Parallel Numerical Processing of Linear Systems with Irregularly Sparse Coefficient Matrix

研究了非结构网格离散化所产生的大型线性方程组的并行求解问题。虽然预条件共轭梯度法（PCG）适用于正对称方程组，但其有效性主要取决于预条件对收敛速度的加速和预条件的并行性。本文提出了一种利用几何结构自动生成多重网格和求解椭圆型偏微分方程非正则有限元方程组的算法。我们发现，虽然这种方法是有效的齐次问题，收敛速度不够快的非齐次问题。对于非齐次问题，提出了一种同时利用代数信息的自动半收敛方法。我们已经表明，我们的方法给出了几乎相同的性能的ICCG（不完全乔列斯基共轭梯度）方法。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。对于非均匀性大于等于100的问题，我们的方法不如ICCG。我们希望继续我们的研究对AMG，代数多重网格方法，及其应用作为一个预处理。

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.