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了解领域论文（摘要翻译）
1. 液晶动力学 Beris-Edwards 系统的全离散有限元方法的收敛性
Convergence of a fully discrete finite element method for the Beris-Edwards system of liquid crystal dynamics
作者: Gonzalo A. Benavides, Franziska Weber
类别: math.NA, math.AP
发布日期: 2026-09-23
arXiv ID: 2609.28444v1
链接: http://arxiv.org/abs/2609.28444v1
摘要翻译
中文标题: 液晶动力学 Beris-Edwards 系统的全离散有限元方法的收敛性
摘要翻译: 我们提出并分析了一种用于向列相液晶动力学的 Beris-Edwards 系统的全离散有限元格式。在该系统中，不可压缩 Navier-Stokes 方程与 Landau-de Gennes Q-张量的梯度流耦合。该格式将时间上的线性隐式、形式二阶精度的向后微分公式与用于不可压缩约束的增量 Chorin 投影步、空间中的协调有限元以及用于非线性体势的质量集中不变能量二次化方法相结合。每个时间步需要求解一个线性系统和一个 Poisson 问题。我们证明该格式唯一可解，保持离散 Q-张量和分子场的对称性和无迹结构，并且对时间步长没有任何限制地满足离散能量定律。我们的主要结果是：当网格尺寸 $h$ 和时间步长 $Δt$ 在满足 $h^{2} = o(Δt)$ 的条件下趋于零时，近似解沿子序列收敛到 Beris-Edwards 系统的弱解。收敛性证明处理了两个困难：投影方法产生两个速度近似，其中只有一个在 $L^2(0,T;H^1_0(Ω))$ 中一致有界；并且耦合项 $\mathcal{H}\nabla Q$ 需要 $\nabla Q$ 的强收敛，我们是从 $\mathcal{H}$ 方程的结构中得到这一点，而不是从任何离散 $H^2$ 界中得到。二维数值实验显示出空间和时间上近似二阶收敛，并再现了 $+1$ 点缺陷分裂为两个 $+1/2$ 缺陷，以及由恒定压力梯度诱导的斯格明子的输运和变形。
2. Vlasov 方程的二阶保结构与保正性凸限制方法
A second-order structure- and positivity-preserving convex limiting method for the Vlasov equations
作者: Katharina Kormann, Murtazo Nazarov, Junjie Wen
类别: math.NA
发布日期: 2026-09-23
arXiv ID: 2609.28412v1
链接: http://arxiv.org/abs/2609.28412v1
摘要翻译
中文标题: Vlasov 方程的二阶保结构与保正性凸限制方法
摘要翻译: 本文引入了一种新的二阶、保正有限元方法，用于 Vlasov 方程，采用凸限制算法。该方法采用强稳定性保持（SSP）Runge-Kutta 时间积分以及相空间网格的张量积构造，以实现高效的高维实现。凸限制算法将稳健的一阶保正图黏性方法与高阶基于残差的黏性稳定化相结合，从而获得高阶保正格式。文中提出了适用于诸如 Vlasov 系统等高维问题的新型一阶和高阶方法。此外，我们针对 Maxwell 方程提出一种散度清理技术，以确保电磁场的散度约束得到满足。数值实验用于证明所提方法的精度和鲁棒性。
3. 单纯形上离散de Rham复形的高阶双正交函数*
High order biorthogonal functions for the discrete de Rham complex on simplices*
作者: Tim Haubold, Sven Beuchler, Joachim Schöberl
类别: math.NA
发布日期: 2026-09-23
arXiv ID: 2609.28345v1
链接: http://arxiv.org/abs/2609.28345v1
摘要翻译
中文标题: 单纯形上离散de Rham复形的高阶双正交函数*
摘要翻译: 众所周知，在hp-FEM中，基函数的选择极大地影响了获得近似解的稳定性和计算成本。对于二维和三维空间中的单纯形单元，由具有不同权重的Jacobi多项式构建的张量积型基函数在条件数和稀疏性方面具有最优性质。在本文中，我们为Nédélec空间以及Raviart-Thomas和Brezzi-Douglas-Marini空间构造了这样的高阶基，修改了Zaglmayr的现有定义，使得这些函数属于第一类Nédélec空间和Raviart-Thomas空间。这些基的设计使得第二类Nédélec空间和BDM空间分别是各自其他空间的扩展。在本文的第二部分，我们引入了$H(\text{div}, Ω)$的双正交基函数，延续了先前对$H^1(Ω)$和$H(\text{curl}, Ω)$的研究。这些函数可以以Jacobi多项式张量积之和的闭式形式表达，从而允许快速计算$L^2$-投影。
4. phi-FD：用于由水平集函数定义的几何区域的二阶有限差分格式——Neumann 情形
phi-FD, a second order finite difference scheme for geometries defined by a level-set function: the Neumann case
作者: Michel Duprez, Vanessa Lleras, Alexei Lozinski, Vincent Vigon, Lisl Weynans
类别: math.NA
发布日期: 2026-09-23
arXiv ID: 2609.28313v1
链接: http://arxiv.org/abs/2609.28313v1
摘要翻译
中文标题: phi-FD：用于由水平集函数定义的几何区域的二阶有限差分格式——Neumann 情形
