gain significant experience in classical calculus of variations, non-equilibrium thermodynamics, statistical physics, computer simulations, and machine learning approaches.

1. Palacci, J. et al. Science 339, 936 (2013).

2. Falk, M. J. et al. Physical Review Research 3 (2021).

3. Davis, L. K. et al. Physical Review X 14 (2024).

4. Cichos, F. et al. Nat. Mach. Intell. 2, 94 (2020).

5. Engel, M. et al. Physical Review X 13 (2023).

6. Leung, N. et al. Physical Review A 95 (2017).

Mathematics presents a profound dichotomy: the flexibility and pragmatism of calculus and the continuum, alongside the hard-edged certainty of the discrete. Moreover, experience shows that simple polynomials and their power series extensions offer a natural bridge between both sides. Superpositions of simple monomial terms, for example, organise much of the classical special-function landscape of mathematical physics, underpin explicit constructions in group representation theory, and furnish the algorithmic machinery of approximation via orthogonal families, quadrature rules, and spectral discretisations.

Scientifically, the continuum and the discrete interrogate each other through two common pursuits: (i) starting with a continuous model (e.g., a PDE) and discretising to obtain approximate numerical solutions; and (ii) starting with a concrete discrete model and taking its “continuum limit” to obtain an analytical tool with simplified explanatory power. In many modern cases, it’s unclear which side better represents the underlying reality. Hydrodynamics is an effective continuum theory emergent from discrete constituents. Alternatively, quantum fields are altogether another matter, with their true nature still one of the most consequential theoretical questions of our time.

This project will investigate these themes through focused case studies. One direction is the formulation of rigorous lattice models of chiral gauge theories with computational relevance for the Standard Model. A second is the development of numerical algorithms for semiclassical contour integration using discrete complex analysis. A third direction involves explaining established models, such as the Quantum Mechanical Hydrogen atom, in terms of elementary combinatorial bijections. Methods that may draw on enumerative combinatorics, applied category theory, representation theory, Clifford algebras, umbral calculus, probabilistic and variational methods, as well as aspects of PDE theory and numerical analysis. The precise trajectory will be developed jointly with the student, with scope to pivot as insights emerge. The project will be embedded in applied mathematics while remaining directly relevant to the mathematical-physics community. It will include active collaboration with established colleagues in that community to ensure both depth and connectivity.

Potential collaborators: Davide Marenduzzo (Physics, UoE).

Concrete Mathematics – A foundation for computer science, Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren.

Orthogonal Polynomials, Szego, Gabor.

Enumerative Combinatorics, Richard P. Stanley

Tensor calculus in spherical coordinates using Jacobi polynomials. Part-I: Mathematical analysis and derivations. G Vasil, et al. Journal of Computational Physics

Many physical systems — ranging from optical waveguide arrays and power grids to vascular and neuronal networks — can be idealised as metric graphs, where thin “wires” (edges) support wave or diffusion processes coupled by matching conditions at the junctions (vertices). On such graphs, even simple linear equations already exhibit rich behaviour: interference, multiple scattering, and subtle dependence on connectivity and boundary conditions. Moreover, introducing nonlinearity expands the possibilities dramatically, with solitons, shocks, and pattern formation, as well as direct feedback with the underlying network.

Examples of possible nonlinear model equations include the Korteweg-de Vries (KdV) type equations for unidirectional dispersive waves, nonlinear Schrödinger (NLS) equations for envelope solitons and wave packets, and reaction–diffusion or conductance-based neuronal models for excitable media. Questions of particular interest include how nonlinear waves scatter at graph junctions, how coherent structures propagate through complex connectivity, and how long it takes for initially localised disturbances to “forget” their origin and become effectively randomised by dispersion and multiple scattering. Alongside these fundamental issues, there is a compelling applied angle: whether appropriately designed graph motifs can function as wave-based computing elements, implementing logic gates, switches, or memory using purely physical dynamics.

