Parallel Numerical Processing of Linear Systems with Irregularly Sparse Coefficient Matrix

研究了不规则结构非结构网格系统离散化产生的大规模线性系统的并行求解问题。预条件共轭梯度法(PCG)虽然适用于正对称方程，但其有效性主要取决于预条件的收敛速度和预条件的并行化程度。本文提出了一种利用几何结构自动生成多重网格和用不规则有限元方法求解椭圆型偏微分方程组的算法。我们发现，虽然这种方法对齐次问题是有效的，但对于非齐次问题，收敛速度不够快。对于非齐次问题，我们提出了一种同时利用代数信息的自动半归约方法。我们已经证明，我们的方法与ICCG(不完全Cholesky共轭梯度)方法具有几乎相同的性能。由于ICCG不能并行化，我们相信我们的方法可以应用于实际问题。对于100个或更多的非齐次问题，我们的方法不如ICCG。我们希望继续研究AMG，代数多重网格法，并将其应用于预条件。

Parallel solution of large-scale linear systems which arise from the discretization of unstructured grid systems with irregular structures is studied. Although preconditioned conjugate gradient (PCG) methods are applicable to positive symmetric equations, the effectiveness of the PCG critically depends on the acceleration of convergence by the preconditioning and the parallelizability of the precontitioning. In this study, an algorithms to generated automatically generate the miltigrid using geometrical structures and to solve the systems of equations given by irregular finite element method for elliptic partial differential equations proposed. We found that although this method is effective for homogeneous problems, the convergence is not fast enough for problems with inhomogeneity. We proposed a kind of automatica semi-coursening method using an algebraic information at the same time for inhomogeneous problems. We have shown that our method gives almost the same performance as the ICCG (Incomplete Cholesky Conjugate Gradient) method. Since the ICCG cannot be parallelized, we believe our method can be applied to practical problems. For problems with the inhomogeneity of 100 or more, our method is inferior to the ICCG.We would like to continue our research toward the AMG, algebraic multi-grid method, and its application as a preconditioning.