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了解领域论文（摘要翻译）

1. 超声速与高超声速流动中薄板的保结构降阶模型

Structure-preserving reduced-order models of thin panels in supersonic and hypersonic flow

作者: Miguel A. Aguilo

类别: cs.CE, cs.LG, math.NA

发布日期: 2026-10-22

arXiv ID: 2610.31504v1

链接: http://arxiv.org/abs/2610.31504v1

摘要翻译

中文标题: 超声速与高超声速流动中薄板的保结构降阶模型

摘要翻译: 我们为 AePW-4 高速工作组的两个挑战问题构建保结构降阶模型。这两个问题分别是 Mach 1.92 下的固支薄板 (RC-19) 和 Mach 5.85 下的悬臂板 (HyMAX)。每个案例已发表的数据均不包含时间分辨响应，也不包含重复运行。因此，物理机理必须由模型的形式来承载。一种构造为每个问题给出两个模型。一个 Ritz-Galerkin 参考模型描述该板。一个学习模型保留参考模型的精确势能，拟合其余广义力，并且只训练不能增加能量的耗散。其能量平衡由构造方式保证对任意权重成立。在训练后的模型上，最坏相对残差为 2.2e-15。在 RC-19 上，激波关系式预测板两侧的压比为 1.521。压敏涂料测量得到相同的值。学习模型的三十次运行将平均主频置于测量值 236 Hz 的 1.0% 以内。在一个周期工况和一个混沌工况下，学习模型在全部十六次运行中复现了一个已发表降阶模型的响应状态。它做到这一点，且不包含该模型具有的 113 Hz 谱峰，而测量并未显示该峰。测量位移还可以通过一个额外且遵守能量平衡的力来校正混沌响应。我们在实验的 5 kHz 采样率下用合成数据测试了这一点。状态误差在无噪声时为板厚的 1.25%，在存在 10 微米噪声时为 3.5%。在 HyMAX 上，学习模型在 65 ms 内跟踪参考模型的后缘位移，相对误差为 3.7%。这些结果表明，当可用数据无法训练模型时，其结构承载物理的模型仍能匹配测得的颤振行为。

2. Kirchhoff 型方程的守恒有限元方法

Conservative Finite Element Methods for Kirchhoff-Type Equations

作者: Benjamin Dörich, Felix Krumbiegel, Jemal Rogava, Andreas Rupp, Zurab Vashakidze

类别: math.NA

发布日期: 2026-10-22

arXiv ID: 2610.31487v1

链接: http://arxiv.org/abs/2610.31487v1

摘要翻译

中文标题: Kirchhoff 型方程的守恒有限元方法

摘要翻译: 我们构造并分析了一个全离散格式，用于二维空间中带时间依赖系数的 Kirchhoff 型非线性积分微分方程。该格式将 Crank–Nicolson 型的对称三层时间离散、任意多项式次数 $p$ 的协调有限元方法以及针对非线性项的 Picard 迭代相结合。我们证明了 Picard 迭代是良定义的，并且迭代序列在以离散 Laplacian 的分数次幂范数度量下收敛到精确解的 Ritz 投影。在未平移范数中运行相同的能量论证，并在相同假设下，我们进而得到整体误差在 $L^2$ 中的最优阶 $\mathcal{O} ( h^{p+1} + τ^2 )$ 以及在 $H^1$ 中的 $\mathcal{O} ( h^{p} + τ^2 )$。对于常系数的齐次问题，该格式的精确解还恒等地保持一个离散能量。采用制造解的数值实验再现了预测的时间二阶精度以及 $L^2$ 范数下的完整空间阶 $h^{p+1}$。

3. 用于 Monge–Ampère 的凸平衡势：曲率界与后验评估

Convex Equilibrium Potentials for Monge–Ampère: Curvature Bounds and A Posteriori Evaluation

作者: Edison Vázquez, Yingdong Lu, Tomasz Nowicki

类别: math.NA, cs.LG

发布日期: 2026-10-22

arXiv ID: 2610.31386v1

链接: http://arxiv.org/abs/2610.31386v1

摘要翻译

中文标题: 用于 Monge–Ampère 的凸平衡势：曲率界与后验评估

摘要翻译: 平衡神经网络将其隐藏状态定义为压缩映射的不动点，因此其有效深度是在评估时而非由架构设定的。将其用于完全非线性偏微分方程会带来显式网络可避免的两个问题：二阶导数可能随有效深度增长，并且势函数只能被近似评估。我们研究压缩因子为 $q<1$ 的非负 softplus 平衡网络，这是一类适用于 Monge–Ampère 方程的凸网络类。预解式中的正性抵消表明 Hessian 至多以 $(1-q)^{-2}$ 的速率增长，而不是直接估计给出的 $(1-q)^{-3}$；并且一个显式的单神经元族表明该指数是尖锐的；三阶导数满足尖锐的 $(1-q)^{-3}$ 界。基于残差的后验估计进而控制近似平衡点处的状态、其输入导数，以及在二维情形下的 Hessian 行列式。结合定义域完整覆盖上的区间算术，它们给出 Monge–Ampère 残差的严格一致界，并通过比较论证给出固定训练网络的解误差的严格一致界。在单位圆盘上的两个制造 Dirichlet 问题上，所有六个训练模型均被认证，其残差低于 $0.25$，边界误差低于 $0.01$。自适应停止最多将 float64 评估时间减半，而带认证的自适应评估仍然比固定深度成本更高。

4. 基于图划分的自适应局部本征正交分解及有限维函数空间中参数化序列的可证明误差有界有损压缩

Adaptive Local Proper Orthogonal Decomposition via Graph Partitioning and Provable Error-Bounded Lossy Compression for Parametric Series in Finite-Dimensional Function Spaces

