生成时间: 2026年09月16日 06:15:13
重点关注论文数量: 2
了解领域论文数量: 19
不相关论文数量: 45
重点关注论文（完整分析）
1. 基于多点应力离散的非均质弹性波传播的时间显式多尺度框架
A time explicit multiscale framework for heterogeneous elastic wave propagation based on multipoint stress discretization
作者: Xiang Zhong, Eric T. Chung, Shubin Fu
类别: math.NA
发布日期: 2026-09-14
arXiv ID: 2609.14954v1
摘要翻译: 模拟非均质介质中的弹性波传播在数学和计算上均面临重大挑战，因为分辨细尺度材料变化需要极细的空间离散，而应力–位移混合格式通常会产生大型鞍点系统，这类系统并不适合高效的时间积分。此外，传统多尺度约化通常会产生非平凡的粗尺度质量矩阵，从而需要在每个时间步进行额外的线性求解，并削弱显式格式的优势。为解决这些困难，我们发展了一种基于多点应力控制体积离散的时间显式多尺度方法。其核心创新是一种统一的空间–时间约化策略，可同时消除两个主要的代数瓶颈：复杂的全局细尺度混合求解，以及传统多尺度时间步进中出现的重复粗尺度质量求解。更具体地说，应力和辅助旋转变量被局部消去，从而得到一个对称正定约化刚度算子；在此基础上，利用密度加权局部谱问题构造多尺度空间。相应的投影质量双线性形式保持物理质量内积，并在密度正交归一辅助基下给出单位质量矩阵，从而自然地导向显式中心差分格式。我们在粗尺度CFL条件下建立了离散能量稳定性，并分别推导了密度加权位移和局部重构应力的收敛性估计。非均质介质中的数值实验验证了理论结果，并展示了所提方法用于弹性波传播的有效性。
Comment: 无
链接: http://arxiv.org/abs/2609.14954v1
1. 研究对象和背景
本文研究非均质介质中线性弹性波传播的高效数值逼近. 设 $\Omega \subset \mathbb{R}^d$ ($d=2,3$) 为多面体区域, 未知量为对称应力张量 $\sigma(\cdot,t):\Omega\to\mathbb{R}^{d\times d}$, 位移场 $u(\cdot,t)$ 以及弱对称混合格式中的辅助旋转变量 $\gamma$. 控制方程为
$$ A\sigma = \varepsilon(u), \quad \rho \frac{\partial^2 u}{\partial t^2} - \operatorname{div}\sigma = f \quad \text{in } \Omega\times(0,T], $$并配以齐次 Dirichlet 边界条件 $u=0$ 于 $[0,T]\times\partial\Omega$ 和初值 $u(x,0)=u_0(x)$, $u_t(x,0)=v_0(x)$. 这里 $\varepsilon(u)=\frac12(\nabla u+(\nabla u)^T)$ 是线性化应变张量, $A$ 是弹性柔度张量, 定义为
$$ A\tau := \frac{1}{2\mu}\left(\tau - \frac{\lambda}{d\lambda+2\mu}(\operatorname{tr}\tau)I\right), $$其中 $\lambda,\mu$ 为 Lamé 系数, 由 Young 模量 $E$ 和 Poisson 比 $\nu$ 给出. 本文允许 $\lambda,\mu$ 在空间上高度非均质且可能高对比, 也允许接近不可压缩情形 $\lambda\to\infty$ 而 $\mu$ 有界.
在数值框架上, 混合应力-位移格式直接逼近应力, 对近不可压缩材料具有鲁棒性. 强对称格式把离散应力逐点约束在对称张量的 $H(\operatorname{div})$ 空间中, 例如 Arnold-Winther 元及其三维对应; 弱对称格式则在完整的矩阵值 $H(\operatorname{div})$ 空间中逼近应力, 并通过辅助旋转变量以 Lagrange 乘子方式变分地施加对称性, 从而允许更简单的稳定单元. 然而标准混合离散通常产生大型鞍点系统, 不适合高效时间积分. 多点应力混合有限元方法通过局部化应力质量矩阵, 可以局部消去应力和旋转, 得到以单元中心位移为未知量的系统. Fu 和 Zhao 进一步提出了多点应力控制体积方法, 使用分片常数近似且不需要特殊求积规则; 在合适旋转空间下, 应力与旋转均可局部消去, 得到对称正定位移系统, 同时保留局部守恒和廉价应力恢复. 但当介质包含大量未解析细尺度时, 细网格上的全局求解仍然昂贵.
多尺度模型约化通过把细尺度系数信息编码到粗基函数中来降低空间复杂度. 已有方法包括多尺度有限元、广义多尺度有限元、局部正交分解、变分多尺度有限元、数值升尺度、异质多尺度方法等. 基于 GMsFEM 与 LOD 的约束能量最小化广义多尺度有限元在过采样区域上构造局部多尺度基函数, 对高对比介质具有鲁棒性. 这些思想已被推广到弹性与弹性波问题, 包括非均质各向异性弹性波传播的 GMsFEM, 时域弹性波的高阶多尺度基, 以及高对比弹性问题的无锁多尺度混合格式. 但传统 Galerkin 约化通常产生非对角粗质量矩阵, 每个时间步都需要额外线性求解, 从而削弱显式格式的优势. 现有显式多尺度波方法也没有同时利用局部凝聚的多点应力格式, 在约化层面保持物理密度内积, 并提供局部应力恢复. 本文要补的空白正是: 在同一个空间-时间约化框架中, 同时消除全局细尺度混合求解与重复粗尺度质量求解, 构造真正时间显式的多尺度弹性波格式, 并建立稳定性与收敛性分析.
2. 主要定理或主要结果
本文的主要结果是构造并分析了一个基于多点应力控制体积离散的时间显式多尺度弹性波格式. 首先, 从混合应力-位移-旋转弱形式出发, 离散空间取 $\Sigma_h$, $U_h$, $\Gamma_h$, 其中应力分片常数, 位移在宏观单元上分片常数, 旋转在交互区域上分片常数. 利用 (7a) 中应力质量矩阵的块对角结构, 可以局部消去应力, 并根据 (7c) 进一步消去旋转. 由此得到仅含位移的细尺度二阶常微分方程
$$ M_u \frac{\partial^2 u}{\partial t^2} + K u = F, $$其中 $K$ 是由 Schur 补得到的对称正定刚度矩阵. 对应的变分形式为
$$ (\rho \partial_t^2 u_h, v_h) + a(u_h, v_h) = (f, v_h), \quad \forall v_h\in U_h, $$能量双线性形式为
$$ a(u_h, v_h) = (A^{-1}P_\star(u_h), P_\star(v_h)), $$其中 $P_\star$ 是弱对称应力投影, 定义在局部消去旋转之后.
在粗网格 $T_H$ 上, 作者首先在每个粗单元 $K_i$ 上求解密度加权局部谱问题
$$ a_i(\phi_j^i, w) = \frac{\lambda_j^i}{H^2}(\rho \phi_j^i, w), \quad \forall w\in U_h(K_i), $$并取前 $l_i$ 个特征函数张成局部辅助空间 $V_{aux}^i$, 归一化条件为 $(\rho\phi_j^i,\phi_{j'}^i)=\delta_{jj'}$. 全局辅助空间 $V_{aux}$ 为这些局部空间的直和. 随后通过约束能量最小化构造多尺度试探基
$$ \psi_j^i = \arg\min_{\psi\in U_h}\{a(\psi,\psi): \pi(\psi)=\phi_j^i\}, $$并在过采样区域 $K_{i,m}$ 上局部化为 $\psi_{j,m}^i$. 这样得到多尺度试探空间 $V_{ms}$. 论文引用并使用了两个关键性质: 投影算子 $\pi$ 的逼近性质
$$ \|(I-\pi)v\|_{\rho(K_i)}^2 \le \frac{H^2}{\lambda_{l_i+1}^i}\|v\|_{a(K_i)}^2, $$以及多尺度基函数的指数衰减
$$ \|\psi_j^i-\psi_{j,m}^i\|_a \le C\theta^m \|\psi_j^i\|_a. $$这些性质来自 CEM-GMsFEM 框架.
时间离散采用密度适配的 Petrov-Galerkin 投影. 定义双线性形式
$$ \hat\rho(v,w) = (\rho\pi v,\pi w). $$由于辅助基满足 $\rho$-正交归一化, 有
$$ \hat\rho(\psi_{j,m}^i,\psi_{j',m}^{i'}) = (\rho\phi_j^i,\phi_{j'}^{i'}) = \delta_{ii'}\delta_{jj'}. $$因此粗质量矩阵是恒等矩阵. 全离散格式为
$$ \hat\rho(D_{tt}u_{ms}^n, w) + a(u_{ms}^n, w) = (f^n, \pi w), \quad \forall w\in V_{ms}, $$在基坐标下给出显式中心差分更新
$$ U^{n+1} = 2U^n - U^{n-1} + \tau^2(F^n - K_{ms}U^n), $$其中 $(K_{ms})_{\alpha\beta}=a(\psi_\beta,\psi_\alpha)$, $F_\alpha^n=(f^n,\pi\psi_\alpha)$. 这意味着每一步不需要求解线性系统.
稳定性方面, 论文证明了多尺度空间上的逆 Poincaré 不等式: 存在 $\beta>0$ 使得
$$ \|\pi v\|_\rho^2 \ge \beta H^2 \|v\|_a^2, \quad \forall v\in V_{ms}, $$且 $\beta$ 依赖网格正则性和特征值 $\lambda_{\max}$, 但不依赖 $h$, $H$ 和系数对比度. 在此基础上, 定义离散总能量
$$ E_{\pi,\rho}^{n+1/2}(v) := \frac12\|\pi D_t v^{n+1/2}\|_\rho^2 - \frac{\tau^2}{8}\|D_t v^{n+1/2}\|_a^2 + \frac12\|\bar v^{n+1/2}\|_a^2. $$在粗尺度 CFL 条件
$$ \frac{\tau}{2H\sqrt{\beta}} < 1 $$下, 能量满足正定性估计
$$ E_{\pi,\rho}^{n+1/2}(v) \ge \frac12\left(1-\frac{\tau^2}{4\beta H^2}\right)\|\pi D_t v^{n+1/2}\|_\rho^2 + \frac12\|\bar v^{n+1/2}\|_a^2, $$并由此建立离散能量稳定性. 摘要还声称导出了密度加权位移与局部恢复应力的收敛估计, 正文提到 Theorem 4.7 和 Theorem 4.9 分别给出位移和应力的收敛结果. 然而, 在提供的 PDF 前 10 页中, 这两个定理的完整陈述, 具体误差阶, 对正则性假设的依赖以及应力恢复误差的显式表达式均未给出, 因此从现有内容无法确认其精确收敛率. 可确认的是, 稳定性常数和逆 Poincaré 常数不依赖细网格 $h$, 粗网格 $H$ 和系数对比度, 且多尺度基的局部化误差随过采样层数 $m$ 指数衰减.
3. 研究方法、关键技术和核心工具
本文的技术障碍主要有三类. 第一, 原始混合应力-位移-旋转格式是鞍点系统, 不能直接用于显式时间推进; 第二, 传统多尺度 Galerkin 约化会产生非对角粗质量矩阵, 每步都需要质量求解; 第三, 局部化多尺度基后, 需要证明约化格式仍满足粗尺度能量稳定性和粗尺度 CFL 条件. 作者的真正创新是把空间约化与时间约化统一设计, 使得局部应力凝聚和显式时间步进在同一刚度算子上实现, 而不是分别处理后拼接.
第一项关键技术是混合系统的局部静态凝聚. 离散方程 (7a) 中应力质量项是块对角的, 因此可以从 (7a) 解出
$$ \sigma_h = A^{-1}(J(u_h)-A\,as^*(\gamma_h)). $$再代入 (7c) 确定旋转 $\gamma_h(u_h)$, 使其满足局部正交条件
$$ (J(u_h)-A\,as^*(\gamma_h(u_h)), as^*(\xi_h)) = 0, \quad \forall \xi_h\in\Gamma_h. $$由此定义弱对称应力投影
$$ P_\star(u_h) := J(u_h)-A\,as^*(\gamma_h(u_h)), $$它是 $J(u_h)$ 在弱对称应力子空间上关于 $A^{-1}$ 内积的正交投影. 最终能量双线性形式 $a(u_h,v_h)=(A^{-1}P_\star(u_h),P_\star(v_h))$ 正是消去应力和旋转后的 Schur 补能量. 这一步把全局混合求解转化为对称正定的位移问题, 为后续多尺度约化和显式时间积分提供基础.
第二项关键技术是密度加权局部谱与约束能量最小化基. 与标准 GMsFEM 类似, 作者在粗块 $K_i$ 上求解
$$ a_i(\phi_j^i,w)=\frac{\lambda_j^i}{H^2}(\rho\phi_j^i,w), $$但这里的能量形式 $a_i$ 来自多点应力控制体积的 Schur 补, 且内积带密度权 $\rho$. 归一化 $(\rho\phi_j^i,\phi_{j'}^i)=\delta_{jj'}$ 使得后续辅助基是密度正交的. 多尺度试探基通过约束能量最小化定义: 在满足 $\pi(\psi)=\phi_j^i$ 的函数中最小化能量 $a(\psi,\psi)$. 过采样区域 $K_{i,m}$ 上的局部化基 $\psi_{j,m}^i$ 具有指数衰减性质, 这是 CEM-GMsFEM 框架的核心工具. 作者证明该衰减性质可以迁移到多点应力弹性波的能量形式上, 因为约化双线性形式 $a(\cdot,\cdot)$ 对称, 强制, 局部, 并满足与标准椭圆能量形式相同的局部稳定性和逆估计.
第三项关键技术是密度适配的 Petrov-Galerkin 投影. 作者不采用标准 Galerkin 测试, 而是用辅助空间 $V_{aux}$ 作为测试空间, 并引入双线性形式
$$ \hat\rho(v,w)=(\rho\pi v,\pi w). $$因为辅助基是 $\rho$-正交归一的, 粗质量矩阵在基坐标下恰好是恒等矩阵. 这是实现显式中心差分的代数核心. 从分析角度看, 该投影保留了物理质量内积, 使得粗尺度质量矩阵不再需要求逆, 从而消除了传统多尺度时间步进中反复出现的粗质量求解. 作者也指出, 在全局试探空间 $V_{glo}$ 上, 该格式等价于带投影质量的对称格式; 在局部化试探空间 $V_{ms}$ 上, 采用类似标量波 CEM 显式格式的投影质量定义.
第四项关键技术是稳定性分析. 为了在粗尺度 CFL 条件下证明能量稳定, 作者需要多尺度空间上的逆 Poincaré 不等式. 证明思路是: 先对全局多尺度空间 $V_{glo}$ 使用稳定右逆 $Rq$. 对任意 $q\in V_{aux}$, 存在 $Rq\in U_h$ 使得 $\pi(Rq)=q$, 且
$$ \|Rq\|_a^2 \le C_{lift}H^{-2}\|q\|_\rho^2. $$利用 $V_{glo}$ 与核空间 $\ker\pi$ 在能量意义下的正交性, 得到 $\|v\|_a^2 \le C_{lift}H^{-2}\|\pi v\|_\rho^2$. 再对局部化空间 $V_{ms}$ 使用过采样基的指数衰减和有限重叠论证, 得到
$$ \|v_{ms}-v_{glo}\|_a \le \eta_m H^{-1}\|\pi v_{ms}\|_\rho, $$其中 $\eta_m$ 随 $m$ 指数衰减. 结合两者即得
$$ \|\pi v\|_\rho^2 \ge \beta H^2\|v\|_a^2, $$其中 $\beta$ 不依赖 $h$, $H$ 和对比度. 有了该逆不等式, 作者定义包含动能, 罚项和势能的离散总能量, 并证明其在粗尺度 CFL 条件 $\tau/(2H\sqrt{\beta})<1$ 下保持正定, 从而建立能量稳定性. 最后, 收敛性分析使用椭圆投影和时间投影误差估计, 分别处理密度加权位移和局部恢复应力; 但如前所述, 具体估计式在提供的 PDF 前 10 页中未被完整展示.
