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1. 基于优化区域分解方法的混合维Biot问题
Hybrid-Dimensional Biot Problem with an Optimization Based Domain Decomposition Approach
作者: Francesca Marcon, Stefano Scialò
类别: math.NA
发布日期: 2026-07-20
arXiv ID: 2607.17885v1
摘要翻译: 本文提出了一种数值方法，用于求解裂缝性多孔介质中的流固耦合问题，该介质被表示为混合维区域。在该公式中，允许三维网格的单元任意穿过裂缝。通过在三维网格上使用扩展有限元方法（XFEM），位移在裂缝处不连续。力学问题被表述为一个鞍点问题，其中使用拉格朗日乘子来强制裂缝处的位移连续性。所得的拉格朗日乘子表示作用在裂缝表面上的应力场。类似地，通过在不协调网格上使用XFEM公式，允许压力场在裂缝处不连续，并使用专门为混合维问题设计的基于优化的区域分解策略来计算压力场。采用固定应力分裂格式解耦流动和力学子问题，而在每个固定应力迭代中使用共轭梯度（CG）方法求解混合维压力问题。固定应力格式与CG求解器的组合对于这类问题证明是高效的。
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链接: http://arxiv.org/abs/2607.17885v1
1. 研究对象和背景
本文研究多孔介质中流体流动与固体变形的耦合问题, 其背景源于地热能开发, 二氧化碳地质封存, 地下水资源管理及地震活动分析等应用. 研究的几何对象是一个三维多孔基岩域 D, 其中包含一个或多个平面裂缝 F^j (j=1,…,#F), 裂缝被建模为二维流形(离散裂缝-基质模型, DFM). 记 ˚D = D \ ∪j F^j 为不含裂缝的矩阵域. 在裂缝表面, 流场与力学场均可能发生间断. 物理上, 采用线性 Biot 模型描述饱和多孔介质中孔隙压力与骨架变形的耦合: 在 ˚D 上满足\n\n$$\frac{\partial}{\partial t}\left(\frac{1}{M}p_D + \alpha \nabla\cdot \mathbf{u}\right) - \nabla\cdot(K_D \nabla p_D) = g, \qquad (x,t)\in \mathring{D}\times(0,T], \tag{1}$$\n$$\alpha\nabla p_D - \nabla\cdot \boldsymbol{\sigma}(\mathbf{u}) = 0, \qquad (x,t)\in \mathring{D}\times(0,T], \tag{4}$$\n\n其中 p_D 为孔隙压力, u 为位移, M 为 Biot 模量, α 为 Biot-Willis 系数, K_D = k_D/μ 为基质渗透率与流体粘度之比, g 为源汇项. 应力-应变本构为线弹性: σ(u) = C ε(u), C 为四阶弹性张量, ε(u) = (∇u + ∇uᵀ)/2. 在每个裂缝 F^j 上, 二维流动满足 Darcy 定律:\n\n$$-\nabla\cdot(K_{F^j}\nabla p_{F^j}) + K_D\nabla p_{D,+}^j\cdot\mathbf{n}_+^j + K_D\nabla p_{D,-}^j\cdot\mathbf{n}_-^j = 0, \qquad (x,t)\in F^j\times(0,T], \tag{2}$$\n\n以及矩阵-裂缝窜流条件:\n\n$$K_D\nabla p_{D,\pm}^j\cdot\mathbf{n}_\pm^j = -\eta^j(p_{D,\pm}^j - p_{F^j}), \qquad (x,t)\in \omega_\pm^j\times(0,T], \tag{3}$$\n\n这里 η^j = k\perp^j/μ 为裂缝法向传导系数, K_{F^j} = k_\parallel^j/μ 为裂缝切向传导系数, p_{D,\pm}^j 分别表示压力在裂缝两侧的迹. 在力学上, 裂缝两侧应力连续条件由 Lagrange 乘子 Λ^j 表示:\n\n
$$\big(\boldsymbol{\sigma}_\pm^j(\mathbf{u}) - \alpha p_{D,\pm}^j\mathbf{I}\big)\cdot\mathbf{n}_\pm^j = \Lambda^j - \alpha p_{F^j}\mathbf{n}_\pm^j, \qquad (x,t)\in \omega_\pm^j\times(0,T], \tag{5}$$\n\n且位移连续性约束:\n
$$\mathbf{u}_+^j = \mathbf{u}_-^j \quad \text{在} F^j\times(0,T]. \tag{6}$$\n\n边界条件取齐次 Dirichlet/Neumann 条件, 初始条件为静止.\n\n尽管 Biot 方程的数值求解已有广泛研究, 但在含裂缝的混合维情形下, 传统方法通常要求计算网格与裂缝位置匹配(即网格面严格贴合裂缝), 这给复杂裂缝网络(如大量相交裂缝, 非平面裂缝)的网格生成带来巨大困难. 文献中已有的工作, 如[9]使用混合有限元-有限体积法, [7]从降维 Brinkman-Biot 模型出发, [3]结合接触力学条件, [6]使用完全匹配网格并分析固定应力参数敏感性, 均要求匹配网格. 为突破这一限制, 本文在非匹配网格上发展数值方法. 其基本思路是: 利用扩展有限元(XFEM)引入 Heaviside 或 bubble 型富集函数捕捉跨越裂缝的间断; 并通过优化基区域分解策略将全局问题解耦为矩阵三维问题和各个裂缝二维问题, 从而允许对矩阵和裂缝独立划分网格. 此外, 耦合问题通过固定应力分裂格式(见[13])将流动与力学迭代解耦, 每一时间步内交替求解压力方程和力学方程. 本文的目标正是将上述思想系统性地推广到混合维 Biot 问题, 给出完整的空间-时间离散、分裂格式、收敛性证明以及数值验证. 尤其值得指出的是, 文中给出了固定应力分裂在该非匹配混合维离散框架下的严格收敛性分析, 这是以往文献中未曾涉及的.\n\n本文试图填补的空白可概括为: 在无需网格匹配的条件下, 构造并分析一套求解含有多条裂缝的三维-二维 Biot 方程的高效数值方案, 并证明所采用的固定应力迭代的线性收敛性. 该工作为复杂裂缝性多孔介质中的流固耦合模拟提供了兼具灵活性和理论保证的计算框架.
2. 主要定理或主要结果
本文的核心理论结果是关于固定应力分裂格式在混合维非匹配离散下的收敛性证明. 在假设 A.1 (α, λ, μ > 0, 1/M, K_D ≥ 0) 和 A.2 (对每条裂缝 η^j > 0, K_{F^j} ≥ 0) 之下, 作者定义了第 k 时间步内固定应力迭代的误差量(见(39)): e^{k,i}{p_D} = (p_D)^{k,i} - (p_D)^k, e^{k,i}{p_F^j} = (p_F^j)^{k,i} - (p_F^j)^k, e^{k,i}u = u^{k,i} - u^k, e^{k,i}Λ = (Λ^j)^{k,i} - (Λ^j)^k, 其中 (·)^k 表示精确的时空离散解(由 Problem 4 给出), (·)^{k,i} 表示第 i 次迭代解. 通过误差方程(43)–(46)及细致的能量估计, 定理 1 得到以下三个不等式:\n\n1. 力学误差受压力误差控制:\n$$2\mu \|\epsilon(e^{k,i}_u)\|_{0,\mathring{D}}^2 + \lambda \|\nabla\cdot e^{k,i}_u\|_{0,\mathring{D}}^2 \le \frac{\alpha^2}{K_{\text{dr}}} \|e^{k,i}_{p_D}\|_{0,\mathring{D}}^2. \tag{40}$$\n\n2. 裂缝压力误差受其两侧压力跳跃控制:\n$$K_{F^j} \|\nabla e^{k,i}_{p_{F^j}}\|_{0,\omega^j}^2 + 2\eta^j \|e^{k,i}_{p_{F^j}}\|_{0,\omega^j} \le \frac{\eta^j}{\delta_F} \|(e^{k,i}_{p_D})^j_\pm\|_{0,\omega^j}^2, \quad \forall j, \tag{41}$$\n其中 δ_F ≤ 2, |(e^{k,i}{p_D})^j\pm|{0,\omega^j}^2 = \int{\omega^j} ([(e^{k,i}{p_D})+^j]^2 + [(e^{k,i}{p_D})-^j]^2).\n\n3. 压力误差自身满足压缩性(即误差每次迭代按常数因子衰减):\n
$$\left(\frac{1}{M} + \frac{\alpha^2}{2K_{\text{dr}}} + \frac{\Delta t\, K_D}{C_P}\right) \|e^{k,i}_{p_D}\|_{0,\mathring{D}}^2 \le \frac{\alpha^2}{2K_{\text{dr}}} \|e^{k,i-1}_{p_D}\|_{0,\mathring{D}}^2. \tag{42}$$\n这里 K_{\text{dr}} = λ + (2/3)μ 为排水体积模量, C_P 为 Poincaré 常数. 由(42)立即得到压缩因子\n
$$\rho = \frac{\alpha^2/(2K_{\text{dr}})}{1/M + \alpha^2/(2K_{\text{dr}}) + \Delta t\, K_D/C_P} < 1,$$\n从而保证迭代收敛.\n\n在数值实现层面, 3D 压力空间 V^{f,D}_h 通过 XFEM 构造(22): 标准线性 Lagrange 元加上两类富集函数——对完全穿越域 D 的裂缝采用符号函数 H^j, 对终止于域内的裂缝采用 bubble 函数 E^j 与 H^j 的乘积, 并作平移调整以保证分片线性性质. 力学位移空间 V^m_h 按同样方式构造. 每条裂缝上的标量空间(压力和 Lagrange 乘子)采用标准线性有限元.\n\n进一步的数值实验部分(第 6 节, 原文未提供细节)显示, 固定应力迭代与共轭梯度法求解流动子问题的组合对于此类问题非常高效. 但具体的收敛阶, 参数影响和计算时间等定量结果需要查阅原文第 6 节, 此处从已提供文本无法明确给出.