作者: Roberto Nuca, Matteo Parsani, Lisandro Dalcin

类别: math.NA

发布日期: 2026-10-22

arXiv ID: 2610.31375v1

链接: http://arxiv.org/abs/2610.31375v1

摘要翻译

中文标题: 基于图划分的自适应局部本征正交分解及有限维函数空间中参数化序列的可证明误差有界有损压缩

摘要翻译: 我们提出一种空间自适应本征正交分解（POD）方法，用于有限维函数空间中快照数据的误差有界有损压缩。该方法将层次图划分与空间子域上的独立低秩逼近相结合，使局部逼近秩能够适应数据中信息的空间分布。对于在离散内积意义下正交的划分，我们建立了全局平方重构误差到局部贡献的精确分解，并刻画了在固定划分和秩下所得局部 POD 逼近的最优性。这些结果为在子域之间分配给定的全局误差容限并选择最小的可容许局部秩提供了严格机制。我们进一步建立了系数空间归一化均方根误差（NRMSE）的一个充分误差界。一种贪心层次细化算法仅在空间细分能够减少局部低秩因子所需存储且同时保持给定重构误差界时，才接受该空间细分。使用合成数据以及来自二维和三维可压缩流模拟的快照序列对该方法进行了评估。数值结果说明了空间自适应秩和划分相对于全局 POD 和固定空间划分如何改进压缩，特别是当数据表现出空间异质动力学时。

5. 利用衍射波对多边形障碍物的数值重建

Numerical reconstruction of polygonal obstacles by diffracted waves

作者: Xi Chen, Kuangmiao Xiong, Boxi Xu

类别: math.NA, math.AP

发布日期: 2026-10-22

arXiv ID: 2610.31271v1

链接: http://arxiv.org/abs/2610.31271v1

摘要翻译

中文标题: 利用衍射波对多边形障碍物的数值重建

摘要翻译: 我们提出一种用于恢复二维刚性多边形障碍物的数值方法，该方法基于 Bao、Chen、Lu 和 Xiong 的基于衍射的逆散射理论。从点源发射单个脉冲，并在接收曲线上测量所得的波场。在分离出衍射信号后，我们由到达时间恢复可见顶点，并由振幅恢复角点取向和开口角。振幅信息的使用允许在反射阴影区进行局部重建，而在该区域反射波无法到达接收器。振幅还确定了与可见顶点相邻的阴影边的方向，从而实现局部非视线成像。针对凸角、凹角、三角形、五边形和星形障碍物的数值实验证实，可以从衍射信号稳定地恢复边界的可见部分。进一步在单弧 $L_1$ 上带有水平为 $0.2\%$ 的加性噪声的凸角测试证实，尖端和开口仍然可恢复。

6. 具有非线性幂律粘性的旋转 MHD-Boussinesq 流动的有限元逼近

Finite Element Approximation of Rotating MHD-Boussinesq Flows with Nonlinear Power-Law Viscosity

作者: Akash Ashirbad Panda, Evana Islam Sarkar

类别: math.NA

发布日期: 2026-10-22

arXiv ID: 2610.31091v1

链接: http://arxiv.org/abs/2610.31091v1

摘要翻译

中文标题: 具有非线性幂律粘性的旋转 MHD-Boussinesq 流动的有限元逼近

摘要翻译: 我们提出并分析具有非线性幂律（$p$-Laplace）粘性的三维不可压缩旋转 MHD-Boussinesq 系统的一个全离散有限元逼近。对速度和温度采用协调元，对磁感应采用 $\mathbf H(\operatorname{curl})$-协调 Nédélec 元，时间上采用半隐式向后 Euler 格式。该离散保持主要能量结构：Lorentz 项与感应项精确抵消，而 Coriolis 力保持能量中性。由此得到适定性和与网格无关的能量稳定性。非线性幂律扩散带来两个主要分析困难。由于离散速度的一致时间导数估计不可得，强紧性通过时间平移论证和 Simon 紧性准则获得。然后借助基于离散能量 $\limsup$ 不等式的 Minty 论证识别非线性扩散的弱极限。紧性和取极限在整个剪切增稠区域成立，而在 Newtonian 情形以及在三维中高于一个由 Lorentz 力而非对流决定的显式阈值时，可获得真正弱解的识别。我们进一步建立了无需任何网格-时间耦合条件的先验误差估计。三维数值实验证实了分析所预测的稳定性和收敛行为。

7. ADER-DG方法中的刚性源项

Stiff source terms inside ADER-DG method

作者: I. S. Popov, A. P. Popova

类别: physics.comp-ph, math-ph, math.NA, physics.app-ph, physics.flu-dyn

发布日期: 2026-10-22

arXiv ID: 2610.30852v1

链接: http://arxiv.org/abs/2610.30852v1

摘要翻译

中文标题: ADER-DG方法中的刚性源项

摘要翻译: 当用于求解带刚性源项的双曲方程时，采用LST-DG预测器的ADER-DG数值方法表现出IMEX计算格式的特征。随着刚性增加（由参数 $z = ν{Δt}^{n} < 0$ 表征），ADER-DG数值方法的Courant数最大值 $\mathrm{CFL}_{\rm max}$ 增大。在强刚性区域 $|z| \gg 1$，Courant数出现线性增长 $\mathrm{CFL}_{\rm max}(z) \sim |z|$。在小源项区域 $|z| \ll 1$，结果 $\mathrm{CFL}_{\rm max}(z)$ 转化为无源项问题 $\mathrm{CFL}_{\rm max}(0)$ 的已知结果。在过渡区域 $|z| \sim 1$，发现了Courant数 $\mathrm{CFL}_{\rm max}(z)$ 的一个不显著下降区间。在线性情形下，不稳定性恰好在矩阵的某个特征值达到 $λ= -1$ 时被破坏，而与数值解的相位 $θ$ 无关。针对基多项式的任意次数 $N$，构造了刚性问题的ADER-DG数值方法的严格稳定性理论。稳定性条件被简化为求解关于Courant数 $\mathrm{CFL}$ 的多项式方程。获得了ADER-DG方法的Courant数 $\mathrm{CFL}_{\rm max}(N, z)$ 对参数 $z$ 的高精度依赖关系。给出了ADER-DG方法逼近性的严格直接证明。严格推导了任意次数 $N$ 的逼近阶 $p = N+1$。开展了一组数值实验，将ADER-DG方法应用于求解线性平流-反应方程以及双组分反应介质的Euler方程组。这些计算得到的结果很好地验证了理论结果。

