Krylov解的延迟重正交化低同步Gram-Schmidt算法
Low-Synch Gram-Schmidt with Delayed Reorthogonalization for Krylov Solvers
摘要：
Krylov迭代法的并行强标度在很大程度上取决于每次迭代所需的全局约简的数量。GMRES和Krylov-Schur算法采用Arnoldi算法来处理非对称矩阵。底层的正交化方案是左视的，一次处理一列。因此，每次迭代至少需要一次全局约简。用于生成Krylov-Schur算法的正交Krylov基向量的传统算法是经典的Gram Schmidt应用两次再正交化（CGS 2），每步需要三次全局约简。CGS 2的一个新的变种，只需要一个减少每次迭代应用到Arnoldi-QR迭代。给出了求非对称矩阵特征值对的强标度结果。Hernandez等人提出了一个初步的尝试，以获得一个类似的算法（一个减少每个Arnoldi迭代与一个强大的正交化计划）。（2007年）。与我们的方法不同，他们的方法对特征值不是前向稳定的。
英文摘要：
The parallel strong-scaling of Krylov iterative methods is largely determined by the number of global reductions required at each iteration. The GMRES and Krylov-Schur algorithms employ the Arnoldi algorithm for nonsymmetric matrices. The underlying orthogonalization scheme is left-looking and processes one column at a time. Thus, at least one global reduction is required per iteration. The traditional algorithm for generating the orthogonal Krylov basis vectors for the Krylov-Schur algorithm is classical Gram Schmidt applied twice with reorthogonalization (CGS2), requiring three global reductions per step. A new variant of CGS2 that requires only one reduction per iteration is applied to the Arnoldi-QR iteration. Strong-scaling results are presented for finding eigenvalue-pairs of nonsymmetric matrices. A preliminary attempt to derive a similar algorithm (one reduction per Arnoldi iteration with a robust orthogonalization scheme) was presented by Hernandez et al.(2007). Unlike our approach, their method is not forward stable for eigenvalues.
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