Initially, the project will comprise extensive numerical simulations, using the Dedalus scientific software package (https://dedalus-project.org). Dedalus’s flexible, high-level specification of PDEs and boundary/vertex conditions makes it straightforward to prototype new models, change graph topologies, and iterate rapidly on ideas, while still generating highly efficient code suitable for large parameter sweeps. Possible directions include designing and testing nonlinear wave-based logic gates on graphs, quantifying dispersion, energy spreading, and “randomisation times” for waves and diffusion starting from localised initial data, and investigating simple neuronal or excitable media models on graph structures inspired by biological networks. Analytical work (e.g. asymptotics, reduced models, or spectral analysis) may follow from numerical discoveries, but the trajectory will be co-shaped with the student, with ample room to pivot towards the most intriguing dynamical phenomena that emerge.

Potential collaborators: Davide Marenduzzo, (Physics, UoE).

A continuum limit for dense spatial networks, Sidney Holden & Geoff Vasil https://arxiv.org/pdf/2301.07086

K J Burns, G M Vasil, et al, “Dedalus: A Flexible Framework for Numerical Simulations with Spectral Methods,” Physical Review Research. https://doi.org/10.1103/PhysRevResearch.2.023068

Many solid materials form a largely regular crystalline lattice at the atomistic scale.
The macroscopic properties of such materials are not just determined by the deformation properties of such a perfect lattice but are critically changed by imperfections in the otherwise regular lattice which we call crystal defects.
In this project we want to study in detail how such defects interact on the atomistic level.

The project consists of two parts.
An analytical part where we use custom tools inspired by elliptic PDE theory to study the problem and its solutions.
There is existing literature that can help us a lot, e.g. [1].
And the second part is a numerical part where we use this acquired knowledge to develop, study, and implement a computational method for a particular case.

[1] Asymptotic Expansion of the Elastic Far-Field of a Crystalline Defect. Julian Braun, Thomas Hudson, Christoph Ortner, Arch Rational Mech Anal 245, 1437–1490, 2022.

This project concerns Smoluchowski coagulation and sol-gel models. We consider scenarios where particles of different sizes coalesce to form larger particles, including possibly a “gel” state. There are many applications including: aerosols, clouds/smog, clustering of stars/galaxies, schooling/flocking, genealogy, nanostructures on substrates such as ripening or island coarsening, blood clotting and polymer growth, for example, in biopharmaceuticals. The goal is to find analytical solutions as well as construct efficient numerical simulation methods. Determining the dual sol-gel state is an important aspect of these models. Planar tree structures play an important role as well, both at the nonlinear partial differential equation model level, and at the particle model level where, naturally, coalescent stochastic processes represent their overall evolution. Including spatial diffusivity in the model at both these levels, for example to model colloids, adds another complexity. There are many directions to explore, for example: the particle interactions can be much more complex; there is a natural Hopf algebra of planar trees that is likely useful in optimising numerical simulation; investigating the dual reverse time, branching Brownian motion, perspective; and so forth.

Potential collaborators: Anke Wiese (HWU), Seva Shneer (HWU).

Malham, S.J.A. 2024, Coagulation, non-associative algebras and binary trees, Physica D 460, 134054.

Doikou, A., Malham, S.J.A., Stylianidis, I. and Wiese A., 2023, Applications of Grassmannian flows to coagulation equations, Physica D 451, 133771.

This project concerns the integrability of non-commutative nonlinear partial differential equations. In particular, the non-commutative Kadomtsev-Petviashvili (KP) hierarchies, and their modified forms, are very much of interest. Establishing their integrability by direct linearisation would be one goal. An underlying operator algebra, the pre-Poppe algebra, provides a natural context for establishing direct linearisation for these hierarchies. These hierarchies have an incredibly rich structure and many applications, for example, in nonlinear optics, ferromagnetism and Bose-Einstein condensates. They have intimate connections to: string theory and D-branes; Jacobians of algebraic curves and theta functions; Fredholm Grassmannians; the KPZ equation for the random growth off a one-dimensional substrate; and so forth. In addition to these aspects, there are many further directions that could also be explored, for example: the log-potential form; super-symmetric extensions; establishing efficient numerical methods via the direct linearisation approach; etc.

Potential collaborators: Gordon Blower (Lancaster).

Blower, G., Malham, S.J.A. 2025, The noncommutative KP hierarchy and its solution via descent algebra, arXiv:2510.01352, submitted.