4. 与之前工作的比较
与经典混合有限元相比, 本文建立在弱对称混合弹性框架上. 强对称格式, 如 Arnold-Winther 元及其三维推广, 直接约束离散应力到对称张量 $H(\operatorname{div})$ 空间, 能逐点保持应力对称, 但构造复杂. 弱对称格式允许应力取完整矩阵值 $H(\operatorname{div})$ 空间, 通过辅助旋转以 Lagrange 乘子变分施加对称性, 从而允许更简单的稳定单元. 本文继承了这一路线, 但进一步采用多点应力控制体积离散, 使得应力与旋转都能局部消去. 与标准混合有限元相比, 本文避免了全局鞍点求解, 并把问题约化为位移对称正定系统.
与多点应力混合有限元以及 Fu-Zhao 的多点应力控制体积方法相比, 本文的直接前身是 Fu 和 Zhao 的格式. 该格式使用分片常数近似且无需特殊求积规则, 具有局部守恒和廉价应力恢复. 本文在此基础上增加了多尺度模型约化, 处理细尺度非均质导致的高计算量问题. 也就是说, 之前的多点应力控制体积方法虽然局部凝聚了应力和旋转, 但在未解析细尺度介质中仍需细网格全局求解; 本文通过粗网格上的谱基和约束能量最小化基, 把全局演化未知量降到粗尺度, 同时保留局部应力恢复能力. 推进点在于空间-时间联合约化, 而不是单纯空间多尺度.
与传统多尺度方法相比, 本文与 GMsFEM, LOD, CEM-GMsFEM 有密切关系. 多尺度有限元和广义多尺度有限元通过局部细尺度问题构造基函数, 局部正交分解通过局部谱问题构造校正子, 异质多尺度方法和数值升尺度则从不同角度处理尺度分离. 本文借用了 CEM-GMsFEM 的过采样局部化和约束能量最小化思想, 并引用其投影算子逼近性质和基函数指数衰减性质. 与标量椭圆或标量波问题中的 CEM 相比, 本文的能量形式来自多点应力弹性混合离散, 需要处理应力对称性, 旋转消元和局部应力恢复. 因此, 本文把 CEM 型分析从标量波和标准椭圆问题推进到混合弹性波问题.
与已有的弹性多尺度工作相比, 文中提到 GMsFEM 用于非均质各向异性弹性波传播, 时域弹性波的高阶多尺度基, 以及高对比弹性问题的无锁多尺度混合格式. 这些工作分别关注各向异性, 高阶精度和锁定现象. 本文的区别在于同时追求三件事: 局部凝聚多点应力格式, 约化层面保持物理密度内积, 以及局部恢复应力. 摘要明确指出, 现有显式多尺度波方法没有同时利用这些特征. 因此, 本文的定位不是替代所有弹性多尺度方法, 而是针对显式时间推进这一特定目标, 消除粗质量矩阵求逆这一瓶颈.
与显式多尺度波方法相比, 谱多尺度空间结合间断 Galerkin 可产生块对角质量矩阵并支持显式时间步进; LOD 也被用于降低空间复杂度并放宽细尺度 CFL 限制; 在 CEM 框架内还有针对标量波方程的能量守恒显式 Petrov-Galerkin 方法. 本文与后者的相似之处在于也使用密度适配的 Petrov-Galerkin 投影, 并在全局空间上讨论对称化. 但本文的对象是弹性波混合应力-位移-旋转系统, 测试空间由密度加权局部谱构造, 粗质量矩阵在密度正交辅助基下成为恒等矩阵, 从而自然得到中心差分显式更新. 与间断 Galerkin 块对角质量方法相比, 本文不依赖 DG 空间, 而是依赖多点应力局部凝聚和密度正交辅助基. 与 LOD 放宽 CFL 的目标相比, 本文仍受粗尺度 CFL 条件 $\tau/(2H\sqrt{\beta})<1$ 限制, 但该条件基于粗网格 $H$, 而不是细网格 $h$; 这是多尺度显式方法的主要收益.
代价与局限方面, 本文需要离线求解每个粗单元上的局部谱问题, 构造过采样多尺度基, 并选择足够多的过采样层数 $m$ 以控制指数衰减误差. 稳定性常数 $\beta$ 依赖网格正则性和谱问题的最大特征值, 虽然不依赖 $h$, $H$ 和对比度, 但仍需估计. CFL 条件依赖 $H$ 和 $\beta$, 因此粗网格不能任意放大. 论文的构造和分析主要在 $d=2$ 下进行, 并说明向 $d=3$ 的扩展是自然的, 但提供的 PDF 前 10 页未展开三维细节. 收敛性方面, 摘要和正文引出 Theorem 4.7 与 Theorem 4.9, 但提供的摘录没有给出完整定理和误差阶, 因此无法与已有显式多尺度方法的误差率做定量比较. 作者在前 10 页中也没有显式展开与所有前人工作的逐项数值比较, 因此关于计算代价和精度优势的判断主要来自方法结构, 而非完整实验数据.
2. 各向异性多相 Mullins-Sekerka 流的保结构对称 Galerkin 边界元方法
A Structure-Preserving Symmetric Galerkin Boundary Element Method for Anisotropic Multi-phase Mullins-Sekerka Flow
作者: Tokuhiro Eto
类别: math.NA
发布日期: 2026-09-13
arXiv ID: 2609.14742v1
摘要翻译: 针对 $\mathbb R^2$ 中的各向异性多相 Mullins-Sekerka 问题，发展了一种保结构的对称 Galerkin 边界元方法 (SGBEM)。通过由分区域 SGBEM 离散组装得到的离散 Dirichlet-to-Neumann 算子消去体调和问题，从而得到与带三叉结点的多边形曲线网络演化耦合的仅边界提法。证明所得边界问题的 Galerkin 近似唯一可解，并且只要演化的多边形网络保持在容许类中，全离散格式关于时间步无条件稳定。通过在与由三个同心圆组成的相似重缩放精确解进行对比的时空耦合加密下的收敛性测试，验证了所提方法的精度。
Comment: 39 pages, 5 figures, 3 tables
链接: http://arxiv.org/abs/2609.14742v1
1. 研究对象和背景
本文研究 $\mathbb R^2$ 中带三重点的各向异性多相 Mullins-Sekerka 流的数值逼近。设演化曲线网络 $\Gamma(t)=\cup_{i=1}^{I_S}\Gamma_i(t)$ 将平面分割为连通区域 $R_\ell(t)$，$1\le \ell\le I_R$，其中一个区域无界。每个区域被赋予相 $p=0,\dots,I_P-1$，相与区域的关系由映射 $p\mapsto \Omega_p\subset\{1,\dots,I_R\}$ 描述，且无界区域属于相 $0$。曲线网络可以包含三重点 $T_k(t)$，每个三重点由三条开曲线相交而成。未知量是化学势向量 $w(\cdot,t):\mathbb R^2\setminus\Gamma(t)\to\mathbb R^{I_P}$ 与曲线网络 $\Gamma(t)$ 的演化。系统为
$$\Delta w=0 \quad \text{in } \mathbb R^2\setminus\Gamma(t),$$$$w\cdot\llbracket\chi\rrbracket=\kappa_\gamma \quad \text{on } \Gamma(t),$$$$\llbracket\nabla w\rrbracket\vec\nu=-V\llbracket\chi\rrbracket \quad \text{on } \Gamma(t),$$$$\nabla w(\vec x,t)=O(|\vec x|^{-2}) \quad \text{as } |\vec x|\to\infty,$$$$\llbracket w\rrbracket=0,\quad w\cdot\mathbf 1=0,$$$$\sum_{\ell=1}^3 \epsilon_{k,\ell}\vec\xi_{s^k_\ell}(\vec\nu_{s^k_\ell})^\perp=\vec 0 \quad \text{at } T_k(t),$$$$\Gamma(0)=\Gamma_0.$$这里 $\chi$ 是相的特征向量，$\llbracket\cdot\rrbracket$ 是跨界面跳跃，$V$ 是法向速度，$\kappa_\gamma$ 是各向异性曲率。每条曲线 $\Gamma_i$ 可带自己的平方根型各向异性
$$\gamma_i(\vec\nu)=c_i\sum_{l=1}^{L_i}\sqrt{\vec\nu^\top G_{i,l}\vec\nu},$$其中 $G_{i,l}$ 对称正定，$c_i>0$。Cahn-Hoffman 向量与各向异性曲率定义为
$$\vec\xi_i(\vec\nu_i)=\nabla\gamma_i(\vec\nu_i),\quad \kappa_{\gamma_i}=-\nabla_\Gamma\cdot\vec\xi_i(\vec\nu_i),$$各向异性表面能为
$$E_\gamma(\Gamma)=\sum_{i=1}^{I_S}\int_{\Gamma_i}\gamma_i(\vec\nu_i)\,d\mathcal H^1.$$该模型源自扩散控制的形态演化、Ostwald 熟化的尖锐界面描述，并可作为 Cahn-Hilliard 方程的尖锐界面极限。其核心困难在于界面几何与体相调和场之间的非局部耦合：Gibbs-Thomson 关系用曲率给出界面化学势，而化学势法向导数跳跃决定法向速度。因此每个时刻都必须在所有相中求解调和问题。该耦合还导致两个基本结构：各向异性界面能耗散，以及每个有界相的面积守恒；在多相三重点处还需满足 Young-Herring 力平衡。
已有解析工作包括两相流在面积保持曲线缩短框架下的存在性、带表面张力的经典适定性、多相格式的提出，以及最近用位势方法把平面两相问题化为仅界面演化的边界形式。数值方面，相场法演化 Cahn-Hilliard 方程，水平集和阈值动力学隐式表示界面，边界积分法只离散界面并用层位势表示调和场，参数有限元法在各向异性几何演化方程中实现了无条件稳定与结构保持；Stefan 与 Mullins-Sekerka 问题也有耦合参数界面与体有限元场的稳定格式，并扩展到多相 Mullins-Sekerka、带动力学过冷的各向异性变体等。作者此前基于电荷模拟法的配点格式也实现了两相面积保持和多相速度级面积守恒。
本文试图补的空白是：在消去体相自由度的同时，仍保留全离散能量耗散不等式和相面积守恒的速度级约束。为此，作者采用界面 Dirichlet-to-Neumann 算子即 Steklov-Poincaré 映射，将体相调和问题化为边界问题，并用对称 Galerkin 边界元法逐区域组装该算子，从而得到一个仅含界面曲线网络演化的变分格式。
2. 主要定理或主要结果
本文的主要结果可分为连续层面、离散算子层面和全离散格式层面。连续层面，作者定义组装的界面 Dirichlet-to-Neumann 算子
$$\Lambda_\Gamma:=\sum_{r=1}^{I_R}P_{R_r}^*S_{R_r}P_{R_r}:H^{1/2}(\Gamma)\to H^{-1/2}(\Gamma),$$其中 $S_{R_r}$ 是区域 $R_r$ 上的 Dirichlet-to-Neumann 映射。命题 3.1 证明 $\Lambda_\Gamma$ 对称、半正定，且 $\ker\Lambda_\Gamma=\mathrm{span}\{1\}$。其核心恒等式为
$$\langle\Lambda_\Gamma w_2,w_1\rangle_\Gamma=\sum_{r=1}^{I_R}\int_{R_r}\nabla w_1^{(r)}\cdot\nabla w_2^{(r)}\,dL^2.$$命题 3.2 证明光滑解满足各向异性能量耗散
$$\frac{d}{dt}E_\gamma(\Gamma(t))=-\sum_{p=0}^{I_P-1}\langle\Lambda_{\Gamma(t)}w_p,w_p\rangle_{\Gamma(t)}\le 0.$$命题 3.3 证明每个相的面积守恒，即对 $p=1,\dots,I_P-1$，
$$\frac{d}{dt}A_p(t)=0.$$此外，作者给出边界约化混合弱形式：寻求 $(w,\kappa,V)\in W_\Gamma\times K_\Gamma\times X_\Gamma^{-1/2}$ 满足
$$\sum_{p=0}^{I_P-1}\langle\Lambda_\Gamma w_p,\phi_p\rangle_\Gamma-\langle V,D_\Gamma^*\phi\rangle_\Gamma=0\quad\forall\phi\in S_\Gamma^{I_P},$$$$(D_\Gamma^*w-\kappa,\psi)_{L^2(\Gamma)}=0\quad\forall\psi\in K_\Gamma,$$$$\sum_{i=1}^{I_S}\int_{\Gamma_i}\kappa_i\vec\nu_i\cdot\vec\eta_i\,d\mathcal H^1+a_{\gamma,\Gamma}(I,\vec\eta)=0\quad\forall\vec\eta\in X_\Gamma,$$其中 $W_\Gamma$ 用 $g_\Gamma(w)=0$ 固定常数核。
离散层面，作者用逐区域 SGBEM 离散化组装离散界面 Steklov-Poincaré 算子，并在命题 5.6 中证明该离散算子仍对称、半正定，并具有精确常数核。该性质是后续能量耗散与面积守恒证明的关键。全离散格式方面，摘要和引言中指出：对平方根类各向异性，作者构造了一个半隐式全离散格式，并证明定理 6.4：只要演化多边形网络保持在其 admissible class 中，该格式对时间步长无条件稳定，并且每个时间步满足能量耗散不等式。
在相面积守恒方面，命题 6.6 证明每个有界相离散带符号面积的一阶变分消失；命题 6.7 进一步证明多边形面积缺陷恰好等于位移多边形的带符号面积。这给出了速度级面积守恒的离散对应物，而不仅是最终面积的近似守恒。关于可解性，定理 6.10 证明在 solvability-admissible 曲线网络上，带 gauge 边界条件的一步系统唯一可解。
数值实验中，作者用三个同心圆的相似重缩放精确解进行耦合时空加密收敛测试，验证所提方法精度；在三个数量级的时间步扫描中确认能量耗散和面积守恒，并验证面积缺陷恒等式达到机器精度；还测试了各向异性粗化，其中不同相具有不同各向异性，以及三重点网络混合三种对称性的情形。需要指出，提供的 PDF 摘录没有给出定理 6.4 的显式能量不等式、收敛阶或误差估计细节，因此这些具体常数、阶数和 admissible class 的精确定义从现有内容中无法确认。
3. 研究方法、关键技术和核心工具
核心障碍是：Mullins-Sekerka 流本身含体相调和场，若直接离散体会引入大量自由度；若只做界面配点，则通常难以同时保持离散能量耗散和速度级面积守恒。作者的技术路线是用边界积分算子严格消去体相，同时把离散算子设计成具有与连续 Steklov-Poincaré 算子相同的对称性、半正定性和常数核。
第一，连续层面使用区域 Dirichlet-to-Neumann 映射组装界面算子。对每个区域 $R$，令 $u$ 为 Dirichlet 数据 $\phi$ 的调和延拓，定义 $S_R\phi=\partial u/\partial\vec\nu_R$。界面算子由区域算子组装为 $\Lambda_\Gamma=\sum_r P_{R_r}^*S_{R_r}P_{R_r}$。证明对称半正定的关键工具是 Green 恒等式。作者对每个有界区域用
$$\langle S_{R_r}P_{R_r}w_2,P_{R_r}w_1\rangle_{\partial R_r}=\int_{R_r}\nabla w_1^{(r)}\cdot\nabla w_2^{(r)}\,dL^2,$$对无界区域则在截断球 $B_L(\vec 0)$ 上作 Green 恒等式，并利用远场条件使球面积分 $O(L^{-1})$ 趋于零。由此得到 $\langle\Lambda_\Gamma w_2,w_1\rangle_\Gamma$ 等于所有区域上的 Dirichlet 能量内积。常数核的证明也由此推出：若 $\Lambda_\Gamma w=0$，则每个区域的梯度为零，区域上的调和延拓为常数；共享曲线两侧迹相等，曲线网络连通区域链又把这些常数连接起来，故只能是全局常数。
第二，边界积分离散采用对称 Galerkin 边界元法。作者引入单层、双层和超奇异算子
$$(V_R\psi)(\vec x)=\int_{\Gamma_R}G(\vec x-\vec y)\psi(\vec y)\,d\mathcal H^1(\vec y),$$$$(K_R\phi)(\vec x)=\int_{\Gamma_R}\frac{\partial G}{\partial\vec\nu_R(\vec y)}(\vec x-\vec y)\phi(\vec y)\,d\mathcal H^1(\vec y),$$$$W_R\phi=-\frac{\partial}{\partial\vec\nu_R(\vec x)}\widetilde K_R\phi(\vec x),$$其中 $G(\vec x)=-\frac{1}{2\pi}\log|\vec x|$。区域 Dirichlet-to-Neumann 的对称表示来自 Calderón 关系
$$S_R^-=W_R+J_R^*V_R^{-1}J_R,\quad J_R=\frac12 I+K_R.$$作者明确说是基于 Maue 恒等式构建 SGBEM，这使得离散算子的对称性、半正定性和精确常数核在变分实现中透明可见。与普通过程中先离散体问题再消元不同，这里直接从界面变分形式出发，把离散 DtN 算子作为非局部 mobility。
第三，多相与三重点的处理需要几何相容空间。作者定义逐曲线 Sobolev 空间 $X_\Gamma^s=\prod_i H^s(\Gamma_i)$，其中不强制三重点相容性；再定义满足三重点处三条曲线位移相等的几何空间
$$X_\Gamma=\{\vec\eta=(\vec\eta_i)\in\prod_i H^1(\Gamma_i;\mathbb R^2):\vec\eta_{s^k_1}(T_k)=\vec\eta_{s^k_2}(T_k)=\vec\eta_{s^k_3}(T_k)\}.$$相-界面 incidence 算子 $M_p$ 记录每条界面上相 $p$ 在哪一侧，其元素 $m_{p,i}=\llbracket\chi_p\rrbracket|_{\Gamma_i}\in\{-1,0,1\}$。由此定义 $D_\Gamma^*\phi=\sum_p M_pP_\Gamma\phi_p$ 及其伴随 $D_\Gamma$，把界面速度与相跳跃耦合起来。弱形式中的曲率项 $a_{\gamma,\Gamma}(I,\vec\eta)$ 通过对 $\kappa_\gamma$ 分部积分产生，三重点处的边界项正好由 Young-Herring 力平衡消去。
第四，离散结构保持的证明依赖离散算子的三个性质：对称性、半正定性、精确常数核。连续能量耗散恒等式为 $dE_\gamma/dt=-\sum_p\langle\Lambda_\Gamma w_p,w_p\rangle_\Gamma$，离散格式的能量不等式正是这一恒等式的全离散对应物。面积守恒的速度级约束则来自 $\Lambda_\Gamma\mathbf 1=0$：连续情形有 $\langle V\llbracket\chi_p\rrbracket,1\rangle_\Gamma=\langle\Lambda_\Gamma w_p,1\rangle_\Gamma=0$，即 $\int_\Gamma\llbracket\chi_p\rrbracket V\,d\mathcal H^1=0$。离散格式通过保持 $\Lambda_\Gamma$ 的常数核来保持这一约束。命题 6.6 和 6.7 进一步说明离散面积的一阶变分消失以及面积缺陷恒等式，因而守恒不是事后修正，而是格式结构的一部分。
