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1. 基于神经网络的多尺度可混合间断伽辽金方法求解多孔介质中的偏微分方程
A Neural-network-based multiscale Hybridizable Discontinuous Galerkin method for solving PDEs in porous media
作者: Tony Haines, Ke Shi
类别: math.NA
发布日期: 2026-08-26
arXiv ID: 2608.25850v1
摘要翻译: 我们开发了一种神经网络加速的多尺度可混合间断伽辽金方法，用于求解具有非均匀系数的椭圆问题。该方法保留了标准MsHDG的局部到全局结构：粗块上的细尺度HDG问题定义了离散的Dirichlet-to-Neumann算子，这些算子通过标准MsHDG全局骨架方程进行组装。为了降低构造这些局部算子的成本，我们在参考块上的系数场上训练神经网络，并使用预测的算子代替重复的细尺度局部求解。
Comment: 无
链接: http://arxiv.org/abs/2608.25850v1
1. 研究对象和背景
研究对象是描述多孔介质中流动的达西方程。设 $\Omega\subset\mathbb{R}^d$ ($d=2,3$) 为有界多边形或多面体区域, 控制方程为 $-\nabla\cdot(\kappa(x)\nabla u)=f(x)$ ($x\in\Omega$), 边界条件 $u=0$ ($x\in\partial\Omega$), 其中 $\kappa(x)\geq\kappa_0>0$ 为高度异质的渗透率系数。在混合形式下, 引入通量未知量 $\boldsymbol{q}=-\kappa\nabla u$ 或等价地 $\alpha\boldsymbol{q}+\nabla u=0$ ($\alpha=\kappa^{-1}$), 以及质量守恒方程 $\nabla\cdot\boldsymbol{q}=f$。这类问题广泛出现在多孔介质流动、复合材料热传导、非均质扩散等应用中。由于系数场可能在多个空间尺度上变化且对比度很大, 直接采用细网格离散会导致巨大的代数系统。混合可杂交间断伽辽金(HDG)方法通过局部消元将全局未知量限制在网格骨架上, 减少了全局自由度。多尺度HDG(MsHDG)方法进一步将域分解为粗块 $\{\Omega_i\}_{i=1}^{N_c}$, 在每个粗块上用细网格求解局部HDG问题, 从而定义从粗迹到法向通量的离散Dirichlet-to-Neumann(DtN)算子。全局问题仅包含粗骨架上的迹变量 $\mu_H\in M_H$, 通过通量连续性条件组装。然而, 构造这些局部DtN算子需要对每一个粗块反复求解细尺度局部问题, 即对每个粗迹基函数应用细尺度局部HDG矩阵的逆。在参数依赖、时间依赖或非线性问题中, 局部算子可能需要在每一步或每次迭代中重新构造, 导致计算负担占主导。本文旨在填补的空白是: 如何用神经网络高效地近似这些局部算子, 从而避免重复的细尺度局部求解, 同时保持标准MsHDG的全局耦合结构。作者提出训练一个神经网络, 学习从局部系数场到离散DtN矩阵 $\boldsymbol{S}_i(\alpha)$ 和源向量 $\boldsymbol{g}_i(\alpha,f)$ 的映射, 并将预测结果直接组装到标准MsHDG骨架系统中。
2. 主要定理或主要结果
本文未建立新的全局收敛定理, 而是通过数值实验验证所提出的神经网络加速多尺度HDG方法(NN-MsHDG)的精度与在线效率。在理论上, 论文沿用标准MsHDG的适定性结果: 若局部有限元空间满足参考文献[9, Theorem 2.5]的相容性条件, 且稳定化参数在每个细单元的至少一个面及每个粗迹面的非空部分为正, 则MsHDG问题存在唯一解。本文采用细尺度空间 $W(K)=P_1(K)$, $\boldsymbol{V}(K)=[P_1(K)]^d$, $M(F_h)=P_1(F_h)$, 稳定化参数 $\tau=1$ 在每个细面上, 因此适定性成立。关于误差估计, 若粗迹空间为 $M_H(F_H)=P_\ell(F_H)$, 粗块直径量级为1, 且满足参考文献[9, Theorem 3.8]的单面稳定化条件, 则理论误差为 $\|u-u_h\|_{L^2(\Omega)}\leq C\big(H^{\ell+3/2}\max\{1,h^{1/2}H^{-1}\}+h^2\big)$。但本文所采用的 $\tau=1$ 在所有细面上的稳定化不满足单面稳定化假设, 因此不声称该最优收敛阶; 作者明确指出其目标不是证明新收敛理论, 而是检验神经网络替代局部求解器的效果。主要数值结果如下: (i) 对于二维二元渗透率场(即系数在两个常数之间取值的随机场), 在对比度适中的情形下, 神经网络预测的局部算子组装后得到的近似解与标准MsHDG解之间的相对建模误差约为百分之几。当粗迹空间维度变化时, 总在线计算时间(包括局部算子生成与全局组装)相比标准MsHDG减少约5到16倍。(ii) 在高对比度情形下, 选用的多项式粗迹空间 $P_\ell(F_H)$ 本身在标准MsHDG中已产生较大的离散误差, 作者认为需要系数自适应的谱迹空间(如文献[8]中的空间)才能改善。(iii) 学习到的局部算子会引入额外的建模误差, 并且当粗迹空间被富化(即 $\ell$ 增大)时, 该误差变得严重。论文还包含一维固定样本系数分辨率研究, 但具体数值结果在提供的文本中未详细展开。这些结果表明, 神经网络替代物能够加速多尺度HDG计算, 但在高对比度问题中需要改进算子表示并理解误差放大机制。
3. 研究方法、关键技术和核心工具
本文的核心技术障碍是标准MsHDG中局部粗块DtN算子的构造成本过高。对于一个粗块 $\Omega_i$, 局部矩阵 $\boldsymbol{S}_i(\alpha)$ 和源向量 $\boldsymbol{g}_i(\alpha,f)$ 需要求解细尺度HDG问题, 其中涉及对细尺度局部矩阵求逆并施加到所有粗迹基函数上。在时间依赖或非线性问题中, 该过程需要重复进行, 成为瓶颈。作者的真正创新点在于将神经网络替代用在粗块级别, 而不是像文献[6]那样用在细元素级别, 也不是像文献[1]那样用在协调有限元子结构上。具体而言, 作者训练一个神经网络学习映射 $\alpha|_{\Omega_i}\mapsto \boldsymbol{S}_i(\alpha)$, 以及 $\alpha,f\mapsto \boldsymbol{g}_i(\alpha,f)$。网络输入是参考块上系数场的表示(如细网格上的系数值), 输出是 $\boldsymbol{S}_i$ 的独立项(由于对称性等, 可能只输出上三角或独立分量)和 $\boldsymbol{g}_i$ 的分量。训练数据通过在参考块上对随机系数场求解局部HDG问题生成。在线阶段, 只需进行网络前向传播即可获得近似的局部算子, 然后组装到全局骨架系统。作者没有给出理论收敛性证明, 而是通过数值实验考察精度和效率。关键分析工具包括: 标准MsHDG的局部-全局分解、HDG的数值通量定义 $\widehat{\boldsymbol{q}}\cdot\boldsymbol{n}=\boldsymbol{q}\cdot\boldsymbol{n}+\tau(u-\widehat{u})$, 以及将局部问题写为鞍点矩阵形式(见方程(4.6))。技术障碍还包括高对比度下的误差放大: 当系数对比度大时, 多项式粗迹空间本身逼近能力不足, 且学习到的算子误差会放大, 尤其是在迹空间富化时。作者未详细展开网络架构与训练超参数, 因此从提供内容中无法确认具体细节。
4. 与之前工作的比较
本文与已有工作的区别主要体现在替代学习的级别和离散化框架上。首先, 与Du和Stechmann[6]的element-learning框架相比, 他们在标准HDG的每个细元素上学习局部输入-输出算子(例如流入-流出算子), 而本文在粗块上学习DtN算子。在低阶有限元细网格下, 单个细元素的局部求解规模很小, 神经网络替代收益有限; 而粗块包含大量细元素, 其局部算子构造昂贵, 因此本文的替代更有意义。其次, 与文献[1]相比, 他们学习非线性DtN映射并用于协调有限元子结构, 而本文在MsHDG框架中学习线性Darcy模型的系数到算子映射, 全局未知量是粗骨架上的杂交迹。本文的方法保留了HDG的局部消元和通量连续性结构, 预测的算子直接组装进入标准的骨架系统。第三, 与标准MsHDG[9]相比, 本文用神经网络近似替换了细尺度局部求解, 在中等对比度的二维二元渗透率场下相对误差为几个百分点, 在线成本降低5至16倍, 但代价是引入了额外的建模误差。在高对比度情形, 多项式粗迹空间本身误差已较大, 作者认为需要系数自适应谱迹空间[8]来改进, 同时指出学习算子在高对比度下的误差放大机制尚不清楚。此外, 本文没有发展新的误差分析理论, 而是采用数值实验验证, 与[9, Theorem 3.8]的收敛理论对比, 本文的稳定化选择(所有细面 $\tau=1$)不满足单面稳定化假设, 因此无法直接应用最优误差估计。作者未显式展开与其他机器学习加速PDE方法(如DeepONet, FNO)的实验比较, 也未提供网络结构和训练代价的详细讨论, 这些从提供内容中无法确认。