3. 研究方法、关键技术和核心工具
本文的数值方案建立在三个核心技术的有机结合之上: 扩展有限元(XFEM)的间断表示, 固定应力算子分裂, 以及优化基区域分解. 其技术上的主要障碍在于: 如何在不要求网格匹配裂缝的条件下, 既精确捕捉压力和位移在裂缝两侧的跳跃, 又保持混合维耦合的物理一致性, 同时还要在迭代解耦的过程中保证整体收敛. 作者围绕这些障碍展开分析框架.\n\n1. XFEM 间断逼近与富集函数构造.\n非匹配网格上处理裂缝的关键是引入能够表示跨裂缝间断的插值基函数, 且不增加全局自由度总数. 文中对 3D 压力 p_D 和位移 u 均采用 XFEM 空间(22). 定义 P_1( T_D ) 为 3D 网格 T_D 上满足齐次 Dirichlet 条件的标准线性 Lagrangian 有限元空间. 对于每条裂缝 F^j, 作者区分两种情况:\n - 若 F^j 完全穿越计算域, 则使用 Heaviside 函数 H^j( x ) = sign( n_{F^j}·(x - x_0) ) 来构造间断, 相应富集基函数为 H^j_k = ϕ_k(x)(H^j(x) - H^j(x_k)).\n - 若 F^j 终止在域内, 直接使用 H^j 会导致域外的虚假间断, 因此需引入一个在裂缝边界上光滑归零的 bubble 函数 E^j. 其定义基于裂缝内部边缘: 在裂缝局部坐标系下, 对每条内部边缘所在的直线方程 ε_i^j 取乘积, 并通过简单缩放保证在裂缝重心处取值为 1. 最终富集函数为 H^j(x)E^j(x), 并经平移调整为 E^j_k = ϕ_k(x)( H^j(x)E^j(x) - H^j(x_k)E^j(x_k) ). 作者特别指出参数 p 控制函数光滑度, 数值实验中取 p = 1/2. 这一构造保证了在裂缝内部单元上函数具有所需的间断性, 而在远离裂缝的区域其对标准有限元空间无扰动(因为当 x 不在裂缝影响区时 E^j(x)=0). 这种技术使得网格完全不需贴合裂缝, 大大降低了网格生成难度.\n\n2. 固定应力分裂及其收敛性证明.\n耦合系统(29)-(32)是全离散(向后 Euler 时间步 + 空间离散)后所得. 为了避免整体求解的巨大计算量, 采用固定应力迭代: 在第 i 次迭代, 先以已知的位移 u^{k,i-1} 解压力方程, 再以所得压力解力学方程, 并反复迭代至满足容差. 为恢复迭代收敛性, 需要在压力方程中增加稳定项 L(p_D^{k,i} - p_D^{k,i-1}), 其中 L 取物理启发的值 L = α^2/K_{\text{dr}}, 该选择等价于在整个迭代过程中“固定”体积应力平均量(见(33)). 收敛性证明是本文的理论核心. 证明框架如下:\n - 首先写出线性误差满足的方程(43)-(46).\n - 从力学误差方程(45)取试函数 v_h = e^{k,i}u, 并利用约束(47)得到(48). 再结合 Young 不等式和不等式(34), 导出力学能量与压力误差的关系(40).\n - 由裂缝压力误差方程(44)取 q{F^j} = e^{k,i}{p{F^j}}, 同样应用 Young 不等式得到(41).\n - 组合压力误差方程(43)与力学、裂缝误差的能量恒等式, 关键处理项为 ∫ α e^{k,i}{p_D} ∇·(e^{k,i}u - e^{k,i-1}u). 利用(45)的差分形式(57)及 Cauchy-Schwarz 与(34), 将该位移差项用压力差控制(58). 将所有非正定项移至右端后, 反复使用 Young 不等式合并同类项, 最终消去位移和裂缝压力项, 得到仅含压力误差的压缩不等式(59)-(60), 即(42). 整个过程需精细选取 Young 系数 δ_U ∈ [1,2], δ_P ∈ [1,2], δ_F ≤ 2, 保证最后等式左端所有系数为正. 最终收敛因子为 α^2/(2K{\text{dr}}) 与 (1/M + α^2/(2K{\text{dr}}) + ∆t K_D/C_P) 之比, 显然小于 1. 该证明技巧拓展了经典固定应力收敛分析(如[19])至带多条裂缝与非匹配网格的情形, 且不依赖特定的有限元空间, 适用于较为一般的 XFEM 空间.\n\n3. 优化基区域分解用于流动子问题.\n在固定应力迭代的每一步, 需在给定 u^{k,i-1} 下求解三维-二维耦合 Darcy 流问题(35)-(36). 直接整体求解仍涉及三维矩阵与多条裂缝的耦合. 作者借鉴[16]的思想, 通过引入每条裂缝的额外界面变量 Ψ^j+, Ψ^j_-, Ψ^j_F, 将原耦合问题转化为一个等式约束下的最小化问题(61): 最小化 ∑j ( ‖(p_D)+^j - Ψ^j_+‖^2 + ‖(p_D)-^j - Ψ^j-‖^2 + ‖p_{F^j} - Ψ^j_F‖^2 ), 同时满足弱形式(35)-(36)将界面变量替换后得到的约束. 这种等价形式允许分别求解各子域: 固定界面变量时, 3D 压力 p_D 和每个 2D 压力 p_{F^j} 独立求解; 然后按梯度迭代更新界面变量. 算法最终等价于一个界面空间的共轭梯度(CG)迭代, 其中矩阵-向量乘积只需求解解耦的子域问题, 因此实现简单且可扩展. 当界面空间取为裂缝网格上的连续线性有限元或分片常数时, CG 被证明非常高效. 该策略彻底分离了几何复杂性, 使得三维网格生成与裂缝离散完全解耦.\n\n综合而言, 本文的技术创新不在于单一的公式推导, 而在于系统性地整合上述三部分, 为混合维 Biot 问题提供了一套可被严格分析的求解思路. 其中收敛性证明中的误差组合与 Young 不等式参数的选取尤其体现了作者对解耦过程中能量传递的精确控制.
