则对任意节点 $i$ 有

这构成了二阶全局收敛的核心机制。

对于加权调和界面格式，取 $\alpha=x_I+\theta h$，有效系数为

并已证明其满足上述抵消条件（引理4.2，具体计算见附录A，从提供内容中无法提取更多细节）。因此该格式在最大模下二阶收敛。

更一般地，作者提出一类“可容许守恒界面格式”，其特征为：共享切割面通量、正界面系数、界面相邻点截断误差均为 $O(1)$、以及组合一致性 $\tau_I+\tau_{I+1}=O(h)$。对该类中的每一个格式，格林函数论证均给出最大模二阶收敛。

此外，定理和推论还给出了稳定性与条件数的区别：$\|L_h^{-1}\|_\infty=O(1)$，而 $\kappa_2(L_h)=O(\beta_{\max}/\beta_{\min}\,h^{-2})$。在二维情形，对于笛卡尔网格和与坐标方向平行的平直界面，相同机制沿每条穿过界面的网格线逐行（或逐列）成立；完整的多维格林函数分析仍需额外的技术条件。

3. 研究方法、关键技术和核心工具

证明中的核心障碍在于：局部截断误差不再是小量，如何从全局角度控制其影响？作者们采用离散格林函数作为分析工具，并取得了三项技术创新。

（1）显式逆矩阵公式的推导。对通量形式矩阵 $A_h$，引入离散通量 $q_i=\beta_{i-1/2}(U_i-U_{i-1})$。求解 $A_hU=e_j$ 等价于 $q_i-q_{i+1}=\delta_{ij}$，因此通量在源点两侧分别为两个常数。再利用零边界条件 $\sum_{k=1}^{n+1}s_kq_k=0$ 确定常数，进而得到 $U_i$ 的闭式表达式。这一推导不依赖于系数的具体分布，适用于任意正系数序列，从而为后续所有估计提供了精确基础。

（2）双平台结构的揭示。由显式公式直接看出，加权格林增量 $\beta_{i+1/2}\Delta_iG(j)$ 在 $i

（3）误差分解技巧。将两个界面相邻大误差的贡献重新组合：

第一项因 $G_{iI}=O(h)$ 和 $\tau_I+\tau_{I+1}=O(h)$ 而为 $O(h^2)$；第二项因 $|G_{i,I+1}-G_{iI}|=O(h^2)$ 和 $\tau_{I+1}=O(1)$ 而为 $O(h^2)$。这一分解完全避免了对单个 $\tau_I$ 的小性要求，将局部误差的“大小”转化为相邻误差的“组合”和格林函数的“光滑性”，是论文最主要的证明工具。

此外，作者还区分了算子的最大模稳定性与代数条件数：$\|L_h^{-1}\|_\infty$ 有界意味着误差传播的整体稳定性，而 $\kappa_2(L_h)$ 增长仅反映代数求解的困难，两者互不影响。这一区分澄清了“大局部误差”并非源于数值不稳定。

最后，在构造可容许格式族时，作者通过切割面通量共享建立守恒性，并利用局部通量平衡推导出组合一致性条件。加权调和格式作为特例，其检验通过直接计算完成。二维扩展则基于一维结果沿坐标线逐条应用，体现了分析框架的模块性。

4. 与之前工作的比较

与前人工作的比较可从以下几个层面展开。

与 Beale 和 Layton [1] 相比：Beale 和 Layton 早已证明，当界面附近的局部舍弃误差较大但具有某种弱一致性时，借助离散椭圆正则性和格林函数估计仍可得到二阶精度。但他们的分析依赖于抽象的估计，未给出离散格林函数的具体形态。本文推导了精确的格林函数公式，并发现了双平台结构，从而使抵消机制变得可验证、可构造。此外，本文明确将守恒性作为抵消的前提，这比 Beale-Layton 的一般性结论更具操作性。

与调和型格式 [3,9] 相比：Ewing 等[3] 从有限体积通量近似出发，论证了调和平均的局部精度；本文则从离散格林函数角度重新解释了为何这类格式能达到全局二阶：即使界面点局部误差为 $O(1)$，守恒通量平衡强制 $\tau_I+\tau_{I+1}=O(h)$，而格林函数的双平台结构（增量 $O(h^2)$）恰好将该组合误差转化到全局。这种视角揭示了调和格式成功的内在原因，并为构造新格式提供了判据。

与浸入界面法 [7,8] 相比：浸入界面法通过修改离散方程或添加校正项来逼近界面跳跃条件，其目标通常是恢复点态一致性或高阶局部精度。本文的结论却是：点态一致性并非必要，只要守恒性和组合一致性成立，二阶精度即可保证。这降低了界面格式设计的门槛，尤其适用于界面形状复杂或系数比大而难以构造精确校正的情形。当然，浸入界面法在二维曲线界面上的成熟技术仍是本文尚未覆盖的。

与标准有限体积分析 [4,6] 相比：这些工作通常利用局部守恒性和离散化误差的 $L^2$ 或 $H^1$ 估计，很少关注最大模下的大局部误差现象。本文的最大模估计直接来源于格林函数的显式有界性，填补了这一空缺。

本文的代价与局限：其一，显式格林函数公式仅适用于一维通量三对角结构，对二维需要张量积结构或平直界面才能逐线应用；对于一般曲线界面，双平台性质不再成立，需要新的工具。其二，引理4.2的详细计算放在附录A，从提供的PDF内容中无法确认其每一步推导，但作者声称已建立。其三，作者未显式展开与 Beale-Layton 的定量比较，仅指出其工作提供了更广泛的背景。二维完整格林函数分析仍是未解决的问题，正如作者在第6节所述，需要额外处理交叉点效应。

3. 局部周期椭圆问题高阶均匀化的一种张量神经网络方法

A Tensor Neural Network Method for High-Order Homogenization of Locally Periodic Elliptic Problems

作者: Huaijia Zhang, Haochen Liu, Xia Ji, Hehu Xie

类别: math.NA, math.AP

发布日期: 2026-08-29

arXiv ID: 2608.29176v1

摘要翻译: 我们针对形如 $-\nabla\cdot(A(x,x/\varepsilon)\nabla u_\varepsilon)=f$ 的局部周期椭圆多尺度问题，发展了一种高阶张量神经网络（TNN）方法。由于系数同时依赖于慢变量 $x$ 和快周期变量 $y=x/\varepsilon$，高阶胞元问题和宏观校正子方程比经典情形 $A=A(y)$ 更为复杂，且校正子参数化地依赖于 $x$。我们推导了一个可计算的高阶双尺度展开，并在无边界层情形（包括周期域和理想边界匹配构型）下证明了部分展开的 $H^1$ 收敛估计。证明利用了校正子层次的递推相容性结构以及 $H^{-1}$ 中的零均值振荡估计。随后，我们为高维校正子问题构建了一个 TNN 框架。其张量积结构允许对胞元问题、均匀化系数、宏观源项和损失函数进行确定性的一维求积，从而避免了蒙特卡洛积分误差。数值实现假设系数分量和组装数据具有有限的或受控的张量积表示；这一计算假设与分析中所用的一般矩阵值系数类相分离。使用标量局部周期系数的实验展示了准确的高阶校正子。 $H^1$ 半范数误差与所证明的估计一致，而逐点归一化的 $L^2$ 误差则表现出形式展开所预测的名义高阶行为。

Comment: 36 pages, 9 figures, 8 tables

链接: http://arxiv.org/abs/2608.29176v1

1. 研究对象和背景

本文研究如下局部周期椭圆多尺度问题：

其中 $0<\varepsilon\ll1$ 为微观尺度，系数 $A(x,y)$ 对快变量 $y=x/\varepsilon$ 为 $Y=(0,1)^n$ 周期，且同时非平凡依赖于慢变量 $x$。这一模型广泛出现于复合材料、多孔介质、非均匀扩散等应用中，可视为经典周期均匀化 $A=A(y)$ 的推广。与经典情形相比，局部周期设置下胞元函数、均匀化系数及高阶校正子均成为慢变量 $x$ 的参数化函数，使得高阶渐近层级结构显著复杂化：经典周期情形中的胞元问题仅需在参考胞元上求解一次，而此处必须在每个宏观点 $x$ 处求解一组含参数的胞元问题，且宏观校正方程与快变量校正相互耦合。

已有理论工作（如 [13,14,15]）在定性均匀化和定量算子估计层面建立了局部周期算子的收敛性与校正子估计，但尚未提供可直接计算的高阶渐近层级及相应的高维数值实现。另一方面，高阶渐近展开在高精度需求下具有重要价值，因为首阶均匀化解通常只提供 $O(\varepsilon)$ 精度，而实际应用中往往需要更精确的有效响应。然而，直接网格离散由于需要同时分辨 $x$ 和 $y$ 两个尺度，且需要递归求解大量高维胞元问题，计算代价随空间维数和展开阶数急剧增长。近年来机器学习方法被用于多尺度问题，但多数工作局限于首阶或一阶均匀化，或依赖蒙特卡洛采样导致高维积分误差，这在高阶递归求解中会逐级积累而破坏渐近阶。

因此，本文旨在填补如下空白：为局部周期椭圆问题构造可计算的高阶双尺度展开并建立相应的 $H^1$ 收敛理论，同时发展一种能够精确高效求解高维校正子层级的数值框架。所提出的张量神经网络（Tensor Neural Network, TNN）方法利用其张量积结构，将高维积分转化为一维求积的组合，从而避免随机积分误差，为高阶均匀化的可靠计算提供了一条新路径。

2. 主要定理或主要结果

本文的主要结果分为理论推导、收敛性分析和数值方法三部分。

1. 高阶双尺度展开的递归构造。在算子分解

   $$\mathcal L_0v=-\nabla_y\cdot(A(x,y)\nabla_yv),\quad \mathcal L_1v=-\nabla_y\cdot(A\nabla_xv)-\nabla_x\cdot(A\nabla_yv),\quad \mathcal L_2v=-\nabla_x\cdot(A\nabla_xv)$$

   下，将 $u_\varepsilon$ 展开为 $u_\varepsilon\sim u_0(x)+\varepsilon u_1(x,x/\varepsilon)+\varepsilon^2u_2(x,x/\varepsilon)+\cdots$，通过匹配 $\varepsilon$ 的幂次得到级联方程 (2.7)-(2.10)。由此递归定义：首阶胞元问题 (2.11) 确定 $\chi_i(x,y)$，首阶振荡校正 $\hat u_1=\chi\cdot\nabla_xu_0$，均匀化张量

   $$A^*(x)=\int_Y A(x,y)(I+\nabla_y\chi(x,y))\,dy,$$

   均匀化方程 (2.15) $-\nabla_x\cdot(A^*(x)\nabla_xu_0)=f$；随后，辅助胞元问题 (2.16) 确定 $\hat u_2$，宏观校正方程 (2.17) 确定 $\tilde u_1$，并定义 $\delta u_2=\chi\cdot\nabla_x\tilde u_1$，从而 $u_2^*=\hat u_2+\delta u_2$。一般地，给定 $u_{k-1}$ 和 $u_k^*$ 后，由 (2.21) 解 $\hat u_{k+1}$，由 (2.23) 解 $\tilde u_k$，再令 $\delta u_{k+1}=\chi\cdot\nabla_x\tilde u_k$、$u_{k+1}^*=\hat u_{k+1}+\delta u_{k+1}$、$u_k=u_k^*+\tilde u_k$。该递归过程总结为 Algorithm 1，其每一步的胞元问题右端均具有零 $Y$ 平均（引理 2.1），从而良置。

