Adrian Hill

Guillaume Dalle, Adrian Hill

For scientific machine learning tasks with a lot of custom code, picking the right Automatic Differentiation (AD) system matters. Our Julia package DifferentiationInterface$.$jl provides a common frontend to a dozen AD backends, unlocking easy comparison and modular development. In particular, its built-in preparation mechanism leverages the strengths of each backend by amortizing one-time computations. This is key to enabling sophisticated features like sparsity handling without putting additional burdens on the user.

Adrian Hill, Guillaume Dalle

From implicit differentiation to probabilistic modeling, Jacobian and Hessian matrices have many potential use cases in Machine Learning (ML), but they are viewed as computationally prohibitive. Fortunately, these matrices often exhibit sparsity, which can be leveraged to speed up the process of Automatic Differentiation (AD). This paper presents advances in sparsity detection, previously the performance bottleneck of Automatic Sparse Differentiation (ASD). Our implementation of sparsity detection is based on operator overloading, able to detect both local and global sparsity patterns, and supports flexible index set representations. It is fully automatic and requires no modification of user code, making it compatible with existing ML codebases. Most importantly, it is highly performant, unlocking Jacobians and Hessians at scales where they were considered too expensive to compute. On real-world problems from scientific ML, graph neural networks and optimization, we show significant speed-ups of up to three orders of magnitude. Notably, using our sparsity detection system, ASD outperforms standard AD for one-off computations, without amortization of either sparsity detection or matrix coloring.

Nick Brown, Michèle Weiland, Adrian Hill et al.

Large Eddy Simulation is a critical modelling tool for scientists investigating atmospheric flows, turbulence and cloud microphysics. Within the UK, the principal LES model used by the atmospheric research community is the Met Office Large Eddy Model (LEM). The LEM was originally developed in the late 1980s using computational techniques and assumptions of the time, which means that the it does not scale beyond 512 cores. In this paper we present the Met Office NERC Cloud model, MONC, which is a re-write of the existing LEM. We discuss the software engineering and architectural decisions made in order to develop a flexible, extensible model which the community can easily customise for their own needs. The scalability of MONC is evaluated, along with numerous additional customisations made to further improve performance at large core counts. The result of this work is a model which delivers to the community significant new scientific modelling capability that takes advantage of the current and future generation HPC machines.

John Butcher, Adrian Hill, Terrence Norton

The article considers symmetric general linear methods, a class of numerical time integration methods which, like symmetric Runge--Kutta methods, are applicable to general time--reversible differential equations, not just those derived from separable second--order problems. A definition of time--reversal symmetry is formulated for general linear methods, and criteria are found for the methods to be free of linear parasitism. It is shown that symmetric parasitism--free methods cannot be explicit, but a method of order $4$ is constructed with only one implicit stage. Several characterizations of symmetry are given, and connections are made with $G$--symplecticity. Symmetric methods are shown to be of even order, a suitable symmetric starting method is constructed and shown to be essentially unique. The underlying one--step method is shown to be time--symmetric. Several symmetric methods of order $4$ are constructed and implemented on test problems. The methods are efficient when compared with Runge--Kutta methods of the same order, and invariants of the motion are well--approximated over long time intervals.