Past

We propose and analyse two structure preserving finite volume schemes to approximate the solutions to a cross-diffusion system with self-consistent electric interactions introduced by Burger, Schlake & Wolfram (2012). This system has been derived thanks to probabilistic arguments and admits a thermodynamically motivated Lyapunov functional that is preserved by suitable two-point flux finite volume approximations. This allows to carry out the mathematical analysis of two schemes to be compared.

This is joint work with Maxime Herda and Annamaria Massimini.

The goal of the talk is to give an overview of the metric theory of currents by Ambrosio-Kirchheim, together with some recent progress in the setting of Banach spaces. Metric currents are a generalization to the metric setting of classical currents. Classical currents are the natural generalization of oriented submanifolds, as distributions play the same role for functions. We present a structure result for 1-metric currents as superposition of 1-rectifiable sets in Banach spaces, which generalizes a previous result by Schioppa. This is based on an approximation result of metric 1-currents with normal 1-currents. This is joint work with D. Bate, J. Takáč, P. Valentine, and P. Wald (Warwick).

High-order tensor methods for solving both convex and nonconvex optimization problems have recently generated significant research interest, due in part to the natural way in which higher derivatives can be incorporated into adaptive regularization frameworks, leading to algorithms with optimal global rates of convergence and local rates that are faster than Newton's method. On each iteration, to find the next solution approximation, these methods require the unconstrained local minimization of a (potentially nonconvex) multivariate polynomial of degree higher than two, constructed using third-order (or higher) derivative information, and regularized by an appropriate power of the change in the iterates. Developing efficient techniques for the solution of such subproblems is currently, an ongoing topic of research,  and this talk addresses this question for the case of the third-order tensor subproblem. In particular, we propose the CQR algorithmic framework, for minimizing a nonconvex Cubic multivariate polynomial with  Quartic Regularisation, by sequentially minimizing a sequence of local quadratic models that also incorporate both simple cubic and quartic terms.

The role of the cubic term is to crudely approximate local tensor information, while the quartic one provides model regularization and controls progress. We provide necessary and sufficient optimality conditions that fully characterise the global minimizers of these cubic-quartic models. We then turn these conditions into secular equations that can be solved using nonlinear eigenvalue techniques. We show, using our optimality characterisations, that a CQR algorithmic variant has the optimal-order evaluation complexity of $O(\epsilon^{-3/2})$ when applied to minimizing our quartically-regularised cubic subproblem, which can be further improved in special cases.  We propose practical CQR variants that judiciously use local tensor information to construct the local cubic-quartic models. We test these variants numerically and observe them to be competitive with ARC and other subproblem solvers on typical instances and even superior on ill-conditioned subproblems with special structure.

The absolute Galois group of ℚₚ determines its field structure: a field K is p-adically closed if and only if its absolute Galois group is isomorphic to that of ℚₚ. This Galois-theoretic characterisation was proved by Koenigsmann in 1995, building on previous work by Arason, Elman, Jacob, Ware, and Pop. Similar results were obtained by Efrat and further developed in his 2006 book.

Our project aims to provide an optimal proof of this characterisation, incorporating improvements and new developments. These include a revised proof strategy; Efrat's construction of valuations via multiplicative stratification; the Galois characterisation of henselianity; systematic use of the standard decomposition; and the function field analogy of Krasner-Kazhdan-Deligne type. Moreover, we replace arguments that use Galois cohomology with elementary ones.

In this talk, I will focus on two key components of the proof: the construction of valuations from rigid elements, and the role of the function field analogy as developed via the non-standard methods of Jahnke-Kartas.

This is joint work with Jochen Koenigsmann and Benedikt Stock.

Right-angled Artin groups (RAAGs) can be viewed as a generalisation of free groups. To what extent, then, do the techniques used to study automorphisms of free groups generalise to the setting of RAAGs? One significant advance in this direction is the construction of 'untwisted Outer space' for RAAGs, a generalisation of the influential Culler-Vogtmann Outer space for free groups. A consequence of this construction is an upper bound on the virtual cohomological dimension of the 'untwisted subgroup' of outer automorphisms of a RAAG. However, this bound is sometimes larger than one expects; I present work showing that, in fact, it can be arbitrarily so, by forming a new complex as a deformation retraction of the untwisted Outer space. In a different direction, another subgroup of interest is that consisting of symmetric automorphisms. Generalising work in the free groups setting from 1989, I present an Outer space for the symmetric automorphism group of a RAAG. A consequence of the proof is a strong finiteness property for many other subgroups of the outer automorphism group.