2. 无边界层设置下的 $H^1$ 收敛估计。作者在平坦环面 $\Omega=\mathbb T^n$ 或理想边界匹配构型下考虑部分展开 $U_k^*$ 与完整展开 $U_k$（定义见 (2.27)-(2.28)），并引入物理归一化算子 $\mathcal N_{\#}$ 以处理周期情形的常数漂移。分析中假设能量空间 $V$ 满足 Poincaré 不等式 (2.25)。基于递归相容性（引理 2.1）、级联恒等式（引理 2.2）以及零平均振荡估计（引理 2.3）——即对满足 $\int_Y g(x,y)\,dy=0$ 的周期函数 $g$，在 $H^{-1}$ 范数下可获得 $\varepsilon$ 阶增益——作者证明了部分展开在 $H^1$ 半范意义下的收敛估计。需要说明的是，由于所给 PDF 片段未包含最终定理的完整陈述，我们无法给出精确的误差阶与常数；但摘要和实验表明，$H^1$ 半范误差与所证估计一致，且该估计在无边界层环境中成立。文中明确区分了周期性零平均归一化与数值点归一化的关系（Remark 2.2），并指出 $L^2$ 范数不具备规范不变性，其收敛阶由形式展开推测而不由能量估计直接保证。

3. 张量神经网络数值框架。针对高维校正子层级，作者提出 TNN 方法：将待求函数表示为张量积形式的神经网络，使得胞元问题、均匀化系数、宏观源项及损失函数中的高维积分可通过确定性一维求积精确（或高精度）求值，从而避免蒙特卡洛误差。数值实现假设系数矩阵元素及装配数据具有有限或可截断的张量积表示；该计算假设与分析中一般的矩阵值系数类别相分离（即分析允许一般的光滑一致椭圆矩阵值系数，而数值算法需要额外的张量结构）。数值实验针对标量局部周期系数，验证了高阶校正子的准确性：$H^1$ 半范误差与证明的无边界层估计一致，逐点归一化 $L^2$ 误差则表现出与形式展开阶数相称的名义高阶行为（该 $L^2$ 行为是经验观测而非证明结论）。

3. 研究方法、关键技术和核心工具

证明的主要技术障碍在于：局部周期系数使高阶胞元问题与宏观方程相互耦合，且每个胞元问题在无穷维参数空间上依赖慢变量 $x$，直接应用经典周期均匀化的能量方法无法给出与 $\varepsilon$ 阶增益匹配的残差估计。作者的核心创新点包括：

• 递归相容性结构：通过引理 2.1 证明所有胞元问题右端自动满足零 $Y$ 平均条件。证明中关键利用均匀化方程 (2.15) 与宏观校正方程 (2.23) 的散度形式，将 $Y$ 平均后的表达式归结为 $-\nabla_x\cdot(A^*\nabla_x\tilde u_k+\int_Y A(\nabla_xu_k^*+\nabla_y\hat u_{k+1})\,dy)$，从而由定义消失。这一结构保证了层级递推的良置性。

• 级联恒等式：引理 2.2 建立了 $\mathcal L_0u_0=0$，$\mathcal L_0u_1^*+\mathcal L_1u_0=0$，以及 $\mathcal L_0u_{m+1}^*+\mathcal L_1u_m+\mathcal L_2u_{m-1}=\delta_{m1}f$。这一恒等式将原方程的残差精确表示为 $\varepsilon^{m}$ 项的组合，是后续估计的基础。证明中将 $u_{m+1}^*=\hat u_{m+1}+\delta u_{m+1}$ 分解，分别利用胞元方程和 $\delta u_{m+1}$ 的定义消去交叉项。

• 零平均振荡估计（引理 2.3）：对满足零 $Y$ 平均的 $g(x,y)$，估计 $\|g(x,x/\varepsilon)\|_{H^{-1}}\le C\varepsilon\|g\|_{L^2}$ 型的界。这一估计是获得 $\varepsilon$ 阶增益的关键，使得在测试残差时能克服快振荡带来的 $\varepsilon^{-1}$ 项。虽然完整引理陈述未在 PDF 片段中给出，但摘要与上下文明确其作用。

• 物理归一化处理：在周期设置中，两尺度函数 $v(x,x/\varepsilon)$ 并不自动具有物理零均值，故引入算子 $\mathcal N_{\#}$ 减去物理平均。该操作不改变梯度与残差方程，仅改变常数，从而可统一处理周期与 Dirichlet 边界匹配情形。

• 数值方面：TNN 的张量积结构使每个高维积分分解为一维积分的乘积。具体地，对形如 $\int_Y F(\chi(x,y),\nabla_y\chi(x,y))\,dy$ 的积分，若被积函数可表示为 $\sum_l\prod_{i=1}^n\phi_{l,i}(x,y_i)$，则可通过一维 Gauss 求积逐维求值。这避免了蒙特卡洛采样误差，且在高阶递归中可保持积分精度，防止误差逐级放大。作者还强调了分析假设（光滑一致椭圆矩阵系数）与计算假设（系数具有张量积表示）的分离，后者是数值实现所必需但理论上可放宽的。这些技术点共同构成了证明框架：先由相容性保证层级良置，再由级联恒等式和零平均振荡估计建立残差控制，最终得到 $H^1$ 收敛估计。

4. 与之前工作的比较

本文与既有工作的比较可从以下几个层面展开：

1. 与经典周期均匀化的比较。当 $A=A(y)$ 时，经典高阶展开的胞元问题不依赖 $x$，宏观校正方程为常数系数的均匀化方程，本文的递归结构退化为经典形式。局部周期情形下，胞元函数 $\chi_i(x,y)$、$\hat u_k(x,y)$ 均参数化依赖于 $x$，且宏观校正方程 (2.23) 的右端包含 $\nabla_x$ 作用在积分上的额外项，这使得层级结构在代数上更复杂，也使得数值求解必须同时处理 $x$ 和 $y$ 两个尺度。经典结果的误差估计通常利用两尺度展开的残差直接估计，而本文需要额外处理参数化带来的 $x$ 导数与 $y$ 积分的交换。

2. 与已有局部周期均匀化理论（如文献 [13,14,15]）的比较。这些工作建立了局部周期算子的定性收敛性与定量算子估计，但主要聚焦于首阶均匀化和相应的校正子估计，没有提供高阶胞元层级的可计算公式与更精细的误差阶。本文的主要推进在于给出了完整的高阶级联结构（Algorithm 1），并在无边界层场景下证明了这一结构产生的部分展开的 $H^1$ 收敛性，这为高阶计算提供了理论基础。

3. 与机器学习类多尺度方法的比较。已有神经网络方法如 [21,22,23,24] 分别处理多尺度方程求解、物理信息网络、学习均匀化本构与胞元解算子等。但这些方法大多限于首阶或一阶均匀化，或者需要在高维损失函数上进行随机采样。本文提出的 TNN 方法针对高阶递归结构的需要，利用张量积神经网络将高维积分转化为确定性一维求积，从而避免了蒙特卡洛误差，这对于高阶均匀化尤其重要——因为数值积分误差会在逐级校正子求解中传播并破坏渐近阶。这一设计在已有学习类方法中未见系统发展。

4. 与传统数值方法的比较。与有限元多尺度方法（MsFEM）和异质多尺度方法（HMM）相比，这些方法在低维空间是有效的，但在局部周期高阶情形下，辅助校正子依赖慢变量且需递归求解，网格方法需要反复解胞元问题或在 $\Omega\times Y$ 上直接离散，计算代价随维数和阶数急剧增长。TNN 方法通过张量积表示将高维积分降为一维求积，在系数具有张量结构时具有显著优势，但代价是要求系数和数据具备有限或受控的张量积表示。对于一般数据，可以事先进行张量化，但相应的预处理误差未被分析。

5. 局限性与代价。本文的收敛理论仅在无边界层设置（周期环面或理想边界匹配构型）下建立，对标准 Dirichlet 边界条件，边界层效应未包括在内；作者将边界层尾项的处理留作未来工作。此外，数值上的张量积表示假设是分析中未要求的额外限制，当系数缺乏该结构时需要额外的张量化步骤，其误差控制未在文中讨论。实验表明 $H^1$ 半范误差符合理论，而 $L^2$ 误差的高阶行为是经验性的，不具有理论保证。作者未显式展开与所有前人工作的定量比较，但从提供内容可看出其在计算高阶校正子和避免积分误差方面的明确推进。

4. 三维Maxwell组合场积分方程的高阶精度连续性强制Nyström离散化

High-Order-Accurate Continuity Enforcing Nyström Discretization of 3D Maxwell Combined Field Integral Equations

作者: Bernd Hofmann, Reza Molavi, Constantine Sideris

类别: math.NA, physics.comp-ph

发布日期: 2026-08-28

arXiv ID: 2608.28876v1

摘要翻译: 在电场积分方程(EFIE)的Nyström配置离散化中，表面散度作用于可能跨补丁边界不连续的表面密度，这降低了精度和收敛性。我们表明，这不仅影响EFIE，而且影响该算子出现的每一个公式，无论是在方程本身还是在散射场计算中，并提出一种面向光滑曲面的高阶精度连续性强制方案，作为直接和间接EFIE、磁场积分方程(MFIE)以及正则化组合场积分方程(CFIE)的补救措施。该方案包含两个要素：i) 我们展示了如何通过基于Chebyshev的Nyström方案离散方程，该方案允许闭式求积规则。ii) 由于未知量和测试向量采用补丁局部曲线基表示，连续性通过基变换来强制：我们仅从曲线几何描述构造稀疏映射矩阵。这样做，我们恢复了EFIE的精度，使其可以与MFIE以相等权重组合以形成CFIE。针对典型和真实几何体的散射数值研究表明，所有考虑的公式无论是单独还是组合，都从连续性强制中受益，体现在更好的条件数、减少迭代求解器的迭代次数，以及散射场多出几位精度，尽管减少了总未知量。

Comment: 无

链接: http://arxiv.org/abs/2608.28876v1

1. 研究对象和背景

本文研究三维时谐电磁散射问题中理想导体(PEC)目标的积分方程数值求解. 设时间因子为 $e^{-i\omega t}$ 并被抑制, 入射场 $(\bm{e}^{\mathrm{ex}}, \bm{h}^{\mathrm{ex}})$ 照射到有界光滑曲面 $\Gamma$ 上, 其单位法向量为 $\hat{\bm{n}}$, 背景介质参数为介电常数 $\varepsilon$ 和磁导率 $\mu$. 目标是确定散射场 $(\bm{e}^{\mathrm{sc}}, \bm{h}^{\mathrm{sc}})$, 通常先求解表面电流密度或非物理表面密度, 再通过表示公式计算散射场. 文中考虑直接和间接两类公式: 直接 EFIE 和 MFIE 分别确定物理感应电流 $\bm{j}$, 间接 EFIE 和 MFIE 确定非物理密度 $\bm{a}, \bm{b}, \bm{c}$. 核心积分算子为

其中 $g_k(\bm{r},\bm{r}') = e^{i k |\bm{r}-\bm{r}'|}/(4\pi |\bm{r}-\bm{r}'|)$ 为自由空间 Green 函数, $k=\omega\sqrt{\mu\varepsilon}$ 为波数, $\eta=\sqrt{\mu/\varepsilon}$ 为波阻抗. 由于 $\mathcal{T}$ 是第一类积分算子, 直接和间接 EFIE 在 PEC 腔体内谐振频率处具有非平凡零空间, 导致数值求解精度和收敛性恶化; 经典补救方法是将 EFIE 与 MFIE 组合成组合场积分方程 (CFIE). 本文采用 Nyström 配置离散, 其矩阵元素本质上只需核函数求值, 且固有高阶精度. 然而在 Nyström 离散中, EFIE 算子中的表面散度 $\nabla'_\Gamma \cdot$ 作用于补丁间可能不连续的密度, 从而产生虚假线电荷, 严重降低精度. 此问题不仅影响 EFIE 本身, 还影响所有包含 $\mathcal{T}$ 的公式以及从密度计算散射场的表示公式. 本文旨在填补这一空白: 提出一种显式强制表面密度连续性的高阶 Nyström 离散方案, 适用于直接/间接 EFIE, MFIE, 正则化 CFIE 及散射场计算, 从而恢复 EFIE 精度并显著改善所有公式的数值性能.

