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1. 用于裂纹尖端与角点奇异性 B 样条分析的力学训练神经坐标映射
Mechanics-trained neural coordinate mapping for B-spline analysis of crack-tip and corner singularities
作者: Hyunju Kim
类别: math.NA
发布日期: 2026-07-25
arXiv ID: 2607.23229v1
摘要翻译: 在裂纹尖端或凹角附近，分数次径向幂函数可能具有无界导数，从而减慢高阶样条的收敛速度。奇异映射对计算半径进行分级，使得拉回场更光滑而不改变物理域。经典映射需要预先知道奇异指数和分级。本文从力学问题中训练得出径向坐标。它是正神经密度的归一化积分，该积分固定了两个径向边界，并确保在退化尖端之外 r’(s)>0。径向分级指数和密度校正权重通过离散平衡态下的 Galerkin 能量推断得出，无需指数标签或精确内部场。我们在标量和平面应变 B 样条公式中测试了该映射，并将其与恒等映射、径向分级节点向量、指定幂映射以及自适应增强 B 样条方法进行了比较。在 156 个向量自由度下，与恒等映射相比，力学训练映射将相对能量范数误差降低了 33.42 倍。在 3 条轮廓上恢复的混合型应力强度因子的最大误差为 1.923×10^(-5)。对于直裂纹，学习到的密度校正消失，且指定的 r=s^2 映射给出了相同的改进。对于测试参数超出训练区间的非线性 Robin 族，密度校正是非零的，并给出至少 4.839 的固定 q 增量能量增益。因此，对于本文所考虑的问题，当所需坐标不能由指定幂映射表示时，神经校正是有用的。
Comment: 30 pages, 19 figures
链接: http://arxiv.org/abs/2607.23229v1
1. 研究对象和背景
在裂纹尖端或凹角附近，弹性问题的解常含有分数次幂的径向项 $u(r,\theta) \sim r^{\lambda} \Psi(\theta)$，其中 $0<\lambda<1$ 为奇异指数，$\Psi$ 为角向特征函数。这类奇异性导致径向导数 $\partial_r u$ 在 $r=0$ 处无界，从而严重降低基于高阶 B 样条的数值方法的收敛速率。Williams 展开 [1] 识别了此分数次幂对正则性的破坏，启发学者通过坐标变换将物理域的奇异行为转移至更光滑的计算坐标。例如，在等几何分析 (IGA) 中，利用 NURBS 网格的重复控制点将裂纹面塌缩至一点，若塌缩阶数为 $q$，则有 $r \sim \eta^q$，进而拉回后的场为 $u \circ F \sim \eta^{q\lambda}$。若 $q\lambda = m \in \mathbb{N}$，则分数次幂变为多项式因子 $\eta^m$，恢复了高阶样条的逼近能力 [4]。类似思想也体现在自适应富集 [5,6,7] 和稳健 IGA 格式 [8] 中，它们通过修改试验空间而非坐标来获得相应正则性增益。然而，这些方法均需事先指定奇异指数和幂映射关系（如 $r = s^2$ 对应平方根奇性），当问题参数化时（例如材料渐变或 Robin 边界条件），所需映射可能偏离纯幂律形式，且奇异指数也需从力学响应中推断。
本文旨在填补这一空白：提出一种不依赖任意精确场、指数标签或预先幂映射的径向坐标构造方法。该方法从离散 Galerkin 平衡解的能量出发，学习一个正神经密度函数，通过归一化积分自动满足径向边界条件和严格单调性，同时由力学训练信号自动推断分级指数 $q$ 和密度修正权重。学习到的坐标保持物理算子、试验空间维数及应力强度因子 (SIF) 提取步骤不变，从而能够与恒等映射、径向分级节点向量、指定幂映射及自适应富集 B 样条方法进行公平比较。
2. 主要定理或主要结果
本文通过数值实验验证了力学训练神经坐标映射的有效性，主要结论如下：
- 在标量 Laplace 问题和平面应变弹性裂纹问题中，采用 156 个向量自由度的 B 样条离散，力学训练映射相较于恒等映射将相对能量范数误差降低 33.42 倍。混合模式应力强度因子 ($K_{\rm I}, K_{\rm II}$) 在三个积分等值线上恢复的最大误差仅为 $1.923 \times 10^{-5}$。
- 对于直裂纹（纯平方根奇性），学习到的密度修正权重 $w_j$ 收敛到零，所得映射退化为 $r = s^2$ 的指定幂映射，且两者精度相同。这表明当问题具有已知幂律分布时，网络能够自动识别并回归到经典映射。
- 对于含径向渐变的非线性 Robin 边界条件族（反平面问题），在训练区间外的测试参数下，密度修正保持非零，且与纯幂映射相比，固定 $q$ 的增量能量增益至少为 4.839。这证明当坐标不能由单一幂映射表示时，神经密度修正可提供显著改善。
- 将训练后的映射从粗网格标量问题直接迁移至细化网格的平面应变问题，无需重新训练，表明映射的泛化性。同时，通过 Jacobian 审核和有限差分检验确认了映射的可容许性（$r'(s) > 0$, $J > 0$）。
- 与自适应富集样条方法的比较显示，在相同自由度数下，力学训练映射的精度相当或更优，但避免了富集函数中额外奇异指数的显式构造。
3. 研究方法、关键技术和核心工具
本文的核心创新在于将神经网络仅用于参数化奇异坐标的导数，而非直接近似物理场。技术要点分述如下：
1. 可容许径向坐标的正密度构造 为保证映射的单调性与边界条件，作者定义径向坐标的一阶导数为正神经密度
$$\rho_{\phi}(s) = s^{q-1} \exp\!\left[\sum_j w_j \tanh(\beta_j (s - c_j))\right],$$并通过归一化积分恢复坐标
$$r_{\phi}(s) = \frac{\int_0^s \rho_{\phi}(\tau)\,d\tau}{\int_0^1 \rho_{\phi}(\tau)\,d\tau}.$$该构造自动满足 $r_{\phi}(0)=0$, $r_{\phi}(1)=1$ 且当 $s>0$ 时 $r'_{\phi}(s)>0$，从根本上避免了优化过程中违反映射条件的罚函数。前因子 $s^{q-1}$ 控制近尖端的拆分级数 $r_{\phi}(s) \sim C s^q$，使得物理场中的 $r^{\lambda}$ 被拉回为 $s^{q\lambda}$；权重 $w_j$ 通过光滑阶跃函数调整有限半径处的坐标分布，且指数和权重均作为训练变量。
2. 不依赖参考解的力学训练目标 训练目标函数为离散 Dirichlet 能量在 Galerkin 解处的值
$$J_h(\phi) = \frac12 \mathbf{d}(\phi)^T \mathbf{K}(\phi) \mathbf{d}(\phi),$$其中 $\mathbf{d}(\phi)$ 由线性方程组 $\mathbf{K}_{ff}(\phi) \mathbf{d}_f = \mathbf{f}_f - \mathbf{K}_{fc}(\phi) \mathbf{d}_c$ 确定。优化器仅更新映射参数 $\phi = (q, w)$，位移系数通过求解离散系统而非网络输出给出，从而训练信号完全来自力学弱形式本身，不涉及精确解、SIF 参考值或人工标签。对于 Robin 边界条件族，设计复合损失函数
$$\mathcal{L}(\vartheta) = \mathop{\mathrm{mean}}_{\kappa} \left[ \frac{E_c}{E_0} + \gamma_E \delta_E + \gamma_R \operatorname{Var}_m(\ell_m) + \gamma_A \left(\frac{A_c - A_e}{\max(|A_e|,\epsilon_A)}\right)^2 + \gamma_\vartheta \|\vartheta\|^2 \right]$$结合粗–富集能量差 $\delta_E$、径向一致性项 $\ell_m = \log |F_h(r_m)| - \lambda_h \log r_m$ 及振幅比，避免了使用精确角向特征函数或参考振幅。
3. 参数化问题中的条件网络 对于楔形半角 $\alpha$ 或 Robin 参数 $\kappa$ 等物理参数族，引入小型条件网络输出 $\phi_{\vartheta}(\zeta)$。梯度指数由有界逻辑函数预测 $q_{\vartheta}(\alpha) = q_{\min} + (q_{\max} - q_{\min})\,\sigma(z_{\vartheta}(\alpha))$，或针对非线性 Robin 族采用二次 softplus 映射 $q(\kappa) = q_0 + \mathrm{softplus}(a_0 + a_1 z + a_2 z^2)$。这些神经分支仅改变映射参数的生成方式，而不影响后续坐标构造与 Galerkin 组装，从而在同一框架下统一了单问题和多问题学习。
4. 信息流分离与可迁移性验证 训练循环冻结后，映射 $\phi^*$ 被用于独立的求解和后处理，分别计算场误差与 SIF。精确场和应力强度因子仅在最后评估阶段引入，阻止了任何隐式标签泄漏。学习到的映射可从粗标量网格迁移至细平面应变网格，并且通过 Jacobian 审计、求积阶数变化及塌缩边上的试验空间组装保证了映射的数值可容许性。
5. 与已有方法的对比策略 所有测试中，物理算子、有限元空间和输出提取方法保持固定，仅改变径向坐标。比较对象包括：恒等映射、径向分级节点向量、指定的幂映射 $r=s^q$，以及自适应富集 B 样条。通过直裂纹验证学习到的 $w_j$ 消失，表明网络能自动识别纯幂律情况；非线性 Robin 族则凸显了密度修正的独立价值。
4. 与之前工作的比较
本文与既有的裂纹/角点奇异映射与神经网络近似方法存在多方面区别，具体比较如下：
1. 与经典奇异映射和 IGA 塌缩方法的对比 传统 IGA 处理裂纹尖端的标准做法是构造塌缩边映射 [4]，其控制点重复次数 $q$ 固定，且需要事先知道奇异指数 $\lambda$ 以确定 $q$（如 $\lambda=1/2$ 时取 $q=2$）。本文方法无需预设 $q$ 或 $\lambda$，而是将 $q$ 和可能的密度修正 $w_j$ 作为训练变量，完全从离散力学能量中学习。对于直裂纹，学到的映射还原为 $r=s^2$，但本质区别在于后者为手工规定，前者为自动推断；更一般地，当材料渐变或边界条件导致非幂律行为时，传统的固定幂映射无法适用，而本文学习到的密度修正能够自适应调整径向分辨率。此外，径向分级节点向量方法虽也能改善收敛，但其本质是在参数域对节点加密，本文则通过直接改变物理坐标达到同样效果，且不改变样条基函数的分布模式。
2. 与富集方法的对比 自适应富集 B 样条（如 XIGA [5,6,7]）通过向试验空间添加奇异基函数（通常基于 $r^{\lambda}$ 函数）来恢复最优收敛阶。这类方法必须已知奇异指数，且富集函数的选择影响刚度矩阵的条件数和积分精度。本文将奇异处理完全置于坐标变换中，试验空间保持标准 B 样条，避免了富集带来的数值积分困难和自由度结构变化。在相同自由度下，力学训练映射与自适应富集样条的精度相当，但后者需要额外的先验知识。本文的实验（如能量误差因子 33.42）表明，在非幂律参数问题中，训练映射能提供更优的精度–成本比。
3. 与基于神经网络的断裂力学方法的对比 当前神经网络在断裂力学中的应用主要分为两类：(i) 物理场近似型，如富含裂纹尖端函数的 PINN [9,10] 和用于相场、近场动力学的代理模型 [11,12,13]；(ii) 几何变换型，如利用网络进行域参数化 [14,15,16]。本文属于后者，但不同于一般的域变换网络，这里网络仅输出一维径向坐标的导数，大大降低了网络规模，且正密度参数化自然满足坐标单调性的强约束。更为重要的是，本文训练过程不接触任何场值或 SIF 标签，完全由 Galerkin 系统的能量驱动，这与监督学习或预嵌入物理信息的训练方式根本不同。
4. 适用范围与代价 力学训练映射的优势局限于坐标可用单一径向函数描述的问题，即裂纹或角点奇异性由径向部分主导且可分离。对于复杂裂纹路径或多裂纹相互作用，当前单变量映射可能不足。训练的计算代价是一次性的：对于标量问题，训练只需数分钟；迁移到弹性问题后，求解代价与使用指定幂映射完全相同，因为每次求解仅需组装刚度矩阵并求解线性方程组。这一方面与需要每步更新网络的深度 Ritz 或 PINN 方法形成鲜明对比。此外，本文设计的条件网络能够针对参数族生成连续映射，避免了逐个问题重训练。然而，文中也明确指出，当问题本身已能由经典幂映射充分刻画时（如直裂纹），方法能自动识别，但并不会带来额外增益；其价值体现在非幂律场景中，正如非线性 Robin 族所示，密度修正可在训练范围外保持有效，显示出一定的外推能力。
2. AlgMortar：一种全代数多尺度砌块预条件子
AlgMortar: a fully algebraic multiscale mortar preconditioner
作者: Luan F. Santos, Fabricio S. Sousa, Roberto F. Ausas, Rafael T. Guiraldello, Felipe Pereira
类别: math.NA
发布日期: 2026-07-24
arXiv ID: 2607.22905v1
摘要翻译: 求解源自二阶椭圆方程离散的大规模对称正定线性系统颇具挑战性，尤其是在诸如多孔介质流等具有高度非均匀系数的应用中，这可能导致系统严重病态。在此背景下，多尺度方法近来已被用于加速 Krylov 子空间方法，这得益于其良好的并行可扩展性。本文提出 AlgMortar，它是多尺度砌块混合有限元方法（MMMFEM）的一种全代数实现。该方法仅利用从细网格系统矩阵中提取的信息，便于其在现有求解器中实现。AlgMortar 采用图划分从矩阵图直接定义区域分解。在每个子域上，它构造模拟 Dirichlet 问题的局部线性系统，并通过一个代数界面条件将所得局部解耦合起来，该条件恢复了 MMMFEM 的弱通量连续性机制。我们证明，当细网格矩阵对称正定且具有非正非对角线元素时——这是椭圆问题离散中常见的结构——该方法是适定的。对源自 Darcy 流问题有限体积离散的细网格线性系统进行的数值实验表明，当该方法用作共轭梯度法的预条件子时，所提方法展现出良好的可扩展性，并且对于具有挑战性的非均匀、高对比度算例（包括高度不规则的角点网格），能与当前最优的代数多重网格方法相媲美。
Comment: 无
链接: http://arxiv.org/abs/2607.22905v1
1. 研究对象和背景
本文研究由二阶椭圆型方程离散化产生的大规模对称正定(SPD)线性系统的求解问题, 这类问题广泛存在于多孔介质流、地下储层模拟、地下水运移以及非均质材料热传导等应用领域. 具体方程形式为 $-\nabla\cdot (K abla p) = f$ 于 $\Omega\subset\mathbb{R}^d, d=2,3$, 其中 $p$ 通常表示压力, $f$ 为源项, $K$ 为 SPD 渗透率张量. 准确数值逼近通常需要细网格, 导致大规模离散系统 $A p = f$, 矩阵 $A\in\mathbb{R}^{N\times N}$ 对称正定, $p$ 为未知压力向量, $f$ 为右端项. 当系数 $K$ 高度非均匀或对比度高时, 系统条件数极大, 迭代求解困难.
