## MAC-MIGS Projects

**MAC-MIGS Projects 2024**

The application process for the MAC-MIGS 2024 PhD programme is centred around available research areas/projects, henceforth collectively termed simply as “Projects”.

Each Project designates an academic Supervisor and, in some cases, one or more Co-supervisors.

**Each applicant may apply to a maximum of two Projects.**

**Deadline for Applications: 22 January 2024**

Applicants who are interested in applying for **one Project only**, they should complete one application form to either University of Edinburgh (UoE) or Heriot-Watt University (HWU), according to the respective university portal at the end of the Project description.

Applicants who are interested in applying for **two Projects **should fill in one or two application forms, depending on the Projects selected as follows:

- If
**both Projects designate application to University of Edinburgh**, the applicant should complete**one application**through the UoE application system, mentioning both Projects in the respective section of the application. - If
**both Projects designate application to Heriot-Watt University**, the applicant should complete**one application**through the HWU application system, mentioning both Projects in the respective section of the application. - If the
**two Projects designate application to different universities**, the applicant should complete**two applications**: one application for each each Project at the Project’s designated University. Moreover, the applicant in the free text/comment section,**should also mention the other project**(the project at the other university) they are applying for in each application form.

## The table below lists the available projects for 2024 intake.

For project-specific enquiries, you may contact each Project’s Supervisor.

General and application-related enquiries should be directed to info@mac-migs.ac.uk

Stability of Artificial Intelligence Algorithms

Data-driven analysis of complex dynamical systems

Ecological and epidemiological models of wildlife and livestock systems

Finite elements, discrete geometry and geometric flows

Finite element exterior calculus on graphs

Uncertainty quantification for multiscale partial differential equations

Stochastic particle optimization methods and applications to machine learning

Multiscale methods and Multiscale Interacting particle systems

Interacting Particle systems and Stochastic Partial Differential Equations

Hybrid numerical methods for high frequency waves

Modelling of fluid droplets

Modelling, parameterization, and control of interacting particle systems

Voigt waves in bianisotropic materials

Large-scale structure formation in turbulent fluid and plasma flows

Fast Numerical Solvers for Discontinuous Galerkin Methods on General Meshes

Inferring ocean turbulence properties from Lagrangian data

Interaction of Two Atomic Crystal Defects: Analysis and Computation

Mathematical modelling of cerebrospinal fluid production and drainage

Modern numerical linear algebra techniques for efficient learning and optimization

Anomalous diffusion: analysis and fast computation

Structured reduced-order deep learning for scientific and industrial applications

Fast Numerical Linear Algebra for Huge-Scale Optimization and Control of PDEs

Hydrodynamical modelling of microplastic transport

Mathematical study of optimization and sampling algorithms for high dimensional data science

Hierarchical Methods for Stochastic Partial Differential Equations

Efficient computation of Rare-risk measures

Statistical Aspects of Bayesian Neural Networks

Kalman Filtering for Parameter Estimation

Mathematical modelling and analysis of plant-soil interactions: slope stabilities and coastal erosion

Multiscale modelling and analysis of mechanical properties of biological tissues

A numerical analysis perspective on Gaussian process regression

Stochastic models of growing cell populations

Mathematical models of gene expression and their integration with single-cell data

Automatic differentiation in the search for simple invariant solutions in vortical and stellar dynamics

Minimal turbulence in viscoelastic fluids

Stochastic models of DNA damage and repair in cancer

Measuring and modelling the variability in DNA copy number alterations over time

Project title | Stability of Artificial Intelligence Algorithms |

Supervisor | Des Higham (UoE) |

Project description | Although responsible for spectacular success stories, tools in Artificial Intelligence (AI) are known to to have significant vulnerabilities. For example, convolutional neural networks performing image classification can be fooled by extremely small perturbations to the input that are imperceptible to the human eye. This project would look at one or both of two general questions: (1) how can concepts and techniques from numerical analysis (including condition numbers and backward errors) be used to understand and quantify these instabilities, (2) under appropriate assumptions, what is the trade-off between accuracy and stability for various AI systems? |

Related references | On adversarial examples and stealth attacks in artificial intelligence systems, Adversarial ink: componentwise backward error attacks on deep learning,L. Beerens and D. J. Higham, IMA Journal of Applied Mathematics, 2023.The feasibility and inevitability of stealth attacks, I. Y. Tyukin, D. J. Higham, A. Bastounis, E. Woldegeorgis and A. N. Gorban, O. J. Sutton, Q. Zhou, I. Y. Tyukin, A. N. Gorban, A. Bastounis, D. J. Higham, |

Where to apply? | University of Edinburgh (UoE) website |

Project title | Data-driven analysis of complex dynamical systems |

Supervisor | Stefan Klus (HWU) |

Project description | The main focus of this project is the data-driven analysis of complex dynamical systems exhibiting multiple time scales. Based only on simulation or measurement data, dominant spatio-temporal patterns can be extracted, which are then, for instance, often 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 insight into the characteristic properties of the underlying dynamical system. Of particular interest are molecular dynamics problems (analysis of protein folding processes), fluid flow problems (detection of coherent sets), quantum physics problems (stochastic formulations of quantum mechanics), and graph theory problems (random walks and spectral clustering).Desired qualifications: – experience in the simulation of complex dynamical systems (ODEs, SDEs, PDEs) – interest in data-driven methods and machine learning as well as applications such as molecular dynamics, fluid dynamics, quantum mechanics, or graph theory – programming skills in Matlab or Python – ideally experience with kernel-based approaches or deep learning |

Related references | S. Klus, F. Nüske, et al.: Data-Driven Model Reduction and Transfer Operator Approximation. Journal of Nonlinear Science (2018). https://doi.org/10.1007/s00332-017-9437-7 S. Klus, I. Schuster, and K. Muandet: Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces. Journal of Nonlinear Science (2020). https://doi.org/10.1007/s00332-019-09574-z |

Where to apply? | Heriot-Watt University (HWU) website |

| |

Project title | Ecological and epidemiological models of wildlife and livestock systems |

Supervisor | Andy White (HWU) |

Project description | Mathematical models are key tools to understand the population and infectious disease dynamics of natural systems. Results from model studies have been used to guide policy decisions and shape conservation strategies to protect endangered species. Models typically focus on pairwise interactions, such as predator-prey or host-disease dynamics, but there is now evidence that the ecological community composition emerges through complex interactions where, for example, the interplay between competition, predation, disease transmission, seasonality and spatial structure can all play a key role. Examples include how the shared pathogen, squirrelpox, and the shared predator, the pine marten, can alter the outcome of species competition between red and grey squirrels (Tompkins et al. 2003; Slade et al. 2022, 2023) and how the re-introduction of a native predator species, wolves, can reduce the prevalence of tuberculosis in wildlife prey species, such as wild boar and deer, and thereby reduce the chance of disease spillover to livestock populations (Tanner et al. 2019). This project would aim to develop new models and theory that capture the complexity of the real world by examining complex species interactions which integrate the effects of competition, predation and disease across trophic levels. The models will be developed in collaboration with biologists with expertise in the red and grey squirrel, squirrelpox and pine marten case study system in the UK and Ireland, with biologists who examine pathogen diversity at the interface between wildlife and livestock populations in Spain and in collaboration with the theoretical ecology group at UC Berkeley, USA. |

Related references | Slade, A., White, A., Lurz, P.W.W., Shuttleworth, C. & Lambin, X. 2022. A temporal refuge from predation can change the outcome of prey species competition. Oikos, e08565. Slade, A., White, A., Lurz, P.W.W., Shuttleworth, C.M., Tosh, D.G. & Twinning, J.P. 2023. Indirect effects of pine marten recovery result in benefits to native prey through suppression of an invasive species and a shared pathogen. Ecological Modelling 476. Tanner, E., White, A., Acevedo, P., Balseiro, A., Marcos, J. & Gortazar, C. 2019. Wolves contribute to disease control in a multi-host system. Scientific Reports. 9, 7940 Tompkins, D.M. White, A & Boots, M. 2003. Ecological replacement of native red squirrels by invasive greys driven by disease. Ecology Letters. 6, 189-196. |

