生成时间: 2026年08月24日 05:07:20

重点关注论文数量: 0

了解领域论文数量: 1

不相关论文数量: 11

了解领域论文（摘要翻译）

1. 弹性梁网络中波传播的杂交化不连续伽辽金方法

A Hybridizable Discontinuous Galerkin Method for Wave Propagation in Elastic Beam Networks

作者: Moritz Hauck, Joseph Holten, Axel Målqvist, Andreas Rupp, Lucia Swoboda

类别: math.NA

发布日期: 2026-08-21

arXiv ID: 2608.21091v1

链接: http://arxiv.org/abs/2608.21091v1

摘要翻译

中文标题: 弹性梁网络中波传播的杂交化不连续伽辽金方法

摘要翻译: 本文研究了弹性波在网络中传播的数值解，该网络由每条边上的弹性动力学方程建模，并通过适当的传输条件在节点处耦合。我们提出并分析了一种杂交化不连续伽辽金方法，该方法利用网络结构将每个时间步的全局问题简化为一个线性系统，其大小仅取决于网络节点数，而与离散化的多项式次数无关。将其与能量守恒的隐式时间离散相结合，我们导出了空间和时间上的最优阶先验误差估计。隐式时间离散避免了由纤维段长度的大变化引起的严重CFL限制。为了有效求解所得通常病态的全局系统，我们引入了一种两级重叠加性Schwarz预处理器。在对网络的适当假设下，我们建立了所得预条件共轭梯度法的均匀收敛性。数值实验证实了理论发现。

不相关论文（基本信息）

1. 现代Hopfield检索动力学的保盆离散化：能量单元、耗散与注意力极限

Basin-Preserving Discretizations of Modern Hopfield Retrieval Dynamics: Energy Cells, Dissipation, and the Attention Limit

作者: Francisco R. Villatoro

类别: math.NA, cs.NE

发布日期: 2026-08-21

arXiv ID: 2608.21304v1

链接: http://arxiv.org/abs/2608.21304v1

摘要: The retrieval dynamics of a modern Hopfield network is the gradient flow of a log-sum-exp energy, while the attention update is its exact difference-of-convex minimization step. We study which time discretizations preserve not only energy decay and equilibria but also basins of attraction. We introduce energy cells, connected components of sublevel sets containing one attractor and no other critical point. Our main theorem shows that every finite energy cell below the escape energy is contained simultaneously in the basin of the continuous flow, every relaxed attention map $Ψ_θ=(1-θ)\,\mathrm{id}+θ\,\mathrm{attention}$ for $0<θ<2$, and implicit Euler throughout its uniqueness regime. A parameter-uniform unit-curvature majorant yields unconditional dissipation and a monotone interpolation of each discrete step. We also derive explicit local contraction bounds near well-separated patterns, with a certified optimal slight overrelaxation; characterize proximal tunneling and overshoot beyond the preservation regimes; compare first-order error constants; establish an order barrier for scalar reparametrizations of the relaxed family; construct a second-order scalar-auxiliary-variable scheme; and extend cell preservation to damped difference-of-convex iterations in Bregman geometry, including a certified overrelaxed window under bounded asymmetry. Nine numerical campaigns test the bounds and failure mechanisms. In two-dimensional basin experiments, all observed disagreements between continuous and discrete retrieval occur above the attractor-specific numerically inferred escape level.

2. 有限精度算术中Lanczos算法的简单稳定性分析

A simple stability analysis of the Lanczos algorithm in finite precision arithmetic

作者: Tyler Chen

类别: math.NA

发布日期: 2026-08-21

arXiv ID: 2608.21268v1

链接: http://arxiv.org/abs/2608.21268v1

摘要: We give a self-contained finite-precision analysis of the symmetric Lanczos algorithm without reorthogonalization. In particular, we derive the perturbed three-term recurrence, Paige’s loss-of-orthogonality identity, containment of all computed Ritz values, and localization of stabilized Ritz values. We then prove a Greenbaum-type backward stability result, exhibiting a nearby problem on which exact Lanczos produces the computed tridiagonal matrix. Our proofs simplify those of Paige and Greenbaum, at the cost of hiding polynomial factors in the iteration count.