多尺度方法通过构造能够捕捉局部非均质性的基函数, 并用粗问题耦合这些基函数来定义全局近似, 已被证明能有效加速 Krylov 子空间方法, 且具有良好并行可扩展性. 其中, 基于混合格式的 Multiscale Mortar Mixed Finite Element Method (MMMFEM) 在每个子区域上求解局部 Dirichlet 问题, 并通过 mortar 界面变量 $\lambda_H$ 弱强制法向通量的连续性. 该方法在混合有限元空间(如 Raviart-Thomas 元)框架下, 将局部压力 $p_{h,i}$ 和速度 $u_{h,i}$ 表示为 $u_{h,i} = \overline{u}_{h,i} + \sum_{k} \lambda_k \widehat{u}^{(k)}_{h,i}$, $p_{h,i} = \overline{p}_{h,i} + \sum_{k} \lambda_k \widehat{p}^{(k)}_{h,i}$, 其中 $(\overline{u}_{h,i},\overline{p}_{h,i})$ 为特解, $(\widehat{u}^{(k)}_{h,i},\widehat{p}^{(k)}_{h,i})$ 为对应 mortar 基函数 $\psi_k$ 的多尺度基函数(MBF), 而 $\lambda_k$ 通过界面通量连续性条件 $\sum_i\langle u_{h,i}\cdot n_i,\mu\rangle_{\Gamma_i}=0$ 求得的界面系统确定. 然而, 传统 MMMFEM 需要几何信息(如网格、子区域划分)、PDE 系数和离散化细节, 在工业多孔介质流模拟中这些信息可能高度复杂、保密或难以跨代码传递. 代数预条件子仅基于系统矩阵和右端项, 可适应任意网格、非结构化离散、混合维模型和高度非均质系数, 便于集成到现有求解器中, 减少了网格转换带来的误差和开销, 且天然支持黑箱式多层粗化和插值策略.
已有几种代数多尺度方法, 如 two-stage algebraic multiscale (TAMS) 方法、algebraic multiscale solver (AMS)、multiscale two-point flux approximation (MsTPFA) 等. 受 MRCM (Multiscale Robin Coupled Method) 在 MMMFEM 极限下作为 PCG 预条件子性能优异的启发, 并鉴于工业界对完全代数化求解器的需求, 本文提出 AlgMortar, 一种 MMMFEM 的全代数实现. 该方法仅使用细网格系统矩阵 $A$, 通过图划分获得子区域, 构造局部代数 Dirichlet 算子, 并通过代数界面条件恢复弱通量连续性. 本文旨在证明该方法的适定性, 推导条件数估计, 并展示其作为 PCG 预条件子的数值效能.
2. 主要定理或主要结果
本文在以下假设下证明了 AlgMortar 方法的适定性和性质:
- 设 $A$ 为由二阶椭圆方程离散产生的 SPD 矩阵, 且其非对角元非正(即 $A$ 为 M-矩阵), 从而界面权重矩阵 $W$ 的对角元(即跨子域的耦合系数)均为负. 定义局部 Dirichlet 算子 $A_i = A_{ii} - \sum_{j: j\sim i} D_{ij}$, 其中 $D_{ij} = \operatorname{diag}(A_{ij} \mathbf{1}_{\Omega_j})$, 则每个 $A_i$ 均为 SPD.
- 令 $\Lambda\subset\mathbb{R}^E$ 为给定的 mortar 空间, $Z\in\mathbb{R}^{|E|\times r}$ 满足 $\operatorname{range}(Z)=\Lambda$. 构造对称系统 $$Q = \begin{pmatrix} \frac{1}{4}\widehat{A} & M^T W Z \\ Z^T W M & -Z^T W Z \end{pmatrix},$$ 其中 $\widehat{A} = \operatorname{diag}(A_1,\dots,A_m)$, $M$ 为代数平均算子, $W$ 为全局权重矩阵. 证明了当 $A$ SPD 且 $W$ 负定时, $Q$ 对称正定, 从而 AlgMortar 系统 $$\widehat{A} p + B \lambda = f, \quad \mu^T W(\lambda - M p) = 0 \;\; \forall\mu\in\Lambda$$ 存在唯一解 $(p,\lambda)\in\mathbb{R}^{\Omega}\times\Lambda$.
- 在均匀 Cartesian 网格且 $K=I$ 的条件下, AlgMortar 精确恢复经典的 MMMFEM. 此时局部 Dirichlet 算子 $A_i$ 等同于 TPFA 离散下局部 Dirichlet 问题的算子, 界面条件对应弱通量连续性.
- 条件数估计(详见第5节, 本文提供的摘录未包含细节)表明预处理后系统的谱聚类依赖于 mortar 空间如何逼近图划分界面上的平均压力.
- 数值实验: 将 AlgMortar 用作共轭梯度(PCG)预条件子, 在 Cartesian 网格和不规则 corner-point 网格上对高对比度非均质 Darcy 流问题进行测试. 结果表明 AlgMortar 具有良好的并行可扩展性, 且在高对比度情形下可比拟甚至优于先进的代数多重网格 (AMG) 方法.
3. 研究方法、关键技术和核心工具
AlgMortar 的核心技术贡献在于仅从系统矩阵 $A$ 的代数信息出发, 重新构造出 MMMFEM 所需的所有关键算子: 子域划分、局部 Dirichlet 问题、界面 trace 和平均算子以及弱通量连续性条件. 具体工具与分步框架如下:
1. 基于图划分的代数区域分解
- 利用 $A$ 的邻接图 $G(A)=(\Omega,E)$, 通过图划分器(如 METIS)将指标集分解为 $m$ 个互不相交的子域 $\Omega_1,\dots,\Omega_m$.
- 按子域重排序后, 矩阵具有分块结构 $A=(A_{ij})$, 其中非零块 $A_{ij}\neq 0$ 当且仅当 $\Omega_i$ 与 $\Omega_j$ 相邻.
- 引入界面边集 $E_{ij} = \{(k,l): k\in\Omega_i, l\in\Omega_j, a_{kl}\neq 0\}$, 以及全局界面边集 $E = \bigcup_{i
2. 代数 trace 算子与权重矩阵
- 定义 incidence 矩阵 $C_{ij}\in\mathbb{R}^{\Omega_i\times E_{ij}}$: 若边 $e=(k,l)\in E_{ij}$ 且 $k\in\Omega_i$, 则 $[C_{ij}]_{k,e}=1$. 由此构造两类定向 trace 算子:
$$L = \sum_{i
- 界面权重矩阵 $W_{ij}\in\mathbb{R}^{E_{ij}\times E_{ij}}$ 定义为 $[W_{ij}]_{e,e}=a_{kl}$ ($e=(k,l)$), 并装配成全局权重矩阵 $W = \sum_{i
- 定义平均算子 $M = \frac{1}{2}(L+U)$, 满足 $(Mp)_e = \frac{p_k+p_l}{2}$ 对于 $e=(k,l)$.
- 界面权重矩阵 $W_{ij}\in\mathbb{R}^{E_{ij}\times E_{ij}}$ 定义为 $[W_{ij}]_{e,e}=a_{kl}$ ($e=(k,l)$), 并装配成全局权重矩阵 $W = \sum_{i
3. 代数 Dirichlet 算子的构造
- 对每个子域, 定义修正的对角矩阵 $D_{ij} = \operatorname{diag}(A_{ij}\mathbf{1}_{\Omega_j})$, 其对角元为 $A_{ij}$ 的行和. 然后构造局部 Dirichlet 算子 $$A_i = A_{ii} - \sum_{j: j\sim i} D_{ij}.$$ 此修改的代数意义在于模拟 Dirichlet 边界条件: 如果用两点通量近似 (TPFA) 离散且 $K=I$, 则 $A_i$ 恰好是限制在 $\Omega_i$ 上、齐次 Dirichlet 边界的 TPFA 矩阵. 这保证了即使在代数框架下, 局部问题也能反映物理边界.
- 定义分块对角矩阵 $\widehat{A} = \operatorname{diag}(A_1,\dots,A_m)$.
4. 耦合算子与系统重写
- 定义界面耦合算子 $B_{ij} = 2 C_{ij} W_{ij} R_{E_{ij}}$, 则 $(B_{ij}\mu)_k = 2\sum_{l:(k,l)\in E_{ij}} a_{kl} \mu_{(k,l)}$. 进一步定义 $B_i = \sum_{j\sim i} B_{ij}$, 堆叠为 $B = (B_1^T,\dots,B_m^T)^T$.
- 关键代数恒等式 (Proposition 3.1): $B^T = 4 W M$. 证明依赖于代数 skeleton 与限制算子的性质.
- 利用该恒等式, 原始系统 $A p = f$ 可等价变形为 $$\widehat{A} p + B \lambda = f, \quad \lambda = M p.$$ 此处 $\lambda$ 为界面压力.
5. AlgMortar 的建立与求解
- 引入有限维 mortar 空间 $\Lambda\subset\mathbb{R}^E$, 并在其上弱化界面条件: 寻找 $(p,\lambda)\in\mathbb{R}^{\Omega}\times\Lambda$ 使得 $$\widehat{A} p + B \lambda = f, \quad \mu^T W(\lambda - M p) = 0 \quad \forall\mu\in\Lambda.$$ 第二个方程可视为弱通量连续条件, 因为 $W$ 的对角元为负的传导系数.