2. 具有双通道传输条件的快速收敛有限元区域分解方法
Rapidly Convergent Finite-Element Domain Decomposition Method With Two-Channel Transmission Conditions
作者: Furkan Şık, Fernando L. Teixeira, Balasubramaniam Shanker
类别: physics.comp-ph, math.NA
发布日期: 2026-08-25
arXiv ID: 2608.25041v1
摘要翻译: 针对麦克斯韦方程组的求解，提出了一种新颖的对偶-原始有限元撕裂与互联 (FETI-DP) 区域分解方法 (DDM)。该方法基于一种双通道传输条件 (TC)，该条件在子域界面上同时强制执行切向场和法向通量的连续性。这两个界面通道分别由法拉第方程和安培-麦克斯韦方程构造。与广泛使用的 Robin 传输条件相比，同时强制执行切向场和法向通量的连续性可将界面跳跃减少几个数量级。由此产生的全局界面方程展现出显著改善的迭代收敛性。针对不同划分策略、子域数量和体积、网格分辨率以及几何形状的数值实验表明，迭代次数大幅减少，同时保持了准确性、大规模并行性和可扩展性。
Comment: 9 pages, 4 figures
链接: http://arxiv.org/abs/2608.25041v1
1. 研究对象和背景
本文研究时谐 Maxwell 方程组的三维有限元求解问题. 在非重叠区域分解框架下, 将计算域 $\Omega$ 划分为 $N_s$ 个非重叠子域 $\Omega_i$, $i=1,\dots,N_s$, 界面为 $\Gamma_{ij}=\partial\Omega_i\cap\partial\Omega_j$. 在每个子域内, 电场 $\mathbf{E}_i$ 与磁感应强度 $\mathbf{B}_i$ 满足 Faraday 定律与 Ampère-Maxwell 定律:
$$\nabla\times\mathbf{E}_i=-j\omega\mathbf{B}_i,\qquad \nabla\times\mathbf{H}_i=\mathbf{J}_i+j\omega\mathbf{D}_i,$$其中 $\mathbf{D}_i=\varepsilon_i\mathbf{E}_i$, $\mathbf{H}_i=\mu_i^{-1}\mathbf{B}_i$, $\omega$ 为角频率, $\mathbf{J}_i$ 为外加电流密度. 作者采用混合 E-B 有限元离散, 即 $\mathbf{E}_i\in W^1(\Omega_i)\subset H(\mathrm{curl};\Omega_i)$ 以 Whitney 棱元展开, $\mathbf{B}_i\in W^2(\Omega_i)\subset H(\mathrm{div};\Omega_i)$ 以 Whitney 面元展开, 未知系数向量记为 $\bar{\mathbf{x}}^{(i)}=(\bar{\mathbf{b}}^{(i)T},\bar{\mathbf{e}}^{(i)T})^T$. 这种离散直接保留了两个一阶旋度方程的结构, 而不是化为关于 $\mathbf{E}$ 的二阶矢量波动方程.
区域分解方法的目的是将一个大规模体积问题转化为定义在子域界面上的全局界面方程, 进而用 Krylov 子空间方法迭代求解. 在 FETI-DP 方法中, 子域界面上的连续性通过 Lagrange 乘子强制, 而角边未知量作为 primal 变量保留, 形成全局粗空间. 传输条件 (TC) 的选择决定了子域问题的适定性和界面方程的迭代收敛性. 目前广泛使用的 Robin TC 在每个界面上引入两个乘子 (FETI-DP-2$\lambda$), 但作者指出, Robin TC 只能阻尼界面误差模式中的传播谱, 无法同时阻尼 TE 型和 TM 型倏逝模; 不存在单一 Robin 系数能同时做到这一点. 二阶传输条件 (SOTC) 通过引入切向导数修正来针对倏逝模, 但其辅助界面电流和电荷在角边上出现冗余, 导致近奇异矩阵而需要专门的角边惩罚; 并且在深倏逝波区域, SOTC 的阻尼效果退化, 趋近于 Robin 行为. 因此, 现有 TC 在大规模问题 (子域数多、角边多、含谐振、多尺度特征、深倏逝波) 下可能导致收敛缓慢甚至停滞, 这促使一些实践者转向基于直接解法的 DDM, 但直接解法牺牲了并行性和可扩展性. 本文的目标是构造一种新的 TC, 使得全局界面方程能够通过稳健的迭代求解器快速无停滞收敛, 同时保持大规模并行与计算可扩展性.
这种空白正是本文试图填补的: 作者从混合 E-B 公式出发, 分别为 Faraday 和 Ampère-Maxwell 方程构造一个通道, 形成两通道传输条件, 同时强制切向场与法向通量在界面上连续, 从而消除 Robin TC 和 SOTC 的关键缺陷.
2. 主要定理或主要结果
本文的主要结果分为算法构造与数值验证两部分.
在算法层面, 作者提出的两通道 TC 为: 在界面 $\Gamma_{ij}$ 上, 从子域 $\Omega_i$ 侧定义 Faraday 通道
$$\Lambda_{ij}^1=\hat{\mathbf{n}}_i\times\mathbf{E}_i+a_{B,i}\,\hat{\mathbf{n}}_i(\hat{\mathbf{n}}_i\cdot\mathbf{B}_i),$$和 Ampère-Maxwell 通道
$$\Lambda_{ij}^2=\hat{\mathbf{n}}_i\times\mathbf{H}_i+a_{D,i}\,\hat{\mathbf{n}}_i(\hat{\mathbf{n}}_i\cdot\mathbf{D}_i),$$其中 $\hat{\mathbf{n}}_i$ 为外法向, $a_{B,i}$ 与 $a_{D,i}$ 为按局部波速 $v_i=1/\sqrt{\mu_i\varepsilon_i}$ 选取的系数. 每个通道在界面两侧各配一个 Lagrange 乘子 $\lambda_{ij}^1,\lambda_{ij}^2$ 及其相邻侧对应乘子, 因此每个界面共有四个乘子, 即 FETI-DP-4$\lambda$ 方法. 对每个子域施加边界条件 $\Lambda_{ij}^q+\lambda_{ij}^q=0$ 后, 得到分块局部系统 (9):
$$\overline{\overline{A}}^{(i)}\bar{\mathbf{x}}^{(i)}+\overline{\overline{G}}^{(i)}\bar{\boldsymbol{\lambda}}^{(i)}=\bar{\mathbf{f}}^{(i)},$$其中 $\overline{\overline{A}}^{(i)}$ 的具体形式由 (10) 给出, 包含面元质量矩阵 $\overline{\overline{M}}_B^{(i)}$, 棱元质量矩阵 $\overline{\overline{M}}_E^{(i)}$, 离散旋度耦合矩阵 $\overline{\overline{C}}^{(i)}$ 以及法向算子 $\overline{\overline{N}}_B^{(i)}$, $\overline{\overline{N}}_D^{(i)}$. 这些法向算子直接来自混合 E-B 离散, 不引入额外界面未知量. 将各子域系统与界面交换条件组装, 得到全局鞍点系统 (13), 再消去子域内部未知量 $\bar{\mathbf{x}}_r$ 和 primal 角边未知量 $\bar{\mathbf{x}}_c$, 得到作用于乘子向量 $\bar{\boldsymbol{\lambda}}$ 的全局界面方程 (15):
$$\overline{\overline{F}}_{4\lambda}\bar{\boldsymbol{\lambda}}=\bar{\mathbf{d}}_{4\lambda},$$该方程用 GMRES 求解, 每次矩阵-向量积只需一次子域级稀疏分解和一次小规模角边粗空间求解.
