迈向预处理CG-like特征值求解器的真实效率和聚类鲁棒性
Toward genuine efficiency and cluster robustness of preconditioned CG-like eigensolvers
AI总结：
本文提出了一种改进的LOBPCG方法，通过渐近等价的二项递推式和及时修正的搜索方向，提高预处理CG-like特征值求解器的效率和鲁棒性。
摘要：
特征值问题求解器（特征值求解器）的性能取决于多种因素，例如预处理和特征值分布。开发稳定且快速收敛的向量特征值求解器是提高其块状实现整体效率的关键步骤。本文关注的是局部最优块预处理共轭梯度（LOBPCG）方法在Hermitian特征值问题中的应用，并受到两个最近提出的替代方法的启发，这些方法用于其单向量版本LOPCG。这些特征值求解器的共同基础是著名的用于线性系统求解的CG方法。然而，CG搜索方向的最优性无法完全转移到CG-like特征值求解器中。特别是，在计算聚类特征值时，LOPCG及其替代方法会频繁出现延迟，导致收敛行为呈阶梯状，无法用现有估计解释。考虑到这一点，我们构建了一类聚类鲁棒的向量迭代方法，其中LOPCG被渐近等价的二项递推式取代，并通过选择一个远早的迭代作为增强来及时修正搜索方向。新方法显著减少了所需步骤的数量和总计算时间。
英文摘要：
The performance of eigenvalue problem solvers (eigensolvers) depends on various factors such as preconditioning and eigenvalue distribution. Developing stable and rapidly converging vectorwise eigensolvers is a crucial step in improving the overall efficiency of their blockwise implementations. The present paper is concerned with the locally optimal block preconditioned conjugate gradient (LOBPCG) method for Hermitian eigenvalue problems, and motivated by two recently proposed alternatives for its single-vector version LOPCG. A common basis of these eigensolvers is the well-known CG method for linear systems. However, the optimality of CG search directions cannot perfectly be transferred to CG-like eigensolvers. In particular, while computing clustered eigenvalues, LOPCG and its alternatives suffer from frequent delays, leading to a staircase-shaped convergence behavior which cannot be explained by the existing estimates. Keeping this in mind, we construct a class of cluster robust vector iterations where LOPCG is replaced by asymptotically equivalent two-term recurrences and the search directions are timely corrected by selecting a far previous iterate as augmentation. The new approach significantly reduces the number of required steps and the total computational time.
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