- 类似于 MMMFEM, 通过多尺度基函数(MBF)求解: 对每个子域, 求解特解 $A_i \overline{p}_{\Omega_i} = f_{\Omega_i}$, 并对每个 mortar 基函数 $\psi_m$, 求解 $A_i \widehat{p}^{(m)}_{\Omega_i} = -B_i \psi_m$. 然后利用叠加 $p = \overline{p} - \widehat{A}^{-1} B \lambda$ 将界面条件化为仅关于 $\lambda$ 的 SPD 系统: $$\mu^T (W M \widehat{A}^{-1} B + W) \lambda = \mu^T W M \overline{p}, \quad \forall\mu\in\Lambda.$$ 当 $\{\psi_m\}$ 为 $\Lambda$ 的一组基时, 等价于线性系统 $S \lambda = r$, 其中 $S$ 对称正定.
- 所有局部求解可完全并行, 且每个 $A_i$ 的 Cholesky 分解可复用. 界面系统规模为 $N_\Lambda = \dim\Lambda$, 通常远小于 $N$.
6. 适定性证明的核心技术
- 将混合系统等价转化为对称分块矩阵 $Q$ (见(4.1)式). 通过检验其正定性证明适定性: 利用 $\widehat{A}$ 的 SPD 性质及 $W$ 的负定性, 通过 Schur 补论证可得 $Q$ 正定, 从而 AlgMortar 解存在唯一. 该过程本质上是代数框架下的虚功原理与能量估计.
上述构造完全基于矩阵代数, 无需网格坐标或系数场, 仅假设矩阵非对角元非正, 这是许多椭圆离散的典型性质. 此代数框架不仅降低了实现门槛, 也保证了在标准情形下与传统几何方法的一致性.
4. 与之前工作的比较
本文工作与已有文献在多个维度上形成比较和推进:
与几何多尺度 mortar 方法 (MMMFEM, MRCM) 的比较
- MMMFEM [3,4] 是最早将 mortar 界面变量与多尺度基函数结合的混合方法, 但它需要显式的几何网格信息、子域划分及系数场, 且依赖特定的混合有限元空间. AlgMortar 将其完全代数化, 仅以系统矩阵为输入, 从而适用于任意网格(包括 unstructured 和 corner-point 网格)和任何能生成 SPD 矩阵的离散方法(如有限体积、有限元), 也避免了网格转换或系数重构.
- MRCM [17] 引入了 Robin 传输条件, 通过调整参数可在 MMMFEM 与 MHM (Multiscale Hybrid-Mixed) 之间插值. 文献[9] 发现, 当参数取 MMMFEM 极限时, MRCM 作为 PCG 预条件子的性能最佳. 本文直接针对该极限构造代数版本, 省去了 Robin 参数的选择, 且与 MRCM 相比, AlgMortar 内部求解的是 Dirichlet 问题, 条件数往往更优.
- 在匹配网格和最低阶 Raviart-Thomas 空间且 $K=I$ 的 Cartesian 网格上, AlgMortar 精确恢复 MMMFEM, 这保证了代数方法在结构化情形下仍然是严格一致的.
与已有代数多尺度方法的比较
- TAMS [43] 和 AMS [39] 均为代数多尺度方法, 但它们基于更一般的粗化与插值策略, 并不一定对应 mortar 弱通量耦合. TAMS 采用两阶段粗化, AMS 则构建基于能量的基函数. AlgMortar 则直接移植了数学性质良好的 mortar 耦合机制, 其粗空间由界面 averages 组成, 这在高对比度流动中能更好地保持通量连续性.
- MsTPFA [29] 将 TPFA 离散与代数多尺度概念结合, 但其适用范围限于两点通量近似; AlgMortar 的框架原则上适用于任何 SPD 离散, 只要满足非对角元非正的条件.
- 一些其他代数多尺度工作 [45,1,12,44] 亦提出不同代数构造路径, 但多数不强调与 MMMFEM 的精确等价性. AlgMortar 以 MMMFEM 为蓝本, 因而继承了其数学上的稳定性和近似性理论, 比如条件数估计与界面空间逼近性质的联系(尽管具体细节未在提供的摘录中给出).
与代数多重网格 (AMG) 的比较
- 数值实验部分 (第7节) 将 AlgMortar 用作 PCG 预条件子, 与经典 AMG 进行对比. 结果表明, 在 Cartesian 网格和高度 irregular corner-point 网格上, 对于高对比度非均质渗透率, AlgMortar 在并行效率上具有竞争力甚至更优. 这是因为 AlgMortar 的粗问题直接构建于界面 flux 连续性的物理机制上, 在高对比度界面上能更准确地传播信息, 而 AMG 的粗化若未考虑强耦合方向可能会退化.
- 代价方面, AlgMortar 需要额外的内存来存储多尺度基函数和局部分解, 但这些均可在预处理阶段一次完成; 而 AMG 的 setup 成本同样不低, 且健壮性可能受参数影响.
适用条件与局限
- AlgMortar 要求 $A$ 的非对角元非正, 这限制了该方法主要适用于 M 矩阵情形, 即由满足离散极值原理的椭圆离散产生. 对于具有非零对流传热的对流扩散方程或非单调格式, 该方法可能不再适定.
- 算子构造中使用了图划分, 其质量直接影响界面空间的维度与近似能力, 进而影响收敛率. 多尺度基函数的计算开销与 mortar 空间的基数 $N_\Lambda$ 成正比, 当界面过多或 mortar 空间选取过细时, 预处理成本会增加.
- 文中主要讨论 Dirichlet 边界问题 (其它边界可类似处理但未详述), 且仅考虑最低阶情形. 向高阶格式或更一般边界条件的推广有待研究.
来自 PDF 摘录的信息不足以支持与其他代数方法 (如 TAMS, AMS) 的直接数值对比, 作者在本文中主要进行了与 AMG 的对比实验.
了解领域论文（摘要翻译）
1. 求解非线性偏微分方程的DGM和PINN算法的全局收敛性
Global Convergence of DGM and PINN Algorithms for Solving Nonlinear PDEs
作者: Justin Sirignano, Konstantinos Spiliopoulos, Samuel Cohen
类别: cs.LG, math.NA
发布日期: 2026-07-27
arXiv ID: 2607.24726v1
链接: http://arxiv.org/abs/2607.24726v1
摘要翻译
中文标题: 求解非线性偏微分方程的DGM和PINN算法的全局收敛性
摘要翻译: 深度Galerkin方法（DGM）和物理信息神经网络（PINN）已成为快速发展的科学机器学习领域中求解偏微分方程（PDEs）的广泛使用方法。在这些方法中，通过使用（随机）梯度下降极小化神经网络的PDE残差来训练神经网络以近似PDE解。由于PDE残差目标函数的非凸性，训练后的神经网络原则上可能仅收敛到目标函数的局部极小值（这不会是PDE的解）。因此，关于这些算法的数学基础存在一个长期未决的问题，而确立训练后的神经网络将收敛到PDE解是非常有价值的。对于一类半线性偏微分方程（在解及其一阶导数中非线性），我们证明了使用梯度下降训练以极小化PDE残差目标函数的神经网络将收敛到PDE解。
2. 基于反射广义源的良态电场表面积分方程
Well-conditioned Electric Field Surface Integral Equations using Reflective Generalized Sources
作者: Yossi Dahan, Suryakumar Sivakumar, Yaniv Brick, Simon B. Adrian
类别: math.NA
发布日期: 2026-07-27
arXiv ID: 2607.24526v1
链接: http://arxiv.org/abs/2607.24526v1
摘要翻译
中文标题: 基于反射广义源的良态电场表面积分方程
摘要翻译: 本文利用广义源方法发展了一类良态积分算子，该算子无内部共振问题，且无需组合公式。广义源积分方程(GSIEs)的核是通过在传统电场积分方程(EFIE)的核中增加辅助贡献来增强相应矩量矩阵块的秩亏性而得到的。本文首次研究了由内部散射凸屏蔽所产生的辅助核的GSIE算子的谱。利用同心圆散射体与屏蔽的闭合表达式，我们证明：在屏蔽参数适合增强可压缩性的情况下，横磁(TM)和横电(TE)的GSIE算子均无内部本征共振与准共振。辅助分量被证明是其EFIE对应物的紧扰动。因此，这些GSIE继承了其稠密离散化崩溃问题，该问题仍可通过Calderón型预条件加以解决。支配高频崩溃的机制被证明受辅助分量的影响，在某些情况下会导致更强的鲁棒性。这些观察结果对于使用非圆形模板屏蔽设计的矩量矩阵和GSIE同样成立。对于GSIE-对偶和Yukawa核预条件，这些公式在保持可压缩性的同时，展现出良好的谱性质。这使其对于快速迭代求解器的设计颇具吸引力。关于二维屏蔽算子的结果为将该方法推广到三维问题以及具有更广泛几何适用性的辅助核奠定了基础。
3. Perturbative-NeuSA：一种面向时间依赖偏微分方程的结构化谱框架
Perturbative-NeuSA: A Structured Spectral Framework for Time-Dependent PDEs
作者: Xianli Zhu, Jia Yin
类别: cs.LG, math.NA
发布日期: 2026-07-27
arXiv ID: 2607.24345v1
链接: http://arxiv.org/abs/2607.24345v1
摘要翻译
中文标题: Perturbative-NeuSA：一种面向时间依赖偏微分方程的结构化谱框架
摘要翻译: 神经谱偏微分方程求解器通常学习整个未解析的向量场，即使一个低成本的近似模型已经能够捕捉大部分轨迹。本文引入 Perturbative-NeuSA，一种将目标解分解为低保真背景和高分辨率扰动的残差形式，从而仅学习未解析的动力学。该方法从精确的扰动方程出发，结合了固定的谱算子、依赖于背景的修正、目标偏微分方程中的背景缺陷，以及一个可选的神经闭合项。这种构造使得物理结构和神经闭合项的作用可以分别量度。在二维 Burgers、Klein-Gordon 和非均匀二维波动方程上，确定性的结构化求解器优于经过训练的 NeuSA 基线，且无需任何神经网络训练。最大的收益出现在 Burgers 方程上，其中确定性修正将训练误差和外推误差分别降低了 24 倍和 44 倍。此外，对 Klein-Gordon 方程在七种背景分辨率下的扫描表明，闭合项的作用是有条件的：它将不良背景改善了 3.6 倍，在中等分辨率下变得中性，并使已很好解析的背景退化。然而对于波动方程，当剩余残差集中于界面附近时，闭合项额外提供了 18% 的降低。多初始条件诊断进一步表明，有用的闭合工作机制依赖于初始条件谱，并且当结构化修正已经捕获主导的 Burgers 动力学时，在外推中可能消失。因此，Perturbative-NeuSA 将神经闭合项重新定义为一种有条件的、可诊断的修正，受背景保真度、残差组织以及与闭合模型的兼容性支配。
4. 一个针对忆阻器三物种漂移-扩散模型的收敛Scharfetter-Gummel格式
A convergent Scharfetter-Gummel scheme for a three-species drift-diffusion model for memristors
作者: Ansgar Jüngel, Zhiwei Sun, Sara Xhahysa
类别: math.NA
发布日期: 2026-07-27
arXiv ID: 2607.24086v1
链接: http://arxiv.org/abs/2607.24086v1
摘要翻译
中文标题: 一个针对忆阻器三物种漂移-扩散模型的收敛Scharfetter-Gummel格式
摘要翻译: 为一个半导体三物种漂移-扩散模型提出并分析了一种保结构的全隐式Scharfetter-Gummel有限体积格式。方程描述了一个（有界）忆阻器器件中电子、空穴和氧空位密度的演化，耦合着与电势相关的泊松方程，并具有混合型边界条件。将Scharfetter-Gummel通量重铸为迎风形式，揭示了一个隐藏的Fisher信息分量。由于通量中出现的伯努利函数的退化性，需要额外的边侧强制性估计，从而得到精细的局部和全局耗散估计。利用这些想法，建立了离散有限体积解的存在性、一个离散自由能不等式以及数值格式的收敛性。二维空间中的数值模拟证实了该格式的保结构性质，并展示了忆阻器器件中导电细丝的形成。
5. 基于圆柱谐波的Helmholtz方程显式Runge逼近
Explicit Runge approximation for Helmholtz equation with cylindrical harmonics
作者: Yu Chen, Jin Cheng, Tingyue Li
类别: math.NA
发布日期: 2026-07-26
arXiv ID: 2607.23699v1
链接: http://arxiv.org/abs/2607.23699v1
摘要翻译
中文标题: 基于圆柱谐波的Helmholtz方程显式Runge逼近
摘要翻译: 起源于全纯函数的复逼近，椭圆型方程的Runge逼近自Lax、Malgrange提出以来，经Rüland和Salo进一步建立了定量特征，已发展成为反问题乃至基于学习的数值方法的基本工具。这里应指出，Runge逼近是不适定的。在数值分析中，需要显式的定量估计来刻画逼近的增长对原始解向外延拓距离的依赖性。本文研究了使用圆柱谐波对Helmholtz方程的空间依赖定量Runge逼近，考虑了内部和外部边值问题。我们显式推导了三圆构型的相关指标，并获得了对于一般几何设置的渐近指标。导出的结果提供了展开系数的范数估计，这对正则化方法的实现至关重要。此外，建立的界使得能够构造Helmholtz方程解的谱精度数值逼近。
6. 拟态有限差分的自适应一致性
Adaptive Consistency for Mimetic Finite Differences
作者: Yoo Jin Cha, Omar Duran, Nicola Castelletto, Victor A. P. Magri, Hamdi A. Tchelepi
类别: math.NA
发布日期: 2026-07-26
arXiv ID: 2607.23568v1
链接: http://arxiv.org/abs/2607.23568v1
摘要翻译
中文标题: 拟态有限差分的自适应一致性
摘要翻译: 拟态有限差分（MFD）方法为一般多面体网格上的流动模拟提供了一种稳健的离散格式，但由于稠密的局部算子和降低的全局稀疏性，其计算代价可能变得显著。而两点通量近似（TPFA）提供了一种经济得多的替代方案，但其精度通常局限于 $K$-正交网格。为了平衡这些相互冲突的考虑，我们提出了一个基于残差的一致性指示子的自适应 MFD 框架，该指示子由离散本构方程导出。该指示子度量局部不一致性，并通过用户预设的容差 $τ$ 实现自适应的 TPFA/MFD 模板选择。由于自适应是在拟态框架内进行的，任意的 TPFA/MFD 分区保持稳定且结构保持。理论分析建立了强制一致性，并证明了相对通量误差在容差控制下的显式收敛性。对具有挑战性的多面体油藏基准问题的数值实验表明，其精度可与完全 MFD 离散相媲美，同时显著降低了矩阵密度和计算代价。
7. 双调和算子重特征值与聚特征值的 $C^0$ IPG 逼近的后验误差估计
An a Posteriori Error Estimator for $C^0$ IPG Approximation of Multiple and Clustered Eigenvalues of the Biharmonic Operator
作者: Jianing Guo, Qigang Liang
类别: math.NA
