Slip and twinning in single crystals and polycrystals

Supervisor: John Ball

Description: The aim of the project is to analyse the interaction between slip (related to plasticity) and twinning (deformations in which the crystal lattices on either side of an interface are reflected with respect to some direction) in single crystals and polycrystals. This has proved important in recent work of materials scientists (see [2,3]). An initial step could be to extend some of the results in [1], based on the Ericksen energy-well picture, to situations allowing solid phase transformations. More generally the study of resulting microstructures leads to deep issues in the calculus of variations related to quasiconvexity, but progress should be possible in some interesting simplified situations.

The project offers training in modern techniques of nonlinear partial differential equations and the calculus of variations applied to materials science.

References:

[1] J. M. Ball, Slip and twinning in Bravais lattices, J. Elasticity 155, 763–785, 2024.

[2] H. Seiner, P. Sedlák, M. Frost and P. Sittner, Kwinking as the plastic forming mechanism of B19 ′ NiTi martensite. International Journal of Plasticity, 168, 103697, 2023.

[3] T. Inamura, Geometry of kink microstructure analysed by rank-1 connection, Acta Materialia, 173, 270-280, 2019.

Nonlinear elasticity and computer vision

Supervisor: John Ball

Description: The project concerns the comparison of images by minimizing a functional depending on a map taking one image to the other and on features of the images. Part of the functional is the same as that for nonlinear elasticity. In previous work (see [1,2]) some basic properties of this model, such as existence of minimizers, have been established, and conditions found under which for linearly related images the minimization delivers the corresponding linear map as the unique minimizer. The project will extend this work in several directions, in particular testing the minimization algorithm numerically, and considering the effect of adding a term depending on second derivatives of the deformation to the functional.

The project offers a training in nonlinear analysis, especially the calculus of variations, in related numerical methods, and in nonlinear elasticity itself.

References:

[1] J. M. Ball and C. L. Horner, Image comparison and scaling via nonlinear elasticity. In L. Calatroni,

M. Donatelli, S. Morigi, M. Prato, and M. Santacesaria, editors, Scale Space and Variational Methods

in Computer Vision, pages 565–574, Cham, 2023. Springer International Publishing.

[2] J. M. Ball and C. L. Horner, A nonlinear elasticity model in computer vision, submitted, arXiv:2408.17237.

Random graphs and networks: limits, approximations, and applications

Supervisors: Fraser Daly and Seva Shneer

Description: We will consider some models for random graphs where some nodes may form more connections than others. Such models include non-homogeneous graphs and configuration models. We aim to study various measures of connectedness of such graphs, for instance the probabilities of randomly chosen nodes forming cliques or other subgraphs.

We aim to find asymptotics as well as approximations for these measures of connectedness as the size of the graph is going to infinity. Techniques which could be applied here include Stein's method for probability approximations. Extensions of this work include analogous results for dynamic random graphs evolving in time and for stochastic processes on random graphs.

Approximations for Random Sums with Dependence

Supervisors: Fraser Daly and Seva Shneer

Description: Sums of a random number of random variables have applications in many areas, including insurance, where they can be used to represent the total claim amount received within a given year: a random number of claims is received, each of which is for a random amount. Classically, these individual claim amounts are assumed to be independent and identically distributed, and independent of the number of claims received. This allows us to derive approximations for the distribution of the total claim amount, for example a Gaussian approximation using the central limit theorem. However, these assumptions of independence are unrealistic, and we would like to relax them.

The aim of this project is to derive and investigate explicit distributional approximations for sums of a random number of random variables with dependence, using Stein's method for probabilistic approximation and other relevant tools.   

References:

[1] L. H. Y. Chen, L. Goldstein and Q.-M. Shao (2011). Normal Approximation by Stein's Method. Springer, Berlin. 

[2] F. Daly (2021). Gamma, Gaussian and Poisson approximations for random sums using size-biased and generalized zero-biased couplings. Scandinavian Actuarial Journal, to appear.

Last passage percolation in directed random graphs.

Supervisor: Sergey Foss

Description: The aim of the project is to study various properties of the maximal path length (or weight) in growing random directed graphs where the vertices are partially ordered and edges may exist only from smaller to bigger vertices.

Prospective results include (but are not limited to) analysis of performance characteristics, various limit theorems and simulation algorithms for the growth rate and variance, with a number of applications in biology and computer sciences.

