Abstract: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 use of the sketched residual norm as stopping criterion as well as the general competitiveness of sketched Krylov methods for ODEs in comparison to other Krylov-based methods.
From: Marcel Schweitzer [view email]
[v1]
Mon, 20 Oct 2025 13:42:13 UTC (422 KB)
[v2]
Thu, 12 Mar 2026 10:05:41 UTC (516 KB)
[v3]
Thu, 25 Jun 2026 10:54:34 UTC (499 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。