论文标题
$ 2 \ times 2 $ block-diagonal预处理的注释
A note on $2\times 2$ block-diagonal preconditioning
论文作者
论文摘要
对于2x2块矩阵，众所周知，具有精确的Schur补体（反逆）的块三角形或块-LDU预处理在最多两次迭代中收敛于定点或最小溶性方法。同样，对于零（2,2）块的鞍点矩阵，块 - 障碍前的预处理最多可以在三个迭代中收敛于最小值的方法，尽管它们可能会在定次迭代方面发挥作用。但是，对于具有确切的Schur补体的非辅助点矩阵和块 - diagonal预处理会发生什么？本说明证明，将最小的残基方法应用于一般的2x2块矩阵，该矩阵用块 - diagonal的预处理（包括精确的Schur补体）进行了预处理，不会（一定）（一定）在固定数量的迭代中收敛。此外，构建了示例，其中（i）具有确切的schur补体的块 - diagonal预处理不比矩阵的对角块块构成比块 - 二基因的预处理的速度，而（ii）通过近似schur互补作为相应的块状块细胞构图快速地汇总。本文通过讨论中性粒子传输中的一些实际应用，引入了一种算法，其中块三角形或块-LDU的预处理优于块 - 基准，而第二个算法则在速度和简单性方面具有优越性。
For 2x2 block matrices, it is well-known that block-triangular or block-LDU preconditioners with an exact Schur complement (inverse) converge in at most two iterations for fixed-point or minimal-residual methods. Similarly, for saddle-point matrices with a zero (2,2)-block, block-diagonal preconditioners converge in at most three iterations for minimal-residual methods, although they may diverge for fixed-point iterations. But, what happens for non-saddle-point matrices and block-diagonal preconditioners with an exact Schur complement? This note proves that minimal-residual methods applied to general 2x2 block matrices, preconditioned with a block-diagonal preconditioner, including an exact Schur complement, do not (necessarily) converge in a fixed number of iterations. Furthermore, examples are constructed where (i) block-diagonal preconditioning with an exact Schur complement converges no faster than block-diagonal preconditioning using diagonal blocks of the matrix, and (ii) block-diagonal preconditioning with an approximate Schur complement converges as fast as the corresponding block-triangular preconditioning. The paper concludes by discussing some practical applications in neutral-particle transport, introducing one algorithm where block-triangular or block-LDU preconditioning are superior to block-diagonal, and a second algorithm where block-diagonal preconditioning is superior both in speed and simplicity.
