A new deflated block GCROT($m,k$) method for the solution of linear systems with multiple right-hand sides

TL;DR

This paper introduces a novel deflated block GCROT method utilizing modified block Arnoldi deflation to efficiently solve linear systems with multiple right-hand sides, improving convergence and computational performance.

Contribution

The paper proposes a new deflated block GCROT($m,k$) method with modified block Arnoldi deflation, enhancing efficiency and robustness in solving large linear systems with multiple right-hand sides.

Findings

The new method effectively addresses linear dependence during block Arnoldi iterations.

The deflation procedure ensures non-increasing singular values of the block residual.

Numerical examples demonstrate improved efficiency and convergence.

Abstract

Linear systems with multiple right-hand sides arise in many applications. To solve such systems efficiently, a new deflated block GCROT($m,k$) method is explored in this paper by exploiting a modified block Arnoldi deflation. This deflation strategy has been shown to have the potential to improve the original deflation which indicates an explicit block size reduction. Incorporating this modified block Arnoldi deflation, the new algorithm can address the possible linear dependence at each iteration during the block Arnoldi procedure and avoids expensive computational operations. In addition, we analyze its main mathematical properties and prove that the deflation procedure is based on a non-increasing behavior of the singular values of the true block residual. Moreover, as a block version of GCROT($m,k$), the new method inherits the property of easy operability. Finally, some numerical examples also illustrate the effectiveness of the proposed method.