第五，全离散一步系统的唯一可解性使用带 gauge 的 bordered 系统。由于 $\Lambda_\Gamma$ 有常数核，必须用 $g_\Gamma(w)=0$ 固定加性常数，类似混合有限元中的鞍点系统。定理 6.10 的假设是曲线网络属于 solvability-admissible 类。作者还要求多边形网络保持 admissible，以保证无条件稳定性。需要强调，提供的 PDF 摘录只到 Calderón 关系附近，未给出 SGBEM 矩阵的具体组装、离散 Maue 恒等式细节、能量不等式的显式形式以及唯一可解性证明步骤，因此这些技术细节从现有内容中无法完整确认。
4. 与之前工作的比较
与参数有限元法 PFEM 的结构保持结果相比，本文的主要推进在于消去体相自由度。PFEM 文献 [10] 对 Stefan 和 Mullins-Sekerka 问题引入耦合参数界面与体有限元场的稳定离散，[29] 发展了多相 Mullins-Sekerka 的保结构格式，[31] 处理带动力学过冷的各向异性变体，[30] 处理密切相关的多相 Stefan 问题。这些方法基于体或全域弱形式，因而需要离散整个相区域或至少保存体网格；本文则通过界面 Steklov-Poincaré 算子把调和问题严格消去，只演化曲线网络，同时仍证明全离散能量耗散和速度级面积守恒。代价是分析限制在二维曲线网络，并需要 admissible 类与 solvability-admissible 类假设；三维推广、拓扑变化与更复杂奇异性未在提供内容中讨论。
与一般边界积分法相比，已有工作 [1,13,14,18,53] 用层位势只离散界面，建立在界面动力学的边界积分与梯度流框架 [37,43] 上，后来扩展到多组分流体与 Ostwald 熟化 [38]，并通过递归压缩逆预条件 RCIP 处理含三重点的边界椭圆问题 [34,35]。这些工作展示了边界积分法的能力，但本文的侧重点不同：作者把 SGBEM 放在变分 Galerkin 框架中，显式证明离散 DtN 算子对称半正定且具有精确常数核，并以此推出无条件能量耗散和速度级面积守恒。RCIP 文献处理三重点的方式与本文的曲线网络三重点 Young-Herring 弱形式不同；提供内容没有给出与这些边界积分法在数值精度、条件数或计算代价上的直接比较。
与作者自己此前的边界-only 配点格式相比，本文的推进是把方法提升为变分 Galerkin 框架。作者在引言中说，基于电荷模拟法的配点格式已近似两相和多相流，两相具有面积保持性质 [27]，多相在速度级达到机器精度的相面积守恒 [28]。本文把这一边界-only 观点放入 SGBEM 变分框架，获得无条件稳定性，并首次给出速度级面积守恒的离散恒等式。与配点法相比，SGBEM 的对称变分结构更适合证明结构保持；但代价是需要处理单层、双层、超奇异算子的离散化及其 Calderón 关系，实施复杂度更高。
与解析边界-only 方法相比，最近平面两相 Mullins-Sekerka 问题已被用位势方法重写为仅界面演化并证明适定性 [25]。本文在思想上与这一边界-only 视角对应，但对象是多相、各向异性、带三重点，并进一步构造可计算的全离散 SGBEM 格式。本文的数值实验包括三同心圆相似重缩放精确解、三个数量级时间步的耗散与守恒验证、机器精度的面积缺陷恒等式，以及不同相不同各向异性与混合三对称的粗化算例。不过，提供内容没有给出与 PFEM 或相场/水平集方法的定量比较，也没有给出收敛阶的具体数值；摘要只说明做了耦合时空加密的收敛测试。
假设与适用范围方面，本文各向异性限定为平方根类 (2.3)，曲线网络需保持在 admissible 类中；一步系统唯一可解性还要求 solvability-admissible 曲线网络。与 PFEM 文献中可能允许更一般 anisotropy 或 kinetic undercooling 的设置相比，本文没有处理动力学过冷、三维或拓扑变化。与相场和水平集方法相比，本文是尖锐界面方法，天然保持界面几何和相面积，但需要对三重点和曲线网络几何作显式追踪，因而对拓扑变化和网格质量有额外要求。与基于体有限元的方法相比，本文优势是降维和结构保持；潜在代价是非局部界面算子稠密、需要组装超奇异算子，以及 admissible 类条件对界面运动的限制。作者在引言中明确承认这些工作属于参数有限元结构保持结果的对应发展，但没有在提供内容中显式展开与所有前人工作的逐项代价比较，因此部分比较只能从方法结构推断。
了解领域论文（摘要翻译）
1. 一个用于捕捉不可压缩湍流宏观行为的无导数框架
A derivative-free framework for capturing macroscopic behavior of incompressible turbulent flows
作者: Jihun Han, Yoonsang Lee
类别: physics.flu-dyn, math.NA
发布日期: 2026-09-15
arXiv ID: 2609.17323v1
链接: http://arxiv.org/abs/2609.17323v1
摘要翻译
中文标题: 一个用于捕捉不可压缩湍流宏观行为的无导数框架
摘要翻译: 无导数损失方法（DFLM）是一种无网格神经网络方法，通过模拟随机游走子而非直接计算导数来求解偏微分方程。尽管 DFLM 此前已被应用于 Navier–Stokes 方程，但将其推广到湍流会揭示其最简单实现的两个局限。这些游走子未考虑流动的旋涡状、方向拉伸运动，而用于评估目标的游走子的随机采样会给学习到的解引入噪声。我们通过游走子局部运动的解析近似来解决这两个局限，该近似捕捉方向拉伸并消除这种采样噪声，同时保持计算高效。我们研究所得方法作为一种非侵入式多尺度求解器，能够学习宏观流动行为，而无需在细网格上分辨每个尺度，也无需显式耦合粗尺度与细尺度模型。对于两个二维湍流流动，所提方法比标准方法更准确地重现完全解析的参考模拟的宏观能谱，尤其是在方向拉伸最强的地方，证实了 DFLM 作为湍流流体系统多尺度求解器的有效性。
2. 高维输运方程的分段常数稀疏网格间断 Galerkin 方法的尖锐 CFL 条件
The sharp CFL condition of the piecewise constant sparse grid discontinuous Galerkin method for high-dimensional transport equations
作者: Juntao Huang
类别: math.NA, physics.comp-ph
发布日期: 2026-09-15
arXiv ID: 2609.17312v1
链接: http://arxiv.org/abs/2609.17312v1
摘要翻译
中文标题: 高维输运方程的分段常数稀疏网格间断 Galerkin 方法的尖锐 CFL 条件
摘要翻译: 我们建立了采用前向 Euler 时间步进的分段常数稀疏网格间断 Galerkin (DG) 方法应用于任意维周期区域上常系数输运方程时的尖锐 CFL 条件。对于输运速度 $\boldsymbol c=(c_1,\ldots,c_d)$ 和尺寸为 $h$ 的一致网格，我们证明该格式是 $L^2$ 稳定的当且仅当 $Δt \leq {h}/{\max_{1\leq \ell\leq d}|c_\ell|}$，而相应的全网格迎风格式众所周知需要 $Δt\leq h/\sum_{\ell=1}^d |c_\ell|$。因此，稀疏网格离散将容许时间步长放大了因子 ${(\sum_{\ell=1}^d |c_\ell|)}/{(\max_{1\leq \ell\leq d}|c_\ell|)}$，该因子介于 $1$ 和 $d$ 之间，并在各向同性输运时达到 $d$。充分性的证明依赖于多级 Haar 分解的投影泄漏恒等式，这些恒等式使得能量估计中的混合方向项可被稀疏网格投影所丢弃的能量吸收。尖锐性的证明来自一维最细层级上的交替模式。作为副产品，我们得到了放大算子的 $L^2$ 算子范数和谱半径的显式公式。对于一般向下封闭指标集上的空间，我们推导了一个显式充分 CFL 条件和其尖锐性的几何准则。二维和四维数值实验证实了理论结果。
3. Interface Control Domain Decomposition (ICDD) 方法的代数收敛性分析
Algebraic convergence analysis for the Interface Control Domain Decomposition (ICDD) method
作者: Marco Discacciati, Paola Gervasio, Alfio Quarteroni
类别: math.NA
发布日期: 2026-09-15
arXiv ID: 2609.17271v1
链接: http://arxiv.org/abs/2609.17271v1
摘要翻译
中文标题: Interface Control Domain Decomposition (ICDD) 方法的代数收敛性分析
摘要翻译: 我们发展了 Interface Control Domain Decomposition (ICDD) 方法的收敛性分析。ICDD 是一种基于最优控制框架的重叠型区域分解方法，使用 Dirichlet 界面控制函数和界面观测。我们考虑可能具有不连续系数的椭圆问题，并用 $hp-$FEM 在每个子区域中进行近似。当离散化在二维区域之间的重叠区域上是协调的时，我们给出了求解与 ICDD 相关的非对称界面 Schur 补系统所需的 GMRES 迭代次数的理论估计。我们的结果是通过结合 ICDD 的 Schur 补矩阵的新谱估计和经典 GMRES 收敛理论得到的。我们证明收敛速率表现为 $\mathcal{O}(δ^{-1} p^{3/2}\log p)$，其中 $δ$ 表示重叠宽度，$p$ 表示局部多项式次数，同时保持与网格尺寸 $h$ 无关。数值实验验证了理论预测，并表明 ICDD 在计算域中存在大系数跳跃时的有效性。由于在协调情形下，所考虑的 ICDD 格式与子结构限制加性 Schwarz (SRAS) 方法一致，该分析也为二维中的 SRAS 提供了收敛估计，并且通过其已知的等价性，为限制加性 Schwarz (RAS) 方法提供了收敛估计。
4. 高阶全局多迹积分方程的准 Helmholtz Calderón 乘性预条件
Quasi-Helmholtz Calderón Multiplicative Preconditioning for Higher-Order Global Multi-Trace Integral Equations
作者: Cedric Münger, Alessandro Zuccotti, Van Chien Le, Kristof Cools
类别: cs.CE, math.NA
发布日期: 2026-09-15
arXiv ID: 2609.17252v1
链接: http://arxiv.org/abs/2609.17252v1
摘要翻译
中文标题: 高阶全局多迹积分方程的准 Helmholtz Calderón 乘性预条件
摘要翻译: 本文提出了一种用于复合物体时谐电磁散射的高阶全局多迹积分方程。该高阶多迹格式采用一个使用高阶准 Helmholtz 投影算子的 Calderón 乘性预条件子进行预处理。高阶准 Helmholtz 投影算子将基函数的无散分量和非无散分量分离。对 Helmholtz 分量的分别访问避免了对病态混合 Gram 矩阵的显式求逆。这使得无需细化网格和借助对偶基函数即可应用 Calderón 乘性预条件。此外，由于无散分量和非无散分量可以分别重新缩放，它还实现了低频稳定化。高阶准 Helmholtz 投影算子通过迭代计算，使迭代求解器能够高效求解预条件后的矩阵系统，在几十次迭代内获得准确解。针对复合介质体进行了数值实验，证实了所提出的预条件子对于采用高阶基函数离散的全局多迹格式在稠密网格和极低频率下的有效性。
5. 通过有限测试与神经代理从边界数据潜空间反演材料系数
Latent Inversion of Material Coefficients from Boundary Data via Finite Tests and Neural Surrogates
作者: Erik Burman, Mats G. Larson, Karl Larsson, Jonatan Vallin
类别: math.NA
发布日期: 2026-09-15
arXiv ID: 2609.17137v1
链接: http://arxiv.org/abs/2609.17137v1
摘要翻译
中文标题: 通过有限测试与神经代理从边界数据潜空间反演材料系数
摘要翻译: 我们由有限多个边界激励重构标量椭圆方程中空间变化的材料系数，每个激励产生一个完整的 Dirichlet 迹。为缓解不适定性和重复求解 PDE 的代价，我们将系数限制在一个低维族中，该族由解析方式指定或从样本中学习得到，并在其潜坐标中求解反问题。我们考虑一个具有 (m) 个潜坐标且在参考点处导数为满秩的 (C^1) 参数化。若连续线性化 Neumann-to-Dirichlet 映射在相应切空间上单射，则至多 (m) 个激励就足以保证局部单射性和 Lipschitz 稳定性。系数灵敏度的收敛性进而将该稳定性传递到协调有限元离散。对足够细的网格，稳定性常数和邻域可选取为与网格尺寸无关。在局部残差比较条件下，代理前向映射的一致精度给出系数误差界，将表示误差、数据噪声、有限元误差和代理误差分离。导数精度还保持代理自身的局部稳定性。所有稳定性陈述都相对于参考系数是局部的。二维数值实验将夹杂和类裂纹系数的解析表示与学习表示同神经前向代理相结合。它们展示了潜空间重构、降低的在线成本，以及来自可选的基于 FEM 的细化带来的进一步改进。
6. 多边形网格上 Reissner–Mindlin 板问题的一种基于 BGG 的任意阶离散格式
An arbitrary-order BGG-based discrete scheme for the Reissner–Mindlin plate problem on polygonal meshes
作者: Arax Leroy
类别: math.NA
发布日期: 2026-09-15
arXiv ID: 2609.17122v1
链接: http://arxiv.org/abs/2609.17122v1
摘要翻译
中文标题: 多边形网格上 Reissner–Mindlin 板问题的一种基于 BGG 的任意阶离散格式
摘要翻译: 我们设计并分析了一种用于一般多边形网格上 Reissner–Mindlin 板问题的任意阶数值格式。该格式由与离散 Bernstein–Gelfand–Gelfand (BGG) 扭曲复形相关联的 Hodge–Laplacian 导出，并利用了一种基于离散 $H_2$ 的横向位移构造。我们为底层离散 de Rham 方法 (DDR) 空间建立了一个离散 Korn 不等式，并证明了该方法的收敛性。在最低阶时，该分析给出了一个关于板厚度一致的误差估计，表明该格式是无闭锁的。在若干族多边形网格上的数值实验支持了理论结果，并说明了横向位移离散化增强连续性带来的优势。
7. 一种用于 Navier-Stokes 方程变分初始状态数据同化的 Reynolds 半鲁棒、全局散度为零的 E-HDG/IMEX-SAV 方法
A Reynolds-Semi-Robust, Globally Divergence-Free E-HDG/IMEX-SAV Method for Variational Initial-State Data Assimilation of the Navier-Stokes Equations
作者: Ya Min, Xian Zhang, Xiaoping Xie
类别: math.NA
发布日期: 2026-09-15
arXiv ID: 2609.17096v1
链接: http://arxiv.org/abs/2609.17096v1
摘要翻译
中文标题: 一种用于 Navier-Stokes 方程变分初始状态数据同化的 Reynolds 半鲁棒、全局散度为零的 E-HDG/IMEX-SAV 方法
摘要翻译: 本文针对由非定常不可压缩 Navier-Stokes 方程控制的变分初始状态数据同化，提出了一种嵌入杂交间断 Galerkin (E-HDG) 方法，该方法与一阶隐式-显式标量辅助变量 (IMEX-SAV) 时间离散相结合。我们采用先优化后离散策略。空间离散对单元速度和压力分别使用次数为 $k$ 和 $k-1$ 的不连续分片多项式，对速度迹使用连续 $k$ 次多项式，对压力迹使用不连续 $k$ 次多项式。所得状态速度和伴随速度以及重构的初始速度均为全局散度为零。正向 IMEX-SAV 状态格式是无条件能量稳定的。在适当的正则性和局部轨道假设（这些假设的界不引入黏度的显式负幂）、依赖于网格的条件 $Δt\lesssim h^2$ 以及 Tikhonov 参数上的压缩性条件下，我们建立了全离散 OTD 最优性系统的局部存在性和唯一性，以及针对状态、伴随和重构初始速度的 Reynolds 半鲁棒 $L^2$ 误差估计，阶为 $O(h^k+Δt)$；其中的常数不含黏度的显式负幂。数值实验在光滑测试上证实了收敛性，并展示了达到机器精度的离散不可压缩性、有效的非线性求解器行为，以及小黏性区域中的稳定性能。
8. 超线性随机反应-扩散方程的适定性、正则性与强逼近
Well-posedness, Regularity, and Strong Approximations of Superlinear Stochastic Reaction-Diffusion Equation
作者: Zhihui Liu
类别: math.NA, math.PR
发布日期: 2026-09-15
arXiv ID: 2609.16979v1
链接: http://arxiv.org/abs/2609.16979v1
摘要翻译
中文标题: 超线性随机反应-扩散方程的适定性、正则性与强逼近
摘要翻译: 本文为具有超线性漂移和扩散系数的随机反应–扩散方程 (SRDE) 的适定性、正则性和强逼近建立了一个一般框架。我们首先将 \emph{W. Liu and M. Röckner, J. Funct. Anal., 2902–2922, 2010} 和 \emph{W. Liu, J. Differential Equations, 572–592, 2013} 中的适定性结果推广到 Gelfand 三元组 (V \hookrightarrow H \hookrightarrow V^*) 中的超线性扩散情形，其中 (V) 配备范数 (|\cdot|_V)，并通过为 (|X|_V^p) 建立一个新的 Itô 公式（对一般 (p \ge 2)）导出一个矩估计。然后，我们将这一抽象结果应用于 SRDE，只要初始数据属于同一 Sobolev 空间，就对任意 (γ\in [0,1]) 建立更高的空间正则性 (\dot H^{1+γ})，并获得时间 Hölder 正则性。最后，我们在关于 tamed 函数的一般假设下，为 SRDE 构造了一族 tamed 有限元方法 (tamed-FEMs)，推导了其长时间无条件稳定性，并建立了最优强收敛阶。据我们所知，这是首个针对具有超线性扩散系数的 SPDE 的强逼近结果。
9. Hellan-Herrmann-Johnson 方法的局部化最大范数误差估计
Localized maximum-norm error estimates for the Hellan-Herrmann-Johnson method
作者: Yuwen Li, Zhuoran Teng
类别: math.NA
发布日期: 2026-09-13
arXiv ID: 2609.14602v2
链接: http://arxiv.org/abs/2609.14602v2
摘要翻译
中文标题: Hellan-Herrmann-Johnson 方法的局部化最大范数误差估计
摘要翻译: 我们推导了针对固支 Kirchhoff 板问题、由 Hellan-Herrmann-Johnson 方法计算得到的弯矩的局部化最大范数界。主要困难在于将离散应力局部化，同时不离开 HHJ 空间或不违反其核约束。利用对称旋度势，我们构造了一种保持核的局部化，并将局部 Green 应力与一个全局离散 Green 应力联系起来。该论证将局部插值误差与一个较弱的全局污染项分离。在辅助问题的显式正则性假设下，所得估计对弯矩值和逐单元一阶导数给出最优逐点收敛性。对正多项式次数，不出现对数损失；而最低阶的值估计仍保留一个对数因子。在附加的反射对称性下，对称恢复对偶多项式次数改善弯矩值，对奇次数改善一阶导数。所证明的增益分别为三分之一阶和二分之一阶。数值实验证实了这种奇偶依赖性，并显示出接近整整一阶的增益，超过了理论上所证明的增益。