4. 与之前工作的比较
本文的工作与既有有限元法及区域分解文献在多个维度上可进行比较, 以下分若干方面具体说明.\n\n1. 与匹配网格方法的比较.\n已有的含裂缝多孔介质 Biot 问题数值方法大多要求计算网格与裂缝几何一致. 例如, [9] 研究 3D-2D 非线性 Biot 模型时采用混合有限元-有限体积离散, 并假定网格在裂缝处匹配; [3] 将 3D Biot 方程与接触力学条件结合, 同样使用匹配网格; [6] 在 3D 基质和 2D 裂缝上均采用标准有限元, 网格完全符合裂缝, 并针对固定应力稳定参数进行敏感性分析. 这些方法对于少量简单裂缝是成功的, 但当裂缝数量众多、相互交错或者非平面时, 生成匹配网格变得极其困难甚至无法实现. 本文提出的 XFEM + 优化域分解方法的核心优势正是在于完全摒弃了这一要求: 3D 网格 T_D 可以独立于裂缝生成(例如一个简单的背景四面体网格), 而裂缝网格 T_{F^j} 也各自独立, 仅需知道裂缝与背景网格的相交信息以构造 XFEM 富集函数. 这带来了极大的网格划分灵活性, 尤其在裂缝网络几何复杂时更为突出. 但是, 这种灵活性以引入额外界面变量和迭代求解子问题为代价: 优化域分解的每个 CG 迭代需要求解解耦的 3D 和 2D 问题, 这可能比直接整体求解增加了一次迭代的算术强度, 然而作者声称 CG 收敛足够快, 整体效率依然良好.\n\n2. 与之前 XFEM 及区域分解工作相比.\n将 XFEM 用于多孔介质流固耦合问题并非首次出现, 但本文的独特之处在于与优化域分解的结合以及给予固定应力迭代的收敛性分析. 在纯 Darcy 流方面, [16] 提出了优化基区域分解处理 3D-2D 耦合问题并允许多条裂缝, 本文正是将该思想从 Darcy 流推广到 Biot 孔力学, 并加入了力学方程的 XFEM 离散. 之前的 [10] 也曾用 Lagrange 乘子处理裂缝两侧位移连续性, 但网格是匹配的; 本文则在非匹配网格下同时处理力学与流动. 在 XFEM 相关文献中, 作者提到 [20] 的 bubble 函数构造方法, 但本文将其移植到混合维多裂缝的情形, 并针对完全穿越与内部终止两种裂缝类型分别处理, 这提升了方法的通用性. 此外, 在固定应力分裂的收敛分析方面, 早期工作 [13] 提出了该分裂, 并在均质弹性无裂缝情形证明了收敛; [19] 给出了更一般框架下的收敛证明; [5] 将其推广到弹塑性问题. 本文的收敛证明(定理 1)沿着 [5,19] 的路线, 但首次将其扩展到带有多条裂缝且采用 XFEM 和优化区域分解的混合维问题. 证明中额外需要处理裂缝界面项 e_{p_{F^j}} 和 e^{k,i}Λ, 以及 XFEM 间断带来的能量恒等式调整. 作者巧妙利用位移约束(47)消去 Lagrange 乘子项, 并通过 Young 不等式将裂缝能量吸收至右端, 最终仍得到只关于 p_D 的压缩估计. 这体现了该分析框架的强健壮性.\n\n3. 与固定应力分裂中常规稳定化比较.\n在经典固定应力分裂中, 稳定参数 L 的选择直接影响收敛速度. 文献中常见 L = α^2/K{\text{dr}} (物理驱动)和 L = α^2/(K_{\text{dr}} ^u) (基于不排水体积模量)等. 本文采用前者, 并在收敛证明中有效地利用了 (34) 式将体积应变能的系数与 K_{\text{dr}} 联系起来, 从而导出精确的压缩因子. 对于多裂缝情况, 该参数依然适用且与裂缝参数无关, 这简化了实际计算. 不过, 文中收敛因子与 Δt、K_D 和 Poincaré 常数 C_P 有关: 当 Δt 较大或 K_D 较大(高渗透性)时, 收敛变慢, 但与裂缝有关的量未出现在压缩因子中, 说明裂缝的存在并未恶化固定应力迭代的收敛性质, 这是本文的一个重要观察.\n\n4. 数值效果方面.\n由于本文的第 6 节未在提供内容中详细展现, 无法明确给出收敛阶和计算效率数据. 从摘要和结论性语句可知, 固定应力与 CG 的组合对于此类问题非常有效. 但具体与匹配网格方法的精度、效率定量对比, 以及 XFEM 富集函数中参数 p 的影响、界面离散的选择等细节, 需要读者查阅原文完整版本. 在此, 我们只能指出这些潜在的比较维度, 而无法给出确切的数值结论.\n\n总结而言, 本文相对于先前工作, 主要推进了三点: 将非匹配网格处理能力从 Darcy 推广到 Biot 方程; 给出了该框架下固定应力分裂的严格收敛性及其误差估计; 通过优化区域分解实现了压力问题的稳健且解耦的求解. 其假设条件(A.1, A.2)与经典 Biot 问题一致, 未附加额外的光滑性或参数限制, 适用范围与匹配网格方法相当. 局限性则在于: 当前分析限于线弹性、小变形和线性 Darcy 流, 且不考虑裂缝的接触和滑移(位移连续性条件(6)为强约束), 未涉及非线性渗流或材料非线性; 此外 XFEM 的实现相对复杂, 需要几何相交检测和特殊积分. 这些将是未来可能的扩展方向.
了解领域论文（摘要翻译）
1. 自适应Mamba神经算子
Adaptive Mamba Neural Operators
作者: Zeyuan Song, Zheyu Jiang
类别: cs.LG, math.AP, math.NA
发布日期: 2026-07-20
arXiv ID: 2607.18043v1
链接: http://arxiv.org/abs/2607.18043v1
摘要翻译
中文标题: 自适应Mamba神经算子
摘要翻译: 在任意几何形状和各种网格上精确求解偏微分方程（PDEs）是科学与工程应用中的重要任务。本文提出自适应Mamba神经算子（AMO），它通过为状态空间模型（SSMs）集成再生核而非SSMs的核积分表示来实现。这是通过构建PDEs的Takenaka-Malmquist系统实现的。AMO提供了与自适应傅里叶分解（AFD）理论相契合的新表示，并能逼近各种几何形状和网格上的PDEs解流形。在流体物理、固体物理和金融领域的多个具有挑战性的基准PDE问题中，包括点云、结构化网格、规则网格和不规则域，AMO在相对$L^2$误差方面一致优于最先进的求解器。总体而言，本工作为设计可解释的神经算子框架提供了一个新范式。
2. 演化曲面上抛物型偏微分方程的后验误差估计
A posteriori error estimates for parabolic PDEs on evolving surfaces
作者: Michael Lantelme
类别: math.NA
发布日期: 2026-07-19
arXiv ID: 2607.17416v1
链接: http://arxiv.org/abs/2607.17416v1
摘要翻译
中文标题: 演化曲面上抛物型偏微分方程的后验误差估计
摘要翻译: 我们针对封闭演化曲面上的抛物型曲面偏微分方程，导出了基于残量的后验误差估计。主要贡献在于证明了所提误差指示子的有效性与可靠性，该指示子从上（全局）界住误差量，并从下（空间全局、时间局部）界住。我们将稳态曲面上抛物型偏微分方程的适应性方法推广至演化曲面，允许非平凡的粗化。文中给出多个数值实验，展示了误差的渐近行为以及加密与粗化策略的有效性。
3. FEVessel: 基于无标签预训练有限元方法的三维压力容器网格无关分析
FEVessel: Mesh-Independent Analysis of 3D Pressure Vessels with the Label-Free Pretrained Finite Element Method
作者: Yipin Sun, Yizheng Wang, Yuzhou Lin, Baiyang Zheng, Xiaoying Zhuang, Timon Rabczuk
类别: math.NA
发布日期: 2026-07-19
arXiv ID: 2607.17318v1
链接: http://arxiv.org/abs/2607.17318v1
摘要翻译
中文标题: FEVessel: 基于无标签预训练有限元方法的三维压力容器网格无关分析
摘要翻译: 在化工、核能及新能源工业中，压力容器分析需要在众多材料、几何和载荷下求解相同的弹性问题，其中网格质量和重复求解决定了精度与成本。有限元方法(FEM)无法分摊这种重复成本，并且在退化网格上失效，而旨在替代它的神经算子仍然需要由FEM生成的带标签数据。本文提出FEVessel，将预训练有限元方法(PFEM)推广至三维压力容器，并验证了其在上述两个限制上的四项能力。FEVessel i) 将每个容器编码为带有坐标、材料和载荷通道的点云，ii) 在总势能上预训练Transolver算子，而非使用FEM标签，iii) 利用其预测结果对迭代求解器进行热启动。单一模型泛化到不同材料、几何和边界条件时，相对位移误差为$1.35\%$，其$2.07\%$的应变误差比监督式傅里叶神经算子的$9.72\%$低约$4.7$倍，后者的结构化网格无法保持沿厚度方向的应变。其热启动将代数多重网格迭代次数从$195$次降至$18$次，在$10^{-3}$工程容差下实现了$9.2$倍的端到端墙上时钟加速。模型可在不用重新训练的情况下跨网格分辨率迁移，当训练点密度仅为$30\%$时，误差保持约$3\%$。在FEM失效的反转和扁片网格上，误差保持在$3.66\%$以下。据我们所知，这是首次对具有退化网格的工业相关三维压力容器进行网格无关求解的系统性研究。由于训练无需标签，FEVessel恰好在FEM无法提供标签的场景下发挥作用，从分析流程中消除了手动网格修复。
4. 扭曲四边形网格上带几何修正的Raviart–Thomas元
Raviart–Thomas Elements with Geometric Correction for Distorted Quadrilateral Meshes
作者: So-Hsiang Chou
类别: math.NA
发布日期: 2026-07-19
arXiv ID: 2607.17285v1
链接: http://arxiv.org/abs/2607.17285v1
摘要翻译
中文标题: 扭曲四边形网格上带几何修正的Raviart–Thomas元
摘要翻译: 基于Raviart–Thomas空间的混合有限元方法广泛用于通量形式的二阶椭圆问题的数值逼近。然而，在四边形网格上，从参考元到实际元的双线性映射引入了空间变化的雅可比行列式，这可能会违反标准Raviart–Thomas空间的包含性质 $\mathrm{div}\,V_h \subset W_h$。本文中，我们提出了四边形网格上经典Raviart–Thomas元的一个简单修正。该修正包括在局部基函数中添加几何驱动的修正项，以补偿双线性映射引入的几何扭曲。得到的空间保持了与经典Raviart–Thomas元相同的维数和自由度，同时恢复了相容性性质。我们给出了构造此类修正空间的一般框架，并通过对最低阶和次低阶Raviart–Thomas元发展修正版本来示例该方法。理论分析在四边形网格的标准形状正则性假设下建立了最优逼近性质。在扭曲网格上的数值实验证实了预测的收敛率，并表明随着几何扭曲增加，修正元始终比经典Raviart–Thomas格式提供改进的精度。
5. Ricci流的有限元方法收敛性
Convergence of Finite Element Methods for Ricci Flow
作者: Guangwei Gao, Evan S. Gawlik, Buyang Li
类别: math.NA
发布日期: 2026-07-19
arXiv ID: 2607.17051v1
链接: http://arxiv.org/abs/2607.17051v1
摘要翻译
中文标题: Ricci流的有限元方法收敛性
摘要翻译: 证明了二维Ricci流有限元离散的收敛性。在该方法中，二维曲面上的Ricci流被表述为解驱动的度量演化，度量演化由Gauss曲率驱动。Gauss曲率满足一个抛物方程，该方程又依赖于度量，从而增强了问题的抛物结构。解驱动的度量演化形式通过有限元方法离散，有限元逼近的收敛性通过采用文献中最初为研究外蕴曲率流中的解驱动曲面演化而发展的矩阵–向量形式来证明。除了收敛性，所提方法还在离散层面上保持了Ricci流的重要几何结构，例如面积守恒和Gauss–Bonnet定理。给出了广泛的数值实验，以展示所提方法的收敛性以及Ricci流的模拟。
6. 基于混合数据集的神经算子预处理求解亥姆霍兹方程：在经颅超声中的应用
Neural operator preconditioning from mixed dataset for the Helmholtz equations: Application to transcranial ultrasound
作者: Yanfei Xiang
类别: math.NA
发布日期: 2026-07-18
arXiv ID: 2607.16906v1
链接: http://arxiv.org/abs/2607.16906v1
摘要翻译
中文标题: 基于混合数据集的神经算子预处理求解亥姆霍兹方程：在经颅超声中的应用
摘要翻译: 本研究针对经颅超声应用中，二维亥姆霍兹方程离散化所产生的线性方程组序列，提出了一种神经算子预处理子空间方法。该问题涉及强烈非均匀、依赖患者的声速场，这些速度场会引起严重的波形畸变，对标准迭代求解器构成重大挑战。基于Giraud等人（HAL RR-9593, 2025）的神经网络预处理框架和Stanziola等人（JCP 441, 2021）用于学习优化器的理想化颅骨数据集，我们在六个混合声速-源数据集上训练了神经算子预处理器，这些数据集结合了随机源配置和基于理想化颅骨的速度场以及随机噪声。所提出的混合数据集策略旨在提高计算效率和不同配置下的泛化能力。神经算子在粗网格上进行训练，使用基于离散亥姆霍兹方程相对残差的物理信息损失函数，并作为非线性预处理器嵌入到灵活GMRES（FGMRES）中。数值实验表明，所得混合方法能够高效求解实际的经颅超声问题，网格尺寸比训练时所用网格大64倍，而经典GMRES和学习优化器在相当的计算预算内均未能收敛。此外，所提方法实现了任意精度的解，并在各种源和速度场配置下表现出强大的分布外泛化能力。本工作强调了科学机器学习中数据集设计的重要性，并为将无矩阵神经算子预处理与Krylov子空间方法相结合以求解实际的大规模亥姆霍兹问题提供了一个实用框架。
不相关论文（基本信息）
1. anyakrakusuma：用于理想几何上的熵正则化薛定谔桥的 Python 库
anyakrakusuma: A Python Library for Entropic Schrödinger Bridges on Idealized Geometries
作者: Sandy Hardian Susanto Herho, Dasapta Erwin Irawan, Agus Wahyu Jatmiko, Sito Fossy Biosa, Candrasa Surya Dharma, Edi Riawan, Astyka Pamumpuni, Rendy Dwi Kartiko, Rusmawan Suwarman, Deny Juanda Puradimaja
