基于带状Lanczos方法的块MINRES算法
A block MINRES algorithm based on the banded Lanczos method
摘要：
提出了一种求解对称不定矩阵的分块最小残差（MINRES）算法.这个版本是建立在带Lanczos方法，产生一个基向量的块Krylov子空间每次迭代，而不是在块Lanczos过程中的整个块。然而，我们修改了方法，使得最昂贵的操作仍然以块的方式执行。使用带Lanczos方法的好处是，可以从计算中产生的标量值检测故障，从而可以直接实现故障的处理。我们推导出一个渐进的制定的MINRES方法的基础上带Lanczos过程，并给出了一些实施细节。具体来说，对步骤进行简单的重新排序使我们能够在块级别执行许多操作，以利用块Lanczos过程提供的通信效率。这是下一代超级计算应用程序中的一个重要问题。我们还提出了一种技术，使我们能够保持块的大小，通过替换依赖的Lanczos向量与预生成的随机向量，其对所有Lanczos向量的正交性保持。数值结果说明了一些样本问题的性能。我们目前的实验表明，右手边之间的关系可以影响该方法的性能。
英文摘要：
We develop a block minimum residual (MINRES) algorithm for symmetric indefinite matrices. This version is built upon the band Lanczos method that generates one basis vector of the block Krylov subspace per iteration rather than a whole block as in the block Lanczos process. However, we modify the method such that the most expensive operations are still performed in a block fashion. The benefit of using the band Lanczos method is that one can detect breakdowns from scalar values arising in the computation, allowing for a handling of breakdown which is straightforward to implement. We derive a progressive formulation of the MINRES method based on the band Lanczos process and give some implementation details. Specifically, a simple reordering of the steps allows us to perform many of the operations at the block level in order to take advantage of communication efficiencies offered by the block Lanczos process. This is an important concern in the context of next-generation super computing applications. We also present a technique allowing us to maintain the block size by replacing dependent Lanczos vectors with pregenerated random vectors whose orthogonality against all Lanczos vectors is maintained. Numerical results illustrate the performance on some sample problems. We present experiments that show how the relationship between right-hand sides can effect the performance of the method.
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