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1. 光子晶体中能带结构的混合非拟合有限元逼近

Mixed Unfitted finite element approximation of band structures in photonic crystals

作者: Yuyan Zou, Luke G. Bennetts, Hailong Guo

类别: math.NA

发布日期: 2026-10-19

arXiv ID: 2610.25598v1

摘要翻译: 光子晶体是周期介电结构，其光学性质由具有不连续材料系数的 Maxwell 特征值问题刻画。对于任意形状的夹杂物，由于几何变化时网格生成与重新划分代价高昂，其数值求解仍然具有挑战性。本文针对这一挑战，发展了基于 Floquet-Bloch 变换的混合非拟合有限元方法。我们的方法使用与夹杂物几何无关的均匀背景网格，从而避免界面拟合网格划分，并允许我们通过直接识别自由度来施加周期边界条件。我们构造的一个关键特征是，所得有限元对保持移位算子的 de Rham 交换关系。该结构使我们能够建立关于切割位置一致成立的 inf-sup 条件，并特别排除伪模。我们进一步建立了特征值和特征空间的最优收敛性，其中常数不依赖于网格尺寸和界面位置。最后，我们给出二维和三维的大量数值实验，以验证理论结果并展示所提方法的鲁棒性。

Comment: 无

链接: http://arxiv.org/abs/2610.25598v1

1. 研究对象和背景

本文研究周期性介电结构中的光子晶体能带计算。物理模型为无自由电荷、无电流、线性非色散介质中的时谐 Maxwell 特征值问题：

$$ \nabla\times\left(\varepsilon^{-1}\nabla\times H\right)=\omega^2 H,\qquad \nabla\cdot H=0\quad \text{in }\mathbb{R}^d,\ d=2,3, $$

其中 $H$ 为磁场, $\omega\ge 0$ 为角频率, 介电常数 $\varepsilon(x)$ 按 Bravais 格 $\mathcal{A}$ 周期排列, 并在基体 $\Omega^+$ 与夹杂 $\Omega^-$ 中分别取正常数 $\varepsilon^+$ 与 $\varepsilon^-$, 在材料界面 $\Sigma=\partial\Omega^-$ 上间断。界面条件要求切向磁场与切向电场连续, 即 $[[\gamma_t(H)]]=0$ 与 $[[\gamma_T(\varepsilon^{-1}\nabla\times H)]]=0$。由 Floquet–Bloch 理论, 全场可写成 $H(x)=e^{\mathrm{i}k\cdot x}u_k(x)$, 其中 $u_k$ 按 $\mathcal{A}$ 周期, 准动量 $k$ 位于第一 Brillouin 区 $\mathcal{B}$。于是原问题化为单位胞 $\Omega$ 上的一族准周期界面特征值问题:

$$ \nabla_k\times\left(\varepsilon^{-1}\nabla_k\times u_k\right)=\omega^2 u_k,\qquad \nabla_k\cdot u_k=0, $$

其中移位算子为 $\nabla_k:=\nabla+\mathrm{i}k$, 且界面条件也通过 $\nabla_k\times$ 进入。对每个固定的 $k$, 特征值 $0\le \omega_1(k)^2\le \omega_2(k)^2\le\cdots\to\infty$ 给出色散关系, 全体 $\omega_n(k)$ 构成能带结构 $\sigma=\bigcup_{k\in\mathcal{B}}\bigcup_{n\ge1}\{\omega_n(k)\}$, 其带隙决定电磁波能否传播。因此能带的高精度数值计算是光子晶体设计中的核心问题。

该问题的数值计算有三类本质困难: (i) 系数 $\varepsilon$ 在 $\Sigma$ 上间断, 解只有分片正则性; (ii) Floquet–Bloch 边界条件 $u_k(x+R)=u_k(x)$ 虽在变换后变为周期条件, 但 quasi-momentum 通过移位算子 $\nabla_k=\nabla+\mathrm{i}k$ 进入方程与界面条件; (iii) $\nabla\times\nabla\times$ 算子具有无穷维核, 必须正确处理无散约束, 否则离散谱会出现伪模。现有方法中, 平面波展开法在尖锐界面处因 Gibbs 现象收敛慢, 时域有限差分法用阶梯近似曲界面, 精度受限; 拟合网格有限元方法理论成熟但每变一次夹杂几何都要重新生成界面贴合网格, 代价高。非拟合有限元方法允许界面任意切割背景网格, 可重复使用同一均匀网格, 但已有非拟合 Maxwell 界面工作多针对源问题, 而非特征值问题; 从源问题到谱问题需要离散核的一致控制和排除谱污染。对于光子晶体的准周期 Maxwell 界面特征值问题, 据作者所述, 现有唯一非拟合方法为二维 Petrov–Galerkin 界面有限元, 但缺少严格理论分析。本文试图填补的空白正是: 在二维和三维中建立非拟合离散的严格谱收敛理论, 证明 inf–sup 条件关于界面切割位置一致, 排除伪模, 并给出特征值与特征空间的最优收敛估计。

在弱形式层面, 作者引入标量 Lagrange 乘子 $p$ 处理无散约束, 在分片空间 $V=H_{\mathrm{per}}(\mathrm{curl};\Omega^+\cup\Omega^-)$ 与 $P=H^1_{\mathrm{per}}(\Omega^+\cup\Omega^-)$ 中求 $(\omega^2,u,p)$ 使得

$$ a(u,v)+b(p,v)=\omega^2(u,v),\qquad b(q,u)=0\qquad \forall v\in V,\ q\in P, $$

其中 $a(u,v)=\sum_{m=\pm}\int_{\Omega^m}\varepsilon_m^{-1}(\nabla_k\times u)\cdot(\nabla_k\times v)\,dx$, $b(p,v)=-\sum_{m=\pm}\int_{\Omega^m}p(\nabla_k\cdot v)\,dx$。这一混合形式既保留无散约束, 又为后续构造保持 de Rham 结构的离散空间提供框架。

2. 主要定理或主要结果

作者的主要结论可概括为: 对每个固定 quasi-momentum $k\in\mathcal{B}\setminus\{0\}$, 在界面 $\Sigma$ 足够光滑、背景网格 $\mathcal{T}_h$ 均匀且形状正则、$h$ 足够小、Nitsche 参数与 ghost penalty 参数固定且不依赖于 $h$ 和切割位置的假设下, 所构造的混合非拟合有限元问题适定且稳定, 并具有关于界面切割位置一致的 inf–sup 条件; 由此离散核得到一致控制, 伪模被排除, 特征值与特征空间满足最优谱收敛。摘要与引言明确声称特征值和特征空间具有二阶收敛, 且误差常数不依赖于网格尺寸 $h$ 与界面相对于背景网格的位置。

具体离散框架为: 取不贴合界面的均匀网格 $\mathcal{T}_h$, 定义界面单元集 $\mathcal{T}_\Sigma=\{K\in\mathcal{T}_h: K\cap\Sigma\neq\emptyset\}$; 通过扩展虚拟区域 $\Omega^\pm_h$ 和两套分支构造扩展空间 $V_h^{\mathrm{ext}}\subset V$ 与 $P_h^{\mathrm{ext}}\subset P$。修正 Nédélec 空间与修正 $P^2$ 空间定义为

$$ V_h^m=\{v_h^m\in H(\mathrm{curl};\Omega_h^m): e^{\mathrm{i}k\cdot x}v_h^m|_K\in \mathrm{NE}_1(K),\ \forall K\cap\Omega^m\neq\emptyset\}, $$

$$ P_h^m=\{q_h^m\in H^1(\Omega_h^m): e^{\mathrm{i}k\cdot x}q_h^m|_K\in P^2(K),\ \forall K\cap\Omega^m\neq\emptyset\}, $$

其中 $\mathrm{NE}_1$ 为第二族一阶 Nédélec 边元。全局空间取 $V_h^{\mathrm{ext}}=\{(v_h^+,v_h^-):v_h^m\in V_h^m\}$ 及其周期子空间 $V_{h,\mathrm{per}}^{\mathrm{ext}}$, 对 $P_h^{\mathrm{ext}}$ 同理。周期边界条件通过对相应自由度直接认同实现。界面条件用 Nitsche 方法弱施加, 并用基于面的 ghost penalty 稳定任意小切割单元。

离散形式为: 求 $(\omega_h^2,u_h,p_h)\in\mathbb{R}^+\times V_{h,\mathrm{per}}^{\mathrm{ext}}\times P_{h,\mathrm{per}}^{\mathrm{ext}}$, $u_h\neq0$, 使得

$$ A_h(u_h,v_h)+b_h(p_h,v_h)=\omega_h^2(u_h,v_h)\quad \forall v_h\in V_{h,\mathrm{per}}^{\mathrm{ext}}, $$

$$ b_h(q_h,u_h)-S_h(p_h,q_h)=0\quad \forall q_h\in P_{h,\mathrm{per}}^{\mathrm{ext}}, $$

其中 $A_h=a_h+g_u$ 包含 Nitsche 界面项与磁场 ghost penalty, $S_h=J_h+g_p$ 包含 Lagrange 乘子的界面跳量稳定与 ghost penalty。等价地可写成全局形式 $\mathcal{A}_h[(u_h,p_h),(v_h,q_h)]=\omega_h^2(u_h,v_h)$。

在稳定性方面, 作者首先证明修正空间保持移位算子的 de Rham 交换关系。定义修正插值算子 $\Pi_h^{\mathrm{curl}}=\Phi_k^{-1}\tilde{\Pi}_h^{\mathrm{curl}}\Phi_k$, $\Pi_h^{\mathrm{grad}}=\Phi_k^{-1}\tilde{\Pi}_h^{\mathrm{grad}}\Phi_k$, 其中 $\Phi_k u=e^{\mathrm{i}k\cdot x}u$ 为 Bloch 映射。关键引理为

$$ \Pi_h^{\mathrm{curl}}\nabla_k=\nabla_k\Pi_h^{\mathrm{grad}}, $$

且对任意 $q_h\in P_{h,\mathrm{per}}^{\mathrm{ext}}$ 有 $\Pi_h^{\mathrm{curl}}(\nabla_k q_h)=\nabla_k q_h$ 与 $\nabla_k q_h\in V_{h,\mathrm{per}}^{\mathrm{ext}}$。该性质保证离散标量空间的移位梯度落入离散 Nédélec 空间, 是后续 inf–sup 分析的核心。

另一个关键稳定性工具是切割单元迹不等式: 对每个界面单元 $K\in\mathcal{T}_\Sigma$ 与任意 $v_h\in V_h^m$, 存在与 $h$ 和切割位置无关的常数 $C$ 使得

$$ \|\nabla_k\times v_h^m\|_{0,\Sigma_K}^2\le C h^{-1}\|\nabla_k\times v_h^m\|_{0,K}^2. $$

作者利用 $e^{\mathrm{i}k\cdot x}v_h^m|_K\in\mathrm{NE}_1(K)=[P^1(K)]^d$, 故 $e^{\mathrm{i}k\cdot x}\nabla_k\times v_h^m=\nabla\times(e^{\mathrm{i}k\cdot x}v_h^m)$ 在 $K$ 上为常数, 再结合切割单元迹不等式得到。该估计的常数不依赖切割位置。

在源问题误差估计与谱逼近方面, 作者先建立离散问题的适定性与关于切割位置一致的稳定性, 再得到源问题误差估计; 随后应用 Babuška–Osborn 谱逼近理论, 证明特征值与特征空间收敛, 并给出二阶收敛结论。由于摘要与引言只给出收敛阶和一致性声明, 提供内容未展开具体定理编号、误差泛函的精确形式以及常数所依赖的假设细节, 故无法从现有文本确认各定理的完整陈述。数值实验方面, 作者在二维和三维中进行了广泛实验, 声称验证了理论收敛率并展示了方法对界面位置的稳健性; 具体算例、误差表和收敛阶数据在提供内容中未出现, 无法进一步核对。

3. 研究方法、关键技术和核心工具

本文的技术核心不是简单地把源问题的非拟合有限元搬到特征值问题, 而是构造一个在移位算子 $\nabla_k$ 下仍保持离散 de Rham 复形的混合有限元对, 并用它来获得关于切割位置一致的稳定性和无伪模的谱逼近。主要障碍有三点。第一, 准动量 $k$ 不仅进入方程, 还通过 $\nabla_k\times$ 进入界面跳量条件, 标准未移位 de Rham 复形不能直接给出离散核控制。第二, 界面任意切割背景网格会产生任意小的切割单元, 标准迹估计和逆估计的常数会依赖切割位置, 从而破坏 inf–sup 条件的一致性。第三, Maxwell 特征值问题的谱污染不能仅靠源问题稳定性排除, 必须控制离散核并证明离散解算子按合适意义收敛。作者的真正创新点在于: 通过 Bloch 映射共轭构造修正空间与修正插值, 使第二族 Nédélec 边元与 $P^2$ Lagrange 元在移位算子下形成交换图; 用 Nitsche 方法弱施加界面条件, 用基于面的 ghost penalty 分别稳定两个材料分支上的场与乘子; 并建立与切割位置无关的 inf–sup 条件, 从而排除伪模。

具体证明框架可分为以下层次。第一层是修正 de Rham 结构。作者定义 Bloch 映射 $\Phi_k:u\mapsto e^{\mathrm{i}k\cdot x}u$, 利用 $\nabla_k u=\Phi_k^{-1}\nabla\Phi_k u$ 与 $\nabla_k\times u=\Phi_k^{-1}\nabla\times\Phi_k u$, 把移位问题共轭变换为未移位问题。修正 Nédélec 空间中的基函数形如 $e^{-\mathrm{i}k\cdot x}\phi$, 其中 $\phi$ 遍历标准第二族 $\mathrm{NE}_1$ 基; 修正 $P^2$ 空间同理。修正插值算子通过共轭定义。于是标准未移位交换关系 $\tilde{\Pi}_h^{\mathrm{curl}}\nabla=\nabla\tilde{\Pi}_h^{\mathrm{grad}}$ 被提升为

$$ \Pi_h^{\mathrm{curl}}\nabla_k=\nabla_k\Pi_h^{\mathrm{grad}}. $$

更重要的是, 对离散标量 $q_h\in P_{h,\mathrm{per}}^{\mathrm{ext}}$, 插值固定 $q_h$, 因而 $\nabla_k q_h\in V_{h,\mathrm{per}}^{\mathrm{ext}}$。这一步直接解决了移位梯度与离散 Nédélec 空间不匹配的问题, 是 inf–sup 证明的骨架。

第二层是界面切割的稳健估计。作者采用扩展虚拟区域 $\Omega_h^\pm$, 在每个切割单元上保留两套独立自由度, 并用权重平均 $\{v\}_\theta=\theta_K^+v^++\theta_K^-v^-$, 其中 $\theta_K^\pm$ 由 $\alpha^\pm|K^\pm|$ 定义, $\alpha^\pm=1/\varepsilon^\pm$。Nitsche 项包含 $\int_\Sigma\{\varepsilon^{-1}\nabla_k\times u_h\}_\theta\cdot[[v_h\times n]]$ 及其共轭项和罚项 $\gamma h^{-1}\int_\Sigma[[u_h\times n]]\cdot[[v_h\times n]]$。对 Lagrange 乘子, 弱形式含 $\int_\Sigma[[p_h]]\{v_h\cdot n\}_\theta$ 与界面跳量稳定 $J_h(p_h,q_h)=\gamma_q h^{-1}\int_\Sigma[[p_h]]\cdot[[q_h]]$。为处理小切割单元, 作者引入基于 ghost 面的罚项 $g_u$ 与 $g_p$: $g_u$ 惩罚 $\nabla_k\times u_h^m$ 在 ghost 面上的跳量, $g_p$ 惩罚 $\nabla_k p_h^m$ 及其法向导数在 ghost 面上的跳量。这些罚项分别作用于两个材料分支, 不跨越材料界面耦合, 从而在保持一致性的同时稳定任意小切割。切割单元迹不等式 $\|\nabla_k\times v_h^m\|_{0,\Sigma_K}^2\le C h^{-1}\|\nabla_k\times v_h^m\|_{0,K}^2$ 的常数与切割位置无关, 是证明 Nitsche 项和 ghost penalty 项可控的关键。

第三层是离散稳定性与 inf–sup。作者定义能量半范数

$$ |v_h|_{V_h}^2=\sum_{m=\pm}\|\varepsilon_m^{-1/2}\nabla_k\times v_h\|_{0,\Omega_h^m}^2+\sum_{K\in\mathcal{T}_\Sigma}\frac{\gamma}{h}\|[[v_h\times n]]\|_{0,\Sigma_K}^2+g_u(v_h,v_h), $$

以及 $\|q_h\|_{Q_h}^2=\sum_{m=\pm}\|\nabla_k q_h\|_{0,\Omega_h^m}^2+\sum_{K\in\mathcal{T}_\Sigma}h^{-1}\|[[q_h]]\|_{0,\Sigma_K}^2+g_p(q_h,q_h)$。利用修正交换图和迹估计, 作者证明离散混合形式在乘积范数下满足关于切割位置一致的 inf–sup 条件。该一致性进一步用于控制离散核 $\{v_h:\nabla_k\times v_h=0\text{ 分片}\}$ 及其与梯度空间的交互, 从而排除零特征值附近的伪模。

第四层是谱逼近。作者先对固定右端项的源问题建立稳定性与误差估计, 再验证 Babuška–Osborn 理论所需的条件: 离散解算子一致收敛、离散核维数稳定、特征值隔离等。由于离散问题一致稳定且无伪模, 可得到特征值与特征空间的收敛; 作者进一步声称达到二阶收敛, 且常数不依赖 $h$ 与界面位置。这里需要区分: 从提供内容可以确认作者明确使用了 Babuška–Osborn 框架并声明二阶谱收敛, 但具体定理的假设、误差范数、特征值误差公式以及证明中各项估计的完整细节未在摘录中展开, 无法核实其精确形式。