在数值结果方面, 作者通过一系列三维波导算例展示收敛性改善. 对于均匀划分的 WR-90 波导 (40 个子域, 39 个界面), 在相对残差 $10^{-6}$ 和 $10^{-10}$ 容差下, 与 Robin TC 相比, GMRES 迭代次数分别从 188 和 301 降至 55 和 68, 即减少因子分别为 3.42 和 4.42; 在 $10^{-10}$ 容差下, 壁钟时间从 380.7 s 降至 189.3 s, 约 2 倍加速. 对于 30 个不规则宽度子域的同一波导 (宽度 $\Delta x\in[0.8,18.8]$ mm), 收敛迭代次数减少约 9 倍. 对于脊加载波导和 METIS 划分的 WR-90 波导 (16 个不规则子域, 37 个界面), 在 $10^{-10}$ 容差下迭代次数分别减少 3.39 倍和 9.53 倍. 对 L 形波导在 40 和 100 个子域情形, 以及 X 波段带通滤波器 (METIS 划分为 110 个子域) 的算例中, 也观察到数倍至近 6 倍的迭代次数下降 (滤波器: 180 次 vs 1020 次). 同时, 精度没有损失: 例如均匀波导中, 提出 TC 与 Robin TC 的电场解相对 $L^2$ 差仅为 $3.2\times10^{-7}$, 两种 TC 均与解析解符合到约 99.25%; 不规则划分时相对 $L^2$ 差约为 0.0001%. 特别地, 界面连续性显著改善: 在 METIS 划分波导的 $10^{-6}$ 停止容差下, 最大法向磁通跳变 $|\Delta(\hat{\mathbf{n}}\cdot\mathbf{B})|$ 从 Robin TC 的 $9.1\times10^{-7}$ 降至提出 TC 的 $2.2\times10^{-12}$, 降低近六个数量级; 最大切向场跳变 $|\Delta(\hat{\mathbf{n}}\times\mathbf{E})|$ 从 $1.4\times10^{-8}$ 降至 $1.2\times10^{-10}$, 降低两个数量级以上. 这些结果表明, 在数值上, 两通道 TC 能够在大范围子域划分、子域数目和体积、网格分辨率以及几何复杂度下保持快速收敛.
需要说明的是, 文中没有给出迭代收敛率的理论估计或误差阶定理; 上述结论均来自数值实验. 关于系数 $a_{B,i}$ 和 $a_{D,i}$ 的选取, 作者说明是按照用于设计 SOTC 和分析 Robin TC 的 Fourier 分析来设置的, 但本文未展开该 Fourier 分析的细节.
3. 研究方法、关键技术和核心工具
本文面对的核心技术障碍有二: 一是 Robin TC 的单一系数无法同时为 TE 和 TM 型倏逝模提供阻尼; 二是 SOTC 在角边处产生冗余辅助变量, 且深倏逝波下其阻尼退化为 Robin. 针对这些障碍, 作者的真正创新在于从混合 E-B 公式的一阶旋度方程出发, 为每一种旋度方程单独构造一个通道, 从而将两种倏逝波族分配到不同的通道, 每个通道有独立的阻尼系数. 具体地说, 通道定义 (4)-(5) 中, Faraday 通道包含切向电场 $\hat{\mathbf{n}}\times\mathbf{E}$ 和法向磁通 $\hat{\mathbf{n}}(\hat{\mathbf{n}}\cdot\mathbf{B})$, 而 Ampère-Maxwell 通道包含切向磁场 $\hat{\mathbf{n}}\times\mathbf{H}$ 和法向电通 $\hat{\mathbf{n}}(\hat{\mathbf{n}}\cdot\mathbf{D})$. 由于 $\mathbf{B}$ 在混合 E-B 离散中直接以面元展开, $\hat{\mathbf{n}}\cdot\mathbf{B}$ 是天然自由度, 这使得法拉第通道可以直接构造而不需要额外辅助未知量; 同理, $\hat{\mathbf{n}}\cdot\mathbf{D}$ 通过切向电场迹的表面 grad-div 算子 $\overline{\overline{N}}_D$ 表示. 这样就在有限元层面同时实现了切向场与法向通量的强制连续.
关键的技术步骤包括:
- 弱形式与界面数据的处理. 对 (1) 和 (2) 分别用 $\mu_i^{-1}\mathbf{W}_f^{(2)}$ 和 $\mathbf{W}_a^{(1)}$ 检验, 得到界面配对 (3). 将通道边界条件 (6) 代入后, 界面数据被重写为乘子和法向通量项的线性组合, 从而导出局部矩阵中的 $\overline{\overline{N}}_B$ 和 $\overline{\overline{N}}_D$. 这里 $\overline{\overline{N}}_B$ 是面元法向迹的 Gram 矩阵, 由于 $\mathbf{B}$ 本身是面元, 该矩阵可直接在界面网格上集成; $\overline{\overline{N}}_D$ 则通过表面上的 grad-div 算子将切向 $\mathbf{E}$ 转化为 $\mathbf{D}$ 的法向通量. 两个算子都只在界面上活动, 不增加新的界面自由度.
- 全局装配. 在界面 $\Gamma_{ij}$ 上, 两个子域的通道必须相等. 作者通过邻居侧边界条件 $\lambda_{ij}^q+\Lambda_{ij}^q[\mathbf{E}_j,\mathbf{B}_j]=0$ 和局部侧条件 (6) 消去相邻侧切向迹, 得到离散交换关系 (12), 其中 $\overline{\overline{P}}_{ij}^q$ 为方向重定向矩阵, $\overline{\overline{T}}_{ji}^q$ 为迹限制. 这样每个界面有两个通道交换方程, 配合局部系统构成鞍点系统 (13).
- 双重 Schur 补约简. 首先对每个子域进行稀疏分解, 消去内部未知量 $\bar{\mathbf{x}}_r$, 得到 (14) 所示的 Schur 补量; 然后再消去 primal 角边未知量 $\bar{\mathbf{x}}_c$, 得到全局界面方程 (15). 后者在 GMRES 迭代中每次矩阵-向量积需要一次每个子域的三角回代以及一个以角边为未知量的粗空间求解. 粗空间在 FETI-DP 中负责全局信息传播, 这是并行可扩展性的关键.
- 系数选取与 Fourier 分析. 作者将 $a_{B,i}$ 和 $a_{D,i}$ 设置为与局部波速 $v_i$ 相关, 并声称依据 Fourier 分析确定具体数值 (参考文献 [14], [17], [27]). 然而, 论文并未给出 Fourier 分析的显式推导或最优公式, 也没有给出严格的收敛率证明. 因此, 本文的理论贡献主要体现在构造性算法和数值验证, 而不是数学证明.
证明框架的推进主要体现在: 作者通过数值实验展示, 在深倏逝波区, 本文 TC 的阻尼是单调的, 而 Robin 停滞、SOTC 反弹. 这与文献中关于 Robin/SOTC 的 Fourier 分析结果一致, 但本文没有提供新的理论分析.