Where to apply? | Heriot-Watt University (HWU) website |

Project title | Finite elements, discrete geometry and geometric flows |

Supervisor | Kaibo Hu (UoE) |

Project description | Geometric flows have important applications in both pure mathematics and other fields, such as computer sciences. In the realm of numerical solutions, most finite element literature focuses on extrinsic geometry, such as mean curvature flows. In finite element exterior calculus, there has been a growing interest in discretizing intrinsic geometry, such as Ricci curvature or the Einstein tensor. In particular, Regge calculus, originally proposed as a scheme for quantum and numerical general relativity, has been interpreted as a finite element. This research establishes a link between finite elements and discrete differential geometry. This project has a dual focus. On one hand, it further explores the connections between finite elements, discrete differential geometry, and their practical applications. On the other hand, it investigates the numerical computation and analysis of geometric flows, such as Ricci flows, and other geometric partial differential equations. These equations can also serve as model problems within the context of numerical relativity. Prerequisite: The topic may involve numerical methods (finite element methods, numerical linear algebra), partial differential equations, and differential geometry. Motivated candidates should be able to pick up missing backgrounds during the course. |

Related references | ‘- Douglas N. Arnold, Finite element exterior calculus. Society for industrial and applied mathematics, 2018. – Lek-Heng Lim. “Hodge Laplacians on graphs.” SIAM Review 62, no. 3 (2020): 685-715. – Oliver Knill, Annie Rak, “Differential equations on graphs”, https://people.math.harvard.edu/~knill/pde/ – Douglas N. Arnold, and Kaibo Hu. “Complexes from complexes.” Foundations of Computational Mathematics 21, no. 6 (2021): 1739-1774. – Snorre H. Christiansen “On the linearization of Regge calculus.” Numerische Mathematik 119 (2011): 613-640. – Nathaniel Trask, Andy Huang, and Xiaozhe Hu. “Enforcing exact physics in scientific machine learning: a data-driven exterior calculus on graphs.” Journal of Computational Physics 456 (2022): 110969. |

Where to apply? | University of Edinburgh (UoE) website |

Project title | Finite element exterior calculus on graphs |

Supervisor | Kaibo Hu (UoE) |

Project description | Finite elements discretize PDEs on triangulations; while graphs or networks directly establish discrete models for a wide range of applications. It turns out that some ideas and techniques from the two areas are closely related. This project investigates the application of numerical PDE ideas (especially from Finite Element Exterior Calculus and Discrete Exterior Calculus) to graphs. The large picture of this topic is making effort towards the following questions: – How to establish PDEs, models, and geometric and topological structures on graphs and what are their properties? The topic has interactions with several other areas, such as discrete (continuum) mechanics, discrete differential geometry, Topological Data Analysis and Data Geometry. Prerequisite: The topic may involve numerical methods (finite element methods, numerical linear algebra), partial differential equations, algebraic topology and differential geometry. Motivated candidates should be able to pick up missing backgrounds during the course. |

Related references | ‘- Douglas N. Arnold, Finite element exterior calculus. Society for industrial and applied mathematics, 2018. – Lek-Heng Lim. “Hodge Laplacians on graphs.” SIAM Review 62, no. 3 (2020): 685-715. – Oliver Knill, Annie Rak, “Differential equations on graphs”, https://people.math.harvard.edu/~knill/pde/ – Douglas N. Arnold, and Kaibo Hu. “Complexes from complexes.” Foundations of Computational Mathematics 21, no. 6 (2021): 1739-1774. – Snorre H. Christiansen “On the linearization of Regge calculus.” Numerische Mathematik 119 (2011): 613-640. – Nathaniel Trask, Andy Huang, and Xiaozhe Hu. “Enforcing exact physics in scientific machine learning: a data-driven exterior calculus on graphs.” Journal of Computational Physics 456 (2022): 110969. |

Where to apply? | University of Edinburgh (UoE) website |

Project title | Uncertainty quantification for multiscale partial differential equations |

Supervisor | Lorenzo Pareschi (HWU) |

Project description | The main objective of the PhD project is the development of advanced methods for the numerical solution of systems governed by multiscale partial differential equations (PDEs) that can depend on various parameters, such as initial and boundary conditions or external sources which may not be precisely known. In addition to classical fields like fluid dynamics and kinetic theory, PDEs with multiple space-time scales play a significant role in many modern applications of mathematics, including environmental modeling, epidemiology, finance, and social sciences. The uncertainty in the parameters is more pronounced in this latter class of models, as they are often constructed based on empirical considerations rather than starting from first principles. In such cases, one usually relies on experimental data to infer statistical information about the model’s parameters, which can then be used to estimate how uncertainty in the data impacts the solution. One of the challenges faced is the so-called “curse of dimensionality”, due to the increase in problem dimensions as a result of uncertainty. Prerequisites for the Ph.D. project include knowledge of at least one course in numerical analysis, covering numerical methods for ordinary differential equations, some familiarity with PDEs, such as a course on the mathematical analysis of PDEs or one on the fundamental equations of mathematical physics, and one course in probability theory. |

Related references | ‘- Giacomo Dimarco, Lorenzo Pareschi, Multi-scale variance reduction methods based on multiple control variates for kinetic equations with uncertainties, Multiscale Model. Simul. 18 (2020), no. 1, 351-382. – Giacomo Dimarco, Lorenzo Pareschi, Numerical methods for kinetic equations. Acta Numerica 23, 369-520, 2014. – Andrea Medaglia, Lorenzo Pareschi, Mattia Zanella, Stochastic Galerkin particle methods for kinetic equations of plasmas with uncertainties, J. Comp. Phys. Volume 479, 112011, 2023 – Lorenzo Pareschi, An introduction to uncertainty quantification for kinetic equations and related problems, in Trails in kinetic theory: foundational aspects and numerical methods, SEMA-SIMAI Springer Series 25:141-181, 2021. |

Where to apply? | Heriot-Watt University (HWU) website |

Project title | Stochastic particle optimization methods and applications to machine learning |

Supervisor | Lorenzo Pareschi (HWU) |

Project description | The relationship between optimization processes and systems of interacting stochastic particles has its roots in the field of “Swarm Intelligence”, where the coordinated behavior of agents interacting locally with their environment leads to the emergence of global patterns. Stochastic particle dynamics are typically guided by heuristics, and the resulting methods often excel at solving complex optimization problems where conventional deterministic methods fall short. Gradient-based optimizers are effective at finding local minima for high-dimensional, convex problems; however, most gradient-based optimizers struggle with noisy, discontinuous functions and are not designed to handle discrete and mixed discrete-continuous variables. Such high-dimensional problems arise in many areas of interest in machine learning applied to fields like engineering, finance, healthcare, and more. The main goal of this Ph.D. project lies at different levels, ranging from the development of a robust mathematical framework for popular metaheuristics like simulated annealing, particle swarm optimization, genetic algorithms, and ant colony optimization, to their convergence analyses and applications to machine learning problems. Prerequisites for the Ph.D. project include knowledge of at least one course in numerical analysis, covering multidimensional optimization methods, and some familiarity with PDEs, such as a course on the mathematical analysis of PDEs or one on the fundamental equations of mathematical physics. Knowledge of probability theory is also recommended. |

Related references | ‘- Alessandro Benfenati, Giacomo Borghi, Lorenzo Pareschi, Binary interaction methods for high dimensional global optimization and machine learning, Applied Math. Optim. 86(9):1-41, 2022. – Giacomo Borghi, Michael Herty, Lorenzo Pareschi, Constrained consensus-based optimization, SIAM J. Optimization 33(1):10.1137, 2023. – Massimo Fornasier, Hui Huang, Lorenzo Pareschi, Philippe Sünnen, Consensus-based Optimization on the Sphere: Convergence to Global Minimizers and Machine Learning, J. Machine Learning Research 22(237):1−55, 2021. – Sara Grassi, Lorenzo Pareschi, From particle swarm optimization to consensus based optimization: stochastic modeling and mean-field limit, Math. Mod. Meth. App. Sci. 31(8):1625–1657, 2021. |

