通过低秩自适应常微分方程实现矩阵的稳定化
Stabilization of a matrix via a low rank-adaptive ODE
AI总结：
本文针对不稳定结构矩阵的稳定化问题，提出一种基于矩阵常微分方程驻点的两级方法，并利用自适应秩积分器保持低秩结构，从而高效处理高维问题。
摘要：
设 $A$ 为具有给定结构（例如实矩阵、稀疏模式、Toeplitz 结构等）的方阵，并假设它是不稳定的，即至少有一个特征值位于复右半平面。稳定化 $A$ 的问题在于计算一个矩阵 $B$，使得其特征值具有负实部，并且扰动 $Δ=B-A$ 具有最小范数。结构化稳定化进一步要求该扰动保持 $A$ 的结构模式。我们通过一个两级过程来解决这一非凸问题，该过程涉及计算一个矩阵常微分方程的驻点。我们利用该问题潜在的低秩特征，使用一种自适应秩积分器，它严格跟随解的秩。我们在若干数值算例中展示了低秩设置带来的好处，这些算例也使得处理高维问题成为可能。
英文摘要：
Let $A$ be a square matrix with a given structure (e.g. real matrix, sparsity pattern, Toeplitz structure, etc.) and assume that it is unstable, i.e. at least one of its eigenvalues lies in the complex right half-plane. The problem of stabilizing $A$ consists in the computation of a matrix $B$, whose eigenvalues have negative real part and such that the perturbation $Δ=B-A$ has minimal norm. The structured stabilization further requires that the perturbation preserves the structural pattern of $A$. We solve this non-convex problem by a two-level procedure which involves the computation of the stationary points of a matrix ODE. We exploit the low rank underlying features of the problem by using an adaptive-rank integrator that follows slavishly the rank of the solution. We show the benefits derived from the low rank setting in several numerical examples, which also allow to deal with high dimensional problems.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
