Stabilization of BiCGSTAB by the generalized residual cutting method

TL;DR

This paper introduces a method to enhance the robustness and convergence stability of the BiCGSTAB iterative solver for large sparse linear systems by applying the generalized residual cutting (GRC) technique.

Contribution

The paper demonstrates that GRC can stabilize BiCGSTAB, reducing convergence issues and failures in solving large sparse nonsymmetric linear systems.

Findings

GRC stabilizes BiCGSTAB, improving convergence behavior.

Application of GRC reduces stagnation and breakdown failures.

Enhanced BiCGSTAB shows more reliable performance in numerical experiments.

Abstract

The residual cutting (RC) method has been proposed as an outer-inner loop iteration for efficiently solving large and sparse linear systems of equations arising in solving numerically problems of elliptic partial differential equations. Then based on RC the generalized residual cutting (GRC) method has been introduced, which can be applied to more general sparse linear systems problems. In this paper, we show that GRC can stabilize the BiCGSTAB, which is also an iterative algorithm for solving large, sparse, and nonsymmetric linear systems, and widely used in scientific computing and engineering simulations, due to its robustness. BiCGSTAB converges faster and more smoothly than the original BiCG method, by reducing irregular convergence behavior by stabilizing residuals. However, it sometimes fails to converge due to stagnation or breakdown. We attempt to emphasize its robustness by further stabililizing it by GRC, avoiding such failures.