Where to apply? | Heriot-Watt University (HWU) website |

Project title | Multiscale methods and Multiscale Interacting particle systems |

Supervisor | Michela Ottobre (HWU) |

Project description | Many systems of interest in the applied sciences share the common feature of possessing multiple scales, either in time or in space, or both. Some approaches to modelling focus on one scale and incorporate the effect of other scales (e.g. smaller scales) through constitutive relations, which are often obtained empirically. Multiscale modelling approaches are built on the ambition of treating both scales at the same time, with the aim of deriving (rather than empirically obtaining) efficient coarse grained models which incorporate the effects of the smaller/faster scales. Multiscale methods have been tremendously successful in applications, as they provide both underpinning for numerics/simulation algorithms and modelling paradigms in an impressive range of fields, such as engineering, material science, mathematical biology, climate modelling (notably playing a central role in Hasselmann’s programme, where climate/ whether are seen as slow/fast dynamics, respectively), to mention just a few. In this project, which is in the field of applied stochastic analysis, we will consider systems that are multiscale in time, with particular reference to multiscale interacting particle systems. We will try to understand how the multiscale approximation interacts with the mean field approximation (produced by letting the number of particles in the systems to infinity so to obtain a PDe for the evolution of the density of the particles). The motivation for this project comes especially from models in mathematical biology, but the applicabity of the framework we will investigate is broader. |

Related references | https://arxiv.org/abs/2305.04632 https://arxiv.org/abs/2204.02679 https://arxiv.org/abs/2001.03920 |

Where to apply? | Heriot-Watt University (HWU) website |

Project title | Interacting Particle systems and Stochastic Partial Differential Equations |

Supervisor | Michela Ottobre (HWU) |

Project description | Many interesting systems in physics and in the applied sciences consist of a large number of particles or agents, (e.g. individuals, animals, cells, robots) that interact with each other. When the number of agents/particles in the system is very large the dynamics of the full Particle System (PS) can be rather complex and expensive to simulate; moreover, one is quite often more interested in the collective behaviour of the system rather than in its detailed description. In this context, the established methodology in statistical mechanics and kinetic theory is to look for simplified models that retain relevant characteristics of the original PS by letting the number N of particles to infinity; the resulting limiting equation for the density of particles is a low dimensional, (in contrast with the initial high dimensional PS) non-linear partial differential equation (PDE), where the non-linearity has a specific structure, commonly referred to as a McKean-Vlasov nonlinearity. Beyond an intrinsic theoretical interest, such models were proposed with the intent to efficiently direct human traffic, to optimize evacuation times, to study rating systems, opinion formation, animal navigation strategies or, noticeably, in control engineering (e.g. collective flight of drones in artificial fireworks displays); and in all these fields they have been incredibly successful.In this project we will consider PSs modelled by Stochastic Differential Equations (SDEs) whose limiting behaviour is described by either a deterministic PDE or a stochastic PDE (SPDE) (note the greatness of this: depending on the nature of the stochasticity in the PS, the limiting equation can be either deterministic or stochastic!). Either way, the limiting equation is of McKean-Vlasov type. The overall aim of the project is the comparison of the ergodic and dynamic properties of the particle system and of the limiting PDE/SPDE. these results will help inform modelling decisions for practitioners. Keywords for this project are: Stochastic (Partial) Differential equations, McKean Vlasov evolutions, ergodic theory, mean field limits.Prerequisites: good background in either stochastic analysis/probability or analysis |

Related references | https://www.sciencedirect.com/science/article/pii/S0022247X23003049 https://arxiv.org/abs/2003.14230 |

Where to apply? | Heriot-Watt University (HWU) website |

Project title | Hybrid numerical methods for high frequency waves |

Supervisor | Lehel Banjai (HWU) |

Project description | The need for accurate computation of the scattering of acoustic or electromagnetic waves is ubiquitous, and includes diverse applications such as medical therapeutics and diagnostics, radar and antenna design, meteorology etc. While brute force numerical methods can be effective at low frequencies, and asymptotic methods at very high frequencies, in applications it is often the mid-to-high range that is of interest. Here, deep mathematical and numerical analysis needs to be combined with the state-of-the-art computer algorithms to produce effective numerical methods. Projects in related aspects of numerical wave propagation are available: hybrid numerical-asymptotic methods for waves, mitigation of the CFL (Courant-Friedrichs-Lewy) condition, methods based on homogenisation, boundary integral equations, scattering by specially coated materials etc. Depending on the interests and background of the student, the project can be focused more on the mathematical and numerical analysis or more on the scientific computing/high performance aspects. An ideal candidate would have interests and knowledge in both. |

Related references | L. Banjai, F.J. Sayas. Integral Equation Methods for Evolutionary PDE: A Convolution Quadrature Approach, Springer Series in Computational Mathematics (SSCM, volume 59), 2022 Gary Cohen , Sébastien Pernet. Finite Element and Discontinuous Galerkin Methods for Transient Wave Equations, Springer (SCIENTCOMP), 2017. |

Where to apply? | Heriot-Watt University (HWU) website |

Project title | Modelling of fluid droplets |

Supervisor | Ben Goddard (UoE) |

Project description | There are many physical and industrial processes that concern the static or dynamic properties of (complex) fluid droplets. Often such droplets are not formed from a single phase in air (such as pure water rain droplets), but contain suspended particles (e.g., in paints, inks, or coffee), or are a mixture of different fluids, sometimes suspended in another medium (e.g., oil droplets in water). Interest lies in both how these droplets behave in equilibrium, such as determining the shapes of droplets on a surface, and also how they behave dynamically, such as through evaporation or phase separation. |

Related references | [1] https://www.youtube.com/watch?v=ZaCGoSTMHyc [Coffee Ring] [2] https://www.youtube.com/watch?v=SX-fh5MFqpk [Alcohol-oil-water] [3] Classical dynamical density functional theory: from fundamentals to applications, M. te Vrugt, H. Löwen, and R. Wittkowski, Adv. Phys., 69.2, 121-247, 2020 [4] Multi-species dynamical density functional theory, B. D. Goddard, A. Nold and S. Kalliadasis, J. Chem. Phys., 138, 144904, 2013 [5] Changing the flow profile and resulting drying pattern of dispersion droplets via contact angle modification, C. M., Perez, M. Rey, B. D. Goddard, J. H. J. Thijssen, https://arxiv.org/abs/2111.00464 [6] Pseudospectral methods for density functional theory in bounded and unbounded domains, A. Nold, B. D. Goddard, P. Yatsyshin, N. Savva and S. Kalliadasis, J. Comp. Phys., 334, 639-664, 2017 [7] MultiShape: A Spectral Element Method, with Applications to Dynamic Density Functional Theory and PDE-Constrained Optimization, J. C. Roden, R. D. Mills-Williams, J. W. Pearson, B. D. Goddard, https://arxiv.org/abs/2207.05589 |

Where to apply? | University of Edinburgh (UoE) website |

Project title | Modelling, parameterization, and control of interacting particle systems |

Supervisor | Ben Goddard (UoE) |

Project description | Many systems can be accurately modelled as systems of ‘particles’ that interact with themselves and their environment. Typically such systems are described by relatively simple interactions, which then lead to complex static and dynamic behaviour, such as layering of particles against a wall [1], or laning of driven particles [2]. These models are not limited to describing actual, physical particles (such as colloids or granular media), but can also be applied in situations where the ‘particles’ represent people who hold an opinion on a topic [3], cells [4], people infected by a virus [5], cars, pedestrians, etc. Hence there are a wide range of possible interdisciplinary applications. The principal challenges of modelling such systems are: (i) choosing/deriving a suitable, computationally tractable, model; (ii) determining the relevant interactions; (iii) validating the models. For (i), typical models that treat the individual particles scale in computational complexity as the number of particles to a smallish power, such as N^2 or N^3 for N particles. This results in models that become computationally intractable for system sizes that accurately reflect most experimental results. This project will use methods from statistical mechanics, which result in integro-partial differential equation models whose computational complexity is essentially independent of the number of particles [6]. However, this still leaves the two remaining challenges. Many current approaches to (ii) use ‘heuristic’ approaches, which tend to reproduce observed experimental effects qualitatively, but not quantitatively. This is closely linked to (iii); the results are qualitatively, but not quantitatively, accurate. This project will focus on using techniques from (optimal) control, alongside experimental data, to parameterize such statistical mechanical models. The resulting methodologies also allow one to control such systems, driving them towards desired states through, e.g., applying an external potential [7]. A related challenge is to develop robust, accurate, and efficient numerical schemes which can be used to implement these methods; there is an existing codebase upon which this project can be based [8,9]. The particular choice of application and theoretical/numerical focus can be tailored to the interests of the student, and there is an opportunity to collaborate with experimentalists and/or social scientists. |