Blower, G., Malham, S.J.A. 2025, Direct linearisation of the non-commutative Kadomtsev-Petviashvili equations, Physica D 481, 134745.

In tokamak experiments a phenomenon can be observed in which the plasma transitions from a low (L) confinement state to a high (H) confinement state as the input heating power is increased beyond a certain threshold. This regime of high confinement, known as H-mode, is characterised by a suppression of turbulent transport and steepening of the pressure profile in a narrow region at the edge of the plasma called the pedestal. Future reactors will almost certainly operate in H-mode however there is currently no theory-based model that can reproduce this L-H transition. This makes development in the understanding of the transition physics a key step towards improving predictions of fusion power plants.

Experimentally, two density branches of the transition have been observed: (a) the low density branch, in which the threshold power decreases with increasing plasma density, and (b) the high density branch, in which the converse is true. This observation suggests something changes in the L-mode edge depending on the local conditions giving rise to multiple avenues of enquiry, such as:

1) Does the nature of the turbulent transport change when going from the low density branch to the high density branch in the L-mode phase?

2) In the resulting H-mode, is there a difference in the transport in a pulse from the low density branch versus one from high density?

3) What signatures of the preceding L-mode transport regime can be identified from the H-mode transport?

4) How are the above influenced by the presence of different isotope masses?

This project will address these questions via high fidelity nonlinear gyrokinetic simulations using the GENE gyrokinetic code, with a high field, high current JET dataset in Deuterium, Tritium and ~50/50 D-T mixtures forming the basis of the simulations. To analyse the turbulent simulation data and place it in the context of the JET experimental results, the student will use data decomposition techniques such as dynamic mode decomposition to gain greater understanding of the spatio-temporal structure of the turbulent fluctuations, elucidating fundamental turbulence phenomena and helping to develop the theory-based understanding of the L-H transition.

Programming experience in python is essential, and a background in plasma physics, specifically kinetic theory is desirable. We encourage applications from candidates who have no fusion experience.

This studentship is co-funded by the Fusion Power CDT (https://fusion-cdt.ac.uk/) through a Community Studentship. For the first six months, the student will be staying in York to participate in the taught programme of the CDT, aspects of which may take place at some of the CDT partner universities. After the taught programme, this project will be mostly based at UKAEA in Culham, Oxfordshire, with occasional visits to York or the CDT partner institutions. The student will also spend some time in Edinburgh, Scotland, and will be affiliated with the School of Mathematics.

This project is partially funded by UKAEA, and has an earlier application deadline of January 9th.

B. Chapman et al., Composition of electron temperature gradient driven plasma turbulence in JET-ILW tokamak plasmas, Phys. Rev. Research 7, L012004 (2025) https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.7.L012004

D.R. Hatch et al., A gyrokinetic perspective on the JET-ILW pedestal, Nucl. Fusion 57 036020 (2017) https://iopscience.iop.org/article/10.1088/1741-4326/aa51e1

E.R. Solano et al., L-H transition studies in tritium and deuterium–tritium campaigns at JET with Be wall and W divertor, Nucl. Fusion 63 112011 (2023) https://iopscience.iop.org/article/10.1088/1741-4326/acee12

P.J. Schmid, Dynamic mode decomposition of numerical and experimental data, J. Fluid Mech. 656, 5-28 (2010) https://doi.org/10.1017/S0022112010001217

N. Kutz et al., Dynamic Mode Decomposition: Data-Driven Modeling of Complex Systems, SIAM (2016) https://epubs.siam.org/doi/book/10.1137/1.9781611974508

Modelling the dynamics of the ocean is one of the most pressing scientific questions of our time. Large-scale numerical simulations are an essential tool in this endeavour. However, the dynamical scales of the ocean span many orders of magnitude, from slowly evolving planetary-scale gyres and overtunings, through highly turbulent eddies ranging from tens to hundreds of kilometres, and all the way down to centimetre dissipation. Moreover, the ocean critically interacts with all the other major Earth systems, including the atmosphere, cryosphere, biological ecosystem, and the global economy.

Capturing the entire vastness of scale and complexity in a computer will remain out of reach for the foreseeable future. However, we currently have extensive knowledge about how to utilise the best computer hardware and numerical software tools available, pushing the limits of spatiotemporal resolution to the extreme with modern high-performance computing platforms. We also have the advantageous ability to extract and simplify various key processes and study them under multiple controls.

