This is part of the Oxford - RAL Computational Mathematics and Applications seminar series. For the full list see Oxford's website.

Upcoming Seminars

Tobias Weinzierl

Modern tasking approaches to simulate black holes (and other interesting phenomena): How can we make them fit to modern hardware?

30 Apr 2026, 2 p.m. - 30 Apr 2026, 3 p.m. at RAL

Over the past decade, my team has developed a simulation code for binary black hole mergers that runs on dynamically adaptive Cartesian meshes. Its dynamic adaptivity, coupled with multiple numerical schemes operating at different scales and non-deterministic loads from puncture sources, makes task-based parallelisation a natural choice: Task stealing across fine-grained work units balances the load across many CPU cores, while treating tasks as atomic compute units should---in theory---allow us to deploy seamlessly to accelerators. In practice, it is far from straightforward. Fine-grained tasks clash with accelerators, which thrive on large, homogeneous data access patterns; task bursts on the CPU overwhelm tasking systems and produce suboptimal execution schedules; and when tasks span address spaces, expensive memory movements kill performance. Surprisingly, many mainstream tasking frameworks even lack the features our domain demands, i.e. to express key task concepts. Our application serves as a powerful lens for examining these challenges. While our code base extends to other wave phenomena, Lagrangian techniques, and multigrid solvers, they all reveal the same fundamental tension: modern hardware increasingly struggles to accommodate modern HPC concepts, and it even challenges the notion that one solution fits all hardware components. The talk proposes practical workarounds and solutions to these shortcomings, while all solutions are designed, wherever possible, to be upstreamed into mainstream software building blocks or at least decoupled from our particular PDE solver, making them broadly applicable to the community.

Estefania Loayza Romero

A Riemannian Approach for PDE-Constrained Shape Optimization Using Outer Metrics

5 Feb 2026, 2 p.m. - 5 Feb 2026, 3 p.m. at RAL

In PDE-constrained shape optimisation, shapes are traditionally viewed as elements of a Riemannian manifold, specifically as embeddings of the unit circle into the plane, modulo reparameterizations. The standard approach employs the Steklov-Poincaré metric to compute gradients for Riemannian optimisation methods. A significant limitation of current methods is the absence of explicit expressions for the geodesic equations associated with this metric. Consequently, algorithms have relied on retractions (often equivalent to the perturbation of identity method in shape optimisation) rather than true geodesic paths. Previous research suggests that incorporating geodesic equations, or better approximations thereof, can substantially enhance algorithmic performance. This talk presents numerical evidence demonstrating that using outer metrics, defined on the space of diffeomorphisms with known geodesic expressions, improves Riemannian gradient-based optimisation by significantly reducing the number of required iterations and preserving mesh quality throughout the optimisation process.

Past Seminars

Paul Goulart

Interior-point optimisation for quadratic programs with conic constraints.

23 Oct 2025, 2 p.m. - 23 Oct 2025, 3 p.m. at RAL

The talk will present the open-source convex optimisation solver Clarabel, an interior-point based solver that uses a novel homogeneous embedding technique offering substantially faster solve times relative to existing open-source and commercial interior-point solvers for some problem types. This improvement is due to both a reduction in the number of required interior point iterations as well as an improvement in both the size and sparsity of the linear system that must be solved at each iteration. For large-scale problems we employ a variety of additional techniques to accelerate solve times, including chordal decomposition methods, GPU sub-solvers, and custom handling of certain specialised cones. The talk will describe details of our implementation and show performance results with respect to solvers based on the standard homogeneous self-dual embedding.

Colin Cotter

HSS iteration for solving the indefinite Helmholtz equation by multigrid with standard components

9 Oct 2025, 2 p.m. - 9 Oct 2025, 3 p.m. at RAL

We provide an iterative solution approach for the indefinite Helmholtz equation discretised using finite elements, based upon a Hermitian Skew-Hermitian Splitting (HSS) iteration applied to the shifted operator, and prove that the iteration is k- and mesh-robust when O(k) HSS iterations are performed. The HSS iterations involve solving a shifted operator that is suitable for approximation by multigrid using standard smoothers and transfer operators, and hence we can use O(N) parallel processors in a high performance computing implementation, where N is the total number of degrees of freedom. We argue that the algorithm converges in O(k) wallclock time when within the range of scalability of the multigrid. We provide numerical results verifying our proofs and demonstrating this claim, establishing a method that can make use of large scale high performance computing systems.