4. 与之前工作的比较

与已有光子晶体计算方法相比, 本文的推进主要体现在四个方面。其一, 相对于平面波展开法, 本文不将场展开为截断 Fourier 级数, 因而避免了尖锐介电界面处的 Gibbs 现象和慢收敛; 相对于时域有限差分法, 本文不通过阶梯近似曲界面, 因而不会因界面几何近似而损失精度。作者在引言中明确把这两类方法作为背景, 但未给出与它们在同一算例上的定量比较, 因此无法从提供内容判断本文方法在具体计算成本上相对 PWE 或 FDTD 的优势幅度。

其二, 相对于拟合网格有限元, 本文使用与夹杂几何无关的均匀背景网格, 同一网格可用于一系列形状、朝向和填充率变化的光子晶体, 避免每次几何变化都重新生成界面贴合网格。拟合网格方法包括自适应 $h$-FEM、谱元法和间断 Galerkin 方法, 它们理论成熟, 但网格生成成本高。本文的代价是: 在切割单元上需要保留两个材料分支的独立自由度, 扩展空间 $V_h^{\mathrm{ext}}\times P_h^{\mathrm{ext}}$ 的自由度数量高于单分支拟合网格; 同时需要选择 Nitsche 罚参数与 ghost penalty 参数。作者未在提供内容中给出自由度计数或与拟合网格的计算量对比, 故无法量化这一代价。

其三, 相对于已有非拟合 Maxwell 界面源问题方法, 本文的关键差别是离散 de Rham 结构。作者明确指出, 已有自适应浸入有限元、浸入虚拟元、高阶非拟合有限元、界面罚方法以及混合非拟合方法等主要处理源问题, 而不是 Maxwell 特征值问题。尤其是文献 [69] 的混合非拟合方法使用第二族一阶 Nédélec 元与 $P^1$ Lagrange 元, 对源问题收敛, 但其标量空间不是本文谱分析所需的 $P^2$, 因而不能实现兼容序列; 当用于 Maxwell 特征值问题时仍会出现伪特征值。本文改用第二族 $\mathrm{NE}_1$ 与 $P^2$ 的兼容对, 并通过 Bloch 共轭保持移位算子的交换关系, 从而建立关于切割位置一致的 inf–sup 条件并排除伪模。这是从源问题稳定性到谱问题无污染的关键推进。

其四, 相对于已有非拟合光子晶体能带方法, 作者称文献中唯一可用的非拟合方法是 [67] 的 Petrov–Galerkin 有限元界面方法。该方法在二维 Maxwell 界面特征值问题中使用 Lagrange 型单元, 直接施加准周期边界条件而不使用 Floquet–Bloch 变换, 且主要关注数值格式与实验, 未提供严格理论分析。本文则在二维和三维中结合 Floquet–Bloch 变换与非拟合混合有限元, 给出严格谱收敛理论、一致 inf–sup 条件和无伪模结论。与 [67] 相比, 本文的适用范围从二维扩展到二维和三维, 从数值实验推进到理论分析; 但本文假设界面光滑、$h$ 足够小、介电常数为两相分片常数, 未在提供内容中处理更一般的色散、损耗或非光滑界面情形。

其五, 相对于 Guo 在 $H^1$ 椭圆型方程上的非拟合 Floquet–Bloch 谱分析, 本文处理的是 $H(\mathrm{curl})$ 中的 Maxwell 特征值问题。后者具有无穷维核、缺乏 $H^1$ 椭圆性, 且需要保持 de Rham 复形才能获得无伪模离散, 因此 Guo 的 $H^1$ 框架不能直接推广。本文通过修正 Nédélec–$P^2$ 混合对和移位交换图填补了这一理论空白。作者在引言中明确声称, 据其所知, 准周期 Maxwell 界面特征值问题的非拟合离散严格谱收敛理论此前尚不存在; 这一声明是作者自述, 提供内容未给出完整文献检索证明。

局限与未确认之处包括: 作者未在提供内容中给出与拟合网格有限元、PWE、FDTD 或 [67] 的具体数值对比; 未给出三维复杂夹杂的详细算例数据; 未讨论参数 $\gamma,\gamma_q,\gamma_u,\gamma_p$ 的最优选取; 未说明误差常数对介电比 $\varepsilon^+/\varepsilon^-$ 的依赖; 也未讨论准动量 $k$ 在 Brillouin 区上整体扫描时的计算成本。由于提供内容仅含论文前 10 页, 后续定理证明与数值表格无法核对, 上述比较中涉及具体收敛阶和实验表现的判断只能依据摘要与引言的作者声明。

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2. 电动-磁流体动力学 Casson 微流动中依赖于屈服的 Taylor 弥散

Yielding-dependent Taylor dispersion in electrokinetic-magnetohydrodynamic Casson microflows

作者: Nanda Poddar, Debabrata Das, Suman Chakraborty

类别: physics.flu-dyn, math-ph, math.NA

发布日期: 2026-10-18

arXiv ID: 2610.25433v1

摘要翻译: 屈服流体中的纵向弥散取决于速度分布以及流动如何维持。我们研究了 Casson 流体在狭缝微通道中压力辅助的电动与磁流体动力学流动中的中性示踪剂。非线性应力与速度方程确定了屈服层、未屈服核心和轴向流动。随后，Mei 的对流-扩散方程多尺度均匀化将横向弛豫、平均漂移和纵向扩展分离，得到 Taylor-Aris 辅助问题和有效扩散率。精确的压力驱动解给出在流动停滞附近平均速度的三次衰减和弥散增强的八次幂衰减。在固定压力和电驱动力下，将屈服参数从 $Y=0$ 增至 $0.7$ 会使最优 Hartmann 数从 $Ha_p=0.696$ 移至 $2.634$，而相应的归一化扩散率从 $D_T^{\mathrm C}=1.454$ 降至 $1.015$。在固定平均流量和 $Ha_p=1$ 时，趋势反转：$D_T^{\mathrm C}$ 从 $1.000433$ 增至 $1.002311$。均匀化能量恒等式将这一差异归因于速度偏离截面平均值，包括未屈服核心内的偏离。解析极限解、直接浓度计算、数值收敛性研究以及耦合 Casson 剪切试验支持简化流动模型的预测。基于电解质的量纲评估将磁最优值识别为代表性水相微通道的理论驱动状态。这些结果对于预测在压力或流量控制下、承载具有屈服应力流体的微通道中的纵向示踪剂扩展具有意义。

Comment: 无

链接: http://arxiv.org/abs/2610.25433v1

1. 研究对象和背景

本文研究对象是高度为 $2h$、宽度为 $W$、长度为 $L$ 的狭缝微通道中不可压缩 Casson 流体的定常完全发展流动与其中性示踪剂的纵向弥散, 典型几何假设为 $2h\ll W$ 且 $h\ll L$。坐标取 $x^*$ 沿流向、$y^*$ 为壁面法向、$z^*$ 为展向; 外加轴向压力梯度、轴向电场 $E_x$、壁面法向磁场 $B_0\mathbf e_y$ 以及展向电场 $-E_z\mathbf e_z$, 两侧壁具有相同 zeta 电位 $\zeta$。速度取单向形式 $\mathbf U=u^*(y^*)\mathbf e_x$, 但 Casson 本构使应力与速度通过屈服面耦合: 在简单剪切中

$$\frac{du^*}{dy^*}=0,\quad |\tau_{xy}|\le\tau_y,\qquad \frac{du^*}{dy^*}=\frac{\mathrm{sgn}(\tau_{xy})}{\mu_p}\left(\sqrt{|\tau_{xy}|}-\sqrt{\tau_y}\right)^2,\quad |\tau_{xy}|>\tau_y.$$

屈服条件 $|\tau_{xy}|\le\tau_y$ 给出未屈服核心, 屈服面 $|\tau_{xy}|=\tau_y$ 是待求自由边界; 在未屈服区应变率为零但示踪剂仍可分子扩散。动量平衡为

$$0=-\frac{dp^*}{dx^*}+\frac{d\tau_{xy}}{dy^*}-\sigma B_0^2u^*+\rho_eE_x+\sigma B_0E_z,$$

其中 $\rho_e$ 为净电荷密度, $\sigma$ 为电导率。低 Reynolds 数与低磁 Reynolds 数、准静态 Ohm 律 $\mathbf J=\sigma(\mathbf E+\mathbf U\times\mathbf B)$ 下得到轴向 Lorentz 力项 $\sigma B_0E_z-\sigma B_0^2u^*$。电场在 Debye-Hückel 近似下满足

$$\frac{d^2\psi}{dy^2}=\kappa^2\psi,\quad \psi(\pm1)=1,\quad \psi=\frac{\cosh(\kappa y)}{\cosh\kappa},$$

其中 $\kappa=h/\lambda_D$。无量纲化后耦合应力-速度方程写为

$$\frac{d\hat\tau}{dy}=-\Pi-\Omega\kappa^2\frac{\cosh(\kappa y)}{\cosh\kappa}-S_pHa_p+Ha_p^2\hat u,$$

$$\frac{d\hat u}{dy}=0,\quad |\hat\tau|\le Y,\qquad \frac{d\hat u}{dy}=\mathrm{sgn}(\hat\tau)\left(\sqrt{|\hat\tau|}-\sqrt Y\right)^2,\quad |\hat\tau|>Y.$$

示踪剂满足对流-扩散方程

$$\frac{\partial C^*}{\partial t^*}+u^*\frac{\partial C^*}{\partial x^*}=D\left(\frac{\partial^2C^*}{\partial x^{*2}}+\frac{\partial^2C^*}{\partial y^{*2}}\right),$$

壁面为零通量, 初始条件为在 $x^*=0$ 处沿截面均匀释放的脉冲。该问题重要, 因为 Taylor-Aris 弥散由轴向对流与横向分子扩散的平衡决定, 而屈服应力流体中未屈服核心以近似刚性塞运动, 但示踪剂仍可穿过并扩散; 若只用剪切层速度而忽略塞速度相对于截面平均的偏离, 会错误估计长时展宽。已有理论分别处理 Newtonian 的 Taylor-Aris 与 Mei 多尺度均匀化、电渗流与电磁流、以及 Casson 屈服应力对压力驱动或电渗弥散的影响, 但屈服面随电磁强迫移动时如何改变磁弥散极值, 以及固定压力与固定平均流量两种运行约束是否给出相反趋势, 尚缺统一分析。本文试图在完全发展狭缝流动极限中同时确定屈服面、未屈服核心、轴向速度与有效弥散系数, 并用精确压力驱动解、线性黏度参考解与近流动停止渐近标度分离流变屈服效应和电磁响应。

2. 主要定理或主要结果

第一, 对任意给定稳态轴向速度 $u(y)$ 或 $\hat u(y)$, 在 $\epsilon=h/L\ll1$、$|Pe|=O(1)$、横向扩散时间 $T_0=O(h^2/D)$ 与轴向对流时间 $T_1=O(L/|u_{HS}|)$、轴向扩散时间 $T_2=O(L^2/D)$ 分离的区分极限下, Mei 多尺度展开给出 $C=C_0+\epsilon C_1+\epsilon^2C_2+\epsilon^3C_3+O(\epsilon^4)$; 在横向模态衰减后 $C_0=C_0(x,t_1,t_2)$ 与 $y$ 无关。一阶横向修正满足辅助边值问题

$$\frac{d^2A_1}{dy^2}=u-\langle u\rangle,\quad \left.\frac{dA_1}{dy}\right|_{y=\pm1}=0,\quad \langle A_1\rangle=0,$$

对 Casson 速度相应有

$$\frac{d^2\hat A_1}{dy^2}=\hat u-\langle\hat u\rangle,\quad \left.\frac{d\hat A_1}{dy}\right|_{y=\pm1}=0,\quad \langle\hat A_1\rangle=0,$$

有效弥散系数为

$$D_T^{\mathrm C}=\frac{D_{\mathrm{eff}}}{D}=1+Pe_0^2\left\langle\left(\frac{d\hat A_1}{dy}\right)^2\right\rangle.$$

因此弥散增强非负, 且由完整截面的速度结构控制; 未屈服核心即使局部剪切率为零, 只要其速度不等于截面平均 $\langle\hat u\rangle$, 仍会通过 $\hat u-\langle\hat u\rangle$ 对 $D_T^{\mathrm C}$ 作贡献。第二, 线性有效黏度参考 $\tau_{xy}=\mu(1+1/\beta)du^*/dy^*$ 给出闭合解

$$u(y)=\frac{\beta_1+S Ha}{m^2(1+1/\beta)}\left[1-\frac{\cosh(my)}{\cosh m}\right]+\frac{1}{(1+1/\beta)}\frac{\kappa^2}{\kappa^2-m^2}\left[\frac{\cosh(my)}{\cosh m}-\frac{\cosh(\kappa y)}{\cosh\kappa}\right],$$

其中 $m^2=Ha^2/(1+1/\beta)$; 在纯压力驱动参考情形下

$$u_{pd}(y)=\frac{\Pi_{\mathrm{ref}}}{Ha^2}\left[1-\frac{\cosh(my)}{\cosh m}\right],\quad m=\frac{Ha}{\sqrt{1+1/\beta}}.$$

精确压力驱动 Casson 极限取 $E_x=E_z=B_0=0$、$P=G>0$, 上半个通道的应力为 $\tau_{xy}=-Ghy$, 定义 $\alpha=\tau_y/(Gh)$、$U_P=Gh^2/\mu_p$, 当 $0\le\alpha<1$ 时中心塞占据 $|y|\le\alpha$, 速度可写为

$$\frac{u^*(y)}{U_P}=F(\alpha),\quad 0\le|y|\le\alpha,\qquad \frac{u^*(y)}{U_P}=F(|y|),\quad \alpha<|y|\le1,$$

$$F(y)=\frac{1-y^2}{2}-\frac{4\sqrt\alpha}{3}(1-y^{3/2})+\alpha(1-y),$$

截面平均为

$$\frac{\langle u^*\rangle}{U_P}=1-\frac{\alpha^3}{3}-\frac{4\sqrt\alpha}{5}(1-\alpha^{5/2})+\frac{\alpha}{2}(1-\alpha^2).$$

令 $\delta=1-\alpha$ 为到完全停止的距离, 在 $\delta\to0^+$ 且 $G,h,\mu_p$ 固定时

$$\frac{\langle u^*\rangle}{U_P}=\frac{\delta^3}{12}+O(\delta^4),$$

摘要进一步报告近流动停止时弥散增强按 $\delta^8$ 衰减; 附录 D 给出该近停止展开, 但提供的前十页文本未显示其显式前因子, 因而不在此确认其精确系数。第三, 电磁强迫下的数值结果表明: 在固定压力和电强迫时, 屈服参数从 $Y=0$ 增至 $Y=0.7$ 会把最优 Hartmann 数从 $Ha_p=0.696$ 移到 $Ha_p=2.634$, 同时对应的归一化弥散从 $D_T^{\mathrm C}=1.454$ 降到 $D_T^{\mathrm C}=1.015$; 在固定平均流量且 $Ha_p=1$ 时趋势反转, $D_T^{\mathrm C}$ 从 $1.000433$ 增至 $1.002311$。作者用均匀化能量恒等式把这一差异追溯到速度相对截面平均的偏离, 包括未屈服核心内的偏离; 解析极限解、直接浓度计算、数值收敛研究和耦合 Casson 剪切测试支持这些约化流动预测。作者还明确区分了理论强迫状态与代表性水溶液微通道的电解质尺度评估: 磁最优值被视为理论强迫状态, 而非直接实验参数。

3. 研究方法、关键技术和核心工具

核心障碍有三点。其一是屈服面 $|\hat\tau|=Y$ 是自由边界, 必须与非线性 Casson 本构和电磁体力同时求解; 其二是未屈服核心的速度不等于截面平均时仍影响 Taylor 弥散, 因此不能把塞区简单当作无贡献区域; 其三是固定压力与固定平均流量对应不同约束, 前者改变流量, 后者需调整压力梯度以维持给定平均速度, 两者不能在同一归一化下混为一谈。作者的技术路线是: 先由 Poisson-Boltzmann 方程的 Debye-Hückel 解得到 $\rho_e$ 与电渗体力, 再对无量纲应力方程

$$\frac{d\hat\tau}{dy}=-\Pi-\Omega\kappa^2\frac{\cosh(\kappa y)}{\cosh\kappa}-S_pHa_p+Ha_p^2\hat u$$

从对称轴 $\hat\tau(0)=0$ 积分, 并用 Casson 逆本构

$$\frac{d\hat u}{dy}=0\ (|\hat\tau|\le Y),\qquad \frac{d\hat u}{dy}=\mathrm{sgn}(\hat\tau)(\sqrt{|\hat\tau|}-\sqrt Y)^2\ (|\hat\tau|>Y)$$

确定屈服面位置 $y_f$、未屈服分数 $f_u$ 与速度剖面。对一般电磁强迫情形, 该耦合系统数值求解; 对纯压力驱动情形, 作者给出精确分段解, 以 $\alpha=\tau_y/(Gh)$ 和 $\delta=1-\alpha$ 为小参数得到近停止标度。

弥散部分使用 Mei 的多尺度均匀化, 而不是直接对有限时间浓度场作矩近似。作者引入 $t_0=t$、$t_1=\epsilon t$、$t_2=\epsilon^2t$ 与 $x=x^*/L$, 把 $\partial_t$ 展开为 $\partial_{t_0}+\epsilon\partial_{t_1}+\epsilon^2\partial_{t_2}$, 并对

$$\frac{\partial C}{\partial t}+\epsilon uPe\frac{\partial C}{\partial x}=\epsilon^2\frac{\partial^2C}{\partial x^2}+\frac{\partial^2C}{\partial y^2}$$

