A Stationary Accumulated Projection Method for Linear System of Equations

TL;DR

This paper introduces a novel iterative method for solving linear systems that does not rely on Krylov subspaces, using accumulated projection techniques to improve convergence over traditional methods like GMRES.

Contribution

The paper proposes a new accumulated projection method (SAP) that overcomes limitations of classical Krylov subspace methods and introduces accelerative schemes (MSAP1 and MSAP2) for better convergence.

Findings

SAP method shows improved convergence behavior.

MSAP schemes outperform GMRES and block-Jacobi in experiments.

Numerical results demonstrate the effectiveness of the proposed methods.

Abstract

It is shown in this paper that, almost all current prevalent iterative \mbox{methods} for solving linear system of equations can be classified as what we called extended Krylov subspace methods. In this paper a new type of iterative methods are introduced which do not depend on any Krylov subspaces. This type of methods are based on the so-called accumulated projection technique proposed by authors. It overcomes some shortcomings of classical Row-Projection technique and takes full advantages of the linear system. Comparing with traditional Krylov subspace methods which always depend on the matrix-vector multiplication with some fixed matrix, the newly introduced method (SAP) uses different projection matrices which differ in each step in the iteration process to form an approximate solution. More importantly some particular accelerative schemes (named as MSAP1 and MSAP2) are introduced to improve the convergence of the SAP method. Numerical experiments show some surprisingly improved convergence behavior; some superior experimental behavior of MSAP methods over GMRES and block-Jacobi are demonstrated in some situations.