Fridays at 12.15 at Wolfson 4W 1.7. All talks will be broadcast on Zoom.
Everyone is welcome at these talks.
|9 Feb 2024
|Behnam Hashemi (Leicester)
RTSMS: Randomized Tucker with Single-Mode Sketching
In this talk we discuss a new algorithm called RTSMS which is designed for approximating low-rank Tucker decompositions of tensors. Our approach uses sketching and a least-squares framework to compute the Tucker decomposition. The single-mode sketching aspect of RTSMS allows utilizing simple sketch matrices which are substantially smaller than alternative methods leading to considerable performance gains. Within its least-squares framework, RTSMS incorporates leverage scores for efficiency with Tikhonov regularization and iterative refinement for stability. The algorithm is demonstrated to be competitive with existing methods, sometimes outperforming them by a large margin. To set the stage for our discussion, we will first provide a brief overview of the preliminaries in tensor decompositions, establishing foundational concepts essential for understanding RTSMS.
|16 Feb 2024
|Andraž Jelinčič (Bath)
SDEs and Brownian Trees: Now also on your GPU
In this talk, we will give a quick overview of the new high order solvers for Stochastic Differential Equations (SDEs) in the Diffrax library. Diffrax is a toolbox for solving and analysing a wide variety of time-evolving differential equations, using sophisticated GPU-acceleration capabilities based on JAX. In particular, ODE/SDE solvers in Diffrax are autodifferentiable and can thus be calibrated to data via gradient-based optimization. To facilitate the use of SDE solvers with adaptive step sizes, Diffrax generates sample paths of Brownian motion using a ‘Virtual Brownian Tree’ (VBT) data structure - which can be queried non-chronologically. We will discuss recent work extending Diffrax’s VBT to generate the additional time integrals of Brownian motion needed for SDE solvers to achieve higher convergence rates. In our numerical experiment, we will show that high order adaptive SDE solvers can achieve state-of-the-art convergence for simulating (underdamped) Langevin dynamics. Whilst this SDE originates as a model in molecular dynamics, it has recently seen applications as an MCMC algorithm for Bayesian inference. Finally, to conclude, we will discuss applications of adaptive solvers in Diffrax to more general classes of SDEs.
|16 Feb 2024
|Guannan Chen (Bath)
Quantum simulation of highly-oscillatory many-body Hamiltonians for near-term devices
We develop a fourth-order Magnus expansion based quantum algorithm for the simulation of many-body problems involving two-level quantum systems with time-dependent Hamiltonians, H(t). A major hurdle in the utilization of the Magnus expansion is the appearance of a commutator term which leads to prohibitively long circuits. We present a technique for eliminating this commutator and find that a single time-step of the resulting algorithm is only marginally costlier than that required for time-stepping with a time-independent Hamiltonian, requiring only three additional single-qubit layers. For a large class of Hamiltonians appearing in liquid-state nuclear magnetic resonance (NMR) applications, we further exploit symmetries of the Hamiltonian and achieve a surprising reduction in the expansion, whereby a single time-step of our fourth-order method has a circuit structure and cost that is identical to that required for a fourth-order Trotterized time-stepping procedure for time-independent Hamiltonians. Moreover, our algorithms are able to take time-steps that are larger than the wavelength of oscillation of the time-dependent Hamiltonian, making them particularly suited for highly-oscillatory controls. The resulting quantum circuits have shorter depths for all levels of accuracy when compared to first and second-order Trotterized methods, as well as other fourth-order Trotterized methods, making the proposed algorithm a suitable candidate for simulation of time-dependent Hamiltonians on near-term quantum devices.
|23 Feb 2024
|Mohammad Sadegh Salehi (Bath)
An adaptively inexact first-order method for bilevel learning
In various domains within imaging and data science, tasks are modelled using the variational regularisation approach. In this framework, manually selecting regularisation parameters poses a cumbersome challenge, especially when employing regularisers involving a large number of hyperparameters. To address this issue, machine learning offers a solution by learning parameters from data. However, the unattainability of exact function values and gradients with respect to hyperparameters requires reliance on inexact evaluations. State-of-the-art approaches often face difficulties in selecting accuracy sequences and determining an appropriate step size due to the unknown Lipschitz constant of the hypergradient. In this talk, we focus on overcoming these challenges through inexact gradient-based methods. We present our algorithm, the ‘Method of Adaptive Inexact Descent (MAID),’ which provides a provably convergent backtracking line search, incorporating inexact function evaluations and hypergradients. This ensures convergence to a stationary point and adaptively determines the required accuracy. We conduct numerical comparisons to assess the effectiveness of our method against various state-of-the-art methods on an image denoising problem. Furthermore, we showcase our method’s robustness across different initial accuracy and step size choices, emphasizing its practical applicability.