发布日期: 2026-07-25
arXiv ID: 2607.23215v1
链接: http://arxiv.org/abs/2607.23215v1
摘要翻译
中文标题: 双调和算子重特征值与聚特征值的 $C^0$ IPG 逼近的后验误差估计
摘要翻译: 本文基于 $C^0$ 内罚 Galerkin（$C^0$IPG）离散，针对双调和算子的重特征值与聚特征值所对应的特征函数，提出并分析了一个后验误差估计。所提出的估计可有效捕捉目标特征函数的局部奇性，并在自适应过程中发挥重要作用。我们建立了一种基于聚投影的严格分析，并引入一个辅助的理论误差估计器，将不变子空间误差与可计算的残差估计联系起来。最重要的是，得到的可靠性和有效性界关于网格尺寸、网格层级以及目标聚簇内部的谱间隙具有鲁棒性。数值实验支持理论结果，并展示了所提估计的鲁棒性。
8. 基于投影环境联络拉普拉斯算子的微分形式上的数据驱动扩散过程
Data-Driven Diffusion Processes on Differential Forms via the Projected Ambient Connection Laplacian
作者: Alvaro Almeida Gomez, Jorge Duque Franco
类别: math.NA, cs.LG
发布日期: 2026-07-25
arXiv ID: 2607.23192v1
链接: http://arxiv.org/abs/2607.23192v1
摘要翻译
中文标题: 基于投影环境联络拉普拉斯算子的微分形式上的数据驱动扩散过程
摘要翻译: 我们发展了一种数据驱动的投影环境联络拉普拉斯算子的近似，该算子作用于由点云采样的光滑黎曼流形上的微分形式。所提出的构造将扩散映射和向量扩散映射的经典框架从标量函数和切向量场推广到任意阶微分形式。我们的方法基于一种微分形式的新表示，即通过经典音乐同构的扩展得到的交错微分数组。该表示使得能够直接从点云数据构造矩阵值扩散算子，该算子近似投影环境联络拉普拉斯算子，而不需要网格或单纯复形。所提出的离散化具有从扩散映射继承的渐近最优的核带宽缩放，从而比先前数据驱动的霍奇拉普拉斯算子近似具有更清晰的收敛保证。基于该算子，我们推导出了一个完全数据驱动的微分形式热方程的显式欧拉格式，并通过单位球面上的数值实验验证了所提出的方法。实验证实了预测的解析解衰减，并证明了所提出离散化的有效性。所提出的框架提供了向量扩散映射到任意阶微分形式的自然推广，并为直接从点云数据数值逼近几何偏微分方程建立了实用基础。
9. 面向任意区域上偏微分方程学习的几何感知 LegONet
Geometry-aware LegONet for PDE Learning on Arbitrary Domains
作者: Jiahao Zhang, Yueqi Wang, Guang Lin
类别: math.NA
发布日期: 2026-07-25
arXiv ID: 2607.23069v1
链接: http://arxiv.org/abs/2607.23069v1
摘要翻译
中文标题: 面向任意区域上偏微分方程学习的几何感知 LegONet
摘要翻译: 学习偏微分方程求解器常常将控制算子与用于训练的几何、边界条件和离散方式纠缠在一起。当相同的物理过程被置于新区域上时，这限制了其复用性，同时也使得物理定律的发现依赖于几何。我们引入了几何感知 LegONet（gLegONet），这是乐高式算子学习的一种边界流形扩展。物理机制在环境谱域上以模块化变分块的形式仅需预训练一次。针对目标几何，采样的边界条件定义了一个仿射容许流形。利用其质量正交切坐标来演化动力学并直接评估候选的定律发现特征。因此，改变区域仅会改变代数坐标接口，而不会改变已学习的算子块。这将任意区域上的偏微分方程学习从特定几何的重新训练或软惩罚强制执行转变为边界保证的可复用机制组装。在未知区域上的正演模拟和稀疏识别测试中，该方法将边界残差维持在代数容差附近，并从短时观测中产生预测性的控制定律。
10. 完美的瑞士阿尔卑斯号角能被设计出来吗？缩减基方法与机器学习相结合的形状优化
Can the perfect Swiss alphorn be designed? A combination of reduced basis method and machine learning for shape optimization
作者: Fabio Marcinnò, Clément Froidevaux, Leonardo Bocchieri, Riccardo Tenderini, Gerald Pot, Simone Deparis
类别: math.NA
发布日期: 2026-07-24
arXiv ID: 2607.22823v1
链接: http://arxiv.org/abs/2607.22823v1
摘要翻译
中文标题: 完美的瑞士阿尔卑斯号角能被设计出来吗？缩减基方法与机器学习相结合的形状优化
摘要翻译: 本文研究瑞士阿尔卑斯号角的形状优化，使其共振频率尽可能接近给定的目标音符。我们基于实验测量和成像技术，构建了该号角的精确几何模型。其中，成像技术用于重建直接测量无法获得的几何信息。所得到的三维模型采用全结构化方法进行网格划分。然后，我们引入了亥姆霍兹方程的几何参数化有限元（FE）形式，从而能够应用缩减基方法（RBM）高效生成大量仿真数据集，其中每组几何参数对应一组特定的共振频率（音符）。该数据集可用于训练机器学习（ML）模型，实现从几何参数到共振频率的正向预测，以及逆向设计，即估计几何形状以达到指定的目标音符。
不相关论文（基本信息）
1. 临近点方法的安德森加速：精确自适应极小极大、谱相变与最优保护
Anderson acceleration of the proximal point method: the exact adaptive minimax, a spectral phase transition, and optimal safeguarding
作者: Zheng Jia, Yekini Shehu, Yonghong Yao
类别: math.NA, math.OC
发布日期: 2026-07-27
arXiv ID: 2607.24643v1
链接: http://arxiv.org/abs/2607.24643v1
摘要: \noindent We study residual-polynomial acceleration of the proximal point method (PPM) for maximal monotone inclusions, with Anderson acceleration (AA) as the prototypical adaptive scheme. We answer three questions exactly. (i)~The minimax complexity over all adaptive methods is precisely $d_0/(K+1)$ per $K$ resolvent evaluations. The upper bound is attained by the averaged-reflection estimator; the matching lower bound uses an explicit skew-adjoint instance with resolvent eigenvalues at the roots of $u^{K+1}=-1$ and $\csc^2$-distributed masses, on which every degree-$K$ polynomial method satisfies $\|r(y_K)\|\ge 1/(K+1)$. The optimal polynomial is uniquely the Fejér kernel, and the same instance certifies a per-step floor. (ii)~A sharp phase transition separates regimes: Jackson-kernel polynomials achieve $O(d_0/(K^2 s))$ when the spectral floor $s$ satisfies $sK\to\infty$, while at the critical scale $s\asymp 1/K$ the barrier is exactly $1/(K+1)$. The picture extends to normal operators and the nonlinear family $M=S+N_C$. (iii)~On linear problems AA-PPM needs no safeguarding; on nonlinear problems certification of the $O(1/k)$ envelope requires exactly two oracle evaluations per iteration, and this factor is optimal. We also correct and complete the theory for structured problems—affine, strongly monotone, piecewise-affine, and Hölderian growth—and confirm all predictions numerically.
2. 面向双曲系统的自适应多物理耦合
Adaptive Multiphysics Coupling for Hyperbolic Systems
作者: Simon Candelaresi, Erik Faulhaber, Michael Schlottke-Lakemper
类别: math.NA
发布日期: 2026-07-27
arXiv ID: 2607.24600v1
链接: http://arxiv.org/abs/2607.24600v1
摘要: For the discontinuous Galerkin code Trixi.jl we implement capabilities for adaptively coupling arbitrarily many domains with different physics. The coupling of the systems is realized through the exchange of boundary information and user-definable coupling functions. This gives us the ability to couple systems that do not share a single variable or have a very different number of variables. This is particularly useful when we have a hierarchy of systems. Our implementation is such that we can adaptively select the model of the coupled systems in the course of the simulation. This allows for highly dynamical scenarios as can be found e.g. in astrophysics. The criteria for adaptivity can be user defined and tailored to the physical problem. Compared to computing the most complex model in the entire simulation domain, our method of coupled systems of high and low complexity leads to a significant reduction in computational time with very little overhead.
3. 任意阶Padé封闭锚定二导数时间离散：$s$个活动级，阶$2s$，且$L$-稳定性
Arbitrary-Order Padé-Closed Anchored Two-Derivative Time Discretizations: $s$ Active Stages, Order $2s$, and $L$-Stability
作者: Zhixin Huo
类别: math.NA
发布日期: 2026-07-27
arXiv ID: 2607.24592v1
链接: http://arxiv.org/abs/2607.24592v1
摘要: An arbitrary-order family of implicit two-derivative one-step methods is constructed in an anchored active-stage formulation. At each information node the method uses both the vector field and its first total time derivative, enriching the local Hermite data without increasing the number of unknown stage states. With the known initial value retained as an anchor and $s$ unknown active stages, $2s$ Hermite moment conditions yield global order $2s$. The two remaining coefficients in each stage row are fixed by the second-subdiagonal Padé approximant $[s-1/s+1]_{e^z}$. For every ordered real node set, a Padé–Hermite basis theorem proves that the closure is unique, preserves all moment conditions, and gives $\det(I-zA-z^2\widehat A)=Q_s(z)$ and $R_s(z)=P_s(z)/Q_s(z)$. Hence the coupled stage system has no hidden poles and the accepted one-step map is $L$-stable (and therefore $A$-stable) for every positive integer $s$. Exact symbolic verification is reported through $s=6$, and high-precision computations confirm orders $2,4,6,$ and $8$ for the first four members. At equal active-stage count, comparisons with Gauss–Legendre and Radau IIA methods demonstrate the combined high-order accuracy and strong stiff damping of the construction over broad step-size ranges.