10. 用于求解偏微分方程的物理信息随机特征神经网络
Physics Informed Random Feature Neural Networks for Solving PDEs
作者: Chi-An Chen, Chunyang Liao, Ming Zhong
类别: math.NA, cs.LG
发布日期: 2026-09-14
arXiv ID: 2609.16406v1
链接: http://arxiv.org/abs/2609.16406v1
摘要翻译
中文标题: 用于求解偏微分方程的物理信息随机特征神经网络
摘要翻译: 近年来，基于机器学习的偏微分方程（PDEs）求解器引起了显著关注。该领域的大部分进展由深度神经网络推动，例如物理信息神经网络（PINNs）和核方法（如物理信息高斯过程）。我们引入一种物理信息随机特征方法，用于对抗基于 PINN 的求解器在某类 PDEs 上面临的部分谱偏差。随机特征方法最初被提出用于近似大规模核机器，并可视为一种专门的随机化神经网络。与其他需要大量配点的最先进的基于 PINN 的求解器相比，我们提出的方法降低了计算复杂度。在本文中，我们发展了一套严格的近似误差分析，并推导了 $H^1$ 范数上的高概率误差界。我们提供了大量数值实验以验证我们关于误差衰减率的理论保证，以及若干比较实验以展示我们所声称的在这些基于深度学习的方法中对抗谱偏差的能力。
11. 周期三次非线性薛定谔（NLS）方程的 Strang 分裂方法的时间误差增长
Temporal Error Growth of Strang Splitting Method for the Periodic Cubic NLS
作者: Yue Feng, Yifei Wu
类别: math.NA
发布日期: 2026-09-14
arXiv ID: 2609.15912v1
链接: http://arxiv.org/abs/2609.15912v1
摘要翻译
中文标题: 周期三次非线性薛定谔（NLS）方程的 Strang 分裂方法的时间误差增长
摘要翻译: 我们研究周期三次非线性薛定谔（NLS）方程的 Strang 分裂方法的时间误差增长。主导误差由受迫线性化方程支配，其增长急剧依赖于维数和非线性项的符号。在一维散焦情形，我们利用全局 Birkhoff 变换以及线性化流由此产生的退化结构，证明了一致二次上界 $C(1+T^2)τ^2$。在高维散焦情形，指数增长通过任意小的不稳定驻波构造得到。此外，为证明高维散焦上界，我们使用 Killip 和 Vişan 的周期 Strichartz 估计证明 $\int_0^T\|u(t)\|_{L^\infty}^2\,dt\le C(u_0)(1+T)$。这给出了不依赖于最终时间的指数速率，尽管高阶 Sobolev 范数可能增长。聚焦情形中的指数下界由平面波附近的不稳定线性化动力学得到，其中 Akhmediev 呼吸子提供了底层机制。此外，对于小初值的一维聚焦情形，局部 Birkhoff 变换的应用保证误差随时间至多二次增长。各种数值实验验证了尖锐的线性、二次和指数误差增长率。
12. 基于张量仿积的积分算子核的正则化与压缩
Regularization and compression of integral operator kernels with tensor paraproducts
作者: Oluwadamilola Fasina, Ronald R. Coifman
类别: math.NA
发布日期: 2026-09-14
arXiv ID: 2609.15822v1
链接: http://arxiv.org/abs/2609.15822v1
摘要翻译
中文标题: 基于张量仿积的积分算子核的正则化与压缩
摘要翻译: 我们提出了一个关于由距离的非线性函数给出的积分算子核的新视角。该视角很有用，因为它允许人们对作用在距离 d(x,y) 的张量 Haar 展开上的非线性变换进行拟线性化。由此得到一种由主项和余项组成的新表示，并具有良好性质；即主分量局部正则性增强（并且在某些情形下具有快速的非对角衰减，从而导致 O(N) 存储），以及由于全局正则性增强而获得具有改进逼近的余项展开。文中包含了对位势核的数值实验。
13. 面向鲁棒 PDE 反问题的物理引导条件流匹配与能量正则化
Physics-Guided Conditional Flow Matching with Energy Regularization for Robust PDE Inverse Problems
作者: Yongsheng Chen, Shuo Lu, Wei Guo, Xinghui Zhong
类别: math.NA
发布日期: 2026-09-14
arXiv ID: 2609.15536v1
链接: http://arxiv.org/abs/2609.15536v1
摘要翻译
中文标题: 面向鲁棒 PDE 反问题的物理引导条件流匹配与能量正则化
摘要翻译: 我们考虑由稀疏、含噪且受污染的观测出发的偏微分方程 (PDE) 反问题，目标是在无网格设置下恢复未知系数场及相应的状态变量。在此类稀疏且受污染的观测下，标准的物理信息方法与生成方法通常不加区分地对待所有样本，因而缺乏一种有原则的机制来协调物理规律与受污染数据。我们通过一个两阶段流匹配框架来解决这一困难。在第一阶段，我们提出物理引导条件流匹配 (PG-CFM)，它通过沿生成轨迹的残差正则化以及间歇性的全局配点约束，将强形式 PDE 信息纳入其中。在第二阶段，我们引入能量正则化流匹配 (ERFM)，它通过从冻结教师模型为每个观测分配一个物理–数据能量分数，并重新加权流匹配目标，以减少高能量、与 PDE 不一致样本的影响，从而微调 Stage-1 模型。我们证明，所得的 Stage-2 目标等价于在教师诱导的重加权数据分布下的流匹配，这给出了鲁棒性机制在总体层面的解释。在若干反问题基准（包括 Poisson 和 Navier–Stokes 问题）上的数值实验表明，所提出的框架比鲁棒 PINN 变体和竞争性生成基线能获得更准确的系数恢复。
14. 不可压缩 Navier-Stokes 方程的强拟涡能稳定积分格式
Strongly enstrophy-stable integrators for the incompressible Navier-Stokes equations
作者: Boris D. Andrews, Matin Shams, Patrick E. Farrell
类别: math.NA
发布日期: 2026-09-14
arXiv ID: 2609.15520v1
链接: http://arxiv.org/abs/2609.15520v1
摘要翻译
中文标题: 不可压缩 Navier-Stokes 方程的强拟涡能稳定积分格式
摘要翻译: 我们提出一种用于不可压缩 Navier-Stokes 方程的混合有限元离散化，它保持能量和拟涡能的演化规律，且这种保持比先前的离散化更强。在二维情形，对于热力学孤立系统，拟涡能的演化规律仅允许耗散，由此得到速度梯度的与 Reynolds 数无关的界，这自然地稳定了格式，即使在严重欠分辨的网格上也是如此。在三维情形，该格式同时保持耗散以及通过涡旋拉伸产生的拟涡能。我们通过系统地在离散化中引入辅助变量来强制这些演化规律。虽然这些格式的协调实现需要具有增强正则性的离散 Stokes 复形，但我们同时引入 (i) 等价重参数化以及 (ii) 罚格式，它们只需要标准离散 de Rham 复形中典型的 curl- 和 div- 协调空间。该格式可处理不同类型的边界条件和弯曲区域。所提格式的稳健稳定性质通过剪切流、球形涡和绕障碍物流动的数值模拟得到展示。我们在数值上观察到，以这种方式保持拟涡能的离散演化对数值解具有很强的稳定作用，尤其是在二维情形。
15. 用于应变梯度弹性的杂交化交错间断 Galerkin–混合有限元方法
A Hybridized Staggered Discontinuous Galerkin–Mixed Finite Element Method for Strain Gradient Elasticity
作者: Bohan Yang, Eric T. Chung
类别: math.NA, math-ph
发布日期: 2026-09-14
arXiv ID: 2609.15186v1
链接: http://arxiv.org/abs/2609.15186v1
摘要翻译
中文标题: 用于应变梯度弹性的杂交化交错间断 Galerkin–混合有限元方法
摘要翻译: 我们提出了一种用于应变梯度弹性模型的杂交化交错间断 Galerkin–混合有限元方法；该模型是由材料长度尺度参数 $ι$ 以及 Lamé 常数 $λ$ 和 $μ$ 支配的四阶奇异摄动问题。通过引入总应力以及缩放后的 Cauchy 应力和超应力作为辅助未知量，我们将模型重写为一阶系统，并用交错间断 Galerkin 空间离散位移和总应力，用 Raviart–Thomas 对离散两个缩放应力，同时对三个应力均强式施加对称性。高阶 Dirichlet 边界条件自然地进入变分形式，从而使该格式避免了限制基于位移方法的数值边界层。通过在对称子空间上建立 inf-sup 条件以及离散 Korn 不等式，我们证明了代数稳定性和最优收敛性，且二者均关于 $ι$ 和 $λ$ 一致。我们进一步发展了一个杂交化格式，其中局部静力凝聚仅将网格骨架上的两个乘子留作全局耦合未知量。数值实验证实了所预测的收敛率和参数鲁棒性。
16. 非全局 Lipschitz 条件下 index-1 随机微分代数方程的随机 theta Milstein 方法强收敛率
Strong convergence rates of stochastic theta Milstein methods for index-1 stochastic differential algebraic equations under non-globally Lipschitz conditions
作者: Caiyuan Zhu, Shiye Wan, Ziheng Chen, Lin Chen
类别: math.NA
发布日期: 2026-09-13
arXiv ID: 2609.14583v1
链接: http://arxiv.org/abs/2609.14583v1
摘要翻译
中文标题: 非全局 Lipschitz 条件下 index-1 随机微分代数方程的随机 theta Milstein 方法强收敛率
摘要翻译: 本文研究了一类具有时间依赖奇异矩阵和非全局 Lipschitz 系数的 index-$1$ 随机微分代数方程（SDAEs）的保结构随机 theta Milstein 方法的强收敛阶。奇异矩阵允许随时间变化，同时保持固定的微分代数分裂，并且漂移和扩散系数可能表现出超线性增长。通过利用精确解的 index-$1$ 代数-微分分解，我们直接在原始 SDAE 变量中确定约化随机微分方程的 Milstein 系数，并建立了所提方法对 $θ\in[1/2,1]$ 的适定性和约束保持性质。在耦合单调性条件和适当的多项式正则性假设下，证明该方法在所有时间层保持代数约束，并在均方根范数下以强一阶收敛。数值实验证实了保结构性质和理论收敛阶。
17. 关于对流扩散方程的一类超收敛 Hermite 型方法
On a class of supraconvergent Hermite-type methods for advection-diffusion equations
作者: Xianyi Zeng
类别: math.NA
发布日期: 2026-09-13
arXiv ID: 2609.14545v1
链接: http://arxiv.org/abs/2609.14545v1
摘要翻译
中文标题: 关于对流扩散方程的一类超收敛 Hermite 型方法
摘要翻译: 我们开发并分析了一类用于一维周期区域上线性对流扩散方程的 Hermite 型方法。这些方法同时演化节点值和单元平均值，因此被称为混合变量 (HV) 方法。利用 Hermite 插值理论，我们将 HV 方法构造到任意精度阶，并证明该方法具有超收敛性，即该方法的空间精度阶高于局部截断误差。在本文第二部分，我们利用正三角级数理论证明，对于线性扩散方程和线性对流扩散方程，所有中心 HV 方法在半离散层面都是稳定的。超收敛性质和中心 HV 格式的稳定性均通过大量数值算例得到验证。
18. Hankel-Koopman 耦合实验数据的有限时域能量分解：一种三阶段数据驱动孪生预测框架
Hankel-Koopman Finite-Horizon Energy Decomposition of Coupled Experimental Data: A Three-Phase Data-Driven Twin Forecasting Framework
作者: Diana A. Bistrian, Marcel Topor
类别: math.NA
发布日期: 2026-09-13
arXiv ID: 2609.14501v1
链接: http://arxiv.org/abs/2609.14501v1
摘要翻译
中文标题: Hankel-Koopman 耦合实验数据的有限时域能量分解：一种三阶段数据驱动孪生预测框架
摘要翻译: 本文提出了一种统一的三阶段数据驱动孪生框架，用于从耦合实验测量中对物理量进行有限时域预测。该框架将一种新的、具有正交模态的 Hankel–Koopman 有限时域能量分解与一个逆校准的带外生输入的多输出非线性自回归模型相结合。非线性动力学在给定校准窗口内于 Hankel 系数空间中被辨识和模拟。随后，在物理测量空间中经过反对角恢复和相应轨迹的重构之后，对候选模型进行评估。由此得到的模拟通道轨迹作为感兴趣量的递归预测模型的外生输入。通过这种方式，所提出的框架集成了降阶表示、非线性动力学辨识、测量重构以及可解释预测。数学分析将 Koopman 延迟坐标理论专门应用于序列化多通道 Hankel 提升，并建立其与移位 Hankel 表示的兼容性。它还证明了模态的正交性以及有限时域模态能量分解，并在明确陈述的假设下证明了所辨识模型的存在性、唯一性、逆稳定性和前向稳定性。在一个太阳能发电厂案例研究中，该框架在所有考虑的预测时域上均产生了一致的高相关性和低相对误差，表明其能够在递归预测过程中保持感兴趣量的时间演化。
19. 线性输运不确定性量化的批量波前扫描
Batched Wavefront Sweeps for Linear Transport Uncertainty Quantification
作者: Tristan Pryer
类别: math.NA
发布日期: 2026-09-13
arXiv ID: 2609.14496v1
链接: http://arxiv.org/abs/2609.14496v1
摘要翻译
中文标题: 线性输运不确定性量化的批量波前扫描
摘要翻译: 我们为线性输运的迎风间断 Galerkin 离散开发了一种图兼容的批量波前表述。对于可扫描离散，在有向迎风依赖图的拓扑排序下，单元方程构成块下三角系统。这种因果结构可以在局部算子、源和入流数据变化的固定源问题族之间共享。我们利用这一观察，将具有共同依赖图的通道分组为图兼容扫描类，并在统一的波前调度中，将样本、右端项、能群和符号兼容的角度坐标作为张量维度承载。我们证明，所得的批量波前算法在代数上等价于对类中每个通道独立进行的经典 DG 扫描，同时在波前单元和通道维度上暴露并行性。我们基于批量单元局部求解将该表述实现为 GPU 张量程序，并量化其工作量、深度和存储需求，包括由样本微批处理引入的吞吐量–内存权衡。我们进一步刻画固定面符号分支上的离散参数到可观测量映射，并表明通过波前算法进行反向模式微分再现相应的离散伴随作用。数值实验验证了预期的 DG 收敛性和伴随一致性，展示了样本批处理带来的显著 GPU 执行增益，并在一个材料遮蔽不确定性量化问题和一个受 C5G7/KAIST 启发的耦合多群幂迭代工作负载中说明了该方法。
不相关论文（基本信息）
1. 关于超临界流体保压力平衡数值通量的存在性
On the Existence of Pressure-Equilibrium-Preserving Numerical Fluxes for Supercritical Fluids
作者: Robin Ben Klein
类别: math.NA
发布日期: 2026-09-15
arXiv ID: 2609.17492v1
链接: http://arxiv.org/abs/2609.17492v1
摘要: In this work we propose a new existence theorem for numerical flux functions for supercritical fluids that are pressure-equilibrium preserving (PEP). In particular, we characterize the existence of consistent numerical flux functions that satisfy an algebraic PEP property. Our theory links the existence of PEP schemes to geometric properties of the equation of state describing the thermodynamics of the fluid. When these geometric properties fail for a pair of states on the same isobar, no PEP schemes of the considered form can exist on a domain containing those states. In our analysis the equation of state itself can be fully general only needing to satisfy some fundamental thermodynamic principles. Recently, PEP compatibility conditions for general equations of state have been derived \cite{channodal} that rely on the existence of certain thermodynamic derivatives which are not guaranteed to be defined under fundamental thermodynamic principles. The geometric conditions in our theory do not depend on the existence of these derivatives and recover the recent compatibility conditions in the case that these derivatives are defined. Finally, using numerical experiments we demonstrate for two supercritical fluids that our existence conditions are restrictive and thus that no PEP schemes of the form we consider exist on the domain we specify for these fluids. Using our geometric perspective, we also shed light on mechanisms by which PEP schemes can develop numerical issues, which we also demonstrate using numerical experiments.