摘要翻译: 我们将 phi-FD 有限差分格式推广到 Neumann 边界条件，该格式处理在笛卡尔网格上由水平集函数描述的区域。该条件通过水平集和一个辅助场在边界附近被放宽；通过将放宽后的条件写在边界节点及其最近的内部节点处，辅助场被消去。该格式仅使用内部节点和水平集：不使用 ghost 值、通量重构或 cut-cell 体积。我们证明了最优相容性，并在非凸区域上展示了 L2、H1 和 Linf 范数下的二阶收敛性。两个基本预防措施——对边界方程进行单独归一化，以及为耦合内部节点的选择制定规则——使条件数为 O(h^-2)，且独立于最小的 cut cell，而无需任何稳定项。
5. 多面体网格上的离散 Sobolev 与 Trudinger 不等式
Discrete Sobolev and Trudinger inequalities on polyhedral meshes
作者: Maxime Herda, Ariane Trescases, Antoine Zurek
类别: math.NA
发布日期: 2026-09-23
arXiv ID: 2609.28152v1
链接: http://arxiv.org/abs/2609.28152v1
摘要翻译
中文标题: 多面体网格上的离散 Sobolev 与 Trudinger 不等式
摘要翻译: 我们重新审视多面体网格上分片常数函数在有界区域上的离散 Sobolev 不等式及相关不等式的推导。我们的结果改进了 Sobolev 常数对 Lebesgue 指数的依赖，并使我们能够恢复连续情形的渐近性。作为推论，我们推导出 Trudinger 不等式的一个离散版本及其变体。我们将结果推广到破碎 Sobolev 空间，并讨论这些新的或精细化的离散泛函不等式在非协调数值方法数值分析中的意义。
6. 一种高效的守恒且保正的空间零维电池模型
An efficient 0D-space conservative and positivity preserving battery model
作者: S. Clain, M. C. Lucas, M. T. Malheiro, C. M. Costa
类别: math.NA
发布日期: 2026-09-23
arXiv ID: 2609.28121v1
链接: http://arxiv.org/abs/2609.28121v1
摘要翻译
中文标题: 一种高效的守恒且保正的空间零维电池模型
摘要翻译: 嵌入式软件旨在实时提供电池单体的主要特性，以管理和优化电池使用。完整的多维模型在就地可用的低计算资源下难以处理。一类空间零维模型由 P2D Doyle-Fuller-Newman 系统导出，并附加了关于函数在空间中形状的假设；它在保持主要特性随时间不变的同时，大幅降低了复杂度。我们提出一个基于 P2D 系统构建的守恒、保正模型，该模型基于 Butler-Volmer 源项在每个子域（阳极、隔膜、阴极）中为常数这一论断。我们给出了数学上严格的构造，以得到该简化模型，并设计了求解该系统的格式。我们强调，该模型保证了守恒性质以及浓度的正性。特别地，我们显式得到了扩散限制电流密度，它表示在仍保持阳离子浓度正性的同时所能使用的最大电流。
7. Kirchhoff 型方程的守恒三层格式
Conservative Three-Layer Schemes for Kirchhoff-Type Equations
作者: Felix Krumbiegel, Jemal Rogava, Andreas Rupp, Zurab Vashakidze
类别: math.NA
发布日期: 2026-09-23
arXiv ID: 2609.27707v1
链接: http://arxiv.org/abs/2609.27707v1
摘要翻译
中文标题: Kirchhoff 型方程的守恒三层格式
摘要翻译: 我们研究二维空间中一类 Kirchhoff 型非线性积分微分方程，其系数允许依赖于时间；针对相应的初边值问题，我们构造了时间上的守恒离散格式。我们考虑两个 Crank–Nicolson 型的对称三层格式，一个是局部线性的，另一个是真正非线性的，并证明它们各自保持常系数齐次问题总机械能的离散类似量。对于非线性格式，我们通过直接使用离散能量建立离散解及其离散速度的一致先验界，而不借助非线性离散 Grönwall 不等式；其中的常数仍随最终时间指数增长；并且我们证明了解及其一阶时间导数的中心差分近似在时间上均具有局部二阶收敛性。每个时间层上出现的非线性系统通过不动点迭代求解：给定满足先验界的前两个层上的迭代值，该方程恰有一个解，且只要时间步长足够小，迭代就以几何速率收敛到该解。由于这些先验界是指标局部的，将它们与该单步求解器交替使用，即可逐步构造轨迹；对时变系数亦然，且是在这些界成立的局部区间上。数值实验在空间离散不贡献误差的情形下进行，展示了离散不变量的守恒性，证实了时间上的二阶收敛，并验证了不动点迭代的几何收敛性。
8. 带 Coriolis 源项的二维双曲系统的半离散 Active Flux 方法
Semi-discrete Active Flux method for two-dimensional hyperbolic systems with Coriolis source terms
作者: Nikhil Manoj, Wasilij Barsukow, Christian Klingenberg
类别: math.NA
发布日期: 2026-09-23
arXiv ID: 2609.27555v1
链接: http://arxiv.org/abs/2609.27555v1
摘要翻译
中文标题: 带 Coriolis 源项的二维双曲系统的半离散 Active Flux 方法
摘要翻译: 我们研究应用于带 Coriolis 源项的二维线性与非线性浅水方程的笛卡尔网格上的半离散 Active Flux 数值方法。这些双曲系统具有由地转平衡支配的非平凡定常解。在此情形下，我们分析半离散 Active Flux 格式的保持定常性的性质。为对线性系统做到这一点，我们对半离散方法应用离散空间 Fourier 变换，并分析所得的演化矩阵。我们证明其离散核是地转平衡的一个非平凡离散化，这意味着该方法保持一个离散地转平衡态。这是 Active Flux 的固有特征，并且无需任何额外修改即可实现。数值实验证实了该方法对线性系统维持地转定常态的能力。此外，我们数值研究了 Active Flux 方法对非线性浅水系统的性能。