逐阶匹配。零阶横向扩散问题给出 $C_0$ 在横向弛豫后与 $y$ 无关; 一阶可解性条件消去长期项并产生 $A_1$ 的 Neumann 辅助问题

$$\frac{d^2A_1}{dy^2}=u-\langle u\rangle,\quad \left.\frac{dA_1}{dy}\right|_{y=\pm1}=0,\quad \langle A_1\rangle=0.$$

该问题的相容性由 $\langle u-\langle u\rangle\rangle=0$ 保证, 解在相差常数下唯一, 常数由 $\langle A_1\rangle=0$ 固定。作者在附录 C 中引入逆 Neumann 算子来处理该辅助问题, 在附录 A 中完成多尺度可解性计算。二阶与三阶展开进一步给出平均漂移、纵向扩散和横向浓度结构; 最终有效弥散由 $D_T^{\mathrm C}=1+Pe_0^2\langle(d\hat A_1/dy)^2\rangle$ 表示。这个框架的关键不是泛泛的“能量估计”, 而是把可解性条件、横向平均和逆 Neumann 算子结合, 将三维或二维对流-扩散问题严格降为截面上的一维辅助问题。

作者还系统处理了归一化与奇异极限。电渗速度尺度 $u_{HS}=-\epsilon\zeta E_x/\mu$ 在 $E_x=0$ 时消失, 因此不能在所有比较中使用同一 $Pe$; 他们引入独立参考速度 $U_0$ 与 $Pe_0=U_0h/D$, 并在无电渗压力驱动比较中引入 $U_{\mathrm{ref}}$、$\Pi_{\mathrm{ref}}$、$Pe_{\mathrm{ref}}$。当 $\Omega\ne0$ 时, 两套无量纲量满足 $Ha=Ha_p$、$\beta_1=\Pi/\Omega$、$S=S_p/\Omega$、$Pe=\Omega Pe_0$、$u=\hat u/\Omega$, 因而对同一物理速度剖面有 $Pe\,u=Pe_0\hat u$ 与 $\hat A_1=\Omega A_1$, 弥散增强保持不变; 但作者强调这一归一化恒等式并不把 Casson 本构与线性本构的预测等同。线性有效黏度解用于基准和分离电磁响应, Casson 精确压力驱动解用于检验非线性屈服面与近停止标度。验证方面, 作者使用直接二维浓度计算、独立矩计算、数值收敛研究以及双分量 Casson 剪切测试来评估轴向剪切近似和屈服面预测; 对磁最优值还做了基于电解质的量纲评估, 以说明其作为理论强迫状态而非直接实验点的含义。作者明确指出, 完全发展单向流动、低 Re 与低 Rm、准静态 Ohm 律、Debye-Hückel 线性化、常数示踪扩散系数以及忽略侧电极附近流动和展向剪切对轴向屈服面的影响等, 都是该分析框架的假设; 双分量剪切测试只用于检验这些约化的敏感度, 并不构成完整三维闭包。

4. 与之前工作的比较

与经典 Taylor-Aris 理论和 Mei 多尺度均匀化相比, 本文的推进在于把屈服应力流体的未屈服核心纳入有效弥散核。经典理论通常假定黏度给定或速度剖面已知, 弥散增强由 $\langle(u-\langle u\rangle)^2\rangle$ 一类截面速度偏离控制; 本文则先由 Casson 本构和电磁动量平衡确定屈服面与塞速度, 再把完整 $\hat u(y)$ 代入 Mei 辅助问题。因此它不只是把 Newtonian 结果套用到非 Newtonian 速度场, 而是显式处理了 $|\hat\tau|=Y$ 的自由边界、塞区 $\hat u-\langle\hat u\rangle$ 的贡献以及压力驱动近停止时 $\langle u^*\rangle/U_P=\delta^3/12+O(\delta^4)$ 与弥散增强 $\delta^8$ 标度。

与作者明确引用的 Ng 关于压力辅助电渗 Casson 狭缝流的工作相比, 本文增加了壁面法向磁场与展向电场导致的 Lorentz 力、长时纵向弥散的多尺度均匀化、磁弥散极值随屈服参数移动的分析, 以及固定压力与固定平均流量两种约束的对照。Ng 的工作主要给出屈服层、未屈服层与压力辅助电渗速度; 本文则把这些速度结构进一步送入 $A_1$ 辅助问题和 $D_T^{\mathrm C}$。与 Dash 等关于 Casson 屈服应力对通道和管中纵向溶质弥散影响的工作相比, 本文的差别在于同时引入电渗与电磁强迫, 并把“移动屈服面”与“磁最优 Hartmann 数”联系起来; 不过作者在提供文本中没有逐项展开与 Dash 等在所有参数区间上的定量比较, 因此不能从现有片段确认两者在全部流变参数下的优劣。与 Casson 电渗弥散、带壁面吸收的多尺度输运以及脉动流相间交换等研究相比, 本文限定于定常完全发展狭缝流, 不含壁面反应、吸收或脉动强迫, 但换来了对屈服面、磁屏遮和两种运行约束的闭合分析。

与 Sen 和 Kairi 的电渗-EMHD Casson 通道研究相比, 本文没有考虑浮力、滑移相关表面电位和壁面反应, 而是聚焦压力辅助、屈服面和纵向弥散; 作者把后者列为背景工作, 但未在提供文本中给出完整的假设差异表, 故其精确适用边界从现有内容无法完全确认。与 Das 等、Sen 等和 Hossain 等关于 Newtonian 或 couple-stress 流体的电磁弥散研究相比, 本文的新意在于 Casson 屈服使速度剖面出现未屈服核心, 并使磁响应不再仅由黏性剪切层控制; 固定压力下最优 $Ha_p$ 从 $0.696$ 移到 $2.634$、$D_T^{\mathrm C}$ 从 $1.454$ 降到 $1.015$, 而固定平均流量且 $Ha_p=1$ 时 $D_T^{\mathrm C}$ 从 $1.000433$ 升到 $1.002311$, 这种趋势反转是本文相对 Newtonian 磁弥散讨论的主要推进之一。与 Boyko 和 Stone 关于剪切率相关流动中的 Taylor-Aris 弥散相比, 本文保持示踪剂扩散系数为常数, 流变依赖性完全通过 Casson 速度场与屈服面进入; 因此它不处理示踪剂扩散率随剪切变化的情形。

在适用范围上, 本文把时间尺度推进到横向扩散已弛豫后的长时 Taylor-Aris 极限, 并用直接有限时间浓度计算检验该极限的可达性; 但有限时间瞬态本身不是本文的主要定理对象。其代价是几何与物理假设较强: 完全发展单向流、狭缝无限宽近似、低 Re 与低 Rm、准静态 Ohm 律、Debye-Hückel 线性化、常数示踪扩散, 以及把轴向剪切律作为忽略展向剪切和侧电极附近流动的近似。作者用双分量 Casson 剪切测试评估了这一近似的敏感度, 但未给出有限宽器件中侧壁电流回路和展向剪切的完整闭包; 磁最优值也仅被标定为理论强迫状态。总体而言, 相对既有工作, 本文放宽了对 Newtonian 黏度和固定速度剖面的依赖, 加入了屈服面自由边界与电磁强迫的耦合, 并把固定压力与固定流量两种微流体运行方式放在同一均匀化框架下比较; 其局限则是结论集中在完全发展狭缝极限和长时有效弥散, 尚不能直接外推到有限宽、强展向剪切或瞬态输运主导的装置。

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3. $A$-调和函数的局部逼近与超局部化猜想

Local Approximation of $A$-Harmonic Functions and the Super-Localization Conjecture

作者: Moritz Hauck, Chupeng Ma, Daniel Peterseim

类别: math.NA

发布日期: 2026-10-18

arXiv ID: 2610.25230v1

摘要翻译: 我们考虑 $\mathbb{R}^d$（$d\ge2$）中具有固定尺寸比的同心立方体上、针对有界、对称、一致椭圆的扩散张量 $A$ 的 $A$-调和函数的局部能量逼近。我们对 Ma [Found.\ Comput.\ Math.\ 26 (2026), 1699–1758] 所猜想的 Kolmogorov $n$-宽度衰减率 $\exp(-cn^{1/(d-1)})$ 给出一个简短证明，其中 $n$ 表示逼近维数。我们将该结果应用于数值均匀化，改进了多尺度谱广义有限元方法的局部逼近估计，并证明了 Hauck 和 Peterseim [Math.\ Comp.\ 92 (2023), 981–1003] 针对内部笛卡尔片所猜想的局部超局部化速率。

Comment: 无

链接: http://arxiv.org/abs/2610.25230v1

1. 研究对象和背景

本文研究粗糙系数椭圆方程所对应的 $A$-调和函数在同心立方体上的局部能量逼近问题. 设 $d\ge 2$, $r>0$, 取同心立方体

$$ \omega:=(-r,r)^d,\qquad \omega^*:=(-3r,3r)^d, $$

并设扩散张量 $A\in L^\infty(\omega^*;\mathbb{R}^{d\times d}_{\rm sym})$ 满足一致椭圆条件

$$ \alpha |\xi|^2\le (A(x)\xi)\cdot\xi\le \beta |\xi|^2 \qquad \text{对几乎处处的 }x\in\omega^*\text{ 与所有 }\xi\in\mathbb{R}^d, $$

其中 $0<\alpha\le\beta<\infty$. 对任意开集 $G\subseteq\omega^*$, 记

$$ a_G(u,v):=\int_G (A(x)\nabla u(x))\cdot\nabla v(x)\,dx,\qquad \|u\|_{a,G}:=a_G(u,u)^{1/2}, $$

并定义 $A$-调和函数空间

$$ H(G):=\left\{u\in H^1(G):\ a_G(u,\varphi)=0\ \forall \varphi\in H^1_0(G)\right\}. $$

等价地, $u\in H(G)$ 是散度型方程 $-\nabla\cdot(A\nabla u)=0$ 在 $G$ 中的弱解. 本文关注的核心对象是限制算子

$$ P:H(\omega^*)\to H(\omega),\qquad Pu=u|_{\omega}, $$

及其在能量半范数下的 Kolmogorov $n$-width

$$ d_n:=\inf_{\substack{V_n\subset H(\omega)\\ \dim V_n=n}}\ \sup_{\substack{u\in H(\omega^*)\\ \|u\|_{a,\omega^*}=1}}\ \inf_{v\in V_n}\|u-v\|_{a,\omega}. $$

这里 $n$ 是局部逼近空间的维数, $d_n$ 刻画了用 $\omega$ 上的 $n$ 维空间去逼近由更大区域 $\omega^*$ 上的 $A$-调和函数限制到 $\omega$ 所得到函数的最优能量误差. 由于 $v=0$ 始终 admissible 且限制不增加能量, 总有 $d_n\le 1$; 真正的问题是 $d_n$ 随 $n$ 如何衰减, 以及衰减指数与空间维数 $d$、系数对比度 $\beta/\alpha$ 的关系.

这个问题在数值均匀化中有直接背景. 多尺度谱广义有限元方法 (MS-GFEM) 由 Babuška 和 Lipton 提出, 其局部逼近空间正是通过上述限制算子的 Kolmogorov $n$-width 与广义特征值问题来刻画. 若局部空间使用 $n$ 个基函数, 则局部逼近误差就是本文的 $d_n$. 因此 $d_n$ 的衰减率直接决定 MS-GFEM 局部基函数的效率. 类似地, Grasedyck-Greff-Sauter 的 AL 基也依赖 $A$-调和函数的局部逼近; 局部正交分解 (LOD) 通过截断指数衰减的修正来构造算子适应基函数, 而子空间分解方法用一致压缩的局部迭代解释这种衰减. 另一方面, 局部逼近估计还可以反过来用于设计具有局部化响应的源项: 若一个归一化粗源在所有 $A$-调和函数上的配对都很小, 则其全局椭圆响应与 patch 局部响应之差也小. Hauck 和 Peterseim 在超局部化正交分解 (SLOD) 中正是利用这一机制优化粗源, 并猜想局部化误差随过采样层数 $\ell$ 呈超指数衰减.

已有理论中, Babuška 和 Lipton 对一般一致椭圆 $A$ 证明了形如 $\exp(-n^{1/(d+1)-\epsilon})$ 的 $n$-width 衰减率, 对任意 $\epsilon>0$. Ma 在统一 MS-GFEM 框架中把指数改进到 $1/d$, 并进一步猜想最优指数应为 $1/(d-1)$. 对 Laplacian, Mohr 证明了与指数 $1/(d-1)$ 相对应的局部 SLOD 衰减估计. 对一般有界、对称、一致椭圆扩散张量, 近期也有证明出现. 本文试图补上的空白正是: 对一般 $A$ 给出一个简短、自包含的证明, 证实 Ma 猜想的 $\exp(-c n^{1/(d-1)})$ 衰减率, 并给出显式常数与改进的系数对比度依赖; 同时把该局部逼近结果应用于 MS-GFEM 与 SLOD, 证明 Hauck-Peterseim 关于内部 Cartesian patch 的局部超局部化率猜想.

2. 主要定理或主要结果

本文的主要严格结果是定理 1 与推论 2. 定理 1 确认了 Ma 猜想的 Kolmogorov $n$-width 衰减率. 设 $d\ge 2$, $r>0$, $\omega=(-r,r)^d$, $\omega^*=(-3r,3r)^d$, $A$ 满足上述一致椭圆条件. 对每个满足

$$ n+1\ge 2(3^d-1) $$

的 $n\in\mathbb{N}$, 定义

$$ d_n:=\inf_{\substack{V_n\subset H(\omega)\\ \dim V_n=n}}\ \sup_{\substack{u\in H(\omega^*)\\ \|u\|_{a,\omega^*}=1}}\ \inf_{v\in V_n}\|u-v\|_{a,\omega}. $$

则

$$ d_n\le \exp\left(-c\sqrt{\frac{\alpha}{\beta}}\,(n+1)^{1/(d-1)}\right), \qquad c:=\frac{1}{128(1+\pi^{-1})}. $$

这里常数 $c$ 是显式的, 且指数中的系数对比度依赖为 $\sqrt{\alpha/\beta}$, 而不是 $\alpha/\beta$ 或其他更差的依赖. 定理还指出 $d_n\le 1$ 是平凡的, 上面的估计在 $n$ 足够大时才有实质意义. 证明针对内部子区域; 若 $\omega^*$ 与区域 $\Omega$ 的边界相交, 则把 $\omega$ 与 $\omega^*$ 替换为 $\omega\cap\Omega$ 与 $\omega^*\cap\Omega$, 并在 $H$ 的定义中对 $\partial\Omega\cap\omega^*$ 施加齐次 Dirichlet 条件, 证明可作与 Ma 等人类似的微小修改后沿用. 通过平移、缩放和有限覆盖, 该结果可推广到紧嵌套的有界 Lipschitz 区域, 但常数会额外依赖几何.

该定理的直接应用是 MS-GFEM 的局部逼近估计. 在 Babuška-Lipton 的原始 MS-GFEM 中, 局部空间由 $\omega^*$ 上的 $A$-调和函数限制到 $\omega$ 构造, 其 $n$ 维最优误差正是 $d_n$. 因此定理 1 给出局部误差上界 $\exp(-c\sqrt{\alpha/\beta}(n+1)^{1/(d-1)})$, 对同心立方体成立; 对紧嵌套 Lipschitz 区域由覆盖论证得到相同率, 常数依赖几何. 这改进了 Ma 先前建立的指数 $1/d$. 对较新版本的 MS-GFEM, 局部空间还要乘以与 $\omega$ 相关的单位分解函数 $\chi$, 相应算子为

$$ P_\chi:H(\omega^*)\to H^1_0(\omega),\qquad P_\chi u:=\chi\,(u|_{\omega}), $$

其中 $H(\omega^*)$ 配以能量范数并识别相差常数的函数. 其 $n$-width 记为 $d_n(P_\chi)$. 文中指出可 straightforwardly 证明 $d_n(P_\chi)$ 被 $d_n$ 的常数倍控制, 常数只依赖 $\chi$, 因而这些方法获得同一衰减率.

第二个主要结果是推论 2, 即 SLOD 的超指数局部化. 取 Cartesian 网格, 网格尺寸 $H>0$, 过采样参数 $\ell\in\mathbb{N}$ 表示一个单元外加周围 $\ell$ 层单元. 将 patch 中心置于原点, 有

$$ \omega^*=\left(-\frac{2\ell+1}{2}H,\frac{2\ell+1}{2}H\right)^d. $$

记 $T_H(\omega^*)$ 为 $\omega^*$ 分成 $(2\ell+1)^d$ 个边长 $H$ 的开 Cartesian 立方体, $Q^0_H(\omega^*)$ 为相应的分片常数函数空间. 定义

$$ \sigma_{H,\ell}:=\inf_{\substack{g\in Q^0_H(\omega^*)\\ \|g\|_{L^2(\omega^*)}=1}}\ \sup_{\substack{v\in H(\omega^*)\\ \|v\|_{H^1(\omega^*)}=1}}\ (g,v)_{L^2(\omega^*)}, $$

其中 $\|v\|^2_{H^1(\omega^*)}:=\|v\|^2_{L^2(\omega^*)}+\|\nabla v\|^2_{L^2(\omega^*)}$. 该量控制 SLOD 的局部化误差. 推论 2 断言: 对每个 $\ell\ge 1$,

$$ \sigma_{H,\ell}\le \frac{(\ell+2)H}{\pi}\sqrt{\frac{\beta}{\alpha}}\,\exp\left(-\frac{c}{8}\sqrt{\frac{\alpha}{\beta}}\,\ell^{d/(d-1)}\right), $$

其中 $c$ 为定理 1 中的常数. 因而局部化误差随 $\ell^{d/(d-1)}$ 超指数衰减, 这证明了 Hauck 和 Peterseim 对内部 Cartesian patch 的局部超局部化率猜想. 作者还指出, 超局部化广义有限元方法的局部化误差贡献同样由 $\sigma_{H,\ell}$ 控制; 在相应定理假设下, 推论 2 给出该贡献随过采样参数 $\ell$ 的超指数衰减. 需要强调的是, 这里的超指数是指 $\exp(-c\ell^{d/(d-1)})$ 对 $\ell$ 的依赖, 而不是简单的 $\exp(-c\ell)$; 这是本文证明的局部 SLOD 率.