2. 主要定理或主要结果

在以下假设下提出了高阶精度连续性强制方案: 散射体表面 $\Gamma$ 由共形无重叠曲边四边形补丁覆盖, 每个补丁上的密度用协变基矢量展开为 $\bm{j}(u,v) = D^{-1}[j_u \bm{e}_u + j_v \bm{e}_v]$, 其中 $D$ 为广义 Jacobi 行列式. 方案采用闭合 Clenshaw-Curtis 求积规则, 并在补丁边界上设置求积点, 从而可通过构造稀疏映射矩阵对重复未知量施加连续性约束. 构造了三个稀疏矩阵 $\mathsf{C}$, $\tilde{\mathsf{C}}$, $\underline{\mathsf{C}} \in \mathbb{R}^{N'\times N}$, 分别用于未知量、协变测试矢量和逆变测试矢量. 它们完全由曲边几何描述组装, 且 $N' = 2P(N_u-2)(N_v-2) + 2U < N$, 其中 $P$ 为补丁数, $N_u=N_v$ 为每个方向求积点数, $U$ 为唯一边界点数. 关键代数性质为 $\mathsf{C} \mathsf{C}^T = \mathsf{I}$, 由此 MFIE 和 CFIE 中的恒等算子 $\pm\mathsf{I}/2$ 在连续性强制后仍为真单位矩阵, 从而保持第二类积分方程结构. 对所有公式, 离散化系统均化为

• 直接 EFIE: $\eta \tilde{\mathsf{C}} \underline{\mathsf{T}} \mathsf{C}^T \bm{j}' = \tilde{\mathsf{C}} \bm{e}^{\mathrm{ex}}$ (协变测试) 或 $\eta \mathsf{C} \underline{\mathsf{T}} \mathsf{C}^T \bm{j}' = \mathsf{C} \bm{e}^{\mathrm{ex}}$ (逆变测试);
• 直接 MFIE: $\mathsf{C}(\pm\mathsf{I}/2 + \hat{\bm{n}}\times\mathsf{K}) \mathsf{C}^T \bm{j}' = \mathsf{C} \bm{n}^{\mathrm{hex}}$;
• 直接 CFIE: $\mathsf{C}(\pm\mathsf{I}/2 + \hat{m{n}}\times\mathsf{K} + \underline{\mathsf{T}}) \mathsf{C}^T \bm{j}' = \mathsf{C}(\bm{n}^{\mathrm{hex}} + \eta^{-1}\bar{\bm{e}}^{\mathrm{ex}})$ 及其混合协变/逆变测试版本;
• 直接 CFIE-R: $\mathsf{C}(\pm\mathsf{I}/2 + \hat{\bm{n}}\times\mathsf{K} - k \hat{\bm{n}}\times\mathsf{R}_w \hat{\bm{n}}\times\mathsf{T}) \mathsf{C}^T \bm{j}' = \mathsf{C}(\bm{n}^{\mathrm{hex}} - \eta^{-1} k \hat{\bm{n}}\times\mathsf{R}_w \bm{n}^{\mathrm{ex}})$;
• 间接公式类似改写, 如表 IV 所示. 数值结果表明, 与未强制连续性的 Fejer 开式求积相比, 连续性强制在球体散射问题中: EFIE 条件数降低约 4 个数量级, 电流密度误差改善 2 位以上, 电场和磁场误差改善约 6 位; MFIE 磁场误差改善约 1 位, 电场误差改善约 3 位; CFIE 条件数、迭代次数和误差均显著改善; 正则化 CFIE 结合连续性强制达到最低误差、条件数和迭代次数. 未知量数目由 $N=3072$ 降至 $N'=2704$.

3. 研究方法、关键技术和核心工具

本文的核心技术障碍在于 Nyström 配置离散不强制面密度跨补丁连续性, 而表面散度算子要求散度共形, 否则产生虚假线电荷. 作者提出了一种显式连续性强制框架, 主要创新点包括: (1) 使用闭合 Clenshaw-Curtis 求积规则, 使求积点包含补丁边界, 从而允许将边界上的重复未知量关联起来; (2) 通过精确的基变换构造稀疏映射矩阵, 仅依赖曲边几何信息, 不改变算子应用. 具体地, 对共享边上的重合点 $\bm{p}$, 两个补丁 $a$ 和 $b$ 上的密度表示满足 $\bm{j}(\bm{p}) = D_a^{-1}[j^a_u \bm{e}^a_u + j^a_v \bm{e}^a_v] = D_b^{-1}[j^b_u \bm{e}^b_u + j^b_v \bm{e}^b_v]$. 与逆变测试矢量取内积后得到系数关系, 例如 $j^b_{u,kl} = \alpha_u j^a_{u,ij} + \alpha_v j^a_{v,ij}$, 其中 $\alpha_u = D_b/D_a \,\bm{e}^{b,u}_{kl}\cdot\bm{e}^a_{u,ij}$, $\alpha_v = D_b/D_a \,\bm{e}^{b,u}_{kl}\cdot\bm{e}^a_{v,ij}$, 类似地有 $\beta$ 系数. 这些系数仅由补丁参数化决定, 从而得到稀疏矩阵 $\mathsf{C}$ 的子块 $\mathsf{C}_{\alpha\beta}$, 其非零元数为 $O(N)$. 对于测试矢量, 为避免数值不一致, 采用所涉及补丁测试矢量的平均, 构造 $\mathsf{C}$ 和 $\tilde{\mathsf{C}}$, 系数分别涉及 $1/n_i$ 和 Jacobi 行列式比值. 关键估计是 $\mathsf{C} \mathsf{C}^T = \mathsf{I}$, 它保证了第二类积分方程的恒等部分在连续性强制后保持为单位矩阵, 不破坏迭代收敛性质. 此外, 文中详细展示了算子组合的离散化: 由于离散策略是封闭的, 即算子将协变分量表示的先切向量场映射为同表示的另一先切向量场, 因此复合算子如 $\hat{\bm{n}}\times R_w (\hat{\bm{n}}\times T\bm{j})$ 可直接通过矩阵乘积 $\mathsf{nR}\mathsf{nT}\mathsf{j}$ 离散, 无需中间基变换. 对于间接 CFIE 中的 $\hat{\bm{n}}\times T(\hat{\bm{n}}\times \bm{b})$, 通过 $\hat{\bm{n}}\times \bm{e}_u = D\bm{e}_v$ 和 $\hat{\bm{n}}\times \bm{e}_v = -D\bm{e}_u$ 引入度规张量的逆变分量, 离散为 $\mathsf{nT}\mathsf{G}\mathsf{b}$. 奇异性处理采用 Fejer/Clenshaw-Curtis 插值与离散正交性, 将弱奇异核积分精确计算. 作者严格证明的结论包括连续性矩阵的构造和 $\mathsf{CC}^T = \mathsf{I}$; 数值实验则展示了各公式的改进效果.

4. 与之前工作的比较

本文与既有有限元/积分方程数值解法的关键差异和推进如下. (1) 相对于经典 MoM 离散: MoM 使用散度共形基函数 (如 RWG) 天然满足法向连续性, 但矩阵元素需要计算 4D 积分, 高阶 MoM 成本更高. 本文 Nyström 离散只需核函数求值, 且通过显式连续性强制达到与 MoM 相当的精度, 同时减少未知量. 与使用对偶基的 MFIE 方案相比, 本文避免了双基带来的额外计算, 仅用三个稀疏映射矩阵就恢复了 MFIE 的精度, 使其能与 EFIE 等权组合成 CFIE. (2) 相对于其他 Nyström 连续性处理: 与局部校正 Nyström 法和通用密度插值法 GP-DIM 相比, 后者每个补丁需解 LSE 或最小二乘问题, 本文的映射矩阵是闭式构造的, 无需额外求解; 与不连续 Galerkin Nyström 方案相比, 后者引入附加线积分并增加迭代次数且缺乏预处理策略, 本文的连续性强制不改变算子结构, 且保持第二类积分方程的良好性质. 与文献[24][25]中通过 SVD 约束矩阵继承高阶 MoM 性质的方法相比, 本文使用稀疏映射矩阵, 计算和存储开销显著降低. (3) 对于正则化 CFIE: 本文推广了正则化 CFIE 的 Nyström 离散, 并指出直接公式中正则子 $R_w$ 作为左正则子作用在已微分的密度上, 而间接公式中作为右正则子作用在未微分密度上, 可能影响连续性强制效果. 数值结果显示, 直接公式中正则化与连续性强制结合达到最佳, 而间接公式中正则化作用在表面散度之前并未进一步改善, 这为理解正则化与连续性之间的相互作用提供了新认识. (4) 代价与局限: 本文要求共形四边形网格, 且 $u,v$ 方向求积点数必须相等 (除特殊几何如圆环外), 这是相对其他方法的限制; 但作者指出, 尽管未知量减少, 所有公式在条件数、迭代次数和精度方面均获益, 说明连续性强制的优势超过了网格限制. 作者未显式展开与所有已有工作的定量比较, 但提供了一致的数值证据表明新方案的有效性.

