In this work, we investigate a block Jacobi-Davidson (J-D) variant suitable for sparse symmetric eigenproblems where a substantial number of extremal eigenvalues are desired (e.g., ground-state real-space quantum chemistry). Most J-D algorithm variations tend to slow down as the number of desired eigenpairs increases due to frequent orthogonalization against a growing list of solved eigenvectors. In our specification of block J-D, all of the steps of the algorithm are performed in clusters, including the linear solves, which allows us to greatly reduce computational effort with blocked matrix-vector multiplies. In addition, we move orthogonalization against locked eigenvectors and working eigenvectors outside of the inner loop but retain the single Ritz vector projection corresponding to the index of the correction vector. Furthermore, we minimize the computational effort by constraining the working subspace to the current vectors being updated and the latest set of corresponding correction vectors. Finally, we incorporate accuracy thresholds based on the precision required by the Fermi-Dirac distribution. The net result is a significant reduction in the computational effort against most previous block J-D implementations, especially as the number of wanted eigenpairs grows. We compare our approach with another robust implementation of block J-D (JDQMR) and the state-of-the-art Chebyshev filter subspace (CheFSI) method for various real-space density functional theory systems. Versus CheFSI, for first-row elements, our method yields competitive timings for valence-only systems and 4-6× speedups for all-electron systems with up to 10× reduced matrix-vector multiplies. For all-electron calculations on larger elements (e.g., gold) where the wanted spectrum is quite narrow compared to the full spectrum, we observe 60× speedup with 200× fewer matrix-vector multiples vs. CheFSI.

中文翻译：

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块Jacobi-Davidson的强大变体，用于提取大量特征对：在基于网格的实空间密度泛函理论中的应用

在这项工作中，我们研究了适合于稀疏对称本征问题的块Jacobi-Davidson（JD）变体，在该问题中，需要大量极值特征值（例如，基态实空间量子化学）。大多数JD算法的变化趋向于随着所需特征对数目的增加而减慢，这是由于频繁对解决的特征向量列表进行正交而导致的。在我们的块JD规范中，算法的所有步骤都是在集群中执行的，包括线性求解，这使我们能够通过块矩阵向量乘法大大减少计算量。另外，我们将内锁外的锁定特征向量和工作特征向量移至正交化，但保留与校正向量的索引相对应的单个Ritz向量投影。此外，我们通过将工作子空间限制在要更新的当前向量和相应的校正向量的最新集合，从而最大程度地减少了计算工作量。最后，我们根据Fermi-Dirac分布所需的精度并入精度阈值。最终结果是，与大多数以前的块JD实现相比，计算工作量显着减少，尤其是随着所需特征对数量的增加而增加。我们将我们的方法与块JD（JDQMR）的另一个健壮实现和用于各种实际空间密度泛函理论系统的最新Chebyshev滤波器子空间（CheFSI）方法进行了比较。与CheFSI相比，对于第一行元素，我们的方法可为纯价系统提供竞争性时序，而对于全电子系统则可产生4-6倍加速，矩阵向量乘数最多可减少10倍。