Complex moment-based methods for differential eigenvalue problems
Complex moment-based methods for differential eigenvalue problems
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DOI：
10.1007/s11075-022-01456-y
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发表时间：
2022-05
影响因子：
2.1
通讯作者：
A. Imakura;K. Morikuni;Akitoshi Takayasu
中科院分区：
文献类型：
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作者：
A. Imakura;K. Morikuni;Akitoshi Takayasu
本文研究了特征值在复平面上某一区域内的微分特征值问题的部分特征对的计算。最近，基于“求解然后离散化”的范例，FEAST方法的操作模拟已被提出用于DEPs而不离散化系数操作。与传统的“离散化然后求解”方法相比，FEAST的算子模拟具有更高的精度;然而，它涉及求解大量的常微分方程（ODE）。在本文中，为了减少计算成本，我们提出了操作类似的樱杉浦型复杂的矩为DEP使用高阶复数矩的特征值求解器，并分析了所提出的方法的误差界。我们发现，要解决的常微分方程的数量可以减少的复杂的时刻的程度的一个因素，而不降低精度，这是由数值结果验证。数值结果表明，所提出的方法是超过5倍的速度相比，运营商模拟FEAST的几个DEP，同时保持几乎相同的高精度。本研究将推动求解离散问题的“求解-离散化”范式，并有助于在实际应用中更快、更准确地求解离散问题。
This paper considers computing partial eigenpairs of differential eigenvalue problems (DEPs) such that eigenvalues are in a certain region on the complex plane. Recently, based on a “solve-then-discretize” paradigm, an operator analogue of the FEAST method has been proposed for DEPs without discretization of the coefficient operators. Compared to conventional “discretize-then-solve” approaches that discretize the operators and solve the resulting matrix problem, the operator analogue of FEAST exhibits much higher accuracy; however, it involves solving a large number of ordinary differential equations (ODEs). In this paper, to reduce the computational costs, we propose operation analogues of Sakurai–Sugiura-type complex moment-based eigensolvers for DEPs using higher-order complex moments and analyze the error bound of the proposed methods. We show that the number of ODEs to be solved can be reduced by a factor of the degree of complex moments without degrading accuracy, which is verified by numerical results. Numerical results demonstrate that the proposed methods are over five times faster compared with the operator analogue of FEAST for several DEPs while maintaining almost the same high accuracy. This study is expected to promote the “solve-then-discretize” paradigm for solving DEPs and contribute to faster and more accurate solutions in real-world applications.