2. 随机 Volterra 方程的马尔可夫逼近在 Hölder 范数下的收敛性
Convergence in Hölder norms for Markovian approximations of stochastic Volterra equations
作者: Noé Corneille, Kristin Kirchner, Pietro Pezzoli Frigerio
类别: math.PR, math.NA
发布日期: 2026-09-15
arXiv ID: 2609.17472v1
链接: http://arxiv.org/abs/2609.17472v1
摘要: We bound the difference between two stochastic Volterra processes with identical Lipschitz coefficients but different kernels. For non-convolution kernels, we establish estimates in $C^0([0,T];L^p(Ω))$, $p\geq 2$, and for convolution kernels in $L^p(Ω;L^q(0,T))$, $q \in [1,p]$, and $C^β([0,T];L^p(Ω))$, $L^p(Ω;C^β([0,T]))$, where the range of the Hölder exponent $β\in (0,1]$ is the maximal permitted by the regularity of the processes. For the fractional kernel, we then construct Markovian approximations whose error we show to decay as $e^{-a\sqrt{N}}$ in the aforementioned norms, using an $N$-node quadrature based on sinc methods. Numerical experiments for the fractional Brownian motion verify our findings.
3. 一种可证明收敛的 MM-GKS 变体，用于大规模反问题
A provably convergent MM-GKS variant for large-scale inverse problems
作者: Mirjeta Pasha, Eric de Sturler, Misha Kilmer
类别: math.NA
发布日期: 2026-09-15
arXiv ID: 2609.17229v1
链接: http://arxiv.org/abs/2609.17229v1
摘要: For high-quality images with sharp edges, a popular choice for edge-preserving regularization is using a general(ized) $\ell_q$-norm of the gradient of the image. This can be implemented efficiently using the $\ell_2$-norm and a sequence of weighted gradients, with weights derived from the current solution estimate. We can solve the resulting sequence of regularized least squares problems using hybrid Krylov subspace methods, which efficiently compute the regularization parameter using the problem projected on the Krylov subspace. However, each update of the regularization operator requires a new Krylov subspace. The majorization-minimization generalized Krylov subspace method (MM-GKS) addresses this problem by using a single, generalized, Krylov subspace (GKS). Unfortunately, for large-scale problems, if convergence is not fast, MM-GKS has overwhelming memory requirements and computational costs. We propose a variant of MM-GKS that alternately compresses and expands the search space while maintaining strict monotonic convergence. We show that our method provably converges to the minimum of the selected functional, even if the search space dimension is kept very small. This substantially improves on previous theoretical results for MM-GKS, where the convergence proof relies on the basis for the solution space (eventually) spanning the full space. We show that our method can solve large-scale problems efficiently both in terms of memory requirements and computational complexity. We further generalize our proposed method to handle streaming problems, where the data is either not all available simultaneously or needs to be treated as such because of the extreme memory requirements. We use numerical examples from image deblurring, dynamic photoacoustic tomography, and streaming X-ray computed tomography (CT) to illustrate the effectiveness of our proposed methods.
4. 迈向 Pólya 猜想：通过能量正交性改进单个 Li-Yau 界
Toward Pólya’s Conjecture: Improving the Individual Li-Yau Bound via Energy Orthogonality
作者: Yifan Wang, Hehu Xie
类别: math.NA, math-ph
发布日期: 2026-09-15
arXiv ID: 2609.17219v1
链接: http://arxiv.org/abs/2609.17219v1
摘要: We establish two complementary lower-bound mechanisms for individual eigenvalues of the Dirichlet Laplacian. First, energy orthogonality yields a frequency-dependent cap on the Fourier density of a finite spectral projection. Combining this cap with the standard $L^2$ Bessel estimate and a radial-capacity bathtub principle gives, on every open set of finite positive measure in $\mathbb R^n$ with $n\geq2$, [ λ_k\geq c_n(2π)^2ω_n^{-2/n}|Ω|^{-2/n}k^{2/n}, \qquad \frac{n}{n+2}<c_n<1. ] The constant $c_n$ is characterized by a scalar equation, with $c_2=0.5383068077\ldots$. This estimate preserves Weyl scaling and strictly improves the individual consequence of the Li-Yau sum inequality, although it does not improve the sharp leading coefficient in that sum inequality. Second, we retain part of the spectral deficit discarded when an eigenvalue sum is bounded by its largest term. A lower bound for the counting function, integrated through the exact first Riesz-mean identity, leads to a strictly monotone scalar equation. Its unique positive root is no weaker than the volume-only bound, and we give necessary and sufficient criteria for strict improvement over both that baseline and any independent lower bound. Quantitative estimates of Jiang-Lin provide an explicit implementation on bounded Lipschitz domains. The final comparisons and numerical example distinguish improvements within this framework from stronger estimates available under additional geometric or spectral assumptions.
5. 时间因果带通小波的框架界与带宽铺砌
Frame bounds and bandwidth tiling for time-causal bandpass wavelets
作者: Jens E. Pedersen, Tony Lindeberg, Peter Gerstoft
类别: eess.SP, math.NA
发布日期: 2026-09-15
arXiv ID: 2609.17201v1
链接: http://arxiv.org/abs/2609.17201v1
摘要: Wavelets and frames that provide strong guarantees for signal representations are a cornerstone in signal processing theory, but have long been restricted to non-causal settings. Recent work extended wavelet analysis to time-causal systems that operate on streaming signals, with exact reconstruction and time-recursive implementations. However, the frame guarantees that make such representations reliable under noise and quantization are uncharacterized. We derive closed-form frame bounds and conditioning on band-limited domains, peak frequencies, constant-Q bandwidth tiling, and spectral decay rates for time-causal bandpass wavelets. Our work provides design rules for the scale ratio and channel count, which we validate numerically against discrete implementations of the causal wavelets in terms of recursive filters.
6. Hamilton–Jacobi–Bellman 方程的边界停止正求积格式的误差界
Error Bounds for Boundary-Stopped Positive Cubature Schemes for Hamilton–Jacobi–Bellman Equations
作者: Haoran Xu, Xingye Yue
类别: math.NA
发布日期: 2026-09-15
arXiv ID: 2609.17186v1
链接: http://arxiv.org/abs/2609.17186v1
摘要: We derive error bounds for a boundary-stopped positive cubature scheme for degenerate parabolic Hamilton–Jacobi–Bellman equations with Dirichlet data on bounded domains. The scheme balances antipodal branches at their first boundary contacts and uses a control-dependent effective time. Its stopped-barrier remainder is (O(τΔt^{γ/2})), and the total mass of interpolated branches is at most (2τ/Δt), where (τ) is the effective time. After division by the effective time, these estimates give a consistency bound that is uniform as (τ\to0). Under a common strict boundary barrier and the stated mesh conditions, interior consistency, coefficient shaking, switching, and localized comparison yield [ |(u-u_h)^+|\infty\le C(Δt^{1/4}+hΔt^{-1/2}), \qquad |(u_h-u)^+|\infty\le C(Δt^{1/10}+h^{1/2}Δt^{-1/4}). ] Thus (Δt\asymp h^{10/7}) gives an (O(h^{1/7})) maximum-norm bound. The analysis applies to any fixed centrally symmetric positive degree-two cubature and permits rank-deficient diffusion. Numerical examples examine parallel heat loss to a cold wall and control-dependent, rank-deficient diffusion.
7. 基于随机化动态模态分解与深度学习的高保真数字孪生数据模型及其在流体力学中的应用
High-Fidelity Digital Twin Data Models by Randomized Dynamic Mode Decomposition and Deep Learning with Applications in Fluid Dynamics
作者: Diana A. Bistrian
类别: cs.LG, math.NA
发布日期: 2026-09-15
arXiv ID: 2609.17101v1
链接: http://arxiv.org/abs/2609.17101v1
摘要: The purpose of this paper is the identification of high-fidelity digital twin data models from numerical code outputs by non-intrusive techniques (i.e., not requiring Galerkin projection of the governing equations onto the reduced modes basis). In this paper the author defines the concept of the digital twin data model (DTM) as a model of reduced complexity that has the main feature of mirroring the original process behavior. The significant advantage of a DTM is to reproduce the dynamics with high accuracy and reduced costs in CPU time and hardware for settings difficult to explore because of the complexity of the dynamics over time. This paper introduces a new framework for creating efficient digital twin data models by combining two state-of-the-art tools: randomized dynamic mode decomposition and deep learning artificial intelligence. It is shown that the outputs are consistent with the original source data with the advantage of reduced complexity. The DTMs are investigated in the numerical simulation of three shock wave phenomena with increasing complexity. The author performs a thorough assessment of the performance of the new digital twin data models in terms of numerical accuracy and computational efficiency.
8. 在HLL型多状态近似Riemann解中加入慢磁声模式
Adding slow magnetoacoustic mode to the HLL-type multi-state approximate Riemann solution
作者: Fan Zhang, Stefaan Poedts, Andrea Lani
类别: physics.comp-ph, astro-ph.IM, math.NA
发布日期: 2026-09-15
arXiv ID: 2609.17049v1
链接: http://arxiv.org/abs/2609.17049v1
摘要: Multi-state HLL-type approximate Riemann solutions of ideal magnetohydrodynamics (MHD) typically assume that the medium within the Riemann fan is incompressible, and thus the slow magnetoacoustic mode cannot be included in their space-time wave configurations for the approximated states. We propose a new strategy to design multi-state HLL-type approximate solutions, allowing for the medium within the Riemann fan to be compressible. In particular, for the complete seven-wave configuration of the MHD Riemann problem, we first estimate the slow magnetoacoustic speeds and a longitudinal flow speed between the slow modes, and then follow the conservation laws and Rankine-Hugoniot jump relations across the fast, Alfvén, and slow waves, to calculate all the intermediate states within the Riemann fan. Moreover, we discuss the solutions when certain wave modes degenerate, ensuring well-posedness and smooth transitions between complete and degenerate wave configurations. Numerical simulations using a Finite Volume (FV) solver show that the proposed approximate Riemann solution is less diffusive than the classic HLLD scheme, particularly for slow mode waves. For example, in a 1D test case with a strong longitudinal magnetic field, the new scheme needs one order of magnitude fewer grid cells compared to the classic HLLD scheme to resolve all wave modes.
9. 近最优非凸矩阵补全
Near-Optimal Nonconvex Matrix Completion
作者: Jian-Feng Cai, Xiliang Lu, Juntao You
类别: math.NA, cs.LG
发布日期: 2026-09-15
arXiv ID: 2609.17048v1
链接: http://arxiv.org/abs/2609.17048v1
摘要: We study nonconvex methods for matrix completion, the problem of recovering a low-rank matrix from a subset of its entries. Convex methods achieve sample complexity linear in the matrix dimension and the rank, up to logarithmic factors, whereas global guarantees for commonly used nonconvex methods require a higher polynomial dependence on the rank. We close this gap by analyzing Riemannian gradient descent (RGD) and Riemannian Gauss–Newton (RGN) methods. For an $n\times n$ matrix of rank $r$ with incoherence parameter $μ$ and condition number $κ$, the two methods achieve exact recovery with high probability from $O(μnr\log n\log(nκ))$ and $O(μnr\log n\log(2μrκ))$ observations, respectively. The methods use a multiscale residual initialization, while the analysis simultaneously controls the spectral error and incoherence. The resulting RGD iterates converge linearly, whereas RGN eventually converges Q-quadratically.