类别: physics.comp-ph, cs.MS, math.NA, math.OC
发布日期: 2026-07-20
arXiv ID: 2607.18184v1
链接: http://arxiv.org/abs/2607.18184v1
摘要: We present anyakrakusuma, an open-source Python library that solves the discrete static Schrödinger bridge problem, the entropically regularized counterpart of optimal transport, through a log-domain Sinkhorn–Knopp iteration and reconstructs the entropic interpolation between two empirical point clouds. The solver is paired with a diagnostic pipeline that characterizes the optimal coupling and the intermediate distributions through information-theoretic and geometric measures. We exercise the library on four idealized planar cases spanning a circle-to-circle dilation, a spiral-to-mixture fragmentation, a rigid reorientation of two moons, and a Lissajous-to-trefoil deformation. The log-domain formulation is necessary rather than merely convenient at the parameters studied, where the cost-to-regularization ratio reaches four hundred and the Gibbs kernel underflows double precision across most of its range; the iteration nonetheless attains a marginal residual of $10^{-9}$ and unit marginal fidelity in every case. Residual histories decay geometrically over approximately eight decades at per-iteration contraction factors between $0.966$ and $0.976$, which are local rates near the fixed point that lie many orders of magnitude below the worst-case Hilbert-metric bound. The covariance analysis recovers an imposed ninety-degree reorientation to within $0.07^\circ$, roughly forty times smaller than its uncertainty, across a masked interval of near-isotropy on which the principal axis is unobservable. The diagnostics are reported with explicit attention to the regimes in which each is well defined, including the differential entropy, which is meaningful only on the open interpolation interval. The presented cases are constructed rather than measured; quantitative application to empirical point clouds requires further study.
2. 探索性均衡中的反馈循环
Feedback Cycles in Exploratory Equilibria
作者: Chen-Hung Wu
类别: math.OC, math.NA, math.PR
发布日期: 2026-07-20
arXiv ID: 2607.18128v1
链接: http://arxiv.org/abs/2607.18128v1
摘要: Entropy regularization smooths equilibrium policies in time-inconsistent stochastic control. At low temperature, the same Gibbs response can strongly amplify errors in learned rewards and dynamics. We show that the derivative of an exploratory equilibrium is governed by a backward Volterra-parabolic resolvent. Along an aligned positive mode, a lower bound has the same exponential order. A block decomposition identifies the source of the amplification: causal paths contribute powers of 1/tau, whereas a positive feedback cycle can produce exponential growth. At fixed temperature, a local equilibrium branch is twice differentiable with respect to finite-dimensional model parameters, which yields a function-valued delta method. A bounded uniformly elliptic diffusion realizes this path-cycle distinction in every finite dimension. Closing one positive cycle changes the root-n linear-response boundary from a power law to order 1/log n; along the cyclic Perron mode, right-endpoint discretization is relatively consistent exactly when N tau^2 -> infinity. An affine model also gives an exact nonlinear transition at the Lambert-W temperature beta T / W(beta T sqrt(n)). Numerical calculations illustrate these rates.
3. Krasnosel’skii-Mann 迭代：超越渐近的组合分析
Krasnosel’skii-Mann iterations beyond asymptotics: a combinatorial analysis
作者: Mario Bravo, Roberto Cominetti
类别: math.NA
发布日期: 2026-07-20
arXiv ID: 2607.18121v1
链接: http://arxiv.org/abs/2607.18121v1
摘要: We revisit the classical Krasnosel’skii-Mann fixed point iteration for contractions and nonexpansive maps in general normed spaces. This iteration is ubiquitous across a wide range of areas, including convex optimization, monotone inclusions, Markov decision processes, under-relaxed methods for nonlinear PDEs, and more. Drawing on a remarkable connection with a Markov chain on $\mathbb{Z}^2$, and using counting arguments from enumerative combinatorics of lattice paths, we derive explicit estimates for the distance between iterates, as well as non-asymptotic error bounds for the fixed point residuals. As the contraction parameter approaches one, these bounds smoothly recover the known estimates for nonexpansive maps. Building upon these estimates, we further derive error bounds for inexact Krasnosel’skii-Mann iterations.
4. 矩阵指数的时间一致实极点有理逼近
Uniform-in-time rational approximation of the matrix exponential with real poles
作者: Stefan Güttel, Shuai Shao
类别: math.NA
发布日期: 2026-07-20
arXiv ID: 2607.18018v1
链接: http://arxiv.org/abs/2607.18018v1
摘要: We propose two new approaches for constructing families of rational functions with shared real poles that nearly uniformly approximate the functions $\exp(-tz)$ for $z\geq 0$ and $t$ in a positive time interval. The first result concerns the case where all real poles coalesce into a single point. With an appropriate choice of a weight function we are able to derive a closed formula for the asymptotically optimal location of such a pole. We then discuss the more general case where all real poles are distinct. Using Zolotarev’s construction of certain optimal rational functions, we present a simple algorithm to derive nearly optimal poles efficiently. We analyze the stability of the numerical evaluation of the resulting rational matrix functions in floating-point arithmetic. By controlling the growth of potential ill-conditioning arising from partial fractions, reliable and highly parallelizable exponential propagators are obtained.
5. Hermite插值的一种直接方法
A Direct Approach to Hermite Interpolation
作者: Simon Bossoney, Marc Troyanov
类别: math.NA
发布日期: 2026-07-20
arXiv ID: 2607.17998v1
链接: http://arxiv.org/abs/2607.17998v1
摘要: We introduce a family of polynomials satisfying the natural duality relations for Hermite interpolation, analogous to the classical Lagrange interpolation polynomials. They yield an explicit closed formula for the Hermite interpolant with arbitrary multiplicities, without recourse to divided differences, recursive corrections, or auxiliary Bézout identities. We also give a detailed account of Hermite’s original approach to the problem, based on an integral formula, which he used both to derive the interpolating polynomial and to estimate the interpolation error for holomorphic data.
