Presentations

We propose the new variant of GMRES($le m_{max}$) method. Our algorithm uses a new parameter for the convergence test of the residualnorm. The restart is also performed by this parameter. From the numerical results for the MIMD parallel computer, COMPAQ Beowulf, we show the effectiveness of our proposed algorithm.

It is important to solve several linear systems of equations in lattice gauge computations in quantum chromodynamics or boundary value problems oftime-dependent partial differential equations. We propose the new procedure for solving the shifted linear systems with an effective preconditioning. We call this new procedure using the parallel GMRES method, Concurrent-GMRES method. Numerical experiments are given for the efficiency of the porposed algorithm.

The solution of large-scale nonsymmetric linear systems of equations: Ax=b, continue to be difficult problems, where A is nonsingular coefficient matrix. These linear equations are often solved by iterative methods due to their moderate computational and storage demands. A main algorithm for solving Ax=b is GMRES method. Recently, GMRES(m) method with some automatic restarting techniques are proposed by several researchers. In this talk, we compare some adaptive techniques for determining automatically when to restart. Several numerical experiments for large-scale problems are presented on the parallel computer Origin 2000.

シュールコンプリメントにより，係数行列を縮小させた行列の新しい前処理行列を反復法を用いて計算する提案を行ったものである

ML(k)BiCGSTAB法の計算量を減らす実装方法について述べたものである。

複素演算を回避したDeflated-GMRESの改良版について述べたものである

We focussed on the GMRES method, one of the most widely utilized Krylov subspace methods. The GMRES method minimizes residual norms at each iteration step. We developed procedure that automatically conductsthe early restarting. The effect of the early restarting is to decrease excessive work. The new proposed algorithm with approximate inverse preconditioning is shown to have good numerical properties for large scale problems. Several numerical experiments are given on the parallel computer Origin 2000.

シフト方程式の近似解法として並列計算可能な前処理を提案し，GMRES(m)法とのカップリングによる高速算法を提案した論文である．

GMRES(m) method is one of the major iterative methods for numerically solving large and sparse nonsymmetric problems of the form Ax=b. However, the information of some eigenvectors that compose the approximation disappears and then the good approximate solution cannot be obtained, because of this restart. Recently, in order to improve such a weak point, some algorithms which named MORGAN, DEFLATION and DEFLATED-GMRES algorithm, have been proposed. Those algorithms add the information of eigenvectors that can be obtained in the previous restart frequency. In this paper, we study those algorithms and also compare their performances. We will also propose the new restarting procedure that accelerates the convergence of their algorithms.

疑似残差法の自動リスタートの算法について述べた．

疑似残差法の一般化について、数値例を交えて解説した.