In stochastic optimal control and conditional generative modelling, a central computational task is to modify a reference diffusion process to maximise a given terminal-time reward. Most existing methods require this reward to be differentiable, using gradients to steer the diffusion towards favourable outcomes. However, in many practical settings, like diffusion bridges, the reward is singular, taking an infinite value if the target is hit and zero otherwise. We introduce a novel framework, based on Malliavin calculus and path-space integration by parts, that enables the development of methods robust to such singular rewards. This allows our approach to handle a broad range of applications, including classification, diffusion bridges, and conditioning without the need for artificial observational noise. We demonstrate that our approach offers stable and reliable training, outperforming existing techniques. 

Understanding the size of the partial sums of the Möbius function is one of the most fundamental problems in analytic number theory. This motivated the 1944 paper of Wintner, where he introduced the concept of a random multiplicative function: a probabilistic model for the Möbius function. In recent years, it has been uncovered that there is an intimate connection between random multiplicative functions and the theory of Gaussian Multiplicative Chaos, an area of probability theory introduced by Kahane in the 1980's. We will survey selected results and discuss recent research on the distribution of partial sums of random multiplicative functions when restricted to integers with a large prime factor.

Quotients of hyperbolic groups (groups that act geometrically on a hyperbolic space) and their generalizations have long been a powerful tool for proving strong algebraic results. In this talk, I will describe the geometry of random quotients of certain of groups, that is, a quotient by a subgroup normally generated by k independent random walks.  I will focus on the class of hierarchically hyperbolic groups (HHGs), a generalization of hyperbolic groups that includes hyperbolic groups, mapping class groups, most CAT(0) cubical groups including right-angled Artin and Coxeter groups, many 3–manifold groups, and various combinations of such groups.  In this context, I will explain why a random quotient of an HHG that does not split as a direct product is again an HHG, definitively showing that the class of HHGs is quite broad.  I will also describe how the result can also be applied to understand the geometry of random quotients of hyperbolic and relatively hyperbolic groups. This is joint work with Giorgio Mangioni, Thomas Ng, and Alexander Rasmussen.

In this talk, we discuss nearly G2 structures, which define positive Einstein metrics, and are, up to scale, critical points of a geometric flow called (modified) Laplacian co-flow. We will discuss a recent joint work with Jason Lotay showing that many of these nearly G2 critical points are unstable for the flow. 

Semidefinite programming (SDP) is a powerful tool in the design of approximation algorithms. After providing a gentle introduction to the basics of this method, I will explore a different facet of SDP and show how it can be used to derive short and elegant proofs of both classical and new estimates related to the MaxCut problem and discrepancy theory in graphs and matrices.

Building on this, I will demonstrate how these results lead to an improved upper bound on the celebrated log-rank conjecture in communication complexity.

I will describe a new method for constructing conformal blocks for the Virasoro vertex algebra with central charge c=1, by "nonabelianization", relating them to conformal blocks for the Heisenberg algebra on a branched double cover. The construction is joint work with Qianyu Hao. Special cases give rise to formulas for tau-functions and solutions of integrable systems of PDE, such as Painleve I and its higher analogues. The talk will be reasonably self-contained (in particular I will explain what a conformal block is).

We are going to study the Ginzburg-Landau energy for two specific geometries, related to the very experiments on fermionic condensates: annuli and strips 

The specific geometry of a strip provides connections between solitons and vortices, called solitonic vortices, which are vortices with a solitonic behaviour in the infinite direction of the strip. Therefore, they are very different from classical vortices which have an algebraic decay at infinity. We show that there exist stationary solutions to the Gross-Pitaevskii equation with k vortices on a transverse line, which bifurcate from the soliton solution as the width of the strip is increased. This is motivated by recent experiments on the instability of solitons by imposing a phase shift in an elongated condensate for bosonic or fermionic atoms.

For annuli, we prescribe a very large degree on the outer boundary and find that either there is a transition from a giant vortex to vortices also in the bulk but tending to the outer boundary.

This is joint work with Ph. Gravejat and E.Sandier for solitonice vortices and Remy Rodiac for annuli.
 