Aretha Teckentrup

Multilevel Monte Carlo Methods with Smoothing

8 May 2025, 2 p.m. - 8 May 2025, 3 p.m. at RAL

Parameters in mathematical models are often impossible to determine fully or accurately, and are hence subject to uncertainty. By modelling the input parameters as stochastic processes, it is possible to quantify the uncertainty in the model outputs. In this talk, we employ the multilevel Monte Carlo (MLMC) method to compute expected values of quantities of interest related to partial differential equations with random coefficients. We make use of the circulant embedding method for sampling from the coefficient, and to further improve the computational complexity of the MLMC estimator, we devise and implement the smoothing technique integrated into the circulant embedding method. This allows to choose the coarsest mesh on the  first level of MLMC independently of the correlation length of the covariance function of the random  field, leading to considerable savings in computational cost.

David Ham

Firedrake: a differentiable programming framework for finite element simulation

20 Mar 2025, 2 p.m. - 20 Mar 2025, 3 p.m. at RAL

Differentiable programming is the underpinning technology for the AI revolution. It allows neural networks to be programmed in very high level user code while still achieving very high performance for both the evaluation of the network and, crucially, its derivatives. The Firedrake project applies exactly the same concepts to the simulation of physical phenomena modelled with partial differential equations (PDEs). By exploiting the high level mathematical abstraction offered by the finite element method, users are able to write mathematical operators for the problem they wish to solve in Python. The high performance parallel implementations of these operators are then automatically generated, and composed with the PETSc solver framework to solve the resulting PDE. However, because the symbolic differential operators are available as code, it is possible to reason symbolically about them before the numerical evaluation. In particular, the operators can be differentiated with respect to their inputs, and the resulting derivative operators composed in forward or reverse order. This creates a differentiable programming paradigm congruent with (and compatible with) machine learning frameworks such as Pytorch and JAX. In this presentation, I will present Firedrake in the context of differentiable programming, and show how this enables productivity, capability and performance to be combined in a unique way. I will also touch on the mechanism that enables Firedrake to be coupled with Pytorch and JAX.

Eric Kerrigan

Integrate your residuals while solving dynamic optimization problems

20 Feb 2025, 2 p.m. - 20 Feb 2025, 3 p.m. at RAL

Many optimal control, estimation and design problems can be formulated as so-called dynamic optimization problems, which are optimization problems with differential equations and other constraints. State-of-the-art methods based on collocation, which enforce the differential equations at only a finite set of points, can struggle to solve certain dynamic optimization problems, such as those with high-index differential algebraic equations, consistent overdetermined constraints or problems with singular arcs. We show how numerical methods based on integrating the differential equation residuals can be used to solve dynamic optimization problems where collocation methods fail. Furthermore, we show that integrated residual methods can be computationally more efficient than direct collocation.

Old seminars

Marta Betcke

Consistent learned reconstruction from limited angle data in Photoacoustic Tomography

24 Oct 2024, 2 p.m. - 24 Oct 2024, 3 p.m. at RAL

Joint work with Bolin Pan. In photoacoustic tomography (PAT) with flat sensor, we routinely encounter two types of limited data. The first is due to using a finite sensor and is especially perceptible if the region of interest is large relatively to the sensor or located farther away from the sensor. In this talk we focus on the second type caused by a varying sensitivity of the sensor to the incoming wavefront direction which can be modelled as binary i.e. by a cone of sensitivity. Such visibility conditions result, in Fourier domain, in a restriction of the data to a bowtie, akin to the one corresponding to the range of the forward operator but further narrowed according to the angle of sensitivity. We show how we can separate the visible and invisible wavefront directions in PAT image and data using a directional frame like Curvelets, and how such decomposition allows for decoupling of the reconstruction involving application of expensive forward/adjoint solvers from the training problem. We present fast and stable approximate Fourier domain forward and adjoint operators for reconstruction of the visible coefficients for such limited angle problem and a tailored UNet matching both the multi-scale Curvelet decomposition and the partition into the visible/invisible directions for learning the invisible coefficients from a training set of similar data.