10. 从基于共识的优化到粒子群优化：漂移-扩散耦合下的收敛性保证
From Consensus-Based Optimization to Particle Swarm Optimization: Convergence Guarantees under Drift-Diffusion Coupling
作者: Franca Hoffmann, Dohyeon Kim, Ritvik Teegavarapu
类别: math.OC, math.NA
发布日期: 2026-09-15
arXiv ID: 2609.17002v1
链接: http://arxiv.org/abs/2609.17002v1
摘要: Particle swarm optimization (PSO) is a widely used algorithm featured in many state-of-the-art optimization tool-kits. However, rigorous performance guarantees are still lacking. The standard PSO dynamics do not admit a natural mean-field description, which would provide an avenue for theoretical analysis. By modifying the PSO formulation, one can recover the consensus-based optimization (CBO) algorithm with memory, which admits a mean-field limit and facilitates rigorous convergence analysis. These theoretical guarantees rely heavily on the fact that for CBO, the drift and noise strengths can be chosen independently, whereas they are coupled for PSO. We analyze how the PSO parameter coupling affects existing convergence guarantees for CBO and its variant with memory effect. We show, by an explicit construction, that the coupling still leaves a non-empty set of admissible parameters for these convergence guarantees to hold. However, the admissible parameter ranges shrink in the limits used to recover PSO. The resulting convergence guarantees from CBO therefore do not directly extend to the classical PSO model. We provide numerical simulations illustrating the parameter tradeoffs shown in the theoretical analysis.
11. 准地转方程的非线性滤波稳定化
Nonlinear filtering stabilizations for the quasi-geostrophic equations
作者: Lander Besabe, Sachin Kumar, Annalisa Quaini
类别: math.NA, physics.comp-ph
发布日期: 2026-09-15
arXiv ID: 2609.16983v1
链接: http://arxiv.org/abs/2609.16983v1
摘要: Numerical simulations of ocean flows typically require fine computational meshes to resolve the Munk scale, leading to high computational costs. Filtering-based large eddy simulation (LES) provides a way to relax the mesh size requirement by modeling the effects of the unresolved scales. For the implementation of this strategy, we propose a three-step algorithm called Evolve-Filter-Relax (EFR) that requires (i) the solution of a QGE problem, (ii) a nonlinear Helmholtz filter for the potential vorticity field leveraging an indicator function, and (iii) a final relaxation step. We show that the EFR algorithm can be interpreted as a splitting scheme for a perturbed QGE problem with additional dissipation and provide a practical choice for the relaxation parameter. For comparison, we also investigate a nonlinear Bardina regularization of the QGE. Numerical results on a classical benchmark show that both the EFR approach and the nonlinear Bardina regularization significantly improve the accuracy and stability of coarse mesh simulations with no LES model. Additionally, the EFR method with a deconvolution-based indicator function delivered the best balance between accuracy, stability, and computational efficiency in a test case involving a more realistic geometry (Mediterranean Sea).
12. 基于SAV方法的能量稳定且保精度的有限体积格式及其在壁面距离计算中的应用
An energy stable and accuracy-preserving finite volume scheme based on the SAV method with application to wall-distance computation
作者: Xiaorui Xu, Qian Wang
类别: physics.comp-ph, math.NA
发布日期: 2026-09-15
arXiv ID: 2609.16885v1
链接: http://arxiv.org/abs/2609.16885v1
摘要: A novel semi-implicit second-order finite volume scheme integrating the scalar auxiliary variable (SAV) approach is proposed for solving the pseudo-time Eikonal equation in wall-distance computation. Unconditional energy stability under zero boundary conditions is rigorously proved, eliminating the dependence of the time step on the grid scale and enabling large-time-step computation. The scheme is a priori accuracy-preserving, and its discretization matrix forms an M-matrix, thereby guaranteeing strict non-negativity of the numerical solution inherently. The framework extends readily to any non-conservative scalar equation and, being independent of the specific finite-volume reconstruction, is compatible with schemes of arbitrary order of accuracy. A vanishing artificial viscosity is introduced to smooth the solution without compromising formal accuracy, and upwinding is incorporated through directional weighting in the weighted least-squares (WLS) reconstruction. Numerical experiments confirm that the scheme achieves the designed accuracy and permits stable computations with uniformly large time steps. For complex configurations such as a three-element airfoil and the three-dimensional ONERA M6 wing, accurate results are obtained on high-aspect-ratio grids, with relative errors in the computed wall distance below 3% relative to the search-based reference, except near geometric singularities.
13. 基于流形插值与子空间更新的在线自适应非侵入式模型降阶：在FSI收敛加速中的应用
Online adaptive non-intrusive model reduction via manifold interpolation and subspace updates: application to FSI convergence acceleration
作者: Azzeddine Tiba, Florian de Vuyst, Iraj Mortazavi
类别: cs.CE, math.NA
发布日期: 2026-09-15
arXiv ID: 2609.16876v1
链接: http://arxiv.org/abs/2609.16876v1
摘要: We introduce a novel online adaptive non-intrusive reduced-order modeling strategy for parameterized dynamical systems involving parameter and time-dependent reduced bases. The proposed framework is based on a unified Grassmann manifold formulation combining three key components: interpolation of local reduced subspaces for unseen parameters, geodesic online subspace updates driven by incoming high-fidelity snapshots, and a latent-space regression strategy relying on Grassmann-distance weighting and Procrustes alignment to consistently aggregate predictions from multiple local models. The adaptive reduced-order model is embedded in a partitioned fluid-structure interaction framework, where it predicts fluid interface forces to provide accurate initial guesses for the nonlinear coupling iterations, thus achieving computational speedups with no loss of accuracy. The reduced basis and the regression operators are adapted independently during the simulation and without requiring the storage of high-dimensional streaming data, preserving computational efficiency while substantially improving predictive capabilities. Numerical results on reference FSI test cases demonstrate superior accuracy with respect to static and global reduced-order models, leading to a significant reduction in the number of fixed-point iterations required for convergence. The proposed framework offers a flexible and fully non-intrusive approach for the efficient simulation of nonlinear parameter-dependent multiphysics problems.
14. 热方程全时 Runge–Kutta 离散的混合精度求解器
Mixed precision solvers for the all-at-once Runge–Kutta discretization of the heat equation
作者: Santolo Leveque, Luca Bergamaschi, Ángeles Martínez, Erin Carson
类别: math.NA
发布日期: 2026-09-15
arXiv ID: 2609.16863v1
链接: http://arxiv.org/abs/2609.16863v1
摘要: We study the effect of mixed precision on the numerical integration of the heat equation discretized with a Runge–Kutta method in time. A full space-time discretization is applied, which results in a very large and sparse linear system to be solved for the numerical approximations and the Runge–Kutta stages of all time steps. The linear system is solved by applying a suitable preconditioned iterative method that can be run in parallel. In order to speed up the solution process, the preconditioner is applied using a mixed precision framework. Sequential results show the robustness of the preconditioner, even when applied in mixed precision. Finally, we present numerical evidence of the improved performance of the mixed precision strategy when applied in a parallel environment, achieving up to a 50% reduction in CPU time.
15. 保持双线性的代数通量修正
Bilinearity Preserving Algebraic Flux Correction
作者: Sarthak Sourav Dash, Abhinav Jha
类别: math.NA
发布日期: 2026-09-15
arXiv ID: 2609.16830v1
链接: http://arxiv.org/abs/2609.16830v1
摘要: Algebraic flux correction schemes for convection–diffusion–reaction equations rely on linearity preservation to avoid unnecessary artificial diffusion: the limiter is inactive whenever the discrete solution is affine. On quadrilateral and hexahedral meshes, this is insufficient, since the finite element space also contains bilinear functions that are reproduced exactly on axis-aligned meshes. We show that a mesh patch admits bilinearity preservation if and only if the central node lies in the convex hull of its neighbours after lifting each neighbour by the product of its coordinate differences, and we derive a sharp explicit value for the limiter parameter on such patches when the cell edges are parallel to the coordinate axes. We further characterize the meshes on which a mapped bilinear finite element space reproduces a bilinear function exactly, thereby establishing a fundamental limitation on what any limiter can achieve on general quadrilaterals. Numerical experiments confirm the theory: on axis-aligned grids, the proposed limiter reproduces a bilinear solution to the accuracy of the nonlinear iteration, whereas the linearity-preserving limiter does not. A sensitivity study shows that the parameter required for bilinearity preservation lies close to the upper end of the range for which the nonlinear problem can be solved by the fixed-point iteration considered here.
16. 状态约束策略迭代的罚参数一致局部化
Penalty-Uniform Localization for State-Constrained Policy Iteration
作者: Yeongjong Kim, Jiwoong Jang, Yeoneung Kim
类别: math.OC, math.NA
发布日期: 2026-09-15
arXiv ID: 2609.16829v1
链接: http://arxiv.org/abs/2609.16829v1
摘要: Penalizing a state constraint creates a singular localization problem: whole-space values may grow like the inverse penalty parameter, while numerical diffusion allows even an inward feedback to cross the boundary. For deterministic discounted optimal control, we show how the penalty itself supplies confinement that offsets this growth. With mesh size $h$ and penalty parameter $\varepsilon$, an inward barrier bounds the additional cost of numerical leakage by $O(h/\varepsilon)$ for a monotone centered-difference scheme with vanishing viscosity. Under $h\le\varepsilon$, an occupation estimate then yields $O(h)$ localization error with a sufficient box margin logarithmic in $1/h$ and independent of $\varepsilon$. We extend the result to bounded squared-distance penalties and combine it with residual-based evaluation bounds and discounted policy-error propagation. Under coupled refinement with vanishing penalization and discretization errors, explicit conditions on evaluation errors and greedy gaps ensure convergence of the neural value approximations to the constrained value, allowing measurable, nonunique greedy selectors. Reference calculations isolate leakage and localization across mesh and penalty scales, and distinguish evaluation, iteration, and approximation errors. An explicit cylindrical state-constraint solution provides a benchmark in arbitrary dimension, tested up to dimension twenty. Paired obstacle-navigation experiments illustrate why policy-value accuracy must be assessed alongside sampled residuals when comparing raw-residual and finite-grid-assisted neural evaluation.
17. 一种适应学习到的多变量结构的加权核逼近方法
A Weighted Kernel Method for Approximation that Adapts to Learned Multivariable Structure
作者: John E. Darges, Laura Weidensager
类别: cs.LG, math.NA
发布日期: 2026-09-15
arXiv ID: 2609.16606v1
链接: http://arxiv.org/abs/2609.16606v1
摘要: Approximating the input-output behavior of a multivariable black-box function from limited data is challenging when blind to the importance of its inputs and their interactions. We introduce total sensitivity kernels (TSKs), a method based on families of weighted ANOVA kernels that learn and adapt to this multivariable structure. TSKs parameterize the weights on each multivariable component of the target function by factors for each input. We propose learning these factors directly from function evaluations by selecting the reproducing kernel Hilbert space (RKHS) in which the target function has minimum norm. Under suitable conditions, we show that this norm-minimization problem admits a unique solution, and we establish consistency of a finite-data formulation based on minimum-norm interpolation. The learned TSK factors characterize the participation of individual inputs across interactions and main effects, providing a kernel-dependent notion of input sensitivity related to total Sobol indices. Numerical experiments demonstrate that adapting the kernel to learned multivariable structure can substantially improve approximation accuracy over a standard product kernel.
18. 具有复合状态方程的可压缩 Euler 方程的保证波速界
Guaranteed wave-speed bounds for the compressible Euler equations with a composite equation of state
作者: Nicolas Favrie, Matthias Maier
类别: math.AP, math.NA, physics.flu-dyn
发布日期: 2026-09-13
arXiv ID: 2609.14287v2
链接: http://arxiv.org/abs/2609.14287v2
摘要: We derive computable upper bounds on the maximum wave speed in the Riemann problem for the compressible Euler equations with a composite equation of state, in which the pressure is the sum of a convex hydrodynamical part and a barotropic correction. The correction may destroy the convexity of the equation of state, so that the Riemann solution can contain composite shock-rarefaction waves. The bounds are obtained by comparison with the auxiliary Riemann problem for the hydrodynamical equation of state alone. The bounds hold irrespective of whether the outer waves are elementary or composite and require no case distinction on the wave type.
19. 结构驱动反演：求解反问题的新范式
Structure-Driven Inversion: A New Paradigm for Solving Inverse Problems
作者: Shengchang Chen
类别: math.NA, physics.geo-ph
发布日期: 2026-09-15
arXiv ID: 2609.16536v1
链接: http://arxiv.org/abs/2609.16536v1
摘要: Inverse problems are predominantly solved within the optimization-driven paradigm, which formulates the problem as objective-function minimization and approaches the solution by iterative search. Though universal, it suffers from limitations in efficiency, interpretability, and multi-parameter decoupling. This paper proposes a new paradigm—Structure-Driven Inversion (SDI). Instead of iterative search, SDI identifies and exploits the intrinsic structure of the problem to construct a solution method. It has two types of structure—mathematical structure and physical structure—and two corresponding driving inversion types: Mathematical Structure-Driven Inversion (MSDI) and Physical Structure-Driven Inversion (PSDI). The current representative method of MSDI is Mathematical Structure-Driven Pseudo-Inverse Inversion (MSDPII), which constructs a pseudo-inverse in the spectral domain via unitary diagonalization. The current representative methods of PSDI are Physical Structure-Driven Waveform Inversion Imaging (PSDWII), which performs inversion through virtual-source projection, and Physical Structure-Driven Back-Propagation (PSDBP), which implements three structural projections for deep neural networks via physical-system analogy. SDI is not a rejection but a complement and extension of Optimization-Driven Inversion (ODI). To the best of the author’s knowledge, no existing study has systematically presented structure-driven inversion as an independent paradigm; this paper aims to fill that gap.
20. 基于动理学缺陷测度的标量双曲守恒律结构信息引导数据驱动降阶建模
Structure-Informed Data-Driven Reduced-Order Modeling of Scalar Hyperbolic Conservation Laws via Kinetic Defect Measure
作者: Marissa Llamas, Jan Fuhg, Hannah Lu
类别: math.NA, physics.comp-ph
发布日期: 2026-09-15
arXiv ID: 2609.16525v1
链接: http://arxiv.org/abs/2609.16525v1
摘要: Reduced-order modeling of transport-dominated systems remains challenging because moving fronts and shocks are poorly represented by low-dimensional linear subspaces. We develop a structure-informed data-driven reduced-order model(ROM) for scalar hyperbolic conservation laws based on the kinetic defect formulation. This formulation separates the nonlinear dynamics into known characteristic transport and a kinetic entropy defect localized on the shock manifold. We exploit this structure by first removing the known transport from the solution snapshots. We then extract and register the remaining defect-driven dynamics in a shock-attached coordinate system. Separate ROMs are used to evolve the shock geometry and the registered defect-driven source. During prediction, the predicted shock geometry is used to inverse-register the learned defect-driven source, which advances the kinetic state and recovers the physical solution. Numerical examples in one and two spatial dimensions demonstrate accurate reconstruction and prediction of nonlinear transport with shocks, including evolution beyond the training interval, while accurately capturing the mass and entropy-dissipation behavior of the reference solution.
21. 变系数双调和问题的无矩阵增广高阶紧致求解器
A Matrix-free Augmented High Order Compact Solver for Variable-Coefficient Biharmonic Problems
作者: Jin Li, Kejia Pan, Xu Qian, Li-Lian Wang
类别: math.NA
发布日期: 2026-09-15
arXiv ID: 2609.16478v1
链接: http://arxiv.org/abs/2609.16478v1
摘要: We propose an augmented high-order compact finite difference method for biharmonic equations with clamped boundary conditions and variable coefficients. Standard mixed-type formulations introduce an auxiliary variable, but its boundary values are unavailable, leaving the resulting discrete systems globally coupled and difficult to solve at large scales. Our key contribution is the development of a new augmented formulation that treats these unavailable boundary values as additional unknowns, reduces the global coupling to a lower-dimensional Schur complement system, and yields decoupled second-order subproblems. The Schur complement is solved by matrix-free GMRES, while the subproblems are handled by FFT-based fast solvers. The method achieves fourth-order accuracy using compact stencils, and has $O(n\log n)$ computational complexity, enabling the solution of the biharmonic equation with $1024^3$ degrees of freedom within several minutes. To the best of our knowledge, this level of computational efficiency has not previously been achieved in either the literature or practice. Using energy estimates and Fourier analysis, we derive a new $L^2$-estimate for Poisson equations with inexact Dirichlet boundary and then prove the convergence of the proposed scheme. We provide ample numerical experiments to confirm the accuracy, efficiency, and further apply the fast and accurate solver to triharmonic equations, high-wavenumber problems, Stokes flow, and plate bending problems.
