Locally optimal and heavy ball GMRES methods

• DOI: 10.1007/s13160-016-0220-1
• Publication Date: 2016-06-09T07:35:19Z
• Authors: Akira Imakura; Ren-Cang Li; Shao-Liang Zhang
• Keywords: Krylov subspace; Ball (mathematics)

The restarted GMRES (REGMRES) is one of the well used Krylov subspace methods for solving linear systems. However, the price to pay for the restart usually is slower speed of convergence. In this paper, we draw inspirations from the locally optimal CG and the heavy ball methods in optimization to propose two variants of the restarted GMRES that can overcome the slow convergence. Compared to various existing hybrid GMRES which are also designed to speed up REGMRES and which usually require eigen-region estimations, our variants preserve the appealing feature of GMRES and REGMRES—their simplicity. Numerical tests on real data are presented to demonstrate the superiority of the new methods over REGMRES and its variants.