On the construction of efficient variable s -step methods for solving linear systems with applications in cyber-physical systems

This paper presents innovative algorithms built upon the extended methods of conjugate residual (CR), generalised conjugate residual (GCR), and Orthomin(k) to solve the linear system of equations Ax = b. Our goal is to improve communication efficiency between memory layers and processors. To achieve this, we introduce the s-step versions of these methods, referred to as s-step CR, s-step GCR, and s-step Orthomin(k). However, selecting the optimal parameter s poses a challenge: a small s may not sufficiently reduce communication, while a large s can slow convergence or even lead to divergence. To address this issue, we employ a variable s-step technique, where s is incrementally increased to minimise the residual norm. This variable s-step approach introduces challenges in defining matrices and vectors that can adapt to size variations across iterations. This complexity affects the selection and utilisation of bases in methods like GCR and Orthomin(k). To tackle these challenges, we incorporate the s-step technique into our algorithms and explore various methods for selecting Krylov bases. Finally, we present numerical experiments to assess the performance and optimality of these algorithms in comparison to traditional methods.