块对角占优性与对称矩阵的块对角预条件

§1.引言 稀疏线性方程组的求解在科学计算与工程应用中非常重要。在材料模拟与设计、电磁场计算、计算流体力学和核爆数值模拟等领域中经常要求解微分方程,并通过有限元或有限差分与有限体积等方法进行离散,化为非线性方程组或稀疏线性方程组。非线性方程组的求解

BLOCK DIAGONAL DOMINANCE AND BLOCK DIAGONAL PRECONDITIONERS TO SYMMETRIC MATRICES

Based on strictly block diagonal dominance of a general matrix, good properties of block diagonal preconditioners are derived in this paper. Complexity of turning a matrix into another one of strictly block diagonal dominance has been analyzed. In order to avoiding this complex problem, we derive a more relaxed definition named block diagonal dominance. Similar to diagonal preconditioner, block variants are also robust and easy to be parallelized. For the motivation of parallelization, size of each block should be almost- the same. But block diagonal dominance should be high to decrease the number of iterations. For these two objectives, we exploit METIS to partition the graph corresponding to a symmetric matrix. The experiments show that there is strict relationship between block diagonal dominance and the number of iterations. When the number of blocks are appropriate, block diagonal preconditioners derived in this paper have higher efficiency than diagonal ones and are more robust than incomplete decompositions.