使用块张量列车格式计算高维中的极值特征值

关于随机对称矩阵极值特征值的比较定理

Comparison theorems for the extreme eigenvalues of a random symmetric matrix

本文建立了一种随机对称矩阵和的极大规模特征值比较定理，强化了此前结果，并应用于谱图理论、量子信息理论等领域，首次完整证明了稀疏随机降维映射的单射性质。

与聚集特征值相对应的 Laplace 特征函数的稳定计算

Stable Computation of Laplacian Eigenfunctions Corresponding to Clustered Eigenvalues

针对区域扰动导致 Laplace 特征值紧密聚集时特征函数难以精确计算的问题，本文利用特征值的形状差商提出一种稳定计算方法。

高维极端特征值问题：低秩张量参数化与优化

High-dimensional extreme eigenvalue problems: low-rank tensor parametrization and optimization

针对高维极端特征值问题，本文提出基于张量列格式的低秩参数化与流形优化方法，显著降低计算成本，并在多个数值实验中达到全空间求解器精度且具备鲁棒性。

对偶Hermitian矩阵的特征值及其在编队控制中的应用

Eigenvalues of Dual Hermitian Matrices with Application in Formation Control

本文提出补充矩阵方法计算对偶Hermitian矩阵特征值，并将其与单位增益图理论结合，为多智能体编队控制中的相对构型可行性问题提供跨学科求解途径。

Fast Toeplitz eigenvalue computations, joining interpolation-extrapolation matrix-less algorithms and simple-loop theory

Under appropriate technical assumptions, the simple-loop theory allows to deduce various types of asymptotic expansions for the eigenvalues of Toeplitz matrices generated by a function $f$. Independently and under the milder hypothesis that $f$ is even and monotonic over $[0,π]$, matrix-less algorithms have been developed for the fast eigenvalue computation of large Toeplitz matrices, within a linear complexity in the matrix order: behind the high efficiency of such algorithms there are the expansions predicted by the simple-loop theory, combined with the extrapolation idea. Here we focus our …

On the extreme eigenvalues and asymptotic conditioning of a class of Toeplitz matrix-sequences arising from fractional problems

The analysis of the spectral features of a Toeplitz matrix-sequence $\left\{T_{n}(f)\right\}_{n\in\mathbb N}$, generated by a symbol $f\in L^1([-π,π])$, real-valued almost everywhere (a.e.), has been provided in great detail in the last century, as well as the study of the conditioning, when $f$ is nonnegative a.e. Here we consider a novel type of problem arising in the numerical approximation of distributed-order fractional differential equations (FDEs), where the matrices under consideration take the form \[ \mathcal{T}_{n}=c_0T_{n}(f_0)+c_{1} h^h T_{n}(f_{1})+c_{2} h^{2h} T_{n}(f_{2})+\cdot…

Eig-PIELM：一种用于高效本征分析的无网格方法，结合物理指导的极限学习机

Eig-PIELM: A Mesh-Free Approach for Efficient Eigen-Analysis with Physics-Informed Extreme Learning Machines

Eig-PIELM通过无网格方法结合物理指导的极限学习机，高效求解线性本征值问题，同时获得本征值和模态形状。

Subspace method for multiparameter-eigenvalue problems based on tensor-train representations

In this paper we solve $m$-parameter eigenvalue problems ($m$EPs), with $m$ any natural number by representing the problem using Tensor-Trains (TT) and designing a method based on this format. $m$EPs typically arise when separation of variables is applied to separable boundary value problems. Often, methods for solving $m$EP are restricted to $m = 3$, due to the fact that, to the best of our knowledge, no available solvers exist for $m>3$ and reasonable size of the involved matrices. In this paper, we prove that computing the eigenvalues of a $m$EP can be recast into computing the eigenvalues …

样本协方差矩阵上特征值算法的普遍性

Universality for eigenvalue algorithms on sample covariance matrices

研究随机正定样本协方差矩阵上幂/逆幂方法和QR特征值算法的迭代次数，证明其收敛性及计算极值特征值的复杂度估计。

一种计算少数稀疏近似特征向量的迭代类Jacobi算法

An iterative Jacobi-like algorithm to compute a few sparse approximate eigenvectors

本文提出一种扩展自Jacobi方法的迭代算法，用于计算对称矩阵的少数极值特征对及其稀疏近似，在稀疏性与精度间实现权衡，并证明了其线性收敛性及在图傅里叶变换和图像分类中的有效性。

超大规模特征值问题中的并行进化计算

Parallel Evolutionary Computation in Very Large Scale Eigenvalue Problems

针对超大规模矩阵特征值问题，提出一种融合进化计算、并行编程和随机搜索的新方法，初步实验验证了其高效性和实用性。