An introduction to the subject may be found in the recent overview paper

S. Foss, T. Konstantopoulos, B. Mallein and S. Ramassami,

``Last passage percolation and limit theorems in Barak-Erdős directed random graphs and related models'',

Probability Surveys, vol. 21, pp. 67--170, 2024.

https://arxiv.org/abs/2312.02884

Heavy-tailed distributions in the SGD algorithms

Supervisor: Sergey Foss

Description: The project relates to developing and analysing of performance and limiting properties of stochastic difference equations in discrete time that contain random components with heavy-tailed distributions.

The problems of this kind do naturally appear in the machine learning literature. It has repeatedly been observed that loss minimization by stochastic gradient descent (SGD) leads to heavy-tailed distributions of neural network parameters.

Basic and advanced properties of heavy-tailed distributions, as well as many exercises, may be found in the book

S. Foss, D. Korshunov and S. Zachary,

An introduction to heavy-tailed and subexponential distributions, 2nd edition.

Springer, 2013.

Epidemics with migration

Supervisors: Sergey Foss and Seva Shneer

Description: We study models where agents may arrive into a system, may change their state (for instance, infected, susceptible, exposed, recovered) and may change their location in the system. We plan to study the stationary regime of such a system. Questions of interest include conditions for extinction/survival of the epidemic and analysis of the effects of mobility on the spread of the epidemic.

References:

[1] F. Baccelli, S. Foss, S. Shneer (2024). Migration-contagion processes. Advances in Applied Probability, 56, 1, 71-105.

Convergence of stochastic processes with applications to computational statistics and machine learning

Supervisor: Mateusz Majka

Description: The project will be concerned with the convergence to equilibrium of several different types of stochastic processes, including solutions of stochastic differential equations (driven by either Brownian motion or jump processes) and interacting particle systems, as well as their discrete-time counterparts (see papers [2] and [4]). We will also explore the connections between stochastic processes and optimization methods on the space of probability measures [3]. Results of such type, besides their theoretical significance, have found numerous applications in computational statistics and machine learning. For instance, by employing the probabilistic coupling technique, one can obtain precise convergence rates of numerous Monte Carlo algorithms that are constructed by utilizing discretisations of stochastic differential equations, and are used in computational statistics for sampling from high dimensional probability distributions [1]. On the other hand, mathematical tools such as functional inequalities provide convergence rates for certain optimization algorithms on the space of probability measures that are directly connected to machine learning applications such as training neural networks [3]. Depending on the candidate's interests, the project can focus either on the numerical/computational aspect or on developing the underlying mathematical theory. For more details about my research interests and my research team, please see my website https://sites.google.com/site/mateuszbmajka

References:

4. L. Liu, M. B. Majka and P. Monmarché, L^2-Wasserstein contraction for Euler schemes of elliptic diffusions and interacting particle systems, Stochastic Process. Appl. 179 (2025), 104504.

3. R.-A. Lascu, M. B. Majka, D. Šiška and Ł. Szpruch, Linear convergence of proximal descent schemes on the Wasserstein space, https://arxiv.org/pdf/2411.15067.

2. M. Liang, M. B. Majka and J. Wang, Exponential ergodicity for SDEs and McKean-Vlasov processes with Lévy noise, Ann. Inst. Henri Poincaré Probab. Stat. 57 (2021), no. 3, 1665-1701.

1. M. B. Majka, A. Mijatović and Ł. Szpruch, Non-asymptotic bounds for sampling algorithms without log-concavity, Ann. Appl. Probab. 30 (2020), no. 4, 1534-1581.

Non-commutative integrable systems

Supervisor: Simon Malham

Description: 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, ferromegnetism 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.

[1] Doikou, A., Malham, S.J.A. and Stylianidis, I. 2021, Grassmannian flows and applications to non-commutative non-local and local integrable systems, Physica D 415, 132744.

[2] Malham, S.J.A. 2022, Integrability of local and nonlocal non-commutative fourth order quintic NLS equations, IMA J. Appl. Math. 87(2), 231-259.

[3] Malham, S.J.A. 2022, The non-commutative Korteweg-de Vries hierarchy and combinatorial Poppe algebra, Physica D 434.

[4] Blower, G., Malham, S.J.A. 2023, The algebraic structure of the non-commutative nonlinear Schrodinger and modified Korteweg-de Vries hierarchy, Physica D 456, 133913.