|23 Feb 2024
|Ben Ashby (Bath)
Structure preserving numerical methods for non-Newtonian fluid flows
In this talk, we will discuss some key structures in non-Newtonian flows and how to design numerical methods that preserve them. The Oldroyd-B model for non-Newtonian fluid flows provides a constitutive relation for the elastic stress. It is typically formulated in terms of the conformation tensor, a quantity motivated by kinematics, and which can be proven to remain positive definite at the PDE level. This property is not guaranteed to be inherited by numerical schemes, and in cases where positivity is lost, numerical solutions can become unstable. We present a discretisation that preserves positive definiteness of the conformation tensor to increase stability of computations. The method consists of a finite element method for the fluid flow coupled to a finite difference method for a Lie derivative formulation of the constitutive law. In this framework, the advection and deformation terms of the upper convected derivative are discretised along fluid particle trajectories in a simple, cheap and cohesive manner, ensuring that the discrete conformation tensor is positive definite. We demonstrate the performance of this method with detailed numerical experiments in a lid-driven cavity setup. Numerical results are benchmarked against published data, and the method is shown to perform well in this challenging case.
|1 Mar 2024
|Emmanuil Georgoulis (Heriot-Watt and National Technical University of Athens)
Hypocoercivity-preserving Galerkin discretisations
Degenerate differential evolution PDE problems are often characterised by the explicit presence of diffusion/dissipation in some of the spatial directions only, yet may still admit decay properties to some long time equilibrium. Classical examples include the inhomogeneous Fokker-Planck equation, Boltzmann equation with various collision kernels, systems of equation arising in micromagnetism or flow vorticity modelling, etc. In the celebrated AMS memoir “Hypocoercivity”, Villani introduced the concept of hypocoercivity to describe a framework able to explain decay to equilibrium in the presence of dissipation in some directions only. The key technical idea involved is to exploit certain commutators to overcome the degeneracy of dissipation. I shall present some results and ideas on the development of numerical methods which preserve the hypocoercivity property upon discretisation. As a result, such numerical methods will be suitable for arbitrarily long-time simulations of complex phenomena modelled by kinetic-type formulations. This will be achieved by addressing the key challenge of lack of commutativity between differentiation and discretisation in the context of mesh-based Galerkin-type numerical methods via the use of carefully constructed non-conforming weak formulations of the underlying evolution problems. Some of the results I plan to present are based on joint work with Zhaonan Dong (INRIA-Paris) and Philip Herbert (Sussex).
|8 Mar 2024
|Boris Shustin (Oxford)
Tractable Riemannian Optimization via Randomized Preconditioning and Manifold Learning
Optimization problems constrained on manifolds are prevalent across science and engineering. For example, they arise in (generalized) eigenvalue problems, principal component analysis, and low-rank matrix completion, to name a few problems. Riemannian optimization is a principled framework for solving optimization problems where the desired optimum is constrained to a (Riemannian) manifold. Algorithms designed in this framework usually require some geometrical description of the manifold, i.e., tangent spaces, retractions, Riemannian gradients, and Riemannian Hessians of the cost function. However, in some cases, some of the aforementioned geometric components cannot be accessed due to intractability or lack of information. In this talk, we present methods that allow for overcoming cases of intractability and lack of information. We demonstrate the case of intractability on canonical correlation analysis (CCA) and on Fisher linear discriminant analysis (FDA). Using Riemannian optimization to solve CCA or FDA with the standard geometric components is as expensive as solving them via a direct solver. We address this shortcoming using a technique called Riemannian preconditioning, which amounts to changing the Riemannian metric on the constraining manifold. We use randomized numerical linear algebra to form efficient preconditioners that balance the computational costs of the geometric components and the asymptotic convergence of the iterative methods. If time permits, we also show the case of lack of information, e.g., the constraining manifold can be accessed only via samples of it. We propose a novel approach that allows approximate Riemannian optimization using a manifold learning technique.
|8 Mar 2024
|Paz Fink Shustin (Oxford)
|15 Mar 2024
|Yury Korolev (Bath)
|15 Mar 2024
|Alex Trenam (Bath)
|22 Mar 2024
|Yue Wu (Strathclyde)
|19 Apr 2024
|Nicolas Boullé (Cambridge)
|26 Apr 2024
|Wei Pan (Imperial College London)
|3 May 2024
|Carlos Jerez Hanckes (Universidad Adolfo Ibáñez)
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Tips for giving talks
Tips for new students on giving talks
Since the audience of the NA seminar contains both PhD students and staff with quite wide interests and backgrounds, the following are some guidelines/hints to make sure people don't give you evil looks at lunch afterwards.
Before too much time passes in your talk, ideally the audience should know the answers to the following 4 questions:
- What is the problem you're considering?
- Why do you find this interesting?
- What has been done before on this problem/what's the background?
- What is your approach/what are you going to talk about?
There are lots of different ways to communicate this information. One way, if you're doing a slide show, could be for the first 4 slides to cover these 4 questions; although in this case you may want to revisit these points later on in the talk (e.g. to give more detail).
- "vertebrate style" (structure hidden inside - like the skeleton of a vertebrate) = good for detective stories, bad for maths talks.
- "crustacean style" (structure visible from outside - like the skeleton of a crustacean) = bad for detective stories, good for maths talks.