# Bath Numerical Analysis Seminar

Fridays at **12.15** at Wolfson 4W 1.7. All talks will be broadcast on Zoom.

** Everyone is welcome at these talks. **

Date | Speaker | Title |

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 |

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) |
## Scalable Gaussian Process Regression with Quadrature-based Features
Gaussian processes provide a powerful probabilistic kernel learning framework, which allows high-quality nonparametric learning via methods such as Gaussian process regression. Nevertheless, its learning phase requires unrealistic massive computations for large datasets. In this talk, we present a quadrature-based approach for scaling up Gaussian process regression via a low-rank approximation of the kernel matrix. The low-rank structure is utilized to achieve effective hyperparameter learning, training, and prediction. Our Gauss-Legendre features method is inspired by the well-known random Fourier features approach, which also builds low-rank approximations via numerical integration. However, our method is capable of generating high-quality kernel approximation using a number of features that is poly-logarithmic in the number of training points, while similar guarantees will require an amount that is at the very least linear in the number of training points when using random Fourier features. The utility of our method for learning with low-dimensional datasets is demonstrated using numerical experiments. |

15 Mar 2024 | Alex Trenam (Bath) |
## Structure-preserving finite element methods for electrolyte models
Structure-preserving numerical methods retain at the discrete level some properties of interest possessed by the continuum model (e.g. conservation of mass, positivity of concentrations, energy dissipation laws). In this talk I will discuss structure preservation in the context of the Poisson-Nernst-Planck system of coupled, non-linear PDEs. This evolution model describes the interaction between concentrations of different charged particles in the presence of a global electric field, which also evolves with the concentrations. In particular, I will consider the use of discontinuous Galerkin finite element methods, with a view towards stable, structure-preserving discretisations of the extended Navier-Stokes-Poisson-Nernst-Planck model. |

15 Mar 2024 | Yury Korolev (Bath) |
## Infinite infimal convolution regularisation
Infimal convolutions of two or more convex regularisers have proven very useful for decomposing an image into a sum of two or more components, each of which has low energy with respect to one of the regularisers. Such components can, e.g., have different smoothness or different dominant directions of anisotropy. We extend this idea further to infimal convolutions of an infinite (continuum) family of functionals. For variational problems regularised by an infinite infimal convolution functional we propose a lifting to the space of vector-valued measures, where we prove a representer theorem (existence of sparse solutions in the case of finite-dimensional measurements) and derive a generalised conditional gradient method. This allows us to select adaptively the parameters of the infinite infimal convolution that best suit a given image. We apply this paradigm for finding dominant directions of oscillation in an image using an infinite infimal convolution of anisotropic fractional Laplacians. Joint work with Kristian Bredies, Marcello Carioni, Martin Holler, and Carola Schönlieb. |

22 Mar 2024 | Yue Wu (Strathclyde) |
## Randomised Numerical Schemes
In this talk, I will introduce randomised numerical schemes for a wide range of differential equations with time-irregular coefficients. The initial scheme considered is a randomised Runge-Kunta for Caratheodory ODEs. Our motivation for studying Caratheodory type initial value problems stems from the fact that certain rough differential equations that are driven by an additive noise can be transformed into a problem of this form. it is well-known that there is lack of convergence for deterministic algorithms in this case, and our proposed method, by incorporating stratified Monte-Carlo simulation into the corresponding deterministic ones, showed that even with very mild conditions, the order of convergence can be at least half with respect to the Lp norm. The idea was later extended to stochastic settings. We proposed a drift-randomized Milstein method to achieve a higher order approximation to non-autonomous SDEs where the standard smoothness and growth requirements of standard Milstein-type methods are not fulfilled. We also investigated the numerical solution of non-autonomous semilinear stochastic evolution equations (SEEs) driven by an additive Wiener noise. Usually quite restrictive smoothness requirements are imposed in order to achieve a high order of convergence rate. To relax such conditions, we proposed a novel numerical method for the approximation of the solution to the semilinear SEE that combines the drift-randomization technique with a Galerkin finite element method. It turns out that the resulting method converges with a higher rate with respect to the temporal discretization parameter without requiring any differentiability of the nonlinearity. Our approach also relaxes the smoothness requirements of the coefficients with respect to the time variable considerably. Recently, we pushed this idea to Caratheodory delay ODEs (CDDEs), where a randomised Euler scheme is proposed to approximate the exact solution. |