This project will simulate the global ocean using the Dedalus scientific software package (https://dedalus-project.org). Dedalus enables the creation of highly efficient code using a compact and high-level representation, easily allowing us to test new techniques on increasingly advanced problems. The complexities of continental coastlines present challenges in a pseudospectral approach. However, pseudospectral algorithms also possess sufficient flexibility to model a wide range of other complex fluid-solid interactions. This project will investigate and apply the best-in-class knowledge to the global ocean and its related aspects.

Fundamentals of Ocean Climate Models, Stephen Griffies, Princeton University Press

K J Burns, G M Vasil, J S Oishi, D Lecoanet, B P Brown, “Dedalus: A Flexible Framework for Numerical Simulations with Spectral Methods,” Physical Review Research,https://doi.org/10.1103/PhysRevResearch.2.023068

The ocean is an extremely complicated fluid dynamical system, with turbulent structures appearing across a broad range of dynamical scales. In practice, however, we are often not interested in the precise details of complicated small-scale turbulent structures, but are instead interested in the behaviour and impact of slowly evolving large-scale fields, such as global-scale ocean currents and overturnings. That is, we are not interested in small-scale eddies, but are instead interested in large-scale averages.

This leads to an apparently basic question: what, precisely, do we mean by ‘average’? It turns out that this is a surprisingly difficult question to answer.

We can imagine, for example, averaging at fixed points in the ocean — conceptually, using fixed moorings — and using this to define what is known as an ‘Eulierian mean’. We might instead imagine averaging along moving trajectories — conceptually, using floating drifters — and using this to define what is known as a ‘Lagrangian mean’. It turns out, perhaps surprisingly, that the comoving ‘Lagrangian mean’ view can be the more useful of the two.

This project will use a recently developed approach for computing Lagrangian-mean quantities, and will apply this to idealized but highly turbulent ocean flows. Key questions to answer include: how do we apply a Lagrangian averaging approach to turbulent flows which evolve on a broad range of time scales? and can we use the Lagrangian mean view to find new and simpler interpretations for the resulting dynamics?

Hossein A. Kafiabad and Jacques Vanneste, Computing Lagrangian means, Journal of Fluid Mechanics (960), 2023, https://doi.org/10.1017/jfm.2023.228

Lois E. Baker, Hossein A. Kafiabad, Cai Maitland-Davies and Jacques Vanneste , Lagrangian filtering for wave–mean flow decomposition, Journal of Fluid Mechanics (1009), 2025,https://doi.org/10.1017/jfm.2025.42

Efficient mixing of fluids is vital in a variety of industrial settings, for example in drug development/discovery or in building ventilation. At low Reynolds numbers one must rely on chaotic advection to mix [1], while mixing in high Reynolds number flows is aided by the presence of turbulence. This project will seek optimal mixing strategies in parallel shear flows using a time-dependent forcing (either pressure gradient or boundary oscillation). The mixing strategies will be determined via a nonlinear optimisation algorithm the student will implement within the Dedalus codebase [2]. There is a strong Dedalus community at the University of Edinburgh and scope for a range of collaborations over the course of this project.

Nonlinear optimisation in fluids has been effective in determining “optimal” initial conditions to trigger transition to turbulence (so-called minimal seeds, [3]). There are many subtleties in extending this approach to mixing, including the definition of an appropriate norm to quantify the mixing [4]. These will be explored by the student over the course of the project. Time permitting, we will also consider the application of these ideas to viscoelastic flows, where turbulent-like states can be triggered in the absence of inertia [5], and where the existence of chaos presents an exciting opportunity for new mixing strategies in micro-flow devices. Moreover the methods being explored can be used to calculate the optimal flows for triggering dynamo action — the mechanism for generating the magnetic fields of the Earth and Sun.