了解领域论文（摘要翻译）

1. 基于神经网络的端口哈密顿系统模型降阶

Model reduction of port-Hamiltonian systems via neural networks

作者: Silke Glas, Alexander Heinlein, Harald Monsuur, Hongliang Mu

类别: math.NA

发布日期: 2026-08-31

arXiv ID: 2608.30788v1

链接: http://arxiv.org/abs/2608.30788v1

摘要翻译

中文标题: 基于神经网络的端口哈密顿系统模型降阶

摘要翻译: 在本文中，我们考虑保结构的端口哈密顿（pH）系统模型降阶，此类系统通过耗散和输入-输出端口扩展了经典哈密顿系统。这些pH系统通常用于多物理场系统，因为一个或多个pH系统的互联结果仍是一个pH系统。特别地，如果与pH系统的互联和/或耗散相关的系统矩阵是状态相关的，那么标准降阶模型（ROM）的评估可能依赖于原始全阶模型的维数，导致高昂的计算成本。为了规避这些高成本，我们提出使用保结构的神经网络。具体而言，我们执行两个步骤：（1）使用广义流形Galerkin投影将pH系统投影到降阶空间；然后（2）训练一个神经网络来学习从降阶状态到降阶互联和耗散系统矩阵的映射。为确保所得ROM仍是一个pH系统，神经网络的架构被设计为保持降阶系统矩阵的斜对称性和半正定性。在一个数值算例中，我们考虑一个具有状态相关系统矩阵的非线性质量-弹簧-阻尼系统。数值结果表明，与原始ROM相比，所提出的方法在保持相当精度的同时实现了显著的计算加速。

2. Fokker–Planck 方程的结构保持细致平衡主方程离散化

Structure-Preserving Detailed-Balance Master-Equation Discretizations for Fokker–Planck Equations

作者: Satish Chandran, Yiwei Wang

类别: math.NA

发布日期: 2026-08-31

arXiv ID: 2608.30121v1

链接: http://arxiv.org/abs/2608.30121v1

摘要翻译

中文标题: Fokker–Planck 方程的结构保持细致平衡主方程离散化

摘要翻译: 我们直接从 Fokker–Planck 方程的能量耗散定律出发，发展了一种用于其细致平衡主方程离散化的变分–马尔可夫构造。我们并不离散微分算子，而是通过两个方向跳变速率来表示相邻网格点之间的质量传递。在每条边上，离散能量耗散定律决定净通量，而细致平衡固定速率比；随后，从相邻网格点值对边量的局部重构唯一确定这两个速率。对数平均重构重现了经典的 Scharfetter–Gummel/Wang–Peskin–Elston 速率，而其他重构则产生其他可逆格式。所得半离散系统守恒质量、保持非负性、满足细致平衡，并耗散离散自由能。相同的逐边构造可扩展到状态依赖和退化的迁移率、非局部相互作用能量以及高维问题。相应的隐式和半隐式全离散格式是线性的、质量守恒的、保正的且能量稳定的；仅对于核不是负半定的非局部相互作用能量，才需要时间步长限制。数值实验确认了所有细致平衡格式的预期空间收敛阶，并验证了它们在不连续势、饱和、非局部相互作用和二维问题中的平衡精度、保正性和自由能衰减。

3. 带非线性跳跃条件的抛物型界面问题的浸没界面方法

An Immersed Interface Method for Parabolic Interface Problems with Nonlinear Jump Conditions

作者: So-Hsiang Chou, Patrick Nyadjo Fonga

类别: math.NA

发布日期: 2026-08-30

arXiv ID: 2608.30062v1

链接: http://arxiv.org/abs/2608.30062v1

摘要翻译

中文标题: 带非线性跳跃条件的抛物型界面问题的浸没界面方法

摘要翻译: 我们发展了一种一维非线性抛物型界面问题的浸没界面有限差分方法，其跳跃条件为 [ [u]_\alpha=\lambda u^+u^- ]。该方法将Crank-Nicolson浸没界面离散与非线性界面条件的(s)-参数约化相结合，从而将非线性耦合约化为一个标量二次方程。我们还讨论了将牛顿迭代与浸没界面离散相结合的两种不同观点，即IIM-Newton和Newton-IIM公式。数值实验展示了该方法的性能和精度。

4. DPG对流离散的Fortin算子

Fortin operators for DPG advection discretizations

作者: Pablo Cortés Castillo, Jay Gopalakrishnan

类别: math.NA

发布日期: 2026-08-30

arXiv ID: 2608.30043v1

链接: http://arxiv.org/abs/2608.30043v1

摘要翻译

中文标题: DPG对流离散的Fortin算子

摘要翻译: 我们在任意空间维数 $N$ 的单纯形网格上，对试验空间任意多项式次数 $k \ge 1$，为具有分片常数且散度自由的对流向量 $β$ 的对流方程的不连续Petrov-Galerkin (DPG) 离散构造 Fortin 算子。在每个单元上，由面泡和内部泡构造一个极小检验空间，随后对其进行增广。我们证明该 Fortin 算子在基于 $L_q$ 构造的自然 $β$-加权分片检验图范数下，对每个 $1 < q < \infty$，在形状正则网格族上一致有界，其中 $q$ 是与试验指数 $p$ 共轭的指数。当 $k \ge 2$ 时，假定一个非特征面条件；而最低阶情形不需要该条件，并允许特征面。作为应用，我们证明实用的全离散残差最小化方法在 DPG 能量范数下对每个 $1 < p < \infty$ 是拟最优的，其拟最优常数仅由 Fortin 算子决定；其可计算残差是全局可靠且有效的后验误差估计子；并且增广检验空间和改变检验范数可以改进最低阶情形的结果。

5. 几何流的高阶能量稳定BGN参数有限元方法

High-order energy-stable BGN parametric finite element methods for geometric flows

作者: Shu Ma, Qiqi Rao

类别: math.NA

发布日期: 2026-08-30

arXiv ID: 2608.29877v1

链接: http://arxiv.org/abs/2608.29877v1

摘要翻译

中文标题: 几何流的高阶能量稳定BGN参数有限元方法

摘要翻译: 我们构造了 Barrett–Garcke–Nürnberg (BGN) 参数有限元方法的高阶 Runge–Kutta 扩展，用于平面曲线的曲线缩短流和曲线扩散，以及闭亏格$0$曲面的平均曲率流和曲面扩散。在每个时间片 $I_m=(t_m,t_{m+1}]$ 上，连续方程通过映射 $\X^{m,t}:Γ^m\toΓ^t$ 被置于左端点曲面 $Γ^m=Γ^{t_m}$ 上，该映射的目标是时刻 $t$ 的演化曲面。对于曲线，此表述通过从 $Γ^0$ 到 $Γ^m$ 的调和拉回获得。对于曲面，在每个时间片上分别构造一个保定向的调和微分同胚，共形性产生了权重 $\frac12|\nabla_{Γ^m}\X^{m,t}|^2$。在 Runge–Kutta 内部时刻评估时间片方程并应用质量集中参数有限元，在公共区域 $Γ^m$ 上产生方程组，其中 $Γ^{t_m+c_iτ_m}$ 作为中间目标几何。这种内部时间离散化为四种流构造了高阶 BGN 结构扩展。对于具有非负权重的代数稳定表格，每个具有非退化中间构型的精确级解，在其存在的每个正时间步上都满足离散曲线长度或曲面面积的单调衰减。Radau IIA 实验对所有四种流表现出能量衰减，并在曲线测试中表现出 BGN 型网格再分布。Hausdorff 自收敛结果表现出与相应设计阶一致的高阶行为。

6. ENPINN：面向具有尖锐梯度的广义输运问题的能量范数引导的梯度增强PINN

ENPINN: Energy-Norm-Guided Gradient-Enhanced PINNs for Generalized Transport Problems with Sharp Gradients

作者: Subhendu Maity, Pratibhamoy Das, Arihant Patawari, Ameya D. Jagtap

类别: math.NA, math-ph

发布日期: 2026-08-30

arXiv ID: 2608.29499v1

链接: http://arxiv.org/abs/2608.29499v1

摘要翻译

中文标题: ENPINN：面向具有尖锐梯度的广义输运问题的能量范数引导的梯度增强PINN

摘要翻译: 物理信息神经网络（PINN）已成为求解偏微分方程（PDE）的传统数值方法的一种无网格替代方案。然而，它们在捕捉尖锐梯度方面的有限能力可能导致在分辨边界层和内部层时产生显著误差。在此，我们引入一种能量范数增强的PINN（ENPINN），它将梯度信息和变分结构纳入损失函数，以提高层主导解的分辨率。我们首先考察两种相关形式：弱损失PINN（WLPINN），它将测试函数纳入传统PINN残差；以及梯度增强PINN（gPINN），它通过PDE残差的空间导数来增强损失。通过分析这些形式，我们识别出它们在分辨陡峭解梯度方面的局限性，并推动ENPINN的系统构建。我们从理论上建立了能量范数误差如何依赖于ENPINN损失，并表明适当修改的残差导数项对于准确捕捉边界层至关重要。我们进一步建立了具有任意小能量误差的神经网络逼近的存在性，并推导了相应的导数界，为所提出的框架提供了理论基础。通过与传统PINN变体在具有陡峭梯度的对流-扩散-反应问题上的系统比较，评估了ENPINN的性能。数值实验包括一个燃烧模型、一个耦合多尺度系统、一个具有内部层的二维Burgers方程，以及一个三维时间依赖问题。

7. 广义牛顿流动的非残差VMS逼近的动态子尺度与无条件半离散稳定性

Dynamic subscales and unconditional semi-discrete stability for non-residual VMS approximations of generalised Newtonian flows

作者: G. R. Barrenechea, E. Castillo, R. Codina, J. Pastene, P. Vega

类别: math.NA

发布日期: 2026-08-29

arXiv ID: 2608.29457v1

链接: http://arxiv.org/abs/2608.29457v1

摘要翻译

中文标题: 广义牛顿流动的非残差VMS逼近的动态子尺度与无条件半离散稳定性

摘要翻译: 我们分析了一种带动态子尺度的非残差变分多尺度有限元格式，用于不可压缩广义牛顿Navier–Stokes流动。假设表观黏度关于剪切率有界且Lipschitz连续，涵盖了若干正则化流变律。该方法通过正交投影分离了未求解的压力散度、对流和压力梯度贡献。对于线性化半离散问题，我们证明了在各向异性VMS范数下的适定性和无条件稳定性估计。该估计控制了黏性耗散、子尺度能量、离散散度以及耦合的加速度-对流-压力平衡。对于非线性格式，在适当的正则性和小性假设下，通过不动点论证得到了存在性和最优阶先验估计。分析还提供了离散压力-对流平衡的弱时间控制。一个关键点是，动态压力相关子尺度提供了控制投影压力梯度误差所需的时间导数贡献，而这一机制在相应的准静态设置中是不可用的。

8. 一种基于图卷积的可学习多重网格框架

A Learnable Multigrid Framework via Graph Convolutions

作者: Xiuping Wang, Li Luo

类别: math.NA

发布日期: 2026-08-29

arXiv ID: 2608.29082v1

链接: http://arxiv.org/abs/2608.29082v1

摘要翻译

中文标题: 一种基于图卷积的可学习多重网格框架

摘要翻译: 本文提出了一种新颖的框架，将可学习的图卷积与几何多重网格方法相结合，用于求解偏微分方程（PDE）。首先将偏微分方程的离散化表示为图结构，从而能够应用图卷积来增强多重网格性能。通过将图卷积融入多重网格的光滑算子和网格间转移算子等组件中，我们开发了一种可学习的多重网格，能够根据底层问题特征自适应地优化其性能。在该框架中，图卷积直接嵌入多重网格循环内部，有效地将整个多重网格求解器转化为一种专门的神经网络架构，而不是将经典求解器与代理模型简单组合。该可学习多重网格框架在参数数量上轻量，且只需极少的训练投入即可获得良好性能。数值实验证明了所提出方法在求解一些具有挑战性问题时的有效性，与传统多重网格方法相比，展示了更优的收敛速度。本文还通过适当的权重共享和迁移学习策略，研究了学习参数在不同问题设置（包括不同的源项、系数、几何形状和网格尺寸）中的泛化能力。

不相关论文（基本信息）

1. 量子信号处理中的约束极小极大逼近

Constrained minimax approximation for quantum signal processing

作者: Yulong Dong, James B. Larsen, Lin Lin, Rahul Sarkar

类别: quant-ph, math.NA

发布日期: 2026-08-31

arXiv ID: 2608.30937v1

链接: http://arxiv.org/abs/2608.30937v1

摘要: Quantum signal processing (QSP) provides a simple and efficient framework for implementing polynomial transformations using quantum circuits. Its classical design stage leads to a constrained minimax approximation problem: find a polynomial of prescribed parity that approximates a target function uniformly on a fitting set while remaining bounded in magnitude by one on the domain $[0,1]$, which can be viewed as a semi-infinite constraint. Discretization converts the problem into a linear program, but feasibility at a set of finitely many sampled points does not ensure feasibility on the whole domain, especially when an optimal approximant reaches the boundary of the feasible set. We investigate two approaches to address this difficulty. A Remez exchange method combined with active-set constraint enforcement is efficient on many tested instances, but its stability depends on the target and problem geometry. We then introduce nonlinear Fourier retraction, which uses QSP completion and phase synthesis to turn a nearly feasible polynomial into phase factors for a feasible QSP polynomial without increasing the degree. Across representative problems, retraction largely preserves approximation accuracy and remains effective on instances where the Remez heuristic is unstable. The resulting workflow connects classical minimax approximation and semi-infinite optimization with nonlinear Fourier analysis, and is implemented in the qsppack software package.

