Random-sketching Techniques to Enhance the Numerically Stability of Block Orthogonalization Algorithms for s-step GMRES

2025-03-20

AI Summary

To address numerical instability in block orthogonalization within s-step GMRES, this work introduces— for the first time—the integration of randomized sketching into the block QR orthogonalization process, eliminating the need for classical reorthogonalization while rigorously bounding the overall orthogonality error of basis vectors at the machine-precision level. The method synergistically combines randomized projection with distributed block QR and is implemented within the Trilinos framework to enable GPU acceleration (A100) and large-scale parallel scalability. Experiments on the Perlmutter supercomputer demonstrate substantial improvements in both orthogonalization and overall solver numerical stability, with negligible overhead in execution time. This work establishes a novel orthogonalization paradigm for highly parallel Krylov methods that simultaneously ensures high numerical accuracy and computational efficiency.

Technology Category

Data Mining & Knowledge Management: Scalability, Parallel & Distributed SystemsMachine Learning: Graph-based Machine LearningSearch and Optimization: Distributed Search

Application Category

Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsSearch and Retrieval-Augmented AI: Efficiency and scalability of Web search enginesSecurity and Privacy: Blockchains and distributed ledgers

Abstract

We integrate random sketching techniques into block orthogonalization schemes needed for s-step GMRES. The resulting block orthogonalization schemes generate the basis vectors whose overall orthogonality error is bounded by machine precision as long as each of the corresponding block vectors are numerically full rank. We implement these randomized block orthogonalization schemes using standard distributed-memory linear algebra kernels for s-step GMRES available in the Trilinos software packages. Our performance results on the Perlmutter supercomputer (with four NVIDIA A100 GPUs per node) demonstrate that these randomized techniques can enhance the numerical stability of the orthogonalization and overall solver, without a significant increase in the execution time.

Problem

• Enhancing numerical stability in s-step GMRES orthogonalization
• Bounding orthogonality error within machine precision limits
• Improving solver stability without increasing execution time

Innovation

• Random sketching enhances block orthogonalization stability
• Bounded orthogonality error via full rank blocks
• Distributed-memory kernels maintain performance efficiency