22. 超越相位约化：耦合振子最优控制中的振幅坍缩
Beyond Phase Reduction: Amplitude Collapse in Optimal Control of Coupled Oscillators
作者: Faranak Rajabi, Frédéric Gibou, Jeff Moehlis
类别: math.OC, eess.SY, math.NA
发布日期: 2026-09-15
arXiv ID: 2609.16449v1
链接: http://arxiv.org/abs/2609.16449v1
摘要: We solve the four-dimensional Hamilton-Jacobi-Bellman (HJB) equation for two diffusively coupled Stuart-Landau-like oscillators to obtain full-state optimal feedback control. A sweep over coupling strength reveals a sharp change in the numerically optimal strategy: below a threshold coupling value, the controller steers the phase difference toward anti-phase while keeping both oscillators near the limit cycle, as reduced-order models would suggest. Above this threshold, the HJB solution changes qualitatively; the controller transiently collapses one oscillator’s amplitude to near zero, thereby enabling large phase repositioning near the origin before rebuilding its amplitude. Direct gradient-based and stochastic optimization do not recover this lower-cost collapse trajectory from the initializations considered, suggesting that it occupies a region of the control landscape that is difficult to access by direct search. A joint sweep over nonisochronicity and coupling shows that collapse can occur even for an isochronous oscillator: phase repositioning near the origin can favor an off-cycle strategy. Nonisochronicity provides an additional energetic benefit through a phase-velocity surplus at small amplitude, quantitatively accounting for the observed reduction in control cost. Comparisons with uncoupled and coupled phase-reduced baselines show that phase models become increasingly inaccurate and cost significantly more energy for strong coupling. Results for coupled Van der Pol oscillators further demonstrate that exploitation of off-cycle dynamics is not specific to the Stuart-Landau-like oscillators.
23. 带 BGK 碰撞的 Vlasov-Poisson 方程的特征映射方法
Characteristic Mapping Method for Vlasov-Poisson with BGK-collisions
作者: Xi-Yuan Yin, Philipp Krah, Zetao Lin, Jean-Christophe Nave, Kai Schneider
类别: physics.plasm-ph, math.NA, physics.comp-ph
发布日期: 2026-09-14
arXiv ID: 2609.16328v1
链接: http://arxiv.org/abs/2609.16328v1
摘要: This work presents the first steps for simulating kinetic plasmas with collisions using the characteristic mapping method (CMM). The CMM is a semi-Lagrangian method that explores a semi-group structure to store diffeomorphic flow maps efficiently. Using the semi-group structure, individual submaps can be composed to relate the flow backward in time to its initial food point. The novelty of the presented work is handling the source term by storing sub-integrals that correspond to the individual sub-maps and allow efficient integration. Furthermore, we use the Lagrangian structure to avoid implicit time integration in the hydrodynamic regime, which is known to be stiff. We benchmark our method on the Boltzmann-BGK and Vlasov-Poisson-BGK equations and consider different test cases, the Sod shock tube problem and nonlinear Landau damping for different Knudsen numbers. We show third-order spatial and temporal convergence and illustrate the fine-scale zoom property of CMM for the bump-on-tail instability.
24. 具有驾驶员依赖加速度的Aw-Rascle-Zhang型模型的微观到宏观推导与稳定性分析
Micro-to-macro derivation and stability analysis of an Aw-Rascle-Zhang-type model with driver-dependent acceleration
作者: Stephan Gerster, Giuseppe Visconti
类别: math.NA
发布日期: 2026-09-14
arXiv ID: 2609.15934v1
链接: http://arxiv.org/abs/2609.15934v1
摘要: We introduce a new traffic flow model. The main idea is to describe acceleration effects in the Aw-Rascle-Zhang model through an additional driver-dependent equation, thereby allowing individual driver characteristics to evolve dynamically rather than being prescribed solely by the traffic density. The resulting system provides a hyperbolic description of traffic dynamics and admits an explicit analysis of its characteristic structure, Riemann problem, and entropy properties. To investigate the stability of the proposed model, we perform a Chapman-Enskog expansion up to second order and derive the corresponding diffusive and higher-order corrections. This analysis reveals how the stability properties depend on the underlying velocity and pressure functions. Several examples are presented to illustrate the theoretical results.
25. 用于波浪能的短行程海水泵式动力输出装置的分块模块化灰箱辨识：损耗分解与多工况验证
Block-modular grey-box identification of a short-stroke seawater-pump power take-off for wave energy: loss decomposition and multi-regime validation
作者: Mahdi Tayyebati, Amir Hamza Siddiqui, Mian Masoud, Kristian Glejbøl, Christian Berggreen
类别: cs.CE, math.NA, physics.app-ph
发布日期: 2026-09-14
arXiv ID: 2609.15835v1
链接: http://arxiv.org/abs/2609.15835v1
摘要: Physically interpretable power take-off models support wave-energy development at every stage. For the seawater-pump class, few have been identified on a real machine and validated at the system level. This paper presents a block-modular grey-box model of a short-stroke seawater pump, identified from bench data alone. The pump decomposes into chamber, relief valve, piston seal and tip check valve, each structured by component physics, and the few parameters are identified block by block from terminal pressure, displacement, force and discharged-mass measurements, with a closed-cycle mass balance that decouples the degenerate seal and valve losses and closes to within 1% of the inflow. Over the sinusoidal campaign, the pressure-energised piston seal carries 84% of the lost mass and 97% of the lost energy, and the pump delivers about four-fifths of its input as high-pressure flow. Generalisation is demonstrated rather than assumed. With parameters fixed on ramps and sinusoids, the model reproduces 360 s sea-state records in free run to about 10% of range, captures 80% of several thousand peaks within 5 bar, transfers to a re-sprung valve on cracking strokes once its crack pressure is identified, follows thousand-cycle endurance records that drift below 0.1 bar, and reproduces the force and cycle-mean absorbed power, 0.1 to 5.7 kW, to a 4% median error. A practical-identifiability and uncertainty analysis confirms the terminal data resolve the retained parameters; thirty-seven repeated records set the experimental floor. The result is a compact, physically labelled model for design, energy assessment and model-based control of seawater-pump power take-offs.
26. 中心有限差分逼近的超收敛性
Superconvergence of Centered Finite Difference Approximations
作者: Mario J. Bencomo, Joseph Igot, Emma Maltes
类别: math.NA
发布日期: 2026-09-14
arXiv ID: 2609.15831v1
链接: http://arxiv.org/abs/2609.15831v1
摘要: The finite difference (FD) method is commonly used to approximate derivatives of smooth functions, with accuracy typically determined by stencil size and derivative order. However, certain centered stencils exhibit unexpectedly higher accuracy, a phenomenon known as superconvergence, which has been observed in practice but lacks rigorous explanation. We present a mathematical framework for superconvergence in centered FD approximations based on Taylor expansions of the truncation error and the resulting linear system for the FD coefficients. By analyzing symmetry properties of these coefficients and their interaction with the parity of the derivative order, we identify conditions under which higher-order error terms cancel. We show that superconvergence occurs for odd-order derivatives with even centered stencils and for even-order derivatives with odd centered stencils, while no superconvergence occurs for even derivatives with even centered stencils. Numerical experiments in MATLAB confirm the predicted convergence rates.
27. 球面上的嵌套 QMC 设计
Nested QMC designs on spheres
作者: Hao-Ning Wu, Xiaosheng Zhuang
类别: math.NA, math.CA, math.ST
发布日期: 2026-09-14
arXiv ID: 2609.15767v1
链接: http://arxiv.org/abs/2609.15767v1
摘要: Nested cubature rules, in which each refinement retains all previously used nodes and thus reuses earlier function evaluations, are natural in multilevel and adaptive integration. For every fixed $s>d/2$, we show that the equal-weight QMC integration rate on $\mathbb S^d$ is compatible with such nested point sets. In the subcritical range $d/2
28. 一种用于时变区域的非拟合网格 φ-FEM 方法
A φ-FEM approach for time-dependent domains with unfitted meshes
作者: Michel Duprez, Johan Marguet, Alexei Lozinski, Igor Voulis
类别: math.NA
发布日期: 2026-09-14
arXiv ID: 2609.15630v1
链接: http://arxiv.org/abs/2609.15630v1
摘要: In this work, we propose an unfitted finite element scheme to approximate the solution of the heat equation on moving domains. We use the φ-FEM paradigm in which the computational domain is described implicitly by a level-set function φ. This function is incorporated into the variational formulation in order to enforce the boundary conditions, which allows the use of unfitted meshes in space. Such a strategy avoids the need for remeshing at each time step and makes it possible to handle complex geometrical evolutions. Moreover, the φ-FEM approach has the advantage of being simple to implement within standard finite element libraries. We introduce a fully discrete scheme that combines the φ-FEM spatial discretization with the lowest-order discontinuous Galerkin method in time. Under regularity assumptions on the level-set function and a mild restriction linking the time step to the mesh size, we establish an optimal a priori error estimate in the L2(0,T;H1) norm. Finally, we present several numerical experiments that confirm this convergence rate, exhibit a second-order convergence in the L{\infty}(0,T;L2) norm, and illustrate the robustness and accuracy of the proposed method.
29. 共享内存与分布式内存中的异步 Jacobi 与随机 Gauss–Seidel 方法：统一的收敛速率分析
Asynchronous Jacobi and randomized Gauss–Seidel methods in shared and distributed memory: A unified convergence-rate analysis
作者: Erin Carson, Yuxin Ma
类别: math.NA
发布日期: 2026-09-14
arXiv ID: 2609.15605v1
链接: http://arxiv.org/abs/2609.15605v1
摘要: Asynchronous iterative methods are attractive for large-scale parallel computing because they reduce synchronization and communication overhead. Existing convergence-rate analyses, however, have primarily focused on shared memory implementations, whereas distributed memory systems introduce more general and potentially inconsistent communication delays. In this work, we revisit asynchronous Jacobi and randomized Gauss–Seidel (RGS) methods for symmetric positive definite linear systems from a unified perspective. We first introduce a general asynchronous model that encompasses both shared memory and distributed memory settings and expresses the two methods through a common coordinate-update framework. We then establish linear convergence in expectation under an explicit stability condition. The resulting convergence bound depends algebraically on the delay through the quantity $\sqrt{ρτ}+ρτ$, where $τ$ is the maximum communication delay and $ρ$ reflects the communication pattern of the underlying parallel implementation. In particular, under an appropriate scaling regime with $ρτ=O(1)$, the guaranteed per-iteration convergence rate has the same asymptotic order as that of synchronous RGS. These results provide a unified framework for quantifying the effect of asynchronicity on asynchronous Jacobi/RGS methods in both shared and distributed memory environments.
30. 快速离散三角变换在参数化数值积分中的效果如何？
How well do fast discrete trigonometric transforms work for parametric numerical integration?
作者: Gerlind Plonka, Daniel Potts, Manfred Tasche
类别: math.NA
发布日期: 2026-09-14
arXiv ID: 2609.15577v1
链接: http://arxiv.org/abs/2609.15577v1
摘要: The fast Fourier transform (FFT) and its real counterparts, the fast algorithms of the discrete cosine transform (DCT) and the discrete sine transform (DST), are frequently employed for the numerical evaluation of the Fourier cosine and Fourier sine transform, thereby acting as parametric quadrature rules based on finitely many samples of a given continuous function. But how accurate are the obtained results? In this paper we study the error occurring if the DCT of type I (DCT-I) and the DST of type I (DST-I) are applied to compute $\int_0^{\infty} f(x)\cos(2πxv)\d x$ and $\int_0^{\infty} f(x)\sin(2πxv)\d x$ for $f \in L^1(\Rp) \cap C(\Rp)$ and $v \in \Rp$, as well as the error occurring if the Chebyshev coefficients $\frac{2}π\int_0^π h(\cosθ)\,\cos(nθ)\dθ$, $n \in \Np$, of a function $h \in C(I)$ on $I := [-1,1]$ are computed by the DCT-I. We present practicable, explicit error estimates under polynomial, exponential, and mixed decay conditions. For the Fourier cosine and sine transform, rational functions lead to a geometric decay in the frequency parameter $P$ combined with an algebraic plateau of order $L^{-3/2}$ in the sampling parameter $L$, while for meromorphic functions such as $\operatorname{sech}(π\,\cdot)$ the critical geometric rate is attained in both parameters. For Chebyshev approximation of a rational function with simple poles in $\C \setminus I$, the quadrature error has the critical geometric rate $ρ_0^{-P}$ with an explicit constant, where $ρ_0 > 1$ is the parameter of the largest Bernstein ellipse of analyticity. Several numerical examples illustrate the theory.
31. 一致范数下采样宽度的尖锐界：核 $D$-最优设计与过采样
A sharp bound for sampling widths in the uniform norm: kernel $D$-optimal designs and oversampling
作者: Sebastian Neumayer, Tino Ullrich
类别: math.FA, math.NA
发布日期: 2026-09-14
arXiv ID: 2609.15556v1
链接: http://arxiv.org/abs/2609.15556v1
摘要: For every reproducing kernel Hilbert space $\mathcal{H}_K$ with bounded kernel $K$, we prove that the linear sampling widths $g_m^{\text{lin}}$ and Gelfand widths $c_n$ in the uniform norm satisfy
$$ g_m^{\text{lin}}(B_{\mathcal H_K})_\infty \leq \frac{m+1}{m-n+1}\, c_n(B_{\mathcal H_K})_\infty\quad , \quad m\ge n. $$In a certain sense, the leading factor $(m+1)/(m-n+1)$ on the right-hand side is sharp. We transfer the problem into the selection of a maximal volume subset of kernel translates (kernel $D$-optimal design). Apart from basic linear algebra, our proof employs low rank approximation techniques and a version of the Eckart-Young-Mirsky theorem. The same argument gives corresponding bounds for the selection of a reduced basis in Hilbert spaces.
32. Mather 测度与 Mather 集的全离散变分逼近
A Fully Discrete Variational Approximation of Mather Measures and Sets
作者: Fabio Camilli, Cristian Mendico
类别: math.DS, math.AP, math.NA, math.OC
发布日期: 2026-09-14
arXiv ID: 2609.15281v1
链接: http://arxiv.org/abs/2609.15281v1
摘要: We introduce a fully–discrete variational approximation of Mather measures and sets for Tonelli Lagrangians on the flat torus, together with a numerical procedure for approximating the entire Mather set. The scheme is based on a fully–discrete Lax–Oleinik operator with integer winding labels. We prove an $O(τ+h/τ)$ error estimate for the critical value, convergence of critical solutions, and a finite-dimensional characterization of fully–discrete Mather measures. Accumulation points of the reconstructed minimizing measures are continuous Mather measures, while the supports satisfy complementary upper and lower convergence results involving the Mañé and Mather sets. To avoid the selection of only some minimizing components by exact discrete minimizers, we introduce a mass-threshold approximation based on almost-minimizing holonomic measures. This yields a finite-dimensional constrained optimization procedure designed to recover the whole Mather set.
33. 心肌中胶原支架不可逆损伤的数学模型
A mathematical model for irreversible damage of the collagen scaffold in the myocardium
作者: Irena Radišić, Francesco Regazzoni, Luca Dede’, Alfio Quarteroni
类别: math.NA, math-ph
发布日期: 2026-09-14
arXiv ID: 2609.15243v1
链接: http://arxiv.org/abs/2609.15243v1
摘要: We propose a dissipative, irreversible damage model for anisotropic media in large deformations to address the damage process of the myocardium following a cardiac infarction. We model damage to the collagen in the cleavage planes resulting from an increased load to the passive tissue. Starting from variational principles, we derive a quasi-static differential model governing the irreversible evolution of the damage. We apply our model to two test cases. First, a rectangular passive slab geometry to which an indenter-like load is applied. Then, we couple our model with a model for the numerical simulation of left ventricular electromechanics. Our results show that a redistribution of the load within a left ventricle occurring following a cardiac infarction can produce the damage to the collagen scaffold in the infarcted area consistent with experimental results. This is indicative of the fact that passive load-dependent dissipative phenomena are implicated in post-infarction remodelling.