4. 与之前工作的比较
与既有工作的比较可从以下几个维度展开.
与 Robin TC (FETI-DP-2$\lambda$) 相比. Robin TC 在界面上的典型形式为 $\hat{\mathbf{n}}\times\nabla\times\mathbf{E}+\alpha\,\hat{\mathbf{n}}\times(\hat{\mathbf{n}}\times\mathbf{E})=\cdots$, 它只能近似强制切向场连续性, 而没有直接强制法向通量连续性. 本文的双通道 TC 通过额外引入法拉第和安培-麦克斯韦通道, 同时强制 $\hat{\mathbf{n}}\times\mathbf{E}$ 和 $\hat{\mathbf{n}}\cdot\mathbf{B}$ 在界面上连续, 因此界面跳变显著下降. 数值上, 迭代次数减少 3.42 至 9.53 倍, 且精度不损失. 代价是每个界面的乘子数从 2 增至 4, 子域未知量从 $N_e^{(i)}$ 增至 $N_e^{(i)}+N_f^{(i)}$, 但作者指出增加的矩阵都是无旋度的质量矩阵或界面 Gram 矩阵, 每迭代一次的额外成本仅为有界因子, 而迭代次数下降一个数量级, 总时间下降. 例如均匀波导中壁钟时间从 380.7 s 降至 189.3 s.
与二阶传输条件 (SOTC) 相比. SOTC 通过添加切向导数修正项来加速倏逝模收敛, 例如参考文献 [14], [17] 中的方法. 但 SOTC 需要引入额外的辅助界面电流和电荷变量, 导致角边上这些变量线性相关, 使得 DDM 矩阵近奇异; 因此需要专门的角边惩罚技术. 本文的 TC 没有引入辅助变量, 乘子直接作用于通道, 因而角边处理自然, 不需要额外惩罚项. 此外, 在深倏逝波区, SOTC 的切向导数修正项贡献相对减小, 阻尼行为退化为 Robin, 而本文的两个通道直接来自一阶方程, 在深倏逝波区仍是单调阻尼. 数值实验中, SOTC-Full 和 SOTC-TE 的迭代次数都明显高于本文方法 (见图 1 和表 I).
与直接求解器型 DDM 相比. 近年来部分工作转向直接求解器型 DDM, 如文献 [21]-[23], 以避免迭代停滞. 但直接求解器需要更大的内存和通信代价, 牺牲了 FETI 类方法的并行可扩展性. 本文通过改进 TC 使迭代求解器在困难问题上也能快速收敛, 从而保留了并行性和可扩展性, 这是对直接求解器路线的一种替代.
与基于第二阶矢量波动方程的传统 FETI-DP 相比. 传统 FETI-DP 只将电场 $\mathbf{E}$ 作为子域未知量, 并求解 $\nabla\times\mu^{-1}\nabla\times\mathbf{E}-\omega^2\varepsilon\mathbf{E}=-j\omega\mathbf{J}$. 这样的公式无法直接以面元自由度表达 $\mathbf{B}$, 因此难以自然构造同时包含 $\hat{\mathbf{n}}\cdot\mathbf{B}$ 的法拉第通道. 本文采用的混合 E-B 公式 (文献 [24], [25]) 将 $\mathbf{E}$ 和 $\mathbf{B}$ 同时离散, 从而使得法向通量连续性成为可施加的强条件. 代价是子域系统规模和界面乘子数增加, 但换来了更快的迭代收敛和更好的界面连续性.
局限与未展开比较. 本文结果没有提供严格的迭代收敛率理论证明, 也没有公开 Fourier 分析的完整细节, 系数 $a_{B,i}, a_{D,i}$ 的具体取值公式未给出 (只说明按参考文献设置). 所有数值实验限于共形四面体网格的波导结构, 没有测试开放域辐射问题和非共形界面. 此外, 作者没有与其他先进预条件子或优化型粗空间方法进行对比, 也没有讨论调参敏感性. 从提供内容中无法确认作者是否在更复杂的实际天线或散射问题上进行过测试; 这些未展开的部分可能限制其普适性.
了解领域论文（摘要翻译）
1. CARE-SAV：面向梯度流能量稳定模拟的条件感知随机特征框架
CARE-SAV: A Conditioning-Aware Random-Feature Framework for Energy-Stable Simulation of Gradient Flows
作者: Bingcheng Hu, Zhaoxiang Li
类别: math.NA
发布日期: 2026-08-26
arXiv ID: 2608.25300v1
链接: http://arxiv.org/abs/2608.25300v1
摘要翻译
中文标题: CARE-SAV：面向梯度流能量稳定模拟的条件感知随机特征框架
摘要翻译: 梯度流模型具有内在的能量耗散结构，在离散层面忠实地保持这一结构对于稳定可靠的长时间模拟至关重要。为此，我们提出了一种具有标量辅助变量的条件感知表示增强（CARE-SAV）框架，该框架从灵活候选特征中构造紧凑的空间逼近空间，并直接在此空间内演化梯度流动力学。所得到的全离散格式保持离散能量耗散律，同时为传统的预设空间离散提供了一种灵活的替代方案。严格的分析确立了所提方法的逼近能力、可解性、稳定性和收敛性。对代表性梯度流问题的数值实验验证了其准确性、鲁棒性和计算效率。我们相信CARE-SAV能够为梯度流问题的保结构离散提供一种简单、灵活且计算高效的范例。
2. 用于多尺度算子学习的框架核方法
The Frame Kernel Method for Multiscale Operator Learning
作者: Branden Frieden, Ryan Whitehead, M. Keith Ballard, Robert M. Kirby, Varun Shankar
类别: cs.LG, math.NA
发布日期: 2026-08-25
arXiv ID: 2608.25084v1
链接: http://arxiv.org/abs/2608.25084v1
摘要翻译
中文标题: 用于多尺度算子学习的框架核方法
摘要翻译: 我们提出了一种原生多尺度算子学习方法，用于（多尺度偏微分方程（PDE）的数值求解器的）代理建模。我们方法的主要新颖之处在于一种新颖的多尺度核框架函数逼近技术。利用这种新的核框架技术，我们将算子学习问题转化为学习输出函数的框架系数作为输入函数框架系数的函数的问题。然后，泛化步骤自动允许输出函数的多尺度分解。我们的方法适用于张量积网格和点云。我们给出了框架逼近的插值证明、误差估计和数值收敛率。然后，我们展示了我们的方法在固有多尺度PDE代理建模中的适用性。新的多尺度框架核方法在文献中的具有挑战性的问题上比流行的神经算子显著更准确，同时在泛化时允许后验多尺度分解。
不相关论文（基本信息）
1. Cahn-Hilliard-Navier-Stokes方程的一族二阶线性无条件稳定方法
A family of second order, linear, unconditionally stable methods for the Cahn-Hilliard-Navier-Stokes equations
作者: Daozhi Han, Nan Jiang, Jonah H. Nissan, Sayantan Sarkar
类别: math.NA, math-ph
发布日期: 2026-08-26
arXiv ID: 2608.26046v1
链接: http://arxiv.org/abs/2608.26046v1
摘要: We present a family of second-order, linear, unconditionally stable implicit-explicit (IMEX) methods for the Cahn-Hilliard-Navier-Stokes (CHNS) equations modeling matched-density two-phase flows. The proposed semi-discrete scheme combines extrapolation of the nonlinear terms with an auxiliary-variable formulation of the nonlinear free-energy term and a temporal-curvature regularization controlled by a parameter $ε$. We establish a discrete energy estimate showing unconditional long-time stability of the method for $θ\in(1/2,1]$ and $ε\geq0$. The resulting scheme requires only linear solves at each time step. Numerical experiments demonstrate approximately second-order temporal convergence and examine mass conservation, energy dissipation, numerical robustness, and several representative interfacial-flow problems, including spinodal decomposition, droplet shape relaxation, two-phase lid-driven cavity flow, and Rayleigh-Taylor instability.