Related references | [1] Fluid structure in the immediate vicinity of an equilibrium three-phase contact line and assessment of disjoining pressure models using density functional theory, A. Nold, D. N. Sibley, B. D. Goddard and S. Kalliadasis, Phys. Fluids, 26, 72001, 2014 [2] Dynamical density functional theory analysis of the laning instability in sheared soft matter, A. Scacchi, A. J. Archer, and J. M. Brader, 2017, Phys. Rev. E, 96(6), 062616, 2017 |

Where to apply? | University of Edinburgh (UoE) website |

Project title | Voigt waves in bianisotropic materials |

Supervisor | Tom Mackay (UoE) |

Project description | Voigt-wave propagation [1] represents an unusual form of electromagnetic plane-wave propagation that is supported by certain anisotropic dielectric materials. It may be distinguished from the usual form of plane-wave propagation, as encountered in standard textbooks on electromagnetics and optics, on the basis of rate of amplitude decay. That is, the decay of Voigt waves is governed by the product of a linear function and an exponential function of propagation distance whereas in the usual case of plane-wave propagation there is only exponential decay with propagation distance. Bianisotropic materials offer much greater scope for Voigt-wave propagation than do anisotropic materials, because of the intrinsic coupling between electric and magnetic fields and the much larger constitutive parameter space that is associated with bianisotropic materials. Conditions for Voigt-wave propagation will be derived for certain types of physically-realizable bianisotropic materials. These bianisotropic materials will take the form of engineered materials that arise from the homogenization of relatively simple component materials that may not themselves support Voigt-wave propagation. Furthermore, the prospects of controlling the directions in which Voigt waves propagate by means of an applied DC electric field will be investigated for certain bianisotropic materials arising from electro-optic component materials [2]. The prospects of harnessing Voigt waves in bianisotropic materials for optical sensing application will also be investigated [3]. In addition, the prospects of realizing Voigt waves which exhibit a linear gain in amplitude with propagation distance will be investigated for certain bianisotropic materials arising from active component materials [4]. |

Related references | [1] Electromagnetic Anisotropy and Bianisotropy, 2nd edition. T.G. Mackay & A. Lakhtakia, World Scientific (2019) [2] T.G. Mackay, “Controlling Voigt waves by the Pockels effect,” Journal of Nanophotonics, 9 [3] T.G. Mackay, “On the sensitivity of directions which support Voigt wave propagation in infiltrated [4] T.G. Mackay & A. Lakhtakia, “On the propagation of Voigt waves in energetically active materials,”European Journal of Physics 37, 064002 (2016) |

Additional comments | Pre-requisites: Undergraduate level applied mathematics. Some background in undergraduate level physics may be helpful but it is not essential. |

Where to apply? | University of Edinburgh (UoE) website |

Project title | Large-scale structure formation in turbulent fluid and plasma flows |

Supervisor | Moritz Linkmann (UoE) |

Project description | Turbulent flows are examples of multi-scale nonlinear chaotic systems. That is, they are usually highly disordered. However, under certain circumstances order emerges out of such chaotic dynamics in the form of spatiotemporally coherent structures. Examples include large-scale convection cells, very large coherent motions in laboratory and atmospheric boundary layers, vortex condensates in thin fluid layers such as Jupiter’s Red Spot, or, in plasma physics, zonal flows in the core and edge regions of tokamak fusion reactors. Such “turbulent superstructures” influence heat transfer, mixing and trigger extreme events, as sharp gradients occur at the interface of a superstructure and the background flow. In fusion reactors, zonal flows suppress turbulence and thereby affect the level of plasma confinement. Despite their importance for applications, fundamental aspects of superstructure formation and their dynamical and statistical properties are not well understood on a mathematical level. Several projects are available in this area, with each focussing on open questions specific to a given fluid dynamics systems. Examples include, but are not limited to, (data-driven) low-dimensional modelling, modelling and analysis of onset mechanism(s), growth, saturation and decay processes, and effects of magnetic fluctuations. Projects usually involve a combination of analytic and computational work and can be tailored to the match the student’s interest and background. Some projects will involve analyses of experimental data. A background in fluid dynamics, plasma physics or pattern formation would be beneficial. |

Related references | M. Linkmann, G. Boffetta, M. C. Marchetti, B. Eckhardt, Phase Transition to Large Scale Coherent Structures in Two-Dimensional Active Matter Turbulence, Phys. Rev. Lett. 122, 214503 (2019) M. Linkmann, M. Hohmann, B. Eckhardt, Non-universal transitions to two-dimensional turbulence, J. Fluid Mech. 892, A18 (2020) P.H. Diamond et al, Zonal flows in plasma — a review, Plasma Phys. Control. Fusion 47 R35 (2005) |

Where to apply? | University of Edinburgh (UoE) website |

Project title | Fast Numerical Solvers for Discontinuous Galerkin Methods on General Meshes |

Supervisor(s) | First: Emmanuil Georgoulis (HWU) Co-supervisor: John Pearson (UoE) |

Project description | 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. |

Related references | [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. [Pr2] Leveque, S., Pearson, J. W. Parameter-robust preconditioning for Oseen iteration applied to stationary and instationary Navier-Stokes control. SIAM J. Sci. Comput.44(2022), no.3, pp. B694–B722. |

Where to apply? | Heriot-Watt University (HWU) website |

Project title | Inferring ocean turbulence properties from Lagrangian data |

Supervisor(s) | First: Jacques Vanneste (UoE) Co-supervisors: Aretha Teckentrup, (UoE), James R Maddison (UoE) |

Project description | The ocean is a key component of the earth system, exerting a strong control on climate as a large heat and carbon reservoir. Reliable climate predictions therefore rely on the development of accurate models of ocean dynamics. The task is difficult because ocean models are poorly constrained by observational data. A rapidly increasing volume of data is however available in the form of trajectories of drifters and other current-following instruments. Inferring ocean properties from such Lagrangian data is challenging because the data is collected on trajectories rather than at fixed spatial location, is sparse, non-uniform and biased. This project aims at tackling this challenge by developing new mathematical methods to infer basic properties of ocean turbulence (energy spectra, fluxes) from trajectory data using Bayesian tools. This will improve on the methods currently employed by offering a systematic way of extracting as much information as possible from the data and by quantifying uncertainty. The project combines fluid dynamics, in particular the theories underpinning our understanding of ocean turbulence, and Bayesian statistics. It will use both observational data and dedicated numerical simulations of simple ocean models. |

Related references | https://arxiv.org/abs/1812.04264 https://arxiv.org/abs/2201.01581 |

Where to apply? | University of Edinburgh (UoE) website |

Project title | Interaction of Two Atomic Crystal Defects: Analysis and Computation |

Supervisor | Julian Braun (HWU) |

Project description | 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 2 such defects interact. The project consists of two parts. An analytical part where we use 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 the problem. |

Related references | [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 |

Where to apply? | Heriot-Watt University (HWU) website |

Project title | Mathematical modelling of cerebrospinal fluid production and drainage |

Supervisor | Mariia Dvoriashyna (UoE) |

Project description | Cerebrospinal fluid (CSF) is a clear, water-like fluid that surrounds the brain and spinal cord, providing protection and nourishment. It is produced in the brain’s choroid plexus, flows through the brain’s ventricular system, and circulates around the brain and spinal canal. It is drained into the superior sagittal sinus and the venous system. The balance between CSF production and drainage controls the brain’s pressure. When this pressure becomes too high, it is associated with a condition called hydrocephalus, which involves an excessive buildup of CSF. |

Related references | Dvoriashyna, M., Foss, A. J., Gaffney, E. A., & Repetto, R. (2022). A mathematical model of aqueous humor production and composition. Investigative Ophthalmology & Visual Science, 63(9), 1-1. Vallet, A., Del Campo, N., Hoogendijk, E. O., Lokossou, A., Balédent, O., Czosnyka, Z., … & Schmidt, E. (2020). Biomechanical response of the CNS is associated with frailty in NPH-suspected patients. Journal of Neurology, 267, 1389-1400. Linninger, A. A., Tangen, K., Hsu, C. Y., & Frim, D. (2016). Cerebrospinal fluid mechanics and its coupling to cerebrovascular dynamics. Annual Review of Fluid Mechanics, 48, 219-257. |

Where to apply? | University of Edinburgh (UoE) website |

Project title | Modern numerical linear algebra techniques for efficient learning and optimization |