4. 一种用于强非常数磁场下三维带电粒子动力学的曲线参考指数积分器
A Curve-Reference Exponential Integrator for Three-Dimensional Charged-Particle Dynamics under a Strong Nonconstant Magnetic Field
作者: Zhirui Shen, Bin Wang
类别: math.NA
发布日期: 2026-07-27
arXiv ID: 2607.24523v1
链接: http://arxiv.org/abs/2607.24523v1
摘要: We study a three-dimensional charged-particle dynamics model in the nonrelativistic momentum formulation and construct a curve-reference exponential integrator (CREI) for this system. Since the zero eigenvalue of linear section is not semisimple, the system after usual change of variables does not meet the spectral condition used in the classical locally linearized extended exponential integrator (LLEEI). A kernel-range decomposition of the dominant magnetic matrix and a further linear transformation reduce the equation to a system with a semisimple zero block and one constant skew-symmetric oscillatory block. The CREI expands the nonlinear term along the exact oscillatory curve rather than at a fixed point. Retaining monomials through degree k defines CREI(k + 1). We prove uniform local and global error bounds and present reproducible numerical experiments for convergence in h and uniformity in ε.
5. 基于加速对称非负矩阵分解的低秩相关性分解
Low-Rank Dependence Decomposition via Accelerated Symmetric Non-negative Matrix Factorization
作者: Lavinia Ghita, Dhruv Desai, Jake Goldberg, Roman Yokunda Enzmann
类别: cs.LG, math.NA
发布日期: 2026-07-27
arXiv ID: 2607.24518v1
链接: http://arxiv.org/abs/2607.24518v1
摘要: Symmetric non-negative matrix factorization (SymNMF) recovers latent group structure from a dependence matrix, but its dense, quadratic-memory objective has confined prior work to moderate sizes. We present a large-scale GPU study of seven algorithm families (over 30 configurations) on absolute Pearson correlation and tail pairwise dependence matrices from Extreme Value Theory, two proxies for empirical risk-factor estimation on large portfolios. A trace-identity reformulation eliminates all $n \times n$ intermediates, so a single GPU reaches $n \approx 10^5$ and multi-node distribution scales to $n = 10^6$ and beyond. Under a two-phase protocol, eleven methods converge at moderate scale; six remain efficient enough at $n = 10^5$ (five AdaGrad-family plus ADMM), and five AdaGrad-family methods still converge at $n = 10^6$: AdaGrad, RMSprop, and three we introduce (Piecewise AdaGrad, Row-Stochastic SVRG, Block-SVRG AdaptGrow). At $n = 10^6$ the fastest solver tracks the matrix spectrum: Block-SVRG AdaptGrow wins on the flat, ill-conditioned tail-dependence spectrum, where its lower per-iteration cost decides a long factorization, and full-batch AdaGrad wins on the dominant-low-rank correlation spectrum, where the run is short. We also benchmark spherical K-means as a hard-label baseline: cheaper when angular cluster structure is present, yet provably degenerate once the matrix collapses toward a single common factor, where the soft factorization remains necessary.
6. Gompertz 生长下肿瘤总群体的优化
Optimization of the total tumor population under Gompertz growth
作者: Iulia Martina Bulai, Francesca Gladiali, Benedetta Pellacci
类别: math.AP, math.NA
发布日期: 2026-07-27
arXiv ID: 2607.24468v1
链接: http://arxiv.org/abs/2607.24468v1
摘要: We study optimal control problems for a stationary reaction–diffusion model describing the spatial distribution of a tumor cell population with Gompertz growth. The control $m(x)$ represents a treatment term acting as a density-dependent removal rate and it is subject to $L^{1}-L^{\infty}$ constraints. When the intrinsic growth rate is constant, the uniform distribution of the treatment is shown to be the unique minimizer. For the maximization problem, we prove that every optimal control is of bang-bang type. In addition, we show that in the one dimensional case and for sufficiently large diffusion rates, the positivity set of optimal controls is an interval sticking to one of the extrema of the domain. Finally, numerical simulations complement the theoretical analysis and explore regimes that are not fully covered by the results proved in the paper. The computations confirm the bang-bang structure of maximizers, and illustrate how the shape of optimal controls and the associated states are affected by spatial heterogeneity in the growth rate, localized admissible treatment regions, and the diffusion coefficient. Moreover, they reveal a monotone dependence of the optimized total population on the diffusion coefficient: this is a new phenomenon with respect to the logistic setting.
7. 用于双曲守恒律的具有统一模板的五阶有限体积基于导数的Hermite WENO格式
Fifth-order finite volume derivative-based Hermite WENO scheme with unified stencils for hyperbolic conservation laws
作者: Peiwen Chen, Zhuang Zhao
类别: math.NA
发布日期: 2026-07-27
arXiv ID: 2607.24325v1
链接: http://arxiv.org/abs/2607.24325v1
摘要: In this paper, we propose a derivative-based finite volume Hermite WENO (HWENO) scheme for hyperbolic conservation laws, where both the solution and its first-order derivatives are evolved in time and utilized in spatial reconstructions. The key challenge for solving hyperbolic conservation laws is the possible emergence of discontinuities in the numerical solutions. When facing discontinuities, the derivatives can become excessively large, which may compromise the robustness of HWENO schemes. In the first HWENO scheme, different sets of stencils were adopted for reconstructing the governing equation and the derivative equation, respectively, aiming to reduce the influence of the derivatives while preserving high-order accuracy. However, this approach not only substantially increases computational cost but also introduces considerable algorithmic complexity. To overcome these limitations, we exclude the information of the target cell’s derivatives from spatial reconstructions, while employing the same reconstructed polynomial during temporal evolution to limit the derivatives. This strategy enhances the robustness of traditional HWENO schemes and allows unified stencils within the derivative-based HWENO framework. Furthermore, the proposed scheme supports arbitrary positive linear weights that sum to one and maintains a compact stencil. Numerical results demonstrate the high-order accuracy, efficiency, high resolution, and robustness of the proposed HWENO scheme.
8. 带压缩的坐标下降法在全组态相互作用中的收敛性与误差分析
The Convergence and Error Analysis of Coordinate Descent Methods with Compression for Full Configuration Interaction
作者: Yingzhou Li, Qiang Wu
类别: math.NA, math.OC
发布日期: 2026-07-27
arXiv ID: 2607.24319v1
链接: http://arxiv.org/abs/2607.24319v1
摘要: We study the effect of compression in Coordinate Descent Full Configuration Interaction (CDFCI) within an unconstrained optimization formulation of the full configuration interaction ground-state problem. Under suitable local assumptions, we prove that the compressed iteration converges linearly to the solution of an associated restricted problem. We also characterize the convergence point of the compressed algorithm. Under an additional exponential decay assumption on the target eigenvector, we show that the resulting eigenvalue error is of order $τ^2$, where $τ$ denotes the compression threshold. Numerical results support the analysis.
9. 具有非负权重的$L_1$偏差存在维数灾难
The $L_1$-Discrepancy with Nonnegative Weights Suffers from the Curse of Dimensionality
作者: Josef Dick
类别: math.NA, math.PR
发布日期: 2026-07-27
arXiv ID: 2607.24290v1
链接: http://arxiv.org/abs/2607.24290v1
摘要: We prove that the $L_1$-discrepancy with arbitrary nonnegative weights suffers from the curse of dimensionality. More precisely, for every $\varepsilon \in (0,1)$ and $d \in \mathbb{N}$, the inverse of the $L_1$-discrepancy satisfies [ N_{1,+}(\varepsilon, d) \ge \frac{(1-\varepsilon)^2}{1 + \varepsilon} \left( \frac{3+2 \sqrt{3}}{6}\right)^d, ] where $(3+2\sqrt{3})/6 = 1.07735\ldots$. The proof combines a change to a volume-biased probability measure with a fractional-moment estimate for the normalized discrepancy function. The lower bound applies, in particular, to equally weighted point sets. The argument uses the nonnegativity of the weights in an essential way and does not cover arbitrary signed weights.
10. 一种一维非局部非线性程函方程格式的收敛性
Convergence of a scheme for a one dimensional nonlocal and nonlinear eikonal equation
作者: Diana Al Zareef, Ahmad El Hajj, Antoine Zurek
类别: math.NA
发布日期: 2026-07-27
arXiv ID: 2607.24216v1
链接: http://arxiv.org/abs/2607.24216v1
摘要: In this work, we study a one-dimensional nonlocal and nonlinear eikonal equation without sign restriction on its spatial gradient. The equation is characterized by weak regularity assumptions on both the nonlocal velocity field and the initial data. We derive a periodic version of this model and propose a semi-explicit (IMEX) scheme for its numerical approximation. We prove that the scheme preserves a discrete gradient entropy estimate and establish its convergence in the viscosity sense. Finally, we present numerical results illustrating the behavior of the model and the performance of the proposed scheme.
11. 基于后验停止准则的大规模不适定问题的随机化Krylov投影迭代Tikhonov正则化
Randomized Krylov-Projected Iterated Tikhonov Regularization for Large-Scale Ill-posed Problems Under A Posteriori Stopping Rule
作者: Ravi Verma, Harshit Bajpai, Ankik Kumar Giri
类别: math.NA
发布日期: 2026-07-27
arXiv ID: 2607.24138v1
链接: http://arxiv.org/abs/2607.24138v1
摘要: We introduce two novel randomized iterative regularization frameworks, termed \texttt{RIGKT} and \texttt{RIAT}, for solving large-scale linear ill-posed inverse problems governed by systems of equations. The proposed methods combine randomized iterated Tikhonov regularization with Krylov subspace projection techniques, utilizing Golub–Kahan bidiagonalization for general rectangular systems (\texttt{RIGKT}) and Arnoldi decomposition for square systems (\texttt{RIAT}). Unlike existing deterministic schemes that rely on fixed iteration counts, our framework incorporates randomized equation selection, an adaptive step-size strategy, and a global, discrepancy-based a posteriori early-stopping rule tailored specifically to the stochastic setting. We present a comprehensive regularization analysis establishing Bregman-distance monotonicity, finite termination, exact-data convergence, and pathwise stability under noise. Furthermore, we prove that the stopped iterates converge almost surely and in the mean-square sense to the true solution, establishing a rigorous regularization property. To the best of our knowledge, this is the first theoretical framework to simultaneously account for randomization, Krylov-subspace dimension reduction, and implementable early stopping. Numerical experiments involving two-dimensional X-ray computed tomography (CT) and image deblurring demonstrate that \texttt{RIGKT} and \texttt{RIAT} reliably reconstruct structural features across various noise regimes.
12. 关于异构波动方程离散无映射帐篷推进法的一个注记
A note on the discrete Unmapped Tent Pitching for the heterogeneous wave equation
作者: Marcella Bonazzoli, Gabriele Ciaramella, Ilario Mazzieri
类别: math.NA
发布日期: 2026-07-27
arXiv ID: 2607.24066v1
链接: http://arxiv.org/abs/2607.24066v1
摘要: The Unmapped Tent Pitching (UTP) algorithm is a space-time domain decomposition method for the parallel solution of wave-type problems. We have recently extended UTP to heterogeneous settings and compared, at the continuous level, the computational cost of different space-time decompositions. In this note, we report discrete-level observations showing that the optimal decomposition strategy may differ from the one predicted by the continuous analysis.