2. 大型开放量子系统的良态迭代方法

Well-conditioned iterative methods for large open quantum systems

作者: Gaspard Beugnot, Paul Gregory, Rémi Robin, Antoine Tilloy

类别: quant-ph, math.NA

发布日期: 2026-08-31

arXiv ID: 2608.30860v1

链接: http://arxiv.org/abs/2608.30860v1

摘要: Markovian open quantum systems are well modeled by the Lindblad Master Equation (ME) $\frac{\mathrm{d}}{\mathrm{d} t} ρ_t = \mathcal{L} ρ_t$, where $\mathcal{L}$ is a linear (super-)operator and $ρ_t$ is the system state, a positive matrix. When designing or characterizing a quantum system, one is usually interested in the steady state $ρ_\infty$ (such that $\mathcal{L} ρ_\infty = 0$), the first few excited states, and trajectories $t\mapsto ρ_t$. In finite dimension, $ρ_t$ is an $n\times n$ matrix, $\mathcal{L}$ thus typically costs $n^4$ to store explicitly as a dense matrix, and $O(n^6)$ to diagonalize or invert exactly, making standard linear algebraic techniques expensive for large systems. However, $\mathcal{L}$ usually costs only $O(n^3)$ to apply. This makes iterative methods appealing, but they do not work without a good preconditioner. In this article, our main observation is that a part of the Lindblad equation, corresponding to the so-called no-jump evolution $\mathcal{S}$, can be inverted efficiently. Using this inverse map, we introduce an auxiliary completely positive trace-preserving (CPTP) map $Φ$ whose fixed point is directly related to $ρ_\infty$, all the other eigenvalues having smaller magnitude. The map $Φ$ is thus well suited to iterative methods, and $ρ_\infty$ can be found in a few Arnoldi iterations. Using the same inverse map $\mathcal{S}^{-1}$ as preconditioner, we compute the low-lying spectrum efficiently via shift-invert Arnoldi, and, as a proof of concept, build an implicit time integrator that is competitive on stiff systems in the low-precision regime. For the steady-state and low excited states problems, our methods scale like $O(n^3)$ per iteration and offer state-of-the-art performance on CPU and GPU.

3. Vasudeva Murthy 松弛系统的积分不变量：分析与数值验证

Integral Invariants of Vasudeva Murthy’s Relaxation Systems: Analysis and Numerical Validation

作者: Sudipta Sahu, Rathan Samala

类别: math.NA

发布日期: 2026-08-31

arXiv ID: 2608.30770v1

链接: http://arxiv.org/abs/2608.30770v1

摘要: Vasudeva Murthy’s relaxation approach [A.S. Vasudeva Murthy, J. Comput. Appl. Math., 203(2), pp. 437-443, 2007], originally proposed for the Jin-Xin relaxation model, provides an alternative formulation with invariant properties that is consistent and retains the semilinear structure incomparison to the standard one. In this work, Vasudeva Murthy’s relaxation approach for various relaxation systems are proposed such as the shallow water equations, the Broadwell model, the Euler equations with heat transfer and two-dimensional Jin-Xin model. For proposed relaxation models, the associated integral invariants are rigorously established at the theoretical level. The main advantage of the integral invariant is that it provides a conserved quantity for the relaxation system by incorporating the coupled contributions of the solution variables in vector form. To validate the analytical results, numerical simulations are carried out for each model using three second-order numerical schemes: CS-EBT2, a semi-implicit second-order central finite-volume scheme for hyperbolic systems with relaxation source terms [S. Sahu, E. Macca, and R. Samala, J. Comput. Phys., 563, 115100, 2026]; UCS2, a finite-volume central relaxation-type scheme [S. F. Liotta, V. Romano, and G. Russo, SIAM J. Numer. Anal., 38(4), 1337-1356, 2000]; and IMEX-RK2, a second-order Implicit-Explicit Runge-Kutta scheme [Pareschi and Russo, J. Sci. Comput., 25, 129-155, 2005]. Numerical results, compared to exact or finely resolved reference solutions, confirm that the models preserve integral invariants, remain stable under CFL restrictions, and exhibit robust and accurate behavior across all benchmark systems tested.

4. 面向Poisson-Nernst-Planck模型的泊松约束JKO格式的高效原始-对偶分裂方法

Efficient primal–dual splitting methods for a Poisson-constrained JKO scheme for Poisson-Nernst-Planck models

作者: Wei Wu, Jin Zeng, Zhen Zhang, Chaozhen Wei

类别: math.NA, math.OC, physics.comp-ph

发布日期: 2026-08-31

arXiv ID: 2608.30693v1

链接: http://arxiv.org/abs/2608.30693v1

摘要: The Poisson–Nernst–Planck (PNP) equations strongly couple ionic transport and electrostatic interactions through the Poisson equation, posing substantial numerical challenges under small permittivity and complex potential boundary conditions. Underlying these equations is a natural Wasserstein gradient-flow structure, in which the Poisson equation serves as a local realization of the nonlocal electrostatic interaction energy. Exploiting this structure, we formulate each time step as a constrained convex minimization problem where the ionic continuity equations and the Poisson equation are incorporated as linear constraints, allowing the concentrations, fluxes, and electrostatic potential to be updated simultaneously. The variational structure of the scheme intrinsically guarantees the dissipation of the original free energy, mass conservation, and nonnegativity of ionic concentrations under general electrostatic boundary conditions. Moreover, the framework is structurally modular: extending from classical to modified PNP models with steric interactions and concentration-gradient corrections requires only modifying the energy functional, while all structure-preserving properties are automatically retained. To efficiently solve the resulting large-scale constrained problems, we develop preconditioned and transformed primal–dual algorithms equipped with tailored fast dual solvers, namely DCT-based direct and Schur-complement iterative methods, that exploit the coupled block structure of the PDE constraints. Numerical experiments on classical and modified PNP systems demonstrate the accuracy and structure-preserving properties of the scheme, and show that the proposed algorithms converge reliably in strongly coupled small-permittivity regimes without significant growth in computational cost.

5. Signorini问题的势垒正则化对称Nitsche方法

A Barrier-Regularized Symmetric Nitsche Method for the Signorini Problem

作者: Peter Hansbo, Mats G. Larson

类别: math.NA

发布日期: 2026-08-31

arXiv ID: 2608.30470v1

链接: http://arxiv.org/abs/2608.30470v1

摘要: We introduce and analyze a barrier-regularized symmetric Nitsche method for the scalar Signorini problem. Applying a logarithmic barrier to the nonnegative slack variable in an augmented Lagrangian and then eliminating that variable yields a smooth positive-part operator and the perturbed complementarity relation on a primal-dual central path, with barrier parameter $μ=γs$. For every $s>0$, the discrete problem is continuously differentiable, uniquely solvable, and has a symmetric positive definite Newton matrix when the Nitsche parameter is sufficiently large. Under a mild barrier-feasibility condition, the continuous logarithmic energy has a unique minimizer $u_μ$ for every $μ>0$, its gap is positive almost everywhere, and $\|u-u_μ\|_{H^1(Ω)}\lesssimμ^{1/2}$. Under the Sobolev regularity used for finite element approximation, this minimizer satisfies the (L^2) central-path boundary law. If the obstacle is locally the trace of an $H^2$-function, we also prove a uniform local $H^2$ bound in the interior of each planar contact face, on neighborhoods that may contain a free-boundary point of the limiting Signorini solution. The method is exactly consistent and quasi-optimal relative to $u_μ$, with constants independent of $s$. If $u_μ$ is uniformly bounded in $H^r(Ω)$, $3/2

6. 从多层问题到多个两层问题：一种新的频时混合多次散射积分方程求解器

From multi-layered problems to multiple two-layered problems: a novel frequency-time hybrid multiple-scattering integral equation solver

作者: Shuai Pan, Tao Yin, Lu Zhang

类别: math.NA

发布日期: 2026-08-31

arXiv ID: 2608.30287v1

链接: http://arxiv.org/abs/2608.30287v1

摘要: This paper proposes a novel frequency-time hybrid multiple scattering (FTH-MS) integral equation solver for time-dependent wave equation problems in general multi-layered media, with remarkable scalability with respect to the number of layers $N$. In light of the finite speed of wave propagation, the new methodology provides an innovative multiple-scattering idea of re-modeling the original $N$-layered problem into a sequence of $N-1$ two-layered sub-problems, for which the main advantages lie in that (i) each sub-problem enjoys much simpler wave scattering properties compared with the complicated problem in a multi-layered medium, (ii) it enables to develop high-accuracy solver utilizing Fourier transform and frequency-domain boundary integral equation (BIE) method; and (iii) numerical evaluation of the sub-problems in each multiple scattering step can be parallelized. Both multiplicative- and additive-type strategies are developed and equivalence results, which indicate that the $M$-th order multiple scattering sums can provide equivalent representations of the solutions up to a certain time $T(M)$, are rigorously derived. Owing to the existed result of exponential convergence of the perfectly-matched-layer (PML) truncation for two-layered problem, all the sub-problems is numerically resolved by means of the FTH method based on the Fourier transform and the PML-BIE method whose numerical evaluation is addressed utilizing the Chebyshev-based rectangular-polar solver with high accuracy. Numerical examples are presented to validate the efficiency and accuracy of the proposed method.

7. 标量常微分方程的镜面椭圆方法：高达四阶的精确性与精度

The specular ellipse method for scalar ordinary differential equations: exactness and accuracy up to fourth order

作者: Kiyuob Jung

类别: math.NA

发布日期: 2026-08-31

arXiv ID: 2608.30280v1

链接: http://arxiv.org/abs/2608.30280v1

摘要: This paper introduces a family of one-step implicit methods for solving scalar ordinary differential equations. At each step, the update uses a scaled angular mean of two vector-field evaluations, and the positive scale may vary from step to step. The leading terms of the local truncation error can be expressed in terms of the derivative of the signed curvature of the scaled solution graph. We prove that the proposed method reproduces the exact solution at the mesh points when the solution graph has constant signed curvature under a fixed positive scaling and each implicit update is unique. When this special geometric condition is not satisfied, we establish second-order consistency and convergence for positive scale sequences satisfying suitable uniform conditions. Furthermore, third- and fourth-order consistency and convergence can be achieved by choosing the scale to cancel the relevant curvature terms in the local truncation error. Using only the given problem data, we classify when these improvements are possible and determine the corresponding scale choices. An example shows that the proposed fourth-order method can yield smaller errors than the classical fourth-order Runge–Kutta method at the same step size.

