SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 744-770, March 2024.
Abstract. We consider the solution of large stiff systems of ODEs with explicit exponential Runge–Kutta integrators. These problems arise from semidiscretized semilinear parabolic PDEs on continuous domains or on inherently discrete graph domains. A series of results reduces the requirement of computing linear combinations of [math]-functions in exponential integrators to the approximation of the action of a smaller number of matrix exponentials on certain vectors. State-of-the-art computational methods use polynomial Krylov subspaces of adaptive size for this task. They have the drawback that the required number of Krylov subspace iterations to obtain a desired tolerance increases drastically with the spectral radius of the discrete linear differential operator, e.g., the problem size. We present an approach that leverages rational Krylov subspace methods promising superior approximation qualities. We prove a novel a posteriori error estimate of rational Krylov approximations to the action of the matrix exponential on vectors for single time points, which allows for an adaptive approach similar to existing polynomial Krylov techniques. We discuss pole selection and the efficient solution of the arising sequences of shifted linear systems by direct and preconditioned iterative solvers. Numerical experiments show that our method outperforms the state of the art for sufficiently large spectral radii of the discrete linear differential operators. The key to this are approximately constant numbers of rational Krylov iterations, which enable a near-linear scaling of the runtime with respect to the problem size.

中文翻译：

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指数龙格-库塔积分器的自适应有理 Krylov 方法

《SIAM 矩阵分析与应用杂志》，第 45 卷，第 1 期，第 744-770 页，2024 年 3 月。
摘要。我们考虑使用显式指数龙格-库塔积分器求解大型刚性 ODE 系统。这些问题是由连续域或固有离散图域上的半离散半线性抛物线偏微分方程引起的。一系列结果将指数积分器中计算数学函数的线性组合的要求降低到较小数量的矩阵指数对某些向量的作用的近似值。最先进的计算方法使用自适应大小的多项式 Krylov 子空间来完成此任务。它们的缺点是，获得所需容差所需的克雷洛夫子空间迭代次数随着离散线性微分算子的谱半径（例如问题大小）而急剧增加。我们提出了一种利用理性克雷洛夫子空间方法的方法，有望实现卓越的近似质量。我们证明了一种新颖的有理 Krylov 近似的后验误差估计，该近似对单个时间点的矩阵指数对向量的作用而言，它允许采用类似于现有多项式 Krylov 技术的自适应方法。我们讨论极点选择以及通过直接和预处理迭代求解器对移位线性系统的出现序列进行有效求解。数值实验表明，对于离散线性微分算子的足够大的谱半径，我们的方法优于现有技术。其关键是近似恒定数量的理性 Krylov 迭代，这使得运行时间相对于问题大小能够实现近线性缩放。