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 |
| 7 Feb 2025 | Andreas Kyprianou (University of Warwick) |
Proton beam de-energisation and the Bragg peak for cancer therapy via jump diffusion stochastic differential equations
Proton beam therapy is a relatively modern way of treating cancer. In short, protons are accelerated to around 2/3 the speed of light and projected into the body in the direction of cancerous tissues. Subatomic interactions slow the protons down causing energy deposition into tissue. The less energy protons have, the greater the number of subatomic interactions and the greater the rate at which energy is deposed. This results in the proton beam having an approximate “end point” where the majority of the initial energy in the beam is deposited. Rather obviously, this needs to be positioned into cancerous tissues and accuracy is essential. In this talk we discuss a mathematical model of the proton beam, grounded in the subatomic physics, taking the form of a (7+1)-dimensional SDE which has both diffusive and jump components. A fundamental modelling question of this class of SDEs pertains to constructing a well-defined meaning of “rate of energy deposition” in the three dimensions of space and how relates to Monte Carlo simulation. Ultimately this boils down to the existence of an occupation density for the SDE, which, in turn requires us to work with fundamental ideas from Malliavin calculus. This work is part of a larger body of research that is currently being undertaken in collaboration with colleagues in the MaThRad programme grant. |
| 14 Feb 2025 | Colin Cotter (Imperial College London) |
Parallel-in-time methods for atmosphere simulation using time diagonalisation
The goal of parallel-in-time methods is to employ parallelism in the time direction in addition to the space direction, in the hope of obtaining further parallel speedups at the limits of what is possible due to spatial parallelism with domain decomposition alone. Recently diagonalisation techniques have emerged as a way of solving the coupled system for the solution of a differential equation at several timesteps simultaneously. One approach, sometimes referred to as “ParaDiag II” involves preconditioning this “all-at-once” system obtained from time discretisation of a linear constant coefficient ODE (perhaps obtained as the space discretisation of a time dependent PDE) with a nearby system that can be diagonalised in time, allowing the solution of independent blocks in parallel. For nonlinear PDEs this approach can form the basis of a preconditioner within a Newton-Krylov method for the all-at-once system after time averaging the (now generally time dependent) Jacobian system. After some preliminary description of the ParaDiag II approach, I will present results from our investigation of ParaDiag II applied to some testcases from the hierarchy of models used in the development of dry dynamical cores for atmosphere models, including performance benchmarks. Using these results I will identify the key challenges in obtaining further speedups and identify some directions to address these. |
| 21 Feb 2025 | Eike Mueller (University of Bath) |
High-order IMEX-HDG timestepping methods for the incompressible Euler equations
Discretisation methods that are of high order in space and time are highly desirable when solving the equations of fluid dynamics since they show superlinear convergence and make efficient use of modern computer hardware. We develop an efficient timestepping method for the incompressible Euler equations based on combining implicit-explicit (IMEX) time-integrators, high-order Hybridised Discontinuous Galerkin (HDG) discretisations and splitting methods. The two computational bottlenecks are the calculation of a tentative velocity and the solution of a velocity-pressure system to enforce incompressibility. The second solve is preconditioned with a non-nested hybridised multigrid preconditioner as proposed by Cockburn, Dubois, Gopalakrishnan and Tan for the Schur-complement system that arises from static condensation to eliminate the pressure/velocity unknowns in favour of variables on the mesh-facets. The tentative velocity solve is preconditioned with ILU0. With this choice of preconditioners, the number of solver iterations depends only weakly on the polynomial degree and the grid-spacing; in other words, empirically the method is approximately h- and p-robust. As a consequence, the cost of a single timestep is roughly proportional to the total number of unknowns. At high order this allows the computation of highly accurate solutions at a low computational cost. The code has been implemented in Firedrake, using static condensation technology and PETSc to construct a fairly complex solver/preconditioner. Numerical results are presented for the simulation of a two-dimensional Taylor-Green vertex. |
| 21 Feb 2025 | Sam McCallum (University of Bath) |
Efficient gradients for Neural ODEs