TBC

Temperley-Lieb algebras are certain finite-dimensional algebras coming originally from statistical physics and knot theory. Around 2019, they became one of the first examples of homological stability for algebras (homology is here taken to be certain Tor-groups), when Boyd and Hepworth showed that in low dimensions the homology vanishes. We're now able to give complete calculations of their homology, which has a surprisingly rich structure (and in particular is very far from vanishing). This is joint work in progress with Rachael Boyd, Oscar Randal-Williams, and Robin Sroka. Prerequisites will be minimal: it will be enough to know what Tor is.

Keywords: SDE on the plane, Brownian sheet, path by path uniqueness, space time local time integral, Malliavin calculus

In this talk, we discuss the existence, uniqueness, and regularisation by noise for stochastic differential equations (SDEs) on the plane. These equations can also be interpreted as quasi-linear hyperbolic stochastic partial differential equations (HSPDEs). More specifically, we address path-by-path uniqueness for multidimensional SDEs on the plane, under the assumption that the drift coefficient satisfies a spatial linear growth condition and is componentwise non-decreasing. In the case where the drift is only measurable and uniformly bounded, we show that the corresponding additive HSPDE on the plane admits a unique strong solution that is Malliavin differentiable. Our approach combines tools from Malliavin calculus with variational techniques originally introduced by Davie (2007), which we non-trivially extend to the setting of SDEs on the plane.

This talk is based on a joint works with A. M. Bogso, M. Dieye and F. Proske.

A famous theorem of Selberg asserts that $\log|\zeta(\tfrac12+it)|$ is approximately a normal distribution with mean $0$ and variance $\tfrac12\log\log T$, when we sample $t\in [T,2T]$ uniformly. This extends in a natural way to a plethora of other $L$-functions, one of them being Dirichlet $L$-functions $L(s,\chi)$ with $\chi$ a primitive Dirichlet character. Viewing $\zeta(\tfrac12+it)$ and $L(\tfrac12+it,\chi)$ as normal variables, we expect indepedence between them, meaning that for fixed $V_1,V_2 \in \mathbb{R}$: $$\textrm{meas}_{t \in [T,2T]} \left\{\frac{\log|\zeta(\tfrac12+it)|}{\sqrt{\tfrac12 \log\log T}}\geq V_1 \text{   and   } \frac{\log|L(\tfrac12+it,\chi)|}{\sqrt{\tfrac12 \log\log T}}\geq V_2\right\} \sim \prod_{j=1}^2 \int_{V_j}^\infty e^{-x^2/2} \frac{\textrm{d}x}{\sqrt{2\pi}}.$$
    When $V_j\asymp \sqrt{\log\log T}$, i.e. we are considering values of order of the variance, the asymptotic above breaks down, but the Gaussian behaviour is still believed to hold to order. For such $V_j$ the behaviour of the joint distribution is decided by the moments $$I_{k,\ell}(T)=\int_T^{2T} |\zeta(\tfrac12+it)|^{2k}|L(\tfrac12+it,\chi)|^{2\ell}\, dt.$$ We establish that $I_{k,\ell}(T)\asymp T(\log T)^{k^2+\ell^2}$ for $0<k,\ell \leq 1$. The lower bound holds for all $k,\ell >0$. This allows us to decide the order of the joint distribution when $V_j =\alpha_j\sqrt{\log\log T}$ for $\alpha_j \in (0,\sqrt{2}]$. Other corollaries include sharp moment bounds for Dedekind zeta functions of quadratic number fields, and Hurwitz zeta functions with rational parameter. 
    

The cover of a dataset is a fundamental concept in computational geometry and topology. In TDA (topological data analysis), it is especially used in computing persistent homology and data visualisation using Mapper. However only rudimentary methods have been used to compute a cover. In this talk, we formulate the cover computation problem as a general optimisation problem with a well-defined loss function, and use gradient descent to solve it. The resulting algorithm, ShapeDiscover, substantially improves quality of topological inference and data visualisation. We also show some preliminary applications in scRNA-seq transcriptomics and the topology of grid cells in the rats' brain. This is a joint work with Luis Scoccola and Heather Harrington.

Our visual perception of the world heavily relies on sophisticated and delicate biological mechanisms, and any disruption to these mechanisms negatively impacts our lives. Age-related macular degeneration (AMD) affects the central field of vision and has become increasingly common in our society, thereby generating a surge of academic and clinical interest. I will present some recent developments in the mathematical modeling of the retinal pigment epithelium (RPE) in the retina in the context of AMD; the RPE cell layer supports photoreceptor survival by providing nutrients and participating in the visual cycle and “cellular maintenance". Our objectives include modeling the aging and degeneration of the RPE with a mechanistic approach, as well as predicting the progression of atrophic lesions in the epithelial tissue. This is a joint work with the research team of Prof. M. Paques at Hôpital National des Quinze-Vingts.