8. 球面上 Allen-Cahn 方程的广义 MLS 与 LS-SVR

Generalized MLS and LS-SVR for Allen-Cahn Equations on Spheres

作者: Vahid Mohammadi, Mehdi Dehghan, Stefano De Marchi

类别: math.NA

发布日期: 2026-10-22

arXiv ID: 2610.30714v1

链接: http://arxiv.org/abs/2610.30714v1

摘要翻译

中文标题: 球面上 Allen-Cahn 方程的广义 MLS 与 LS-SVR

摘要翻译: 本文将基于广义移动最小二乘（GMLS）逼近的最小二乘支持向量回归（LS-SVR）推广到球面上偏微分方程的数值求解。我们证明，所提方法以其对偶形式构造了一个基于核的逼近器，并建立了其与相应再生核 Hilbert 空间（RKHS）的联系。此外，我们证明通过所提 LS-SVR 框架求解 PDE 等价于在相应 RKHS 中求解一个平滑问题或核岭回归问题。作为应用，该方法被用于求解球面上的 Allen-Cahn 方程及其守恒对应模型。采用隐式一阶算子分裂格式将每个模型分离为线性与非线性子问题。线性子问题采用向后差分公式离散，并在每个时间步使用由基于 GMLS 的配点格式导出的 LS-SVR 进行逼近，而非线性部分则解析处理。所得格式被证明在每个时间步均具有唯一数值解。对于守恒 Allen-Cahn 模型，引入校正过程以保持质量。数值实验验证了所提框架的精度与有效性。

9. 算子适配多重网格中的学习型光滑：受控精度增益与计算限制

Learned smoothing in operator-adapted multigrid: controlled accuracy gains and computational limits

作者: Kumar Kaushik Ranjan

类别: math.NA

发布日期: 2026-10-22

arXiv ID: 2610.30692v1

链接: http://arxiv.org/abs/2610.30692v1

摘要翻译

中文标题: 算子适配多重网格中的学习型光滑：受控精度增益与计算限制

摘要翻译: 学习型多重网格组件通常是在固定迭代预算下评估的，但这种比较不能区分局部逼近增益与实际求解器效率。我们针对非结构有限元网格上的高对比度椭圆问题研究这一区分。一个学习型 V-cycle 使用算子相关转移算子、Galerkin 粗网格算子、近核收缩和能量范数保护机制；仅其光滑子被训练。该架构表示一类指定的固定多项式光滑子，并且关于载荷为正齐次。在新的导电通道问题上，一次学习型遍历相对于使用四步 Jacobi 预条件共轭梯度光滑的同一循环，将能量误差缩小 5.7-6.1 倍；对于绝缘势垒，相应的因子为 1.38-1.45。在通道问题上的另一项固定输入测试中，对于每个位置和训练初始化，细网格层学习型光滑子在 100 个新输入中的 97-100 个上取得了比其自身输入的五维 Krylov 空间上的精确最小值更小的能量范数误差；这并不确立其优于所有多项式光滑循环。当经典光滑预算增加时，配对的几何平均精度在通道问题上每次光滑调用 8 到 14 步之间、在势垒问题上 5 到 8 步之间超过学习型循环的精度。所测量的 CPU 实现相对于所测试的经典求解器没有计算优势，且载荷变化和浮岛进一步限制其精度。

10. 具有紧致显式单步时间离散的高阶全离散熵稳定 DG 方法

High-Order Fully Discrete Entropy-Stable DG Methods with Compact Explicit Single-Step Time Discretizations

作者: Yuchang Liu, Yan Jiang, Mengping Zhang

类别: math.NA

发布日期: 2026-10-21

arXiv ID: 2610.30535v1

链接: http://arxiv.org/abs/2610.30535v1

摘要翻译

中文标题: 具有紧致显式单步时间离散的高阶全离散熵稳定 DG 方法

摘要翻译: 我们为具有紧致显式单步时间离散的高阶全离散熵稳定间断 Galerkin（DG）方法建立了一个统一框架。在每个时间步结束时，将初步的 DG 多项式向其单元平均缩放，以强制满足强单元熵不等式，同时保持守恒性。该修正可行当且仅当更新后的单元平均满足弱单元熵不等式；而标准的三阶强稳定性保持 Runge–Kutta DG 更新在任意小的正 CFL 数下都可能违反该不等式。对于紧致单步格式，包括紧致 Runge–Kutta、Lax–Wendroff 和 ADER-DG 方法，我们在每个界面上推导了可行数值熵通量的显式区间，并证明该区间在 CFL 条件下非空。将一个相容的高阶目标通量投影到该区间上，可保证弱单元熵不等式，并在光滑解上保持单步高阶精度，包括在非退化极值附近。该限制器还同时强制多个熵不等式成立。我们还建立了一个 Lax–Wendroff 型定理。针对标量守恒律和可压缩 Euler 方程的数值实验表明，该方法具有高阶精度、多熵控制和强激波鲁棒性。

11. 双曲守恒律的熵有界紧致模板Runge-Kutta间断Galerkin方法

Entropy bounded compact stencil Runge-Kutta discontinuous Galerkin method for hyperbolic conservation laws

作者: Deepak Kumar, Pratima Rai, Arvind Patel

类别: math.NA, math-ph, math.AP

发布日期: 2026-10-21

arXiv ID: 2610.30496v1

链接: http://arxiv.org/abs/2610.30496v1

摘要翻译

中文标题: 双曲守恒律的熵有界紧致模板Runge-Kutta间断Galerkin方法

摘要翻译: 本文介绍一种用于可压缩气体动力学Euler方程的紧致模板Runge-Kutta间断Galerkin方法，该方法保持类似于Tadmor在Euler方程熵解[4]中引入的最小熵原理。紧致模板RKDG由Chen等人在[15]中引入，是标准RKDG方法的一种变体，以更小的模板尺寸和并行效率著称。主要挑战在于，cRKDG方法必须表示为Butcher形式，而并非该形式下的每个Runge-Kutta方法都是前向Euler步的凸组合。因此，为了建立该性质，我们利用cRKDG格式的保界框架[8]、一个熵有界限制器[6]和经典的保正限制器。最后一节给出数值实验以证明该方法的稳健性。

12. 梯度流自适应 SSAV–BDF2 积分的能量速率等分布

Energy-rate equidistribution for adaptive SSAV–BDF2 integration of gradient flows