Training Neural ODEs requires backpropagating through an ODE solve. The state-of-the-art backpropagation algorithm is recursive checkpointing that balances recomputation with memory cost. In this talk, I will introduce a class of algebraically reversible ODE solvers that significantly improve upon both the time and memory cost of recursive checkpointing. |
| 28 Feb 2025 | Henry Lockyer (University of Bath) |
Learning splitting and composition methods
Splitting and composition methods are ubiquitous in the solution of ordinary and partial differential equations since it is often easier to solve the differential equation under separate flows. This may be due to the availability of exact solutions under sub-flows or due to their efficient approximations that exploit, for instance, the favourable structure, sparsity or smaller spectral radius of the resulting numerical linear algebra problems. Traditionally, splitting and composition methods have been derived using analytic and algebraic techniques. This normally means truncated Taylor series or its Lie algebraic analogue – the Baker-Campbell-Hausdorff formula. Very high order splitting and composition methods have been derived in this way, which achieve high accuracy for small time-steps, and have exact or near conservation of physical properties such as mass, unitarity, and energy. However, in many applications of practical interest a moderate accuracy is acceptable while the limited computational resources necessitate the use of larger time-steps. For these large time-steps, the asymptotically derived high-order splittings typically fail to achieve adequate accuracy. In contrast, machine learning approaches for solving differential equation are often efficient, but come with no convergence guarantees, and often do not conserve physical properties such as energy. In this work we outline a hybrid approach for designing machine learned splitting and composition methods that work effectively for large time steps, and have provably convergence and conservation guarantees. |
| 28 Feb 2025 | Sergey Dolgov (University of Bath) |
Smoothed risk-averse optimisation
We develop an algorithm to solve high-dimensional risk-averse optimization problems governed by differential equations (ODEs and/or PDEs) under uncertainty. As an example, we focus on the so-called Conditional Value at Risk (CVaR), but the approach is equally applicable to other coherent risk measures. To avoid non-smoothness of the objective function underpinning the CVaR, we propose an adaptive strategy to select the width parameter of the smoothed CVaR to balance the smoothing and approximation errors. This enables both Newton’s method for asymptotically quadratic convergence in the number of iterations, and higher-order (such as spectral tensor) approximation of the high-dimensional random fields for exponential convergence in the number of degrees of freedom. |
| 7 Mar 2025 | Eric Hester (University of Bath) |
Towards optimal complexity solvers for multiphase systems: A Cahn-Hilliard case study
In this talk I’ll tie together various themes of my research with a new take on a classic problem: how to efficiently solve the Cahn-Hilliard equation. It’s a perfect archetype for the complexities of multiphase problems: |
| 7 Mar 2025 | Michael Murray (University of Bath) |
How does the sample-to-feature ratio affect overfitting outcomes: benign, tempered, or harmful?
Traditional machine learning wisdom suggests that training a highly flexible model on noisy data without explicit regularization will likely result in harmful overfitting. However, experimentally it has been observed that neural networks trained in this context can generalize well despite achieving near perfect training loss. This unexpected behaviour is referred to as benign or tempered overfitting, depending on whether the overfitting causes a marginal or proportional (to the noise level) drop in test accuracy. In this talk I’ll provide a high-level overview of this topic before presenting a couple of recent results pertaining to the role of the sample-to-feature ratio in driving different training outcomes. |
| 14 Mar 2025 | Sam Flynn (National Physical Laboratory) |
Advanced Numerical Techniques in Primary Standard Calorimetry at the UK's National Standards Institute
Achieving optimal patient outcomes in radiation therapy depends on precise radiation dose measurements. The most accurate form of radiation detector is the calorimeter, which in the UK serve as the primary standard for calibrating clinical dosimetry systems. However, accurate dose determination requires a series of correction factors and conversions, derived from a variety of numerical techniques. This seminar will explore the computational methods underpinning primary standard calorimetry, including thermal modelling and Monte Carlo simulations, to reduce the uncertainty in radiation dose measurements and maximise patient quality of life. |
| 21 Mar 2025 | Jemma Shipton (University of Exeter) |
Time-parallel spectral deferred correction schemes for numerical weather prediction