2. 随机材料中的风险规避优化：算法进展与高性能计算加速
Risk-averse Optimization in Random Materials: Algorithmic Advances and HPC Acceleration
作者: Niklas Baumgarten, Marcel Koch, David Schneiderhan, Tim Schrader, Robin Weiß
类别: math.NA
发布日期: 2026-08-26
arXiv ID: 2608.25967v1
链接: http://arxiv.org/abs/2608.25967v1
摘要: We summarize our advances in the algorithmic development and hardware utilization for risk-averse optimization problems in random materials. This includes risk-averse optimization using the entropic risk measure, as well as recently developed sampling techniques for random materials, that are interoperable with the optimization framework. Furthermore, we discuss recent progress in the efficient utilization of modern hybrid hardware architectures for these methods to solve three-dimensional partial differential equations.
3. 资源约束下的优化多层采样方法
Optimized Multilevel Sampling Methods under Resource Constraints
作者: Niklas Baumgarten
类别: math.NA
发布日期: 2026-08-26
arXiv ID: 2608.25958v1
链接: http://arxiv.org/abs/2608.25958v1
摘要: We present recent developments in multilevel sampling methods under resource constraints. Over the past 15 years, multilevel methods have become widely used for uncertainty quantification. However, scaling them to high-dimensional problems and high-performance computing (HPC) environments remains challenging. In this work, we discuss two algorithms designed to address these issues: the budgeted Multilevel Monte Carlo (MLMC) method and the Multilevel Stochastic Gradient Descent (MLSGD) method. We demonstrate their effectiveness on HPC systems under consideration of the available computational resources for applications in forward uncertainty quantification (UQ) and optimal control (OC) under uncertainty.
4. $L_2$ Marcinkiewicz-Zygmund不等式中的所需点数
Required Number of Points in $L_2$ Marcinkiewicz-Zygmund Inequalities
作者: Felix Bartel
类别: math.NA
发布日期: 2026-08-26
arXiv ID: 2608.25886v1
链接: http://arxiv.org/abs/2608.25886v1
摘要: We determine, up to absolute constants, the worst-case number of point evaluations required for a weighted $L_2$ Marcinkiewicz-Zygmund inequality for an $m$-dimensional complex function space. If $0<\varepsilon<1$ is the relative distortion, this number is
$$Θ\Big(\min\Big\{m^2,\frac{m}{\varepsilon^2}\Big\}\Big),$$and exact discretization has the sharp worst-case value $m^2$. While the upper bounds follow from recent constructions, our contribution is the construction of function spaces that are hard to discretize and yield matching lower bounds. We use a trace-variance inequality for weighted subframes of unit-norm tight frames. One such instance is the complete-graph edge frame, which yields a construction in every dimension. Singer equiangular tight frames improve the constant when $m-1$ is a prime power, while maximal equiangular tight frames give the strongest bound possible using our method whenever they exist. We also derive consequences for the conditioning of weighted least-squares systems and for standard condition-number-based iteration estimates when these systems are solved by LSQR.
5. 完整电极模型下电阻抗断层成像的自适应Nesterov动量方法
Adaptive Nesterov Momentum Method for Electrical Impedance Tomography with the Complete Electrode Model
作者: Kai Zhu, Jijun Liu, Min Zhong
类别: math.NA
发布日期: 2026-08-26
arXiv ID: 2608.25837v1
链接: http://arxiv.org/abs/2608.25837v1
摘要: We apply the adaptive Nesterov momentum (ANM) method [30] to electrical impedance tomography under the complete electrode model. The forward problem is formulated in the variational CEM setting, accounting for finite electrode size, contact impedance, insulating gaps, and the mean-free voltage gauge. The resulting nonlinear inverse problem is treated within a unified dual-to-primal framework using three classes of strongly convex structural penalties: an L2-type penalty, an L1-type penalty promoting sparse deviations from a calibrated homogeneous background, and a TV-type penalty favoring approximately piecewise-constant conductivities with sharp interfaces. The TV class is implemented using both smoothed TV and Huber-TV formulations. The data- misfit gradient is computed through CEM adjoint equations and stabilized by Sobolev smoothing. The method is evaluated on the publicly available KIT4 tank measurement data after calibration of the background conductivity and contact impedance from no-object measurements. The experiments include single, multiple, mixed-conductivity, and geometrically challenging phantom configurations. The L2-type penalty generally produces smooth but diffuse reconstructions, whereas the L1-type penalty yields cleaner backgrounds with occasional geometric distortion. The smoothed TV and Huber-TV penalties provide more spatially coherent localization and exhibit similar reconstruction behavior across most tested configurations. These results demonstrate the practical applicability of the adaptive Nesterov framework to measured CEM-EIT data.
6. 弱非线性耗散微分方程及其推广的快速推进量子算法
Fast-forwarding quantum algorithms for weakly nonlinear dissipative differential equations and beyond
作者: Yixiang Li, Dong An
类别: quant-ph, math.NA
发布日期: 2026-08-26
arXiv ID: 2608.25822v1
链接: http://arxiv.org/abs/2608.25822v1
摘要: We study a fast-forwarded quantum algorithm for solving weakly nonlinear dissipative ordinary differential equations. Our approach is a combination of the Carleman embedding technique and the linear combination of Hamiltonian simulation algorithm for linearized systems with fast-forwarded scaling. The complexity of our algorithm does not explicitly depend on the evolution time $T$, thus greatly improving the previous state-of-the-art $\widetilde{\mathcal{O}}(\sqrt{T})$ to $\mathcal{O}(1)$, and any remaining time dependence enters through the output norm and forcing parameters. We rigorously analyze the performance of this approach by convergence guarantees of the Carleman embedding for time-dependent coefficient matrices and detailed complexity estimates, and improve the realization of the Carleman-embedding-based algorithms by simplifying the post-selection step. In addition, we perform a numerical study on differential equations beyond the weakly nonlinear case, and identify possibility of achieving fast-forwarding scaling for systems with stronger nonlinearity or linear non-resonant effect.
7. 基于GLT分析的全耦合变系数分数阶演化方程的块预处理
Block preconditioning for all-at-once variable-coefficient fractional evolution equations via the GLT analysis
作者: Muhammad Faisal Khan, Stefano Serra-Capizzano
类别: math.NA
发布日期: 2026-08-26
arXiv ID: 2608.25796v1
链接: http://arxiv.org/abs/2608.25796v1
摘要: We study a class of nonlocal evolutionary partial differential equations with weakly singular temporal kernel and spatially variable diffusion coefficient. The model is posed on $Ω\subset \mathbb{R}$, and involves a left-sided Riemann–Liouville fractional derivative in space multiplied by a variable coefficient $a(x)$. The temporal derivative is approximated by an $L1$ type scheme, while the spatial operator is discretized by finite difference techniques, resulting in large scale all at once linear systems with a twolevel Toeplitz like structure. We develop and analyze a block lower triangular strategy that mimics the structure of the coefficient matrix while simplifying its components for computational efficiency. The analysis is carried out at the level of matrix sequences by means of generalized locally Toeplitz (GLT) theory. Within this framework, we characterize the asymptotic spectral distribution of the discretized operators and use the associated GLT symbol to guide the construction of the structured approximation. Numerical experiments using the GMRES solver demonstrate that the proposed preconditioning strategy significantly improves convergence rates, robustness, and scalability for large-scale problems. Open problems and possible extensions are briefly discussed at the end of the present work.
8. 从估计到证明：奇异薛定谔算子的可验证基态能量界
From estimate to proof: certified ground-state energy bounds for singular Schrödinger operators
作者: Xuefeng Liu
类别: math.NA
发布日期: 2026-08-26
arXiv ID: 2608.25760v1
链接: http://arxiv.org/abs/2608.25760v1
摘要: Many predictions in quantum chemistry, materials science, and spectral geometry hinge on the lowest energy levels of a Schrödinger operator, yet standard simulations return approximations with no guarantee of how far they sit from the true value. We present a computational framework that returns mathematically certified bounds—an interval provably containing the exact energy—even for the singular, unbounded, sign-changing potentials of real molecules, where existing certified methods fail. For the hydrogen molecular ion this framework delivers, to our knowledge, the first guaranteed two-sided enclosure of the true infinite-domain ground-state energy: an interval of width below $5\times10^{-4}$ that provably contains the accepted reference value. The key that makes a whole-space guarantee possible is an \emph{explicit}, computable bound on the error of restricting the problem to a finite box—replacing the classical argument that the wavefunction merely decays—which brackets the true energy from both sides. Remarkably, the certified interval reproduces the uncertified numerical value to eleven digits, so mathematical rigor costs almost nothing in accuracy. The result turns eigenvalue computation from an estimate into a proof.