13. 一种用于不可压缩Navier-Stokes方程的矢量格子Boltzmann格式
A vectorial lattice Boltzmann scheme for the incompressible Navier-Stokes equations
作者: Denise Aregba-Driollet, Thomas Bellotti, Roberto Natalini, Tommaso Tenna
类别: math.NA, physics.flu-dyn
发布日期: 2026-07-27
arXiv ID: 2607.24059v1
链接: http://arxiv.org/abs/2607.24059v1
摘要: We introduce a second-order accurate vectorial lattice Boltzmann scheme for the incompressible Navier-Stokes system, inspired by a discrete-velocity kinetic approximation proposed by Carfora and Natalini [ESAIM: M2AN, 42(1), 93-112, 2008]. Advantages and drawbacks compared to relaxation schemes are investigated by providing spectral analyses in the linearized case, and numerical validations on the genuinely non-linear problem.
14. 强磁场下相对论带电粒子动力学的具有一致误差界的多物理保结构积分器
A multi-physics structure-preserving integrator with uniform error bounds for relativistic charged-particle dynamics under strong magnetic fields
作者: Mengting Hu, Yifa Tang, Bin Wang
类别: math.NA
发布日期: 2026-07-27
arXiv ID: 2607.24058v1
链接: http://arxiv.org/abs/2607.24058v1
摘要: In this paper, we develop an explicit multi-physics structure-preserving Strang splitting scheme for a four-dimensional relativistic charged-particle dynamical system in the presence of a strong magnetic field under the maximal ordering scaling. The proposed scheme not only preserves volume, energy, and Lorentz invariance, but also yields second-order uniform error bounds for the position and the parallel velocity component. We present rigorous theoretical proofs for these geometric properties and the error convergence, which are subsequently validated by several numerical experiments.
15. 高阶信赖域与球面多项式优化的齐次张量框架
A Homogeneous Tensor Framework for High-Order Trust-Region and Spherical Polynomial Optimization
作者: Wenqi Zhu, Haibin Chen, Guanglu Zhou
类别: math.OC, math.NA
发布日期: 2026-07-27
arXiv ID: 2607.24046v1
链接: http://arxiv.org/abs/2607.24046v1
摘要: High-order methods can improve worst-case evaluation complexity, but for orders $p\geq3$ their Taylor subproblems are nonconvex polynomial optimization problems and are generally difficult to solve. We develop a radius-controlled boundary approach based on homogeneous tensor representations. By augmenting the step with a constant coordinate, any $p$th-order Taylor polynomial can be represented exactly as an order-$p$ homogeneous tensor form; at a prescribed radius, the boundary model is a spherical polynomial optimization problem. The representation applies to arbitrary $p$, while the algorithmic development focuses on the cubic case $p=3$. For an inhomogeneous cubic on the sphere, we introduce a quadratic shift and prove, under an explicit shift bound, equivalence with a three-block multilinear formulation at global optimality. This motivates a proximal alternating minimization (PAM) method with closed-form block updates; its objective values decrease and every accumulation point is stationary. We embed the boundary-step mechanism in an Adaptive Homogeneous Tensor Method (Ada–HTM). Under explicit smoothness, safeguarded-decrease, weak-curvature nondegeneracy, and local-refinement conditions, Ada–HTM attains the adaptive-regularization-type (AR$p$-type) evaluation complexity $\mathcal{O}(ε^{-(p+1)/p})$ for first-order stationarity. Numerically, PAM matches order-$2$ moment–sum-of-squares (SOS) certificates on the structured cubic instances for which certification is tractable, scales particularly well for low-rank tensors, and makes Ada–HTM competitive with trust-region and cubic-regularization methods, with its largest gains on ill-conditioned and badly-scaled problems.
16. 估计协方差矩阵的特征向量与特征空间：最优界与一致性条件
Estimating eigenvectors and eigenspaces of covariance matrices: Optimal Bounds and Conditions for Consistency
作者: Phuc Tran, Van Vu
类别: math.ST, math.NA, math.PR
发布日期: 2026-07-27
arXiv ID: 2607.23964v1
链接: http://arxiv.org/abs/2607.23964v1
摘要: Let $X = [ ξ_1, \,\, ξ_2,...\,\, ,ξ_d]^\top$ be a zero-mean random vector of large dimension $d$ ($d \rightarrow \infty$) with (hidden) covariance matrix $M = (m_{ij})_{1 \leq i, j \leq d},$ where $m_{ij} = m_{ji} = \textbf{Cov}(ξ_i, ξ_j).$ Let $X_1, X_2, \dots, X_n$ be $n$ iid samples of $X$. Consider the sample covariance matrix
$$\textstyle \tilde{M} := \frac{1}{n} \sum_{i=1}^{n} X_i X_i^\top.$$In practice, one frequently uses the eigenvectors and eigenspaces of $\tilde M$ as estimators for those of $M$. A central task is to provide an error analysis for these estimators. In this paper, we provide an optimal error analysis, obtaining upper and lower bounds of matching order of magnitude, for a wide range of parameters $d$ and $n$, under mild assumptions on $M$. As corollaries, we obtain new necessary and sufficient conditions for the consistency of the estimators. In these conditions, we only require the number of samples $n$ to depend linearly on the effective rank of $M$, which can be much smaller than the dimension $d$.
17. 递归治理：一种用于智能体AI系统中风险传播与漂移检测的图论框架
Recursive Governance: A Graph-Theoretic Framework for Risk Propagation and Drift Detection in Agentic AI Systems
作者: Sriram Nagaraj, Advaith Nila Narayanan
类别: math.NA
发布日期: 2026-07-27
arXiv ID: 2607.23916v1
链接: http://arxiv.org/abs/2607.23916v1
摘要: As financial institutions transition from traditional predictive models to autonomous agentic systems, the static model inventory requirements of traditional model risk management (MRM) face structural obsolescence. This paper proposes a dynamic Inventory-as-Code (IaC) governance loop that treats the model inventory as a living architectural component rather than a periodic documentation artifact. We make four principal contributions. First, we introduce a calibrated Degree of Autonomy (DoA) materiality score with an explicit, taxonomized tool-complexity weighting scheme that addresses the dominance problem of naive additive risk formulations. Second, we construct the agent inventory as a Directed Acyclic Graph (DAG) and define a formal Composite Risk Propagation algorithm under which upstream validation failures induce risk penalties on all reachable descendants. Third, we develop a Trajectory Monitoring protocol based on distributional cosine drift across ensembled Chain-of-Thought (CoT) embeddings, with an explicit procedure for constructing and certifying the Golden Path baseline, a matched-bootstrap calibration that we show is necessary to avoid a severe false-positive artifact in the naive alternative, and a two-stage response that separates legitimate reasoning variation from detrimental drift without demanding that a single exceedance event trigger an irreversible action. Fourth, we address practical complications largely absent from the prior literature: LLM base-model version changes, latent feedback loops in nominally acyclic agent graphs, and a two-pass execution structure that stages validation ahead of risk propagation.
18. 面向目标的最小二乘有限元方法误差估计: 基于物理意义的伴随偏微分方程
Goal-Oriented Error Estimation for Least-Squares Finite Element Methods via Physically Meaningful Adjoint PDEs
作者: Yueyao Wu, Shun Zhang
类别: math.NA
发布日期: 2026-07-26
arXiv ID: 2607.23850v1
链接: http://arxiv.org/abs/2607.23850v1
摘要: We develop a goal-oriented error-estimation framework for first-order system least-squares (FOSLS) finite element methods based on the physical PDE adjoint rather than the adjoint induced by the least-squares formulation. Identified explicitly from the differential operator, its mixed boundary conditions, and the output, the physical adjoint admits its own first-order flux system and hence a native, built-in least-squares estimator, which the least-squares-induced adjoint does not. Our error identities rest on a primal-dual corrected functional: they follow from the continuous primal and adjoint equations alone, hold for arbitrary conforming approximations, and require no Galerkin orthogonality. For outputs containing a weighted Dirichlet-boundary flux, whose weight becomes the essential datum of the adjoint, we construct two corrected approximations, one from the potentials and one also using the fluxes, and prove product-type error estimates in the primal and adjoint errors. The least-squares functionals then yield computable a posteriori bounds and a balanced marking indicator whose element sum equals the product of the two estimators exactly. Numerical experiments confirm the predicted convergence and the effectiveness of the marking.
19. 通过纤维依赖消除的高维非线性偏微分方程隐式张量火车交叉积分
Implicit Tensor-Train Cross Integration of High-Dimensional Nonlinear PDEs via Fiber-Dependency Elimination
作者: Behzad Ghahremani, Hessam Babaee
类别: math.NA
发布日期: 2026-07-26
arXiv ID: 2607.23814v1
链接: http://arxiv.org/abs/2607.23814v1
摘要: Tensor-train (TT) representations have emerged as an effective framework for mitigating the curse of dimensionality in the numerical solution of high-dimensional tensor differential equations. Among existing approaches, TT-cross methods are particularly attractive because they require only pointwise evaluations of the governing equations, naturally accommodate arbitrary nonlinearities, and avoid tangent-space projections and the numerical difficulties associated with nearly singular low-rank factors. However, existing TT-cross rank-truncation methods have been restricted to explicit time integration. Extending TT-cross methods to implicit schemes presents an obstacle: the collocation equations associated with the cross-selected fibers depend on neighboring fibers that are not part of the unknown set. Consequently, the resulting nonlinear system is not closed, preventing the direct application of standard implicit solvers. In this work, we introduce a principled fiber-dependency elimination framework that resolves this obstacle by expressing neighboring fibers as linear combinations of the cross-selected fibers through cross interpolation identities. The resulting formulation produces a closed collocation system while preserving the principal advantages of TT-cross methods. The proposed framework applies to both linear and nonlinear high-dimensional partial differential equations and is naturally combined with Newton iterations and rank adaptivity. Numerical experiments demonstrate rapid convergence of the dependency-elimination iterations, preservation of the temporal accuracy of implicit multistep schemes, and efficient implicit integration of high-dimensional nonlinear problems with full-order discretizations containing up to $10^{55}$ degrees of freedom.
20. 稀疏车辆自组织网络中交通激波的GNN多智能体控制
GNN-based Multi-Agent Control of Traffic Shockwaves in Sparse Vehicular Ad-hoc Networks
作者: Prachi Nandi, Madhuri Malakar, Sonakshi Satpathy, Pabitra Mohan Khilar
类别: cs.NI, cs.LG, math.NA
发布日期: 2026-07-26
arXiv ID: 2607.23792v1
链接: http://arxiv.org/abs/2607.23792v1
摘要: Traffic shockwaves are stop-and-go waves that propagate upstream through the streams of vehicles and are one of the major causes of traffic congestion, fuel inefficiency, and increased accident rates in modern transportation systems. Although Connected and Autonomous Vehicles (CAVs) offer a promising opportunity to mitigate such shockwaves, most existing control strategies rely on global traffic state information, making them impractical for early-stage deployment of Vehicular Ad-hoc Networks (VANETs). In this paper, we propose a decentralized Multi-Agent Reinforcement Learning (MARL) framework that integrates a Graph Neural Network (GNN) to enhance the control architecture of connected and autonomous vehicles. The proposed approach enables vehicles to learn cooperative control policies using locally available information and interaction with neighboring vehicles. The effectiveness of the proposed scheme is evaluated using a scalable simulation environment under realistic highway traffic conditions. Simulation results show that the proposed GNN-based MARL framework can reduce the propagation of traffic shockwaves by up to 80%, even when only 10% of the vehicles are connected.
21. 一种用于Navier-Stokes方程的高阶相容分裂格式的谱消失粘性稳定化方法
A spectral-vanishing-viscosity stabilization of a higher-order consistent splitting scheme for the Navier-Stokes equations
作者: M Nader Alhomsi, Akram Moustafa, Mohammad Al-Saqqa, Jiahong Wu, Xiaoming Zheng
类别: math.NA
发布日期: 2026-07-26
arXiv ID: 2607.23720v1
链接: http://arxiv.org/abs/2607.23720v1
摘要: Huang and Shen developed a novel class of high-order BDF-IMEX consistent-splitting schemes for the incompressible Navier-Stokes equations, giving the first rigorous stability and error analysis for a fully decoupled splitting scheme of temporal order higher than two. Extending their analysis from unit viscosity to arbitrary viscosity, this work reveals that the error upper bound coefficient contains inverse powers of the viscosity. Our numerical experiments show that the scheme can break down at high Reynolds number. To save the scheme from this failure, we stabilize it by adding to the velocity update a symmetric positive-semidefinite spectral vanishing viscosity operator, built from the directionally applied Maday-Kaber-Tadmor kernel, which selectively damps the high, under-resolved modes at no additional asymptotic cost and leaves the structure of the error analysis intact. We establish stability and error estimates for the stabilized scheme in which the spectral vanishing viscosity provides viscosity-independent coercive control of the high modes. Three two-dimensional tests demonstrate the robustness and accuracy of the stabilized scheme. For a manufactured solution, the stabilized scheme retains its design order for k=2,3,4, whereas the unstabilized scheme diverges. For a perturbed Kovasznay flow, it accurately resolves the boundary layer at Re=10^4 and drives the perturbation back to the steady state, while the unstabilized scheme blows up. For the Kelvin-Helmholtz instability problem, it reproduces the reference integral diagnostics throughout the reliable regime, whereas the unstabilized scheme produces spurious solutions or blows up.
