Randomized Krylov subspace methods that employ the sketch-and-solve paradigm to substantially reduce orthogonalization cost have recently shown great promise in speeding up computations for many core linear algebra tasks (e.g., solving linear systems, eigenvalue problems and matrix equations, as well as approximating the action of matrix functions on vectors) whenever a nonsymmetric matrix is involved. An important application that requires approximating the action of matrix functions on vectors is the implementation of exponential integration schemes for ordinary differential equations. In this paper, we specifically analyze randomized Krylov methods from this point of view. In particular, we use the residual of the underlying differential equation to derive a new, reliable a posteriori error estimate that can be used to monitor convergence and decide when to stop the iteration. To do so, we first develop a very general framework for Krylov ODE residuals that unifies existing results, simplifies their derivation and allows extending the concept to a wide variety of methods beyond randomized Arnoldi (e.g., rational Krylov methods, Krylov methods using a non-standard inner product, ...). In addition, we discuss certain aspects regarding the efficient implementation of sketched Krylov methods. Numerical experiments on large-scale ODE models from real-world applications illustrate the quality of the error estimate as well as the general competitiveness of sketched Krylov methods for ODEs in comparison to other Krylov-based methods.

翻译：采用草图求解范式以显著降低正交化成本的随机化Krylov子空间方法，近年来在加速涉及非对称矩阵的核心线性代数任务（如求解线性系统、特征值问题、矩阵方程，以及逼近矩阵函数对向量的作用）计算方面展现出巨大潜力。其中需要逼近矩阵函数对向量作用的重要应用场景，是实现常微分方程的指数积分格式。本文专门从这一视角分析随机化Krylov方法。特别地，我们利用基础微分方程的残差推导出一种新颖可靠的后验误差估计方法，该方法可用于监测收敛性并决定何时停止迭代。为此，我们首先构建了一个非常通用的Krylov常微分方程残差框架，该框架统一了现有结果、简化了推导过程，并将该概念扩展到随机化Arnoldi方法之外的多种方法（如有理Krylov方法、使用非标准内积的Krylov方法等）。此外，我们讨论了草图Krylov方法高效实现的若干关键问题。基于实际应用的大规模常微分方程模型的数值实验表明，该误差估计方法具有优良性能，且草图Krylov方法在常微分方程求解中相较于其他基于Krylov的方法具有普遍竞争力。