9. 实直线上离散多重正交多项式的递推关系的更新与降阶更新
Updating and Downdating the Recurrences of Discrete Multiple Orthogonal Polynomials on the Real Line
作者: Amin Faghih, Marc Van Barel, Raf Vandebril
类别: math.NA
发布日期: 2026-08-26
arXiv ID: 2608.25758v1
链接: http://arxiv.org/abs/2608.25758v1
摘要: We study multiple orthogonal polynomials with orthogonality defined by multiple positive discrete measures on the real line. Focusing on typeI and typeII multiple orthogonal polynomials and their step-line recurrence relations, we consider the reconstruction of the associated banded upper Hessenberg recurrence matrix from given nodes and weights of the measures. We propose efficient updating and downdating procedures that modify an existing recurrence matrix when nodes are added to or removed from the discrete measures. The updating strategy is based on solving an inverse eigenvalue problem, while the downdating procedure relies on an eigenvalue deflation technique inspired by a QR-type of algorithm. Numerical experiments confirm the stability and efficiency of the proposed approaches.
10. 反电导率问题的分散测量误差分析
Error Analysis of the Inverse Conductivity Problem with Scattered Measurements
作者: Bangti Jin, Qimeng Quan, Wenlong Zhang
类别: math.NA
发布日期: 2026-08-26
arXiv ID: 2608.25749v1
链接: http://arxiv.org/abs/2608.25749v1
摘要: In this work, we investigate the inverse problem of recovering the conductivity coefficient in an elliptic equation from noisy measurements collected at finitely many deterministic scattered points in the domain $Ω$, and corrupted by random noise. Inspired by the regularity analysis, we propose a numerical scheme based on the regularized least-squares formulation with a $W^{1,4}(Ω)$ penalty, and discretize the regularized problem using the Galerkin finite element method with continuous piecewise linear elements. Under suitable assumptions on the problem data, we provide an error analysis of the regularized solution and its Galerkin approximation. We establish $L^2(Ω)$ error bounds in a high-probability sense, which depend explicitly on the regularization parameter $γ$, the number $n$ of data points and the mesh size $h$. We also present numerical experiments to illustrate the theoretical findings.
11. Jacobi加权直方插值中的精确对角Gram矩阵
Exactly Diagonal Gram Matrices in Jacobi Weighted Histopolation
作者: Allal Guessab, Federico Nudo, Stefano Serra-Capizzano
类别: math.NA
发布日期: 2026-08-26
arXiv ID: 2608.25714v1
链接: http://arxiv.org/abs/2608.25714v1
摘要: In the current work, we study univariate polynomial weighted histopolation on $[-1,1]$, where the data are weighted integrals over a family of intervals. After choosing a polynomial basis, the weighted moment conditions lead to a histopolation matrix whose structure depends on the weight and on the geometry of the cells. We investigate its nonsingularity, which guarantees unisolvence, together with exact diagonality of its Gram matrix, which allows its singular values and spectral condition number to be determined explicitly. For families of intervals whose endpoints belong to a fixed grid, we characterize unisolvence in terms of the connectedness of the associated endpoint graph. In the unisolvent case, this graph is a tree, and the unique paths joining consecutive grid points provide an explicit expression for the inverse matrix. This identity gives explicit formulas for the singular values of both matrices, and shows that their condition numbers in the two-norm coincide and grow linearly with the matrix size. Moreover, it yields the limiting singular value distributions of the two matrix sequences. We also establish a general diagonalization criterion based on discrete weighted orthogonality. The criterion recovers the first kind Chebyshev construction and leads to a diagonal configuration for the constant weight based on discrete sine orthogonality. For interval families with a connected endpoint graph, the corresponding moment vectors define an inner product on the polynomial space and lead to a monic basis with a diagonal weighted Gram matrix. Finally, we derive reduction formulas for cell moments associated with generalized Jacobi weights and introduce an alternative basis for shifted Jacobi weights. Applied to the Chebyshev weight of the fourth kind, this basis, together with a correction of one nonconstant element, yields an exactly diagonal Gram matrix.
12. 一类标量随机微分方程的一阶显式保正格式分析
Analysis of a first-order explicit positivity preserving scheme for a class of scalar SDEs
作者: Charles-Edouard Bréhier, David Cohen
类别: math.NA, math.PR
发布日期: 2026-08-26
arXiv ID: 2608.25689v1
链接: http://arxiv.org/abs/2608.25689v1
摘要: We propose and analyze a first-order numerical scheme for a class of scalar Itô stochastic differential equations with almost surely positive solutions. We construct a new explicit numerical scheme, such that for any choice of the time-step size, the numerical solution is guaranteed to remain almost surely positive. The main result of this article is the first-order strong convergence of the proposed positivity preserving scheme, which is illustrated with numerical experiments and proved rigorously. It is also shown how to adapt the scheme and the results to Stratonovich stochastic differential equations with positive solutions.
13. 一类随机微分方程的显式保域数值格式
Explicit domain preserving numerical schemes for a class of stochastic differential equations
作者: Charles-Edouard Bréhier, David Cohen
类别: math.NA, math.PR
发布日期: 2026-08-26
arXiv ID: 2608.25685v1
链接: http://arxiv.org/abs/2608.25685v1
摘要: We construct and analyze numerical schemes for systems of stochastic differential equations, which preserve almost surely a given hypercube of arbitrary dimension. We propose a new general class of explicit schemes, such that for any choice of the time-step size the numerical solution takes values in the hypercube. We prove strong and weak convergence results for this general class of domain preserving numerical schemes, with strong order $1/2$ and weak order $1$ in general. We also construct a variant of the scheme which achieves strong order $1$ when the stochastic differential equation is driven by a one-dimensional Brownian motion. The convergence results are illustrated with numerical experiments.
14. 全尺度全球海洋动力学的计算模型
A Computational Model for Global Ocean Dynamics at all Scales
作者: Peter Korn
类别: math.NA
发布日期: 2026-08-26
arXiv ID: 2608.25679v1
链接: http://arxiv.org/abs/2608.25679v1
摘要: The dynamics of the world’s ocean spans a breathtaking range of scales. The circulation carries processes from thousands of kilometres down to the millimetre, and from seconds to millennia. Our understanding of global ocean dynamics is to a large extent based on the analysis of ocean model simulation data; computational models provide spatial and temporal coverage that observations can not deliver. However, the scales a computational ocean model can simulate are limited by the available computational capacity. Smaller scales, anticipated as out of reach, have been excluded a priori by shortcuts in the dynamical equations; examples are the hydrostatic and Boussinesq approximations. Hydrostatic Boussinesq models have been used since \citet{Bryan1969} up to now to simulate the planetary scale down to the mesoscale eddy resolving scale of the order of kilometers, and the hydrostatic approximation holds for most oceanic phenomena at the $\sim$1–100,km resolutions that have been computationally affordable. The effect of removed scales on represented scales–the convective scale in the case of the hydrostatic approximation-had to be substituted by ad hoc parametrizations.
