MPiK: Mixed Precision in Krylov Subspace Methods

Context. Scientific simulations play a crucial role in many fields, from engineering to physics, from biology to medicine. The ever-increasing scale of computational problems demands ever more powerful supercomputers. To remain sustainable while delivering their record-breaking performance, modern supercomputers rely heavily on accelerators – specialized hardware designed to achieve peak performance in specific tasks. A computer can only use limited precision. When the result of a mathematical operation cannot be represented exactly with the limited precision available, the computer will round it, introducing a tiny error. Nowadays, numerical simulations involve trillions of individual operations, and these rounding errors can quickly accumulate over the course of a computation, easily contaminating the final result. This is especially true for accelerators, as they use extremely low precision, which in turn magnifies rounding errors. Challenge. Bridging the gap between high-performance, low-precision hardware and the high-accuracy requirements of scientific computing is a timely challenge that can be tackled by developing mixed-precision methods. These methods strategically combine low- and high-precision computations to solve scientific problems as accurately – but more efficiently – than existing alternatives, which typically use high precision throughout. Mixed-precision algorithms have been developed for a few numerical linear algebra problems, where they have been shown to improve performance and lower carbon footprint. Krylov subspace methods underpin a significant portion of the workload of modern supercomputers. These methods are used to solve large, linear eigenvalue problems and to compute the action of a matrix function on a vector – two operations that are crucial in physics and engineering simulations, as well as in numerical weather and climate prediction. Currently, Krylov subspace methods assume that high precision is used throughout, and, therefore, are unable to exploit low-precision accelerators. This makes current implementations inefficient and unsustainable. Aims and objectives. The goal of this project is to design and implement robust, reliable, and scalable mixed-precision Krylov subspace methods for eigenvalue problems and matrix functions. We will develop a precision-aware error analysis of the Arnoldi iteration, the key component of any Krylov subspace methods. This analysis will guide the design of our new mixed-precision algorithms We will work with our partners to maximise the impact of the project. We will implement the new algorithms as part of two well-established high-performance computing libraries: the Software for Linear Algebra Targeting Exascale (SLATE) and the Scalable Library for Eigenvalue Problem Computations (SLEPc). We will work with the core developers of these libraries, to ensure that our algorithms become available to the user community without requiring major changes to existing code. We will also contribute to the development of a fully functional numerical weather prediction model capable of using mixed precision and, therefore, of fully exploiting modern supercomputers. Applications and benefits. Krylov methods are a crucial tool in density functional theory, lattice quantum chromodynamics, and numerical weather prediction, to name a few. More generally, they play a key role in methods for the numerical integration of partial differential equations, and they are at the heart of a sizeable portion of the workload of modern supercomputers. This project will allow existing codes to make full use of modern, energy-efficient hardware and therefore run on current and future supercomputers. This will result in faster and more energy-efficient runs, and will allow researchers to tackle ever larger problems, accelerating the pace of scientific discovery.