作者: Suki Shergill, JF Williams

类别: math.NA

发布日期: 2026-10-21

arXiv ID: 2610.30454v1

链接: http://arxiv.org/abs/2610.30454v1

摘要翻译

中文标题: 梯度流自适应 SSAV–BDF2 积分的能量速率等分布

摘要翻译: 我们为梯度流中结合变步长 BDF2 离散的标量辅助变量（SAV）格式发展了一种自适应时间步进策略。该策略根据瞬时能量耗散率选取步长，并通过步长拒绝机制强制一个可计算的对称能量具有单调性。我们证明，实际使用的步长规则对应于沿能量曲线对监视函数进行等分布，从而将能量自适应时间步进与 r-自适应网格方法的经典理论联系起来。监视函数 $g = |\dot{E}|$ 是渐近最优的，因为它在给定步数下最小化主导阶的最大对称能量偏差。一种在平衡附近切换到 $g = \sqrt{|\dot{E}|}$ 的组合监视函数限制了步长增长，且不降低该误差界。先前工作中使用的弧长参数 $β$ 被证明是最大允许步长与能量误差容限之比。该控制器与 SSAV–BDF2 离散相结合。被接受的步满足对称能量的单调性，由此我们推出物理能量的一个关于步数一致的界，$\sup_n E(u^n) \le E(u^{0}) + τ_1 + O(Δt_{\max}^2)$，并且在标准步长比限制下，该格式达到二阶全局精度。在 Cahn–Hilliard、Ohta–Kawasaki、Phase–Field Crystal 和 functionalized Cahn–Hilliard 模型上的数值实验证实了该理论并展示了该方法的效率。

13. 一般函数空间的分部求和算子：通量差分、熵稳定性与精度

Summation-by-parts operators for general function spaces: Flux differencing, entropy stability, and accuracy

作者: Henry M. Haase, Andrew R. Winters, Jan Glaubitz

类别: math.NA

发布日期: 2026-10-21

arXiv ID: 2610.30272v1

链接: http://arxiv.org/abs/2610.30272v1

摘要翻译

中文标题: 一般函数空间的分部求和算子：通量差分、熵稳定性与精度

摘要翻译: 分部求和（SBP）算子是双曲守恒律的可证明稳定高阶方法的基石。对于非线性问题，Fisher 和 Carpenter 的通量差分框架将此类 SBP 算子与熵守恒两点通量相结合，得到在 Lax–Wendroff 意义下守恒、且离散熵守恒或稳定的半离散化。迄今为止，该框架仅针对对多项式精确的算子发展。函数空间 SBP（FSBP）算子在可针对具体问题定制的一般有限维函数空间上精确，从而解除了这一限制。本文中，我们将通量差分推广到适用于一般（可能非多项式）函数空间的 FSBP 算子。我们证明：只要该算子具有对角范数且其函数空间包含常数，所得半离散化就在 Lax–Wendroff 意义下守恒、在单个单元内半离散熵守恒，并且跨单元界面熵稳定。我们还给出了 FSBP 算子的首个理论精度分析：我们通过底层函数空间上的代数准则刻画其设计阶，证明合适的非多项式空间达到与同维多项式空间相同的设计阶，并表明通量差分半离散化保持该阶。值得注意的是，熵守恒两点通量原样复用；仅底层函数空间与多项式情形不同。针对一维和二维 Burgers 方程与可压缩 Euler 方程的数值实验验证了这些性质。

不相关论文（基本信息）

1. 从随机特征到完全可训练网络的相变

Phase Transition from Random Features to Fully Trainable Networks

作者: Xin He, Huan Lao, Xiliang Lu, Shijun Zhang

类别: math.NA, cs.LG, math.ST, stat.ML

发布日期: 2026-10-22

arXiv ID: 2610.31663v1

链接: http://arxiv.org/abs/2610.31663v1

摘要: Constructing accurate and computationally tractable approximation spaces is a fundamental task in scientific computing. Partially trainable neural networks bridge fixed random features and fully adaptive representations. This paper quantifies a phase transition pattern in approximation bounds and observes analogous behavior in training errors: a rapid but smooth initial improvement followed by diminishing returns, with a sharper transition under greater initialization mismatch. Under polynomial spectral decay with exponent $α>1$ and leverage compatibility, we establish a squared $L_2$ approximation rate $\mathcal{O}((κ_P\log(m/δ)/m)^α)$ for two-layer networks of width $m$ and trainable proportion $P$, where $κ_P=(P/κ_1^*+(1-P)/κ_0^*)^{-1}$. The bound holds uniformly over the unit ball of a reproducing kernel Hilbert space (RKHS) with probability at least $1-δ$. The constants $κ_0^*$ and $κ_1^*$ quantify the fixed and trainable groups’ sampling mismatches relative to the target RKHS. For spherical biased-ReLU features in dimension $d$, we verify these assumptions with $α=(d+2)/(d-1)$ and $κ_0^*=4\sqrt{2}τ(d+1)/d$ for initialization scale $τ\ge1$, and $κ_1^*=1$ under ridge leverage sampling. For $d=5$ and $τ=10$, a trainable proportion of approximately $4.1\%$ captures $90\%$ of the reduction in the approximation bound from $P=0$ to $P=1$. A structured three-layer example illustrates how a few trainable first-layer neurons exploit fixed downstream blocks through nonlinear composition. Numerical experiments across diverse settings, including networks with up to 15 layers, exhibit similar phase transitions in training errors.

2. 凸Bernstein极限与扩张区间上的锐利恢复

Convex Bernstein limits and sharp recovery on expanding intervals

作者: Matthew J. Colbrook, Catherine Drysdale

类别: math.NA, math.CA

发布日期: 2026-10-22

arXiv ID: 2610.31546v1

链接: http://arxiv.org/abs/2610.31546v1

摘要: We determine when approximation by convex polynomials on expanding intervals converges to best approximation on the real line by convex entire functions of exponential type at most $σ>0$. For intervals of radius $(n-γ_n)/σ$, where $0\leγ_n