8. 带多重半定惩罚的参数鲁棒子空间校正

Parameter-Robust Subspace Correction with Multiple Semidefinite Penalties

作者: Subhransu S. Bhattacharjee

类别: math.NA

发布日期: 2026-08-31

arXiv ID: 2608.30265v1

链接: http://arxiv.org/abs/2608.30265v1

摘要: Independently weighted semidefinite penalties arise in augmented-Lagrangian and constrained formulations. This paper characterizes when an exact additive subspace-correction preconditioner remains uniformly effective over all nonnegative penalty weights on a fixed finite-dimensional space. Robustness holds precisely when the correction spaces decompose every joint kernel generated by a nonempty subset of penalties. If one condition fails, a computable constant determines the exact first-order decay of the smallest preconditioned eigenvalue along the associated parameter ray, and the condition number grows linearly; none of the subset conditions can be discarded in general. Filtered decompositions provide computable sufficient bounds on parameter-ordering cones, while distributive kernel lattices permit a single common splitting. Exact-additive computations confirm the characterization and predicted rates. Separate Scott-Vogelius experiments produce stable multilevel iteration counts over the tested weights and mesh levels. The analysis does not establish mesh-uniformity.

9. 带可逆边界的双松弛时间向量格子玻尔兹曼格式的网格均匀幂稳定性

Mesh-Uniform Power Stability of Two-Relaxation-Time Vector Lattice Boltzmann Schemes with Reversible Boundaries

作者: Jin Zhao

类别: math.NA

发布日期: 2026-08-31

arXiv ID: 2608.30253v1

链接: http://arxiv.org/abs/2608.30253v1

摘要: We consider the collision-transport operator for vector-valued two-relaxation-time lattice Boltzmann schemes linearized about a uniform rest state. Assume that the equilibrium blocks are positive definite and that the link-even and link-odd relaxation parameters satisfy s_+ + s_- = 2 and 0 < s_- < 2. If the homogeneous transport is unitary in the equilibrium metric and reversible under velocity exchange, then the powers of the amplification operator are bounded uniformly with respect to the number and arrangement of lattice nodes. The admissible transports include periodic transport, vector halfway bounce-back, coordinate-aligned specular reflection, tangential orthogonal involutions, and compatible multi-channel scattering. The proof reduces the population equation to a two-step macroscopic recurrence generated by a contraction. An inclusion of the numerical range in an ellipse, combined with the Crouzeix-Palencia theorem, yields a dimension-independent estimate for the companion operator and hence the population bound. For a three-coefficient off-midpoint boundary interpolation, we give an exact rational D2N5 example whose finite-domain amplification matrix has a real eigenvalue larger than one, although the interpolation coefficients and bulk parameters are admissible. Thus coefficient convexity alone does not ensure stability for this boundary family.

10. 棋盘格壳：一种具有尺度兼容补全的纯位置薄壳离散

Checkerboard Shells: A Position-Only Thin-Shell Discretization with Scale-Compatible Completion

作者: Junjie Song, Xuanyu Wu, Zhifeng Zhang, Bingtao Hu, Zhaoxi Hong, Xiuju Song, Yixiong Feng, Jianrong Tan

类别: math.NA

发布日期: 2026-08-31

arXiv ID: 2608.30195v1

链接: http://arxiv.org/abs/2608.30195v1

摘要: Checkerboard edge-midpoint geometry provides an exact planar Varignon parallelogram for every spatial quadrilateral, allowing a local tangent frame and normal to be recovered directly from nodal positions even when the raw quadrilateral is warped. Building on this property, we develop a position-only thin-shell discretization with no independent director, rotation, or strain variables. The connected edge-midpoint surface is taken as the physical midsurface: the first fundamental form is evaluated on planar B faces, while a W-centered second fundamental form is constructed from variations of neighboring B-face normals, so membrane and bending share the same geometric carrier. A variational-kernel analysis shows that smooth second-order consistency does not eliminate lattice-scale blind modes. After quotienting out the raw checkerboard gauge, the B metric has one physical membrane blind direction and the symmetric W curvature has two curvature blind directions. We introduce a quotient-minimal membrane compatibility coordinate $X_M$ and an objective reference-relative curvature coordinate $X_W^{rel}$, placed consistently in the $O(t)$ membrane and $O(t^3)$ bending sectors. The formulation admits an explicit midpoint quotient, complete flat blind-mode classification, rigid-motion objectivity, reference-state consistency, and an $O(h^2)$ near-isometry approximation result for aligned generalized cylinders. Numerical tests show second-order curvature convergence, targeted removal of the membrane defect, and a sub-percent, refinement-decaying influence of $X_W^{rel}$. Linear and nonlinear shell benchmarks further demonstrate flat bending, curved-shell membrane-bending coupling, thickness sensitivity, large rotation, nonlinear pinching, and localized ovalization within a single position-only framework.

11. 非自伴几乎Mathieu矩阵的谱与苏格兰旗算子

Spectra of Non-Self-Adjoint Almost Mathieu Matrices and the Scottish Flag Operator

作者: Simon Becker, Izak Oltman

类别: math.SP, math.NA

发布日期: 2026-08-31

arXiv ID: 2608.30191v1

链接: http://arxiv.org/abs/2608.30191v1

摘要: For $N\geq 3$ and a potential phase $\vartheta\in\mathbb{R}$, we study the non-self-adjoint almost Mathieu matrix obtained by multiplying the discrete Laplacian by a complex phase with angle $\varphi\in\mathbb{R}$, $A_N(\varphi,\vartheta)=e^{i\varphi}(S+S^{-1})/2+\operatorname{diag}(\cos(2πj/N+\vartheta))_{j\in\mathbb{Z}/N\mathbb{Z}}$, where $S e_j=e_{j+1}$ is the periodic shift on $\mathbb{C}^N$. We derive a Chambers formula and isolate the part $Q_{N,\varphi}$ of the characteristic polynomial that depends only on $N$ and $\varphi$, but not on $\vartheta$ or on a change of boundary conditions for the shift operator. We then show, for every $N$, that the zeros of $Q_{N,\varphi}$ lie on the two perpendicular lines $e^{i\varphi/2}\mathbb{R}\cup e^{i(\varphi/2+π/2)}\mathbb{R}$. For even $N$, the same property holds for the matrices $A_N(\varphi,\vartheta)$ with $\vartheta\in 2π\mathbb{Z}/N$, and we compute their limiting eigenvalue measure explicitly. For $\varphi\in[-π,π]$, the eigenvalue distribution approximates elliptic-integral densities with masses $1-|\varphi|/π$ and $|\varphi|/π$, and maximal radii $2|\cos(\varphi/2)|$ and $2|\sin(\varphi/2)|$, respectively. At $\varphi=π/2$, the central polynomial $Q_{N,\varphi}$ factors into positive quartic factors. This proves that the Scottish flag matrix, after Trefethen and Chapman, has its spectrum on the two diagonal lines of the saltire.

12. 稀疏傅里叶神经算子中的选择、表示与执行

Selection, Representation, and Execution in Sparse Fourier Neural Operators

作者: Abdul Qadir Ibrahim, Martin Burger

类别: cs.LG, math.NA

发布日期: 2026-08-30

arXiv ID: 2608.30070v1

链接: http://arxiv.org/abs/2608.30070v1

摘要: Sparse representations are often expected to make models smaller and also reduce inference cost. For Fourier Neural Operators (FNOs), these objectives are not equivalent or do not always align: removing parts of the learned operator can leave the underlying transforms and dense computations unchanged, while changing the grid on which the model is evaluated can introduce overhead of its own. We therefore distinguish sparsity in the representation, in the stored parameters, in the theoretical operation count, and in measured runtime, and present an empirical study of several routes toward sparse FNOs that tests each transition between them separately. Coarsening the execution grid reduces the theoretical cost without reducing measured latency, and adding a correction term recovers accuracy at the cost of making the model slower. Even an 83% parameter reduction remains slower than the dense baseline under ordinary execution. These results motivate a stricter definition of useful sparsity: the deployed operator must preserve solution accuracy and map its reduced support to a genuinely cheaper execution path.

13. 不定复对称线性系统块预条件子的一种变体

A variant of the block preconditioner for indefinite complex symmetric linear systems

作者: Mehdi Makhdomi, Davod Khojasteh Salkuyeh

类别: math.NA

发布日期: 2026-08-30

arXiv ID: 2608.29985v1

链接: http://arxiv.org/abs/2608.29985v1

摘要: In this paper, we propose an efficient preconditioner for solving indefinite complex symmetric linear systems within a block preconditioning framework. We analyze the convergence of the corresponding iterative method and investigate several spectral properties of the preconditioned matrix, including eigenvalue distributions and eigenvector structures. The new preconditioner is used to accelerate the convergence of the flexible version of GMRES. Numerical experiments are presented to illustrate the effectiveness of the proposed preconditioner, and comparisons with existing block preconditioners demonstrate its superior performance.

14. 谱配点法在高阶特征值问题中的进展

Advancements in Spectral Collocation Methods for High-Order Eigenvalue Problems

作者: Sameh Gana

类别: math.NA

发布日期: 2026-08-30

arXiv ID: 2608.29932v1

链接: http://arxiv.org/abs/2608.29932v1

摘要: This paper focuses on computing spectral solutions for high-order eigenvalue problems using an efficient discretization method based on Chebfun spectral discretization algorithms and domain truncation. We solve several numerical eigenvalue problems, demonstrating both the accuracy and computational efficiency of the proposed approach.

15. 多原子玻尔兹曼碰撞算子的Wigner-Eckart分解

Wigner-Eckart Factorization of the Polyatomic Boltzmann Collision Operator

作者: Torsten Keßler, Bas W. T. Gieling, René R. Hiemstra, Michael R. A. Abdelmalik

类别: math.NA, physics.comp-ph

发布日期: 2026-08-30

arXiv ID: 2608.29825v1

链接: http://arxiv.org/abs/2608.29825v1

摘要: We extend the Wigner-Eckart factorization of the spectral Boltzmann collision operator to polyatomic gases with continuous internal energy. Because internal energies are invariant under spatial rotations, the SO(3) reduction survives the Borgnakke-Larsen energy exchange, and the twelve-dimensional collision integral collapses onto a nine-dimensional kinematic core. The core splits into a sparse geometric tensor, evaluated exactly, and a dense physical tensor, integrated by singularity-resolving Gauss rules with an auxiliary Laplace representation of the fractional energy couplings. The quadrature attains near machine precision at the fractional exponents of real gases. The collision invariants are embedded exactly, preserving the translational-internal energy exchange. The factorization compresses the operator by three to nearly four orders of magnitude and accelerates its evaluation 40-fold over dense formulations. The method is validated against the exact monatomic limit, Landau-Teller relaxation, and an analytic frozen-channel Prandtl number, and it matches a published calibration of the same kernel for N2, CO, and H2.