Supervisor(s) | First: Francesco Tudisco (UoE) Co-supervisor: John Pearson (UoE) |

Project description | Deep learning ultimately boils down to solving optimization problems with functional constraints. In a variety of modern application These PDE-inspired neural network architectures yield several advantages over standard deep learning models as one can, for instance, exploit the broad literature on dynamical systems and PDEs to analyze their stability and to design more efficient learning algorithms that use fast and robust numerical solvers. In this project we will combine state-of-the-art deterministic and randomized linear algebra techniques for parallel-in-time integration and optimization, including problems with PDE constraints, to improve the efficiency of learning algorithms. We will consider applications to computer vision, language models, and scientific simulations. A wide variety of problems can be cast as optimization problems with constraints defined in terms of (partial) differential equations, motivating the combination of linear algebra solvers for optimization with deep learning techniques. The supervisors have a range of experience in numerical linear algebra-inspired algorithms for machine learning [6-8], and fast and robust linear algebra solvers for optimization problems constrained by PDEs [9,10]. The student on this project should have experience with computational mathematics, for example numerical linear algebra, numerical solution of PDEs, optimization, and coding, as well as an interest in applying such techniques to modern machine learning algorithms. |

Related references | [1] E. Lorin. Derivation and analysis of parallel-in-time neural ordinary differential equations, Annals of Mathematics and Artificial Intelligence 88, 1035–1059, 2020. [2] E. M. Turan, J. Jäschke. Multiple shooting for training neural differential equations on time series, IEEE Control Systems Letters 6, 1897-1902, 2022. [3] S. Dutta, T. Gautam, S. Chakrabarti, T. Chakraborty. Redesigning the transformer architecture with insights from multi-particle dynamical systems, Advances in Neural Information Processing Systems (NeurIPS), 2021. [4] Z. Hao, C. Ying, H. Su, J. Zhu, J. Song, Z. Cheng. Bi-level physics-informed neural networks for PDE constrained optimization using Broyden’s hypergradients, The Eleventh International Conference on Learning Representations (ICLR), 2023. [5] J. Barry-Straume, A. Sarshar, A. A. Popov, A. Sandu. Physics-informed neural networks for PDE-constrained optimization and control, Computational Science Laboratory Report CSL-TR-22-2, Department of Computer Science, Virginia Tech, 2022. [6] F. Tudisco, A. R. Benson, K. Prokopchik. Nonlinear higher-order label spreading, Proceedings of The Web Conference (WWW), 2021. [7] S. Schotthöfer, E. Zangrando, J. Kusch, G. Ceruti, F. Tudisco. Low-rank lottery tickets: finding efficient low-rank neural networks via matrix differential equations, Advances in Neural Information Processing Systems (NeurIPS), 2022. [8] D. Savostianova, E. Zangrando, G. Ceruti, F. Tudisco. Robust low-rank training via approximate orthonormal constraints, Advances in Neural Information Processing Systems (NeurIPS), 2023. [9] J. W. Pearson, J. Gondzio. Fast interior point solution of quadratic programming problems arising from PDE-constrained optimization, Numerische Mathematik 137, 959–999, 2017. [10] 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, B694-B722, 2022. |

Where to apply? | University of Edinburgh (UoE) website |

Project title | Anomalous diffusion: analysis and fast computation |

Supervisor | Lehel Banjai (HWU) |

Project description | In recent years there has been growing interest in non-local, integro-fractional-differential equations modelling anomalous diffusion. Anomalous diffusion is a term used to describe the dynamics of diffusing particles that do not follow the traditional Brownian model. Applications are wide ranging, including pattern formation in biology, therapeutic ultrasound in medicine, anomalous diffusion in finance and engineering etc. While much has been done in the analysis, computation, and modelling, there are still many avenues that are unexplored and questions to be answered. In this project the student would work on efficient numerical and probabilistic methods for fractional differential equations, extensions to non-linear problems, and analysis and numerical analysis of problems with multiple scales. The optimal background for the student would include basic courses on PDE analysis, numerical analysis, probability, and an ability to program in Python or Matlab. |

Related references | Banjai, L., Melenk, J.M., Nochetto, R.H., Otárola, E., Salgado, A.J., Schwab, C. Tensor FEM for spectral fractional diffusion, Foundations of Computational Mathematics volume 19, 901–962(2019) Banjai, L, López-Fernández, M. Efficient high order algorithms for fractional integrals and associated fractional differential equations, Numer. Math., 141(2), (2019) Sposini, V., Krapf, D., Marinari, E. et al. Towards a robust criterion of anomalous diffusion. Commun Phys 5, 305 (2022). |

Where to apply? | Heriot-Watt University (HWU) website |

Project title | Structured reduced-order deep learning for scientific and industrial applications |

Supervisor | Francesco Tudisco (UoE) |

Project description | Deep learning has revolutionized various domains, including computer vision, language processing, and scientific computing. However, the growing complexity of neural network architectures alongside the ever-growing size of the available data, has led to computational challenges, making it increasingly important to develop techniques that reduce the number of parameters while maintaining performance, robustness, and model-specific underlying structures. This project topic proposes to investigate structure-preserving parameter reduction methods for neural networks with applications in computer vision, language models, reinforcement learning, and scientific simulation. The project will combine state-of-the-art deterministic and randomized techniques in computational mathematics, including numerical optimization on smooth manifolds, bi-level optimization, and numerical integration of matrix and tensor differential equations, to analyze performance, bias, robustness, and generalization of modern deep learning architectures and design new efficient learning algorithms that reduce memory requirements and training time. Specific objectives of the project topic are: The student on this project should have experience and be passionate about one or more of the following: computational mathematics, optimization, scientific coding, deep learning |

Related references | L. Franceschi, M. Donini, P. Frasconi, M. Pontil. Forward and reverse gradient-based hyperparameter optimization. International Conference on Machine Learning, 2017. R. T. Chen, Y. Rubanova, J. Bettencourt, D. K. Duvenaud. Neural ordinary differential equations. Advances in neural information processing systems (NeurIPS), 2018. T. Wang, J. Zhu, A. Torralba, A. Efros. Dataset distillation, 2018. arXiv:1811.10959. S. Arora, N. Cohen, W. Hu, Y. Luo. Implicit regularization in deep matrix factorization. Advances in Neural Information Processing Systems. 2019 S. Schotthöfer, E. Zangrando, J. Kusch, G. Ceruti, F. Tudisco. Low-rank lottery tickets: finding efficient low-rank neural networks via matrix differential equations. Advances in Neural Information Processing Systems, 2022. H.H. Chou, C. Gieshoff, J. Maly, H. Rauhut. Gradient descent for deep matrix factorization: Dynamics and implicit bias towards low rank. Applied and Computational Harmonic Analysis. 2023. N. Boumal. An introduction to optimization on smooth manifolds. Cambridge University Press; 2023. |

Where to apply? | University of Edinburgh (UoE) website |

Project title | Fast Numerical Linear Algebra for Huge-Scale Optimization and Control of PDEs |

Supervisor(s) | First: John Pearson Co-supervisors: Ben Goddard (UoE) and Stefan Güttel (Manchester) |

Project description | PDE-constrained optimization and optimal control are classes of problems with significant practical relevance, for example to fluid flow control, image processing including medical imaging, and chemical and biological systems. Such problems are also highly applicable to industrial processes, including model predictive control problems from transport or thermodynamics, and optimal sensor placement on mechanical structures. The key challenge this project seeks to address is the fast and robust numerical solution of problems which, upon discretization, lead to huge-scale systems of equations to be solved. These lead to substantial theoretical and algorithmic challenges, as they are so dependent on efficient and reliable computation. A particular focus of this work is that of iterative methods accelerated by powerful preconditioners, to enable us to reliably tackle much larger problems than would otherwise be possible. Some particular challenges include: – Investigating how preconditioned iterative methods may be applied in parallel over many computational units, in particular working alongside or within parallel-in-time methods. Such methods include multiple shooting, domain decomposition, and space–time multigrid methods. – The combination of the above techniques with a recently-devised multiple saddle point framework, which has to date been focused on steady optimization problems and PDEs, and with which there is an excellent opportunity to tackle complicated time-dependent systems. – The incorporation of cutting-edge linear algebra within solvers for discretizations using space–time Galerkin finite element methods, including discontinuous Galerkin methods. Some work previously undertaken on numerical linear algebra for PDE-constrained optimization, with PhD students at Edinburgh, is provided in the references below. 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. 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. Interest in high-performance computing is desirable but not essential. |