3. 采用顶点片光滑的混合精度合并访存无矩阵多重网格

Mixed-Precision Coalesced Matrix-Free Multigrid with Vertex-Patch Smoothing

作者: Michał Wichrowski

类别: math.NA

发布日期: 2026-10-22

arXiv ID: 2610.31496v1

链接: http://arxiv.org/abs/2610.31496v1

摘要: I present a geometric multigrid solver for the Poisson equation with continuous cubic elements in three dimensions, smoothed by a multiplicative vertex-patch Schwarz method with exact local solves. All fields are kept in persistent cell-wise storage, where every cell holds its own copy of each shared degree of freedom. The patch smoother yields a continuous solution without any exchange between cells. On the GPU, the patch update is computed in registers, one cell per thread, and the cells of a patch exchange data by warp shuffles, so shared memory is avoided almost entirely. Despite the low occupancy that this design implies, the arithmetic of the patch update runs close to the FP32 peak. Fast diagonalization, merged transforms, and a reflection symmetry of the patch then reduce the arithmetic until a sweep moves its data at $87\%$ of the bandwidth of a plain device copy. The hierarchy therefore runs in single precision inside a double-precision outer iteration. Rounding the unassembled residual destroys the cancellation between opposite-sign copies of shared degrees of freedom; exchanging the copies between cells in double precision performs this cancellation before rounding, without changing the dual vector they represent. With this \emph{rebalancing}, the cycle converges as in double precision and independently of the mesh size, the error converges at the optimal rate down to the precision of the solve, and the solver reaches over $400$ million degrees of freedom per second on a single NVIDIA A100 GPU.

4. T-product 下四元数张量奇异值分解的快速算法

A Fast Algorithm for Quaternion Tensor Singular Value Decomposition under the T-product

作者: Daochang Zhang, Yue Zhao, Jingqian Li

类别: math.NA

发布日期: 2026-10-22

arXiv ID: 2610.31451v1

链接: http://arxiv.org/abs/2610.31451v1

摘要: Quaternion tensor singular value decomposition (T-SVD) under the T-product provides a framework for analyzing multidimensional quaternion data. However, the T-SVD construction of Pan and Ng (SIAM J. Matrix Anal. Appl., 2024) \cite{PanNg2024} contains a structural gap: an unrestricted quaternion matrix SVD need not preserve the paired Fourier block structure required for tensor factor recovery, potentially leading to loss of T-unitarity and inaccurate reconstruction. To address this issue, we develop a structure-preserving construction that transforms each paired quaternion block into a complex matrix and maps its complex SVD factors back to quaternion factors with the required structure. This construction corrects the proof of \cite[Theorem 4]{PanNg2024} and leads to a corrected version of \cite[Algorithm 2]{PanNg2024}, hereafter referred to as the ``corrected algorithm’’, ensuring T-unitary tensor factors and an f-diagonal core for arbitrary third-order quaternion tensors. A numerical counterexample demonstrates the failure of \cite[Algorithm 2]{PanNg2024} and verifies the corrected algorithm at machine precision. Notably, we propose a fast structure-preserving algorithm, Algorithm~\ref{alg:t-fast}, and compare it numerically with the corrected algorithm. The experimental results show that Algorithm~\ref{alg:t-fast} reduces the decomposition time while maintaining near-machine-precision accuracy. Furthermore, its application to color video encryption yields high reconstruction quality, favorable ciphertext statistics, and stability under input perturbations.

5. 针对 Rayleigh-Bénard 对流的带松弛零能量变量的二阶线性解耦格式

A second-order linear decoupled scheme with a relaxed zero-energy variable for Rayleigh-Bénard convection

作者: Matthew McCurdy

类别: physics.flu-dyn, math.NA

发布日期: 2026-10-22

arXiv ID: 2610.31364v1

链接: http://arxiv.org/abs/2610.31364v1

摘要: We construct a second-order, linear, and fully decoupled scheme for Rayleigh–Bénard convection between Navier-slip walls. The nonlinear terms are treated explicitly and multiplied by a scalar with exact value $1$ and no energy contributed to the continuous problem. As a result, each BDF2 step, with a rotational pressure correction, solves only constant-coefficient linear problems. For the time discretization, we prove that, for any time-step, the energy decays when the walls have equal temperatures, under a condition on the thermal weight, and the energy is bounded on finite time intervals when heated from below. Relaxing the scalar towards $1$ preserves both estimates. Energy stability does not control the accuracy of the scalar; the velocity and temperature fields are computed with its unrelaxed value, whose deviation from $1$ scales like $(E/C)(δt/τ)^3$ for kinetic energy $E$, time-scale $τ$ of the flow, and weight $C$ of the scalar equation. In our tests, choosing $C$ comparable to $E$ keeps this deviation small and doubles the largest time-step for which steady rolls are reproduced to a relative difference of $10^{-8}$. With Taylor–Hood elements, we observe second-order convergence in time, recover the critical Rayleigh numbers, and match published Nusselt numbers to within $0.5\%$. A three-dimensional implementation passes corresponding validation tests.

6. 用于无穷维随机 SVD 的希尔伯特空间外草图

Sketching Outside the Hilbert Space for the Infinite-Dimensional Randomised SVD

作者: Morgan Lynch-Davies, Marcus Webb

类别: math.NA

发布日期: 2026-10-22

arXiv ID: 2610.31333v1

链接: http://arxiv.org/abs/2610.31333v1

摘要: In randomised numerical linear algebra, common choices for distributions of sketch vectors $ω\in \mathbb R^n$ are isotropic, so that $\mathbb E[ωω^T]=I_n$. Unfortunately, the natural extension of an isotropic vector in $(\mathbb R^n, \|\cdot \|_{2})$ to an infinite-dimensional separable Hilbert space $\mathcal H$ may not exist as an element of the space, posing an obstacle for any attempt to lift the randomised SVD to infinite-dimensional target operators $A$. Current workarounds either forego isotropy and take $ω$ to be a continuous Gaussian process with trace-class covariance operators, or leave $ω$ implicit by instead sampling $Aω$ as if $ω$ were isotropic Gaussian. We unify their approaches by introducing “sketch functionals”, an example of cylindrical random variables (Vakhania–Tarieladze–Chobanyan 1987) which allow us to treat general, non-Gaussian distributions with bounded covariance operators. To demonstrate the utility of the new theory, we apply it to develop novel bounds for the infinite-dimensional randomised SVD with a general class of sub-Gaussian sketch distributions.

