论文标题
在低同步中使用混合精度重新定向块经典革兰am-schmidt
Using Mixed Precision in Low-Synchronization Reorthogonalized Block Classical Gram-Schmidt
论文作者
论文摘要
在降低计算成本和通信成本方面，在算法中使用较低的精度可能是有益的。在此的推动下，我们旨在进一步开发和分析迭代方法的混合精度变体。在这项工作中，我们专注于具有重新安装的低同步经典革兰氏式schmidt的块变体，我们称之为BCGSI+LS。我们证明，这种正交方案产生的正交性的丧失可能超过$ o（u）κ（\ Mathcal {x}）$，其中$ u $是单位的圆形，$κ（\ nathcal {x}）$是矩阵的条件数量是正交的，因此我们无法实现这一目标。然后，我们开发了该算法的混合精度变体，称为BCGSI+LS-MP，该算法在计算的某些部分中使用了更高的精度。我们通过实验证明，对于许多具有挑战性的测试问题，我们的混合精度变体成功地将正交性丧失在$ O（u）κ（\ Mathcal {x}）$以下。这表明我们可以实现一个向后稳定的块GMRE算法，该算法仅需要一个迭代中的一个同步。
Using lower precision in algorithms can be beneficial in terms of reducing both computation and communication costs. Motivated by this, we aim to further the state-of-the-art in developing and analyzing mixed precision variants of iterative methods. In this work, we focus on the block variant of low-synchronization classical Gram-Schmidt with reorthogonalization, which we call BCGSI+LS. We demonstrate that the loss of orthogonality produced by this orthogonalization scheme can exceed $O(u)κ(\mathcal{X})$, where $u$ is the unit roundoff and $κ(\mathcal{X})$ is the condition number of the matrix to be orthogonalized, and thus we can not in general expect this to result in a backward stable block GMRES implementation. We then develop a mixed precision variant of this algorithm, called BCGSI+LS-MP, which uses higher precision in certain parts of the computation. We demonstrate experimentally that for a number of challenging test problems, our mixed precision variant successfully maintains a loss of orthogonality below $O(u)κ(\mathcal{X})$. This indicates that we can achieve a backward stable block GMRES algorithm that requires only one synchronization per iteration.