6. 基于有理逼近的Cubature
Cubature from rational approximation
作者: Gentian Zavalani
类别: math.NA
发布日期: 2026-07-20
arXiv ID: 2607.17851v1
链接: http://arxiv.org/abs/2607.17851v1
摘要: We present a numerical construction of cubature rules for area integrals of analytic functions over planar domains with rectifiable Jordan boundary. The starting point is the Cauchy–Green identity. Given a weight $w$, we choose a $\bar\partial$-antiderivative $W$ and reduce the area integral to a contour integral involving the boundary values of $W$. These values are then approximated by a rational function with free poles, computed by the AAA algorithm. The poles inside the domain become cubature nodes, the corresponding residues become weights, and the boundary residual controls the error through an a posteriori estimate, rigorous once the continuous boundary residual is bounded. The same rule admits a dual reading, as the exact integral of a rational interpolant to the integrand, the area analogue of the one-dimensional interpolatory viewpoint. The numerical examples recover the disk mean-value rule and the focal-segment rule of the ellipse to machine precision, reproduce the exact finite quadrature identities of quadrature domains with both separated and confluent nodes, and evaluate logarithmic and Cauchy volume potentials from boundary data alone. The interior poles trace analytic skeletons that we identify tentatively with the mother bodies of potential theory, along with image points that appear without being imposed; for the square the observed convergence is root-exponential.
7. 矩阵$\varphi$-函数作用的增强Krylov方法的值域分析
Field-of-values analysis of augmented Krylov methods for matrix $\varphi$-function actions
作者: Xiaobo Liu, Marcel Schweitzer
类别: math.NA
发布日期: 2026-07-20
arXiv ID: 2607.17819v1
链接: http://arxiv.org/abs/2607.17819v1
摘要: We revisit established Krylov subspace methods for linear combinations of matrix $\varphi$-function actions from the viewpoint of the block triangular formulation of Al-Mohy and Liu [SIAM J. Sci. Comput., 48 (2026), pp. A726–A747]. In algorithms such as KIOPS [J. Comput. Phys., 372 (2018), pp. 236–255], one uses an augmentation approach based on evaluating the exponential of a slightly larger matrix that contains the operant vectors in its off-diagonal block, and its field of values may therefore grow substantially with these vectors. Typical convergence estimates for Krylov subspace methods result from bounding the error of polynomial approximations for the exponential on the field of values, so that only very pessimistic convergence estimates are available for these methods, in spite of their good practical performance. In contrast, the larger block formulation established by Al-Mohy and Liu involves an operator whose field of values is independent of the operant vectors, leading to more favorable convergence bounds. We work out the details of how these two approaches are connected to each other, which allows us to transfer the convergence bounds from the latter to the former, thus better explaining the observed performance.
8. 带有超线性漂移的随机微分方程的逐路径斜对称离散化
Pathwise skew-symmetric discretisation for SDEs with superlinear drift
作者: Yuga Iguchi, Samuel Livingstone, Giorgos Vasdekis, Rui-Yang Zhang
类别: math.NA, stat.CO
发布日期: 2026-07-20
arXiv ID: 2607.17735v1
链接: http://arxiv.org/abs/2607.17735v1
摘要: The skew-symmetric discretisation has recently been proposed as a new robust simulation method for weakly approximating stochastic differential equations (SDEs) with non-globally Lipschitz drift. This work develops a pathwise version of the scheme by representing the noise increment as a skew-normal distribution and coupling it with the driving Brownian increments, thereby enabling its use in the multilevel Monte Carlo (MLMC) framework. Under suitable conditions, we establish strong convergence of order 1/2 in $L^2$. Subsequently, the associated MLMC estimator is shown to have computational complexity $\mathcal{O} \bigl(\varepsilon^{-2} (\log (1/\varepsilon))^2 \bigr)$ to achieve a mean-squared error $\varepsilon^2$. We then analytically compare the proposed scheme with the tamed Euler scheme, another benchmark for robust discretisation. Under a strong inward-drift regime with the current state being far from the stable region, we show that the probability of moving in the wrong direction tends to vanish in the skew-symmetric scheme, whereas the tamed Euler scheme makes such moves with a non-trivial probability. Furthermore, in the MLMC setting, employing a one-dimensional stochastic Ginzburg-Landau model, we specify the range of step sizes for which the asymptotic variance of the coupled level difference obtained via the pathwise skew-symmetric scheme is lower than that obtained via the tamed Euler scheme. Numerical experiments on several model examples support the theoretical rate of strong convergence and demonstrate the stability and effectiveness of the resulting MLMC in the superlinear drift setting.
9. 区域分解的在线学习增强的非线性消除预条件子
A domain decomposition online-learning-enhanced nonlinear elimination preconditioner
作者: Pai Zhang, Linyan Gu, Li Luo
类别: math.NA
发布日期: 2026-07-20
arXiv ID: 2607.17723v1
链接: http://arxiv.org/abs/2607.17723v1
摘要: Nonlinearly preconditioned inexact Newton methods form an effective class of solvers for large-scale nonlinear algebraic systems arising from the discretization of partial differential equations. A central challenge in nonlinear elimination (NE) preconditioning is the reliable identification of the slowly converging components to be eliminated. Existing selection strategies often rely on problem-specific physical information or user-tuned thresholds applied directly to the raw nonlinear residual, which may contain irregular oscillatory structures near stagnation regions, making the selected bad subset highly sensitive to threshold parameters. In this work, we propose an online-learning-enhanced NE preconditioner that identifies the bad subset from the dominant structure of the nonlinear residual rather than from the raw residual itself. Residual snapshots are collected online during the stagnation phase of the current Newton solve, and an unsupervised extraction model is trained to capture the principal nonlinear imbalance. We consider both a linear extractor based on principal component analysis and nonlinear extractors based on autoencoder neural networks. Moreover, we integrate the approach into a parallel domain decomposition framework, which trains a local extraction model independently on each subdomain. The learned residual reconstruction is then used to define the bad subset and guide the nonlinear elimination process. Numerical experiments on lid-driven cavity flows at Reynolds numbers up to 10,000 show that the proposed method produces more reliable and coherent bad subsets, is robust with respect to both NE and learning parameters, and outperforms the baseline NE preconditioner in terms of the convergence.
10. 超图$p$-拉普拉斯算子的算子分裂算法及其在缺失数据恢复中的应用
An operator-splitting algorithm for the hypergraph $p$-Laplacian with applications to missing data recovery
作者: Kehan Shi, Jin Liu, Martin Burger
类别: math.NA
发布日期: 2026-07-20
arXiv ID: 2607.17606v1
链接: http://arxiv.org/abs/2607.17606v1
摘要: Hypergraph $p$-Laplacian regularization is a fundamental model in data analysis with successful applications in various tasks. It aims to minimize a nonsmooth and typically large-scale objective function defined as the sum of the $p$-th powers of the Lipschitz regularization over hyperedges. In this paper, we propose an operator-splitting algorithm for the hypergraph $p$-Laplacian that allows us to handle hyperedges separately in a Gauss-Seidel fashion. Each subproblem can be viewed as a generalized graph Lipschitz learning on a hyperedge, for which we introduce an auxiliary variable to overcome the nonsmoothness and solve it with one step of the alternating direction method of multipliers (ADMM). The resulting algorithm performs proximal ADMM updates sequentially over the hyperedges, and its convergence is proven. We test the algorithm on missing data recovery problems, including image sparse inpainting and semi-supervised learning, to demonstrate that it is faster than existing methods.
11. FlowSonic：通过高阶轨迹积分实现稳定的零样本音乐编辑
FlowSonic: Stable Zero-Shot Music Editing via High-Order Trajectory Integration
作者: Ali Boudaghi, Hadi Zare
类别: cs.SD, cs.LG, eess.AS, eess.SP, math.NA
发布日期: 2026-07-20
arXiv ID: 2607.17526v1
链接: http://arxiv.org/abs/2607.17526v1
摘要: Zero-shot text-guided editing of real-world music recordings requires balancing semantic modification with faithful preservation of the original musical structure. Although recent diffusion transformers trained with rectified flow have achieved remarkable success in text-to-music generation, extending them to edit existing recordings remains challenging because editing requires accurate deterministic inversion, reliable structural preservation, and numerically stable integration throughout the inversion and generation processes. We present FlowSonic, a zero-shot music editing framework built upon a pretrained diffusion transformer trained with rectified flow. FlowSonic first deterministically inverts a real-world recording into the latent space and preserves its musical structure during editing by reusing cross-attention representations extracted during inversion. To improve the numerical reliability of inversion-based editing, we introduce a high-order ODE solver and systematically investigate how different numerical integration schemes influence trajectory stability, structural preservation, and semantic controllability. Comprehensive experiments on timbre-transfer and genre-modification tasks demonstrate that FlowSonic consistently outperforms existing music editing methods across semantic alignment, harmonic preservation, structural consistency, and perceptual audio quality. We further provide geometric and empirical analyses showing how the proposed numerical integration strategy improves latent trajectory stability and leads to more reliable music editing.