In numerical weather prediction and climate modelling, semi-implicit time-stepping methods have long been favoured for their ability to take large time steps without excessively damping slow-moving waves. Fast-Wave Slow-Wave Spectral Deferred Correction (FWSW-SDC) methods offer an attractive alternative, achieving arbitrary-order accuracy by iteratively solving a collocation problem, akin to implicit Runge-Kutta methods. Similar to semi-implicit methods, FWSW-SDC improves stability compared to fully explicit schemes, enabling large time steps while retaining accuracy. Additionally, the rich literature on parallel-in-time SDC methods presents opportunities for both parallelization within the correction process and across time steps. |
| 28 Mar 2025 | Poppy Nikou (UCL Hospitals NHS Foundation Trust) |
Modelling anatomical changes of head and neck cancer patients during radiotherapy treatment
Progressive anatomical changes are often observed in head and neck cancer patients over the course of radiotherapy. To create proton therapy treatment plans that are robust against these changes, these potential anatomical changes must be known in advance. In this work we present a novel methodology for developing statistical population models that capture progressive anatomical changes and their variability. The models are continuous in time, and can be built from a population of patients with a varying number and frequency of images over treatment. The statistical population models are based on principal component analysis of the progressive anatomical changes, and can generate scenarios of many potential anatomical changes. In this work two principal components were used to generate nine scenarios of potential anatomical change. The evaluation found, on average, one scenario that matched their progressive anatomical changes better than by just considering the average change. |
| 28 Mar 2025 | Pablo Arratia López (University of Bath) |
Dynamic CT image reconstruction from highly undersampled measurements using neural fields
Dynamic inverse problems present unique challenges: as the quantity to be discovered evolves over time, the motion of the object during acquisition of measurements together with hardware limitations results in highly undersampled measurements in space. Not accounting for the dynamics of the process leads to a poor reconstruction with non-realistic motion and lack of temporal coherence. Traditional variational approaches, which employ grid-based discretisations and penalize time evolution, have been proposed to address these issues. |
| 4 Apr 2025 | Cristopher Salvi (Imperial College London) |
Quantum Signature Kernels
I will formulate randomised path developments within the framework of matrix models in quantum field theory. I will then explain how to derive the corresponding loop equations, and examine the associated kernels. If time allows, I will interpret these randomised path developments as time evolution operators on a Hilbert space of qubits, leading to the construction of a quantum algorithm for computing the kernels. Joint work with Samuel Crew and Will Turner. |
| 4 Apr 2025 | Maud Lemercier (University of Oxford) |
A High Order Solver for Signature Kernels
Signature kernels are at the core of several machine learning algorithms for analysing multivariate time series. The kernels of bounded variation paths, such as piecewise linear interpolations of time series data, are typically computed by solving a linear hyperbolic second-order PDE. However, this approach becomes considerably less practical for highly oscillatory inputs, due to significant time and memory complexities. To mitigate this issue, I will introduce a high order method which involves replacing the original PDE, which has rapidly varying coefficients, with a system of coupled equations with piecewise constant coefficients. These coefficients are derived from the first few terms of the log-signatures of the input paths and can be computed efficiently using existing Python libraries. |
| 25 Apr 2025 | Ben Tapley (SINTEF, AI and Analytics group) |
Structure-Preserving Neural Networks for Dynamical Systems
Incorporating known physics and geometry constraints into machine learning models is an effective strategy for improving generalisation, interpretability, and data efficiency in modelling dynamical systems. Popular models such as physics-informed neural networks (PINNs) can enforce physical laws in a weak sense via penalty terms in the loss function. However, a stronger inductive bias arises when these laws are embedded directly into the neural network architecture. Such models are often called “structure-preserving” and offer an alternative approach to exploiting known physics and geometry such as energy-preservation, a preserved symplectic form, variational structure, dissipation, Lyapunov functions and so forth. In this talk, we will review the motivation for structure-preserving modelling, summarise some popular structure-preserving models, how they guard against spurious phenomena, and show how to extend these methods to capture realistic dissipative effects. |
| 2 May 2025 | Xinyi Wu (University of Birmingham) |
Stochastic Norton Dynamics: An Alternative Approach for the Computation of Transport Coefficients in Dissipative Particle Dynamics