9. 通过残差最小化的稳定时间推进：瞬态抛物问题的有限元与神经网络近似
Stable time-stepping via residual minimization: finite-element and neural-network approximations for transient parabolic problems
作者: Judit Muñoz-Matute, Sergio Rojas
类别: math.NA
发布日期: 2026-09-23
arXiv ID: 2609.27548v1
链接: http://arxiv.org/abs/2609.27548v1
摘要翻译
中文标题: 通过残差最小化的稳定时间推进：瞬态抛物问题的有限元与神经网络近似
摘要翻译: 我们提出一个用于瞬态强制变分问题的时间推进最小残差（MinRes）框架，适用于有限元与神经网络试探近似。对于 Backward Euler 格式，我们在每个时间层上，以由稳态检验空间范数诱导、并按时间步长缩放的对偶范数来度量残差。该选择给出稳定性估计，控制一个离散抛物能量量，直至相差一个时间离散化缺陷项。我们发展了协调与破碎检验格式，并将分析具体应用于扩散-对流-反应问题。对于有限维检验空间，我们通过用一个补充贡献来补足可计算离散残差，从而推导出全离散可靠性估计；该补充贡献考虑了检验空间无法分辨的残差分量。我们随后将该框架推广到神经网络试探类，并得到协调与破碎多项式检验空间的可计算残差分解。数值实验证实了预期的有限元收敛率，考察了全局时空神经网络近似中检验空间加密的影响，并展示了对具有移动局部特征的瞬态问题进行残差驱动的空间加密。
10. 球面上轴对称核的形变方法：球谐耦合与正则性
A Deformation Approach to Axially Symmetric Kernels on the Sphere: Harmonic Coupling and Regularity
作者: Janin Jäger, Fabian Stallbauer
类别: math.NA
发布日期: 2026-09-23
arXiv ID: 2609.27419v1
链接: http://arxiv.org/abs/2609.27419v1
摘要翻译
中文标题: 球面上轴对称核的形变方法：球谐耦合与正则性
摘要翻译: 本文构造球面上的轴对称核。我们首先取一个各向同性核，并通过一个特定形变将其变换为轴对称核。对于一般的形变，未必能获得显式的球谐表示。对于我们的形变，可以用原各向同性核的展开系数得到变形后核的这种表示。我们通过显式描述该形变对球谐基的影响来推导这一表示。由此，推导出耦合系数的显式有限和公式，并证明对于每个固定输入模态，这些系数以由我们的形变参数控制的速率几何衰减。我们进一步指出，如果各向同性核的原生空间与某个 Sobolev 空间范数等价，那么变形后的核具有相同的原生空间，并具有等价范数。在数值实验中，我们表明当底层随机场具有轴对称协方差时，我们的形变可以改进预测。
11. 面向椭圆界面问题的混合迭代深度 Ritz 方法
A Hybrid Iterative Deep Ritz Method for Elliptic Interface Problems
作者: Tianhao Hu, Bangti Jin, Fengru Wang, Yifeng Xu
类别: math.NA, cs.LG
发布日期: 2026-09-23
arXiv ID: 2609.27325v1
链接: http://arxiv.org/abs/2609.27325v1
摘要翻译
中文标题: 面向椭圆界面问题的混合迭代深度 Ritz 方法
摘要翻译: 本文针对一类二阶椭圆算子界面问题，提出了一种混合迭代深度 Ritz 方法（H-IDRM）。该方法基于该问题的一种新混合形式，并涉及求解一系列凸极小化问题。我们采用水平集神经网络架构，其特征是界面的水平集表示，以适应该问题解与通量的分片光滑性。该方法仅涉及体积表示，而不是界面上的对偶配对，并避免了对于复杂界面几何而言不方便的显式界面采样。此外，我们给出了该方法的分析，包括由神经网络逼近、Monte Carlo 逼近、迭代格式和惩罚参数引起的误差。数值实验表明，在高维区域、复杂界面几何和较低子区域正则性的问题上，H-IDRM 优于现有的神经求解器。
12. 基于物理信息神经网络的面向气候感知数字孪生的稀疏观测大气热力预报
Sparse-Observation Atmospheric Thermal Forecasting with Physics-Informed Neural Networks for Climate-Aware Digital Twins
作者: Tannaz Goodarzvand Chegini, Elyas Shivanian, Behzad Karimi, Faraz Dadgostari
类别: cs.AI, math.AP, math.NA
发布日期: 2026-09-23
arXiv ID: 2609.27290v1
链接: http://arxiv.org/abs/2609.27290v1
摘要翻译
中文标题: 基于物理信息神经网络的面向气候感知数字孪生的稀疏观测大气热力预报