12. 分裂场完美匹配层时域有限差分求解器的定量基准测试：狭缝衍射，以及完美电导体与介质圆柱体的散射
Quantitative Benchmarking of a Split-Field PML FDTD Solver: Slit Diffraction, and Scattering from PEC and Dielectric Cylinders
作者: Sabrina Saima, Tasin Intisar
类别: physics.optics, eess.SP, math.NA, physics.comp-ph
发布日期: 2026-07-19
arXiv ID: 2607.17360v1
链接: http://arxiv.org/abs/2607.17360v1
摘要: This paper presents a two-dimensional TMz finite-difference time-domain (FDTD) solver based on Yee’s scheme for modeling radiation from an infinitely long z-directed line current, with the open region truncated by a Berenger split-field perfectly matched layer (PML). After validating cylindrical-wave propagation and negligible late-time reflections in free space, the solver is applied to three inhomogeneous configurations: (i) diffraction through a one-cell-thick perfectly electrically conducting (PEC) sheet with single and double slits; (ii) scattering from infinitely long PEC cylinders of circular and rectangular cross section; and (iii) scattering from infinitely long dielectric cylinders of varying cross section and permittivity. Beyond qualitative field maps, the diffraction case is characterized quantitatively: a steady-state phasor extracted by a running discrete Fourier transform yields the transmitted intensity, from which the fringe visibility and the far-field pattern are computed and compared against the closed-form Fraunhofer prediction. The single- and double-slit cases are cleanly separated by a visibility that rises from near zero to near unity, and the double-slit interference maxima agree with the grating condition arcsin(m λ_0 / d) to within a fraction of a degree. For dielectric cylinders, the field penetrates the obstacle with the expected reduced internal wavelength λ_0 / \sqrt{ε_r}, and the scattered field strength grows with permittivity contrast. A reference-subtraction method isolates the scattered field throughout. The results confirm that the FDTD-PML framework accurately captures open-region diffraction and geometry- and material-dependent scattering.
13. 用于图像复原问题的迭代图拉普拉斯算子
Iterated graph Laplacian for image restoration problems
作者: Stefano Aleotti, Davide Bianchi, Florian Bossmann, Marco Donatelli, Pietro Maurino
类别: math.NA
发布日期: 2026-07-19
arXiv ID: 2607.17313v1
链接: http://arxiv.org/abs/2607.17313v1
摘要: We study the graph Laplacian operator as a regularizer in a generalized Tikhonov framework for linear ill-posed problems. The Laplacian is updated iteratively from the current reconstruction, so that progressively sharper structural information about the solution is fed into the regularization term. We introduce three schemes: a standard one that rebuilds the Laplacian from each new iterate; an error-equation scheme that, following the error-based formulation of iterated Tikhonov regularization, builds the Laplacian from an estimate of the reconstruction error rather than of the image itself; and a mixed scheme combining the two. We establish convergence of all three schemes for noisy data under a priori parameter and stopping rules as the noise level tends to zero. Numerical experiments in two-dimensional computed tomography and image deblurring show consistent gains in reconstruction quality and sharper recovery of fine details.
14. 低剂量计算机断层成像中平行束几何下正则化重建方法的稳定性与鲁棒性分析
Stability and Robustness Analysis of Regularized Reconstruction Methods for Low-Dose Computed Tomography in Parallel-Beam Geometry
作者: Mohamed Berrada
类别: math.NA, eess.IV
发布日期: 2026-07-19
arXiv ID: 2607.17298v1
链接: http://arxiv.org/abs/2607.17298v1
摘要: Low-dose computed tomography (LDCT) reduces radiation exposure but increases the ill-posedness of the reconstruction problem due to noise and sparse data. While regularized methods like Tikhonov and Total Variation (TV) improve image quality, their performance depends heavily on noise characteristics, sampling conditions, and parameter selection. This study presents a systematic stability and robustness analysis of Filtered Back Projection (FBP), Tikhonov regularization, and TV minimization within a 2D parallel-beam CT framework. A unified simulation pipeline based on the Radon transform is developed and evaluated using both the modified Shepp-Logan phantom and a clinical thorax image. Reconstruction behavior is investigated under multiple degradation scenarios involving Gaussian, Poisson, and mixed noise models, across baseline (180 projections) and sparse-view (60 projections) acquisition geometries. To ensure a fair comparison, regularization parameters are optimized for each scenario through an exhaustive SSIM-based grid-search. Quality is assessed via RMSE, PSNR, and SSIM, while robustness is quantified through an empirical Stability Factor S measuring perturbation amplification from measurement to image space. The results show that FBP is highly sensitive to noise and undersampling. Tikhonov regularization improves structural fidelity compared with FBP but remains more sensitive to perturbation than TV. Conversely, TV provides the best compromise between noise suppression, edge preservation, accuracy, and numerical stability. These findings highlight the stability-resolution trade-off in LDCT and demonstrate that the proposed Stability Factor S offers valuable complementary information to conventional metrics.
15. Stiefel流形上用于Wasserstein梯度流的动态近似格式
A Dynamical Approximation Scheme on the Stiefel manifold for Wasserstein Gradient Flows
作者: Isabella Carla Gonnella, Olga Mula, Federico Pichi, Gianluigi Rozza
类别: math.NA
发布日期: 2026-07-19
arXiv ID: 2607.17235v1
链接: http://arxiv.org/abs/2607.17235v1
摘要: We propose a meshless Lagrangian dynamical method for approximating Wasserstein gradient flows (WGFs). The evolving measure is represented as the pushforward of the initial measure $μ_0$ through a transport map in the weighted Hilbert space $L^2_{μ_0}$. We approximate this map in time-dependent linear subspaces of $L^2_{μ_0}$, whose orthonormal frames are evolved by a Dirac–Frenkel dynamical principle on a Stiefel manifold constrained to a finite-dimensional background space, adaptively constructed via local approximations of the WGF velocity field. We prove that the resulting transport map induces an absolutely continuous curve of probability measures in Wasserstein space, whose velocity is obtained by projecting the exact WGF velocity onto the background space, and we show that the approximation preserves the energy dissipation structure up to the projection error of the velocity. Moreover, for geodesically convex energies, we derive an a posteriori estimate controlling such projection error through the adaptive construction of the background space, yielding as well a bound on the approximation error of the pushforward measure in the Wasserstein metric. Numerical experiments on linear and nonlinear Fokker–Planck equations, porous-medium diffusion, and interaction energies demonstrate the accuracy of the method, its energy-dissipation properties, and the advantages of the adaptive construction.
16. 第二类Fredholm积分方程的有理离散配点法
A Rational Discrete Collocation Method for Second Kind Fredholm Equations
作者: Domenico Mezzanotte, Donatella Occorsio, Mario Pezzella, Woula Themistoclakis
类别: math.NA, math.FA
发布日期: 2026-07-19
arXiv ID: 2607.17179v1
链接: http://arxiv.org/abs/2607.17179v1
摘要: In this work we present a novel discrete collocation method for the numerical solution of Fredholm integral equations of the second kind in the space of continuous functions equipped with the uniform norm. The method is based on a rational interpolation scheme recently developed within the general framework of reproducing kernel Hilbert spaces. This rational approximation has no real poles, interpolates the target function at arbitrary Jacobi nodes and exhibits uniformly bounded Lebesgue constants. Moreover, it converges uniformly for all continuous functions at a rate at least equal to that of the best uniform polynomial approximation. These interesting properties are inherited by the resulting numerical method, for which stability, convergence and good conditioning are established under minimal assumptions on the integral kernel. A series of numerical experiments confirm the theoretical findings and indicate that, in the presence of particularly challenging kernels, the proposed approach provides a robust and effective alternative to Nyström-type methods.
17. 一种用于应变梯度弹性的带有三阶张量变量的最低阶稳健混合有限元方法
A Lowest-Order Robust Mixed Finite Element Method with a Third-Order Tensor Variable for Strain Gradient Elasticity
作者: Xuehai Huang, Zheqian Tang
类别: math.NA
发布日期: 2026-07-19
arXiv ID: 2607.17134v1
链接: http://arxiv.org/abs/2607.17134v1
摘要: A lowest-order mixed finite element method is developed for the strain gradient elasticity (SGE) model in arbitrary dimensions. We take the physically meaningful third-order double stress tensor $\boldsymbolΦ:=ι^2\operatorname{grad}\boldsymbolσ(\boldsymbol{u})\in\mathbb{S}\otimes\mathbb{R}^d$ as a primary variable and derive a distributional mixed formulation. The double stress is approximated by an $\mathbb{S}\otimes\mathbb{R}^d$-valued extension of the lowest-order Raviart–Thomas element, while the displacement is approximated by the vector-valued linear Crouzeix–Raviart element. Thus, the method avoids both high-degree bubble enrichment and a Nitsche-type treatment of the higher-order boundary condition. We establish parameter-robust discrete stability, an optimal first-order error estimate for fixed parameters, and a complementary parameter-uniform $\mathcal{O}(ι^{1/2}+h)$ error estimate with constants independent of both the size parameter $ι$ and the Lamé coefficient $λ$. In the boundary-layer regime $ι^{1/2}\lesssim h$, the latter retains a first-order convergence rate in $h$. We also develop a local quadratic post-processing and a hybridized formulation. Numerical experiments in two and three dimensions support the theoretical results.