34. Reynolds 流与 Stokes 流中空化的保结构增广拉格朗日有限元方法
Structure-Preserving Augmented Lagrangian Finite Element Methods for Cavitation in Reynolds and Stokes Flows
作者: Peter Hansbo, Mats G. Larson
类别: math.NA
发布日期: 2026-09-14
arXiv ID: 2609.15223v1
链接: http://arxiv.org/abs/2609.15223v1
摘要: In this paper we propose augmented Lagrangian finite element methods for cavitation in the Reynolds and Stokes models of lubrication, for which the discrete complementarity conditions hold exactly and the augmentation parameter drops out of the method. For the Reynolds equation this follows from nodal quadrature in a mixed piecewise linear method, for which we prove the classical first-order error estimate. We also determine the computable stability threshold of a multiplier-free stabilised alternative. For Stokes flow we show that the constrained scalar is the mechanical pressure when the deviatoric stress, with zero bulk viscosity, is used, but not with the customary incompressible stress. We discretise with a jump-stabilised Crouzeix-Raviart element and piecewise constant pressure, and prove well-posedness, stability in two and three dimensions, and a first-order error estimate. We present numerical examples which verify these properties and show that close pressure profiles in the two models do not imply close cavity predictions, the latter also being sensitive to the end conditions of the computational domain.
35. 采用非拟合有限元的多相水平集拓扑优化的形状微积分与自动微分
Shape calculus and automatic differentiation for multi-phase level-set topology optimisation with unfitted finite elements
作者: Zachary J. Wegert, Martin Berggren, Vivien J. Challis
类别: math.NA, math.OC
发布日期: 2026-09-14
arXiv ID: 2609.15084v1
链接: http://arxiv.org/abs/2609.15084v1
摘要: We present shape calculus techniques and a scalable automatic shape differentiation framework for multi-phase topology optimisation on unfitted discretisations defined by several level-set functions. First we establish general, exact shape calculus expressions in the discrete case for multi-phase systems by leveraging concepts from convex geometry. To complement this theoretical foundation, we introduce an open-source multi-phase automatic shape differentiation framework based on polytopal cutting. This computational framework is validated against both finite differences and our established exact expressions, matching the latter to near machine precision. Furthermore, the proposed automatic shape differentiation is scalable across distributed computing environments, demonstrating near ideal weak scaling up to 1.65 billion finite elements across 13,824 computer cores. We demonstrate our implementation by solving unfitted multi-phase topology optimisation problems for anisotropic diffusion, linear elasticity, and fluid–structure interaction. Together, these theoretical and computational contributions provide a robust and accessible foundation for advancing multi-phase topology optimisation using unfitted finite element methods. In particular, the methods enable the solution of topology optimisation problems involving multi-phase and multi-physics systems with non-trivial boundary conditions. The open-source software is available at https://github.com/zjwegert/GridapTopOpt.jl.
36. 全度量变形网格上的高效无矩阵高斯-牛顿走时层析成像
efficient matrix-free gauss-newton traveltime tomography on full-metric deformed grids
作者: Zuwei Huang, Pingchuan Ma, Peng Yu, Takao Koyama, Luolei Zhang, Chongjin Zhao, Mingxin Chu
类别: math.NA
发布日期: 2026-09-14
arXiv ID: 2609.14982v1
链接: http://arxiv.org/abs/2609.14982v1
摘要: First-arrival traveltime tomography provides a computationally efficient means of imaging subsurface velocity structure, while boundary-conforming deformed grids allow rugged topography to be represented without abandoning a logically structured mesh. In matrix-free Gauss-Newton inversion, however, the linearized full-metric Eikonal transport and its transpose must be applied repeatedly at a fixed background model. Directional sweeping consequently revisits the same state-dependent dependency structure for every Krylov right-hand side. We recast the frozen tangent transport as a directed graph and distinguish loss of the background-traveltime ordering from genuine algebraic cyclicity. Strongly connected components identify the irreducible part of the transport, whereas the remaining dependencies admit exact scalar substitution after reordering. The condensation graph and local block factorizations are constructed once for each frozen source state and then reused for both primal and transpose applications. Numerical experiments on deformed grids and a three-dimensional tomography problem show that the resulting block-triangular formulation preserves the frozen sensitivity action while substantially reducing the repeated transport cost. By eliminating this dominant inner-iteration cost, the proposed method markedly accelerates Gauss-Newton computation and makes full-metric matrix-free inversion on deformed grids practical at much lower cost.
37. 学习求解具有未知漂移和运行奖励的随机控制：理论、算法与收敛性
Learning to Solve Stochastic Controls with Unknown Drifts and Running Rewards: Theory, Algorithms and Convergence
作者: Jin Ma, Gaozhan Wang, Jianfeng Zhang, Xunyu Zhou
类别: cs.LG, math.NA, math.OC
发布日期: 2026-09-14
arXiv ID: 2609.14972v1
链接: http://arxiv.org/abs/2609.14972v1
摘要: We study continuous-time and possibly high-dimensional stochastic control problems where drift coefficients and running reward functions are unknown. Due to these missing model primitives, we take the exploratory, reinforcement learning (RL) framework of Wang, Zariphopoulou, and Zhou(2020) with relaxed controls and entropy regularization. The objective is to develop theoretically grounded, efficient and scalable RL algorithms to learn both the optimal value functions (which also solve the exploratory HJB equation) and optimal exploratory feedback control policies. When the diffusion coefficients do not contain control, we employ probabilistic representations of both the optimal value function and its gradient based on an auxiliary state process depending only on the diffusion part of the original dynamics. With a delicate analysis on some properly defined mappings and their fixed points, this leads to the introduction of our policy iteration algorithms and their convergence. We demonstrate the performance of our algorithms through various numerical examples. Finally, we study a special control-dependent diffusion case where probability representation of the Hessian is called for.
38. 洛伦兹镜面模型中的边界透射与偏移几何：一项数值研究
Boundary transmission and excursion geometry in the Lorentz mirror model: a numerical study
作者: Jian Gu, Qin Hao
类别: math-ph, math.NA
发布日期: 2026-09-14
arXiv ID: 2609.14949v1
链接: http://arxiv.org/abs/2609.14949v1
摘要: We study boundary transmission and excursion geometry in the planar Lorentz mirror model through finite-volume numerical experiments. Cylinder penetration quantiles grow approximately linearly with circumference over the simulated range, with a systematic upward drift in the normalized quantiles when vacancies are present. Estimated slab-transmission probabilities yield exponential confinement bounds with quantified statistical confidence at the tested widths. Planar excursions conditioned to reach a distant boundary exhibit transverse ranges proportional to the exit radius, while their repeated-visit fractions vary strongly with mirror density. Endpoint-label collision probabilities substantially exceed a general gluing lower bound, although they account for only part of the observed same-column crossing probability. Finally, rerouting at repeated vertices preserves the exit while substantially shortening the sampled paths. These findings characterize finite-volume transport and excursion geometry without determining an asymptotic scaling exponent or resolving infinite-plane localization.
39. 矩阵接近问题：实输入是否容许实解？
Matrix nearness problems: Do real inputs admit real solutions?
作者: Vanni Noferini, Matvei Zhukov
类别: math.NA, math.OC
发布日期: 2026-09-13
arXiv ID: 2609.14828v1
链接: http://arxiv.org/abs/2609.14828v1
摘要: Given a real square matrix $A$ and a nonempty closed target set of complex matrices $\mathcal E$, invariant by complex conjugation and containing real matrices, does the subset of $\mathcal{E}$ consisting of the matrices nearest to $A$ in the Frobenius distance always contain a real matrix? We give negative answers when $\mathcal{E}$ is either the set of normal matrices (solving a question posed by N. Higham) or the set of matrices whose eigenvalues lie in the closed left half-plane (solving a question posed by the first author and F. Poloni). For the latter problem, we also argue that the ratio between the real and complex distances is unbounded in every dimension $n\geq 3$, and this holds for every distance induced by a unitarily invariant norm. For both problems and $n \geq 3$, we show that a real input $A$ uniformly drawn from the unit sphere $\|A\|_F=1$ has no real minimizer over $\mathcal{E}$ with probability strictly between $0$ and $1$. The paper is complemented by some further results that are valid for more general target sets $\mathcal{E}$.
40. 通过在线空间滤波学习流体流动的长期稳定算子推断降阶模型
Learning Long-Term Stable Operator Inference Reduced-Order Models of Fluid Flows through Online Spatial Filtering
作者: Ian Moore, Ping-Hsuan Tsai, Anthony Gruber, Ionuţ Farcaş, Christopher Wentland, Irina Tezaur, Traian Iliescu
类别: math.NA, physics.flu-dyn
发布日期: 2026-09-13
arXiv ID: 2609.14812v1
链接: http://arxiv.org/abs/2609.14812v1
摘要: This paper introduces an online evolve–filter–relax (EFR) strategy for long-term stability of Operator Inference (OpInf) reduced-order models (ROMs) of complex fluid flow simulations. The main novelty of the new EFR-OpInf strategy is the use of online (i.e., at the learned ROM online evaluation level) spatial filtering inspired from large eddy simulation to significantly improve long-term stability and predictive performance of standard OpInf. Furthermore, the EFR-OpInf strategy reduces, and in some cases even eliminates, the need for standard $L^2$ regularization, while providing physical interpretability for the OpInf hyperparameters. The EFR-OpInf framework is modular, readily integrated into existing OpInf workflows, and accommodates a user-selected ROM filtering strategy. We demonstrate the EFR-OpInf’s effectiveness using a fully non-intrusive projection-based ROM filter, a ROM differential filter, and a hybrid projection-differential ROM filter. The new EFR-OpInf models are evaluated on a high-Péclet-number convection–diffusion–reaction problem that embeds the variation in one parameter and two unsteady Navier–Stokes problems that focus on predictions beyond a training horizon: a transitional two-dimensional flow past a cylinder and a three-dimensional turbulent minimal channel flow. Across these three scenarios, EFR-OpInf can reduce prediction errors by up to an order of magnitude relative to standard OpInf. Moreover, EFR-OpInf remains stable over long prediction horizons in cases where standard OpInf diverges. Depending on the ROM filter used, EFR-OpInf’s computational cost is comparable to that of standard OpInf.
41. 计算机辅助的正多边形关于扭转刚度的局部极大性
Computer-assisted local maximality of regular polygons for torsional rigidity
作者: Beniamin Bogosel
类别: math.NA, math.OC
发布日期: 2026-09-13
arXiv ID: 2609.14785v1
链接: http://arxiv.org/abs/2609.14785v1
摘要: We study torsional rigidity as a function of the labeled vertices of a convex polygon. Starting from the distributed second shape derivative, we derive the Hessian with respect to vertex coordinates. At a regular polygon, dihedral symmetry makes this matrix block circulant in radial-tangential coordinates, reducing its spectrum to the eigenvalues of Hermitian matrices of order two. We also derive an exact second-variation Galerkin identity and guaranteed functional residual majorants. Finite elements approximate the PDE solutions entering the Hessian, and FLINT/Arb provides the interval arithmetic needed for certification. In the scale-invariant setting, we certify exactly four zero eigenvalues generated by similarities and $2n-4$ strictly negative eigenvalues for $5\leq n\leq25$. The regular polygons in this range are therefore strict local maximizers, modulo similarities, of torsional rigidity divided by area squared. The observed Hessian error decreases nearly quadratically with the mesh size; the transmission regularity needed to prove this rate is stated separately as a conjecture.
42. 分数阶边值问题的随机 Galerkin 方法：收敛性分析与数值处理
Stochastic Galerkin Method for Fractional Boundary Value Problems: Convergence Analysis and Numerical Treatment
作者: Abhishek Kumar Singh, Dhyan Divakar Laad, Vaibhav Mehandiratta, Roland Pulch
类别: math.NA
发布日期: 2026-09-13
arXiv ID: 2609.14669v1
链接: http://arxiv.org/abs/2609.14669v1
摘要: We study two-point fractional boundary value problems with uncertain input data, where randomness may enter through the coefficients and boundary conditions. To quantify the resulting uncertainty in the solution, we employ the generalized polynomial chaos (gPC) framework and develop a stochastic Galerkin formulation of the problem. A particular focus of this work is the convergence analysis of the resulting approximation. Rather than imposing assumptions directly on the stochastic coefficients appearing in the gPC representation, we introduce minimal regularity assumptions on the input data and use them to establish the properties required for the convergence analysis. Based on these results, we prove the convergence of the stochastic Galerkin approximation to the corresponding gPC solution. Numerical experiments are presented to illustrate the theoretical findings and to investigate the influence of random coefficients and boundary conditions on the statistical behavior of the solution.
43. t-乘积下张量最小二乘问题的带重球动量的随机平均块坐标下降法
Randomized average block coordinate descent method with heavy-ball momentum for tensor least squares problem under the t-product
作者: Li-Lin Ji, Ni-Hong Ke, Jun-Feng Yin
类别: math.NA
发布日期: 2026-09-13
arXiv ID: 2609.14516v1
链接: http://arxiv.org/abs/2609.14516v1
摘要: A tensor randomized average block coordinate descent method with heavy-ball momentum is proposed for the tensor least squares problem with respect to the t-product. Theoretical analysis for tensor block coordinate descent method is established and average techniques are applied to avoid the computation of tensor Moore–Penrose inverse. To further accelerate convergence, a heavy-ball momentum scheme is incorporated into the block coordinate descent method, where the step size and momentum parameters are adaptively determined by a two-dimensional minimal residual projection. Theoretical analyses give the convergence of the new method and provide an improved bound on the linear convergence rate. Numerical experiments further verify the efficiency of the proposed method in terms of the number of iterations and the better video recovery performance.
44. 区域稳定性与新的特征值扰动界
Regional Stability and New Eigenvalue Perturbation Bounds
作者: Phuc Tran, Van Vu
类别: math.NA, math.PR
发布日期: 2026-09-13
arXiv ID: 2609.14303v1
链接: http://arxiv.org/abs/2609.14303v1
摘要: Let $A$ be an $n\times n$ symmetric matrix with eigenvalues $λ_1\geq\cdots\geqλ_n$. Let $\tilde A:=A+E$, where $E$ is a symmetric noise matrix, and denote the eigenvalues of $\tilde A$ by $\tildeλ_1\geq\cdots\geq\tildeλ_n$. Bounding the perturbation $|\tilde λ_i -λ_i| $ is a central problem in linear algebra and numerical analysis. In this paper, we prove new perturbation results by exploring the actual interaction between $E$ and the eigenvectors of $A$. In the setting where $E$ does not act adversarially with respect to these vectors (for instance, if $E$ is random), we obtain a considerable improvement over Weyl’s inequality. One can routinely extend these results to the Hermitian and rectangular settings. We obtain the new bounds as corollaries of a regional stability result, which provides a sufficient condition for a region on the real line to be stable (containing the same number of eigenvalues) after the perturbation. This result is of independent interest. We prove our regional stability result using contour integral analysis. Our main new technical ingredient here is the \textit{double-jump} argument, which is robust and could be useful in many other situations involving Neumann series.
45. 受控 B-级数、受控 Runge-Kutta 方法与受控粗糙微分方程
Controlled B-series, controlled Runge-Kutta methods and controlled rough differential equations
作者: Xinyuan An, Xingya Fan, Xing Gao
类别: math.NA
发布日期: 2026-09-13
arXiv ID: 2609.14196v1
链接: http://arxiv.org/abs/2609.14196v1
摘要: We develop controlled B-series and controlled Runge-Kutta methods as extensions of classical B-series and Runge-Kutta theory to controlled-driven rough differential equations. The driving signal is a path controlled by an underlying $α$-Hölder step-$2$ rough path, with $α\in(1/3,1/2]$, as arises when the output of one rough system drives another. Controlled B-series retain the classical rooted-tree and elementary-differential structure, while incorporating driver-dependent coefficients defined recursively through controlled rough integration. We establish finite-order expansions of the exact solution and its controlled Runge-Kutta approximation, and derive corresponding tree-based order conditions. Under suitable smoothness, solvability, stability, and boundedness assumptions, matching through tree order $p$ yields local error $O(|t-s|^{(p+1)α})$ and global error $O(h^{(p+1)α-1})$, provided $(p+1)α>1$. The standard rough differential equation formulation is recovered when the controlled driver is the reference path itself, and the classical time-driven case recovers the usual B-series and Runge-Kutta methods. We further construct a simplified method using only increments of the controlled driver. Segmentwise canonical lifting of its piecewise linear interpolation gives explicit tree coefficients and reduces the third-order conditions to the classical ones. Assuming a Wong-Zakai solution-approximation rate $r_0>0$, the simplified method has global error $O(h^{\min\{r_0,(p+1)α-1\}})$. Numerical experiments with fractional Brownian controlled drivers illustrate the resulting convergence behaviour.