Stable Iterative Solvers for Ill-conditioned Linear Systems

TL;DR

This paper introduces new algorithmic frameworks that modify Krylov subspace iterative methods to ensure stability and prevent divergence when solving large, ill-conditioned linear systems, enhancing their practical applicability.

Contribution

The paper proposes general frameworks for stabilizing Krylov subspace methods, with specific application to SciPy implementations, improving their robustness for ill-conditioned systems.

Findings

Frameworks successfully prevent divergence in ill-conditioned systems.

Numerical experiments show improved stability across synthetic and real-world data.

Enhanced iterative methods expand applicability in high-performance computing.

Abstract

Iterative solvers for large-scale linear systems such as Krylov subspace methods can diverge when the linear system is ill-conditioned, thus significantly reducing the applicability of these iterative methods in practice for high-performance computing solutions of such large-scale linear systems. To address this fundamental problem, we propose general algorithmic frameworks to modify Krylov subspace iterative solution methods which ensure that the algorithms are stable and do not diverge. We then apply our general frameworks to current implementations of the corresponding iterative methods in SciPy and demonstrate the efficacy of our stable iterative approach with respect to numerical experiments across a wide range of synthetic and real-world ill-conditioned linear systems.