[5] Blower, G., Malham, S.J.A. 2024, Direct linearisation of the non-commutative Kadomtsev-Petviashvili equations, submitted.

Coagulation sol-gel phenomena

Supervisor: Simon Malham

Description: 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.

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

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

Markov Chain Monte Carlo with applications to Computational Imaging and Machine Learning

Supervisors: Konstantinos Zygalakis (UoE), Paul Dobson (HWU)

Description: Modern data science relies strongly on probability theory to solve challenging problems. In this context, probabilistic models represent the raw data observation process and the prior knowledge available; and solutions are then obtained by performing (often Bayesian) statistical inference analyses. From a computation viewpoint, these analyses are conducted using Markov chain Monte Carlo (MCMC) algorithms, stemming from the theory of Markov processes. Constructing efficient MCMC algorithms for high-dimensional problems is difficult (e.g., problems related to image processing and machine learning), and this has stimulated a lot of research on efficient high-dimensional MCMC algorithms and theoretical structures (relying mostly on the theory of Markov processes) to understand, analyse and quantify their efficiency. In particular, MCMC algorithms based on SDEs have received a lot of attention lately, leading to some major developments in highly efficient MCMC methodology.

In this project, we will be looking to combine ideas from different areas of applied probability and machine learning to develop further highly efficient MCMC algorithms for solving high dimensional arising in computational imaging and machine learning.

In addition to joining the Maxwell Institute this PhD is also related to the Probabilistic AI hub a collaboration between 6 leading UK universities.

References:

A. Durmus, G. O. Roberts, G. Vilmart, K. C. Zygalakis, Fast Langevin based algorithm for MCMC in high
dimensions Ann. App. Prob., 27(4), 2195-2237, (2017).

J. M. Sanz Serna, K. C. Zygalakis, Wasserstein distance estimates for the distributions of numerical
approximations to ergodic stochastic differential equations. J. Mach. Learn. Res., 22, 1–37, (2021).

T. Klatzer, P. Dobson, Y. Altmann, M. Pereyra, J. M. Sanz-Serna, and K. C. Zygalakis Accelerated Bayesian
imaging by relaxed proximal-point Langevin sampling. SIAM J. Imaging Sci. 17(2) 1078-1117, (2024).

The s-numbers of higher order Sobolev embeddings

Supervisor: Lyonell Boulton

Description: The theory of embeddings between function spaces began with the need to prove existence and uniqueness results about the solution of partial differential equations. The theory florished as the demand for solving more complicated equations drew a need to suply sharper inequalities. This symbiosis is illustrated by the classical Sobolev inequality discovered in the 1930s. In many applications it is enough to show existence of the constant, but for some other applications the precise value of the minimal constant is required. One example of this is in the theory surounding the so-called Euclidean isoperimetric inequality.

The optimal constant for the first order Sobolev inequality on a segment with zero boundary conditions was computed by Talenti in 1976. Since then, rather little has been studied about the case of Sobolev spaces of higher order. An exception to this is the case of second order embeddings examined recently in [1], using tools from [3]. The purpose of this project will be a thorough investigation of properties of the optimal constant, and the so-called singular numbers for higher order embeddings.

A good starting point to understand the background of this project, is the

book [2].

References:

[1] L. Boulton and J. Lang, Nonlinear Analysis. 236 (2023), 113362.

[2] D.E. Edmunds and J. Lang, Eigenvalues, Embeddings and Generalised

Trigonometric Functions. Springer-Verlag, Berlin, 2011.

[3] D.E. Edmunds, P. Gurka and J. Lang, J. Approx Theo. 164 (2012), 47-56.

Are regular polygons optimal in relativistic quantum mechanics?

Supervisor: Lyonell Boulton

Description: Among all polygons with the same perimeter, regular polygons are known to minimise the ground energy of the Dirichlet Laplacian. This is also the case, if we consider the area as the fixed geometrical invariant rather than the perimeter. In the language of quantum mechanics, this can be re-interpreted by saying that the non-relativistic Schrodinger operator on boxes with the same perimeter or area, attains its minimal energy when the box has a regular base. These results can be traced back to the work of Polya in the 1950s and, arguably, they have enabled the rise to a whole new area of research in Geometrical Spectral Theory which is still an active subject of enquiry with strong links to Mathematical Physics.

