A block MINRES algorithm based on the band Lanczos method
A block MINRES algorithm based on the band Lanczos method
复制标题
基于带Lanczos方法的分块MINRES算法
DOI：
10.1007/s11075-014-9907-z
复制
发表时间：
2013
影响因子：
2.1
通讯作者：
Kirk M. Soodhalter
中科院分区：
文献类型：
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作者：
Kirk M. Soodhalter
我们为对称不定矩阵开发了块最小残差（MINRES）算法。该版本基于带 Lanczos 方法构建，该方法每次迭代生成块 Krylov 子空间的一个基向量，而不是像块 Lanczos 过程中那样生成整个块。然而，我们修改了该方法，使得最昂贵的操作仍然以块方式执行。使用频带 Lanczos 方法的好处是，可以根据计算中出现的标量值检测故障，从而允许对故障进行直接实施的处理。我们基于 Band Lanczos 过程推导了 MINRES 方法的渐进公式，并给出了一些实现细节。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，以便利用块 Lanczos 进程提供的通信效率。这是下一代超级计算应用背景下的一个重要问题。我们还提出了一种技术，允许我们通过用预先生成的随机向量替换依赖的 Lanczos 向量来保持块大小，该随机向量与所有 Lanczos 向量的正交性得以保持。数值结果说明了一些示例问题的性能。我们提出的实验表明右侧之间的关系如何影响该方法的性能。
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 affect the performance of the method.
DOI：
10.1137/1.9780898718003
发表时间：
2003-05
期刊：
--
影响因子：
--
作者：
Y. Saad
通讯作者：
Y. Saad