3. SO(3)上向量场与微分算子的调和框架

A Harmonic Framework for Vector Fields and Differential Operators on SO(3)

作者: Ralf Hielscher, Erik Wünsche

类别: math.NA

发布日期: 2026-08-21

arXiv ID: 2608.21235v1

链接: http://arxiv.org/abs/2608.21235v1

摘要: We present a comprehensive framework for tangent vector fields and differential operators on the rotation group $\mathrm{SO}(3)$ using harmonic series expansions, which provides a mathematical foundation for their implementation in the crystallographic texture analysis software MTEX. The central idea is to employ the standard left- and right-invariant frames as global orthonormal frames of the tangent bundle, thereby avoiding the numerical instabilities of the classical tangent space basis derived from the Jacobian of the Euler angle parametrization. While the choice of frames is primarily motivated by their geometric and numerical properties, their full potential emerges in the harmonic setting. Representing tangent vector fields through harmonic expansions of their frame components, we derive explicit frequency domain formulas for the gradient, divergence, and curl, allowing these operators to be applied directly to the harmonic coefficients. Moreover, we show that these differential operators preserve harmonic band-limitedness. In addition, the left- and right-invariant representations of tangent vector fields can be transformed into one another directly in the frequency domain, with an increase in harmonic bandwidth of at most one degree.

4. 高等线性代数及其应用——第一部分（面向PDE、机器学习和数据同化的数值线性代数）

Advanced Linear Algebra with Applications - Part I (Numerical linear algebra for PDEs, machine learning, and data assimilation)

作者: Victorita Dolean, Jemima Tabeart

类别: math.NA, cs.LG

发布日期: 2026-08-21

arXiv ID: 2608.21234v1

链接: http://arxiv.org/abs/2608.21234v1

摘要: These lecture notes form the first part of a master’s-level course on advanced numerical linear algebra. Their aim is not only to present the classical algorithms, but to show why the subject has become considerably more central than it was a generation ago. Numerical linear algebra grew up alongside the numerical solution of partial differential equations, and for a long time that is where its large sparse systems came from. Ranking the nodes of a network, assimilating observations into a weather forecast, and fitting a model to a large noisy data set now lead to problems of the same kind: too large to factorise, structured, and accessible only through matrix-vector products. Strikingly few ideas are needed for all of them. Each chapter therefore develops a standard topic and then puts it to work outside its original setting. We treat norms, factorisations, conditioning and floating-point arithmetic; sparse matrices arising from finite differences, from graphs and from machine learning; stationary iterations and the smoothing property; the conjugate gradient and Lanczos methods, with spectral clustering and regularisation by early stopping; Arnoldi and GMRES, with PageRank and large least squares; and finally preconditioning, Schwarz domain decomposition and multigrid. We assume a first course in linear algebra. Every section closes with a summary of what should be retained and every chapter with exercises, several drawn from past examinations. Accompanying Python code reproduces the numerical illustrations.

5. 锚定正则化直接最小二乘法 (ARDLS)：在层次分析法中集成既定优先级算子以进行偏好提取

Anchored Regularized Direct Least Squares (ARDLS): Integrating Established Prioritization Operators for Priority Elicitation in the Analytic Hierarchy Process

作者: Kevin Kam Fung Yuen

类别: math.OC, cs.AI, math.NA

发布日期: 2026-08-21

arXiv ID: 2608.21187v1

链接: http://arxiv.org/abs/2608.21187v1

摘要: Pairwise reciprocal matrices are fundamental to the Analytic Hierarchy Process (AHP), a decision-making model. While the Direct Least Squares (DLS) method provides an intuitive mechanism for deriving priority vectors without complex transformations, the DLS provides multiple solutions. Under high levels of inconsistency, such as cyclic contradictions, this non-convexity yields multiple distinct global minima, resulting in unstable priority rankings that critically depend on initial algorithmic guesses. To overcome this structural deficiency, this paper introduces the Anchored Regularized Direct Least Squares (ARDLS) optimization model. ARDLS integrates uniquely determined established prioritization operators, such as normalization techniques, the Eigenvector method, Singular Value Decomposition, Cosine Maximization, and the Pseudo-Inverse Gram Matrix (the closed-form solution of Weighted Least Squares), as theoretical anchors within a regularization penalty. This integration systematically breaks mathematical symmetries, tilting the optimization landscape to guarantee convergence upon a single, unique global minimum. Comprehensive numerical experiments and simulations validate that the ARDLS framework successfully reduces root mean square error among established priority operators, while guaranteeing strict mathematical uniqueness. The proposed ARDLS may be the ideal alternative for the AHP applied to many application domains.