The aim of this PhD project is to develop research in the following direction. Suppose that we replace the Schrodinger operator with the free Dirac operator and pose the same question now in the relativistic setting. Will the box with a regular base be the optimal minimising energy shape (in absolute value) among all others?

Recent progress on this problem include the paper [1], which considers general regions and [2], where this question is posed, but not solved. It appears that, even when the region is rectangular, an answer as to whether the square is indeed the optimal shape is not so easy to tackle.

A concrete â€¯initial stage of the project will be the analytic investigation of the problem on quadrilateral regions, trying to identify a shape for which the problem can be reduced and treated with exact formulas. Depending on progress we might consider numerical investigations along the lines of [3], in order to gain an insight on further lines of enquiry. The initial phase of the project might lead towards different directions, including the addition of magnetic or electic fields.

References:

[1] Ann. Henri PoincarÃ© 19, 1465â€“1487 (2018)

[2] J. Math. Phys. 63, 013502 (2022)

[3] App. Num. Math. 99, 1â€“23 (2016)

The Laplacian eigenvalues on regions with symmetries.

Supervisors: Lyonell Boulton and Beatrice Pelloni

Description: The eigenvalues of the Laplacian on a rectangle can easily be found in terms of trigonometric functions. On an ellipse they can be found in terms of Bessel functions. From these two examples, we might be tempted to assume that then it is possible to find the eigenvalues on other simple regions (for example by arguments involving symmetry). Indeed they can be computed exactly on a straight isosceles triangle in terms of those on the square. In general, however, this is not true. Even for a generic triangle (the simplest possible 2D region), it is not true in that we can always find a close expression for the smallest eigenvalue or the corresponding eigenfunction.

Regions for which we know the exact eigenvalues include the equilateral triangle. A list of these was first computed by Gabriel Lamé in the 19th Century and the arguments for the calculation are very sophisticated, involving a tessellation of the plane by means of parallelepipeds and studying various symmetry groups. It is remarkable that it was only in 1985, that the first full proof that Lamé’s list was complete was found [1].

Recently, a new approach by Fokas and Kalimeris [3], seems to provided an effective mechanism to compute the full list of eigenvalues for the equilateral triangles with Dirichlet, Neumann and other natural boundary conditions. This techinique appears to be very promising and it is yet to be tested on other regions.

This PhD project will begin by analysing and comparing the proofs made in [1] and [3], alongside with a simpler proof found in [2], that Lamé's list of eigenvalues on the triangle is complete. Then, we will move onto investigating the following problem. Can we compute explicitly the eigenvalues on a regular hexagon? Then, what about other regular polygons in general? For this, the case of mixed boundary conditions (Dirichlet and Neumann) on triangles might be a natural line of enquiry.

See [4] for a full list of references on the subject.

References:

[1] M. Pinsky, SIAM J. Math. Anal. 16 (1985) 848-851.

[2] B.J. McCartin, SIAM Rev. 45 (2003), 267-287.

[3] T. Fokas and M. Kalimeris, Comp. Methods and Funct. Theo. 14 (2014) 10-33.

[4] D.S. Grebenkov and B.-T. Nguyen, SIAM Rev. 55 (2013) 601-667.

Travelling-wave solutions for models of growth, interaction, depletion and diffusion

Supervisor: Seva Shneer

Description: We will study interacting-particle systems relevant for the study of scheduling mechanisms in stochastic networks such as redundancy and load balancing. The models are also of interest in other application areas ranging from biology to communication networks. The evolution of states of particles consists of mechanisms of growth, interaction, depletion and diffusion. We will aim at characterising their scaling limits, in particular in terms of travelling waves.

Semi-discrete optimal transport theory: Numerical methods and applications

Supervisor: David Bourne

Description: Optimal transport theory goes back to 1781 and the French engineer Gaspard Monge, who wanted to find the optimal way of transporting soil for building earthworks for Napoleon's troops. While Leonid Kantorovich made some progress on the problem in the 1940s with the invention of linear programming, the problem remained unsolved for over 200 years. In fact it was not even known whether there existed a solution until some big mathematical breakthrough in the 1980s and 1990s. These theoretical advances opened the floodgates to applications. Optimal transport theory is now applied to PDEs, geometry, economics, image processing, crowd dynamics, statistics, machine learning, and the list goes on. In July 2018 the Italian mathematician Alessio Figalli won a Fields Medal for his work in optimal transport and PDEs.