Related references | 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. L. Bergamaschi, J. Gondzio, Á. Martínez, J.W. Pearson, S. Pougkakiotis. A new preconditioning approach for an interior point–proximal method of multipliers for linear and convex quadratic programming, Numerical Linear Algebra with Applications 28(4), e2361, 2021. 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. S. Leveque, J.W. Pearson. Fast iterative solver for the optimal control of time-dependent PDEs with Crank–Nicolson discretization in time, Numerical Linear Algebra with Applications 29(2), e2419, 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. |

Where to apply? | University of Edinburgh (UoE) website |

Project title | Hydrodynamical modelling of microplastic transport |

Supervisor | Cathal Cummins (HWU) |

Project description | The global oceans contain an estimated 5.25 trillion plastic pieces (Salvador Cesa et al., 2017) and are annually polluted with about 1.5 million tonnes of microplastics. Notably, the ocean’s surface demonstrates a size-based selection in plastic removal (Kooi et al., 2017). Particle properties significantly influence microplastics’ underwater movements (Kreczak et al., 2021), yet key considerations such as inertial effects, geometric effects, and the presence of coastal structures have not been considered in any great detail. Considering microplastics range from 1micron to 5mm in diameter (Jambeck et al., 2015), a comprehensive analysis involving these effects is essential. This project seeks to investigate these effects on microplastics’ oceanic migration. |

Related references | Jambeck, J. R., Geyer, R., Wilcox, C., Siegler, T. R., Perryman, M., Andrady, A., Narayan, R., & Law, K. L. (2015). Plastic waste inputs from land into the ocean. Science, 347(6223), 768–771. https://doi.org/10.1126/SCIENCE.1260352 Kooi, M., Nes, E. H. van, Scheffer, M., & Koelmans, A. A. (2017). Upsand Downs in the Ocean: Effects of Biofoulingon Vertical Transport of Microplastics. Environmental Science & Technology, 51(14), 7963. https://doi.org/10.1021/ACS.EST.6B04702 Kreczak, H., Willmott, A. J., & Baggaley, A. W. (2021). Subsurface dynamics of buoyant microplastics subject to algal biofouling. Limnology and Oceanography, 66(9), 3287–3299. https://doi.org/10.1002/LNO.11879 Salvador Cesa, F., Turra, A., & Baruque-Ramos, J. (2017). Synthetic fibers as microplastics in the marine environment: A review from textile perspective with a focus on domestic washings. In Science of the Total Environment. https://doi.org/10.1016/j.scitotenv.2017.04.172 Sevilla, C. (2021). Basset-Boussinesq history term in ocean wave induced microplastic pollution. Heriot-Watt Unviersity. |

Where to apply? | Heriot-Watt University (HWU) website |

Project title | Mathematical study of optimization and sampling algorithms for high dimensional data science |

Supervisor | Benedict Leimkuhler (UoE) |

Project description | Optimization strategies lie at the heart of modern data science. A typical challenge is to explore the high dimensional parameter space of a large scale model in order to identify parameters that are relevant for some set of known data (and, ideally, which generalise to many similar data sets). For very large applications, e.g. large language models as used in chatGPT, the parameters may number in the billions and the training, even on multi-GPU systems, may take days, weeks or months. Given the scale of the challenge, and its evident commercial relevance, innovative algorithms which improve training efficiency by even a few percent can have the potential for high impact. The most popular training algorithms are built on a dynamical systems foundation. For example stochastic gradient descent is based on numerical discretization of gradient flow. The term “gradient” refers to the gradient of the objective function which is typically defined in terms of the empirical loss. By substituting carefully designed dynamical systems one can construct methods which converge more rapidly or have desirable properties in relation to generalisation performance or stability with respect to gradient noise. Examples in this direction explored in the group include Adaptive Langevin methods, multirate methods and, more recently, Friction Adaptive Descent. We are also interested in developing sampling procedures for high dimensional distributions which work with stochastic or approximate gradients. PhDs from my group working in this area have studied Langevin dynamics algorithms and generalised Langevin dynamics, tempering schemes, ensemble quasi-newton methods, randomised Hamiltonian Monte Carlo, and, most recently, unbiased estimation based on ideas from multilevel Monte Carlo. It is worth noting that all references in the list below, and indeed nearly all of the research work in my group, involves PhD students as co-authors. As this field changes very rapidly, interested students are invited to contact Prof. Leimkuhler at an early stage to discuss ideas for projects and topics which would be suitable for PhDs. |

Related references | • B. Leimkuhler, M. Sachs, G. Stoltz, Hypocoercivity properties of adaptive Langevin dynamics, SIAM J. Appl. Math. 80, 1197-1222, 2020. https://doi.org/10.1137/19M1291649 • Tiffany J Vlaar, Benedict Leimkuhler Proceedings of the 39th International Conference on Machine Learning, PMLR 162:22342-22360, 2022. https://proceedings.mlr.press/v162/vlaar22b.html • A. Karoni, B.Leimkuhler and G.Stoltz, Friction-adaptive descent: A family of dynamics-based optimization methods, J. Comp. Dyn. to appear 2023. https://doi.org/10.3934/jcd.2023007 • B. Leimkuhler, C. Matthews, G. Stoltz, The computation of averages from equilibrium and nonequilibrium Langevin molecular dynamics, IMA Journal of Numerical Analysis 36, Pages 13–79, 2016. https://doi.org/10.1093/imanum/dru056 • B. Leimkuhler, D. Paulin and P. Whalley, Contraction rate estimates of stochastic gradient kinetic Langevin integrators, SIAM J. Numerical Analysis, to appear, 2023. https://arxiv.org/pdf/2306.08592.pdf • B. Leimkuhler and M. Sachs, Efficient numerical algorithms for the generalized Langevin equation, SIAM J. Num. Anal. 44, A364-A388, 2022. https://arxiv.org/abs/2012.04245 • B. Leimkuhler, C. Matthews and J. Weare, Ensemble preconditioning for Markov chain Monte Carlo simulation, Statistics and Computing 28, 277-290, 2018. https://arxiv.org/abs/1607.03954 |

Where to apply? | University of Edinburgh (UoE) website |

Project title | Hierarchical Methods for Stochastic Partial Differential Equations |

Supervisor | Abdul-Lateef Haji-Ali (HWU) |

Project description | Partial Differential Equations (PDEs) are important versatile tools for modelling various phenomena, like fluid dynamics, thermodynamics, nuclear waste, etc… Stochastic Partial Differential Equations (SPDEs) generalize PDEs by introducing random parameters or forcing. One is then interested in quantifying the uncertainty of outputs of such models through the computations of various statistics. Accurate computations of such statistics can be costly as it requires fine time- and space-discretization to satisfy accuracy requirements. Several hierarchical methods were developed to address such issues and applied successfully to Stochastic Differential Equations (SDEs) and in this project we will extend these works to deal with the more complicated SPDEs. |

Related references | Haji-Al, Abdul-Lateef, and Andreas Stein. “An Antithetic Multilevel Monte Carlo-Milstein Scheme for Stochastic Partial Differential Equations.” arXiv preprint arXiv:2307.14169 (2023). |

Where to apply? | Heriot-Watt University (HWU) website |

Project title | Efficient computation of Rare-risk measures |

Supervisor | Abdul-Lateef Haji-Ali (HWU) |

Project description | Certain rare events have high cost, both humanitarian and financial, which make them significant events that industries and governments must plan for. Taking measures to reduce or mitigate the risks of such events is the goal of risk management which requires accurate assessment of such risks. This project’s goal is to speed up computations of accurate risk measures of rare events to ensure effective risk management. The goal will be achieved by developing novel computational methods which utilize approximation properties of the underlying stochastic models and which are based on Monte Carlo and random sampling methods which are easily parallelizable and fully exploit the increased availability of computational resources. |

Related references | ‘- Haji-Ali, Abdul-Lateef, Jonathan Spence, and Aretha L. Teckentrup. “Adaptive multilevel monte carlo for probabilities.” SIAM Journal on Numerical Analysis 60.4 (2022): 2125-2149. – Papaioannou, Iason, and Daniel Straub. “Combination line sampling for structural reliability analysis.” Structural Safety 88 (2021): 102025. |