6. 李雅普诺夫算子对称极大元猜想的一个反例

A counterexample to the symmetric-maximizer conjecture for Lyapunov operators

作者: Daniel Kressner, Bart Vandereycken

类别: math.NA

发布日期: 2026-08-21

arXiv ID: 2608.20875v1

链接: http://arxiv.org/abs/2608.20875v1

摘要: It has been conjectured that the operator norm of the Lyapunov operator induced by the Frobenius norm is always attained at a symmetric matrix. The conjecture is known to hold for all matrices of order at most five. We give an integer matrix of order seven for which the skew-symmetric restricted norm is strictly larger than the symmetric restricted norm. A rational separator and exact-arithmetic certificates establish the strict inequality without relying on floating-point computations. A direct-sum construction yields counterexamples in every order $n \geq 7$; the case $n = 6$ remains open.

7. 无网格方法中的点云质量

Point Cloud Quality for Meshfree Methods

作者: Mohsen Abdolahzadeh, Oleg Davydov, Isabel Michel, Pratik Suchde

类别: math.NA, cs.CE

发布日期: 2026-08-21

arXiv ID: 2608.20872v1

链接: http://arxiv.org/abs/2608.20872v1

摘要: Mesh quality is very well studied and widely used to quantify a good mesh. In contrast, a systematic study of quality of meshfree point clouds is lacking. This gap makes it difficult to substantiate the common claim that generating a good-quality point cloud is easier than generating a good mesh. Various definitions of point cloud quality have been proposed, some of which have theoretical significance for proving convergence and error bounds, while others are used in computational studies. In this work, we compare and contrast existing point cloud quality metrics and introduce a few new ones. We conduct extensive numerical tests with a meshfree collocation method across a wide range of scenarios, including both elliptic and hyperbolic equations, 2D and 3D domains, and variations in parameters of the numerical method. Based on these tests, we assess which quality metrics best correlate with numerical error. Our findings reveal six metrics that consistently serve as reliable indicators of point cloud quality, while also demonstrating that several widely used metrics are poor predictors of accuracy.

8. 随机选主元Cholesky算法的一种新分析

A new analysis of the randomly pivoted Cholesky algorithm

作者: Ethan N. W. Epperly

类别: math.NA, cs.DS, stat.CO

发布日期: 2026-08-21

arXiv ID: 2608.20633v1

链接: http://arxiv.org/abs/2608.20633v1

摘要: The randomly pivoted Cholesky algorithm is one of the leading methods for computing a low-rank approximation to a large positive-semidefinite matrix. However, while it consistently achieves accuracy comparable to or better than competing methods of its type in experiments, its theoretical analysis lags somewhat behind other methods. This paper closes this gap, proving that randomly pivoted Cholesky produces an approximation with expected error within a $1+\varepsilon$ factor of the optimal rank-$r$ approximation in $\mathcal{O}(r/\varepsilon + r\sqrt{\log r})$ steps. This result nearly matches the optimal complexity $Θ(r/\varepsilon)$ for any low-rank approximation method based on a partial Cholesky decomposition (also known as a column Nyström approximation). The paper also presents bounds on the randomly pivoted Cholesky trace and spectral-norm errors that hold with high probability. The mathematical argument is largely due to GPT 5.6-Sol (Pro), with some refinements by the author.

