ABSTRACT. Project NEPTUNE approaches one of the grand challenge problems in fusion - modelling plasma in the edge region of a tokamak. In this edge region, hot plasma interfaces with the reactor wall and cold neutral gas resulting in a computationally expensive multi-scale problem that is considered an exascale challenge. The NESO (Neptune Exploratory SOftware) implementation is an intermediate project to evaluate the suitability and performance of proposed techniques and algorithms for NEPTUNE.

The plasma use case requires discretisations that represent distributions over both physical space and over the product of physical space and velocity space. In NESO we apply finite elements as a discretisation over physical space and use a particle representation as a so called "kinetic" representation in velocity space. In this talk we discuss how NESO tightly couples finite element and particle approaches within the same simulation domain. This coupling requires efficient bookkeeping operations to transport particles over unstructured (and potentially high-order meshes) and to convert between particle and finite element representations.

ABSTRACT. We present an efficient inverse homogenisation framework for the design of metamaterials with prescribed non-linear structural behaviour. In particular, two and three-dimensional periodic microstructures are designed to exhibit targeted stress responses when subjected to finite strains.

The framework relies on the Firedrake package for solving the underlying non-linear Partial Differential Equations (PDEs), namely the hyperelastic constitutive model, which permits the consideration geometric non-linearities during the design of metamaterials. A gradient-based optimisation methodology is utilised for the design of the microstructures, where the sensitivities required are straightforwardly computed using the adjoint capabilities provided by Firedrake Adjoint.

ABSTRACT. Optimal transport (OT) is a theory built around the question of finding a path of minimal energy connecting two distributions of mass. OT theory has found nowadays a wide range of applications since this simple question (or some variant of it) emerges naturally in many fields ranging from physics, and in particular fluid mechanics, to economics and social sciences. When tackling OT numerically one often needs to deal with large and singular optimisation problems, which require ad hoc discretization approaches and are generally expensive to solve accurately. In this talk I will describe some finite element strategies to solve OT problems, and discuss the main challenges related to the construction of robust discretizations, the design of efficient algorithms and their implementation in Firedrake.

ABSTRACT. Liquid crystals possess material properties that are intermediate between those of liquids and solid crystals. Because they are controllable by electric fields, they have a wide range of potential scientific and industrial uses. Several families of finite element methods have been proposed and developed in recent years to model equilibrium states of different types of liquid crystals. Among common liquid crystal phases, smectic phases are distinguished by their orientational and partial translational order, forming distinct layers at equilibrium. In comparison to the better-studied nematic case, there has been little work in developing finite-element simulation tools for smectic liquid crystals, primarily due to the complex nature of their governing free-energy functionals. In this talk, we present a new mixed finite element formulation for a recently-proposed model of smectic-A liquid crystals. We devise effective nonlinear and linear solvers for this formulation, combining nested iteration and monolithic multigrid methods. Numerical results are implemented using Firedrake for the finite element discretization and PETSc for the nonlinear and linear solvers.

ABSTRACT. PDEs are central to describing and modelling complex physical systems that arise in many disciplines across science and engineering. However, in many realistic applications, PDE modelling provides an incomplete description of the physics of interest. Machine learning techniques have become increasingly popular in filling the knowledge gap between the fundamental physical laws, expressed as differential equations, and the real-world phenomena studied by scientists and engineers. The emergence of this approach urges the need for scientific simulation frameworks that allow for the efficient development and deployment of models coupling PDEs and ML.

We employ a differentiable programming approach to build a highly efficient and composable interface that provides researchers, engineers, and domain experts with diverse backgrounds with a highly productive way to run high-performance simulations coupling PDEs, implemented using the finite element method (FEM) in Firedrake, and machine learning models, specified in PyTorch. The resulting framework maintains separation of concerns while only requiring trivial changes to existing code.

ABSTRACT. Numerical simulations play a central role in understanding the impact and risks of pressing global problems, particularly to the scale-up challenges of energy generation from complex, non-linear sources such as wind and tidal. Effectively discretizing the extreme spatial scale variability inherent in such geophysical fluid dynamics problems can come at a high computational cost when targeting a reasonable level of accuracy for meaningful results. Mesh adaptation is the process of modifying the discretized structure to improve the accuracy of numerical simulations. In addition to mesh adaptation, identifying opportunities to augment the numerical methods, particularly iterative methods, with machine learning workflows has potential to further reduce computational overhead by automating the process and incorporating prior knowledge. In this talk, we review our work applying machine learning for mesh adaptation in numerical simulations motivated by tidal energy applications. We focus the discussion on trade-offs between accuracy preservation and efficiency gain for machine learning-based mesh adaptation methods that utilizes a combination of supervised and unsupervised learning techniques. A series of simple benchmark applications will be presented.