摘要翻译: 短期大气温度预报对于支撑气候感知数字孪生系统是必要的，但此类预报必须在热力观测不完整的区域生成。本研究评估了一个用于位温预报的物理信息神经网络，该网络受气压坐标热力学平流-源方程以及一个非绝热源闭合关系约束，该闭合关系由前12小时时段拟合得到，并在未来时刻训练之前冻结。利用三个气压层上的逐小时ERA5再分析资料，将该模型作为条件后报进行评估，提前时间为1、2、3小时，并与持续性、局部趋势以及两个匹配的神经网络基线进行比较；其中一个基线接收与PINN相同的未来气象强迫，从而有助于区分物理约束与对未来强迫的获取。在俄克拉何马州开发案例中，相对于最强基线的平均RMSE改进从1小时的8.1%增长到3小时的23.8%；在观测密度扫描降至候选位置的5%时，这一3小时优势仍保持在14.6–16.9%，且没有证据表明更低密度会改善性能。在将固定协议迁移至具有三种虚拟观测布局的阿拉巴马州高温事件时，3小时改进范围为19.7-24.4%，并在各起报时刻均一致胜出。一项并行的蒙大拿州压力测试中，固定气压层与复杂地形相交，导致3小时性能下降约17.5%，从而识别出该方案与地形相关的适用性限制。总之，这些结果表明，物理约束的收益随预报时效增长，在严重观测稀疏下依然存在，并可跨区域迁移，但受限于固定垂直坐标表示在复杂地形上的有效性；这些证据与气候感知预报和数字孪生系统中的物理约束组件相关。
13. 多组分反应流离散化中的逐层离散证书
Layer-wise discrete certificates in multicomponent reacting-flow discretizations
作者: Jingchao Zhang
类别: math.NA
发布日期: 2026-09-23
arXiv ID: 2609.27272v1
链接: http://arxiv.org/abs/2609.27272v1
摘要翻译
中文标题: 多组分反应流离散化中的逐层离散证书
摘要翻译: 多组分反应流离散化必须守恒总组分质量、保持质量分数为正，并耗散一个凸混合势。然而，这些性质在求解器的不同阶段受到控制，在一个阶段建立的证书未必能延续到下一阶段。我们针对混合平均扩散、非正交有限体积重构以及保正 Patankar 时间积分研究这一兼容性问题。该分析是等温的，使用给定密度，并使用合成细致平衡反应网络进行化学测试。在面级，熵变量跳跃被证明等于一个对数平均 Gibbs Hessian 精确作用于质量分数跳跃的结果。要求单纯形切空间上的质量修正项具有守恒性和熵中性，则得到唯一逐分量正的面组分以及精确的两点熵证书。算术权重面准则也由生产的中性空气/碳数据和准中性双极性空气对照所满足，但面级证书不会自动延伸到后续的离散化层。在所测试原型中，加权最小二乘非正交修正可以将一个耗散的两点算子转化为一个反耗散的多点算子。同样，Patankar 重缩放可以在增加自由能的同时保持正性和守恒性；一个显式有限步有理反例在不依赖系综统计的情况下确立了这一失败。一个全离散双单元例子进一步表明，正的面熵裕量并不保证全离散更新的正性。因此，守恒性、正性和自由能耗散是各层特有的性质，并且需要不同的离散证书。
不相关论文（基本信息）
1. Hutch#：最优非自适应 Frobenius 范数估计
Hutch#: Optimal non-adaptive Frobenius norm estimation
作者: Tyler Chen, Diana Halikias, Christopher Musco, David Persson
类别: math.NA, cs.DS
发布日期: 2026-09-23
arXiv ID: 2609.28472v1
链接: http://arxiv.org/abs/2609.28472v1
摘要: The Girard–Hutchinson estimator provides an extremely simple randomized estimate of the Frobenius norm of a matrix $A$ that can only be accessed implicitly via matrix-vector products. In particular, if $Ω$ is a random Gaussian matrix with $r = O(1/\varepsilon^2)$ columns, than $\frac{1}{r}\|AΩ\|_F^2$ provides a $(1\pm \varepsilon)$ multiplicative approximation to $\|A\|_F^2$ with high probability. In this work, we introduce a closely related estimator, given by \begin{align*} {\frac{1}{r}|AΩ|_F^2 + \frac{1}{r}|Ψ^T A|_F^2 - \frac{1}{r^2}|Ψ^T AΩ|_F^2}, \end{align*} where $Ψ$ is a second, independent random Gaussian matrix with $r$ columns. We prove that this estimator yields a $(1\pm\varepsilon)$ multiplicative approximation to $\|A\|_F^2$ when $r = O(1/\varepsilon)$, a quadratic improvement over Girard–Hutchinson. This dependence on $\varepsilon$ is optimal. Our method, which we call Hutch# (pronounced ``Hutch sharp’’), matches the complexity of the Hutch++ algorithm [Meyer, Musco, Musco, Woodruff, 2021]. However, unlike Hutch++, Hutch# uses only \textit{non-adaptive} matrix-vector products with $A$ and $A^T$ and requires no orthogonalization or other adaptive linear algebra steps. Thus, Hutch# combines the simplicity of the Girard–Hutchinson estimator and the optimal query complexity of Hutch++.
