Systems modelled by partial differential equations (PDEs) are ubiquitous in science and engineering. They are used to model problems including structures, fluids, materials, electromagnetics, wave propagation and biological systems, and in areas as varied as aerospace, image processing, medical therapeutics and economics. PDEs comprise a forward model for predicting the response of a system, but are also a key component in the solution of inverse problems, for design optimisation, uncertainty quantification and data science applications, where the forward computation is repeated many times with different inputs.

The numerical simulation of complex systems modeled by PDEs is a challenging topic. It involves the choice of underlying equations, the selection of suitable numerical solvers, and implementation on specific hardware. Over the decades numerous software libraries have been developed to support this task. But adapting these libraries to the specific model and combining the various components in a low-level high-performance programming language requires a major development effort. This required effort has become significantly more challenging with the advent of heterogeneous mixed CPU/GPU devices on the path to exascale systems. Implementations need to be adapted for each individual device type in order to achieve good performance. As a consequence, developing new simulations at scale has become an ever more costly and time-intensive task.

In this project we propose a different simulation paradigm, based on the use of high-productivity languages such as Python to describe the problem, and automatic code generation and just-in-time compilation to translate the high-level formulations into high-performance exascale-ready code. Based on the experience with the component software libraries Firedrake, FEniCS and Bempp, the investigators will build a toolchain for complex exascale simulations of PDEs on unstructured grids, using state of the art finite element and boundary element technologies. The research will include mathematical and algorithmic underpinnings, concrete software development for automatic code generation of low-level CPU/GPU kernels, high-productivity language interfaces, and the application to 21st century exascale challenge problems in the areas of battery storage systems, net-zero flight, and high-frequency wave propagation.