A general framework for Krylov ODE residuals with applications to randomized Krylov methods

TL;DR

This work develops a general framework for Krylov ODE residuals that unifies residual-based error monitoring across matrix-function Krylov methods, including randomized sketching. It proves an a posteriori error bound showing that the sketched residual norm controls the true error up to a factor depending on the embedding quality $oldsymbol{
}$, enabling reliable stopping criteria for the sketched Arnoldi method in exponential integration contexts. The framework accommodates rational Krylov and non-standard inner products and integrates practical implementation strategies, such as efficient compressed-matrix updates and residual-time restarting, to boost stability and efficiency. Numerical experiments on large-scale convection-diffusion, Maxwell, and wave-equation models demonstrate the approach’s reliability and competitiveness against state-of-the-art Krylov-based methods, validating the practical impact for high-dimensional ODE simulations.

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 quality of the error estimate as well as the general competitiveness of sketched Krylov methods for ODEs in comparison to other Krylov-based methods.

Figures (4)

• Figure 1: Execution time for 3d convection diffusion with $t = 1$.
• Figure 2: Convergence curves for 3d convection diffusion problem considered in \ref{['subsec:conv-diff']} with $N = 70$ and $t = 1$.
• Figure 3: Convergence curves for photonic crystal problem on $80 \times 80 \times 80$ grid with $t=10$.
• Figure 4: Convergence curves for the vibrating membrane problem.

Theorems & Definitions (11)

• Theorem 3.1
• Proof 1
• Corollary 3.2
• Proof 2
• Example 3.3
• Example 3.4
• Example 3.5
• Example 3.6
• Example 3.7
• Example 3.8