[Paper Review] An Error-Resilient Redundant Subspace Correction Method

Tao Cui, Jinchao Xu+1 more|arXiv (Cornell University)|Sep 1, 2013

Matrix Theory and Algorithms38 references4 citations

TL;DR

This paper proposes a redundant subspace correction (RSC) method to enhance fault tolerance in high-performance scientific computing by enabling iterative linear solvers to maintain convergence under detectable errors. The method uses redundant subspace projections to ensure convergence even if one processor fails, with minimal computational overhead and low communication cost, significantly improving mean time to failure in exascale systems.

ABSTRACT

As we stride toward the exascale era, due to increasing complexity of supercomputers, hard and soft errors are causing more and more problems in high-performance scientific and engineering computation. In order to improve reliability (increase the mean time to failure) of computing systems, a lot of efforts have been devoted to developing techniques to forecast, prevent, and recover from errors at different levels, including architecture, application, and algorithm. In this paper, we focus on algorithmic error resilient iterative linear solvers and introduce a redundant subspace correction method. Using a general framework of redundant subspace corrections, we construct iterative methods, which have the following properties: (1) Maintain convergence when error occurs assuming it is detectable; (2) Introduce low computational overhead when no error occurs; (3) Require only small amount of local (point-to-point) communication compared to traditional methods and maintain good load balance; (4) Improve the mean time to failure. With the proposed method, we can improve reliability of many scientific and engineering applications. Preliminary numerical experiments demonstrate the efficiency and effectiveness of the new subspace correction method.

Motivation & Objective

• Address the growing challenge of hardware and soft errors in exascale supercomputing, which threaten reliability and system uptime.
• Improve fault tolerance in iterative linear solvers—critical for large-scale PDE simulations—without incurring high computational or communication costs.
• Develop a scalable, algorithm-level solution that maintains convergence even when errors occur during computation.
• Ensure minimal performance degradation in error-free scenarios while maintaining good load balance and low point-to-point communication.
• Enhance the mean time to failure (MTTF) of scientific applications by integrating redundancy at the algorithmic level.

Proposed method

• Formalize a general framework of redundant subspace corrections (RSC) based on additive Schwarz and subspace correction methods.
• Construct a successive redundant subspace correction (SRSC) method that uses duplicated subspaces to maintain convergence when errors occur.
• Define a preconditioner $ B_{\text{SRSC}} $ such that $ \|I - B_{\text{SRSC}}A\|_A \leq \|I - B_{\text{SSC}}A\|_A $, ensuring convergence even under errors.
• Use a multiplicative combination of two incomplete subspace correction preconditioners ($ B_{\text{PSC}}^c $ and $ \tilde{B}_{\text{PSC}}^c $) to form a redundant preconditioner $ B_{\text{PRSC}} $, improving robustness.
• Leverage domain decomposition and overlapping subspaces to enable parallel computation with localized communication and balanced work distribution.
• Apply the method to standard PDEs (Poisson, Maxwell, elasticity) using finite element discretizations and METIS-based domain partitioning.

Experimental results

Research questions

• RQ1 Can a redundant subspace correction method maintain convergence of iterative linear solvers when hardware or soft errors occur during computation?
• RQ2 What is the computational overhead of the proposed method in the absence of errors, and can it be kept minimal compared to standard methods?
• RQ3 How does the method compare to traditional subspace correction in terms of communication cost and load balancing in large-scale parallel systems?
• RQ4 To what extent does the method improve the mean time to failure (MTTF) in exascale HPC environments?
• RQ5 Can the method be applied effectively to real-world PDE-based scientific simulations without degrading performance?

Key findings

• The redundant subspace correction (RSC) method maintains convergence even when errors occur, with the convergence rate bounded by the standard subspace correction (SSC) method: $ \|I - B_{\text{SRSC}}A\|_A \leq \|I - B_{\text{SSC}}A\|_A $.
• In error-free scenarios, the convergence rate is bounded by the product of two SSC rates: $ \|I - B_{\text{SRSC}}A\|_A \leq \|I - B_{\text{SSC}}A\|_A \cdot \|I - \tilde{B}_{\text{SSC}}A\|_A $, ensuring stability.
• The method achieves convergence twice as fast as the standard SSC method under ideal conditions, as shown by theoretical analysis.
• The method requires only small, localized communication, maintaining good load balance and scalability on large-scale systems.
• Preliminary numerical experiments on Poisson, Maxwell, and elasticity equations confirm the method's efficiency and effectiveness in both error-free and erroneous scenarios.
• The approach is robust even when multiple processing units fail, as long as at least one unit per redundant pair remains functional, demonstrating strong fault resilience.