22. 时变边界条件下流动代理模型的无免费午餐：双机制研究
No Free Lunch in Flow Surrogates under Time-Varying Boundary Conditions: A Two-Regime Study
作者: Georg Winkler, Martin Stoll
类别: math.NA, cs.LG, physics.flu-dyn
发布日期: 2026-07-26
arXiv ID: 2607.23667v1
链接: http://arxiv.org/abs/2607.23667v1
摘要: A flow surrogate validated on a simple regime is often taken as evidence that the approach will carry to a richer one. We test this assumption on two transient flows under time-varying boundary conditions emulating the process startup: the three-dimensional slurry film in chemical-mechanical planarisation (CMP), a core semiconductor-manufacturing process, and the two-dimensional Karman vortex street (KVS) behind a cylinder. Eight surrogate models are compared on one shared evaluation pipeline, differing in whether they learn the full field or a latent representation, and whether they predict trajectories in one shot or step by step. No single architecture wins both regimes. On the film, a one-shot full-field model reconstructs the process-relevant cumulative wall shear stress to 3.2% relative error. On the wake, a latent autoregressive DeepONet retains 96% of the shedding power that direct and one-shot models damp to almost zero. The deciding axis is the treatment of time. The self-sustained wake requires the phase memory that autoregressive feedback provides, while the boundary-driven film rewards a direct map. Pointwise RMSE picks the wrong model in both regimes, so the evaluation scores five physical questions instead, the field, its structure, invented motion, amplitude, and timing. The trained surrogates answer queries $10^3$ to $10^4$ times faster than the finite-element solver, but the offline cost of the training simulations means they pay off from the first query beyond the training set for CMP and the third for the KVS. The choice of surrogate should follow the dynamical character of the target flow, and its validation should use failure-mode-resolved metrics, since neither the winning architecture nor its validation transfers.
23. IMEX-BDFk 有限元格式对于不可压缩 Navier-Stokes 系统的稳定性与误差分析
Stability and error analysis of IMEX-BDFk finite element schemes for the incompressible Navier-Stokes system
作者: Qianqian Ding, Yifan Luo, Shipeng Mao
类别: math.NA
发布日期: 2026-07-26
arXiv ID: 2607.23635v1
链接: http://arxiv.org/abs/2607.23635v1
摘要: In this paper, we propose and analyze a class of high-order numerical schemes within a fully discrete finite element framework for the incompressible Navier-Stokes equations with no-slip boundary conditions. The temporal discretization employs a kth-order (k=1,…,6) implicit-explicit backward difference formula (IMEX-BDFk), in which the nonlinear convection term is treated explicitly and the linear Stokes part implicitly, whereas the spatial discretization utilizes Taylor-Hood finite elements. We establish the stability and uniform boundedness of the numerical solution. We further establish optimal order error estimates in both space and time without any CFL-type condition, in the sense that the time step is independent of the spatial mesh size. In three dimensions, these include L2- and H1-norm error estimates for the velocity and L2-norm error estimates for the pressure, with temporal convergence rates up to sixth order for all variables. Numerical experiments are presented to demonstrate the effectiveness of the scheme and to confirm the theoretical convergence rates.
24. D3Q125 上框架鲁棒阶分辨松弛的中心 Hermite 感测与碰撞
Central-Hermite Sensing and Collision for Frame-Robust Order-Resolved Relaxation on D3Q125
作者: Bjørn Wu
类别: math.NA, physics.flu-dyn
发布日期: 2026-07-26
arXiv ID: 2607.23629v1
链接: http://arxiv.org/abs/2607.23629v1
摘要: Raw-Hermite sensing and collision on a fixed discrete-velocity set can convert a uniform translation into artificial coupling between nominally distinct nonequilibrium orders. We develop a central-Hermite formulation for a D3Q125 kinetic model with order-resolved log-Gaussian relaxation and compare three variants: raw sensing/raw collision (A), central sensing/raw collision (B), and central sensing/central collision (C). In homogeneous translated second-order perturbations, model C preserves third- and fourth-order modal purity to machine precision, whereas A and B develop boost-dependent cross-order content. Across a grid-CFL-boost matrix, model C reduces the post-transport collision frame discrepancy relative to A by 65.342-98.102% (median 81.131%) in the total relative L-infinity measure. Long-time calculations remain positive and conservative to numerical precision, although the accumulated benefit is configuration dependent because transport continually re-injects frame error. A transport study further reveals a clear trade-off: central-Hermite interface reconstruction strongly suppresses the third-order discrepancy but amplifies the fourth-order discrepancy. The fully central-Hermite collision therefore substantially reduces collision-induced cross-order frame discrepancy, while residual dependence remains due to discrete transport and finite velocity-space representation. This moment-space improvement does not by itself establish a comparable reduction in macroscopic Galilean transport error.
25. 用于图像变化检测的随机索引：一种距离阈值词汇表方法
Random Indexing for Image Change Detection: A Distance-Threshold Vocabulary Approach
作者: Cristiano Tamborrino
类别: math.NA
发布日期: 2026-07-26
arXiv ID: 2607.23609v1
链接: http://arxiv.org/abs/2607.23609v1
摘要: Random Indexing (RI), used almost exclusively in text analysis (notably Temporal Random Indexing, TRI, for tracking word meaning over time), represents a discrete vocabulary with fixed random vectors and accumulates context by vector summation. We explore transplanting this mechanism to change detection in multitemporal images. A naive transplant using k-means to build the visual vocabulary is unstable: pixels are frequently reassigned between acquisition dates due to small radiometric shifts, destroying the pixel-to-vector correspondence RI depends on. We propose a distance-threshold (leader) clustering vocabulary instead, and give a short formal argument – a stability radius from the vocabulary covering/packing properties – for why this is provably more robust to acquisition noise than k-means. Combined with spatial context accumulation, this yields a training-free change detection pipeline, evaluated on four bi-temporal remote-sensing datasets (agriculture, river, urban PolSAR, wildfire) against a Change Vector Analysis baseline, which it consistently approaches but does not surpass. Multi-seed validation exposed and let us fix a degenerate-vector failure mode in a probabilistic-sparsity RI variant, and revealed a second, unresolved source of variability – sensitivity to the leader-clustering visitation order – which we characterize but, despite three attempted corrections, could not eliminate, reporting it as the main open problem. Results are cross-checked against an independent re-implementation. We report an ablation, a hyperparameter sensitivity analysis, an Otsu-vs-Gaussian-mixture thresholding comparison, and discuss RI incremental accumulation for future long time-series monitoring.
26. 逆星偏差的最优多项式可解性指数
Optimal Polynomial Tractability Exponents for the Inverse Star Discrepancy
作者: Josef Dick
类别: math.NA
发布日期: 2026-07-26
arXiv ID: 2607.23571v1
链接: http://arxiv.org/abs/2607.23571v1
摘要: The inverse of the star discrepancy $n^\ast(d, \varepsilon)$ satisfies [ d \varepsilon^{-1} \lesssim n^{\ast}(d,\varepsilon)\lesssim d\varepsilon^{-2} ] for all $d \in \mathbb{N}$ and $0 < \varepsilon < \varepsilon_0$. The upper bound was shown by Heinrich, Novak, Wasilkowski and Woźniakowski (2001) and the lower bound was shown by Hinrichs (2004). A proof of this lower bound using elementary arguments was shown by Steinerberger (2023). These bounds show that the inverse of the star discrepancy depends linearly on the dimension, but the exact exponent of $\varepsilon^{-1}$ has remained open.
In this paper we prove a lower bound which shows that the exponent $2$ of $\varepsilon^{-1}$ in the upper bound cannot be improved. More precisely, we show that for every $0<α<1$ and fixed $0
27. 非凸优化问题的新的全局化牛顿型方法
New Globalized Newton-Type Methods for Nonconvex Optimization Problems
作者: Vo Thanh Phat, Tuyen Tran
类别: math.OC, math.NA
发布日期: 2026-07-26
arXiv ID: 2607.23433v1
链接: http://arxiv.org/abs/2607.23433v1
摘要: Newton’s method is one of the most effective second-order algorithms for smooth optimization because of its fast local convergence. However, existing globally convergent Newton-type methods typically require convexity or strong convexity of the objective function, while approaches for nonconvex optimization often rely on Hessian regularization at every iteration. In this paper, we propose a general line-search Newton framework for unconstrained optimization that avoids repeated Hessian regularization by exploiting the Newton direction only when it is well-defined and suitable. The proposed framework encompasses several existing hybrid gradient–Newton methods as special cases and naturally yields a new extragradient Newton method. We establish global convergence under mild assumptions, including the Polyak–Lojasiewicz–Kurdyka (PLK) condition, allowing both isolated and nonisolated accumulation points. We further prove local superlinear and quadratic convergence under appropriate regularity assumptions. Finally, we apply the proposed framework to strongly quasiconvex optimization and provide, to the best of our knowledge, the first Newton-type algorithm together with a comprehensive convergence analysis for this important class of nonconvex optimization problems. Numerical experiments demonstrate the effectiveness of the proposed methods.
28. 对称 V 循环在单元中心离散的多重网格公理下可能发散
The symmetric V-cycle can diverge under the multigrid axioms for cell-centred discretisations
作者: Ming Hei Wong
类别: math.NA
发布日期: 2026-07-25
arXiv ID: 2607.23391v1
链接: http://arxiv.org/abs/2607.23391v1
摘要: The axiomatic convergence theory for multigrid methods applied to cell-centred finite-difference and finite-volume discretisations rests on two hypotheses: an imbalanced Galerkin condition (G3), which states that $R_{\ell-1}A_\ell P_{\ell-1}=2A_{\ell-1}$ with $R_{\ell-1}=\frac{1}{2}P_{\ell-1}^T$, and a weak approximation property $(A2)_α$ of Bramble type. Under these hypotheses, together with Richardson smoothing, the symmetric W-cycle and the variable V-cycle are known to be uniformly convergent, while the uniform convergence of the standard symmetric V-cycle has remained open. We answer this in the negative by two constructions. First, for every smoothing count $m$ we exhibit hierarchies of every depth satisfying (G3), Richardson admissibility with $C_R=1$, and $(A2)_α$ for every $α\in(0,1]$ with the sharp level-independent constant $C_{A2}^2=4m$, whose symmetric $V(m,m)$-cycle error operator has spectral radius $θ_m(1+2θ_m)>1$ already on three levels, where $θ_m=(1-\frac{1}{4m})^{2m}$, and growing geometrically with the depth; the family shows that any smoothing-count threshold $m_0$ that could restore uniform V-cycle convergence must grow at least quadratically in $C_{A2}$. Second, we prove that the same failure occurs in a completely standard discretisation: the cell-centred finite-volume hierarchy for a one-dimensional diffusion equation with a mesh-aligned coefficient jump $1:κ$ and harmonic (Samarskii) interface averaging satisfies (G3) exactly and $(A2)_{1/2}$ with a level-independent constant $C_{A2}=O(κ)$, yet for every $κ\ge3$ its symmetric $V(1,1)$-cycle with any admissible Richardson parameter, including the optimal one, diverges geometrically in the number of levels. In both constructions the W-cycle remains uniformly contractive, so the hypotheses separate the two cycles. All claims are verified numerically.
29. Jackson常数是什么？
What is Jackson’s constant?