[1] Aref, “Stirring by chaotic advection”, J. Fluid Mech. 143 (1984)

[2] Burns et al, “Dedalus: A flexible framework for numerical simulations with spectral methods”, Phys. Rev. Research 2 (2020)

[3] Kerswell, “Nonlinear nonmodal stability theory”, Ann. Rev. Fluid Mech. 50 (2018)

[4] Foures et al, “Optimal mixing in two-dimensional plane Poiseuille flow at finite Peclet number”, J. Fluid Mech. 748 (2014)

[5] Groisman & Steinberg, “Elastic turbulence ina polymer solution flow”, Nature 405 (2000)

The dynamics at the ocean surface are central to many oceanographic concerns such as air–sea exchanges and remote sensing. Sea-surface dynamics are dominated by the interaction between currents and surface waves. This is two-way interaction: currents affect surface waves and surface waves affect currents. To represent this two-way interaction requires coupling two types of models: a model for the currents, which solves standard fluid equations augmented by terms capturing the impact of surface waves, and a phase-averaged surface wave model which describes the evolution of wave energy in position–wavenumber space. Models in current use, e.g. by weather forecasting centres, have been coupled in an ad hoc way. As a result, they do not respect fundamental properties including conservation of energy and momentum. This is problematic.

This project centres on a new class of coupled models constructed to satisfy energy and momentum conservation. This is achieved using a variational (least-action) formulation of the dynamical equations and a systematic approximation procedure. The study of this class of models has just started and can be pushed in several directions: derivation of models accounting for effects such as the finite depth of the ocean and large-scale deformations of the ocean surface, analysis of the stability of various steady states, numerical implementation of coupled models, study of the energy and momentum exchanges between winds, surface waves and currents, and development of simplified models suitable for operational forecasting.

The project will appeal to students interested in fluid mechanics, classical mechanics (Lagrangian/Hamiltonian systems) and applications to the environment. It will involve a combination of analytical and numerical work in proportions that can be tailored to the student’s predilections. The project has the potential to impact physical oceanography, weather forecasting and climate modelling. It is part of a collaboration with W R Young (Scripps, UCSD).

Potential collaborators: James Maddison, Lois Baker (UoE), Steve Tobias (Physics, UoE).

Vanneste J & Young W R: Stokes drift and its discontents, Phil. Trans. R. Soc. Lond. A, 380, 20210032 (2022).

Wang H, Villas Boas A B, Young W R & Vanneste J: Scattering of swell by currents, J. Fluid Mech., 975, A1 (2023).

Wang H, Villas Boas A B, Vanneste J & Young W R: Scattering of surface waves by ocean currents: the U2H map, J. Fluid Mech., 1005, A12 (2025).

Context:
Turbulence—the chaotic motion of fluids—acts to transport and mix energy, momentum, and other material properties in many systems of interest, ranging from pipe flows to Jupiter’s atmosphere. Turbulence is not just chaotic but also multi-scale, and the transfer (or ‘cascade’) of energy across scales is a central aspect of turbulent flows. It determines the size and lifetime of flow structures, which in turn affect transport and mixing. Despite its central role in the few predictive turbulence theories that exist, a dynamical understanding of what determines the cascade of energy (and its direction in scale) has yet to be established [1]. This is particularly evident in geophysical and astrophysical flows, where the energy cascade can have a variety of unexplained behaviors depending on parameters such as rotation rate or aspect ratio.

Aims and approaches:
This project aims to study the statistics of energy transfers and cascades in turbulent flows using a combination of direct numerical simulations, simplified numerical models of turbulence, and stochastic models. The work will focus on the role that so-called ‘triads’ (the interaction of flow structures at three different scales) play in the energy cascade. Triads can be regarded as the smallest dynamical system representing cascade dynamics [2]. Students will be performing large-scale simulations of turbulence on a computing cluster, either on multiple processors in parallel or on a single Graphical Processing Unit (GPU), analyzing the statistical properties of triads and working towards the creation of simplified models of their statistics and energy transfers [3], eventually paving the way for a predictive theory of the energy cascade in turbulent systems.

Useful background:
Interested students should have a strong background in fluid dynamics and scientific computation (experience with python as a minimum, but preferably numerical PDEs). Experience with dynamical systems/mathematical modelling is also encouraged, but not necessary.

Candidates are encouraged to contact Dr. Benavides to discuss directions and ideas.

[1] Alexakis, A., and L. Biferale. “Cascades and transitions in turbulent flows.” Physics Reports 767: 1-101 (2018).

