矩阵φ函数作用的增广Krylov方法的值域分析
Field-of-values analysis of augmented Krylov methods for matrix $φ$-function actions
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中文总结 AI 辅助
该研究从块三角公式角度重新审视矩阵φ函数作用的Krylov方法，对比KIOPS等算法与Al-Mohy和Liu方法，前者值域因运算向量增长致收敛估计悲观，后者收敛界更有利，研究两者联系以转移收敛界解释性能。
中文摘要 AI 辅助
我们从Al-Mohy和Liu [SIAM J. Sci. Comput., 48 (2026), pp. A726--A747]的块三角公式的角度重新审视用于矩阵φ函数作用线性组合的既定Krylov子空间方法。在诸如KIOPS [J. Comput. Phys., 372 (2018), pp. 236--255]等算法中，使用了一种增广方法，该方法基于评估一个稍大矩阵的指数，该矩阵在其非对角块中包含运算向量，因此其值域可能会随着这些向量大幅增长。Krylov子空间方法的典型收敛估计是通过在值域上对指数的多项式逼近误差进行界定得到的，所以尽管这些方法在实际中表现良好，但只能得到非常悲观的收敛估计。相比之下，Al-Mohy和Liu建立的更大块公式涉及一个值域与运算向量无关的算子，从而导致更有利的收敛界。我们详细阐述了这两种方法如何相互联系，这使我们能够将后者的收敛界转移到前者，从而更好地解释观察到的性能。
英文摘要
We revisit established Krylov subspace methods for linear combinations of matrix $φ$-function actions from the viewpoint of the block triangular formulation of Al-Mohy and Liu [SIAM J. Sci. Comput., 48 (2026), pp. A726--A747]. In algorithms such as KIOPS [J. Comput. Phys., 372 (2018), pp. 236--255], one uses an augmentation approach based on evaluating the exponential of a slightly larger matrix that contains the operant vectors in its off-diagonal block, and its field of values may therefore grow substantially with these vectors. Typical convergence estimates for Krylov subspace methods result from bounding the error of polynomial approximations for the exponential on the field of values, so that only very pessimistic convergence estimates are available for these methods, in spite of their good practical performance. In contrast, the larger block formulation established by Al-Mohy and Liu involves an operator whose field of values is independent of the operant vectors, leading to more favorable convergence bounds. We work out the details of how these two approaches are connected to each other, which allows us to transfer the convergence bounds from the latter to the former, thus better explaining the observed performance.