15. 基于迭代区域分解法的组合高维与降阶建模
Combined High-Dimensional and Reduced-Order Modeling Based on an Iterative Domain Decomposition Method
作者: Taiji Saito, Tomoshi Miyamura, Yasunori Yusa
类别: math.NA
发布日期: 2026-08-26
arXiv ID: 2608.25567v1
链接: http://arxiv.org/abs/2608.25567v1
摘要: A method is developed for structural analysis in which parts of an analysis domain are modeled by reduced-order models (ROMs) and the remaining part is modeled by a high-dimensional model (HDM). The multiple ROMs and HDM are combined based on an iterative domain decomposition method (DDM) with an arbitrary number of non-overlapping subdomains. A diagonal preconditioner is derived for the iterative DDM. The method is implemented based on a framework of the existing DDM-based analysis code. The ROM is implemented on GPU. Snapshot generation methods for the ROM are also proposed. Static structural analyses of nuclear power plant models are conducted using the combined multiple ROMs and HDM with multiple-subdomains model. The analyses are performed in parallel by the hybrid MPI-CUDA implementation. The convergence property of the iterative DDM is improved by the preconditioner.
16. 扩散界面肿瘤生长模型的预处理
Preconditioning for Diffuse Interface Tumor Growth Models
作者: Jessica Bosch, Pelin Çiloğlu, Chen Greif, Martin Stoll
类别: math.NA
发布日期: 2026-08-26
arXiv ID: 2608.25537v1
链接: http://arxiv.org/abs/2608.25537v1
摘要: Computational models of tumor growth form a contemporary area of study. Diffuse interface models have been proposed as an important tool for cancer modeling. In this work, we consider preconditioners for numerical simulations of two tumor growth models based on diffuse interface models. For numerical efficiency, we propose block preconditioners that rely on using effective Schur-complement approximations. We show that these methods combined with state-of-the-art adaptive finite element discretizations lead to robust simulations in two and three dimensions. In extensive numerical experiments, we show that the proposed methods show robust convergence behavior.
17. 面向大规模最小二乘问题的带充分性监控的自适应混合子空间莱文贝格-马夸特算法
Adaptive Hybrid Subspace Levenberg Marquardt Algorithm with Adequacy Monitor for Large Scale Least Squares Problems
作者: M. Duc Hoang, Timothy J. Lewis
类别: math.NA, cs.LG, math.OC
发布日期: 2026-08-26
arXiv ID: 2608.25524v1
链接: http://arxiv.org/abs/2608.25524v1
摘要: The Levenberg-Marquardt (LM) algorithm is the most widely used method for solving nonlinear least-squares problems, as it combines the robustness of steepest descent with the fast local convergence of the Gauss-Newton method. However, its computational cost can become prohibitive for large-scale problems because each iteration requires solving a large damped linear system, and conventional step acceptance strategies may require repeated solves as the damping parameter is adjusted. Despite this computational challenge, many large-scale least-squares problems exhibit effective low-dimensional structure, with only a small number of parameter-space directions strongly informed by the data. We propose an adaptive hybrid subspace Levenberg-Marquardt (HSLM) algorithm that constructs a low-dimensional subspace from complementary sources of gradient, memory, Krylov-subspace, and randomized curvature information and computes a spectrally damped LM step within this subspace. A distinguishing feature of the method is a deterministic adequacy monitor that quantifies how much descent information is captured by the reduced space and adaptively enriches the subspace when necessary. Step acceptance is decoupled from damping adjustment: Armijo backtracking determines the accepted step length, while the ratio of actual to predicted reduction is used solely to update the damping parameter, thereby avoiding repeated damped-system solves during step acceptance. For the HSLM algorithm, we establish global convergence to stationarity and prove local linear and superlinear convergence. Numerical experiments on neural-network training problems show that HSLM achieves convergence behavior comparable to classical and Krylov subspace LM (KSLM) while substantially reducing per-iteration computational cost, with increasing advantages observed as the parameter dimension grows.
18. 随机化学反应网络的数据驱动有效建模
Data-driven Effective Modeling of Stochastic Chemical Reaction Networks
作者: Yuan Chen, Weize Mao, Dongbin Xiu
类别: math.NA, cs.LG, math.DS
发布日期: 2026-08-26
arXiv ID: 2608.25421v1
链接: http://arxiv.org/abs/2608.25421v1
摘要: The Stochastic Simulation Algorithm (SSA), widely considered an exact algorithm for stochastic chemical reaction networks, suffers from high computational cost. In this work, we propose a data-driven effective model that operates on a user-defined coarse time step independent of the underlying microscopic reaction-event scale. This is accomplished by directly approximating the finite-time transition kernel of the continuous-time Markov chain induced by SSA, using a generative machine learning model trained on short bursts of SSA simulation data. The trained model constructs a stochastic propagator that recursively generates statistically consistent trajectories at the constant coarse time step, with significantly reduced computational cost. In this paper, we employ conditional normalizing flow as the stochastic propagator. A comprehensive set of numerical examples is presented to demonstrate the accuracy and efficiency of the proposed method.
19. 闵可夫斯基时空双曲叶状结构上的三维分部求和格式
A 3D Summation-by-Parts scheme on a Hyperboloidal Foliation of Minkowski
作者: Shalabh Gautam
类别: gr-qc, math-ph, math.AP, math.NA
发布日期: 2026-08-26
arXiv ID: 2608.25363v1
链接: http://arxiv.org/abs/2608.25363v1
摘要: This paper summarises our previous work on a fully $3$D Summation-by-Parts scheme, derived for a class of linear wave equations on hyperboloidal slices on a fixed Minkowski background. The scheme is derived in spherical polar coordinates, and allows having grid points at the origin and on the $z$-axis, despite coordinate singularities, and at infinity, by introducing compactification followed by rescaling, and is proved to be stable. Reducing it to the standard Cauchy problem, or to finite spacelike slices with an outer boundary, will follow a similarly. Second-order accurate finite-difference methods are used to implement this scheme numerically, but higher-order finite-difference or spectral methods could also be used. Kreiss-Oliger dissipation operators are generalized to curvilinear coordinates and are defined everywhere in the domain, including at the boundary points, such that they satisfy the dissipative property in the energy norms. We also propose new norm convergence tests that include all the grid points at all resolutions and produce more accurate results. Promising results are obtained, giving hope for application to fully nonlinear systems, like the Einstein Field Equations, and extracting the resulting gravitational waves free of systematic errors or gauge ambiguities.
20. 动力学Langevin动力学的标准Strang分裂的可证明非加速性
Provable Non-Acceleration of Standard Strang Splittings of Kinetic Langevin Dynamics
作者: Nawaf Bou-Rabee
类别: math.PR, math.NA, stat.CO, stat.ML
发布日期: 2026-08-26
arXiv ID: 2608.25279v1
链接: http://arxiv.org/abs/2608.25279v1
摘要: The OBABO and BAOAB schemes and the other standard Strang splittings of kinetic (underdamped) Langevin dynamics are widely used Markov chain Monte Carlo algorithms. Under a suitable friction scaling, the underlying diffusion relaxes on a ballistic time scale, suggesting that these discretizations, suitably tuned, sample targets with condition number $κ$ in $O(\sqrtκ)$ iterations. We prove that no fixed choice of step size and friction, based only on the curvature bounds and the dimension, achieves this acceleration: total variation mixing time lower bounds for OBABO show that ballistic cold-start mixing fails uniformly over the smooth strongly convex class, and the lower bounds extend, with the same orders, to BAOAB and the other four Strang splittings. The proof transfers non-acceleration from optimization to sampling. Eliminating velocity gives an exact noisy heavy-ball recursion, and by the non-acceleration theorem of Goujaud, Taylor and Dieuleveut, for every tuning either some Gaussian target has a mode with relaxation time at least of order $κ$, or an attracting cycle exists on a smooth potential; dilating such a potential as $U_R(x)=R^2U(x/R)$ preserves its curvature bounds and produces metastability for a number of steps exponential in $R^2$, from an initial state at Wasserstein distance $O(R)$ from equilibrium. Using contraction estimates of Leimkuhler, Paulin and Whalley and a Wasserstein-to-total-variation regularization estimate, we prove a complementary upper bound of $O(κ)$ steps, up to logarithmic factors, for a fixed-parameter OBABO tuning. Hence, among fixed-parameter OBABO tunings, the optimal condition-number dependence of cold-start total variation mixing over this class is linear, up to logarithmic factors. A direct Gaussian calculation also rules out fixed-parameter acceleration for the left-endpoint exponential integrator.
