A Task Parallel Orthonormalization Multigrid Method For Multiphase Elliptic Problems

2025-12-09

AI Summary

Traditional K-cycle orthogonalization multigrid methods suffer from poor scalability on large-scale parallel systems due to synchronization bottlenecks inherent in the bulk-synchronous parallel (BSP) paradigm, particularly when solving discretized multiphase elliptic PDEs. To address this, we propose an asynchronous, task-parallel orthogonalization multigrid method. This is the first approach to integrate a task-parallel programming model with fine-grained asynchronous scheduling into the orthogonalization multigrid framework, synergistically combining multigrid preconditioning with Krylov subspace residual minimization while preserving convergence guarantees. Experimental results demonstrate that the proposed method significantly improves parallel efficiency and strong/weak scalability for high-order and strongly anisotropic problems on modern HPC architectures, substantially reducing solution time. Both time and space complexity achieve theoretical optimality.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

Multigrid methods have been a popular approach for solving linear systems arising from the discretization of partial differential equations (PDEs) for several decades. They are particularly effective for accelerating convergence rates with optimal complexity in terms of both time and space. K-cycle orthonormalization multigrid is a robust variant of the multigrid method that combines the efficiency of multigrid with the robustness of Krylov-type residual minimalizations for problems with strong anisotropies. However, traditional implementations of K-cycle orthonormalization multigrid often rely on bulk-synchronous parallelism, which can limit scalability on modern high-performance computing (HPC) systems. This paper presents a task- parallel variant of the K-cycle orthonormalization multigrid method that leverages asynchronous execution to improve scalability and performance on large-scale parallel systems.

Problem

• Develops a task-parallel K-cycle orthonormalization multigrid method
• Enhances scalability for multiphase elliptic PDEs on HPC systems
• Addresses limitations of bulk-synchronous parallelism in traditional implementations

Innovation

• Task-parallel K-cycle orthonormalization multigrid method
• Asynchronous execution for improved scalability
• Optimized for large-scale parallel HPC systems