随机化Jacobi-Davidson方法
Randomized Jacobi-Davidson method
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中文总结 AI 辅助
提出随机化Jacobi-Davidson方法，以廉价随机正交化替代经典正交化，在保持局部二次收敛的同时降低计算成本，实验验证其可靠性与经典方法相当。
中文摘要 AI 辅助
Jacobi-Davidson方法是一种广泛使用的子空间方法，用于计算大型稀疏非Hermitian矩阵中离目标最近的少数特征对。与其他子空间方法一样，它对每个新的扩展向量相对于整个搜索基进行正交化，其成本随子空间维数二次增长，并且在分布式环境中，每次迭代都需要全局同步。我们提出了一种随机化Jacobi-Davidson方法，用更廉价的随机化正交化过程替代这种正交化，该过程不需要涉及全维向量的内积。我们证明，在均匀分离条件下，随机化方法保留了经典Jacobi-Davidson方法在非Hermitian问题上的局部二次收敛性。关键步骤是证明随机化修正方程的精确解是任意位移下草图逆迭代的一步，这使我们能够将分析扩展到草图提取的调和与精化变体，这些变体更适合于内部特征值。在真实和合成的非Hermitian特征值问题上的数值实验证实了预测的收敛阶，并表明随机化方法在降低正交化成本的同时，达到了与经典Jacobi-Davidson方法相当的可靠性。
英文摘要
The Jacobi-Davidson method is a widely used subspace method for computing a few eigenpairs of a large, sparse, non-Hermitian matrix closest to a target. Like other subspace methods, it orthogonalizes each new expansion vector against the whole search basis, at a cost that grows quadratically with the subspace dimension and, in a distributed setting, requires a global synchronization at every iteration. We introduce a randomized Jacobi-Davidson method that replaces this orthogonalization with a much cheaper randomized orthogonalization process, which requires no inner products involving full-dimensional vectors. We prove that, under a uniform separation condition, the randomized method retains the local quadratic convergence of classical Jacobi-Davidson for non-Hermitian problems. The key step is showing that an exact solution of the randomized correction equation is one step of sketched inverse iteration for an arbitrary shift, which lets us extend the analysis to harmonic and refined variants of the sketched extraction, better suited to interior eigenvalues. Numerical experiments on real and synthetic non-Hermitian eigenvalue problems confirm the predicted convergence order and show that the randomized method matches the reliability of classical Jacobi-Davidson while reducing orthogonalization cost.
发表机构
- EPF Lausanne（洛桑联邦理工学院）
- PSI Center for Scientific Computing, Theory and Data, PSI（瑞士保罗谢勒研究所科学计算、理论与数据中心）
机构由 AI 辅助整理，请以论文原文为准。