3. 研究方法、关键技术和核心工具

证明的核心技术障碍在于: 由 Babuška-Lipton 的谱刻画, $d_n$ 等于限制算子 $P:H(\omega^*)\to H(\omega)$ 的奇异值序列中第 $n+1$ 个奇异值, 即

$$ d_n=\sqrt{\lambda_{n+1}}, $$

其中 $\lambda_1\ge\lambda_2\ge\cdots\ge 0$ 是 $P^*P$ 在能量内积下的特征值, 相应特征函数 $u_i\in H(\omega^*)$ 满足

$$ a_{\omega}(u_i,v)=\lambda_i a_{\omega^*}(u_i,v)\qquad \forall v\in H(\omega^*). $$

因此要证明 $d_n$ 的指数衰减, 必须控制第 $n+1$ 个特征值. 已有证明往往依赖边界谱估计或较复杂的局部化迭代; 本文的真正创新是采用 Li-Wang 的能量行列式思想, 把问题转化为固定有限维空间 $V=\operatorname{span}\{u_1,\dots,u_{n+1}\}$ 上不同区域能量矩阵的行列式比较, 并只使用局部 Poincaré 不等式与 Hadamard 不等式. 这避免了边界谱估计, 给出显式常数, 并把指数中的系数对比度依赖改进为 $\sqrt{\alpha/\beta}$.

具体框架如下. 首先把 $\omega^*$ 与 $\omega$ 之间的环带分成 $K$ 层. 取

$$ K:=\left\lfloor\left(\frac{n+1}{2(3^d-1)}\right)^{1/(d-1)}\right\rfloor,\qquad h:=\frac{2r}{K}, $$

并定义

$$ \omega_k:=(-r-kh,r+kh)^d,\qquad k=0,\dots,K, $$

于是 $\omega_0=\omega$, $\omega_K=\omega^*$. 将 $\omega^*$ 分成边长 $h$ 的开 Cartesian 立方体, 记第 $k$ 层为 $S_k:=\omega_k\setminus\omega_{k-1}$, 其包含的单元数为

$$ m_k=(K+2k)^d-(K+2k-2)^d,\qquad \sum_{k=1}^K m_k=(3^d-1)K^d. $$

这是关键的维数计数: 每层新增 $m_k$ 个单元, 所有层的单元总数为 $(3^d-1)K^d$, 而 $K$ 的选择保证该总数约为 $(n+1)/2$.

其次, 对每层定义子空间

$$ W_k:=\left\{v\in V:\ \int_T v\,dx=0\quad \forall T\in T_h(S_k)\right\}, $$

即 $V$ 中在每个第 $k$ 层单元上均值为零的函数. 这是 $m_k$ 个线性条件, 故

$$ \dim W_k\ge (n+1-m_k)_+. $$

对 $v\in W_k$, 在每个小单元上用均值为零的 Poincaré 不等式, 常数在 $(0,1)^d$ 上为 $\pi^{-1}$, 缩放并求和得到

$$ \|v\|_{L^2(S_k)}\le \pi^{-1}h\|\nabla v\|_{L^2(S_k)}. $$

再取 cutoff 函数

$$ \eta_k(x):=\min\left\{1,\max\left\{0,\frac{r+kh-\|x\|_\infty}{h}\right\}\right\}, $$

它满足在 $\omega_{k-1}$ 上为 $1$, 在 $\omega_k$ 外为 $0$, 且 $|\nabla\eta_k|\le h^{-1}$ a.e. 因 $\eta_k v\in H^1_0(\omega^*)$, 用弱调和性 $a_{\omega^*}(v,\eta_k v)=0$ 可把 $\omega_{k-1}$ 上的能量与层 $S_k$ 上的能量联系起来:

$$ \|v\|^2_{a,\omega_{k-1}} \le \int_{\omega^*}\eta_k(A\nabla v)\cdot\nabla v\,dx = -\int_{S_k} v(A\nabla v)\cdot\nabla\eta_k\,dx \le \frac{\sqrt{\beta}}{h}\|v\|_{L^2(S_k)}\|v\|_{a,S_k}. $$

结合 Poincaré 不等式与 $h$ 的定义, 得到层压缩估计

$$ \|v\|^2_{a,\omega_{k-1}}\le \theta\|v\|^2_{a,\omega_k}, \qquad \theta:=\frac{\pi^{-1}\sqrt{\beta/\alpha}}{1+\pi^{-1}\sqrt{\beta/\alpha}}<1. $$

这一步是证明的几何与 PDE 核心: 它说明在每层上均值为零的函数, 其内层能量被外层能量的固定比例 $\theta$ 控制.

第三步是 Li-Wang 的能量行列式比较. 对固定的 $V$, 定义 $(n+1)\times(n+1)$ 矩阵

$$ G_k:=\left(a_{\omega_k}(u_i,u_j)\right)_{i,j=1}^{n+1},\qquad k=0,\dots,K. $$

由特征方程, 端点满足

$$ G_0=\operatorname{diag}(\lambda_1,\dots,\lambda_{n+1}),\qquad G_K=I_{n+1}. $$

固定一层 $k$, 取 $V$ 关于 $a_{\omega_k}$ 的标准正交基 $\{v_1,\dots,v_{n+1}\}$, 并使其前 $\dim W_k$ 个向量张成 $W_k$. 令 $Q_k$ 为相应基变换矩阵, 则

$$ Q_k^\top G_k Q_k=I_{n+1},\qquad B_k:=Q_k^\top G_{k-1}Q_k=\left(a_{\omega_{k-1}}(v_i,v_j)\right)_{i,j=1}^{n+1}. $$

由层压缩估计, $B_k$ 的前 $\dim W_k$ 个对角元不超过 $\theta$; 其余对角元由能量在限制下的单调性不超过 $1$. 于是, 利用行列式乘性与正定矩阵的 Hadamard 不等式,

$$ \frac{\det G_{k-1}}{\det G_k}=\det B_k\le \prod_{i=1}^{n+1}(B_k)_{ii}\le \theta^{\dim W_k}\le \theta^{(n+1-m_k)_+}. $$

将所有层相乘, 并用 $\sum m_k=(3^d-1)K^d$, 得到

$$ \prod_{i=1}^{n+1}\lambda_i=\frac{\det G_0}{\det G_K}\le \theta^{\sum_{k=1}^K(n+1-m_k)_+}\le \theta^{(n+1)K-(3^d-1)K^d}. $$

由 $K$ 的选择, $(3^d-1)K^{d-1}\le (n+1)/2$, 故指数至少为 $(n+1)K/2$. 再由特征值排序 $\lambda_{n+1}^{n+1}\le \prod_{i=1}^{n+1}\lambda_i$, 得到 $d_n=\sqrt{\lambda_{n+1}}\le \theta^{K/4}$. 最后, 由

$$ -\log\theta=\log\left(1+\pi\sqrt{\frac{\alpha}{\beta}}\right) \ge \frac{1}{1+\pi^{-1}}\sqrt{\frac{\alpha}{\beta}} $$

和 $K$ 的下界, 得到

$$ d_n\le \exp\left(-\frac{1}{8(1+\pi^{-1})}\sqrt{\frac{\alpha}{\beta}}\left(\frac{n+1}{2(3^d-1)}\right)^{1/(d-1)}\right), $$

再用 $(2(3^d-1))^{1/(d-1)}\le 16$ 即得定理 1 的常数.

SLOD 部分的技术路线是把局部 $n$-width 估计转化为源项 $\sigma_{H,\ell}$ 的估计. 对 $\ell\ge 7$, 取 $m=\lfloor(\ell-1)/3\rfloor$, 在内部子立方体 $\omega$ 上取最优 $n$ 维空间 $V_n\subset H(\omega)$, 其中 $n=(2m+1)^d-2$, 使分片常数源空间 $Q^0_H(\omega)$ 的维数为 $n+2$. 由于对 $g\in Q^0_H(\omega)$ 要求它与 $V_n+\operatorname{span}\{1\}$ 在 $L^2(\omega)$ 上正交, 至多 $n+1$ 个线性条件, 而未知元有 $n+2$ 个, 故存在非零的 $g$ 满足正交条件, 并归一化为 $\|g\|_{L^2(\omega^*)}=1$. 对任意 $v\in H(\omega^*)$ 与 $q\in V_n$, 利用 $g$ 在 $\omega$ 外为零、与 $q$ 及常数正交、Cauchy-Schwarz 不等式和 Poincaré 不等式, 得

$$ (g,v)_{L^2(\omega^*)}\le \frac{(\ell+2)H}{\pi\sqrt{\alpha}}\|v-q\|_{a,\omega}. $$

对 $q\in V_n$ 取下确界, 用 $V_n$ 的最优性与定理 1, 得到 $\sigma_{H,\ell}$ 的界. 对 $1\le\ell\le 6$ 的小过采样参数, 作者改用两个相邻单元上的正负常数源, 通过 Poincaré 不等式和初等估计单独处理. 这体现了证明的完整性: 大 $\ell$ 用渐近率, 小 $\ell$ 用显式粗糙界.

总体而言, 本文的方法不是泛泛的能量估计加紧性, 而是通过: (i) 用谱刻画把 $n$-width 转化为特征值; (ii) 用同心层的局部 Poincaré 与 cutoff 得到几何压缩; (iii) 用固定有限维空间上的能量矩阵行列式与 Hadamard 不等式把逐层压缩转化为特征值乘积的指数界; (iv) 用维数计数把指数与 $n$ 精确联系起来; (v) 再把该 $n$-width 界用于构造与最优局部空间正交的源, 从而控制 SLOD 的局部化误差. 这些步骤共同给出了显式常数与 $\sqrt{\alpha/\beta}$ 的对比度依赖.

4. 与之前工作的比较

与 Babuška 和 Lipton 的原始 MS-GFEM 理论相比, 本文首先把一般一致椭圆 $A$ 下的 $n$-width 衰减指数从他们证明的 $1/(d+1)-\epsilon$ 推进到 $1/(d-1)$. 原工作建立了局部逼近空间与 Kolmogorov $n$-width 的谱刻画, 并证明近指数收敛; 但其率较弱. 本文不仅给出更优指数, 而且证明是简短的, 常数显式. 代价是定理 1 的陈述针对同心立方体, 需要 $n+1\ge 2(3^d-1)$ 的门槛; 对更一般的区域需通过有限覆盖, 常数依赖几何. 作者没有讨论该率的最优性, 也没有给出匹配下界, 因此这里的“最优指数”应理解为证实 Ma 的猜想率, 而非证明在所有可能逼近空间中的绝对最优性.

与 Ma 的统一框架相比, 本文的直接推进是把指数从 $1/d$ 改进到 $1/(d-1)$, 这正是 Ma 明确提出的猜想. Ma 的工作统一处理 MS-GFEM 与多尺度 PDE 的低秩逼近, 并给出 $\exp(-c n^{1/d})$ 型估计; 本文则给出一个 alternative short proof, 不依赖边界谱估计, 而依赖 Li-Wang 的能量行列式思想、局部 Poincaré 不等式和 Hadamard 不等式. 因此两者在证明技术上差别明显: Ma 的框架更一般、更系统, 本文更专注于局部 $n$-width 的核心机制. 在应用层面, 本文指出原始 MS-GFEM 的局部误差就是 $d_n$, 而带单位分解的变体 $P_\chi$ 的 $n$-width 至多被 $d_n$ 的常数倍控制, 故同样获得 $1/(d-1)$ 率; 这一比较在文中是显式给出的. 但作者未展开 $P_\chi$ 常数对 $\chi$ 的精确依赖, 从提供内容中只能确认“常数只依赖 $\chi$”.

与 Mohr 的工作相比, 本文的关键差异是一般有界、对称、一致椭圆扩散张量 $A$, 而 Mohr 的结果针对 Laplacian. 对 Laplacian 可利用更多的显式调和函数结构与谱信息; 本文的层压缩与行列式论证对粗糙系数仍成立, 因此适用范围更广. 与近期 Jin 对一般可测系数给出的局部逼近宽度证明相比, 本文在摘要与引言中明确承认 [6] 已给出一般 $A$ 的证明, 并说明本文给出的是 short alternative proof, 同样使用 Li-Wang 的能量行列式思想. 两文的详细常数、假设门槛和证明长度在提供内容中未展开比较, 因此无法确认本文在常数或对比度依赖上是否严格优于 [6]; 可以确认的是, 本文强调其证明只依赖局部 Poincaré 与 Hadamard 不等式, 给出显式 numerical constant, 并改进 exponent 中的 contrast 依赖. 这是一个方法性与定量性的推进, 但不应表述为首次证明一般 $A$ 的猜想.

与 Hauck 和 Peterseim 的 SLOD 工作相比, 本文证明了他们关于内部 Cartesian patch 的局部超局部化率猜想. 原工作通过优化粗源并利用与 $A$-调和函数的配对来控制全局响应与 patch 局部响应之差, 猜想误差随过采样参数 $\ell$ 呈超指数衰减; 本文给出

$$ \sigma_{H,\ell}\le \frac{(\ell+2)H}{\pi}\sqrt{\frac{\beta}{\alpha}}\exp\left(-\frac{c}{8}\sqrt{\frac{\alpha}{\beta}}\,\ell^{d/(d-1)}\right), $$

从而严格证实 $\exp(-c\ell^{d/(d-1)})$ 型衰减. 这里的推进是明确的: 把猜想的率变成定理, 并给出显式常数与对比度依赖. 代价是限制在 Cartesian 网格与内部 patch; 作者在文中说更一般的设置可以考虑, 但未证明. 对 $1\le\ell\le 6$ 的小过采样参数, 证明使用粗糙的显式界, 不是渐近率; 因此该推论的实际意义主要在 $\ell$ 较大时.

与 Freese-Hauck-Keil-Peterseim 的超局部化广义有限元方法相比, 本文的贡献是把该方法的局部化误差贡献与 $\sigma_{H,\ell}$ 联系起来, 并代入推论 2 得到超指数衰减. 文中引用 [2, Theorem 5.3] 说明局部化贡献由 $\sigma_{H,\ell}$ 控制, 因此在该定理假设下可直接得到结论. 这意味着本文不是重新分析整个 SLOD-GFEM 的全局误差, 而是补上了其局部化误差衰减率这一关键环节. 与 LOD 和子空间分解类方法 [8,7,12] 相比, 那些工作通过算子适应基函数、指数衰减修正或一致压缩迭代给出局部化估计, 并可导出 $n$-width 界; 本文则直接从 $A$-调和函数的局部 $n$-width 出发, 用谱与行列式论证得到 $1/(d-1)$ 指数, 技术路线不同. 文中没有显式比较本文常数与 LOD 类估计的常数大小, 也没有讨论在更一般网格、边界 patch 或非对称系数下的扩展.

最后需要区分严格结论与未讨论内容. 本文严格证明的是: 同心立方体上定理 1 的 $n$-width 界, 边界情形可经 Dirichlet 修改类似处理; 紧嵌套 Lipschitz 区域经有限覆盖得到同率但几何相关常数; Cartesian 内部 patch 上推论 2 的 $\sigma_{H,\ell}$ 界, 以及 MS-GFEM/SLOD 的相应局部误差率. 文中没有给出下界、没有证明指数 $1/(d-1)$ 的最优性, 没有数值实验, 也没有证明任意非 Cartesian patch 或边界 patch 的 SLOD 超局部化率. 这些应视为该文未覆盖或需未来工作处理的范围, 不能从提供内容中推断为已解决.