This PhD project focusses on an important class of optimal transport problems known as semi-discrete transport problems, which have recently found applications in weather modelling [1], pattern formation [2], microstructure modelling [3], optics, and fluid mechanics. In this project we will explore further applications and develop novel numerical methods.

References:

[1] Bourne, D.P., Egan, C.P., Pelloni, B. & Wilkinson, M. (2022) Semi-discrete optimal transport methods for the semi-geostrophic equations, Calculus of Variations and Partial Differential Equations, 61:39.

[2] Bourne, D.P. & Cristoferi, R. (2021) Asymptotic optimality of the triangular lattice for a class of optimal location problems, Communications in Mathematical Physics, 387, 1549-1602.

[3] Bourne, D.P., Kok, P.J.J., Roper, S.M. & Spanjer, W.D.T. (2020) Laguerre tessellations and polycrystalline microstructures: A fast algorithm for generating grains of given volumes, Philosophical Magazine, 100, 2677-2707.

Elasticity methods in computer vision

Supervisor: John Ball

Description: The project concerns the comparison of images by minimizing a functional depending on a map taking one image to the other and on features of the images (see [1]). Part of the functional is the same as that for nonlinear elasticity. In work with a current student some basic properties of this model, such as existence of minimizers, have been established, and conditions found under which for linearly related images the minimization delivers the corresponding linear map. The project will extend this work in several directions, in particular testing the minimization algorithm numerically, and considering the effect of adding a term depending on second derivatives of the deformation to the functional.

The project offers a training in nonlinear analysis, especially the calculus of variations, in related numerical methods, and in nonlinear elasticity itself.

References:

[1] J. M. Ball, Nonlinear elasticity and image processing, Lecture at Newton Institute, https://talks.cam.ac.uk/talk/index/96634

Equilibrium of liquid crystals in exterior domains

Supervisor: John Ball

Description: The aim of the project is to investigate the equilibrium configurations of nematic liquid crystals in the 3D region outside a finite number of bounded open sets $W_i$, according to the Oseen-Frank theory whose state variable is a unit vector field giving the mean orientation of the rod-like molecules forming the liquid crystal. This problem was studied in 2D in [1], but the 2D theory has a different flavour in that equilibria are smooth, while in 3D they can have singularities (see [2]) such as point defects. Currently there is much interest in liquid crystal colloids, in which the $W_i$ are particles that can move, and the proposed project has possible developments for the study of such dynamical situations.

The project offers training in modern techniques of nonlinear partial differential equations and the calculus of variations adaptable to other situations.

References:

[1] Lu Liu, The Oseen-Frank theory of liquid crystals, Thesis, Oxford, 2019.

[2] Haïm Brezis, Jean-Michel Coron, and Elliott H Lieb. Harmonic maps with defects. Communications in Mathematical Physics, 107(4):649–705, 1986.

The Laplacian eigenvalues on regions with symmetries.

Supervisors: Lyonell Boulton and Beatrice Pelloni

Description: The eigenvalues of the Laplacian on a rectangle can easily be found in terms of trigonometric functions. On an ellipse they can be found in terms of Bessel functions. From these two examples, we might be tempted to assume that then it is possible to find the eigenvalues on other simple regions (for example by arguments involving symmetry). Indeed they can be computed exactly on a straight isosceles triangle in terms of those on the square. In general, however, this is not true. Even for a generic triangle (the simplest possible 2D region), it is not true in that we can always find a close expression for the smallest eigenvalue or the corresponding eigenfunction.

Regions for which we know the exact eigenvalues include the equilateral triangle. A list of these was first computed by Gabriel Lamé in the 19th Century and the arguments for the calculation are very sophisticated, involving a tessellation of the plane by means of parallelepipeds and studying various symmetry groups. It is remarkable that it was only in 1985, that the first full proof that Lamé’s list was complete was found [1].

Recently, a new approach by Fokas and Kalimeris [3], seems to provided an effective mechanism to compute the full list of eigenvalues for the equilateral triangles with Dirichlet, Neumann and other natural boundary conditions. This techinique appears to be very promising and it is yet to be tested on other regions.

