标题： 求解大规模稀疏线性系统的低秩SPIKE框架及其应用 显示英文标题

标题： Low-Rank SPIKE Framework for Solving Large Sparse Linear Systems with Applications

摘要： 求解线性方程组的 SPIKE 方法族通过块三对角划分提供并行性。通常基于 SPIKE 的求解器应用于带状系统，导致结构化的非对角块，其中非零元素限制在相对较小的子矩阵中，这些子矩阵构成了原始矩阵的带状部分。在这项工作中，研究了基于低秩奇异值分解 (SVD) 的非对角块近似方法。这种方法能够更有效地处理具有大而稀疏带状部分的矩阵。实现了一组灵活的分布式求解器，即 LR-SPIKE 变体，它们适用于广泛的应用场景——从用作改进经典块雅可比预条件器的“黑箱”预条件器，到用作专门的“近似直接求解器”。针对选择的 SuiteSparse 矩阵进行了新预条件器有效性的研究，特别关注来自三维有限元模拟的矩阵。此外，SPIKE 近似的线性方程组求解器还与 FEAST 特征值求解器结合使用，在前者快速收敛和后者允许较宽松的线性方程组收敛条件下，这种组合所需的求解器迭代次数非常少。 显示英文摘要

摘要： The SPIKE family of linear system solvers provides parallelism using a block tridiagonal partitioning. Typically SPIKE-based solvers are applied to banded systems, resulting in structured off-diagonal blocks with non-zeros elements restricted to relatively small submatrices comprising the band of the original matrix. In this work, a low-rank SVD based approximation of the off-diagonal blocks is investigated. This produces a representation which more effectively handles matrices with large, sparse bands. A set of flexible distributed solvers, the LR-SPIKE variants, are implemented. There are applicable to a wide range of applications -- from use as a "black-box" preconditioner which straightforwardly improves upon the classic Block Jacobi preconditioner, to use as a specialized "approximate direct solver." An investigation of the effectiveness of the new preconditioners for a selection of SuiteSparse matrices is performed, particularly focusing on matrices derived from 3D finite element simulations. In addition, the SPIKE approximate linear system solvers are also paired with the FEAST eigenvalue solver, where they are shown to be particularly effective due to the former's rapid convergence, and the latter's acceptance of loose linear system solver convergence, resulting in a combination which requires very few solver iterations.