Where to apply? | Heriot-Watt University (HWU) website |

Project title | Statistical Aspects of Bayesian Neural Networks |

Supervisor(s) | First: Neil Chada (HWU) Co-supervisors: Daniel Paulin (UoE), Benedict Leimkuhler (UoE) |

Project description | This project will focus on the development and understanding of Bayesian neural networks (BNN’s), from a statistical perspective. Neural networks have revolutionized machine learning, which has resulted in the new, well-known field of deep learning. Specifically in this project we will focus on BNN’s which provide a way to account for uncertainty that may exist, where our output of a neural network can be constructed through a probabilistic distribution. As a result, this has been applied in various settings and problems of machine learning. However, there still remains many open questions on this, related to asymptotic properties, and how they can be successfully implemented in applications such as molecular dynamics and parameter estimation. The presented project will look to go over and explore these different areas, which will provide a combination of developing methodology, theory and numerical simulations. The project will include some form of programming, ideally in either of Matlab, Julia or Python. |

Related references | https://arxiv.org/pdf/2309.16314.pdf https://arxiv.org/pdf/2007.06823.pdfhttps://arxiv.org/pdf/2006.12024.pdf |

Where to apply? | Heriot-Watt University (HWU) website |

Project title | Kalman Filtering for Parameter Estimation |

Supervisor(s) | First: Neil Chada (HWU) Co-supervisor: Michal Branicki (UoE) |

Project description | Parameter estimation, also known as inverse problems, is a field of mathematics concerned with the recovery, or reconstruction, of a parameter from noisy measurement data. These problems arise in many fields of science, such as geosciences, numerical weather prediction and medical imaging. This project will focus on inverse problems, in the context of a particular methodology which is the ensemble Kalman filter (EnKF). The EnKF is a well-known filter aimed to estimates states of a dynamical process, using measurement data. It was proposed to alleviate computational issues from other well-known filters. Since then it has also been applied to inverse problems. This project will continue in this direction aiming to provide further insights into this methodology. Examples of this would include exploring the effectiveness of this method, compared to other optimizers, as inherently it is a derivative-free optimizer. Another topic of interest is how one can apply well-known machine learning techniques to increase the acceleration of the method. These topics, and others, will be complemented with experiments with real-world data. |

Related references | https://iopscience.iop.org/article/10.1088/0266-5611/29/4/045001 https://link.springer.com/book/10.1007/978-3-642-03711-5 https://link.springer.com/book/10.1007/978-0-387-76896-0 |

Where to apply? | Heriot-Watt University (HWU) website |

Project title | Mathematical modelling and analysis of plant-soil interactions: slope stabilities and coastal erosion |

Supervisor(s) | First: Mariya Ptashnyk (HWU) Co-supervisor: Qingping Zou (HWU) |

Project description | Plants acts as natural (green) coastal protection infrastructure against increasing coastal erosion due to more frequent and severe storms and rising sea level. Water has a significant impact on the soil properties and stability of soil slopes. Thus a better understanding of the impact of plants on the mechanical properties of soil, by modifying the content and flow of water in soil, will allow us to develop strategies for using plants to mitigate the slope instabilities induced by seepage and coastal erosion by severe storms. |

Related references | J. Vandamme, Q. Zou, (2013) Investigation of slope instability induced by seepage and erosion by a particle method, Computers and Geotechnics, 48, 9-20. G.J. Meijer, D.M. Wood, J.A. Knappett, A.G. Bengough, T. Liang (2023) Root reinforcement: continuum framework for constitutive modelling, Géotechnique, 73, 600-613 J.W. Both, M. Borregales, J.M. Nordbotten, K. Kumar, F.A. Radu (2017) Robust fixed stress splitting for Biot’s equations in heterogeneous media, Applied Mathematics Letters, 68, 101-108 |

Where to apply? | Heriot-Watt University (HWU) website |

Project title | Multiscale modelling and analysis of mechanical properties of biological tissues |

Supervisor | Mariya Ptashnyk (HWU) |

Project description | A better understanding of the interplay between mechanical properties and chemical processes in a biological tissue is important for analysis and understanding of development and growth of organs and organisms, as well as of causes for diseases. In humane and plant tissues mechanics and chemistry are interconnected on different spatial and temporal scales and multiscale modelling and analysis are vital for understanding of the nonlinear couplings between different processes and different scales. This project will address the development of multiscale models, model reduction, and multiscale analysis and numerical simulations of model equations, in the form of the nonlinear partial differential equations. The well-posedness analysis of the novel mathematical models and analysis of the numerical schemes will comprise the theoretical part of the project. A possible more computational direction will constitute the development of the efficient computable surrogate models using machine learning techniques. |

Related references | M. Ptashnyk, B. Seguin (2016) Multiscale analysis of a system of elastic and reaction-diffusion equations modelling plant cell wall biomechanics, ESAIM M2AN, 50, 593-631 A. Boudaoud, A. Kiss, M. Ptashnyk (2023) Multiscale modelling and analysis of growth of plant tissues, SIAM J Applied Math, in print A. Saraswathibhatla, D. Indana, O. Chaudhuri (2023) Cell–extracellular matrix mechanotransduction in 3D. Nat Rev Mol Cell Biol 24, 495-516 B. Blanco, H. Gomez, J. Melchor, R. Palma, J. Soler, G. Rus (2023) Mechanotransduction in tumor dynamics modeling, Physics of Life Reviews, 44, 279-301 |

Where to apply? | Heriot-Watt University (HWU) website |

Project title | A numerical analysis perscpective on Gaussian process regression |

Supervisor | Aretha Teckentrup (UoE) |

Project description | 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, 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. |

Related references | ‘- T. Bai, A.L. Teckentrup, K.C. Zygalakis. Gaussian processes for Bayesian inverse problems associated with linear partial differential equations. Available as arXiv preprint arXiv:2307.08343. – A.L. Teckentrup. Convergence of Gaussian process regression with estimated hyper-parameters and applications in Bayesian inverse problems. SIAM/ASA Journal on Uncertainty Quantification, 8(4), p. 1310-1337, 2020. Available as arXiv preprint arXiv:1909.00232. – M.M. Dunlop, M. Girolami, A.M. Stuart, A.L. Teckentrup. How deep are deep Gaussian processes? Journal of Machine Learning Research, 19, 1-46, 2018. Available as arXiv preprint arXiv:1711.11280. |

Where to apply? | University of Edinburgh (UoE) website |

Project title | Stochastic models of growing cell populations |

Supervisor | Tibor Antal (UoE) |

Project description | Stochastic models of growing cell populations. This project will focus on establishing and studying models of growing cell populations. The main motivation is cancer, where there are infinitely many open problems. There is also a (practically) infinite amount of new experimental or clinical data available. I’m particularly interested in initiation, progression and metastasis formation models. I’m also keen to work on modelling the effects of ageing, environment, or the immune system on cancer. I have specific projects in modelling chromosomal instability or multi drug resistance. The project(s) could benefit from existing collaborations with Oxford Oncology and Harvard Medical School, as well as from new collaborations set up specifically for the topic you choose. The tools used are stochastic processes and Markov chain theory, but the approach can range from rigorous results to simulations, data analysis and heuristic derivations. Similar evolutionary principles appear in bacterial or virus evolution, where new theories can be applied and experimentally tested in some cases. Here we’ll benefit from existing collaborations with e.g. Ecology and Evolution at Edinburgh University. For more details please check out my recent papers on my website and email me. |

Related references | From my website please check: “Evolutionary dynamics of cancer in response to targeted combination therapy” and “Cancer recurrence times from a branching process model” for examples. |

Where to apply? | University of Edinburgh (UoE) website |

Project title | Mathematical models of gene expression and their integration with single-cell data |

Supervisor | Nikola Popovic (UoE), Co-supervisor: Ramon Grima (School of Biological Sciences, UoE) |

Project description | A gene regulatory network involves a set of genes interacting with each other to control cellular functions. Mathematical models of stochastic gene expression have provided insight into how “intrinsic” noise due to transcription and translation can be controlled [1]. However, these models ignore important sources of fluctuations such as those due to cell growth, cell division, DNA replication and cell-size dependent transcription. In this project, the student will build on recent advances [2] to construct a detailed stochastic model of gene regulation that includes these noise sources. A first aim is the approximate analytical solution of the model to quantify how each source of noise contributes to emergent phenomena observed at the single-cell level. A secondary aim is a reduction of this detailed model by the modification of recently proposed AI techniques [3]. A final aim involves the use of the analytical solution within a Bayesian inference framework to estimate parameters in gene regulatory networks from single-cell data. The project will give the student a solid foundation in the basic molecular biology of transcription and its modelling through stochastic simulation, the chemical master equation, and techniques from machine learning. No previous background on these topics is assumed, though experience in the analytical and numerical solution of ordinary differential equations and some experience in coding is preferable. |