18. 基于求积节点的数字网：在低维投影上继承求积精度
Digital Nets on Cubature Nodes: Inheriting Cubature Accuracy on Low-Dimensional Projections
作者: Takehito Yoshiki
类别: math.NA
发布日期: 2026-07-19
arXiv ID: 2607.17080v1
链接: http://arxiv.org/abs/2607.17080v1
摘要: Base-2 digital nets are practical high-dimensional integration rules: the sample budget $N=2^m$ can be chosen independently of the ambient dimension, and the generating matrices provide algebraic control of projections and Walsh-dual weights. They are therefore well suited to problems whose error is governed by weighted or low-dimensional projection structure. However, when one restricts attention to a smooth low-dimensional projected component, a low-dimensional cubature rule with a comparable number of nodes can be substantially more accurate than the projected digital-net points. This raises the question of whether low-dimensional cubature accuracy can be inserted into a high-dimensional digital-net rule without forming the full tensor product. We answer this question by a simple coordinate embedding: read the leading $p$ binary digits of each coordinate as an index into $2^p$ equal-weight cubature nodes, and replace the coordinate by the indexed node. When a projection forms the full $p$-bit grid, the transformed rule coincides on that projection with the corresponding product cubature rule; small projected $t$-values provide sufficient conditions for such full-grid recovery. For general integrands, the error separates into the corresponding product cubature error and a residual digital-net term. Experiments with scrambled Sobol’ nets in dimension $50$ illustrate this mechanism and show finite-budget improvements for the smooth low-order and coordinate-decaying test functions considered here.
19. 通过累积分布变换的一维守恒偏微分方程降阶建模
Reduced Order Modeling of One-Dimensional Conservative PDEs via the Cumulative Distribution Transform
作者: Harbir Antil, Rocío Díaz Martín, Ivan V. Medri, Kristofor E. Pas, Gustavo K. Rohde, Aryan Saxena, Sarswati Shah
类别: math.AP, math.NA
发布日期: 2026-07-19
arXiv ID: 2607.17066v1
链接: http://arxiv.org/abs/2607.17066v1
摘要: We propose a reduced order modeling (ROM) framework for 1D conservative PDEs based on the cumulative distribution transform (CDT). The CDT maps nonnegative, equal-mass states into a Hilbert space in which 1D Wasserstein distances become weighted $L^2$ distances and translations become affine shifts. This makes the transform especially suited for transport-dominated dynamics, where Eulerian linear-subspace ROMs often suffer from slow decay of Kolmogorov widths. We study this phenomenon for scalar conservative dynamics by analyzing the solution manifold in CDT coordinates. For linear transport, the transformed solution manifold is contained in the 2-dimensional space spanned by the transformed initial datum and the constant function, and has zero Kolmogorov $2$-width. For nonlinear hyperbolic conservation laws, we prove two complementary types of estimates: robust $O(n^{-1})$ bounds that rely only on the conservative transport structure and remain meaningful after shock formation, and sharper $O(n^{-2})$ bounds in smooth pre-shock regimes. For conservative advection-diffusion, we show that the CDT trajectory remains within distance $O(\sqrt{DT})$ of the pure-transport plane, and we also obtain sharper $O(D^2T^2)$ estimates under additional regularity or away from initial layers. In both cases, the zero 2-width behavior of linear transport is recovered as the diffusion coefficient tends to zero. Motivated by these estimates, we develop a CDT-POD numerical scheme: snapshots are mapped to CDT space, Proper Orthogonal Decomposition (POD) is performed in transformed coordinates, and the inverse CDT is used to reconstruct physical states. Numerical experiments for several transport-dominated dynamics show that CDT-POD can capture solution manifolds with substantially fewer modes than Eulerian POD.
20. 一种用于二维Navier–Stokes方程的具有时间一致稳定性的线性变步长嵌入式ETD格式
A Linear Variable-Step Embedded ETD Scheme with Uniform-in-Time Stability for the 2D Navier–Stokes Equations
作者: Haifeng Wang, Xiaoming Wang, Min Zhang
类别: math.NA
发布日期: 2026-07-19
arXiv ID: 2607.17036v1
链接: http://arxiv.org/abs/2607.17036v1
摘要: We propose a linear variable-step exponential time-differencing method for the incompressible Navier–Stokes equations in vorticity–streamfunction formulation on a two-dimensional periodic box. The method consists of a second-order scheme and an embedded first-order variant, yielding a natural mechanism for adaptive time stepping and a posteriori error control. Each time step requires only uniquely solvable linear problems: two heat equation solves, efficiently handled by Fourier methods in the periodic setting, and one linear scalar auxiliary-variable equation, evaluated via Laplace transform and Talbot’s numerical inverse transform. The construction combines the ETD framework, a mean-reverting scalar auxiliary variable (mr-SAV), and second-order extrapolation of the nonlinear term. The mean-reverting correction enables long-time stability while preserving full linearity, distinguishing the method from related mr-SAV schemes that require nonlinear algebraic solves. We prove unconditional long-time stability: for uniformly bounded $L^2$ forcing, the discrete vorticity remains bounded in $L^\infty(0,\infty;L^2)$ for all Reynolds numbers and time-step sizes. Numerical experiments
21. 基于新型拟线性七阶和九阶格式的双曲守恒律自适应新方法
Novel Adaptive Methods for Hyperbolic Conservation Laws Based on New Quasi-Linear Seventh- and Ninth-Order Schemes
作者: Shaoshuai Chu, Pingyao Feng, Vadim A. Kolotilov, Alexander Kurganov, Vladimir V. Ostapenko
类别: math.NA
发布日期: 2026-07-18
arXiv ID: 2607.16918v1
链接: http://arxiv.org/abs/2607.16918v1
摘要: We develop new adaptive numerical schemes for one- and two-dimensional hyperbolic systems of conservation laws. The methodology relies on the use of a smoothness indicator to automatically partition the computational domain into smooth and nonsmooth (rough) regions. We then follow the scheme adaption strategy recently introduced in [S. Chu, P. Feng, V. A. Kolotilov, A. Kurganov, and V. V. Ostapenko, Commun. Comput. Phys., accepted], but instead of the quasi-linear (QL) fifth-order finite-difference scheme used there, we employ the new QL seventh- and ninth-order schemes in the smooth regions. A series of numerical experiments for the Euler equations of gas dynamics demonstrates that the new adaptive schemes contain a smaller amount of numerical dissipation and achieve higher resolution compared with their counterpart that uses the QL fifth-order scheme in the smooth areas.
22. PolyChopper：一种多面体剖分算法
PolyChopper: a Polyhedron Splitting Scheme
作者: Tommaso Sorgente, Fabio Vicini
类别: math.NA
发布日期: 2026-07-18
arXiv ID: 2607.16887v1
链接: http://arxiv.org/abs/2607.16887v1
摘要: We introduce and implement a novel scheme, called \textit{PolyChopper}, for splitting a convex polyhedron into a finite set of disjoint convex sub-polyhedra. The algorithm cuts the polyhedron with a plane, which can either be prescribed as an external constraint or automatically determined from the inertia tensor. The resulting intersection vertices are then adjusted to control the complexity and quality of the generated elements, introducing additional ``notch wedges’’, tetrahedra and pyramids, according to a user-defined quality parameter. Experimental results demonstrate that the proposed approach is robust and consistently produces sub-polyhedra that are both simpler and of higher quality than the original polyhedron. Furthermore, recursively applying the algorithm generates a hierarchy of polyhedral subdivisions with progressively smaller elements, all guaranteed to remain convex and predominantly tetrahedral.
23. 带两个小参数的四阶奇异摄动边值问题的解析正则性
Analytic regularity for a fourth-order singularly perturbed boundary balue problem with two small parameters
作者: I. Sykopetritou, C. Xenophontos
类别: math.NA
发布日期: 2026-07-18
arXiv ID: 2607.16781v1
链接: http://arxiv.org/abs/2607.16781v1
摘要: We consider a fourth order singularly perturbed boundary value problem with two small parameters, in one dimension, under the assumption of analytic input data. We show that the solution may be decomposed into a smooth part, two different width boundary layers, and a negligible remainder. We provide estimates for arbitrary order derivatives of each term of the decomposition, which are explicit in the differentiation order and the singular perturbation parameters, and are needed for proving the convergence of high order numerical methods, such as the $p/hp$ versions of the Finite Element Method. We also provide classical differentiability results, which show that the solution will be analytic, if the data are analytic, but negative powers of the singular perturbation parameter(s) show up once we start differentiating.
24. 完全非线性抛物型偏微分方程的深度二阶随机残差方法
A Deep Second-Order Stochastic Residual Method for Fully Nonlinear Parabolic PDEs
作者: Zhenhua Zhao, Jihao Long
类别: math.NA, math.OC, stat.ML
发布日期: 2026-07-18
arXiv ID: 2607.16730v1
链接: http://arxiv.org/abs/2607.16730v1
摘要: We introduce the Deep Second-Order Stochastic Residual Method (D2SRM) for high-dimensional, Hessian-dependent fully nonlinear parabolic PDEs. A single scalar space–time network generates derivative-consistent approximations of the solution, gradient, and Hessian, which are trained jointly through second-order Brownian one-step residuals and terminal value and gradient penalties. For globally Lipschitz equations with identity diffusion and sufficiently weak Hessian coupling, we establish well-posedness in a Brownian occupation space and develop a population-level convergence theory. Under additional regularity, an a posteriori estimate bounds the squared full-jet occupation error of any admissible candidate by the time step and its population objective. For approximate population minimizers, the error bound separates time discretization, neural approximation, and population suboptimality; when the latter two terms are $O(h)$, the full-jet occupation norm is $O(h^{1/2})$. Experiments on a 100-dimensional manufactured benchmark compare terminal treatments, probe Hessian couplings inside and outside the proved small-gain range, and show decreasing errors as the time step decreases. The code is available at https://github.com/ZZHPKU/D2SRM.