7. 最近矩形辛矩阵问题的强对偶阈值与最小曲率

Strong Duality Thresholds and Minimum Curvature for the Nearest Rectangular Symplectic Matrix Problem

作者: Yulin Xie, Aiqing Zhu, Yifa Tang

类别: math.OC, math.NA

发布日期: 2026-10-22

arXiv ID: 2610.31298v1

链接: http://arxiv.org/abs/2610.31298v1

摘要: We study the nearest rectangular symplectic matrix problem in the squared Frobenius norm for arbitrary targets $T\in\mathbb R^{2n\times2k}$, with $n\ge k$. We prove that its semidefinite relaxation is exact for every target if and only if $n\ge2k-1$. Under this condition, the primal problem, the relaxation, and both Lagrangian dual problems attain the same optimal value. In this range, we derive the exact minimum Rayleigh quotient of the Riemannian Hessian at every feasible point under the Euclidean induced metric, and prove that all second-order stationary points are global minimizers. We also obtain global objective error bounds. We extend the strong-duality and minimum-curvature results to Hermitian positive definite row weights, proving that strong duality holds for every target if and only if the lowest eigenvalue of the weight has complex multiplicity at least $2k-1$. The dual formulation leads to a method whose main iterations use matrices of order $2k$, with $O(k^2)$ iterative storage apart from the full input and output. Across six tested sizes, mean-time speedups are $2.55$–$12.60$ over a Riemannian first-order method and $4.61$–$43.93$ over Jensen–Zimmermann R–TR2; batching gives a further factor of $3.45$–$23.53$ over sequential calls. The theoretical results complete the critical dimensions not covered by Gram convexification and quadratic matrix programming.

8. 深度 ReLU 网络对连续目标函数可能不具备最佳逼近性质

Deep ReLU Networks Can Fail the Best Approximation Property for Continuous Target Functions

作者: Sebastian Kassing

类别: cs.LG, math.NA, math.OC

发布日期: 2026-10-22

arXiv ID: 2610.31264v1

链接: http://arxiv.org/abs/2610.31264v1

摘要: We construct a continuous function on $[-1,1]^2$ for which the infimum of the $L^2$-approximation error over the class of ReLU networks with two hidden layers and architecture $(2,2,1,1)$ is not attained. This counterexample shows that deep ReLU networks do not satisfy the best approximation property in general, even for continuous target functions. Our findings contrast with existing results for shallow neural networks, where ReLU is one of the few activation functions that guarantee the existence of global minima in typical regression tasks. The proof is based on the fact that the breaklines for neurons in the second hidden layer can bend, which breaks the symmetry arguments used in the one-hidden-layer case.

9. 由 R-函数表示的可行域的序贯蒙特卡罗采样

Sequential Monte Carlo Sampling of Feasible Regions Represented by R-Functions

作者: Sinan Yıldırım, Sergei Kucherenko

类别: stat.CO, math.NA

发布日期: 2026-10-22

arXiv ID: 2610.31042v1

链接: http://arxiv.org/abs/2610.31042v1

摘要: Feasible-region sampling and volume estimation become increasingly inefficient when constraints define only a small fraction of a convenient bounding domain. We combine R-functions, which encode compound feasibility as a continuous signed field, with sequential Monte Carlo (SMC) samplers. The proposed construction exploits not only the sign of the R-function, which determines exact feasibility, but also its continuous negative values outside the feasible set, which provide graded information on constraint violation. Logistic intermediate distributions use this information to guide particles progressively towards feasibility, while the final target distribution remains uniform on the feasible region. Experiments on intersections and unions, including disconnected cases of hyperballs, show substantial variance reductions relative to direct rejection sampling, with the statistical advantage generally increasing as the feasible fraction decreases. For sufficiently small feasible regions, these advantages persist even after accounting for computational cost. To isolate the benefit of continuous R-function information, we also compare the proposed method with an SMC sampler with a region-only (RO) construction that uses the same sequential particle machinery but retains only binary feasible/infeasible information. We demonstrate that the R-function-informed SMC method concentrates particles in the feasible region earlier and produces lower-variance volume estimates than RO.

10. 数据依赖的紧支撑多节点Shepard方法

Data-dependent Compact Multinode Shepard Method

作者: Francesco Dell’Accio, Filomena Di Tommaso, Francesco Larosa, Juan Ruiz-Álvarez, Dionisio F. Yáñez

类别: math.NA

发布日期: 2026-10-22

arXiv ID: 2610.31026v1

链接: http://arxiv.org/abs/2610.31026v1

摘要: We introduce a data-dependent compact multinode Shepard method for the approximation of scattered multivariate data. The proposed approach extends the compact multinode Shepard operator by incorporating nonlinear, data-dependent weights inspired by the weighted essentially non-oscillatory (WENO) methodology while preserving the partition of unity structure and the locality of the approximation. The construction is based on compactly supported multinode weight functions associated with local polynomial interpolants defined on unisolvent stencils. A smoothness indicator is employed to adapt the contribution of each stencil according to the local regularity of the underlying function, thereby reducing spurious oscillations near discontinuities while retaining the optimal approximation order in smooth regions. We establish the theoretical approximation properties of the proposed operator: it achieves the order of accuracy of its linear counterpart for sufficiently smooth functions and, under an explicit indicator-separation hypothesis, the same order is retained at every covered evaluation point near a jump discontinuity. Numerical experiments for local degrees up to four illustrate the performance of the method on both smooth and non-smooth data, on scattered as well as on gridded nodes.

11. 大规模低秩耦合广义Sylvester方程的数值解

Numerical Solution of Large-Scale Low-Rank Coupled Generalized Sylvester Equations

作者: Umair Zulfiqar

类别: math.NA, eess.SY

发布日期: 2026-10-22

arXiv ID: 2610.30804v1

链接: http://arxiv.org/abs/2610.30804v1

摘要: This paper develops projection methods for computing low-rank solutions to large-scale coupled generalized Sylvester equations. The dominant subspaces of these solutions are shown to be spanned by specific resolvent directions, motivating a Petrov–Galerkin projection framework. Conditions for the unique solvability of the projected coupled generalized Sylvester equations are also discussed. Based on these dominant subspaces, two algorithms are proposed: an extended Krylov-based projection algorithm and a rational Krylov-based projection algorithm. The former uses fixed shifts, whereas the latter uses adaptive shifts. Numerical experiments on large-scale benchmark problems demonstrate that the proposed methods achieve high accuracy and fast convergence.