Related references | [1] R. Grima et al “Steady-state fluctuations of a genetic feedback loop: An exact solution.” The Journal of chemical physics 137.3 (2012): 035104; Z. Cao and R. Grima. “Linear mapping approximation of gene regulatory networks with stochastic dynamics.” Nature communications 9.1 (2018): 1-15. [2] C. Jia and R. Grima. “Frequency domain analysis of fluctuations of mRNA and protein copy numbers within a cell lineage: theory and experimental validation.” Physical Review X 11.2 (2021): 021032. [3] J. Qingchao, et al. “Neural network aided approximation and parameter inference of non-Markovian models of gene expression.” Nature communications 12.1 (2021): 1-12 |

Where to apply? | University of Edinburgh (UoE) website |

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Project title | Automatic differentiation in the search for simple invariant solutions in vortical and stellar dynamics |

Supervisor | Jacob Page (UoE), Co-supervisor: Anna Lisa Varri (UoE) |

Project description | The rise of machine learning has had a tremendous impact on computational science, although one under-exploited area is the scope for automatic differentiation (AD) to revolutionise “traditional” numerical solvers (e.g. see [1]). Unlike standard machine learning models, AD has no impact on interpretability but allows us to approach complex nonlinear problems in a new way, with a clear methodology to search for trajectories of interest via gradient-based optimisation. This project will explore the use of AD in building robust low order (N-body) models to explain observations in both large scale turbulence and stellar dynamics. The student will first adapt an N-body point-vortex code, originally used to find vortex crystals in superfluids [2], to explore the role of unstable vortex crystals in upscale energy transfer in full two-dimensional Navier-Stokes turbulence, both in the plane and on the surface of a sphere. The hope would be to build a simple symbolic dynamical picture with a small handful of relevant point-vortex states. Similarly, this approach can be exploited also to search for new equilibria in low-order systems of bodies interacting by gravitational forces. Elegant periodic solutions have been occasionally discovered [3], but further progress has been hampered by the prohibitive computational cost of traditional numerical methods. This project, therefore, carries significant potential to extend to a new regime the insightful analogy [4] between the properties of two-dimensional point vortices and three-dimensional self-gravitating systems, as initiated by Onsager [5] and Chandrasekhar [6]. |

Related references | [1] “Machine learning–accelerated computational fluid dynamics”, D. Kochkov, J. A. Smith, A. Alieva, Q. Wang, M. P. Brenner, and S. Hoyer, PNAS 118 (2021) [2] “Exploring the free-energy landscape of a rotating superfluid”, A. Cleary, and J. Page, Chaos 33 (2023) [3] “A remarkable periodic solution of the three-body problem in the case of equal masses”, A. Chenciner, and R. Montgomery, Ann. of Math. 2, 152 (2000) [4] “Statistical Mechanics of Two-Dimensional Vortices and Collisionless Stellar Sys- tems”, P. Chavanis, J. Sommeria, and R. Robert, Astrophys. J. 471, 1, (1996) [5] “Statistical hydrodynamics”, L. Onsager, Nuovo Cimento Suppl. 6, 279 (1949) [6] “Principles of stellar dynamics”, S. Chandrasekhar, Dover, (1942) |

Where to apply? | University of Edinburgh (UoE) website |

Project title | Minimal turbulence in viscoelastic fluids |

Supervisor | Jacob Page (UoE) |

Project description | Viscoelasticity is a property of a wide range of industrial and biological fluids – polymer solutions, blood and even magma flows behave ‘viscoelastically’ in certain situations. Viscoelastic fluids often defy intuition we have gained from studying Newtonian fluids like air or water (e.g. viscoelastic fluids will climb a rotating rod). From an industrial point of view, the most important behaviour of viscoelastic fluids is their ability to support new types of self-sustaining turbulent motion – even in the absence of inertia. At present there are believed to be three flavours of viscoelastic turbulence: perturbed Newtonian turbulence, two-dimensional elasto-inertial turbulence and inertialess elastic turbulence [1, 2, 3], though our understanding of these regimes and their connections is weak. In this project the student will seek to study the different types of viscoelastic turbulence by identifying “minimal flow units” in both inertia-dominated and inertialess flows. Minimal flow units are the smallest computational domains in which turbulence can be sustained [4]. The approach will be based on a combination of direct numerical simulation and techniques from modern dynamical systems theory, with the aim being to connect the chaotic dynamics to linear instabilities [e.g. see 5]. The project may also involve application of data-driven methods for identifying unstable periodic orbits, in the hope of isolating the self-sustaining processes at play in viscoelastic chaos. A particular focus will be on the importance of a newly discovered ‘polymer diffusive instability’ (PDI) [6] in the bifurcations leading to elastic turbulence. |

Related references | [1] “Machine learning–accelerated computational fluid dynamics”, D. Kochkov, J. A. Smith, A. Alieva, Q. Wang, M. P. Brenner, and S. Hoyer, PNAS 118 (2021) |

Where to apply? | University of Edinburgh (UoE) website |

Project title | Stochastic models of DNA damage and repair in cancer |

Supervisor | Michael Nicholson (UoE, Mathematics), Martin Taylor (UoE, Human Genetics) |

Project description | The DNA in our cells is continuously attacked by a variety of molecular processes, which can lead to a multitude of adverse outcomes, such as the generation of cancer-causing mutations. DNA repair processes exist to minimise these risks. Measuring the mechanisms of DNA repair in human cells, e.g. how frequently repair occurs and under what circumstances, is experimentally challenging. Such measurements have wide implications in understanding cancer initiation, ageing, and drug resistance. To meet this need, in this project stochastic models will be created to map how damage and repair interact in shaping the mutations observed in cancer genomes. Combining these models with DNA sequencing data will allow measurement of these fundamental molecular processes in a statistically rigorous manner. The student will have access to large, state-of-the-art genomics datasets and will be closely integrated with Martin Taylor’s group at the Human Genetics Unit. The interdisciplinary training offered will including probabilistic modelling, simulation, statistical inference, bioinformatics, molecular biology, and will provide an excellent platform for careers in either academia or industry. |

Related references | C. Anderson, L. Talmane et al (2022), Strand-resolved mutagenicity of DNA damage and repair, biorXiv preprint: https://doi.org/10.1101/2022.06.10.495644 [In particular, see the section on modelling transcription-coupled repair] N. Spisak (2023), Disentangling sources of clock-like mutations in germline and soma, https://doi.org/10.1101/2023.09.07.556720 S. Aitken et al (2021), Pervasive lesion segregation shapes cancer genome evolution, Nature |

Where to apply? | University of Edinburgh (UoE) website |

Project title | Measuring and modelling the variability in DNA copy number alterations over time |

Supervisor | Michael Nicholson (UoE, Mathematics), Thomas (Ollie) McDonald (Harvard T.H. Chan School of Public Health) |

Project description | Determining when and why cancer-causing DNA mutations arise is critical for a variety of clinical needs, such as the early detection of cancer. For a particular class of mutations, copy number alterations, it appears that these often occur in a short time window, years before the onset of cancer. Why this happens is unclear. Using a stochastic modelling approach, this project will investigate molecular and evolutionary hypotheses for these observations. The models will be combined with state-of-the-art single cell DNA sequencing datasets. The student will work closely with colleagues at Harvard University. The interdisciplinary training offered will including probabilistic modelling, simulation, statistical inference, bioinformatics, molecular biology, and will provide an excellent platform for careers in either academia or industry. |

Related references | [1] D. Minussi, M.D.Nicholson, H. Ye et al (2021) Breast tumours maintain a reservoir of subclonal diversity during expansion, Nature [In particular, see the section and supplementary information on transient instability] [2] R. Gao et al (2016), Punctuated copy number evolution and clonal stasis in triple-negative breast cancer, Nature Genetics [3] Gerstung et al (2020), The evolutionary history of 2,658 cancers, Nature |

Where to apply? | University of Edinburgh (UoE) website |