一种预处理混合SVD方法用于准确计算大矩阵的奇异三元组
A Preconditioned Hybrid SVD Method for Computing Accurately Singular Triplets of Large Matrices
AI总结：
本文提出一种预处理混合SVD方法，结合预处理和高效求解器，准确计算大矩阵的最大和最小奇异三元组，通过两阶段元方法提升精度和效率。
摘要：
计算大稀疏矩阵的几个奇异三元组是一项具有挑战性的任务，尤其是当需要高精度的最小奇异值时。最近的努力主要通过Lanczos双对角化方法的变种来解决，但即使对于中等规模矩阵也面临困难。我们提出了一种新的SVD方法，可以利用预处理和任何精心设计的求解器来计算最大和最小奇异三元组。通过混合、两阶段元方法PHSVDS，实现了精度和效率的提升。第一阶段，PHSVDS将正则方程求解到最佳精度。如果需要进一步精度，方法会自动切换到带有扩展矩阵的特征值问题。因此，它结合了两个阶段的优点，即更快的收敛速度和精度。对于扩展矩阵，通过合理利用第一阶段的优质初始猜测和高效实现的细化投影方法，简化了内部特征值的求解。我们还讨论了如何预处理PHSVDS以及如何处理由此产生的某些问题。数值实验展示了该方法的效率和鲁棒性。
英文摘要：
The computation of a few singular triplets of large, sparse matrices is a challenging task, especially when the smallest magnitude singular values are needed in high accuracy. Most recent efforts try to address this problem through variations of the Lanczos bidiagonalization method, but they are still challenged even for medium matrix sizes due to the difficulty of the problem. We propose a novel SVD approach that can take advantage of preconditioning and of any well designed eigensolver to compute both largest and smallest singular triplets. Accuracy and efficiency is achieved through a hybrid, two-stage meta-method, PHSVDS. In the first stage, PHSVDS solves the normal equations up to the best achievable accuracy. If further accuracy is required, the method switches automatically to an eigenvalue problem with the augmented matrix. Thus it combines the advantages of the two stages, faster convergence and accuracy, respectively. For the augmented matrix, solving the interior eigenvalue is facilitated by a proper use of the good initial guesses from the first stage and an efficient implementation of the refined projection method. We also discuss how to precondition PHSVDS and to cope with some issues that arise. Numerical experiments illustrate the efficiency and robustness of the method.
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