Scalable s-step Preconditioned Conjugate Gradient with Chebyshev Basis and Gauss-Seidel Gram Solve

Abstract

We present a variant of the s--step Preconditioned Conjugate Gradient (PCG) method that combines a Chebyshev--stabilized Krylov basis with a Forward Gauss--Seidel (FGS) iteration for the solution of the reduced Gram systems. In s--step PCG, multiple search directions are generated per outer iteration, reducing global synchronization costs but requiring the solution of small dense Gram systems whose conditioning is critical for stability. For a raw Chebyshev Krylov block, we derive an exact moment--based representation of the associated Gram matrix in the unpreconditioned setting and discuss its extension to left preconditioning. This representation provides a structural interpretation of the favorable Gram--matrix behavior observed for moderate step sizes. Building on inexact Krylov theory and on the algebraic connection between FGS and Modified Gram--Schmidt (MGS), we derive residual bounds for the reduced solves and provide a rationale for using a fixed, moderate number of FGS sweeps. Numerical experiments assess the resulting accuracy of the Gram solves and the convergence behavior of the outer iteration. Large--scale experiments on modern NVIDIA GPU architectures, including weak-scaling tests with aggregation-based Algebraic MultiGrid (AMG) preconditioning, show that, within the tested configurations, the proposed Chebyshev--stabilized $s$--step PCG method with FGS Gram solves achieves convergence comparable to classical PCG while reducing global synchronization overhead. The results demonstrate its potential as a scalable communication--reducing solver for current and next--generation accelerator systems.