子空间夹角、Ritz值和图拉普拉斯谱变化的主化关系

Majorization bounds for Ritz values of self-adjoint matrices

A priori, a posteriori, and mixed type upper bounds for the absolute change in Ritz values of self-adjoint matrices in terms of submajorization relations are obtained. Some of our results prove recent conjectures by Knyazev, Argentati, and Zhu, which extend several known results for one dimensional subspaces to arbitrary subspaces. In addition, we improve Nakatsukasa's version of the $\tan Θ$ theorem of Davis and Kahan. As a consequence, we obtain new quadratic a posteriori bounds for the absolute change in Ritz values.

混合类型的Rayleigh-Ritz主导误差界

Rayleigh-Ritz majorization error bounds of the mixed type

本文基于向量残差范数和向量夹角，推导了混合类型矩阵Rayleigh商绝对变化的主导误差界，并利用主子空间的奇异值和主角关系，建立了新的特征值误差界。

Rayleigh-Ritz 主化误差界及其在有限元方法中的应用

Rayleigh-Ritz majorization error bounds with applications to FEM

本文针对 Hermitian 算子 A 的 Rayleigh-Ritz 方法，提出了关于 A-不变子空间与任意子空间之间主角的主化误差界，并推广到相对误差和乘积误差，最后应用于有限元方法。

Deep Ritz method for the spectral fractional Laplacian equation using the Caffarelli-Silvestre extension

In this paper, we propose a novel method for solving high-dimensional spectral fractional Laplacian equations. Using the Caffarelli-Silvestre extension, the $d$-dimensional spectral fractional equation is reformulated as a regular partial differential equation of dimension $d+1$. We transform the extended equation as a minimal Ritz energy functional problem and search for its minimizer in a special class of deep neural networks. Moreover, based on the approximation property of networks, we establish estimates on the error made by the deep Ritz method. Numerical results are reported to demonstr…

0<s≤1/2时积分分数阶拉普拉斯算子的Ritz投影的几乎L^∞稳定性

Almost $L^\infty$-stability of the Ritz projection for the integral fractional Laplacian with $0<s\leq \tfrac12$

该研究证明了0<s≤1/2时，满足一致外球条件的有界Lipschitz域上，分段线性有限元空间的\tilde{H}^s投影在最大范数下几乎稳定，且网格域无需与原域重合。

Distributed solution of Laplacian eigenvalue problems

The purpose of this article is to approximately compute the eigenvalues of the symmetric Dirichlet Laplacian within an interval $(0,Λ)$. A novel domain decomposition Ritz method, partition of unity condensed pole interpolation method, is proposed. This method can be used in distributed computing environments where communication is expensive, e.g., in clusters running on cloud computing services or networked workstations. The Ritz space is obtained from local subspaces consistent with a decomposition of the domain into subdomains. These local subspaces are constructed independently of each othe…