A new concept of strong controllability via the Schur complement in adaptive tracking

On the usefulness of persistent excitation in ARX adaptive tracking

The usefulness of persistent excitation is well-known in the control community. Thanks to a persistently excited adaptive tracking control, we show that it is possible to avoid the strong controllability assumption recently proposed in the multidimensional ARX framework. We establish the almost sure convergence for both least squares and weighted least squares estimators of the unknown parameters. A central limit theorem and a law of iterated logarithm are also provided. All this asymptotical analysis is related to the Schur complement of a suitable limiting matrix.

非光滑随机复杂度和流形采样的精确Schur-Sylvester降维

Exact Schur-Sylvester Dimensionality Reductions for Non-Smooth Stochastic Complexity and Manifold Sampling

提出利用块Schur补和Sylvester行列式恒等式，将非光滑估计器NML编码长度计算中投影和体积因子的复杂度从O(N^3)降至O(k^3+N^2k)，并推广到稀疏SVM、弹性网和组Lasso。

A Schur Complement Cheeger Inequality

Cheeger's inequality shows that any undirected graph $G$ with minimum nonzero normalized Laplacian eigenvalue $λ_G$ has a cut with conductance at most $O(\sqrt{λ_G})$. Qualitatively, Cheeger's inequality says that if the relaxation time of a graph is high, there is a cut that certifies this. However, there is a gap in this relationship, as cuts can have conductance as low as $Θ(λ_G)$. To better approximate the relaxation time of a graph, we consider a more general object. Instead of bounding the mixing time with cuts, we bound it with cuts in graphs obtained by Schur complementing out vertices…

Hypocoercivity with Schur complements

We propose an approach to obtaining explicit estimates on the resolvent of hypocoercive operators by using Schur complements, rather than from an exponential decay of the evolution semigroup combined with a time integral. We present applications to Langevin-like dynamics and Fokker--Planck equations, as well as the linear Boltzmann equation (which is also the generator of randomized Hybrid Monte Carlo in molecular dynamics). In particular, we make precise the dependence of the resolvent bounds on the parameters of the dynamics and on the dimension. We also highlight the relationship of our met…

随机Galerkin有限元方法的分层Schur补预条件子

Hierarchical Schur complement preconditioner for the stochastic Galerkin finite element methods

针对随机Galerkin有限元离散产生的大型线性系统，提出一种利用全局矩阵递归分层结构的分层Schur补预条件子，结合内层Krylov迭代和Schur补思想，无需显式形成全局矩阵，并通过数值实验验证性能。

On statistics of bi-orthogonal eigenvectors in real and complex Ginibre ensembles: combining partial Schur decomposition with supersymmetry

We suggest a method of studying the joint probability density (JPD) of an eigenvalue and the associated 'non-orthogonality overlap factor' (also known as the 'eigenvalue condition number') of the left and right eigenvectors for non-selfadjoint Gaussian random matrices of size $N\times N$. First we derive the general finite $N$ expression for the JPD of a real eigenvalue $λ$ and the associated non-orthogonality factor in the real Ginibre ensemble, and then analyze its 'bulk' and 'edge' scaling limits. The ensuing distribution is maximally heavy-tailed, so that all integer moments beyond normali…