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了解领域论文（摘要翻译）

1. 计算非均匀介电环境中电子排斥积分的边界积分方法

A boundary integral method for computing electron repulsion integrals in heterogeneous dielectric environments

作者: Xuanzhao Gao, Malte Rösner, Jason Kaye, Alex Barnett

类别: math.NA, cond-mat.mtrl-sci

发布日期: 2026-10-19

arXiv ID: 2610.26745v1

链接: http://arxiv.org/abs/2610.26745v1

摘要翻译

中文标题: 计算非均匀介电环境中电子排斥积分的边界积分方法

摘要翻译: 我们提出一种边界积分方法，用于在分段常数介电常数的介电环境中计算电子排斥积分 (ERIs)。该方法通过引入结构化介电环境（例如现代二维材料器件中存在的基底和/或封装层）的屏蔽，推广了传统的裸库仑 ERIs。我们将问题表述为介电界面上诱导表面电荷的第二类边界积分方程，并在自适应细化面板上用 Nyström 方法对其进行离散。由每个光滑轨道源密度产生的入射势，在其源盒内或附近通过 Fourier 截断核方法计算，在较远处通过快速多极方法 (FMM) 加速的求积计算。线性系统由同样经 FMM 加速的 Krylov 迭代求解。我们关注长方体器件几何，并通过近接触面板求积在高纵横比下保持效率。我们进一步开发了一个分布式流水线，用于计算嵌入单一介电几何中的完整 ERI 张量，其中包含数千个轨道对。我们在 Si/SiO$_2$ 异质结上的有效三层石墨烯板上演示了该方法，重点关注由 $198$ 个 $p_z$ 型轨道表示的中央石墨烯层，并保留 $198^4$ 大小的 ERI 张量中 $2613^2$ 个不可忽略元素。该模型在 $1.3$ 百万个表面未知量上求解，相对误差为 $0.004\%$，代价为 $22$ 节点小时。

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2. 基于 Whitney 形式的不可压缩磁流体动力学鲁棒有限元方法

A robust finite element method for incompressible magnetohydrodynamics using Whitney forms

作者: Tianwei Yu

类别: math.NA

发布日期: 2026-10-19

arXiv ID: 2610.26606v1

链接: http://arxiv.org/abs/2610.26606v1

摘要翻译

中文标题: 基于 Whitney 形式的不可压缩磁流体动力学鲁棒有限元方法

摘要翻译: 本文提出了一种用于不可压缩磁流体动力学系统的稳定化半离散有限元格式。该格式分别对流体速度和磁感应强度使用 Whitney $2$-形式和 $1$-形式。关键要素是一个磁旋度跳跃罚项，它使我们能够证明阶为 $O(h^{1/2})$ 的预渐近误差估计，其中常数依赖于解范数，但不显式依赖于流体黏度和磁扩散率。

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3. 含角区域上非齐次修正 Helmholtz 方程的闪电算法及其应用

Lightning Algorithm for Inhomogeneous Modified Helmholtz Equations on Cornered Domains and Their Applications

作者: Desong Kong, Jie Shen

类别: math.NA

发布日期: 2026-10-19

arXiv ID: 2610.26540v1

链接: http://arxiv.org/abs/2610.26540v1

摘要翻译

中文标题: 含角区域上非齐次修正 Helmholtz 方程的闪电算法及其应用

摘要翻译: 本文针对二维复杂区域中的非齐次修正 Helmholtz 方程发展了一种高效求解器，特别关注角点奇性的解析。我们将解表示为一个特解与修正 Helmholtz 方程的解之和。基于合适的区域嵌入方法，我们使用 Fourier 谱方法和/或 Legendre–Fourier 谱方法计算该特解。为处理角点奇性，我们发展了一种闪电算法：采用第二类修正 Bessel 函数的展开，并结合渐缩的指数聚簇极点来逼近角点奇性，同时使用第一类修正 Bessel 函数的展开来逼近解的光滑部分。最后，我们通过求解一个最小二乘问题以满足边界条件来确定未知展开系数。数值实验表明，对于含角点奇性的修正 Helmholtz 方程可以实现根指数收敛。为进一步说明所提算法的适用性，该算法已成功应用于求解一些典型的线性与非线性抛物型方程。我们还首次尝试将闪电算法推广到三维修正 Helmholtz 方程。

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4. 混合 Dirichlet-Neumann 边界条件下奇异椭圆 PDE 的混合变分-配点 PINNs 中基于残差的自适应加密

Residual-Based Adaptive Refinement in Hybrid Variational-Collocation PINNs for Singular Elliptic PDEs with Mixed Dirichlet-Neumann Boundary Conditions

作者: Somnath Giri, Muslim Malik, Rishikesh Yadav

类别: math.NA, cs.LG, physics.comp-ph

发布日期: 2026-10-19

arXiv ID: 2610.26458v1

链接: http://arxiv.org/abs/2610.26458v1

摘要翻译

中文标题: 混合 Dirichlet-Neumann 边界条件下奇异椭圆 PDE 的混合变分-配点 PINNs 中基于残差的自适应加密

摘要翻译: 在非凸或含裂纹区域上、具有混合 Dirichlet-Neumann 条件的二阶椭圆 PDE 问题的解通常不属于 $H^2(Ω)$。在凹角、裂纹尖端以及边界条件类型发生改变的点附近，梯度按 $r^{λ-1}$ 增长，其中 $λ<1$。我们研究混合变分-配点 PINNs 中基于残差的自适应加密。一个多尺度 Fourier 特征网络在复合损失上训练。该损失结合了以 Legendre 气泡函数检验的变分残差、强形式配点项以及针对 Dirichlet 和 Neumann 数据的惩罚项。变分项仅需要网络的一阶导数，而配点项需要二阶导数，并提供一个可在任意点求值的残差。在每个训练阶段之后，由该残差选出的候选点被加入配点集。该方法在五个二维基准问题上进行测试，即 Mode~III 裂纹、具有四个不等幅凹角的十字形区域、具有四个局部峰的反应-扩散问题、带边界层的单调半线性问题以及 Motz 问题。在相同的配点与迭代预算下，与均匀插点相比，基于残差的加密使相对 $L^2$ 误差最多降低至原来的 $1/32$，使相对 $L^\infty$ 误差最多降低至原来的 $1/38$，并使相对 $H^1$ 半范数误差最多降低至原来的 $1/15$。

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5. 波动方程的贝叶斯几何反演：适定性与多尺度多模态后验近似

Bayesian Geometric Inversion for Wave Equation: Well-Posedness and Multiscale Multimodal Posterior Approximation

作者: Xiong-Bin Yan, Ming-Jia Pei, Guang-Hui Zheng, Ting Wei

类别: math.NA

发布日期: 2026-10-19

arXiv ID: 2610.26151v1

链接: http://arxiv.org/abs/2610.26151v1

摘要翻译

中文标题: 波动方程的贝叶斯几何反演：适定性与多尺度多模态后验近似

摘要翻译: 几何全波形反演旨在从含噪波形观测中恢复尖锐的地下界面。我们在贝叶斯水平集框架内表述该问题，其中未知界面表示为赋予高斯先验的潜函数的零水平集。尽管水平集到介质参数的映射是不连续的，我们证明水平集到观测的映射关于先验几乎必然连续，从而保证后验测度是适定的，并且在 Hellinger 度量下关于数据局部 Lipschitz 连续。为在反问题的严重非线性及由此产生的多模态性下实现可处理的后验计算，我们提出一种多尺度多模态 Laplace 近似。频率延拓策略从低频到高频逐步引入波形数据，在相继尺度上识别并传播多个候选模态。在最终频率层级，围绕保留的模态构造局部高斯近似，并将其组合为高斯混合，其权重通过最小二乘拟合未归一化后验密度确定。数值实验表明，该方法能够准确重建异常界面，同时有效捕捉后验多模态性以及与不同合理几何形态相关的局部不确定性。

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6. 逆散射问题的迭代截断SVD

Iterative Truncated SVD for Inverse Scattering Problems

作者: Zhinan Han, Ying Liang, Hongkai Zhao

类别: math.NA

发布日期: 2026-10-19

arXiv ID: 2610.25726v1

链接: http://arxiv.org/abs/2610.25726v1

摘要翻译

中文标题: 逆散射问题的迭代截断SVD

摘要翻译: 我们提出一种基于非线性最小二乘形式的 Newton 法的迭代截断 SVD（iTSVD）算法，用于逆散射问题。其关键思想是估计相应线性化算子（或离散化后的 Jacobian 矩阵）的有效秩，该有效秩定义了相关正演算子的信号空间的维数。一旦估计出有效秩，便可在信号空间中基于快速随机 SVD（RSVD）高效计算截断 SVD（TSVD）重构——仅需要矩阵-向量乘法，而该乘法可通过求解多个同类型正演与伴随问题以无矩阵形式实现。我们的分析和数值实验表明，iTSVD 能通过减少迭代次数显著提高计算效率，并通过基于物理/信息的反演增强稳定性。此外，有效秩估计自然基于模型和数据。我们给出了大量实验，包括远场或近场、有限孔径以及无噪声或有噪声的随机测量数据。

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7. 基于微观-宏观分解的高场 Vlasov-Poisson-Fokker-Planck 系统的降阶模型

A Reduced-Order Model for the High-Field Vlasov-Poisson-Fokker-Planck System Based on a Micro-Macro Decomposition

作者: Shiheng Zhang, Jingwei Hu

类别: math.NA

发布日期: 2026-10-18

arXiv ID: 2610.25569v1

链接: http://arxiv.org/abs/2610.25569v1

摘要翻译

中文标题: 基于微观-宏观分解的高场 Vlasov-Poisson-Fokker-Planck 系统的降阶模型

摘要翻译: 我们为高场 Vlasov-Poisson-Fokker-Planck 系统建立了一个基于投影的降阶模型。基于微观-宏观分解，所提出的 Micro-ROM 仅对微观涨落应用本征正交分解（POD）-Galerkin 降阶，同时在完整空间网格上保留宏观密度。依赖于场的 Maxwellian 和碰撞算子引入了额外的计算成本，这些成本无法仅通过 Galerkin 投影消除。为消除在线模拟中对速度网格分辨率的依赖，我们使用局部电场函数的预计算 Chebyshev 展开来近似投影后的 Maxwellian 项，并利用碰撞算子的可分离离散化，由预计算块精确组装降阶碰撞矩阵。采用一阶隐式-显式格式演化降阶系数和密度。所得方法保持离散质量守恒，并在 $\varepsilon\rightarrow 0$ 时形式上恢复极限高场模型的离散化。其每个时间步的在线计算成本与速度网格分辨率无关。数值实验展示了所提出方法在一系列参数（包括高场区域）下的精度和效率。

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8. 用于不可压缩流中最优混合的自监督上下文算子学习

Self-Supervised In-Context Operator Learning for Optimal Mixing in Incompressible Flows

作者: Weiwei Hu, Rongjie Lai, Ziqian Li

类别: math.NA

发布日期: 2026-10-18

arXiv ID: 2610.25556v1

链接: http://arxiv.org/abs/2610.25556v1

摘要翻译

中文标题: 用于不可压缩流中最优混合的自监督上下文算子学习

摘要翻译: 我们开发了一个用于不可压缩流中最优混合的自监督算子学习框架，旨在克服经典 PDE 约束最优控制的计算局限性。目标是在最小化动能的同时，设计能够高效均匀化被动标量的速度场，并通过 $\dot H^{-1}(Ω)$ 混合范数进行量化。经典方法使用迭代前向-伴随过程为每个初始条件求解一个单独的最优控制问题，导致巨大的计算成本和有限的可扩展性。相比之下，我们提出直接学习将初始标量场映射到最优混合控制的解算子。所提出的自监督方法不需要访问监督标签，从而显著降低监督算子学习方法的数据生成负担。我们的框架将输运方程的拉格朗日重构与基于 transformer 的神经算子架构相结合。输运动力学通过特征 ODEs 表示，使得基于粒子的 Monte Carlo 优化成为可能；同时，学习到的速度场在无散度谱基中参数化，精确施加不可压缩性和无穿透条件。在二维和三维空间中对初始条件参数族进行的数值实验表明，该方法对未见情形具有稳健泛化能力，并且混合范数近似指数衰减。学习到的控制也一致地迁移到独立的欧拉有限体积模拟中，验证了学习算子的物理保真度。据我们所知，本文提供了首个用于最优混合的自监督算子学习框架，并且属于完全三维不可压缩流中最优混合的首批数据驱动方法之一。

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9. 熵稳定的单元局部连续-间断残差修正分部求和离散

Entropy Stable Element-Local Continuous-Discontinuous Residual Correction Summation-By-Parts Discretization

作者: Zelalem Arega Worku, David C. Del Rey Fernandez

类别: math.NA

发布日期: 2026-10-18

arXiv ID: 2610.25388v1

链接: http://arxiv.org/abs/2610.25388v1

摘要翻译

中文标题: 熵稳定的单元局部连续-间断残差修正分部求和离散

摘要翻译: 残差修正为在高阶双曲偏微分方程离散中强制熵稳定性提供了一种高效替代两点体积通量差分的方案。我们发展了一种单元局部残差修正分部求和（SBP）方法，该方法在协调网格上耦合连续-SBP（C-SBP）与间断-SBP（D-SBP）。该构造在适当之处保留无需同时逼近项（SATs）的连续组装，将 SATs 限制于选定的间断面片，并允许耦合随时间变化，作为自适应离散的基础。标量熵亏损和作用于守恒变量的基于图的分布向量完全由单元局部数据计算，无需相邻状态、辅助梯度或修正 SATs。同一修正适用于每一种面片划分，包括纯 C-SBP 与 D-SBP 极限。对于固定划分，我们建立了每个单元残差的守恒与熵恒等式、精确全局守恒、至多相差一个显式正则化亏损的熵稳定性，以及设计阶一致性。划分的变化由质量加权正交投影处理，该投影保持守恒矩和凸容许集，不增加熵，并且不放大分片范数中的误差。一至三维数值实验表明，在重复重划分下，该混合方法保持其 D-SBP 端点的光滑解精度。非线性测试表明其鲁棒性与 Hadamard 形式的两点通量离散相当，尤其是与人工耗散耦合时。

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10. 量子化学模拟中的空间离散化误差

Spatial Discretization Error for Quantum Chemistry Simulation

作者: Di Fang, Jiaqi Zhang

类别: quant-ph, math-ph, math.NA

发布日期: 2026-10-18

arXiv ID: 2610.25260v1

链接: http://arxiv.org/abs/2610.25260v1

摘要翻译

中文标题: 量子化学模拟中的空间离散化误差

摘要翻译: 非相对论量子化学基于连续、无界空间中的 Schrödinger 方程。经典计算机与量子计算机上的计算方法通常截断空间区域并引入有限维基，从而将连续问题约化为有限维问题。因此存在一个基本问题：这种近似在多大程度上捕捉了原始的连续问题？在本工作中，我们针对无界 Hamiltonians 的量子模拟，建立了一个用于估计空间离散化误差的一般 Sobolev-Galerkin 框架。我们将该框架应用于多体 Coulomb 系统，并为波函数动力学与基态能量建立了周期化误差估计和 Galerkin 误差估计。对于动力学，精确波函数的有限基投影误差并不能刻画完整的基离散化误差：一个额外的动力学投影误差以不同速率进入我们的界，我们引入内积论证来控制这一贡献。有趣的是，波函数动力学与基态能量的空间收敛速率在定量上不同。对于基态能量，我们在固定元胞尺寸下建立了一个尖锐的 $M^{-1}$ 收敛速率，其中 $M$ 是单粒子基函数的数量。我们还证明，对于相应的一般 $H^2$ 初值的实时波函数误差，一致的 $M^{-1}$ 收敛速率是不可能的。该分析依赖于具有对系统规模的显式多项式依赖的多体 Sobolev 估计。将这些估计与最先进的量子算法相结合，据我们所知，我们得到了分子与电子动力学量子模拟的首个严格端到端复杂度界，且所有空间误差均被显式控制。基函数的数量直接由连续问题确定，无需正则化。

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11. 可压缩反应流动方程的斜对称形式与能量稳定格式

Skew-Symmetric Formulations and Energy-Stable Schemes for Compressible Reactive Flow Equations

作者: Junbo Gong, Weifeng Zhao

类别: math.NA

发布日期: 2026-10-17

arXiv ID: 2610.24524v1

链接: http://arxiv.org/abs/2610.24524v1

摘要翻译

中文标题: 可压缩反应流动方程的斜对称形式与能量稳定格式

摘要翻译: 本文针对反应 Euler 方程和反应 Navier–Stokes 方程发展了斜对称形式与能量稳定的有限差分格式。通过引入适当的变换变量，我们将控制方程改写为斜对称形式，并利用标准的分部积分论证得到连续能量估计。所提出的形式将可压缩流动方程的斜对称框架推广到反应系统；在其中，化学能和组分方程必须纳入变换变量中。基于这些形式，我们利用分部求和（SBP）算子构造半离散格式。对流项由线性迎风 SBP 算子的非耗散部分离散，而非线性 SBP–WENO 算子仅用于在间断附近提供人工耗散。在周期边界条件下，所得格式满足与连续分析一致的半离散能量估计。一维和二维空间中的数值实验验证了所设计的精度阶，并展示了其在含激波、黏性层、爆轰型波和复杂波相互作用问题上的稳定性能。

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12. 神经偏微分方程中的 Segal-Bargmann 变换

The Segal-Bargmann Transform in Neural Partial Differential Equations

作者: Jie Xu, Julio Castrillón-Candás, Mark Kon, Qinxun Bai

类别: math.NA, math.ST

发布日期: 2026-10-17

arXiv ID: 2610.24497v1

链接: http://arxiv.org/abs/2610.24497v1

摘要翻译

中文标题: 神经偏微分方程中的 Segal-Bargmann 变换

摘要翻译: 我们提出了一种新的神经算子方法，该方法基于 Segal-Bargmann 变换和 Bargmann-Fock 空间，用于学习偏微分方程 (PDEs) 的解算子。该方法将数据编码、隐空间表示、算子学习和解码统一到一个框架中。仅从有限多个观测出发，我们建立了阶为 $ L^{-\frac{1}{2}} $ 的 sup 范数重构误差，其中 L 是与原始采样和观测解耦的复采样点的数量。该重构误差与空间维数和原始采样点的数量无关。我们证明，这种新的无网格 Segal-Bargmann 神经算子方法相比现有的 NO 方法（如 FNO、BelNet、GNOT 等）具有更优的性能，特别是在对 PDE $ \mathcal{L}u = f $ 的解的逐点控制 $ \sup_{x}\lvert\hat{u}_{NO} - u_{truth}\rvert $ 方面，这提供了如经典数值求解器般的逐点精度和可解释性。我们还展示了更高维非线性 PDE 的解算子的重构，这表明我们提出的方法在应用于更具挑战性的 PDE 乃至更高空间维度方面具有很大潜力。

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13. 时间依赖 Ginzburg–Landau 方程的规范不变有限元方法

A Gauge-Invariant Finite Element Method for the Time-Dependent Ginzburg–Landau Equations