In this talk, I will introduce the frameworks of globally valued fields (Ben Yaacov-Hrushovski) and adelic curves (Chen-Moriwaki). Both of these frameworks aim at understanding the arithmetic of fields sharing common features with global fields. A lot of examples fit in this scope (e.g. global fields, finitely generated extension of the prime fields, fields of meromorphic functions) and we will try to describe some of them.

Although globally valued fields and adelic curves came from different motivations and might seem quite different, they are related (and even essentially equivalent). This relation opens the door for new methods in the study of global arithmetic. As an application, we will sketch the proof of an arithmetic analogue of Siu inequality in algebraic geometry (a fundamental tool to detect the existence of global sections of line bundles in birational geometry). This is a joint work with Michał Szachniewicz.

In the 1980s, Mazur and Tate proposed refinements of the Birch–Swinnerton-Dyer conjecture that also capture congruences between twists of Hasse–Weil L-series by Dirichlet characters. In this talk, I will report on new results towards these refined conjectures, obtained in joint work with Matthew Honnor. I will also outline how the results fit into a more general approach to refined conjectures on special values of L-series via an enhanced theory of Euler systems. This final part will touch upon joint work with David Burns.

The currently available quantum computers fall into the NISQ (Noisy Intermediate Scale Quantum) regime. These enable variational algorithms with a relatively small number of free parameters. We are now entering the FTQC (Fault Tolerant Quantum Computer)  regime where gate fidelities are high enough that error-correction schemes are effective. The UK Quantum Missions include the target for a FTQC device that can perform a million operations by 2028, and a trillion operations by 2035.

This talk will present the outcomes from assessments of  two quantum linear equation solvers for FTQCs– the Harrow–Hassidim–Lloyd (HHL) and the Quantum Singular Value Transform (QSVT) algorithms. These have used sample matrices from a Computational Fluid Dynamics (CFD) testcase. The quantum solvers have also been embedded with an outer non-linear solver to judge their impact on convergence. The analysis uses circuit emulation and is used to judge the FTQC requirements to deliver quantum utility.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.

This talk introduces an ongoing research project focused on building mechanistic models to study retinal degeneration, with a particular emphasis on the geometric aspects of the disease progression.

As we develop a computational model for retinal degeneration, we will explore how cellular materials behave and how wound-healing mechanisms influence disease progression. Finally, we’ll detail the numerical methods used to simulate these processes and explain how we work with medical data.

Ongoing research in collaboration with the group of M. Paques (Paris Eye Imaging - Quinze Vingts National Ophthalmology Hospital and Vision Institute).

Spectro-microscopy is an experimental technique with great potential to science challenges such as the observation of changes over time in energy materials or environmental samples and investigations of the chemical state in biological samples. However, its application is often limited by factors like long acquisition times and radiation damage. We present two measurement strategies that significantly reduce experiment times and applied radiation doses. These strategies involve acquiring only a small subset of all possible measurements and then completing the full data matrix from the sampled measurements. The methods are data-driven, utilizing spectral and spatial importance subsampling distributions to select the most informative measurements. Specifically, we use data-driven leverage scores and adaptive randomized pivoting techniques. We explore raster importance sampling combined with the LoopASD completion algorithm, as well as CUR-based sampling where the CUR approximation also serves as the completion method. Additionally, we propose ideas to make the CUR-based approach adaptive. As a result, capturing as little as 4–6% of the measurements is sufficient to recover the same information as a conventional full scan.

Nowadays, the 3D ultrastructure of a fibrous tissue can be reconstructed in order to visualize the complex nanoscale arrangement of collagen fibrils including neighboring proteoglycans even in the stretched loaded state [1]. In particular, experimental data of collagen fibers in human artery layers have shown that the f ibers are not symmetrically dispersed [2]. In addition, it is known that collagen f ibers are cross-linked and the density of cross-links in arterial tissues has a stiffening effect on the associated mechanical response. A first attempt to characterize this effect on the elastic response is presented and the influence of the cross-link density on the mechanical behavior in uniaxial tension is shown [3]. A recently developed extension of the model that accounts for dispersed fibers connected by randomly distributed cross-links is outlined [4]. A simple shear test focusing on the sign of the normal stress perpendicular to the shear planes (Poynting effect) is analyzed. In [5] it was experimentally observed that, in contrast to rubber, semi-flexible biopolymer gels show a tendency to approach the top and bottom faces under simple shear. This so-called negative Poynting effect and its connection with the cross-links as well as the fiber and crosslink dispersion is also examined. 