2. 由 Epstein zeta 函数的高阶导数计算各向异性奇异和
Computation of anisotropic singular sums from high-order derivatives of Epstein zeta functions
作者: Andreas A. Buchheit, Jonathan K. Busse
类别: math.NA, cond-mat.str-el, math.NT
发布日期: 2026-09-23
arXiv ID: 2609.28282v1
链接: http://arxiv.org/abs/2609.28282v1
摘要: The precise and efficient evaluation of large-scale lattice sums involving power-law kernels is a fundamental computational problem in the simulation of classical and quantum systems with long-range interactions. While methods for spatially isotropic kernels, some based on Epstein zeta functions, have advanced considerably in recent years, the anisotropic case has lagged behind, despite its broad relevance to both fundamental and effective interactions such as the dipole interaction in magnetic materials. In this work, we solve this issue by defining and analyzing anisotropic Epstein zeta functions for which we derive stably computable representations obtained from wave vector derivatives of lattice sums over isotropic interaction kernels. These functions find direct application in the analytical and numerical study of anisotropically interacting lattice systems. Going further, they provide the correction term in an exact equivalence between discrete lattices and their continuous analogs in a recent generalization of the classical Euler-Maclaurin summation formula to lattices and summands involving power-law kernels. Their connection to high-order derivatives of zeta functions can be used to improve convergence rates of numerical algorithms, for instance in micromagnetics, or to provide rapidly convergent expansions suitable for precomputations of generalized zeta functions. We derive a stably computable representation of anisotropic Epstein zeta functions, including the possibility for analytically removing Rayleigh–Wood singularities, and we develop a numerical algorithm for their stable evaluation for any lattice, power-law decay exponent and anisotropy order. We benchmark the algorithm against closed-form identities, direct summation and multi-precision results, obtaining machine precision across various lattices, power-law exponents, and anisotropy orders.
3. 耦合 Fokker-Planck-Darcy 系统的贝叶斯统计反问题
Bayesian statistical inverse problems for a coupled Fokker-Planck-Darcy system
作者: Grigorios A. Pavliotis, Andrew M. Stuart, Andrea Zanoni
类别: math.ST, math.AP, math.NA
发布日期: 2026-09-23
arXiv ID: 2609.28242v1
链接: http://arxiv.org/abs/2609.28242v1
摘要: We study the nonparametric statistical inverse problem of recovering the space-dependent permittivity in a coupled Fokker-Planck-Darcy system from discrete, noisy observations of the Fokker-Planck solution. We consider the coupled parabolic-elliptic system on a bounded domain with the physically natural no-flux boundary condition for the Fokker-Planck equation and homogeneous Dirichlet boundary conditions for the Darcy equation. The inverse problem is indirect, since the unknown coefficient enters only through the elliptic equation and is observed only through its effect on the density in the parabolic equation. We first develop the analytical theory of the forward problem required for the statistical analysis, establishing well-posedness and a priori bounds uniform over the admissible set of permittivities. We then derive two stability estimates for the inverse problem, namely a Lipschitz-type forward estimate and a generalized backward estimate. Placing a rescaled Gaussian process prior on the log-shifted permittivity, we show that the posterior contracts around the truth at an explicit polynomial rate in the number of observations, with the posterior mean converging at the same rate. Numerical experiments in one and two dimensions, using preconditioned Crank-Nicolson and ensemble Kalman filter algorithms, complement our theoretical results.
