Mauro G. Carrozzo | Searcharxiv

arXiv2026

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

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.

math.NA↗

arXiv2025

Communication-reduced Conjugate Gradient Variants for GPU-accelerated Clusters

Linear solvers are key components in any software platform for scientific and engineering computing. The solution of large and sparse linear systems lies at the core of physics-driven numerical simulations relying on partial differential equations (PDEs) and often represents a significant bottleneck in datadriven procedures, such as scientific machine learning. In this paper, we present an efficient implementation of the preconditioned s-step Conjugate Gradient (CG) method, originally proposed by Chronopoulos and Gear in 1989, for large clusters of Nvidia GPU-accelerated computing nodes. The method, often referred to as communication-reduced or communication-avoiding CG, reduces global synchronizations and data communication steps compared to the standard approach, enhancing strong and weak scalability on parallel computers. Our main contribution is the design of a parallel solver that fully exploits the aggregation of low-granularity operations inherent to the s-step CG method to leverage the high throughput of GPU accelerators. Additionally, it applies overlap between data communication and computation in the multi-GPU sparse matrix-vector product. Experiments on classic benchmark datasets, derived from the discretization of the Poisson PDE, demonstrate the potential of the method.

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