12. 一种具有稳定性保证的改进直接特征值交换算法

A Modified Direct Eigenvalue Swapping Algorithm with Guaranteed Stability

作者: Meiyue Shao, Rentao Xu

类别: math.NA

发布日期: 2026-10-22

arXiv ID: 2610.30795v1

链接: http://arxiv.org/abs/2610.30795v1

摘要: For a given matrix ( A ) in the real Schur form, the eigenvalue swapping algorithm is used to swap two adjacent diagonal blocks. In LAPACK, this operation is performed by the subroutine~\texttt{SLAEXC}/\texttt{DLAEXC}, which is based on a direct swapping algorithm proposed by Bai and Demmel in 1993. In this work, we present a thorough error analysis of a modified direct swapping algorithm. By a carefully designed pivoted Givens QR factorization, this algorithm can achieve an error bound of the form ( O( \boldsymbol{u} \lVert A \rVert ) ) under mild assumptions, where ( \boldsymbol{u} ) is the unit roundoff. We present empirical evidence to illustrate the superior stability of our algorithm over existing approaches.

13. 反向 Ritz 包含：由有限秩压缩的伪谱给出算子伪谱的可计算界

The reverse Ritz inclusion: computable bounds on the pseudospectra of an operator by those of a finite-rank compression

作者: Isaia Nisoli

类别: math.NA

发布日期: 2026-10-21

arXiv ID: 2610.30530v1

链接: http://arxiv.org/abs/2610.30530v1

摘要: We consider a bounded operator on a separable complex Hilbert space, reached only through its action on vectors and that of its adjoint, and split it along a subspace of finite dimension into a compression, a far block acting on the complement, and the two couplings between them. Given upper bounds on the norms of the couplings and of the far block, we show that outside a disc a pseudospectrum of the operator lies in a pseudospectrum of the compression, with an explicit loss in the level; this reverses the classical inclusion of the Ritz values in a pseudospectrum of the operator. The same bounds, with the smallest singular value of the shifted compression, bound the resolvent of the operator, count its eigenvalues with algebraic multiplicity inside every contour outside the disc on which a computed inequality holds, and enclose its spectral projector; no normality, self-adjointness or positivity is used.

14. 利用自适应数值离散发现偏微分方程中的奇点

Discovery of Singularities in Partial Differential Equations Using Adaptive Numerical Discretizations

作者: Allen Yang, Lucas Amoudruz, Petros Koumoutsakos

类别: math.NA, math.AP

发布日期: 2026-10-21

arXiv ID: 2610.30492v1

链接: http://arxiv.org/abs/2610.30492v1

摘要: We compute self-similar singularity profiles of partial differential equations (PDEs) using adaptive numerical discretizations based on optimizing a discrete loss (ODIL). The fields are represented in overlapping local Chebyshev patches joined with a mapped asymptotic patch. After factoring out the leading power laws of solution ansatzes, their limiting amplitudes at infinity are determined by the optimization, eliminating domain truncation and prescribed far-field conditions. Adaptive refinement is guided by Chebyshev-coefficient decay, and the nonlinear least-squares problem is solved using automatically differentiated Gauss–Newton iterations with Levenberg–Marquardt damping. The method achieves O(10^{-14})-level accuracy for Burgers profiles, residuals of O(10^{-14}) to O(10^{-10}) for the generalized De Gregorio family, and of O(10^{-8}) in a local region for the two-dimensional Boussinesq equations. These results demonstrate that ODIL can identify singularity profiles for PDEs with high accuracy using adaptive numerical methods, complementing ongoing efforts that rely on artificial intelligence.

15. 用于二维和三维电磁仿真的梳状张量网络时域有限差分算法

Comb Tensor Network Finite-Difference Time-Domain Algorithms for 2D and 3D Electromagnetic Simulations

作者: Jonathan Tabares, Erika Ye, Yang Liu

类别: math.NA

发布日期: 2026-10-21

arXiv ID: 2610.30486v1

链接: http://arxiv.org/abs/2610.30486v1

摘要: This paper presents efficient 2D and 3D finite-difference time-domain (FDTD) algorithms augmented by the comb tensor network (CTN) representation of the field and auxiliary quantities. The proposed CTN-FDTD algorithm leverages superior compressibility of CTN compared with quantized tensor train (QTT), and rank-reduction strategies including local and global smoothing techniques to achieve logarithmic memory requirement and computation (per time step) cost. Numerical examples validate these complexities and demonstrate the applicability of CTN-FDTD using a variety of source excitation, frequency and permittivity profiles in both 2D and 3D. Remarkably, CTN-FDTD can simulate 35 trillion spatial grid points in 3D with low memory and computation costs on one computation node, whereas standard FDTD would require 281 TB memory to store each of its 31 3D arrays. Moreover, CTN-FDTD was applied to realistic 2D photonic crystal structure simulation with one million grid points and 3D scattering analysis with 8.6 billion grid points.

16. 基于速度法的等几何 Kirchhoff–Love 壳鲁棒拓扑优化

Robust topology optimization of isogeometric Kirchhoff–Love shells by the Speed Method

作者: Philipp Eisenhardt, Fernando Hübner Scherer, Barbara Wohlmuth, Andrei Constantinescu

类别: math.NA, cond-mat.mtrl-sci

发布日期: 2026-10-21

arXiv ID: 2610.30389v1

链接: http://arxiv.org/abs/2610.30389v1

摘要: This paper presents a framework for robust topology optimization of Kirchhoff-Love (KL) multi-patch shells using isogeometric analysis and non-matching meshes and provides a CAD-compatible method. The material distribution is optimized via a NURBS-based level set method, enabling accurate boundary representation and seamless export to standard CAD/CAM workflows. The shape derivative is computed using the speed method, yielding a volume-based formulation that avoids boundary approximations and aligns with the isogeometric principle of coarse, high-order discretizations. This eliminates the need for local refinement, adaptive quadrature, or trimming as the descent direction for the compliance and volume objectives are obtained through a biharmonic equation. The framework is extended to robust optimization under uncertainty, accounting for stochastic loading and material properties via Monte Carlo sampling and Gaussian Random Fields. Numerical examples demonstrate the method’s efficiency and robustness across single- and multi-patch shells, reproducing literature results with fewer elements than traditional FEM or refined IGA approaches. The developed methods and tools can be directly extended to problems involving stress-based optimization and risk-averse metrics. The numerical implementations are open-source and available on GitHub.

