We consider the solution of large stiff systems of ordinary differential equations with explicit exponential Runge--Kutta integrators. These problems arise from semi-discretized semi-linear parabolic partial differential equations on continuous domains or on inherently discrete graph domains. A series of results reduces the requirement of computing linear combinations of $\varphi$-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 Krylov subspace iteration numbers to obtain a desired tolerance increase 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 rational Krylov iteration numbers, which enable a near-linear scaling of the runtime with respect to the problem size.

翻译：我们考虑使用显式指数龙格-库塔积分器求解大型刚性常微分方程组。这些问题源于连续域或固有离散图域上半离散化的半线性抛物型偏微分方程。一系列研究成果将指数积分器中计算φ函数线性组合的需求，简化为对少量矩阵指数作用于特定向量的近似。现有最先进的计算方法采用自适应规模的多项式Krylov子空间来完成此任务，其缺点在于：随着离散线性微分算子（如问题规模）谱半径增大，为达到期望容差所需的Krylov子空间迭代次数会急剧增加。我们提出了一种利用有理Krylov子空间方法的新方案，该方法具有更优的近似特性。针对单个时间点上矩阵指数作用于向量的有理Krylov近似，我们证明了一种新颖的后验误差估计，该估计允许采用与现有多项式Krylov技术相似的自适应方法。我们讨论了极点选择问题，以及通过直接求解器和预处理迭代求解器高效求解所产生的移位线性系统序列的方法。数值实验表明，当离散线性微分算子的谱半径足够大时，我们的方法优于现有技术。其关键在于有理Krylov迭代次数近似恒定，这使得运行时随问题规模呈近线性扩展。