Modelling the earth’s geomagnetic environment on Cray machines using PETSc and SLEPc

The British Geological Survey's global geomagnetic model, Model of the Earth's Magnetic Environment (MEME), is an important tool for calculating the earth's magnetic field, which is continually in flux. Whilst the ability to collect data from ground based observation sites and satellites has grown rapidly, the memory bound nature of the code has proved a significant limitation in modelling problem sizes required by modern science. In this paper we describe work done replacing the bespoke, sequential, eigen-solver with that of the SLEPc package for solving the system of normal equations. This work had a dual purpose, to break through the memory limit of the code, and thus support the modelling of much larger systems, by supporting execution on distributed machines, and to improve performance. But when adopting SLEPc it was not just the solving of the normal equations, but also fundamentally how we build and distribute the data structures. We describe an approach for building symmetric matrices in a way that provides good load balance and avoids the need for close co-ordination between the processes or replication of work. We also study the memory bound nature of the code from an irregular memory accesses perspective and combine detailed profiling with software cache prefetching to significantly optimise this. Performance and scaling characteristics are explored on ARCHER, a Cray XC30, both comparing the new model's SLEPc approach against the previous model's bespoke solver, and also considering much larger system sizes up to 100,000 model coefficients. Some of the challenges of modelling systems of this large scale are explored, and mitigations including hybrid MPI+OpenMP along with the use of iterative solvers are also considered. The result of this work is a modern MEME model that is not only capable of simulating problem sizes demanded by state of the art geomagnetism but also acts as further evidence to the utility of the SLEPc libary.