[2] Waleffe, F. “The nature of triad interactions in homogeneous turbulence.” Phys. Fluids 4, 350-363 (1992).

[3] Benavides, S. J., and Bustamante, M. D. “Phase dynamics and their role determining energy flux in hydrodynamic shell models.” arXiv 2507.03397 (2025). https://doi.org/10.48550/arXiv.2507.03397

Context:
The transition from simple laminar flow to the chaotic, multiscale turbulent state remains one of the great open problems in fluid dynamics. It involves elements of nonlinear dynamics, bifurcation theory, pattern formation, and non-equilibrium phase transitions. What is particularly fascinating is that, in simple flows such as pipes and channels, the transition to turbulence is subcritical: laminar flow remains linearly stable to perturbations, and yet turbulence arises through the local proliferation of large-scale turbulent patches within the domain. In flows between parallel walls, this gives rise to striking phenomena featuring oblique turbulent–laminar patterns (or ‘bands’), whose geometry and dependence on flow speed are not yet understood [1].

Aims and approaches:
This project aims to improve the understanding and modelling of turbulent-laminar bands in planar shear flows through a combination of direct numerical simulations and mathematical modelling. Various possible approaches could be considered, including: numerical experiments and observation using the python-wrapped parallelized PDE solver Dedalus [2]; studying a simplified model of such a transition [3] through the lens of dynamical systems; studying large-scale linear instabilities of the turbulent [4] and model systems; and improving or extending the current simplified model to other contexts.

Useful background:
Interested students should have a strong background in fluid dynamics and scientific computation (experience with python as a minimum, but preferably numerical PDEs). Experience with dynamical systems/mathematical modelling is also encouraged, but not necessary.

Candidates are encouraged to contact Dr. Benavides to discuss directions and ideas.

This is part of an ongoing collaboration with Prof. Dwight Barkley at the University of Warwick.<

[1] Tuckerman, L. S., Chantry, M., and Dwight Barkley. “Patterns in wall-bounded shear flows.” Annu. Rev. Fluid Mech. 52:343-67 (2020).

[2] Burns, K. J., Vasil, G. M., et al. “Dedalus: A flexible framework for numerical simulations with spectral methods.” Phys. Rev. Research 2, 023068 (2020).

[3] Benavides, S. J. and Barkley, D. “Model for transitional turbulence in a planar shear flow.” Proc. R. Soc. A. 481:20250391 (2025).

[4] Kashyap, P. V., Duguet, Y., et al. “Linear instability of turbulent channel flow.” Phys. Rev. Lett. 129, 244501 (2022).

The general concept of geometric phase arises in many different guises in both classical and quantum physics [1]. The manifestation of geometric phase known as Pancharatnam–Berry (PB) phase arises in optics. It provides a measure of the dissimilarity of two electromagnetic plane waves of the same frequency, based on the evolution of their polarization states [2,3]. Complex materials which exhibit anisotropy and/or spatial non-homogeneity provide settings in which the effects of PB phase are most readily appreciated. As well as a topic of fundamental importance in optics in its own right, PB phase is also attracting interest for applications such as wavefront engineering and polarization-dependent lenses. The recent emergence of engineered materials or structures, such as metamaterials or metasurfaces which
can incorporate anisotropy and/or non-homogeneity in a controlled manner, has fuelled this drive towards PB-based applications.Non-singular optical propagation has provided the backdrop for research on PB phase to date. The following fundamental issue has yet to be addressed: How does singular optical propagation effect PB phase? The proposed research project aims to address this issue by investigating PB phase for Voigt waves and Voigt surface waves supported by slabs of anisotropic and/or non-homogeneous materials.

[1] E. Cohen, H. Larocque, F. Bouchard, F. Nejadsattari, Y. Gefen, E. Karimi, “Geometric phase from Aharonov–Bohm to Pancharatnam–Berry and beyond,” Nature Rev. Phys. 1, 437–449 (2019).

[2] M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. Roy. Soc. A 392, 45–54 (1984).

[3] S. Pancharatnam, “Generalized theory of interference, and its applications. Part I: coherent pencils,” Proc. Indian Acad. Sci., Ser. A 44, 247–262 (1956).