17. 一种基于Transformer的可变初始/终止条件实时多智能体路径规划方法

A Transformer-based Approach for Real-time Multi-agent Path Planning with Varying Initial/Final Conditions

作者: Qian Zhang, Shanqing Liu, Jerome Darbon, George Karniadakis

类别: math.OC, cs.LG, math.NA

发布日期: 2026-10-21

arXiv ID: 2610.30388v1

链接: http://arxiv.org/abs/2610.30388v1

摘要: We address the problem of real-time multi-agent path planning under prescribed initial and final configurations, formulated as a high-dimensional optimal control problem subject to inter-agent collision avoidance and obstacle constraints. Classical solvers based on the Hamilton–Jacobi–Bellman equation or Pontryagin’s maximum principle (PMP) become computationally intractable as the number of agents grows, due to the curse of dimensionality and the non-convexity introduced by interaction constraints. We propose OCFormer, a transformer-based neural network that parameterizes the inverse of a coordinate transformation mapping a simple, analytically tractable latent space, in which agents evolve independently under a decoupled Hamiltonian, to the physical space of the coupled, interacting system. The model is trained by minimizing a physics-informed loss comprising the Hamiltonian ODE residual from the PMP and a boundary matching term, without requiring pre-computed trajectory data. The self-attention mechanism endows OCFormer with permutation equivariance over agents, a structural symmetry that prior symplectic network approaches do not natively enforce. When trained on a distribution of boundary conditions, the amortized OCFormer produces near-optimal trajectories for unseen initial and final configurations in a single forward pass ($\sim$220ms), enabling real-time inference without retraining. We further introduce a lightweight transfer learning framework that adapts a pretrained OCFormer to out-of-distribution perturbations in seconds via L-BFGS fine-tuning. Numerical experiments on two- and three-dimensional multi-agent navigation with up to 128 agents demonstrate the scalability and efficiency of the proposed approach.

18. 鞍点系统增广拉格朗日预条件的统一变分框架：在混合维方法与虚拟区域方法中的应用

A unified variational framework for augmented Lagrangian preconditioning of saddle point systems: applications to mixed-dimensional and fictitious domain methods

作者: Michele Benzi, Marco Feder, Luca Heltai, Federica Mugnaioni

类别: math.NA

发布日期: 2026-10-21

arXiv ID: 2610.30357v1

链接: http://arxiv.org/abs/2610.30357v1

摘要: We present a general framework to derive augmented Lagrangian preconditioners for saddle point systems arising from mixed finite element discretizations. Our approach is based on augmenting the variational formulation at the continuous level prior to discretization, providing a unified strategy for designing efficient preconditioners that are robust under mesh refinement for a broad class of problems leading to symmetric saddle point systems. The robustness of the proposed preconditioners is established through a detailed spectral analysis. This framework is applied to different model problems, focusing on mixed-dimensional and fictitious domain formulations involving Lagrange multipliers, with some configurations featuring large coefficient jumps. Extensive numerical experiments on different test cases and geometries are presented, validating the effectiveness and robustness of our preconditioners when used in conjunction with flexible GMRES.

19. GPU 上 $C^1$ Hermite 离散的无矩阵局部多重网格

Matrix-Free Local Multigrid for $C^1$ Hermite Discretizations on GPUs

作者: Michał Wichrowski

类别: math.NA

发布日期: 2026-10-21

arXiv ID: 2610.30308v1

链接: http://arxiv.org/abs/2610.30308v1

摘要: I present a matrix-free local multigrid method for $C^1$ tensor-product Hermite discretizations on adaptively refined Cartesian meshes, using the clamped biharmonic problem as the model problem. Extending a cell-wise primal-dual formulation to Hermite elements, I enforce continuity through a constraint operator that is never formed. At refinement edges, geometric prolongation supplies both the trace and its normal derivative, so the local V-cycle needs no edge operator and no hanging-node machinery. For the cubic element, an even-odd decomposition based on the signed reflection symmetry of the Hermite basis, applied along two axes, splits the element matrix into four dense $16\times16$ blocks and reaches 85% of peak memory bandwidth. Chebyshev-Jacobi and vertex-patch smoothers both preserve conformity by construction. In three dimensions on one NVIDIA A100 in double precision, conjugate gradients preconditioned by the Chebyshev-Jacobi V-cycle need five or six iterations on uniform meshes and solve at up to 4.23 million cells per second. On locally refined meshes the count rises to at most sixteen, ordered by the fraction of cells on a refinement edge and nearly equal for both smoothers.

20. 部分线性贝叶斯反问题的保结构降维与代理模型

Structure-Preserving Dimension Reduction and Surrogates for Partially Linear Bayesian Inverse Problems

作者: Ziheng Zhang, Dingcheng Luo, Thomas O’Leary-Roseberry, Umberto Villa, Omar Ghattas

类别: math.NA

发布日期: 2026-10-21

arXiv ID: 2610.30306v1

链接: http://arxiv.org/abs/2610.30306v1

摘要: Many Bayesian inverse problems are characterized by two parameter vectors with distinct roles. A medium property, operator coefficient, or geometry block enters the forward model nonlinearly, whereas a volumetric, initial condition, or boundary condition source enters linearly. General-purpose nonlinear surrogates must relearn this known source dependence from data and may not preserve the correct source scaling outside the training regime. We introduce a structure-preserving partially linear (SPPL) framework that constructs separate reduced spaces for outputs, (linearly appearing) sources, and (nonlinearly appearing) medium variables and represents the reduced forward map as an affine function of the retained source coordinates, with coefficients that depend on the medium variables. Here, “affine” refers to the retained, mean-centered source coordinates: the physical forward map remains linear in the source, while the affine offset is induced by the source-prior mean and vanishes when that mean is zero. The analysis first identifies the optimal blockwise ridge approximation and then derives the optimal affine-in-source approximation under independent medium and source priors. A direct decomposition of the forward error into output, source, medium, and surrogate contributions yields blockwise spectral criteria and prior-predictive expected-KL control. For a Gaussian source prior, the optimal affine-in-source approximation coincides with the optimal blockwise ridge approximation. In a coseismic numerical example, a three-model controlled ablation on common block-reduced coordinates separates the effects of coefficient-matrix supervision and structure-preserving affine-in-source parameterization. Reduced-space surrogate MCMC provides orders-of-magnitude online acceleration relative to high-fidelity posterior sampling.