We study a novel alternative approach for the computation of transport coefficients at mesoscales. While standard nonequilibrium molecular dynamics (NEMD) approaches fix the forcing and measure the average induced flux in the system driven out of equilibrium, the so-called “stochastic Norton dynamics” instead fixes the value of the flux and measures the average magnitude of the forcing needed to induce it. We extend recent results obtained in Langevin dynamics to consider the generalisation of the stochastic Norton dynamics in the popular dissipative particle dynamics (DPD) at mesoscales, important for a wide range of complex fluids and soft matter applications. We demonstrate that the responses profiles for both the NEMD and stochastic Norton dynamics approaches coincide in both linear and nonlinear regimes, indicating that the stochastic Norton dynamics can indeed act as an alternative approach for the computation of transport coefficients, including the mobility and the shear viscosity, as the NEMD dynamics. In addition, based on the linear response of the DPD system with small perturbations, we derive a closed-form expression for the shear viscosity, and numerically validate its effectiveness with various types of external forces. Moreover, our numerical experiments demonstrate that the stochastic Norton dynamics approach clearly outperforms the NEMD dynamics in controlling the asymptotic variance, a key metric to measure the associated computational costs, particularly in the high friction limit. |
| 9 May 2025 | Patrick Fahy (University of Bath) |
Greedy learning to optimise with convergence guarantees
Learning to optimise is an approach that leverages training data to accelerate the solution of optimisation problems. Many approaches use unrolling to parametrise the update step and learn parameters. Although L2O has shown empirical advantages over classical optimisation algorithms, memory restrictions often greatly limit the unroll length, and learned algorithms usually do not provide convergence guarantees. In contrast, we introduce a novel method employing a greedy strategy that learns iteration-specific parameters by minimising the function value at the next iteration. This enables training over significantly more iterations while maintaining constant GPU memory usage. We parametrise the update such that parameter learning is no more difficult than solving the initial optimisation problem. In particular, we explore an extension of Polyak’s Heavy Ball Method with multiple parametrisations, including a convolutional preconditioner. With our learned algorithms, convergence in the training set is proved even when the preconditioners are not necessarily symmetric nor positive definite. Convergence on a class of unseen functions is also obtained under certain assumptions, ensuring robust performance and generalisation beyond the training data. We test our learned algorithms on a Computed Tomography example, on which we demonstrate improved empirical performance over classical optimisation algorithms such as Nesterov’s Accelerated Gradient Method and L-BFGS. |
| 9 May 2025 | Max Scott (University of Bath) |
Underdamped Langevin MCMC with third order convergence
Markov Chain Monte Carlo methods of the discretised underdamped Langevin SDE allow for the sampling of a given target distribution. This in turn has a wide array of applications, ranging from molecular dynamics to machine learning. However, especially in high dimensional settings, inefficient sampling methodologies can quickly become prohibitively computationally expensive. I will present a new sampling method named QUICSORT with the novel behaviour that it exhibits third order global convergence under the 2-Wasserstein metric. The core components of this theoretical result will be discussed along with examples of it in action. |
| 16 May 2025 | Andraž Jelinčič (University of Bath) |
A Scalable Deep Learning Architecture for Probabilistic Computers: How Denoising Boltzmann Machines Overcome the Mixing-Expressivity Tradeoff
Deep learning often uses probabilistic models to handle the inherent randomness of real-world phenomena. This fact, along with the growing energy demand of deep learning applications, has prompted proposals for specialized probabilistic computers. Despite promising efficiency gains, these proposals have failed to gain commercial traction in part because they rely solely on Energy-Based Models (EBMs), which often struggle to represent real-world datasets. In this work, we address this shortcoming by introducing the Thermodynamic Denoising Models (TDMs), an architecture that merges probabilistic-hardware-accelerated EBMs with Denoising Diffusion Models. TDMs can take advantage of the significant energy efficiency gains of probabilistic computers without making the sacrifices to expressivity associated with EBMs. Simulations suggest that TDMs running on future probabilistic hardware achieve performance parity with GPUs on image generation benchmarks, while using approximately 10000 times less energy. |
| 23 May 2025 | Robert Kirby (Baylor University) |
TBC
TBC |
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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.