This PhD project will begin by analysing and comparing the proofs made in [1] and [3], alongside with a simpler proof found in [2], that Lamé's list of eigenvalues on the triangle is complete. Then, we will move onto investigating the following problem. Can we compute explicitly the eigenvalues on a regular hexagon? Then, what about other regular polygons in general? For this, the case of mixed boundary conditions (Dirichlet and Neumann) on triangles might be a natural line of enquiry.

See [4] for a full list of references on the subject.

References:

[1] M. Pinsky, SIAM J. Math. Anal. 16 (1985) 848-851.

[2] B.J. McCartin, SIAM Rev. 45 (2003), 267-287.

[3] T. Fokas and M. Kalimeris, Comp. Methods and Funct. Theo. 14 (2014) 10-33.

[4] D.S. Grebenkov and B.-T. Nguyen, SIAM Rev. 55 (2013) 601-667.

One-parameter Semigroups on Metric Graphs

Supervisor: Lyonell Boulton

Description: The purpose of this project is to study the time-evolution equation associated to linear differential operators on a graph. We assume that the edges of the graph (of a certain length) are segments and that suitable regularity conditions and boundary conditions are fixed on the nodes.

According to the seminal work of Kramar-Fijavz, in the case of the Laplacian we know necessary and sufficient conditions at the nodes, for the time evolution problem to have a solution for any initial condition which is square integrable - in the Hilbert space $L^2$. This is a universal existence result. The main purpose of this project is to extend the work of Kramar-Fijavz following three main leads.

1- Classify the family of initial conditions for which the evolution problem still has a solution, despite of exhibiting non-universal existence.

2- Consider more general operators, such as those of Sturm-Liouville-type.

3- Consider the more general case of Banach spaces $L^p$.

The project itself will ensure training in the state-of-the-art of non-self-adjoint spectral theory and the theory of one-parameter semigroups on graphs.

The s-numbers of higher order Sobolev embeddings

Supervisor: Lyonell Boulton

Description: The theory of embeddings between function spaces began with the need to prove existence and uniqueness results about the solution of partial differential equations. The theory florished as the demand for solving more complicated equations drew a need to suply sharper inequalities. This symbiosis is illustrated by the classical Sobolev inequality discovered in the 1930s. In many applications it is enough to show existence of the constant, but for some other applications the precise value of the minimal constant is required. One example of this is in the theory surounding the so-called Euclidean isoperimetric inequality.

The optimal constant for the first order Sobolev inequality on a segment with zero boundary conditions was computed by Talenti in 1976. Since then, rather little has been studied about the case of Sobolev spaces of higher order. An exception to this is the case of second order embeddings examined recently in [1]. The purpose of this project will be a thorough investigation of properties of the optimal constant, and the so-called singular numbers for higher order embeddings.

References:

[1] Arxiv 2204.04703

Hierarchical Methods for Stochastic Partial Differential Equations

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.

Supervisor: Abdul-Lateef Haji-Ali

Numerical Methods for Financial Market Models

Description: Many models for the evolution of financial and economic variables, for example interest rates, inflation rates, advanced models of stock prices, have no known closed-form analytical solution. To be able to work with these models, for example to value financial derivative products and their risk management, it is of fundamental importance to design numerical methods that are accurate, efficient, and easily and efficiently adaptable to changing market conditions. Financial derivatives are ubiquitous, they are embedded in many standard financial and insurance products and play an important role themselves in the risk management of companies. In this project, we will apply methods from stochastic analysis and probability theory to models of financial markets to enhance the understanding of their stochastic properties, and to design high-quality fast methods for their numerical treatment.

Supervisor: Anke Wiese

An Optimisation View of Deep Learning Methods for Data Science

Description: Data science transforms data into interpretable information to enable accurate decision-making. Such methods rely on advanced mathematical tools. Optimisation is one of them, and it is broadly used to design robust, fast, and scalable algorithms to minimise given objective functions. Since early 2000’s, proximal methods have become state-of-the-art to solve minimisation problems, in particular in the context of inverse problems. During the last decade, proximal algorithms involving neural networks (NNs) have emerged. Two main classes of such hybrid methods can be distinguished. The first approach consists in unrolling an optimisation algorithm over a fixed number of iterations to build the layers of a NN, leading to unfolded NNs. Unfolded NNs are particular instances of end-to-end NNs, that are directly used to solve inverse problems, processing corrupted data to produce a corrected output. The second approach relies on replacing the denoising steps of an optimisation algorithm by NNs, leading to PnP algorithms.