9. 对称不定矩阵的Nyström方法

Nyström method for symmetric indefinite matrices

作者: Yijia Chen, Yuji Nakatsukasa, Anjali Narendran, Taejun Park

类别: math.NA

发布日期: 2026-08-20

arXiv ID: 2608.20531v1

链接: http://arxiv.org/abs/2608.20531v1

摘要: The Nyström method approximates $A\approx A(\,:\,,I)A(I,I)^{\dagger} A(\,:\,,I)^{\top}=CA(I,I)^{\dagger} C^{\top}$, where $C:=A(:,I)\in\mathbb{R}^{n\times r}$ is a column subset matrix of $A$. When applied to symmetric but indefinite matrices, the Nyström method can fail because the core matrix $A(I,I)$ may severely underestimate the eigenvalues of $A$ and may become (nearly) singular. We address this issue by developing and analyzing an algorithm that carefully chooses $\widehat{M}\in\mathbb{R}^{r\times r}$ in place of $A(I,I)^\dagger$ by solving the two-sided sketched least-squares problem $\min_{M}\|X(A-CMC^{\top})X^{\top}\|_F$, where $X\in\mathbb{R}^{t\times n}$ is a random sketch matrix. We study in detail the cases where $X$ is a Gaussian or a leverage score sampling (LSS) matrix, and show that with oversampling $t>r$ the residual $\|A-C\widehat{M}C^{\top}\|_*$ is comparable to $\min_{M}\|A-CMC^{\top}\|_*$. For the Gaussian sketch, we require $t=\mathcal{O}(r)$ samples; for LSS, we show that $t=\mathcal{O}(r \log r)$ samples suffice for the theoretical guarantee, with the LSS approach carrying the advantage that once a set of $t$ row indices is identified, the approximation requires only $t^{2}$ matrix-entry evaluations to find $\widehat{M}$, given $C$. We illustrate our results with synthetic examples and applications to kernel methods.

10. 加权上确界范数下 Fourier-Laguerre 级数的正则化求和

Regularized summation of Fourier-Laguerre series in the weighted sup-norm

作者: S. G. Solodky, Y. A. Volynets

类别: math.NA

发布日期: 2026-08-20

arXiv ID: 2608.20493v1

链接: http://arxiv.org/abs/2608.20493v1

摘要: We consider the problem of recovering functions defined on the half-line from inexact input information. For the proposed regularizing summation methods based on Laguerre polynomials, we analyze their approximation properties on functions from Wiener classes. We establish conditions under which these methods are stable with respect to small perturbations of the input data and order-optimal not only in accuracy in the weighted $\sup$-norm, but also in the number of Fourier-Laguerre coefficients used.

11. 任意维数下 Neumann–Poincaré 算子的最优等周不等式

A sharp isoperimetric inequality for the Neumann–Poincaré operator in every dimension

作者: Matthew J. Colbrook, Siavash Sadeghi

类别: math.AP, math.NA, math.SP

发布日期: 2026-08-20

arXiv ID: 2608.20466v1

链接: http://arxiv.org/abs/2608.20466v1

摘要: Let $Ω\subset\mathbb{R}^d$, $d\ge2$, be a bounded connected domain with boundary of class $C^{1,α}$, where $0<α<1$. For the adjoint Neumann–Poincaré operator $K^*_{\partialΩ}$, normalised so that its distinguished eigenvalue is $1/2$, let $λ_j^+(Ω)$ denote the upper min–max values on the mean-zero energy space. We prove $\sum_{j=1}^{d}λ_j^+(Ω)\ge \frac{d-2}{2}.$ It follows that $λ_1^+(Ω)\ge \frac{d-2}{2d},$ with equality if and only if $Ω$ is a ball. In dimension three this proves the $1/6$-conjecture of Miyanishi and Suzuki. The proof uses the coordinate boundary-charge densities induced by uniform applied fields. Their energy Gram matrix is the perfect-conductor polarization tensor $M_\infty$. Positivity of a $2d\times2d$ Gram matrix yields the endpoint Hashin–Shtrikman inequality $|Ω|\mathrm{Tr}(M_\infty^{-1})\le1,$ and bounds the trace of the compression of $K^*_{\partialΩ}$ to the applied-field space. Equality in the inverse-trace inequality makes the interior Newtonian potential quadratic, so a converse to Newton’s theorem identifies $Ω$ as an ellipsoid. Equality in the spectral estimate also makes the Hessian of this potential isotropic, which forces the ellipsoid to be a ball.