References 

[1]A. Pukaluk et al.: An ultrastructural 3D reconstruction method for observing the arrangement of collagen fibrils and proteoglycans in the human aortic wall under mechanical load. Acta Biomaterialia, 141:300-314, 2022. 

 [2] G.A. Holzapfel et al.: Modelling non-symmetric collagen fibre dispersion in arterial walls. Journal of the Royal Society Interface, 12:20150188, 2015. 

 [3] G.A. Holzapfel and R.W. Ogden: An arterial constitutive model accounting for collagen content and cross-linking. Journal of the Mechanics and Physics of Solids, 136:103682, 2020. 

 [4] S. Teichtmeister and G.A. Holzapfel: A constitutive model for fibrous tissues with cross-linked collagen fibers including dispersion – with an analysis of the Poynting effect. Journal of the Mechanics and Physics of Solids, 164:104911, 2022. 

[5] P.A. Janmey et al.: Negative normal stress in semiflexible biopolymer gels. Nature Materials, 6:48–51, 2007.

In this talk, we consider the coefficients of the Fourier series obtained by exponentiating a logarithmically correlated holomorphic function on the open unit disc, whose Taylor coefficients are independent complex Gaussian variables, the variance of the coefficient of degree k being theta/k where theta > 0 is an inverse temperature parameter. In joint articles with Paquette, Simm and Vu, we show a randomized version of the central limit theorem in the subcritical phase theta < 1, the random variance being related to the Gaussian multiplicative chaos on the unit circle. We also deduce, from results on the holomorphic multiplicative chaos, other results on the coefficients of the characteristic polynomial of the Circular Beta Ensemble, where the parameter beta is equal to 2/theta. In particular, we show that the central coefficient of the characteristic polynomial of the Circular Unitary Ensembles tends to zero in probability, answering a question asked in an article by Diaconis and Gamburd.

A landmark result in the study of locally compact, abelian groups is the Pontryagin duality. In simple terms, it says that for a given locally compact, abelian group G, one can uniquely associate another locally compact, abelian group called the Pontryagin dual of G. In the realm of C*-algebras, whenever such an abelian group G acts on a C*-algebra A, there is a canonical action of the dual group of G on the crossed product of A by G. In particular, it is natural to ask to what extent one can relate properties of the given G-action to those of the dual action. 

In this talk, I will first introduce a property for actions of locally compact abelian groups called the abelian Rokhlin property and then state a duality type result for this property. While the abelian Rokhlin property is in general weaker than the known Rokhlin property, these two properties coincide in the case of the acting group being the real numbers. Using the duality result mentioned above, I will give new examples of continuous actions of the real numbers which satisfy the Rokhlin property. Part of this talk is based on joint work with Johannes Christensen and Gábor Szabó.

In this talk I will describe a possible strategy to obtain new solution to LMCF in the Kummer K3 surface by a fixed point argument. The key idea is that the regions where curvature concentrates in the Kummer K3 surface are modeled on the Eguchi-Hanson space. 

Property (T) (respectively aTmenability) is equivalent to admitting only a trivial action (respectively, a proper action) on a median space, and is also equivalent to admitting only a trivial action (respectively, a proper action) on a Hilbert space (so some L2). For p>2 I will investigate an analogous equivalent characterisation.

In this talk I will discuss a new lower bound for the off-diagonal Ramsey numbers $R(3,k)$. For this, we develop a version of the triangle-free process that is significantly easier to analyse than the original process. We then 'seed' this process with a carefully chosen graph and show that it results in a denser graph that is still sufficiently pseudo-random to have small independence number.

This is joint work with Marcelo Campos, Matthew Jenssen and Marcus Michelen.

Nichols algebras, also known as small shuffle algebras, are a family of graded bialgebras including the symmetric algebras, the exterior algebras, the positive parts of quantized enveloping algebras, and, conjecturally, Fomin-Kirillov algebras. As the case of Fomin-Kirillov algebra shows, it can be very
difficult to determine the maximum degree of a minimal generating set of relations of a Nichols algebra. 

