Parallel Sparse and Data-Sparse Factorization-based Linear Solvers

2026-02-15

AI Summary

Solving large-scale, ill-conditioned, and indefinite sparse linear systems is critical in multiphysics simulation, machine learning, and related fields, yet poses significant challenges in computational efficiency and scalability. This work proposes an optimized sparse direct solver framework that integrates task and data parallelism with low-rank approximation and hierarchical matrix compression techniques. Implemented on heterogeneous high-performance computing platforms, the approach substantially reduces communication overhead and computational complexity while preserving the numerical robustness inherent to direct solvers. The resulting method achieves marked improvements in both strong and weak scalability, delivering a highly efficient, reliable, and scalable linear solver toolchain tailored for modern heterogeneous architectures.

Technology Category

• Machine Learning: Scalability of ML Systems
• Data Mining & Knowledge Management: Scalability, Parallel & Distributed Systems
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Web applications in cross-disciplinary domains and verticals such as mixed reality, smart cities, and digital health
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs

Abstract

Efficient solutions of large-scale, ill-conditioned and indefinite algebraic equations are ubiquitously needed in numerous computational fields, including multiphysics simulations, machine learning, and data science. Because of their robustness and accuracy, direct solvers are crucial components in building a scalable solver toolchain. In this article, we will review recent advances of sparse direct solvers along two axes: 1) reducing communication and latency costs in both task- and data-parallel settings, and 2) reducing computational complexity via low-rank and other compression techniques such as hierarchical matrix algebra. In addition to algorithmic principles, we also illustrate the key parallelization challenges and best practices to deliver high speed and reliability on modern heterogeneous parallel machines.

Problem

• sparse direct solvers
• ill-conditioned systems
• indefinite algebraic equations
• large-scale linear systems
• heterogeneous parallel computing

Innovation

• sparse direct solvers
• communication reduction
• low-rank compression
• hierarchical matrix algebra
• parallel linear solvers