论文标题
基于LEJA和KRYLOV的迭代方案的指数集成商的比较
A comparison of Leja- and Krylov-based iterative schemes for Exponential Integrators
论文作者
论文摘要
长期以来，基于Krylov的算法已被首选用于计算指数积分器中出现的矩阵指数和指数状函数。最近，这些指数样函数的作用的直接多项式插值已被证明与Krylov方法具有竞争力。我们分析了最先进的Krylov算法，KIOPS的性能以及LEJA点的多项式插值方法，用于许多指数集成符，以解决各种测试问题，并且具有不同量的刚度。此外，我们研究了迭代方案的性能，该方案结合了Kiops和Leja方法，名为Lekry，该方法比Leja-和Krylov基于某些指数积分器的方法都进行了实质性改进。尽管我们确实为我们在这项研究中考虑的每个指数集成符挑选了一个受欢迎的迭代方案，但我们找不到任何确定的证据，可以选择Kiops或Leja对于不同类别的指数集成商。对于大多数（如果不是全部）所考虑的问题，我们无法确定一个卓越的指数积分器，该集成符的性能比所有其他集成符都更好。但是，我们确实发现该性能显着取决于迭代方案与所考虑的特定指数积分器之间的相互作用。
Krylov-based algorithms have long been preferred to compute the matrix exponential and exponential-like functions appearing in exponential integrators. Of late, direct polynomial interpolation of the action of these exponential-like functions have been shown to be competitive with the Krylov methods. We analyse the performance of the state-of-the-art Krylov algorithm, KIOPS, and the method of polynomial interpolation at Leja points for a number of exponential integrators for various test problems and with varying amounts of stiffness. Additionally, we investigate the performance of an iterative scheme that combines both the KIOPS and Leja approach, named LeKry, that shows substantial improvements over both the Leja- and Krylov-based methods for certain exponential integrators. Whilst we do manage to single out a favoured iterative scheme for each of the exponential integrators that we consider in this study, we do not find any conclusive evidence for preferring either KIOPS or Leja for different classes of exponential integrators. We are unable to identify a superior exponential integrator, one that performs better than all others, for most, if not all of the problems under consideration. We, however, do find that the performance significantly depends on the interplay between the iterative scheme and the specific exponential integrator under consideration.