4. 一种 Lloyd 稳定化的 Voronoi 粒子方法
A Lloyd-stabilized Voronoi particle method
作者: Bruno Després, Borjan Geshkovski
类别: math.NA
发布日期: 2026-09-23
arXiv ID: 2609.28135v1
链接: http://arxiv.org/abs/2609.28135v1
摘要: We study the stability and consistency of Lloyd’s algorithm used in combination with the Lagrangian transport of a density on a Voronoï tessellation. We show that relaxation rates as strong as $O(h^{-1/2})$, where $h$ is the mesh size, give convergence to the continuity equation in Wasserstein distance at the rate $O(h^{1/4})$. Thus the mesh can be kept regular by the correction alone, without the remeshing that Lagrangian methods usually require. An application is proposed for the compressible Euler equations.
5. Transformer 键值缓存的张量分解：谱结构与格式比较
Tensor Decomposition of Transformer Key-Value Caches: Spectral Structure and Format Comparison
作者: Rahul Krishnan, Volker Schulz
类别: math.NA, cs.CL, cs.LG
发布日期: 2026-09-23
arXiv ID: 2609.28029v1
链接: http://arxiv.org/abs/2609.28029v1
摘要: The key-value (KV) cache of autoregressive transformers can be viewed as a fourth-order tensor spanning attention heads, tokens, features, and grouped layers. We measure the singular-value spectra of all four mode unfoldings on Mistral-7B-v0.3 and LLaMA-2-13B and compare four standard tensor decompositions: Tucker, CP, tensor train, and t-SVD, at matched storage. The spectra partition the four axes into two classes. The token and feature modes carry low-rank structure, particularly for keys. The head and layer modes are nearly full-rank and resist compression at any practical error level. Among the four decompositions, Tucker achieves the lowest reconstruction error at every compression ratio from $2\times$ to $5\times$, because it can leave the full-rank modes untouched. Comparisons with two-dimensional unfolding baselines show that the preferred representation differs between keys and values: 2D methods achieve lower key error, while four-way Tucker achieves lower value error at matched storage. A mode-pinning theorem certifies the full-rank preservation from the measured spectra alone. Two further spectral properties affect the compressible modes without touching the full-rank ones: values reach a higher error floor than keys at every ratio, and post-RoPE keys lose $41\%$ - $64\%$ of their pre-RoPE compressibility on both models.
6. 一类低参数正交矩阵的黎曼结构与优化
Riemannian Structure and Optimization for a Class of Low-Parametric Orthogonal Matrices
作者: Ali Aliev, Maxim Rakhuba
类别: cs.LG, cs.AI, math.DG, math.NA
发布日期: 2026-09-23
arXiv ID: 2609.27982v1
链接: http://arxiv.org/abs/2609.27982v1
摘要: In this paper, we are concerned with matrices formed by block-diagonal factors interleaved with fixed permutations – a flexible family of structured matrices. This class has recently drawn interest in deep learning architectures for its balanced expressivity-efficiency trade-off, yet efficient computational strategies for working with it remain to be found. We approach this problem through Riemannian geometry and examine under what conditions this class admits a smooth manifold structure. For the practically important case of orthogonal two-factor matrices, we derive the essential Riemannian tools and propose efficient algorithms for their implementation. The algorithms leverage automatic differentiation, support parameter sharing within each factor, and avoid explicit dense matrix construction. We test them within the Riemannian optimization framework on the best matrix approximation problem and for parameter-efficient fine-tuning of large language models. Beyond the two-factor setting, we study the geometric and matrix-theoretic properties of factorizations with a larger number of block-diagonal factors.
7. 聚点对谱有限记忆预条件子中舍入误差与敏感性的影响
The Influence of the Cluster Point on Rounding Errors and Sensitivity in the Spectral Limited-Memory Preconditioner
作者: Hisham Elzayyadi, Jemima M. Tabeart
类别: math.NA
发布日期: 2026-09-23
arXiv ID: 2609.27652v1
链接: http://arxiv.org/abs/2609.27652v1
摘要: The spectral limited-memory preconditioner (sLMP) clusters leading eigenvalues of symmetric positive definite matrices to accelerate conjugate gradient (CG) convergence. In practice, the cluster point is often chosen to be unity. In some cases, however, this choice can fail to accelerate convergence relative to unpreconditioned CG, even when highly accurate spectral information is available. Alternative cluster points have been proposed based on exact-arithmetic convergence analysis, but such analysis does not explain this finite-precision behaviour. We study how the cluster point influences two sources of numerical error in sLMP-preconditioned CG. First, we analyse the propagation of floating-point rounding errors during application of the preconditioner and derive computable relative-error bounds. For the dominant subspace (spanned by the eigenvectors associated with the leading eigenvalues of the unpreconditioned system) and its orthogonal complement (spanned by the remaining eigenvectors), these bounds are minimized by a weighted median and a weighted arithmetic mean of the leading eigenvalues, respectively. Our analysis explains why small cluster points can strongly amplify errors in the dominant subspace. Second, we investigate sensitivity to perturbations in the dominant spectral information when constructing the preconditioner. The resulting perturbation bound is minimized by a weighted median of the perturbed dominant eigenvalues, with weights determined by the eigenvector perturbation magnitudes. Numerical experiments on synthetic problems illustrate the predicted rounding-error and sensitivity behaviour. Together, these results show that cluster-point selection in finite precision should account for exact-arithmetic convergence, rounding errors, and inaccuracies in the available spectral information.
