Fast Interpolation-Based Globality Certificates for Computing Kreiss Constants and the Distance to Uncontrollability

We propose a new approach to computing global minimizers of singular value functions in two real variables. Specifically, we present new algorithms to compute the Kreiss constant of a matrix and the distance to uncontrollability of a linear control system, both to arbitrary accuracy. Previous state-of-the-art methods for these two quantities rely on 2D level-set tests that are based on solving large eigenvalue problems. Consequently, these methods are costly, i.e., $\mathcal{O}(n^6)$ work using dense eigensolvers, and often multiple tests are needed before convergence. Divide-and-conquer techniques have been proposed that reduce the work complexity to $\mathcal{O}(n^4)$ on average and $\mathcal{O}(n^5)$ in the worst case, but these variants are nevertheless still very expensive and can be numerically unreliable. In contrast, our new interpolation-based globality certificates perform level-set tests by building interpolant approximations to certain one-variable continuous functions that are both relatively cheap and numerically robust to evaluate. Our new approach has an $\mathcal{O}(kn^3)$ work complexity and uses $\mathcal{O}(n^2)$ memory, where $k$ is the number of function evaluations necessary to build the interpolants. Not only is this interpolation process mostly “embarrassingly parallel," but also low-fidelity approximations typically suffice for all but the final interpolant, which must be built to high accuracy. Even without taking advantage of the aforementioned parallelism, $k$ is sufficiently small that our new approach is generally orders of magnitude faster than the previous state of the art.

Récupéré en direct depuis OpenAlex et désinversé. Les résumés ne sont pas conservés dans cette base de données : les index inversés représentent 8,6 Go des 9,3 Go de texte de la base, et le serveur dispose de 13 Go libres.