作者: Huadong Gao, Yuhui Huang, Wen Xie

类别: math.NA

发布日期: 2026-10-17

arXiv ID: 2610.24470v1

链接: http://arxiv.org/abs/2610.24470v1

摘要翻译

中文标题: 时间依赖 Ginzburg–Landau 方程的规范不变有限元方法

摘要翻译: 众所周知，Ginzburg–Landau 方程具有规范不变性。实际模拟中最常用的规范选择是时间规范，也称为零电势规范。然而，常用的 Nédélec 近似在具有凹角的区域上可能产生伪巨涡旋。为了解决该问题并同时保留 Nédélec 近似，我们提出一个在离散层面保持时间依赖 Ginzburg–Landau 方程规范不变性的有限元格式。在时间规范下，该方法在三角形和四面体网格上对磁势采用最低阶 Nédélec 棱单元，对序参量采用连续分片线性单元，并得到一个线性解耦格式，该格式在每个时间步仅需求解线性方程组。我们证明该格式具有规范不变性，保持序参量的离散最大值有界原理 $\|ψ_h^n\|_{L^\infty(Ω)}\le 1$，并在温和的步长与网格条件下满足离散能量律。数值实验验证了这些理论性质。特别地，在具有凹角的区域上，该格式计算出正确的物理解，而采用 Nédélec 单元的传统时间规范离散则会产生非物理解。

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14. 抛物型障碍问题的可认证神经网络近似

Certified neural approximations for parabolic obstacle problems

作者: Davood Damircheli

类别: math.NA, cs.LG, math.FA

发布日期: 2026-10-16

arXiv ID: 2610.24202v1

链接: http://arxiv.org/abs/2610.24202v1

摘要翻译

中文标题: 抛物型障碍问题的可认证神经网络近似

摘要翻译: 残差训练的神经代理不提供误差保证，这对相变和接触等自由边界问题构成限制。我们为一阶状态–通量–乘子形式下抛物型障碍问题的神经网络近似推导了有保证的误差界。主要结果用本构残差、初始失配、互补缺陷以及由 $c_F^2/α$ 加权的 PDE 残差来界定网络通量误差及其可行投影 $\max\{v,g\}$ 的能量误差，而无需一阶表述的强制性和范数等价常数。它将热算子的 Apushkinskaya–Repin 上界泛函推广到一般系数、自由乘子和通量界。一个穿透–相容性结果表明，存在具有任意小界的神经三元组。逐单元向外舍入的 Taylor 模型包络将该界转化为在任意自适应剖分上有效的有限计算。对于一维 Stefan 问题，投影输出的通量误差和梯度误差被认证为分别低于解尺度的 $0.27$ 和 $0.33$；在二维扩张圆盘上，经过自适应细化后，全部三个训练网络的这些误差被认证为低于一半。在扩张椭圆上，梯度界仍高于一半；代入诊断将训练网络的下限归因于自由边界附近的投影状态，并将剩余差距归因于包络开销。在没有闭式解的情况下，一个经验证的下界替代解范数；此时包络几乎精确，但训练网络的误差高于解尺度，且未认证出任何有用的精度。这些界控制的是状态–通量范数，而不是乘子或自由边界位置。

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不相关论文（基本信息）

1. 双曲空间上 Lipschitz 测地凸优化的紧下界

Tight lower bounds for Lipschitz geodesically convex optimization on hyperbolic space

作者: Christopher Criscitiello

类别: math.OC, math.DG, math.NA

发布日期: 2026-10-19

arXiv ID: 2610.26815v1

链接: http://arxiv.org/abs/2610.26815v1

摘要: We determine the dimension-independent first-order query complexity of minimizing a Lipschitz geodesically convex (g-convex) function on hyperbolic space. Given a ball of radius $r$ containing a global minimizer, we prove that $Ω(ζ_r/ε^2)$ queries are necessary to attain accuracy $εMr$, where $M$ is the Lipschitz constant and $ζ_r=r/\tanh r$ captures the effect of curvature. This matches the known $O(ζ_r/ε^2)$ upper bound for the Riemannian subgradient method (Zhang and Sra, 2016). This is the first query-complexity result for g-convex optimization that is tight simultaneously in curvature and target accuracy. The standard Euclidean recipe of constructing hard instances as maxima of affine functions does not transfer to hyperbolic space, so the lower bound requires a new mechanism. Our starting point is the worst-function construction of Criscitiello and Boumal (2023, Appendix A). It gives the same lower bound for a subclass of algorithms, including the subgradient method, whose queries remain in prescribed g-convex regions. We retain this geometric setup and carefully tune and augment their hard function to accommodate every deterministic first-order algorithm. The key new mechanism is an advance-or-stall scheme: when a query lies in these regions, the oracle reveals at most one new information direction (advance); when a query leaves them, we show that the oracle can answer without revealing any new hidden direction (stall). Unlike earlier curvature-dependent lower bounds, the proof uses no ball-packing argument.

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2. 用于干燥损伤耐火浇注料热-湿-力相场模拟的高效开源框架

An Efficient Open-Source Framework for Thermohygromechanical Phase-Field Simulation of Drying Damage Refractory Castables

作者: Murilo H. Moreira, Roberto F. Ausas, Stefano Dal Pont

类别: math.NA

发布日期: 2026-10-19

arXiv ID: 2610.26752v1

链接: http://arxiv.org/abs/2610.26752v1

摘要: The water removal from refractory castables is a complex process involving heat and mass transport and thermohygromechanical stresses. In order to reduce the risk of explosive spalling by improving the heat-up curves, numerical models are critical to predict the castable behavior during drying. However, most of the existing models either neglect the thermomechanical stresses or consider them as an \textit{a posteriori} assessment of the likelihood of failure. The present work proposes an open-source thermohygromechanical phase-field model to simulate the drying of refractory castables and predict the occurrence of explosive spalling. The model is based on an adaptive two-scale time-stepping strategy that allows for a significant reduction of the computational cost associated with the simulation. The model was validated against literature results and qualitative comparisons with experimental data were carried out to assess its ability to predict the failure behavior of partially saturated porous media during heating. The numerical simulations were also applied to investigate the impact of different heating methods on the risk of failure during drying. It was found that even at the significantly lower heating rates typical of refractory castable drying - compared to the fire accidents that motivate most thermomechanical studies on concrete structures - thermomechanical stresses still play a significant role in the explosive spalling, a finding well established for concrete under fire but novel for refractory castables. As a consequence, it was also concluded that the use of heterogeneous heating methods can increase the likelihood of explosive spalling. Ultimately, feasible solutions to increase the drying efficiency and safety that are compatible with the industrial setting were proposed and a free, open-source implementation of the model is provided.

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3. Polar Switch：基于多项式电路的快速 Muon 正交化

Polar Switch: Fast Muon Orthogonalization with Polynomial Circuits

作者: Zhehang Du, Weijie Su

类别: math.NA, cs.LG, math.OC

发布日期: 2026-10-19

arXiv ID: 2610.26742v1

链接: http://arxiv.org/abs/2610.26742v1

摘要: Muon, a 2024-born optimizer for language models, differs from the 2015-born spectral gradient descent mainly in its fast, GPU-friendly Newton–Schulz polynomial for orthogonalizing the gradient matrix. Accordingly, a line of research proposes faster polynomial iterations for gradient orthogonalization, such as Polar Express and Gram Newton–Schulz, both built on quintic polynomials. In this paper, we introduce Polar Switch, a new approach to polynomial gradient orthogonalization. Using simple polynomial circuits, Polar Switch reaches a given high polynomial degree at a lower FLOP cost than quintic iterations. We define a circuit as a short sequence of symmetric matrix products on a few square buffers, and circuits with $k$ products form a restricted family of odd polynomials of degree up to $2^{k+1}+1$. We choose the sequence of products in each circuit, their coefficients, and how the circuits are composed by numerically minimizing a worst-case bound on the polar approximation error. This minimization is subject to a FLOP budget that accounts for symmetric matrix multiplication kernels and a rounding constraint for numerical stability in half precision. Our two recommended Polar Switch methods have 15.5–18.7% lower runtime than Polar Express on square matrices and 8.6–16.2% lower runtime than Gram Newton–Schulz on rectangular matrices with aspect ratios 3.25, 3.5, and 8, with comparable approximation accuracy. In pretraining Modded-NanoGPT #91, Qwen3-0.6B, and Gemma 3-1B, Polar Switch reaches final validation losses similar to those of established orthogonalization methods. Across the more than 14 million matrices that each of the two recommended methods orthogonalized in these runs, the estimated output spectral norms never exceed the designed bounds by more than 3%, demonstrating their numerical stability.

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4. 基于张量缩并聚合的同时矩阵乘法

Simultaneous Matrix Multiplication by Tensor Contraction Aggregation

作者: Edgar Solomonik, Markus Bläser

类别: math.NA

发布日期: 2026-10-19

arXiv ID: 2610.26563v1

链接: http://arxiv.org/abs/2610.26563v1

摘要: We present an interpretation of the fast matrix multiplication algorithms of Coppersmith and Winograd as aggregations of tensor contractions. This point of view provides a direct connection to older constructions by Pan and to more recent work on bilinear algorithms for symmetric tensor contraction. Further, we provide a new simple approach for exact aggregation of tensor contractions, each one yielding an independent matrix-matrix product. These constructions and border rank adaptations thereof do not come close to improving the best known bounds on $ω$ (the matrix multiplication exponent), but provide explicit moderate-size fast matrix multiplication algorithms that achieve a better exponent than Strassen’s algorithm and have potential practical value. These results imply improvements to the best known bounds on the rank and border rank of a fixed number of independent square matrix products. In particular, using $\langle m,m,m\rangle$ to denote the matrix multiplication tensor for matrices of dimension $m$, we obtain constructions that give bounds such as the border rank bound [ \underline R!\left(\langle n^2,n^2,n^2\rangle^{\oplus9}\right) \le (n+1)^6+6(n+1)^5, ] and the exact rank bound [ R!\left(\langle n^3,n^3,n^3\rangle^{\oplus14}\right)\le (n+1)^9. ] We quantify the accuracy of such constructions in finite precision, and provide border-rank constructions that reduce cost with limiting retained fractions of accuracy at least $1/2$, $1/3$, or $1/4$ relative to the classical matrix-multiplication approach under the stated error model.

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5. 关于垂心四面体与等角四面体的二面角和的精确范围

On Exact Dihedral Angle Sum Ranges for Orthocentric and Isogonic Tetrahedra

作者: Sergey Korotov

类别: math.NA

发布日期: 2026-10-19

arXiv ID: 2610.26529v1

链接: http://arxiv.org/abs/2610.26529v1

摘要: For a nondegenerate Euclidean tetrahedron T, let Σ(T) denote the sum of its six interior dihedral angles. We establish the exact range [6 arccos(1/3), 3π) for the orthocentric and isogonic classes. The orthocentric part gives a short alternative proof of the sharp lower bound; the main new result is the corresponding isogonic theorem. Starting from classical insphere-contact geometry, we derive a reciprocal half-angle identity and a sharp defect inequality controlled by the scale-invariant dispersion of the reciprocal tangent-length parameters. This quantitative refinement shows that any nonzero dispersion forces a positive dihedral-angle-sum gap, and that near equality forces the normalized reciprocal parameters to be close to equality. Finally, the common kite subfamily realizes every value in the interval and proves sharpness of both endpoints.

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6. 用于光度立体三维重建的 Matlab/Octave 软件包

A Matlab/Octave package for photometric stereo 3D reconstruction

作者: Alessandro Buccini, Elisa Crabu, Federica Pes, Giuseppe Rodriguez

类别: math.NA

发布日期: 2026-10-19

arXiv ID: 2610.26162v1

链接: http://arxiv.org/abs/2610.26162v1

摘要: The photometric stereo model allows the construction of 3D virtual replicas of real objects from images acquired under different light conditions. The assumptions of this model are strongly restrictive and are seldom fully satisfied in applications. Indeed, the surface of the observed objects may deviate from ideal conditions, and the geometry of the scene is often only an approximation of the one assumed by the model. Additionally, real-world surfaces may present reflection phenomena, which introduce distortions in reconstructions. In this paper, we present a revised version of the Matlab package \ttt{\small ps3d}~\cite{cdfrv21}, developed to implement the photometric stereo model under various conditions. The new package contains various extensions, the most notable one is the presence of an alternative model to address the presence of a reflective object surface. In addition, it implements particular metrics to analyze the compliance of a dataset to model assumptions, and it is compatible with GNU Octave. The tool allows users to test the model under different observation configurations, also in presence of reflection, using either synthetic or experimental data.

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7. 对偶四元数离散线性正则变换：基于FFT的快速算法与联合旋转–平移位姿去噪

Dual Quaternion Discrete Linear Canonical Transforms: FFT-Based Fast Algorithms and Joint Rotation–Translation Pose Denoising

作者: Yi Wu, Xuan Chen, Kit Ian Kou

类别: math.NA

发布日期: 2026-10-19

arXiv ID: 2610.26140v1

链接: http://arxiv.org/abs/2610.26140v1

摘要: Rigid-body pose signals couple rotation and translation, yet most existing denoising pipelines treat them separately and analyse them only in the Fourier domain. We show that this separation is unnecessary and often suboptimal. We fuse dual quaternions, which encode rotation and translation in a single singularity-free algebra, with linear canonical transforms, which supply flexible quadratic-phase analysis, to develop one- and two-dimensional dual quaternion discrete linear canonical transforms (DQ-DLCTs) with an explicit kernel ordering. We establish inverse and energy-preservation properties, structured spatial and frequency convolution relations, and, crucially, fast algorithms built on complex FFTs. These algorithms make the DQ-DLCT practical: a two-stage chirp-adaptive scheme then performs joint rotation–translation denoising in a single transform domain. Numerical experiments confirm the accuracy and efficiency of the fast transform and demonstrate strong denoising performance on measured pose sequences and constructed two-dimensional pose fields. The DQ-DLCT thus turns a heavy algebraic construction into a deployable tool for pose signal processing.

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8. 关于信赖域半径更新的结构条件。第二部分：二阶收敛性与半径的渐近行为

Structural conditions on trust-region radius updates. Part II: second-order convergence and asymptotic radius behaviour

作者: Jérémy Rieussec, Fabian Bastin

类别: math.OC, math.NA

发布日期: 2026-10-19

arXiv ID: 2610.26137v1

链接: http://arxiv.org/abs/2610.26137v1

摘要: Continuing Part I (arXiv:2606.30202), we ask not which structural conditions on $\{Δ_k\}$ suffice for global convergence but what the radius does; the answers separate the mechanisms by whether the radius is tied to a criticality measure. We identify three asymptotic regimes: the criticality-anchored rules give $\sum_k Δ_k^2/M_k < +\infty$, hence $\liminf_k Δ_k = 0$, and $Δ_k \to 0$ once the model Hessians are bounded; the step-driven rules inherit the summability of the step series, hence $\sum_k Δ_k < +\infty$; the fixed-factor rules keep $Δ_k$ bounded away from zero. Near a critical point with positive definite Hessian, convergence of a subsequence forces convergence of the full sequence. We then extend the framework to second-order criticality through $τ_k = \max\{\|g_k\|, -λ_k^-\}$, obtaining convergence to second-order critical points with the worst-case complexity $\mathcal{O}(\max\{\varepsilon_g^{-2}\varepsilon_H^{-1}, \varepsilon_H^{-3}\})$, which reduces to $\mathcal{O}(\varepsilon^{-3})$ for a common tolerance and is sharp for quadratic models. Finally we establish eventual inactivity of the constraint and superlinear convergence, isolating the two properties of the radius rule through which the mechanism enters; the rate itself is mechanism-independent. The criticality-anchored and capped gradient-scaled rules reach inactivity only under a side condition, $ζ> \overlineκ$ or $\overlineμ > \overlineκ$, with $\overlineκ$ a constant of the solution that cannot be checked in advance. These conditions are necessary, not artefacts: on $f(x) = \sin(x)/x$, $\overlineκ$ is unbounded over the minimisers, so no parameter value meets the condition at all of them. Among the rules tied to a criticality measure, only those with an unbounded multiplier escape such a constant.

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9. 信赖域算法中半径更新规则的数值研究

A numerical study of radius update rules in trust-region algorithms

作者: Jérémy Rieussec, Fabian Bastin

类别: math.OC, cs.MS, math.NA

发布日期: 2026-10-19

arXiv ID: 2610.26134v1

链接: http://arxiv.org/abs/2610.26134v1

摘要: Part I [arXiv:2606.30202] and Part II establish which trust-region radius update mechanisms can fail and under which hypotheses they must succeed. Neither says where the thresholds sit once a rule runs with real constants, or how the rule interacts with the model Hessian and the subproblem solver. This paper turns them into nine falsifiable claims and settles what it can, on four designed problems where the theory’s quantities are computable and on CUTEst, where only aggregate cost is visible. Every mechanism runs in one Julia package written for this study, so two runs differing in one component differ in nothing else. The rule is worth less than the model and more than nothing. Crossing the mechanisms with four model Hessians puts $96.5$ per cent of the variation on the model and $1.6$ on the rule, and the rules separate most under the most accurate model rather than least. Uncapped R-grad is the robust default among the criticality-anchored rules, though the retrospective rule is ahead of it on this collection. No constant escapes the side conditions. For every cap there is a minimiser at which the constraint stays active, and for every anchoring constant a flatness on which escape takes arbitrarily long. The second of these is proved here as well as measured, and the proof uses no property of the model or of the subproblem solution beyond $\|s_k\| \leq Δ_k$. The radius rule governs the length of the step, and so the cost of a run. The model decides the search direction, and with it whether the run converges at all.