8. 型材挤出模具形状优化的高效几何表示策略
Efficient Geometry Representation Strategies for the Shape Optimization of Profile Extrusion Dies
作者: Jana Sasse, Maximilian Esser, Markus Mügge, Stefan Turek
类别: math.NA, cs.GR, math.OC
发布日期: 2026-09-23
arXiv ID: 2609.27602v1
链接: http://arxiv.org/abs/2609.27602v1
摘要: The design of profile extrusion dies remains a challenging task due to the complex rheological behavior of polymer melts and the geometric intricacies of flow channels. Traditional manual optimization approaches, which rely heavily on human experience, are inefficient and often employ unvalidated heuristics. To address these challenges, we present a deterministic and explainable framework for automatic die design based on adjoint-based shape optimization. This approach enables the computation of sensitivities that directly indicate beneficial modifications to the flow channel geometry. A major difficulty in such optimization processes lies in generating boundary-conforming meshes that evolve consistently with changing geometries. To overcome this issue, we employ non-boundary conforming geometry representation methods that eliminate the need for an explicit surface representation along physical boundaries. A dedicated reconstruction technique is developed to recover accurate sensitivity information at the virtual interface between fluid and solid regions. The proposed algorithm is demonstrated on 3D geometries with varying complexity, including realistic extrusion die flow channels. Several objective functionals relevant to industrial applications, such as flow balance at the outflow, are considered. The results highlight significant improvements in performance metrics while maintaining numerical robustness. This work showcases the potential of adjoint-based techniques for automated die design in a domain still largely governed by manual trial-and-error procedures, establishing a foundation for data-efficient, sustainable manufacturing workflows using computational rheology.
9. 计算奇异 M-矩阵平方根的逐分量精确不动点迭代
Component-wise accurate fixed point iterations for computing the square root of a singular M-matrix
作者: Dario Andrea Bini, Bruno Iannazzo, Beatrice Meini, Jie Meng
类别: math.NA
发布日期: 2026-09-23
arXiv ID: 2609.27520v1
链接: http://arxiv.org/abs/2609.27520v1
摘要: We analyze two fixed-point iterations for computing the principal square root of an M-matrix $A$. Although these iterations, with customary initialization, converge sublinearly when $A$ is a singular M-matrix, we show that, under suitable mild conditions on the initial approximation, the convergence is linear. Moreover, we provide component-wise accurate versions of these iterations, which allow us to approximate the principal square root with a component-wise relative error uniformly bounded by a small multiple of the machine precision. Numerical experiments demonstrating the effectiveness of the proposed algorithms for certain classes of problems are presented.
10. 随机强迫的二维 Navier-Stokes 方程的快速平流极限中平稳测度在大尺度上的集中
Concentration of Stationary Measures onto Large Scales in the Fast-Advection Limit of the Stochastically Forced Two-Dimensional Navier-Stokes Equations
作者: Tobias Rohner
类别: physics.flu-dyn, math-ph, math.NA, physics.comp-ph
发布日期: 2026-09-23
arXiv ID: 2609.27512v1
链接: http://arxiv.org/abs/2609.27512v1
摘要: We investigate the fast-advection limit ($\varepsilon \to 0$) of the stochastically forced two-dimensional incompressible Navier-Stokes equations on the torus. Numerical simulations are performed at resolution $N = 128$ for four different forcing shells on both the square torus and a thin torus, with the advection parameter $\varepsilon$ varied over approximately one decade. We find that the stationary distributions become increasingly concentrated on the lowest Fourier modes as $\varepsilon \to 0$, with the energy-to-enstrophy ratio $E/Ω$ and the fraction $R$ of enstrophy in the lowest modes both increasing monotonically. At finite $\varepsilon$ the degree of concentration depends on the forcing shell through the effective spectral value $B_1/B_0$, in agreement with the condensation bound of Sznitman and Widmayer. Linear extrapolation to $\varepsilon = 0$ suggests that the limiting values may be forcing-independent, with $B_1/B_0$ governing only the rate of convergence. The same qualitative behaviour persists on the thin torus.
11. 算子复合信赖域方法的相对原始–对偶间隙证书
Relative Primal–Dual Gap Certificates for Operator-Composite Trust-Region Methods
作者: Harbir Antil
类别: math.OC, math.AP, math.NA
发布日期: 2026-09-23
arXiv ID: 2609.27348v1
链接: http://arxiv.org/abs/2609.27348v1
摘要: We study trust-region minimization of a smooth, possibly nonconvex functional plus a convex functional composed with a bounded linear operator. A relative primal–dual gap condition controls both the error in an approximate proximal-gradient step and its linear-model decrease. Together with a computable absolute stationarity test, it yields a finite Cauchy search, convergence of the proximal stationarity measure to zero, and an $O(\varepsilon^{-2})$ bound on outer trials. The outer analysis allows the linear operator to take values in a Banach space and does not require dual attainment. When the operator takes values in a Hilbert space and the regularizer is finite and Lipschitz, the dual proximal-gradient method produces finite gaps tending to zero, provided the required proximal maps and functional values can be evaluated. We prove $O(j^{-1})$ gap bounds for both recovered and averaged primal candidates and give a sharper bound on the primal error for exactly recovered points. A semilinear elliptic control problem with unsmoothed total-variation regularization and an $L^2$ control cost illustrates the method in the full $H^1$ metric. Across five meshes, outer and state Newton counts remain constant, while interior-point iteration counts vary mildly.