Building upon Kapranov and Schechtman’s equivalence between the category of perverse sheaves on Sym(C) and the category of graded connected bialgebras,  we describe the geometric counterpart of the maximum degree of a generating set of relations of a graded connected bialgebra, and we show how this specialises to the case o Nichols algebras.

The talk is based on joint work with Francesco Esposito and Lleonard Rubio y Degrassi.
 

We identify a three-dimensional system that exhibits long-range entanglement at sufficiently small but nonzero temperature--it therefore constitutes a quantum topological order at finite temperature. The model of interest is known as the fermionic toric code, a variant of the usual 3D toric code, which admits emergent fermionic point-like excitations. The fermionic toric code, importantly, possesses an anomalous 2-form symmetry, associated with the space-like Wilson loops of the fermionic excitations. We argue that it is this symmetry that imbues low-temperature thermal states with a novel topological order and long-range entanglement. Based on the current classification of three-dimensional topological orders, we expect that the low-temperature thermal states of the fermionic toric code belong to an equilibrium phase of matter that only exists at nonzero temperatures. We conjecture that further examples of topological orders at nonzero temperatures are given by discrete gauge theories with anomalous 2-form symmetries. Our work therefore opens the door to studying quantum topological order at nonzero temperature in physically realistic dimensions.

Agents in social networks with threshold-based dynamics change opinions when influenced by sufficiently many peers. Existing literature typically assumes that the network structure and dynamics are fully known, which is often unrealistic. In this work, we ask how to learn a network structure from samples of the agents' synchronous opinion updates. Firstly, if the opinion dynamics follow a threshold rule where a fixed number of influencers prevent opinion change (e.g., unanimity and quasi-unanimity), we give an efficient PAC learning algorithm provided that the number of influencers per agent is bounded. Secondly, under standard computational complexity assumptions, we prove that if the opinion of agents follows the majority of their influencers, then there is no efficient PAC learning algorithm. We propose a polynomial-time heuristic that successfully learns consistent networks in over 97% of our simulations on random graphs, with no failures for some specified conditions on the numbers of agents and opinion diffusion examples.

A subset $A$ of $A_n$ is $k$-product-free if for all $a_1,a_2,\dots,a_k\in A$, $a_1a_2\dots a_k$ $\notin A$.
We determine the largest $3$-product-free and $4$-product-free subsets of $A_n$ for sufficiently large $n$. We also obtain strong stability results and results on multiple sets with forbidden cross products. The principal technical ingredient in our approach is the theory of hypercontractivity in $S_n$. Joint work with Peter Keevash.

Given a space with some kind of geometry, one can ask how the geometry of the space relates to its homology. This talk will survey some comparisons of geometric notions of complexity with homological notions of complexity. We will then focus on hyperbolic 3-manifolds and the main result will replace a spectral gap problem related to torsion in homology with a geometric version involving geodesic length and stable commutator length. As an application, we provide "bad" examples of hyperbolic 3-manifolds with bounded geometry but extremely small (1-form) spectral gaps.

Scaling arguments give a natural guess at the regularity condition on the noise in a stochastic PDE for a local solution theory to be possible, using the machinery of regularity structures or paracontrolled distributions. This guess of ``subcriticality'' is often, but not always, correct. In cases when it is not, a the blowup of the variance of certain nonlinear functionals of the noise necessitates a different, multiplicative renormalisation. This led to a general prediction and the first results in the case of the KPZ equation in [Hairer '24]. We discuss recent developments towards confirming this prediction. Based on joint works with Fabio Toninelli and Yueh-Sheng Hsu.

I will present some ongoing work on solving parametric linear systems arising from the application of the finite elements method on elliptic partial differential trial equations. The focus of the talk will be on leveraging randomised numerical linear algebra to solve these equations in high-dimensional parameter spaces with special emphasis on the multi-query context where optimal sampling is not practical. In this context I will discuss some ideas on choosing a suitable low-dimensional approximation of the solution, as well as reducing the variance of the sketched systems. This research aims at exploring the potential of randomisation as a probabilistic framework for model order reduction, with potential applications to online simulations, uncertainty quantification and inverse problems, via the research grant EPSRC EP/V028618/1

Bio: Nick Polydorides is a professor in computational engineering at the University of Edinburgh and has interests in randomised numerical linear algebra, inverse problems and edge computing. Previously, he was a faculty at the Cyprus Institute, and a postdoctoral fellow at MIT’s lab for Information and Decision Systems. He has a PhD in Electrical Engineering from the University of Manchester.