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10. 迭代拟可逆性方法的有限元离散研究——Laplacian 数据同化情形

Study of the finite element discretization of the iterated quasi-reversibility method – The Laplacian data assimilation case

作者: Jérémi Dardé, Jérémy Heleine

类别: math.NA

发布日期: 2026-10-19

arXiv ID: 2610.26117v1

链接: http://arxiv.org/abs/2610.26117v1

摘要: We investigate the finite element discretization of the iterated quasi-reversibility method, in the context of the data assimilation problem for the Laplacian. We provide error estimates explicit in all the parameters appearing in the model and the method (i.e. the amplitude of noise, the size mesh, the regularization parameter and the number of iterations of the method). This allows to devise strategies to choose the aforementioned parameters, for example to guarantee convergence of the method when the level of noise goes to zero. Numerical experiments on a two-dimensional problem are provided, illustrating the relevance of the obtained estimates.

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11. 面向等离子体约束的 Vlasov–Poisson 系统的全时域磁控制

Full-horizon magnetic control of the Vlasov–Poisson system for plasma confinement

作者: Giacomo Albi, Giacomo Dimarco, Federica Ferrarese, Lorenzo Pareschi

类别: math.NA

发布日期: 2026-10-19

arXiv ID: 2610.26110v1

链接: http://arxiv.org/abs/2610.26110v1

摘要: We study the full-horizon control of a two-dimensional Vlasov–Poisson system through an externally imposed magnetic field. The control is represented by time-dependent coefficients associated with smooth, spatially localized basis functions, and is designed to steer selected position and velocity moments of the distribution function while limiting particle dispersion and the control effort. Starting from the kinetic constrained optimization problem, we derive the continuous first-order optimality conditions, including a backward Vlasov equation, an elliptic adjoint equation, and the variational inequality associated with bounded controls. We then project this system onto a particle–grid representation and identify the resulting particle adjoints and control gradient. The full-horizon field is then computed on a reduced particle ensemble, initialized by an instantaneous control, and subsequently applied to a larger set of particles. Numerical experiments indicate that the instantaneous control field already provides an effective baseline, while the full-horizon correction can further reduce the relative thermal energy near the boundaries and improve confinement over the prescribed time interval.

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12. 论降秩回归

On Reduced Rank Regression

作者: Th{é}o Rudkiewicz, Manon Verbockhaven

类别: math.NA

发布日期: 2026-10-19

arXiv ID: 2610.26051v1

链接: http://arxiv.org/abs/2610.26051v1

摘要: We propose a decoupled solution to the Reduced Rank Regression (RRR) problem, i.e. a pair of matrices (A*, B*) whose product AB minimizes the least squares error under a rank constraint. Using our formalism, we rewrite two other solutions from the literature and compare them with ours. In particular, we prove that all these formulations are equivalent. We then analyze their time complexity, and compare their running time and numerical precision on synthetic data, from which we derive practical recommendations. An open-source implementation of the algorithms and experiments is available here: https://gitlab.inria.fr/trudkiew/reduced_rank_regression.

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13. 一种用于具有连续内能的多原子 Boltzmann 碰撞算子的快速 Laguerre–Fourier 谱方法

A fast Laguerre–Fourier spectral method for the polyatomic Boltzmann collision operator with continuous internal energ

作者: Pei Zhang, Yanli Wang, Zhenning Cai

类别: math.NA

发布日期: 2026-10-19

arXiv ID: 2610.25921v1

链接: http://arxiv.org/abs/2610.25921v1

摘要: The polyatomic Boltzmann equation extends the classical kinetic description to gases whose molecules possess internal energy in addition to their translational motion. The distribution function then has one more variable than in the monatomic case, and the collision operator becomes an eight-fold integral whose evaluation dominates the cost of solving the equation. In this work, we propose a fast Laguerre–Fourier spectral method for the polyatomic Boltzmann equation with continuous internal energy, which evaluates the collision operator in $O(K^2MN^4\log N + K^3MN^4)$ online operations after a solution-independent precomputation, where $N$ is the number of Fourier modes in each velocity direction, $K$ is the number of Laguerre modes and $M$ is the number of quadrature points on the sphere. Specifically, the distribution function is expanded in generalized Laguerre functions in the internal energy and in Fourier modes in velocity. A sequence of changes of variables then separates the internal-energy integrals from the velocity convolutions, so that the former no longer involve the distribution function and are computed once and stored. Numerical experiments are presented for both the spatially homogeneous and the spatially inhomogeneous Boltzmann equation. They confirm the spectral accuracy of the proposed method and its agreement with the direct simulation Monte Carlo method.

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14. 正则化 Landau 粒子动力学的交替 score-系统迭代的收敛性

Convergence of alternating score-system iterations for regularized Landau particle dynamics

作者: Yeoneung Kim, Donghyun Lee, Sung-Jun Son

类别: math.NA, math-ph, math.AP

发布日期: 2026-10-19

arXiv ID: 2610.25780v1

链接: http://arxiv.org/abs/2610.25780v1

摘要: We analyze an alternating iteration for score-based particle approximations of the homogeneous Landau equation. Each step computes a regularized score along the previous trajectory and recomputes the trajectory with that score frozen. First, at fixed positive regularization, we prove existence, uniqueness, and $W_1$-Lipschitz stability of a Tikhonov score minimizer in a vector-valued reproducing kernel Hilbert space embedded in $C_b^2$. For bounded Lipschitz collision matrices, this yields factorial convergence to the coupled regularized dynamics on every finite interval, with constants independent of particle number. Global objective gaps and system residuals control inexact solves, and structural kernel assumptions ensure momentum and energy conservation. The mean-field and alternating-iteration limits commute at fixed regularization in this regime. Second, score-difference cancellation extends the theory to far-field-truncated Landau kernels with $γ\geq -2$ and the uncut case $γ=-2$. For the uncut three-dimensional Coulomb kernel, a separate continuum construction gives local factorial convergence from bounded, compactly supported densities. Third, we characterize the obstruction to removing regularization at fixed particle number and establish a separate conditional density-level convergence criterion under uniform regularity and approximation assumptions, using relative entropy. A heat-equation counterexample shows that an unregularized density-level Picard iteration can diverge. Atomic Coulomb and simultaneous particle-zero-regularization limits remain open.

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15. 二次方程组线性化的一种方法

A Method for Linearization of Systems of Quadratic Equations

作者: A. A. Pak, A. S. Pak

类别: math.NA

发布日期: 2026-10-18

arXiv ID: 2610.25506v1

链接: http://arxiv.org/abs/2610.25506v1

摘要: Integrating physical conservation laws into deep neural architectures remains a central SciML challenge. Soft-penalty approaches, including many PINNs, may be sensitive to loss weighting and optimization; generic DEQ and implicit-layer solvers may require instance-dependent iteration counts. We study a norm-linearization scheme for networked systems with linear continuity constraints and nonlinear quadratic or power-law energy balances. We derive sharp universal and finite-$n$ bounds on relative variance and, via the weighted Kantorovich inequality, relate them to worst-case single-step squared-norm distortion when target and operating states have a bounded multiplicative ratio. Euler’s theorem shows that the scheme is Newton’s method for degree-one norm differences $u_i(\mathbf{x})-v_i(\mathbf{x})=0$. Recasting raw degree-$α$ balances as norm differences removes radial curvature, yields a degree-zero homogeneous Jacobian, eliminates the affine Newton term, and row-equilibrates the loop Jacobian at equilibrium. Under regularity and nonsingularity assumptions, this gives local quadratic convergence for Darcy-Weisbach ($α=2$) and Hazen-Williams ($α=1.852$) models. We implement dynamic sign partitioning in a FlowDEQ layer using a reduced incidence matrix and implicit-function adjoint differentiation.

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16. KernelPack-JAX：一个用于节点生成及正向与反向PDE问题的无网格可微GPU框架

KernelPack-JAX: a meshless differentiable GPU framework for node generation and forward and inverse PDE problems

作者: Alberto Cattaneo, M. Keith Ballard, Robert M. Kirby, Grady B. Wright, Varun Shankar

类别: math.NA, cs.MS

发布日期: 2026-10-18

arXiv ID: 2610.25484v1

链接: http://arxiv.org/abs/2610.25484v1

摘要: We present KernelPack-JAX, a meshless, differentiable, GPU framework in JAX for solving forward and inverse PDE problems in domains whose boundaries, including internal obstacles, are given only by samples of their smooth pieces. Our node generator is based on a new stacked representation of piecewise-smooth boundaries, in which a parametric partition of unity of each smooth piece induces the supports of a curl-free implicit partition of unity of the whole boundary. This representation keeps the edges and corners of the boundary and classifies every point of space without a global fit. Boundary and interior nodes are then selected by a parallel greedy independent-set algorithm that reproduces sequential Poisson-disk sampling exactly. We discretize elliptic and parabolic problems with radial basis function-generated finite differences (RBF-FD) on stencils selected by a new visibility heuristic to avoid crossing the boundary, with ghost nodes for the boundary conditions. In addition, our framework exposes a GPU sparse direct solver to JAX as a differentiable linear solve, so that Jacobians of time-dependent problems cost a small multiple of a forward solve. We derive error estimates that separate consistency from stability and bound the parameter error by the forward error. We demonstrate high-order convergence for Darcy flow, reaction-diffusion, and advection-diffusion in 2D and 3D domains, including one with an internal obstacle, and show that the recovered parameters largely follow the forward errors. We also present timings for every stage.

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17. 通过对偶 Padé 有理逼近推进 Sheffer 序列理论

Advancing the Theory of Sheffer Sequences via Dual Padé Rational Approximants

作者: Ujair Ahmad, Subuhi Khan

类别: math.CA, math.NA

发布日期: 2026-10-18

arXiv ID: 2610.25457v1

链接: http://arxiv.org/abs/2610.25457v1

摘要: Sheffer sequences offer an operational framework that unifies a wide range of classical polynomial families. In this article, a rational approximation is developed for both the amplitude and the delta-operator symbol in the exponential generating function. The resulting Padé Sheffer polynomials are shown, within the umbral calculus framework, to retain the Sheffer structure. Explicit operational representations are obtained, together with finite-order differential recurrences and closed-form connection constants characterizing the deviation from the underlying polynomial family. An important consequence of the construction is that rational generating indicators reduce the usual transcendental Lagrange inversion problem to a finite multinomial computation. The framework also suggests an interpretation in terms of weighted finite automata, linking polynomial constructions with state-space dynamics for unbounded combinatorial partitions.

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18. 玻尔兹曼方程的一个守恒约化碰撞模型

A conservative reduced collision model for the Boltzmann equation

作者: Pei Zhang, Yanli Wang

类别: math.NA

发布日期: 2026-10-18

arXiv ID: 2610.25398v1

链接: http://arxiv.org/abs/2610.25398v1

摘要: The evaluation of the quadratic collision operator is a major computational bottleneck in solving the Boltzmann equation owing to the high dimensionality and the nonlinear structure of the collision term. In this work, we develop a conservative reduced collision (RC) model within the discrete velocity framework. The bilinear collision operator is approximated through a kernel tensor preassembled using problem-dependent reduced bases for the distribution function and the collision term. Consequently, online evaluation of the quadratic collision operator is reduced to tensor contractions. Exact discrete conservation of mass, momentum, and energy is enforced by constructing the collision basis in the orthogonal complement of the space spanned by the discrete collision invariants. The resulting model can be incorporated into existing deterministic Boltzmann solvers. Numerical experiments on the spatially homogeneous relaxation, the steady channel and lid-driven cavity flows, and an unsteady diffusion problem show that the proposed RC model attains the accuracy of the fast Fourier spectral method (FSM) at moderate ranks. Since the kernel tensor is assembled once in the offline stage, the online cost of each solve is reduced by orders of magnitude, so that the total cost is substantially lower even when the offline stage is included.

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19. 基于双指数公式计算矩阵对数的改进算法

Improved algorithms for computing the matrix logarithm based on the double-exponential formula

作者: Tomoaki Okayama, Tsubasa Setogawa

类别: math.NA

发布日期: 2026-10-18

arXiv ID: 2610.25361v1

链接: http://arxiv.org/abs/2610.25361v1

摘要: Tatsuoka et al. proposed an efficient algorithm for computing the matrix logarithm using the double-exponential (DE) formula. Its discretization error decreases exponentially as the number of quadrature nodes grows, and its nested grids allow previously computed values to be reused when the mesh is refined. They also reported that their approach is more robust than Gauss–Legendre quadrature for ill-conditioned matrices. We propose two algorithms based on the DE formula. The first adaptively refines the mesh while reusing previously computed values and requires no matrix norm computations. The second uses a rigorous error bound to select the number of quadrature nodes a priori. Numerical experiments show that both methods attain lower errors than Tatsuoka et al.’s method at comparable elapsed times over the tested ranges.

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20. 用格规则逼近高维积分的浅显导引，以及 QMC 积分相关函数空间的或许并不那么浅显的导引

A gentle introduction to approximating high-dimensional integrals with lattice rules and a maybe-not-so-gentle introduction to the relevant function spaces for QMC integration

作者: Dirk Nuyens

类别: math.NA

发布日期: 2026-10-18

arXiv ID: 2610.25331v1

链接: http://arxiv.org/abs/2610.25331v1

摘要: These lecture notes give both a practical and theoretical introduction to high-dimensional integration with lattice rules. We introduce lattice points and rules in combination with accompanying Julia code. Next we introduce the typical function spaces used in the literature. We compare classical Sobolev spaces with Sobolev spaces of dominating mixed smoothness. We derive lower bounds and show that the curse of dimensionality is unavoidable for these unweighted function spaces. Next we switch to weighted periodic spaces and derive a constructive upper bound using the component-by-component construction for the generating vector of a lattice rule. We pick periodic spaces because they are a perfect match with lattice rules. The upper bound shows that under suitable conditions on the weights we obtain dimension-independent worst-case error bounds. We then use embedding constants to transfer these bounds to some common norms for these periodic spaces.

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21. 多边形区域上随机抛物型方程有限元方法的极大 $L^p$-$L^q$ 正则性

Maximal $L^p$-$L^q$ regularity of finite element methods for stochastic parabolic equations in polygonal domains

作者: Genming Bai, Jianbo Cui, Buyang Li

类别: math.NA

发布日期: 2026-10-18

arXiv ID: 2610.25275v1

链接: http://arxiv.org/abs/2610.25275v1

摘要: We prove that a spatial semidiscrete finite element method preserves the stochastic maximal regularity of the heat equation on bounded convex polygonal and polyhedral domains in dimensions two and three. A main difficulty is to establish suitable Green-function estimates for stochastic convolutions, whose paths generally lack classical time derivatives. Besides, the failure of the Burkholder-Davis-Gundy (BDG) inequality at the endpoint $p=\infty$ creates additional difficulties. Our approach leverages a hidden symmetry structure of the discrete Green’s function, which enables the use of a key integration-by-parts type formula. Then the discrete stochastic maximal $L^p$-$L^p$ regularity with $p\in [2,\infty)$ is obtained from the discrete local energy estimates and interpolation inequalities. To extend further the discrete stochastic maximal $L^p$-$L^q$ regularity with the full range of integrability exponents, i.e., $p\in (2,\infty), q\in [2,\infty)$ or $p=q=2$, we also show that the discrete convolution kernel satisfies the stochastic Hörmander’s condition, allowing the application of a stochastic version of the Calderón-Zygmund theory.

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22. 双调和方程有限元方法的尖锐逐点误差估计

Sharp Pointwise Error Estimates of Finite Element Methods for the Biharmonic Equation

作者: Buyang Li, Zhenglei Li, Qigang Liang, Xuejun Xu

类别: math.NA

发布日期: 2026-10-18

arXiv ID: 2610.25115v1

链接: http://arxiv.org/abs/2610.25115v1

摘要: We prove sharp pointwise error estimates for finite element approximations of the clamped biharmonic equation on convex polygonal domains. For $C^1$-conforming elements, we establish $W^{2,p}$-stability for all $2\le p\le \infty$, together with best-approximation type estimates in $W^{1,\infty}$ and $L^\infty$. For the nonconforming Morley element, we derive optimal estimates in $W^{2,p}$, $W^{1,\infty}$, and $L^\infty$; in particular, the $L^\infty$-error is of order $O(h^2)$ without an additional logarithmic factor. The analysis is based on three ingredients: local energy estimates for conforming and nonconforming plate elements, a removal of the local $L^2$-error term in these estimates, and Hölder estimates for derivatives of the Green’s function. These tools are combined with a dyadic decomposition argument for regularized Green’s functions. The results remove logarithmic losses present in earlier pointwise estimates for fourth-order problems and yield sharp rates for classical plate elements such as the Argyris, Bogner–Fox–Schmit (BFS), Hsieh–Clough–Tocher (HCT), and Morley elements.

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23. 辛神经网络的可积构建模块：学习型层、保纤维层以及单层能做什么

Integrable building blocks for symplectic neural networks: learned-type layers, fiber-preserving layers, and what a single layer can do

作者: Andrew Qing He

类别: math.NA, cs.LG, math.SG

发布日期: 2026-10-18

arXiv ID: 2610.25023v1

链接: http://arxiv.org/abs/2610.25023v1

摘要: Exactly symplectic neural networks are built almost exclusively from shear maps and gain expressivity only through depth. We ask what else a single explicit symplectic layer can be. Observing that every layer in use is the exact flow of a Hamiltonian that is a function of commuting integrals, we obtain three families of layers: a learned-type layer, in which every conjugate pair is rotated, squeezed or sheared by a state-dependent amount and the type is found by training; moving-plane shears, residual symplectic maps outside the known ridge-function class; and fiber-preserving layers, which combine an unconstrained point transformation with a fiber translation at the cost of one linear solve. We characterize residual symplectic layers $x \mapsto x + J\nabla H(x)$: in one degree of freedom they are exactly the shears along a line, while from two degrees of freedom on a strictly larger class exists. Conjugating a learned-type layer by a symplectic network yields an exact one-parameter group with an exactly conserved learned Hamiltonian. In experiments on sixteen systems with one to forty degrees of freedom, against SympNets, HénonNets and learned-plane shear networks, layers built on nonlinear integrals are far more accurate when the Hamiltonian contains functions of those integrals. On nonlinear Schrödinger lattices with six to forty sites, twist layers alternating with linear symplectic maps are over a hundred times more accurate than every shear-based network, and with an on-site potential of learned form three to four orders of magnitude more accurate. On chains of Tao oscillators with the step size as an input, stretch layers are 45 to 70 times more accurate than learned-plane shears and recover the Hamiltonian to 1%. On generic non-separable systems, fiber layers improve on learned-plane shears by at most a factor of two, and widening the networks does not help any architecture.