12. 从变分优化到基于流的传输：面向病态数据同化的后验几何正则化
From Variational Optimization to Flow-Based Transport: Posterior-Geometry Regularization for Ill-Conditioned Data Assimilation
作者: Siming Liang, Feng Bao, Hristo G. Chipilski, Peter Jan van Leeuwen, Guannan Zhang
类别: math.NA
发布日期: 2026-09-23
arXiv ID: 2609.27243v1
链接: http://arxiv.org/abs/2609.27243v1
摘要: Variational data assimilation computes a Bayesian update through optimization on a fixed posterior geometry, whose conditioning is determined by the prior covariance and the observation operator. In strongly anisotropic, weakly observed, or highly informative regimes, this geometry can become severely ill conditioned, leading to substantial sensitivity to solver design and preconditioning. This work investigates an alternative computational route based on flow-based transport, using the ensemble score filter as a training-free realization. Rather than repeatedly optimizing over the full posterior geometry, the flow-based update transports the predictive distribution through a sequence of intermediate distributions whose covariance and curvature are progressively regularized before reaching the target posterior. In the linear-Gaussian setting, we characterize this mechanism through the evolution of the diffused posterior covariance and the associated time-dependent curvature, revealing a continuation from a near-isotropic reference geometry to the ill-conditioned posterior geometry targeted by variational assimilation. Numerical stress tests show that this geometric regularization makes the flow-based method substantially less sensitive to severe ill-conditioning than the variational method, while a heterogeneous $10^4$-dimensional Lorenz-96 benchmark demonstrates improved robustness and lower delivered computational cost. These results identify posterior-geometry regularization as a key mechanism distinguishing transport-based from optimization-based data assimilation.
13. KATOsuper：基于灵敏度一致的傅里叶神经算子的代理加速神经拓扑优化
KATOsuper: Surrogate-accelerated neural topology optimization with sensitivity-consistent Fourier neural operators
作者: Shengyu Yan, Jasmin Jelovica
类别: cs.CE, cs.AI, cs.LG, math.NA
发布日期: 2026-09-23
arXiv ID: 2609.27216v1
链接: http://arxiv.org/abs/2609.27216v1
摘要: Topology optimization (TO) remains computationally intensive due to repeated finite element analysis (FEA) evaluations required at each iteration. While neural network-based surrogates offer potential acceleration, existing approaches often suffer from gradient inconsistency between predicted objectives and sensitivities, leading to optimization instability. This work presents KATOsuper, an objective-agnostic framework that couples neural-reparameterized topology optimization with a Sensitivity-Consistent Fourier Neural Operator (SC-FNO). The framework employs the forward_split architecture, which derives deployed sensitivities via automatic differentiation through the predicted objective field and thereby preserves consistency between the predicted objective and the gradient used for optimization. The case studies include three 2D benchmark problems and three 3D structures considering compliance or stress minimization. A physics-informed multi-channel input encoding with Fourier position embedding enables resolution-invariant learning, supporting zero-shot extrapolation beyond the training resolution, with useful performance at moderate scaling factors and topology-preserving exploration at up to 64x without retraining. The framework extends to 3D through KATO3D, featuring novel KANConv3D blocks with learnable B-spline activations. KATOsuper demonstrates 15–110x deployment-time speedup over MATLAB baselines while maintaining competitive optimality, with the clearest gains observed in complex 3D and stress-optimization cases. The insight that sensitivity direction matters more than magnitude enables robust optimization even with approximate physics evaluation, extensible to other differentiable physics-driven design objectives.
14. mKdV 呼吸子的鲁棒性：数值视角
Robustness of mKdV Breathers: A Numerical Perspective
作者: Chandler Haight, Svetlana Roudenko, Diana Son, Kai Yang
类别: nlin.PS, math.AP, math.NA
发布日期: 2026-09-22
arXiv ID: 2609.27080v1
链接: http://arxiv.org/abs/2609.27080v1
摘要: We systematically investigate breather solutions in the modified Korteweg-de Vries (mKdV) equation via numerical simulations. We show that the breather solutions are stable under a variety of perturbations, including amplitude changes, modifications of internal parameters, and perturbations of the nonlinearity of the equation. Our results are not only consistent with the known analytical studies on the stability of breathers in integrable systems, extending them further, but also provide numerical evidence that such breather-type structures can persist and remain stable over the simulated time scales in some non-integrable settings with symmetric potentials.