Research related to optimisation-based NNs for inverse imaging problems is relatively recent, and devoted methods are evolving fast. Some of the challenges of interest in this field are related to (i) theoretical understanding, (ii) design of new methods (including optimisation algorithms, sampling methods, NNs, etc.), (iii) applications (e.g., medical, astronomical, photon imaging).

Possible research directions for this project include (but are not restricted to):

(1) Theoretical guarantees of hybrid optimisation-NN methods: Although convergence of PnP algorithms has started to be better understood recently, many questions remain unanswered (or partially answered). For instance, to unsure convergence of PnP algorithms, NNs must satisfy some technical conditions. How to build such NNs? Similarly, when unrolling a fixed number of iterations of an optimisation algorithm, all the theoretical guarantees are lost. What type of guarantees do unfolded NNs offer?

(2) Building flexible NNs for inverse problems: In PnP methods, the involved NNs depend on the underlying statistical models (e.g., higher noise level on the measurements requires stronger denoisers). Hence different NNs must be trained depending on the inverse problem' statistical model, which is computationally prohibitive. How to build more flexible NNs that can be adapted to multiple statistical models?

(3) Improve NN efficiency using optimisation: Optimisation methods benefit from numerous acceleration strategies. Can (unfolded) NNs benefit from such acceleration techniques to design more powerful networks?

Supervisors: Audrey Repetti and Prof. Jean-Christophe Pesquet (University of Paris-Saclay, CentraleSupelec)

Interacting particle systems and SPDEs of McKean-Vlasov type

Supervisor: Michela Ottobre

Description: The study of Interacting Particle Systems (IPSs) and related kinetic equations has attracted the interest of the mathematics and physics communities for decades. Such interest is kept alive by the continuous successes of this framework in modelling a vast range of phenomena, in diverse fields such as biology, social sciences, control engineering, economics, game theory, statistical sampling and simulation, neural networks etc.

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 stochastic analysis 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, typically, 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. We will consider PS where the dynamics of each particle is described by a Stochastic Differential Equation (SDE). When N tends to infinity, depending on the specific nature of the noise, one can obtain either a PDE for the particle density (i.e. a deterministic equation, which is the classical setting) or a Stochastic PDE. It is well known that, even in the classical case in which the limit is a PDE, the particle system and the PDE can have very different properties, raising questions about whether the PDE is a good approximation of the initial PS. The case in which the limit is stochastic is by far less investigated and this is the regime on which this project will focus.

Analysis of Multiscale problems: stochastic averaging and homogenization

Supervisor: Michela Ottobre

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 empirical constitutive relations. 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 (CG) models which incorporate the effects of the smaller/faster scales. Obtaining such CG descriptions in a principled way helps one strike a compromise between microscopic models, accurate but computationally expensive, and macroscopic models, which are less accurate but simpler. Multiscale methods play a fundamental role across science, providing both underpinning for numerics/simulation algorithms and modelling paradigms in an impressive range of fields, such as engineering, material science, mathematical biology, social sciences, 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.

This project is primarily concerned with the study of stochastic systems which possess multiple time-scales and that are modelled by stochastic differential equations (SDEs). In their simplest form, the systems we consider are made of two components, commonly referred to as the fast and slow scale. In this case, assuming the fast process (FP) evolves towards a (unique) equilibrium, so called equilibrium measure (EM), established methodologies allow one to obtain an effective dynamics by substantially ‘replacing’ the FP with its behaviour in equilibrium (intuitively, due to the large time-scale separation, the FP will immediately reach its equilibrium state). In this context, the method of multiscale expansions provides a way to formally derive the CG dynamics, while stochastic averaging (and homogenization) techniques provide analytical tools for rigorous proofs.

When using any of these techniques a key assumption is that the dynamics for the FP (more precisely, the so-called ‘frozen process’) has a unique EM (that is, that it is ergodic). Without such an assumption even formal multiscale expansions seem to be no longer useful. Nonetheless many systems in the applied sciences do exhibit multiple equilibria. In particular, when the fast process has multiple EMs, the procedure we have informally described above can no longer be used as is, and even producing an ansatz for the reduced description of the dynamics becomes nontrivial. This is the starting point for this project.

Departments

Actuarial Mathematics and Statistics

Computer Science

Mathematics