25. 面向简化Ericksen–Leslie向列型液晶模型的增广拉格朗日预处理
Augmented Lagrangian preconditioning for a simplified Ericksen–Leslie model of nematic liquid crystals
作者: Yanying Li, Xu Qian, Jingmin Xia
类别: math.NA
发布日期: 2026-07-18
arXiv ID: 2607.16628v1
链接: http://arxiv.org/abs/2607.16628v1
摘要: The numerical solution of the simplified Ericksen–Leslie model for nematic liquid crystals is challenging because the flow and director equations are strongly coupled and because incompressibility and the unit-length condition must be enforced simultaneously. A Lagrange multiplier formulation avoids a small Ginzburg–Landau parameter, but the Newton systems have a double saddle-point structure. We develop an augmented Lagrangian block preconditioner in which both constraints are augmented while their discrete enforcement remains multiplier based. After finite element discretization and backward Euler time integration, the Newton increments are grouped into velocity–director and pressure-multiplier variables. A block-diagonal approximation of the coupled velocity-director block then leads to separate, physically scaled approximations of the pressure and director-multiplier Schur complements. Manufactured-solution tests show the expected spatial accuracy and first-order temporal convergence for the primary variables; the multiplier error reaches a spatial-error floor on the fixed mesh used in the temporal study. In the reported parameter ranges, the outer FGMRES iteration counts are nearly mesh independent, remain stable under time-step and viscosity variation, and improve as the augmentation parameters increase. A smooth benchmark also exhibits monotone decay of the computed total energy.
26. 正分配伴随预报器用于非线性动力学及其有限差分诊断
Positive-Allocation Companion Predictors for Nonlinear Dynamics and Their Finite-Difference Diagnostics
作者: Cynthia V. Flores, James E. Pascoe
类别: math.DS, math.NA
发布日期: 2026-07-17
arXiv ID: 2607.16529v1
链接: http://arxiv.org/abs/2607.16529v1
摘要: We introduce a positive-allocation companion construction for Koopman-inspired finite-dimensional prediction of nonlinear dynamical systems. The method determines recurrence coefficients by representing a target observable snapshot as a nonnegative, normalized combination of earlier training snapshots. These coefficients define a companion matrix whose spectral structure is induced by the allocation constraints at the construction stage. We prove that normalized positive allocation places the companion spectrum in the closed unit disk and, because the coefficients sum to one, includes $1$ as an eigenvalue. Additionally, we develop modal and non-modal diagnostics for the resulting model trajectory. When the companion matrix is diagonalizable, the modal representation shows that first and second finite differences act as spectral filters through factors of $λ_\ell-1$ and $(λ_\ell-1)^2$. We also derive $C$-based finite-difference bounds that avoid diagonalization and can be evaluated directly from the training data and companion matrix. Numerical experiments on the FitzHugh–Nagumo and susceptible–infectious–recovered (SIR) models illustrate the behavior of the construction in oscillatory and transient dissipative settings. The examples demonstrate both the interpretability of the companion recurrence and its limitations, particularly when pointwise trajectory agreement degrades while finite-difference and modal diagnostics remain informative.
27. 有限元指数积分用于非线性波动方程的粗糙解。第一部分：多边形与多面体域上的Dirichlet边界条件
Finite element exponential integration for rough solutions of nonlinear wave equations. Part I: Dirichlet boundary conditions on polygonal and polyhedral domains
作者: Jiachuan Cao, Benjamin Dörich, Marlis Hochbruck, Buyang Li
类别: math.NA
发布日期: 2026-07-17
arXiv ID: 2607.16511v1
链接: http://arxiv.org/abs/2607.16511v1
摘要: We study a fully discrete scheme for nonlinear wave equations on general bounded polygonal/polyhedral domains with initial data $(u^0,v^0)\in H^γ(Ω)\times H^{γ-1}(Ω)$, $0<γ\le 1$, subject to the natural compatibility condition associated with the homogeneous Dirichlet boundary condition. The scheme combines an exponential Euler time integrator with a finite element spatial discretization. In contrast to existing low-regularity error analyses, which are mostly based on Fourier spectral discretizations, our approach applies to general bounded domains and finite element spatial discretizations. We prove rigorous error estimates for low-regularity solutions. The analysis is based on a frequency decomposition of the underlying elliptic operator, used solely as an analytical regularization device and not in the actual implementation, which allows low-regularity techniques to be extended beyond the Fourier spectral framework. The results also indicate that higher-order finite element methods remain advantageous in spatial approximation even for solutions of limited Sobolev regularity. Numerical experiments on different domains and with different polynomial degrees confirm the predicted convergence behavior.
28. 非侵入式POD-GPR峰值热力场机器学习仿真器的替代模型-代码验证及其在碳碳飞机刹车盘中的应用
Surrogate-to-code verification of a non-intrusive POD-GPR machine-learning emulator of peak thermomechanical fields, with application to a carbon-carbon aircraft brake disc
作者: Franklin Kamche
类别: math.NA, cs.CE
发布日期: 2026-07-17
arXiv ID: 2607.16447v1
链接: http://arxiv.org/abs/2607.16447v1
摘要: While non-intrusive reduced-order emulators successfully substitute for costly finite-element simulations, certification-grade applications demand a rigorous error assessment relative to the solver’s own discretization uncertainty. This work introduces a verification framework designed to evaluate prediction accuracy directly against this numerical uncertainty floor, rather than physical observation. We demonstrate this approach on a carbon-carbon aircraft brake disc during a rejected take-off scenario, predicting peak temperature and peak von Mises stress fields. Operating over a fixed mesh of 60914 nodes, the emulator relies on a pipeline coupling proper orthogonal decomposition (POD) and Gaussian process regression (GPR) parameterized by 5 inputs. We formulate an algebraically exact, a posteriori error budget that isolates the emulator’s signed error into distinct truncation and regression components at the extremum. This error is then evaluated against a discretization floor established via mesh-convergence studies on the peak quantities of interest. On a completely independent test set, our emulator matches the high-fidelity solver’s peak values within its own discretization uncertainty for both fields, a benchmark we define as surrogate-to-code fidelity. Notably, the residual error stems primarily from linear reduction limits rather than regression inaccuracies. Beyond the main framework, we present two secondary contributions: the concurrent, non-intrusive prediction of the thermal and mechanical peaks, and a probabilistic approach to spatial zone localization for these extrema. This methodology applies to non-intrusive reduced-order models of parameterized finite-element simulations wherever a discretization-error estimate for the quantity of interest can be obtained.
29. 带跳随机比例时滞Mckean-Vlasov模型的截断Euler-Maruyama算法的收敛性与稳定性
Convergence and stability of truncated Euler-Maruyama algorithm for stochastic proportional delay Mckean-Vlasov models with jump process
作者: Amr Abosenna, Zhuoqi Liu
类别: math.NA
发布日期: 2026-07-17
arXiv ID: 2607.16438v1
链接: http://arxiv.org/abs/2607.16438v1
摘要: Stochastic Mckean-Vlasov models have a substantial importance in different fields such as finance, biology and control. This paper puts the light on stochastic proportional delay Mckean-Vlasov model with Lévy jump where the non-jump coefficients are granted the permission to grow beyond linearity. The truncated Euler-Maruyama algorithm is then applied to our addressed model where the convergence rate and almost sure exponential stability of the aforementioned numerical algorithm are being investigated. Finally, numerical examples are presented to foster the theoretical analysis done throughout the paper
30. 一类用于Biot固结模型的混合-混合强守恒有限元方法
A family of mixed-mixed strongly conservative finite element methods for Biot’s model of consolidation
作者: Qingguo Hong, Johannes Kraus, Maria Lymbery
类别: math.NA
发布日期: 2026-07-17
arXiv ID: 2607.16418v1
链接: http://arxiv.org/abs/2607.16418v1
摘要: We consider a four-field formulation of Biot’s quasi-static model of consolidation with the (effective) elastic stress, the solid displacement, the fluid flux and the fluid pressure as unknown physical quantities of interest. The weak form of this system composed of the momentum and mass balance equations complemented by the stress-strain relation and Darcy’s law yields a two-fold perturbed saddle-point problem. Its well-posedness is proven for a natural choice of inifinite-dimensional Hilbert spaces, where the displacement-flux pair is sought in a novel Hilbert space. Next, based on recent works on $H(\rm div \, \rm div)$-conforming finite elements, we propose a family of mixed-mixed finite element methods that preserve the angular momentum as well as the fluid mass balance pointwise, i.e., in a strong sense. We prove the well-posedness of the arising discrete problem and establish optimal a priori error estimates for the related family of fully conservative mixed-mixed finite element methods.