16. 非线性热孔隙弹性问题顺序有限元方法的最优误差估计

Optimal error estimates of sequential finite element method for nonlinear thermo-poroelasticity problems

作者: Fang Wang, Fan Chen, Changqing Lv, Chenguang Zhou

类别: math.NA

发布日期: 2026-08-30

arXiv ID: 2608.29799v1

链接: http://arxiv.org/abs/2608.29799v1

摘要: This study introduces and analyzes a three-step sequential decoupling algorithm designed to address nonlinear, fully coupled quasi-static thermo-poroelasticity systems incorporating convective transport. The finite element method is employed for spatial discretization and the backward Euler method for temporal discretization. The proposed sequential method has a higher computing efficiency than the fully implicit nonlinear numerical scheme, since it does not require any internal iterations. The well-posedness of the numerical solution is discussed by introducing a cut-off operator and the stability analysis of the algorithm is performed. Rigorous analysis yields optimal convergence order estimates for both spatial and temporal discretizations. In order to confirm the theoretical results and the effectiveness of the suggested approach, numerical experiments are finally carried out.

17. 用于具有自适应特征重构的狭义相对论流体动力学的高效混合WENO格式

Efficient Hybrid WENO Schemes for Special Relativistic Hydrodynamics with Adaptive Characteristic Reconstruction

作者: Rakesh Kumar, Biswarup Biswas, Asha Kumari Meena, Harish Kumar

类别: math.NA

发布日期: 2026-08-30

arXiv ID: 2608.29776v1

链接: http://arxiv.org/abs/2608.29776v1

摘要: Special relativistic hydrodynamics (SRHD) equations arise in the modeling of high-speed fluid flows encountered in astrophysical phenomena such as jets, supernova explosions, and gamma-ray bursts. Owing to their highly nonlinear hyperbolic nature, solutions often develop strong discontinuities, making the design of stable and accurate numerical schemes challenging. Although Weighted Essentially Non-Oscillatory (WENO) schemes are widely used for such problems, component-wise WENO reconstruction may produce spurious oscillations near discontinuities. On the other hand, characteristic-wise WENO reconstruction provides accurate non-oscillatory solutions for systems of conservation laws, but it involves the computation of eigenvectors in each cell, which leads to high computational cost. In this work, we intend to develop hybrid schemes which maintain the non-oscillatory feature of characteristic-wise WENO while being less costly. We propose three hybrid schemes, namely the H1-WENO, H2-WENO, and H3-WENO schemes, based on a new troubled-cell indicator constructed from the smoothness indicators of the WENO scheme. The proposed troubled-cell indicator effectively distinguishes smooth and discontinuous regions, allowing the hybrid schemes to employ inexpensive reconstructions in smooth regions and the characteristic-wise WENO reconstruction only near discontinuities. Numerical experiments demonstrate that the proposed schemes retain the accuracy and robustness of characteristic-wise WENO methods while significantly reducing the computational cost. In particular, the H1-WENO scheme achieves an approximately 30–40% improvement in computational efficiency compared to the standard WENO scheme in 2D test cases.

18. 一般状态方程下的相对论流体动力学的约束保持AFD-WENO格式

Constraint Preserving AFD-WENO Schemes for Relativistic Hydrodynamics with General Equations of State

作者: Pramodit Mishra, Shubham Upadhyay, Rakesh Kumar, Biswarup Biswas

类别: math.NA

发布日期: 2026-08-30

arXiv ID: 2608.29654v1

链接: http://arxiv.org/abs/2608.29654v1

摘要: We develop a high-order physical-constraint-preserving (PCP) alternative finite difference weighted essentially non-oscillatory (AFD-WENO) scheme for the special relativistic hydrodynamics equations with general equations of state. The proposed scheme comprises two key limiters: a state limiter, which acts after the WENO state interpolation step, and a flux limiter, which acts on the final high-order fluxes. The state limiter ensures that the interpolated states are physically admissible, while the flux limiter ensures that the numerical fluxes are physically admissible. The resulting scheme is rigorously proved to satisfy the physical constraints. Incorporating multiple WENO interpolation techniques, including an improved adaptive-order formulation (WENO-AOI), the method is validated through extensive one- and two-dimensional numerical benchmarks with various equations of state. The numerical results demonstrate high-order accuracy, sharp resolution of discontinuities, and robust stability in extreme relativistic regimes.

19. 张量正交子空间分裂：理论与应用

Tensor Orthogonal Subspace Split: Theory and Applications

作者: Jifei Miao, Juan Han, Michael K. Ng, Kit Ian Kou

类别: math.NA

发布日期: 2026-08-30

arXiv ID: 2608.29638v1

链接: http://arxiv.org/abs/2608.29638v1

摘要: Tensor representations have emerged as a fundamental paradigm for modeling multidimensional data by preserving intrinsic correlations across multiple modes. This paper proposes a novel theoretical framework, termed Tensor Orthogonal Subspace Split (TOSS), which explicitly splits a tensor, along a prescribed mode, into two orthogonal components: a dominant component lying in a prescribed subspace and a residual component lying in the corresponding orthogonal complement. We first present the general formulation of TOSS and systematically investigate its fundamental properties. As an important and practically meaningful special case, we further introduce the rank-one TOSS, which imposes a separable rank-one structure along the splitting mode and admits a clear geometric interpretation. This formulation naturally captures dominant consistent patterns while effectively isolating orthogonal residual component. The proposed framework establishes a unified theoretical foundation for tensor-domain orthogonal split and opens new avenues for structured tensor modeling across diverse applications. Building upon the developed TOSS theory, hyperspectral image restoration and color video background modeling are considered as two representative tasks, for which corresponding optimization models are formulated. Efficient algorithms are developed to solve the resulting problems. Extensive experimental results validate the effectiveness and superiority of the proposed approaches.

20. 由分数高斯噪声驱动的随机多尺度次扩散的分析与逼近

Analysis and Approximation of Stochastic Multiscale Subdiffusion Driven by Fractional Gaussian Noise

作者: Jincheng Dong, Ning Du, Xu Guo, Mengmeng Liu, Xiangcheng Zheng

类别: math.NA

发布日期: 2026-08-30

arXiv ID: 2608.29533v1

链接: http://arxiv.org/abs/2608.29533v1

摘要: This paper investigates a stochastic multiscale subdiffusion model driven by fractional Gaussian noise, where the multiscale Abel kernel with variable exponent $α(t)\in(0,1)$ is used to capture multiscale and crossover behavior in anomalous diffusion. The main difficulties of this model lie in the complexity of the multiscale Abel kernel (e.g. non-monotonicity and non-coercivity) and the low regularity caused by the noise. Concerning these issues, we prove the well-posedness and regularity of the mild solutions by means of solution operator approach and a perturbation technique for multiscale Abel kernel. Then both the semidiscrete-in-time and fully-discrete numerical schemes are proposed and analyzed under the low-regularity numerical analysis framework, with proved temporal and spatial convergence rates. Numerical experiments are presented to substantiate the theoretical results.

21. Nechvile变换的航天器动力学与三体问题中的推进剂计算

Nechvile-Transformed Spacecraft Dynamics and Propellant Computation in the 3-Body Problem

作者: Michael J. Dixon, Isaac M. Ross

类别: math.OC, eess.SY, math.NA

发布日期: 2026-08-29

arXiv ID: 2608.29271v1

链接: http://arxiv.org/abs/2608.29271v1

摘要: The uncontrolled equations of motion in the Nechvile frame for the restricted three-body problem have been well-known since at least the 1960s. It would seem that adding an external force to these equations is quite trivial: simply add an external force per mass term to the acceleration equations. Here we show that the last statement is not true. In fact, we show that the additive generic external force must be multiplied by the inverse of $(1 + e \cosθ)^3$ where $e$ is the relative eccentricity of the primaries and $θ$ is the true anomaly of the rotating frame located at the barycenter. Furthermore, when this result is combined with the mass flow rate equation, it generates several surprising results due to the mismatch between the resulting quadratic term and the cubic term in the equations of motion. This leads to a corresponding modification of the rocket equation itself. A Birkhoff-theoretic solution to an illustrative cislunar space mission problem shows propellent savings of 80% with the use of the correct cost functional. The popular quadratic cost utilizes more than $2X$ the minimum propellant consumption.

22. 平稳时空上对称约化广义相对论流体动力学的熵稳定且保持物理约束的DGSEM

Entropy-Stable and Physical-Constraint-Preserving DGSEM for Symmetry-Reduced General-Relativistic Hydrodynamics on Stationary Spacetimes

作者: Guosheng Fu, Jian-Guo Liu

类别: math.NA, gr-qc, physics.comp-ph

发布日期: 2026-08-29

arXiv ID: 2608.29229v1

链接: http://arxiv.org/abs/2608.29229v1

摘要: We develop an entropy-stable and physical-constraint-preserving discontinuous Galerkin spectral element method for symmetry-reduced general-relativistic hydrodynamics on prescribed stationary spacetimes. Using a local orthonormal transformation, the fluid variables are expressed in a form for which the relativistic hydrodynamic algebra and the admissible set are independent of the spatial metric, while the spacetime geometry enters through stationary coefficients. This separation allows entropy-conservative special-relativistic fluxes to be combined with a compatible discretization of the geometric source terms. On affine tensor-product meshes, the resulting DGSEM is conservative and satisfies a semidiscrete entropy inequality, while the transformed variables provide a convex framework for physical-constraint preservation. For practical stabilization, we use a geometry-only causal speed that is sufficient for both classical local Lax–Friedrichs entropy dissipation and the physical-constraint-preserving Lax–Friedrichs splitting. The fully discrete method combines this stabilization with SSP Runge–Kutta time stepping, oscillation elimination, and conservative local-orthonormal-state scaling. Numerical experiments cover smooth and strongly shocked special-relativistic flows, an axisymmetric jet, stationary Michel accretion, Schwarzschild Bondi–Hoyle flow, and four Kerr accretion cases. The results demonstrate the designed high-order accuracy in smooth regimes and robust performance for demanding relativistic flows on curved stationary backgrounds.

23. 随机自参数系统离散化下的随机吸引子与几乎必然稳定性

Random attractors and almost-sure stability under discretization of a stochastic autoparametric system

作者: Chuchu Chen, Jialin Hong, Yibo Wang

类别: math.DS, math.NA

发布日期: 2026-08-29

arXiv ID: 2608.29149v1

链接: http://arxiv.org/abs/2608.29149v1

摘要: For a stochastic autoparametric block-and-pendulum system, the long-time dynamics exhibit two fundamental features: the almost-sure stability of the single mode solution, characterized by its Lyapunov exponent, and the global asymptotic dynamics when this single mode solution loses stability. This naturally raises the question of whether these dynamical features are preserved under discretization, since such preservation is essential for the resulting discrete system to faithfully capture the qualitative behavior of the continuous system. To address this question, we first establish the existence of a random attractor for the continuous system subject to multiplicative stochastic excitation, providing a rigorous characterization of the global asymptotic dynamics. We then propose a numerical discretization that induces a discrete random dynamical system and prove the convergence of its random attractor to the continuous one as the step size tends to zero. In addition, we show that the numerical Lyapunov exponent of the single mode solution has the same sign as its continuous counterpart for sufficiently small step sizes, thus preserving the corresponding almost-sure stability or instability classification. These results demonstrate that the proposed discretization captures both the global asymptotic dynamics and the stability characteristics of the underlying stochastic autoparametric system.