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24. LowNet：具有24个可学习参数、用于求解Poisson方程的循环小波神经网络

LowNet: Looped Wavelet Neural Networks with 24 Learnable Parameters for Solving Poisson Equations

作者: Yuanhao Gong

类别: cs.AI, cs.CV, cs.LG, eess.IV, math.AP, math.NA

发布日期: 2026-10-18

arXiv ID: 2610.24941v1

链接: http://arxiv.org/abs/2610.24941v1

摘要: Poisson equations frequently appear in various fields, such as mathematics, physics and informatics. Solving Poisson equation, however, is challenging. Traditional methods are not data-specific, although have high certainty. They require a large number of iterations due to the CFL numerical stability. In contrast, deep learning methods are data-driven and more efficient, but require accurate training data and suffer from heavy computation. To tackle these issues, we propose a wavelet-inspired Poisson solver with 24 trainable scalars. The proposed network has several advantages. First, it is compact. It has four independent $5\times5$ kernels, each represented by six parameters, leading to 24 scalars in total. Such compact format can be deployed on most devices. Second, thanks to the loop design, it achieves higher accuracy than previous methods. A residual loop reuses the wavelet block without increasing its parameters. Numerical experiments confirm that such loop indeed improves the solving accuracy. On 10000 images, one, two, and three loops of the wavelet block lead to PSNRs of $47.27$, $74.14$, and $100.74$ dB, respectively. Third, the proposed method is computationally fast. Thanks to the wavelet transform, the proposed method has linear computation complexity, leading to high computation performance. In short, we propose a novel looped wavelet neural network with compactness, accuracy and efficiency for solving Poisson equations. Our method can be applied in various fields, such as computational fluid dynamics, image reconstruction, biomedical engineering, electromagnetic field simulation.

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25. 具有二阶离散误差的随机梯度 Leimkuhler-Matthews 积分器

Stochastic Gradient Leimkuhler-Matthews Integrator with Second Order Discretization Error

作者: Xuda Ye, Yuliang Wang, Guang Lin, Cindy Xiangrui Kong

类别: math.NA, math.PR

发布日期: 2026-10-17

arXiv ID: 2610.24781v1

链接: http://arxiv.org/abs/2610.24781v1

摘要: Stochastic gradient Langevin dynamics (SGLD) has a discretization error of first order in the step size $h$, even with the full gradient. We study the stochastic gradient Leimkuhler-Matthews (SGLM) integrator, which has the same cost as SGLD, one stochastic gradient per step, and reuses each Gaussian vector in consecutive steps. With the full gradient, SGLM is the Leimkuhler-Matthews (LM) integrator, whose discretization error for the invariant distribution is of second order. For a potential strongly convex outside a ball, we prove the bound $\mathcal O(\frac{1}{Nh} + Σ^2 h^2 + h^4)$ on the mean square error (MSE) of time averages of SGLM over $N$ steps, where $Σ$ bounds the variance of the gradient noise. The proof uses a modified class that absorbs the non-Markovian part of SGLM. The discretization error enters this bound as $h^4$ and the MSE bound of SGLD as $h^2$, while a larger batch reduces only the term $Σ^2 h^2$ in both bounds. In Bayesian logistic regression on MNIST, SGLM has a smaller estimated MSE than SGLD. The coefficient of $h$ in the numerical bias of SGLM is given explicitly by the covariance of the gradient noise and the continuous Poisson equation. The coefficient for SGLD contains in addition the Hessian of the potential, which comes from the discretization error. Through Riesz representors, the coefficients of the numerical bias determine the coefficients of the Wasserstein-1 errors of SGLM and SGLD.

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26. 具有声学边界条件的非线性耦合热弹性系统的数值分析

Numerical Analysis for a Nonlinear Coupled Thermoelastic System with Acoustic Boundary Conditions

作者: Adriano A. Alcântara, Bruno A. Carmo, Juan B. Límaco, Carlos M. Guzmán

类别: math.NA

发布日期: 2026-10-17

arXiv ID: 2610.24704v1

链接: http://arxiv.org/abs/2610.24704v1

摘要: This paper presents the first $L^2$-norm error analysis for a nonlinear thermoelastic system with acoustic boundary conditions. The model involves three coupled variables: the acoustic potential $u$, the temperature $θ$, and the normal displacement to the boundary $δ$, with nonlinearities $|u|^d u$ in the domain and $g(\cdot,u')$ on the boundary, in addition to a nonlocal term $β(\int_Ωθdx)Δθ$ in the heat equation. We establish error estimates for the fully discrete problem, using the Crank-Nicolson method for time discretization and finite element method with basis of degree $r \geqslant 1$ for space discretization. We demonstrate that the domain-boundary coupling induces a half-power loss in the spatial convergence order for the variable $u$, and this effect propagates to the remaining variables due to the thermoelastic and boundary couplings, resulting in the final error estimate $\mathcal{O}(h^{r+\frac{1}{2}} + τ^2)$. A bootstrap argument is employed to ensure the uniform boundedness of $\|U_h^n\|_{L^\infty}$, which is essential for handling the nonlinearity. Numerical results using Lagrange finite element basis of degrees $ r = 1, 2, 3 $ (linear, quadratic, and cubic) corroborate the theoretical estimates and illustrate the transition from suboptimal to optimal convergence orders when the exact solution possesses additional $L^\infty$ regularity.

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27. 多变量 Vandermonde 矩阵的谱与高维超分辨问题的极小极大误差界

Spectra of Multivariate Vandermonde Matrices and Min-max Error Bounds for the High-dimensional Super-resolution Problem

作者: Nuha Diab, Dmitry Batenkov

类别: math.SP, cs.IT, math.NA

发布日期: 2026-10-17

arXiv ID: 2610.24670v1

链接: http://arxiv.org/abs/2610.24670v1

摘要: We study multivariate rectangular Vandermonde matrices $U_N$ whose rows are indexed by the grid $G_N=\{-N,\ldots,N\}^d, $ and whose columns correspond to irregularly spaced points on the unit $d$-dimensional torus. Motivated by the problem of super-resolution (recovery of spike trains from low frequency noisy measurements), we consider configurations in which the points form several clusters, with points within each cluster separated by at most $h$ and distinct clusters are well-separated. We show that the column subspaces associated with different clusters are nearly orthogonal, i.e. for $N \rightarrow \infty, Nh \ll 1$, the minimal principal angle between any such pair is at least $ {π\over 2} - \frac{C_1}{N} - C_2Nh - \frac{C_3}{\sqrt{N}}, $ for some constants $C_1,C_2,C_3$ independent of $N$ and $h$. This near-orthogonality enables the spectral analysis of such $U_N$ to be reduced to the analysis of each individual cluster. As a consequence, we derive sharp scaling for all singular values of $U_N$, revealing how the geometry of clustered points controls the conditioning of the high-dimensional super-resolution problems. Then we establish min-max error bounds for the recovery of clustered point sources, requiring only an upper bound on the largest cluster size and no knowledge of the geometry within the clusters. These bounds coincide with the optimal rates known in the one-dimensional setting. Furthermore, when the clustered points are in generic position (i.e. do not lie on a low-dimensional algebraic variety) our results indicate that super-resolution in high dimensions is more stable than in one dimension.

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28. 闭合轮廓序列的能量极小化插值B样条放样

Energy-minimized interpolatory B-spline lofting for sequences of closed contours

作者: Shutao Tang, Zixuan Zhao

类别: cs.GR, math.NA

发布日期: 2026-10-17

arXiv ID: 2610.24569v1

链接: http://arxiv.org/abs/2610.24569v1

摘要: B-spline lofting through closed cross-sections is common in reverse engineering, but interpolating each section on its own knot vector can produce an unnecessarily large control net. We construct a common-knot method for closed section curves of odd and even degree by minimizing the strain energy of each section subject to its interpolation constraints. Feasibility for closed section data and uniqueness of the resulting saddle-point solution require full row rank of the periodic collocation matrices, a property that a previously proposed periodic Schoenberg-Whitney window condition does not always guarantee. We establish the exact parity distinction: the condition is sufficient at odd point counts but admits singular configurations at every admissible even point count. For the odd and even degrees used here, the natural and shifting anchor knots are nonsingular; at odd point counts the window condition certifies the selected candidate, whereas at even point counts a rank test with an anchor fallback is applied. The lofting method therefore retains a nonsingular square subsystem for each section and uses nestedness to transfer full row rank to the rectangular common-knot system. Across six closed-contour models the common-knot method uses 83-98% fewer control points than traditional interpolating lofting, with section energies no larger than those of a previous knot-selection method within numerical precision.

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29. 非均匀介质中空间依赖变指数次扩散反势问题的分析与逼近

Analysis and approximation for inverse potential problem of space-dependent variable-exponent sub-diffusion in heterogeneous medium

作者: Zhiyuan Li, Chunlong Sun, Xiangcheng Zheng

类别: math.NA

发布日期: 2026-10-17

arXiv ID: 2610.24531v1

链接: http://arxiv.org/abs/2610.24531v1

摘要: Anomalous transport in complex heterogeneous medium is often characterized by non-Markovian memory effects and spatially varying trapping properties. From the point of view of continuous-time random walks, such heterogeneity can be described by a position-dependent heavy-tailed waiting-time density $q(t \mid x)\sim C(x)t^{-1-α(x)}$, where $0<α(x)<1$ is the local anomalous exponent. This microscopic interpretation naturally leads to the sub-diffusion with a space-dependent variable exponent $α(x)$. Compared with constant-exponent models, the variable-exponent fractional operators generally do not commute with the spatial Laplacian, which makes both the forward and inverse analysis substantially more delicate. This paper investigates the inverse problem of recovering a spatially dependent radiative or potential coefficient in variable-exponent sub-diffusion from terminal observation data. We formulate the inverse problem as a nonlinear fixed-point equation and prove that the associated fixed-point operator is monotone such that the fixed-point iteration converges decreasingly to the unique solution. We further establish a Lipschitz-type conditional stability estimate for the terminal observation map, which also implies uniqueness of the inverse problem for sufficiently large terminal time. For numerical reconstruction, we develop a fully discrete finite element scheme and derive an error estimate for the inverse problem. Numerical simulations are conducted to validate the effectiveness of the method, even in the presence of noisy data.

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30. Transformer 作为跨任务的自适应核学习器以实现上下文学习

Transformers as Adaptive Kernel Learners across Tasks for In-Context Learning

作者: Zhaiming Shen, Alexander Cloninger, Rongjie Lai, Wenjing Liao

类别: cs.LG, math.NA, math.ST

发布日期: 2026-10-17

arXiv ID: 2610.24441v1

链接: http://arxiv.org/abs/2610.24441v1

摘要: Transformers have emerged as a dominant architecture in modern machine learning, with in-context learning (ICL) enabling models to adapt to new tasks by conditioning on prompts, without any parameter updates. To understand the attention mechanism in transformer blocks, a growing line of theoretical work interprets ICL through the lens of classical statistical estimators. Recent works have established a connection between the attention mechanism and kernel methods, while the kernel function is usually assumed to be fixed. In practice, the query and key matrices in the attention mechanism are trained, which allows the attention mechanism to serve as an adaptive kernel method. In this work, we demonstrate that transformers can act as adaptive anisotropic kernel learners. Specifically, we study ICL for low-rank linear and multi-index nonlinear regressions, and conduct numerical simulations to show that the learned attention scores closely resemble the oracle kernels for these function classes. We then develop a general framework to analyze the generalization error of ICL by connecting transformers to the oracle kernel estimator for each function class. Our theory gives an explicit and constructive implementation of the oracle kernel estimators for these function classes using a wide and shallow transformer architecture, illustrating how the prediction is carried out by the interaction between the query token and the context tokens. Our work complements existing studies by providing numerical and theoretical evidence of how the attention mechanism adapts to anisotropic kernels in both linear and nonlinear settings.

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31. 关于夹持 B 样条的矩形 Schoenberg-Whitney 定理的一点注记

A note on the rectangular Schoenberg-Whitney theorem for clamped B-splines

作者: Shutao Tang, Zixuan Zhao

类别: math.NA, cs.GR

发布日期: 2026-10-17

arXiv ID: 2610.24425v1

链接: http://arxiv.org/abs/2610.24425v1

摘要: Wang et al.(Computer-Aided Design 40 (2008) 999-1008) proposed a knot-subsequence characterization of full row rank for rectangular B-spline collocation matrices. We identify an incorrect inequality in their Appendix proof, which invalidates the argument as written, and make explicit the endpoint and multiplicity conventions on which the criterion depends. The defect lies in the proof, not in the underlying result, which is valid under suitably stated hypotheses; in the lofting construction of Wang et al. these hypotheses are implicitly satisfied, so the lofted curves are unaffected. We state the corrected result and give a new proof, independent of the disputed argument, for clamped B-splines of arbitrary degree with continuous interior multiplicities. In our proof, sufficiency follows from inclusion of spline spaces and knot insertion; necessity selects independent columns and builds the reduced knot vector from them. The proof uses no recursive column-selection construction.

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32. 实 Grothendieck 不等式的带证书 Fourier 舍入

Certified Fourier rounding for the real Grothendieck inequality

作者: Bo Peng

类别: math.FA, math.NA

发布日期: 2026-10-17

arXiv ID: 2610.24417v1

链接: http://arxiv.org/abs/2610.24417v1

摘要: We give a computer-assisted proof that the real Grothendieck constant is strictly smaller than $1.77987164$. The proof provides a complete interval-arithmetic realization of Ren’s five-variable construction. A finite Fourier calculation encloses every scalar coefficient through degree $403$; a fourth-derivative estimate on the entire unit circle bounds the remaining coefficient sum by $10^{-4}$. Exact rational arithmetic then certifies the final inequality. We include the transfer to all finite real matrices and arbitrary vector dimensions, the analytic quadrature errors, the rational parameters, and source code for checking and regenerating the certificate.

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33. 用于零阶优化的球面设计预言机：确定性、无方差梯度估计

Spherical-Design Oracles for Zeroth-Order Optimisation: Deterministic, Variance-Free Gradient Estimation

作者: Amir Ali Farzin, Yuen-Man Pun, Philipp Braun, Iman Shames

类别: math.OC, cs.GT, math.NA

发布日期: 2026-10-17

arXiv ID: 2610.24373v1

链接: http://arxiv.org/abs/2610.24373v1

摘要: In this work, we develop a zeroth-order (ZO) method that estimates the gradient with minimal bias and variance. Instead of the random directions often used in the ZO literature, we consider central differences with directions given by a fixed spherical $t$-design, a finite point set that integrates every polynomial of degree at most $t$ exactly. The oracle is deterministic and thus does not induce any variance.We also propose a general finite difference method in the higher-order-smoothness regime and show the bias of both finite difference methods. We then show the convergence of the proposed ZO method via the input-to-state stability concept. Numerical experiments show the superiority of our proposed method over the existing random ZO methods.

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34. 用于固定算子有限差分 PDE 求解器的可复用 QUBO 导出 Block-Schur 逆因子

Reusable QUBO-Derived Block-Schur Inverse Factors for Fixed-Operator Finite-Difference PDE Solvers

作者: Arash Fath Lipaei, Özgür E. Müstecaplıoğlu, Özlem Salehi, Selin Aslan

类别: math.NA, quant-ph

发布日期: 2026-10-16

arXiv ID: 2610.24158v1

链接: http://arxiv.org/abs/2610.24158v1

摘要: Many simulations of partial differential equations require the repeated solution of linear systems whose matrix remains fixed while the righthand side changes. We present a hybrid offline online framework for this setting. The discretized operator is organized into small coupled blocks and reduced through block Schur complements. Instead of expressing the complete PDE solution as one large binary optimization problem, quadratic unconstrained binary optimization is used only to construct reusable local inverse factors. These factors are computed during an offline stage, decoded and stored, and then reused in an ordinary classical block solve for subsequent righthand sides. The optimizer facing problem size is governed by the local block size and chosen numerical precision, rather than by the dimension of the full spatial grid. We develop the mathematical formulation, analyze the accuracy and resource requirements of the binary representation, and examine the method on representative finite difference PDE problems, including execution of the local optimization stage on quantum optimization hardware. The resulting architecture provides a reusable interface between conventional numerical linear algebra and QUBO based quantum optimization.

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35. 对称经典Dunkl正交多项式的时间-频带限制

Time-and-band limiting for symmetric classical Dunkl orthogonal polynomials

作者: M. M. Castro, F. A. Grünbaum

类别: math.CA, math.NA, math.SP

发布日期: 2026-10-16

arXiv ID: 2610.24100v1

链接: http://arxiv.org/abs/2610.24100v1

摘要: The problem of ‘’time and band limiting’’ has been previously considered for expansions in orthonormal polynomials that are common eigenfunctions of some second order differential operator. In this paper we consider the situation of orthonormal polynomials that are eigenfunctions of some second order differential-difference operator as introduced by Ch. Dunkl. These polynomials appear already in the work of G. Szegő and at the same time play a role in supersymmetric quantum mechanics.

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