ABSTRACT. In this talk we will present Hydrogym, a reinforcement learning platform for flow control connecting flow control benchmark problems implemented in Firedrake with state-of-the-art reinforcement learning. We will present a set of initial environments, as well as the seamless scalability of HydroGym scaling from laptops to high-performance computing resources. We will conclude by going into HydroGym’s core design with its extensible interface enabling scientists to easily implement their own environment and benefit from HydroGym’s scaling for their own flow control problems.

ABSTRACT. We have employed Firedrake to develop an ensemble-based data assimilation methodology for stochastic PDEs. The algorithm presented here is a particle filter framework which combines tempering, jittering and nudging. All the filtering steps are implemented using ensemble parallelism. This framework is demonstrated on the stochastic Camassa-Holm and quasi-geostrophic models.

ABSTRACT. Modern high performance computing architectures are massively parallel, so efficient use of HPC resources requires algorithms that scale to very large core counts. For time-dependent problems, speedups are limited once spatial parallelism is saturated, at which point the time-to-solution increases with the number of timesteps. This barrier can be alleviated with parallel-in-time (PinT) algorithms that increase the problem size by solving for multiple timesteps simultaneously.

ParaDiag is a PinT algorithm that solves multiple timesteps simultaneously by preconditioning the system Jacobian with a block diagonalisable matrix which can be solved efficiently in parallel.

In this talk we will explain the ParaDiag algorithm and introduce asQ (github.com/firedrakeproject/asQ), a library implementing this method for finite-element models with Firedrake and PETSc. Following Firedrake, asQ aims to be flexible enough to be a sandbox for method development, whilst also being performant enough to scale to large core counts on realistic problem sizes. We will discuss our efforts towards achieving these aims, including the design of the library interface and requirements for efficient parallel performance.

The parallel performance of asQ will be demonstrated on test cases from weather and climate, which is one of the primary target application for PinT methods.

ABSTRACT. We consider a new time-stepping approach for the rotating shallow water equations on the sphere. We first solve separately for the convective parts of the equations using a DG-space for the velocity and after we solve for the remaining part of the equations with the velocity in H(div). We present a first and second order scheme and discuss our latest numerical results.

ABSTRACT. We present a phase-averaging framework for the rotating shallow-water equations and a time-integration methodology for it. Phase averaging consists of averaging the nonlinearity over phase shifts in the exponential of the linear wave operator. Phase averaging aims to capture the slow dynamics in a solution that is smoother in time (in transformed variables), so that larger timesteps may be taken. In our numerical implementation, the averaging integral is replaced by a Riemann sum, where each term can be evaluated in parallel. This creates an opportunity for parallelism in the timestepping method. Here we used the “ensemble parallelism” capability of Firedrake, which was implemented for this project.

In this talk, we will show proof-of-concept results and analyse their errors in order to examine the impact of the averaging on the rotating shallow-water solution. An evaluation of the parallel aspects will follow in later work.

ABSTRACT. In this talk we propose an adaptive algorithms which allows for an accurate representation of physics between scales, with an aim for developing an implementation in Firedrake. For exposition, we focus primarily on adaptivity in time as this is the novel (and difficult to implement) aspect of this algorithm.

ABSTRACT. Gusto is a toolkit using Firedrake for modelling geophysical fluids using compatible finite element methods. It focuses particularly on numerical methods for the dynamical cores that lie at the heart of numerical weather prediction and climate models. Gusto is designed to facilitate rapid prototyping of new numerical schemes, which can often be difficult to implement in operational models. It also provides a user-friendly sandbox for exploring research questions relating to numerical methods, or the underlying fluid equations. This is all enabled by Firedrake's high-level interface.

This talk will give an overview of some of the work done using Gusto, capabilities recently added to Gusto and our future plans.

[Note to organisers: I may only be able to attend the workshop from Thursday lunchtime onwards]

ABSTRACT. I will give a mini introduction to shape optimization and then discuss the latest developments of firedrake's shape optimization toolbox fireshape. I will conclude with the latest developments on an on-going industrial collaboration.

ABSTRACT. Continuing on from my presentation at Firedrake '22, I have been developing methods to more systematically assess the numbers of corners in shapes resulting from shape optimisation. This has raised both theoretical and practical questions as to how to detect a corner. I will discuss briefly the problem and solutions that have caused this question to arise and then discuss what definition we can use to to count corners, along with the programming considerations in implementing them, and some issues I ran in to with Firedrake whilst doing so.