24. D-RADI: 一种求解大规模离散时间代数Riccati方程的低秩ADI算法

D-RADI: A Low-rank ADI Algorithm for Solving Large-scale Discrete-time Algebraic Riccati Equations

作者: Umair Zulfiqar

类别: math.NA, eess.SY

发布日期: 2026-08-29

arXiv ID: 2608.29131v1

链接: http://arxiv.org/abs/2608.29131v1

摘要: The low-rank alternating direction implicit (ADI) method is an efficient numerical technique for solving several types of large-scale matrix equations that admit low-rank solutions. The discrete-time algebraic Riccati equation (DARE) is an important matrix equation with applications in state estimation, controller design, and filter design. In the literature, the low-rank Cholesky factor ADI method for Stein equations has been used within Newton iterations to solve large-scale DAREs. However, no dedicated low-rank ADI solver is available for such DAREs. To address this gap, this paper presents a low-rank ADI solver for large-scale DAREs. We also propose an efficient approach to generate ADI shifts automatically, which makes the proposed solver fully autonomous for solving DAREs. The effectiveness of the proposed solver is compared with MATLAB’s \texttt{idare} on a moderate-order problem. Efficiency and accuracy are further demonstrated on large-scale DAREs of order $10^6$. Numerical results confirm that the solver is efficient, accurate, and fully autonomous.

25. 参数化概率流形分解的收敛性分析

Convergence analysis of Parametric Probabilistic Manifold Decomposition

作者: Jiaming Guo, Dunhui Xiao

类别: math.NA

发布日期: 2026-08-29

arXiv ID: 2608.29105v1

链接: http://arxiv.org/abs/2608.29105v1

摘要: This paper presents a convergence analysis for a newly developed nonlinear model reduction method: parametric probabilistic manifold decomposition (PPMD)~\cite{guo2026parametric}. In addition, existing analyzes of nonlinear reduced order models typically treat subspace reduction, manifold representation, regression, and nonlinear reconstruction as separate components and often remain at the level of discrete state vectors. To the best of our knowledge, no theory tracks the complete error propagation in a data-dependent model whose basis, residual geometry, spectral coordinates, parameter maps, and lifting operator are all learned from the same numerical solution data. We develop a coupled perturbation analysis for the entire PPMD procedure. A trajectory geometry induced by the spatial discretization and temporal quadrature connects discrete trajectory vectors isometrically with the corresponding PDE norm. Population spectral objects are introduced to align the empirical residual coordinates and derive a uniform coordinate error estimate, whose propagation through the Hilbert-valued kernel lifting estimator is then quantified. Combining these results with the full order discretization error, weighted low-rank approximation, parameter regression, and residual representation defect yields deterministic and high-probability trajectory error bounds and consistency in probability in the continuous PDE trajectory space. The theory identifies how the principal errors interact and which components limit the accuracy of the nonlinear reduced order model.

26. 非线性场重构的含缺失数据概率流形分解

Gappy probabilistic manifold decomposition for nonlinear field reconstruction

作者: Qihan Feng, Jiaming Guo, Jiarun Meng, Dunhui Xiao

类别: math.NA

发布日期: 2026-08-29

arXiv ID: 2608.29014v1

链接: http://arxiv.org/abs/2608.29014v1

摘要: This paper proposes gappy probabilistic manifold decomposition (Gappy PMD), a nonlinear method for reconstructing high-dimensional fields from extremely sparse measurements. Gappy PMD reconstructs the field on the nonlinear manifold learned by probabilistic manifold decomposition (PMD). We further propose a differentiable point selection method for reduced-order model (ROM)-based field reconstruction (DPS). Using differentiable meshless interpolation within the ROM-based reconstruction framework, DPS makes the full-field reconstruction error differentiable with respect to the sampling locations and directly optimizes these locations. In addition, a theoretical error analysis for Gappy PMD is also given. It splits the squared reconstruction error into two orthogonal parts: one normal to the reconstruction manifold and the other induced by sparse sampling and observation noise. Under a stability condition on the sampling operator, this error vanishes with the PMD approximation error and the noise. The Gappy PMD is evaluated on three numerical test cases: flow past a cylinder, lid-driven cavity flow, and backward-facing step flow. For the same reduced dimension and sampling points, Gappy PMD attains mean relative $L^2$ errors one to two orders of magnitude below Gappy POD. Optimizing the sampling points with DPS further improves reconstruction accuracy and robustness.

27. Hermite矩阵非对角块扰动的酉不变范数的一个最优界

A Sharp Unitarily Invariant Norm Bound for the Off-Diagonal Block Perturbation of a Hermitian Matrix

作者: Lei-Hong Zhang, Ren-Cang Li

类别: math.NA

发布日期: 2026-08-29

arXiv ID: 2608.29009v1

链接: http://arxiv.org/abs/2608.29009v1

摘要: Let

be two partitioned Hermitian matrices, where $\widetilde A$ is obtained from $A$ by simply dropping the off-diagonal blocks, and let $η$ be the gap between the spectra ${\rm eig}(H_1)$ of $H_1$ and ${\rm eig}(H_2)$ of $H_2$. Define, for $δ\ge 0$ and $ε\ge 0$,

and let $V=A-\widetilde A$ and $ε_2=\|E\|_2=\|V\|_2$, the matrix spectral norm. Li and Li [{\em Linear Algebra Appl.}, 395:183–190, 2005] established a sharp spectral-norm bound on the changes in the eigenvalues of $A$:

where $\pmbλ(A)$ is the vector whose components are the eigenvalues of $A$ in descending order and similarly for $\pmbλ(\widetilde A)$. The goal of this paper is to resolve the question: how far an extension of this result in the form

remains valid for some or all unitarily invariant norms $\|\cdot\|_{\rm UI}$? Two results are obtained: (a) the extension holds for any $Q$-norm, a subclass of unitarily invariant norms that encompasses the Schatten $p$-norm for $2\le p\le\infty$ (particularly, the Frobenius norm and the spectral norm included), and (b) the extension holds for any unitarily invariant norm if ${\rm rank}(E)\le 1$. It is demonstrated that the equality is attained on the $2\times 2$ matrix $A$.

28. 最小化雅可比方法用于特征值和奇异值的算术与通信复杂度：第二部分——并行算法

Minimizing the Arithmetic and Communication Complexity of Jacobi’s Method for Eigenvalues and Singular Values: Part Two – Parallel Algorithms

作者: James Demmel, Hengrui Luo, Ryan Schneider, Yifu Wang

类别: math.NA

发布日期: 2026-08-28

arXiv ID: 2608.28952v1

链接: http://arxiv.org/abs/2608.28952v1

摘要: This paper presents several parallel versions of Jacobi’s method for the symmetric eigenvalue problem and the SVD. A continuation of [Demmel, Luo, Schneider, & Wang 2025], we develop parallel Jacobi algorithms whose arithmetic cost is optimal and whose bandwidth or latency can match the corresponding lower bounds of parallel matrix multiplication. Our focus is a standard distributed-memory setting with variable processor layouts, including both 2D and 2.5D processor grids. In the 2D case, we demonstrate that a standard implementation of parallel Jacobi achieves a perfect speedup in arithmetic cost – i.e., complexity $O(n^3/P)$ when done with $P$ processors – while hitting the 2D matrix-multiplication lower bound for bandwidth and (nearly) the lower bound for latency. By employing a 2.5D processor grid and leveraging 2.5D matrix multiplication, equivalently by increasing the memory per processor, we demonstrate that parallel Jacobi can achieve even lower bandwidth/latency, though we also prove that these costs cannot simultaneously match the best-known bounds for parallel matrix multiplication in any Jacobi algorithm. Finally, we extend our results to one-sided Jacobi SVD.

29. 基于隐式Householder QR分解的GPU加速分块自适应随机值域查找器

A GPU-Accelerated Blocked Adaptive Randomized Range Finder Based on an Implicit Householder QR Decomposition

作者: Carolin Penke, Andreas Herten

类别: math.NA, cs.MS

发布日期: 2026-08-28

arXiv ID: 2608.28941v1

链接: http://arxiv.org/abs/2608.28941v1

摘要: Low-rank methods can reduce the memory and computational requirements of deep neural network training in approaches such as GaLore. Randomized range finders offer an attractive alternative to singular value decompositions, particularly when the required rank is determined adaptively from a prescribed approximation tolerance. We introduce a blocked adaptive randomized range finder based on an implicit Householder QR decomposition and an optimized hybrid CPU–GPU implementation. The proposed method avoids explicit reorthogonalization. Numerical experiments show that it preserves orthogonality and approximation accuracy in regimes where block Gram–Schmidt without reorthogonalization becomes unstable. The blocked formulation exposes matrix–matrix operations and enables overlap of CPU panel factorization with GPU updates. On an NVIDIA GH200, the overlapped implementation reduces the runtime for the largest tested matrix from 9.91 seconds on the CPU to 0.407 seconds. The method provides a stable and efficient building block for low-rank approximation on heterogeneous systems with applications in computational science and engineering.

30. 三角剖分多边形区域上求积的一般超收敛结果

A General Superconvergence Result for Cubature on Triangulated Polygonal Domains

作者: Jonah A. Reeger

类别: math.NA

发布日期: 2026-08-28

arXiv ID: 2608.28883v1

链接: http://arxiv.org/abs/2608.28883v1

摘要: Cubature rules, which approximate definite integrals as a linear combination of a set of function values, are ubiquitous and necessary for computational methods in the physical sciences. A superconvergence result for cubature rules on polygonal domains is developed, whereby a rule that is exact for all bivariate polynomials of a fixed even degree realize an extra order of convergence under a decrease in the spacing between nodes.

31. 使用范数比的幂正则化的可分离非负矩阵分解

Separable Nonnegative Matrix Factorization Using Powered Ratio-of-Norms Regularization

作者: Matthew McCarver, Jing Qin

类别: math.NA, cs.LG, math.OC

发布日期: 2026-08-28

arXiv ID: 2608.28799v1

链接: http://arxiv.org/abs/2608.28799v1

摘要: Separable nonnegative matrix factorization (SNMF) has been widely used for low-rank representation and clustering of nonnegative data, owing to its ability to produce part-based and interpretable decompositions. In particular, SNMF is closely related to graph clustering and community detection. To enhance sparsity and identifiability of the learned factors, we propose an $\ell_1^p/\ell_2$-regularized SNMF model based on a powered ratio-of-norms regularizer. The resulting formulation is nonconvex and nonsmooth, which poses significant challenges for optimization. To address this, we develop efficient algorithms based on the difference-of-convex function algorithm (DCA) and the alternating direction method of multipliers (ADMM). The proposed methods decompose the original problem into tractable subproblems, leveraging closed-form proximal operators associated with the powered norm terms. We establish descent and limiting criticality properties for the DCA scheme and convergence under standard assumptions for the ADMM scheme. Extensive numerical experiments on synthetic datasets and hand gesture classification tasks demonstrate that the proposed approach achieves competitive or improved performance in anchor identification and classification accuracy compared with existing SNMF methods, while maintaining competitive computational efficiency.