21. “截断傅里叶滤波”方法用于一般域中积分与卷积的快速高阶求值
“Truncated Fourier Filtering” method for fast and high-order evaluation of integrals and convolutions in general domains
作者: Oscar P. Bruno, Ambuj Pandey, Krishna Y. Poojara
类别: math.NA
发布日期: 2026-08-26
arXiv ID: 2608.25264v1
链接: http://arxiv.org/abs/2608.25264v1
摘要: This paper introduces and analyzes a novel algorithm—Truncated Fourier Filtering (TFF)—for the fast, high-order accurate evaluation of standard integrals and convolutions involving piecewise-smooth (possibly discontinuous) integrands over general $m$-dimensional domains ($m \ge 1$) employing an $m$-dimensional Cartesian grid. For an $N$-point discretization, the method runs at a computational cost of $\mathcal{O}(N)$ operations for standard integrals and $\mathcal{O}(N \log N)$ operations for convolutions, following, in either case, a one-time $\mathcal{O}(N \log N)$ precomputation step (not required in dimension $m = 1$). The core idea underlying TFF is to approximate the characteristic function of the integration domain by a truncated Fourier expansion of it over a suitably extended periodic domain, and to evaluate the resulting integrals via trapezoidal quadrature on a Cartesian grid with an appropriately chosen discretization size. Despite its conceptual simplicity, TFF attains high-order accuracy even for complex, possibly non-smooth or even non-Lipschitz geometries. A complete theoretical analysis is provided that establishes the superalgebraic convergence (i.e., convergence faster than any negative power of $N$) of the overall approach.
22. 哈密顿非线性波与三维流的双向耦合
Hamiltonian Two-Way Coupling of Nonlinear Waves and 3D Flows
作者: Sinan Wang, Ruicheng Wang, Taiyuan Zhang, Fan Feng, Jinjin He, Yuchen Sun, Zhiqi Li, Bo Zhu
类别: cs.GR, math.NA, physics.flu-dyn
发布日期: 2026-08-25
arXiv ID: 2608.25203v1
链接: http://arxiv.org/abs/2608.25203v1
摘要: Simulating large-scale free-surface water by coupling a localized 3D fluid solver to a cheaper 2D surface model has long faced a mismatch in wave dynamics: efficient 2D wave models used in graphics are typically either linear or non-dispersive. These models are fast, simple, and accurate for calm, small-amplitude seas, but coupling them with strongly nonlinear 3D solvers produces visible reflections and artifacts at the 2D–3D interface. We address this problem by introducing a nonlinear and dispersive 2D wave model based on the canonical Zakharov formulation. Its Hamiltonian structure, in which the surface elevation and surface potential form a canonical pair ($η$, $ψ$) governed by the wave energy, enables a canonically consistent two-way coupling scheme, allowing information to pass smoothly across the 2D–3D interface. Our 2D solver reduces mean wave-height error by 1.7–5$\times$ over SWE, BEM, and Airy baselines while running more than $10^3\times$ faster than BEM; it achieves greater nonlinear accuracy and coupling fidelity than SWE and Airy, with minor losses in speed and stability. Coupling it with a 3D Navier–Stokes solver yields a full system that suppresses visible seam artifacts across a range of experiments, including dispersion-matching and Kelvin-wake tests, and runs over 4$\times$ faster than a pure GPU NB-FLIP simulation on the same domain.
23. ROMNet：一种混合降阶建模与机器学习的波形反演方法
ROMNet: a hybrid reduced order modeling and machine learning approach to waveform inversion
作者: Liliana Borcea, Alexander Mamonov, Kui Ren, Haizhao Yang, Chugang Yi
类别: math.NA, cs.LG, physics.geo-ph
发布日期: 2026-08-25
arXiv ID: 2608.25160v1
链接: http://arxiv.org/abs/2608.25160v1
摘要: Waveform inversion seeks to estimate the wave speed of a heterogeneous, inaccessible medium, from time-resolved measurements of the waves at user controlled sensors. We consider this inverse problem for acoustic waves and an active array of source/receiver sensors that emit probing signals and measure the generated pressure waves. The forward map, from the wave speed to the measurements, is nonlinear and oscillatory. The oscillations cause cycle skipping, the main impediment to using the standard, nonlinear least-squares data fitting formulation, known as full waveform inversion (FWI). A recently introduced alternative waveform inversion approach computes from the measurements an algebraic surrogate of the wave operator, a reduced order model (ROM) matrix, which is then used to estimate the wave speed. The mapping from the measurements to the ROM is nonlinear, but well understood. It is computed efficiently, in a non-iterative manner. The nonlinear mapping from the ROM to the wave speed is less understood, and its approximation involves time-consuming optimization. Our goal in this paper is to use a neural network to map the ROM matrix to a nearby one, that has a simpler and explicit dependence on the wave speed. This simplifies and reduces the computational cost of the ROM-based waveform inversion. We introduce the methodology, called ROMNet, and test it with numerical simulations, using two training data sets: The first set consists of random media with variations of the wave speed modeled by a superposition of Gaussians with random amplitudes and standard deviations. The second is the publicly available GeoFWI dataset introduced for benchmarking FWI using deep learning. We compare the performance of ROMNet with the direct ROM-based inversion and with two representative deep learning approaches to FWI: Fourier-DeepONet" and InversionNet".
24. 可压缩Navier-Stokes方程ALE格式的熵稳定运动壁边界条件
Entropy-stable moving-wall boundary conditions for the ALE formulation of the compressible Navier-Stokes equations
作者: Luca Galimberti, Roberto Nuca, Lisandro Dalcin, Alberto Guardone, Matteo Parsani
类别: math.NA, math.AP
发布日期: 2026-08-25
arXiv ID: 2608.25146v1
链接: http://arxiv.org/abs/2608.25146v1
摘要: We present a high-order entropy-stable framework for the compressible Euler and Navier-Stokes equations on moving domains. The space-time mapping describing the domain motion is recast in an arbitrary Lagrangian Eulerian (ALE) formulation, in which the physical inviscid fluxes and the contributions induced by mesh motion are treated in a unified manner. At the continuous level, we prove that the proposed moving-wall boundary conditions are entropy conservative for the Euler equations and entropy stable for the Navier-Stokes equations. The no-slip condition is formulated in terms of the velocity relative to the moving wall, yielding a bounded inviscid contribution to the entropy balance, while the viscous terms contribute only entropy dissipation. Using diagonal norm summation-by-parts (SBP) operators together with appropriate numerical fluxes, these properties are extended to the semi-discrete formulation, resulting in nonlinear stability in the $L^2$ sense. The accuracy, robustness, and scalability of the proposed method are demonstrated in practice through an extensive set of numerical experiments, ranging from canonical two-dimensional verification cases to large-scale turbulent and supersonic simulations involving moving boundaries and fluid-structure interaction. The results confirm the suitability of high-order entropy-stable schemes for complex moving-domain problems across a broad range of flow regimes and multiphysics applications. Because the analysis relies on the SBP property rather than on a particular discretization, the framework naturally extends to a broad class of methods based on diagonal-norm SBP operators, including finite volume, finite element, and flux reconstruction schemes.
25. 关于由前$N$个Rademacher函数张成的空间中$L_2$范数的精确离散化
On exact discretization of the $L_2$-norm in the space spanned by the first $N$ Rademacher functions
作者: Anna Kazakova
类别: math.FA, math.NA
发布日期: 2026-08-25
arXiv ID: 2608.25058v1
链接: http://arxiv.org/abs/2608.25058v1
摘要: We study the exact discretization of the $L_2$-norm in the space spanned by the first $N$ Rademacher functions. It is shown that the sufficient number of nodes for discretization with the minimal number of nodes is equal to $N$ or $N+1$ and depends on the dimension $N$. The connection with Hadamard matrices and the Hadamard conjecture is demonstrated.