SIAM Journal on Matrix Analysis and Applications, Volume 42, Issue 2, Page 578-607, January 2021.
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.

中文翻译：

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用于计算 Kreiss 常数和不可控距离的基于快速插值的全局性证书

SIAM 矩阵分析与应用杂志，第 42 卷，第 2 期，第 578-607 页，2021 年 1 月。
我们提出了一种新方法来计算两个实变量中奇异值函数的全局最小值。具体来说，我们提出了新算法来计算矩阵的 Kreiss 常数和线性控制系统的不可控距离，两者均达到任意精度。先前针对这两个量的最先进方法依赖于基于解决大特征值问题的 2D 水平集测试。因此，这些方法代价高昂，即 $\mathcal{O}(n^6)$ 使用密集特征求解器工作，并且在收敛之前通常需要多次测试。已经提出了分而治之的技术，将工作复杂度降低到平均 $\mathcal{O}(n^4)$ 和最坏情况下的 $\mathcal{O}(n^5)$，但是这些变体尽管如此，它们仍然非常昂贵，并且在数值上可能不可靠。相比之下，我们新的基于插值的全局性证书通过构建对某些单变量连续函数的插值近似来执行水平集测试，这些函数相对便宜且数值稳健以进行评估。我们的新方法具有 $\mathcal{O}(kn^3)$ 工作复杂度并使用 $\mathcal{O}(n^2)$ 内存，其中 $k$ 是构建插值所需的函数评估次数. 这种插值过程不仅主要是“令人尴尬的并行”，而且低保真近似值通常足以满足除最终插值之外的所有要求，必须以高精度构建。即使不利用上述并行性，$k$ 也足够小我们的新方法通常比以前的最先进方法快几个数量级。