19 Apr 2024 | Nicolas Boullé (Cambridge) |
## Elliptic PDE learning is provably data-efficient
PDE learning is an emerging field at the intersection of machine learning, physics, and mathematics, that aims to discover properties of unknown physical systems from experimental data. Popular techniques exploit the approximation power of deep learning to learn solution operators, which map source terms to solutions of the underlying PDE. Solution operators can then produce surrogate data for data-intensive machine learning approaches such as learning reduced order models for design optimization in engineering and PDE recovery. In most deep learning applications, a large amount of training data is needed, which is often unrealistic in engineering and biology. However, PDE learning is shockingly data-efficient in practice. We provide a theoretical explanation for this behaviour by constructing an algorithm that recovers solution operators associated with elliptic PDEs and achieves an exponential convergence rate with respect to the size of the training dataset. The proof technique combines prior knowledge of PDE theory and randomized numerical linear algebra techniques and may lead to practical benefits such as improving dataset and neural network architecture designs. |

26 Apr 2024 | Wei Pan (Imperial College London) |
## Numerics and applications of Stochastic Advection by Lie Transport
In the prediction of geophysical fluid dynamics (GFD) such as ocean and weather phenomena, two main types of uncertainty arise. The first concerns the transport uncertainty, which dictates the path of fluid flow. The second pertains to the uncertainty in the thermodynamic behaviour. “Stochastic advection by Lie transport” (SALT) addresses the first type of uncertainty. SALT serves as a stochastic parameterisation for fluid transport, yielding stochastic partial differential equations (SPDE) models that preserve circulation. In this presentation, first we provide an overview of the application of SALT-type SPDEs in high dimensional data assimilation. Then we introduce the recently developed SALT thermal-quasigeostrophic model. We discuss the consistency of our implementation and demonstrate how alpha regularisation can be employed to control inherent system instabilities. |

3 May 2024 | Carlos Jerez Hanckes (Universidad Adolfo Ibáñez) |
## Multiple Traces Formulation: a marriage between Boundary Integral and Domain Decomposition Methods
For over a decade boundary integral formulations dubbed Multiple Traces Formulations (MTFs) were developed aiming originally to tackle wave scattering by heterogenous objects. These formulations regard solution traces over each subdomain as independent unknowns and impose adequate transmission conditions weakly. Such an approach is reminiscent to the more established Domain Decomposition methods (DDMs) but portraying several novelties. Specifically, under general conditions, MTFs can be shown to be well posed, robust, parallelizable and either well conditioned or easy to precondition with Krylov methods converging quickly. In this talk, we will explain the construction of MTFs, their intrinsic properties and present an overview of its numerical implementations and applications in electromagnetics, acoustics, bioengineering as well as their limitations and recent advances. |

## Subscribe to seminar calendar

You can subscribe to the NA calendar directly from your calendar client, including Outlook, Apple’s iCalendar or Google calendar. The web address of the calendar is this ICS link which you will need to copy.

To subscribe to a calendar in Outlook:

- In Calendar view, select “Add Calendar” (large green +)
- Select “From Internet”
- Copy paste the ICS link, click OK, and click Yes to subscribe.

To subscribe to a calendar in iCalendar, please follow these instructions. Copy paste the ICS link in “web address”.

To subscribe to a calendar in Google Calendar:

- Go to link.
- On the left side go to "Other Calendars" and click on the dropdown.
- Choose "Add by URL".
- Copy paste the ICS link in the URL of the calendar.
- Click on "Add Calendar" and wait for Google to import your events. This creates a calendar with a somewhat unreadable name.
- To give a readable name to the calendar, click on the three vertical dots sign next to the newly created calendar and select Settings.
- Choose a name for the calendar, eg. Numerical Analysis @ Bath, and click back button on top left.

## How to get to Bath

See here for instructions how to get to Bath. Please email James Foster (jmf68@bath.ac.uk) and Aaron Pim (arp46@bath.ac.uk) if you intend to come by car and require a parking permit for Bath University Campus for the day.## 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).

Remember:

- "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.