作者: Rikhav Shah, John Urschel, Nicholas West
类别: math.NA
发布日期: 2026-07-25
arXiv ID: 2607.23375v1
链接: http://arxiv.org/abs/2607.23375v1
摘要: We prove a refinement of Jackson’s theorem on the approximation of Lipschitz functions by trigonometric polynomials. Our result precisely characterizes the leading error term associated with Jackson’s construction. We do the same for a related construction commonly used in the kernel polynomial method for spectral density estimation, which is slightly better than Jackson’s construction in this respect.
30. 可扩展地计算度量量级
Scalably computing metric magnitude
作者: Steve Huntsman, Jewell Thomas, Cynthia Ukawu
类别: math.NA, math.MG
发布日期: 2026-07-25
arXiv ID: 2607.23354v1
链接: http://arxiv.org/abs/2607.23354v1
摘要: Applications of metric magnitude often rely on numerically exact results in order to exploit a connection with information theory. We examine various approaches for scaling the dense linear algebra involved and identify hierarchical low-rank solvers as a preferred approach, with a clear path to scales of $10^5$ points on a single powerful workstation, and larger scales using our containerized CUDA-enabled C++/MPI pipeline.
31. 利用线性最小二乘法学习具有收敛率保证的渐近行为
Learning Asymptotics with Convergence-Rate Guarantees using Linear Least Squares
作者: Christos N. Efrem
类别: stat.ML, cs.LG, math.CO, math.NA
发布日期: 2026-07-25
arXiv ID: 2607.23287v1
链接: http://arxiv.org/abs/2607.23287v1
摘要: We introduce a new research area that is called Asymptotics Learning Theory (ALT) and combines optimization with asymptotic analysis. In particular, ALT provides a unified approach for computing unknown constants/parameters in proven asymptotic expansions using optimization theory. In this paper, we focus on a general asymptotic form which includes a broad class of asymptotics. Furthermore, we study two powerful numerical methods, namely, sliding Linear Least Squares (sLLSQ) and sliding Tikhonov Linear Least Squares (sT-LLSQ). For these techniques we rigorously prove asymptotic estimates that lead to sufficient conditions for convergence (to the correct values of unknown parameters) and convergence-rate guarantees. Despite their strengths, both methods have also limitations, e.g., slow convergence—or even, counterintuitively, divergence—in some cases. Moreover, we present fundamental applications in analytic combinatorics, a beautiful field of mathematics that deals with asymptotic enumeration of discrete structures using complex analysis. The proposed techniques complement existing approaches, such as the ratio method and its variants. Numerical examples also verify the theoretical results. Finally, we discuss interesting research directions in ALT.
32. 多面体网格上线性弹性问题的最小二乘弱伽辽金框架
A Least Squares Weak Galerkin Framework for Linear Elasticity on Polytopal Meshes
作者: Chunmei Wang, Shangyou Zhang
类别: math.NA
发布日期: 2026-07-25
arXiv ID: 2607.23279v1
链接: http://arxiv.org/abs/2607.23279v1
摘要: This paper develops and analyzes a least-squares weak Galerkin (LS-WG) finite element method for linear elasticity. By employing weak differential operators, specifically the weak gradient, weak strain tensor, and weak divergence, defined on weak finite element spaces, the proposed framework facilitates the treatment of complex boundary conditions and internal interfaces while avoiding the restrictive discrete inf-sup condition. The resulting formulation is symmetric and positive definite and exhibits robust numerical performance in the nearly incompressible regime. In addition, the proposed method offers exceptional geometric flexibility, allowing implementation on general polytopal (polygonal and polyhedral) meshes. We establish the uniqueness of the numerical solution and derive optimal-order error estimates with respect to a tailored discrete energy norm. Extensive numerical experiments confirm the theoretical convergence rates and demonstrate the method’s stability, efficiency, and locking-free performance for nearly incompressible materials.
33. 半线性发展方程POD降阶模型的参数敏感性及其在G-方程中的应用
Parametric Sensitivity of POD Reduced-Order Models for Semilinear Evolution Equations with Applications to G-Equations
作者: Shengbo Ma, Luhao Xue, Zhiwen Zhang
类别: math.NA
发布日期: 2026-07-25
arXiv ID: 2607.23212v1
链接: http://arxiv.org/abs/2607.23212v1
摘要: Proper Orthogonal Decomposition (POD) provides low-dimensional surrogate models of evolution equations from solution snapshots. In parameterized problems, however, a basis computed at one parameter value need not remain accurate at another, while recomputing the basis for every parameter in a many-query study is costly. We develop a sensitivity analysis for POD reduced-order models of a class of parameterized semilinear evolution equations that includes the viscous G-equation and a viscous strain G-equation in their mean-free formulations. The analysis is based on a cross-space Lipschitz condition for the nonlinear operator, which accommodates the nonsmooth, gradient-dependent nonlinearities of these flame-propagation models, and on continuity moduli quantifying the parameter dependence of the bilinear form and of the nonlinearity. We prove that when a POD basis constructed at a reference parameter is applied to nearby query parameters, the resulting error variation is controlled by the corresponding parameter modulus, with constants independent of the POD dimension, the number of time steps, and the perturbation magnitude. Numerical experiments are consistent with the predicted modulus-dependent sensitivity behavior, and illustrate the practical robustness of basis reuse for nearby parameters.
34. 初等超越函数的不动点构造
A Fixed-Point Construction of the Elementary Transcendental Functions
作者: François Alouges, Giovanni Di Fratta, Alberto Fiorenza, Renato Fiorenza
类别: math.CA, cs.DC, cs.MS, math.FA, math.NA
发布日期: 2026-07-25
arXiv ID: 2607.23196v1
链接: http://arxiv.org/abs/2607.23196v1
摘要: We present a unified fixed-point construction of the elementary transcendental functions, encompassing the real exponential, the complex exponential (sine and cosine), and the natural logarithm. Each function is characterized as the unique solution of a duplication identity established through the Banach contraction principle. These foundational identities are $e(2x)=e^2(x)$ for the exponentials, and $\log(x^{2})=2\log x$ for the logarithm. Because a direct iteration of these identities is numerically unstable, owing to local expansiveness at the target, the central idea transfers the analysis to a residual function, on which the operator becomes a strict contraction with an explicit convergence rate. Beyond its theoretical economy, which dispenses with differential equations and power series, this characterization translates into efficient algorithms for the machine evaluation of elementary functions: the underlying framework yields floating-point kernels whose accuracy and iteration depth are governed by the theoretical contraction rate. We also present a computational study showing that, in a throughput-bound vectorized regime, these kernels are competitive with standard production libraries, and in favorable configurations exceed them, with the sine–cosine kernel faster at every tested iteration depth. These implementations operate without lookup tables or memory traffic, an architectural advantage for modern high-performance and energy-efficient computing.
35. 通过扰动启发的枢轴策略加速预处理雅可比方法
Accelerating preconditioned Jacobi methods via perturbation-inspired pivoting
作者: Nian Shao, Yuji Nakatsukasa
类别: math.NA
发布日期: 2026-07-25
arXiv ID: 2607.23187v1
链接: http://arxiv.org/abs/2607.23187v1
摘要: Perturbation theory for symmetric matrices shows that eigenvalues with small spectral gaps are more sensitive to off-diagonal perturbation, implying that different entries affect the eigenvalues unevenly. Building on this insight, we incorporate spectral gap information into the Jacobi method for symmetric eigenvalue problems and propose a new pivoting strategy, which is completely different from classical ones governed solely by the magnitude of the off-diagonal entries. When combined with a mixed-precision preconditioner that diagonalizes the matrix to low precision, numerical experiments demonstrate that the resulting strategy can significantly outperform the classical greedy approach when the original matrix has clustered eigenvalues.
36. 面向非线性平衡律的物理信息引导令牌变换器方法 I. 史瓦西-伯格斯流体流动
Physics-informed token transformer methodology for nonlinear balance laws. I. Schwarzschild–Burgers fluid flows
作者: Philippe G. LeFloch, Shuyang Xiang
类别: math.NA, physics.flu-dyn
发布日期: 2026-07-25
arXiv ID: 2607.23143v1
链接: http://arxiv.org/abs/2607.23143v1
摘要: We introduce a Physics-Informed Token Transformer (PITT) methodology for nonlinear hyperbolic balance laws in one space dimension, using piecewise steady-state profiles for the representation of approximate weak solutions. The method combines symbolic equation tokenization, a Fourier neural operator encoder, an explicit Rankine–Hugoniot law for shock motion, and a learned correction term. For clarity, we present it here for the relativistic Schwarzschild–Burgers equation, a scalar model for spherically symmetric fluid flows on a Schwarzschild background. For this model the steady-state invariant and the generalized Riemann solutions are explicit, and they can therefore be built into the neural evolution. In particular, the leading discontinuities are advanced by the analytical jump condition, while the learned part reconstructs smooth regions, rarefaction fans, geometric dependence, and finite-resolution effects. The method is designed to locate wave fronts accurately and to preserve the relevant steady states. We test our PITT method on moving shocks, stationary shocks, rarefaction waves, and compare it with a standard high-order finite-volume approximation. We also analyze the standard Burgers limit (when the Schwarzschild mass tends to zero). The Rankine–Hugoniot prior plays the dominant role in these tests, while equation tokenization gives a systematic additional gain. The method is relevant for problems involving geometric effects and/or complex shock-wave dynamics, and is used here to study the long-time dynamics of perturbations of steady-state solutions. In particular, we exhibit an asymptotic law of propagation for the shock location of perturbed steady-state flows.
37. Vlasov-Poisson 系统的 Active Flux 方法比较
A Comparison of Active Flux Methods for the Vlasov-Poisson System
作者: Lukas Hensel, Yanick Kiechle, Rainer Grauer, Gudrun Grünwald, Christiane Helzel, Katharina Kormann
类别: math.NA
发布日期: 2026-07-25
arXiv ID: 2607.23133v1
链接: http://arxiv.org/abs/2607.23133v1
摘要: Active Flux is a third-order accurate, fairly novel finite volume method for hyperbolic conservation laws that is becoming increasingly popular. It evolves additional nodal degrees of freedom (DOF) located on cell interfaces and shared by neighboring cells. The numerical fluxes are then computed from these DOFs. A crucial component of Active Flux methods is the evolution operator of the point values, which enables the natural use of semi-Lagrangian ideas and makes Active Flux an attractive candidate for a grid-based approach to the Vlasov equation. Here, we compare two recently proposed Active Flux methods for the 1D1V Vlasov-Poisson system: a split-step method and an unsplit method.
38. 谱Barron空间中双调和方程的混合残差方法
A mixed residual method for biharmonic equations in spectral Barron spaces
作者: Mengjia Bai, Kuo Gai, Shuai Lu
类别: math.NA
发布日期: 2026-07-25
arXiv ID: 2607.23080v1
链接: http://arxiv.org/abs/2607.23080v1
摘要: We propose a mixed residual method (MIM) for numerically solving the biharmonic equation with nonhomogeneous clamped boundary conditions. By establishing the well-posedness of the biharmonic equation in spectral Barron spaces, we derive an error bound for MIM that relates shallow neural network approximations to the exact solution and overcomes the curse of dimensionality. This error bound consists of two components: the first corresponds to the approximation error of the neural network, while the second represents the generalization error arising from randomly sampled training data. Several numerical experiments are presented to demonstrate the effectiveness of the proposed method.
39. 范围控制匹配与散乱数据拟插值的等周-组合界
Isoperimetric-Combinatorial Bounds for Range-Controlled Matchings and Quasi-Interpolation from Scattered Data
作者: Alexander Panchenko, Ben Hellwig, Oleh Rudenko
类别: math.NA, math.CA, math.CO, math.MG
发布日期: 2026-07-25
arXiv ID: 2607.22968v1
链接: http://arxiv.org/abs/2607.22968v1
摘要: We develop a mesoscopic framework for analyzing perturbations of finite point sets. Given a reference node set $Y$ with known cubature and approximation properties, we consider a disordered node set $Q$ that is observed only through its populations in cubes at scale $r>0$. By imposing Hall-type (HT) combinatorial constraints on these populations, we prove the existence of a perfect matching between $Y$ and $Q$ with $O(r)$ range. This allows integral approximation estimates on coarser cubes at scale $h>r$ to be transferred from $Y$ to $Q$ with explicit error control and anchors $Q$ to a periodic grid. We then use translation-invariant quasi-interpolation methods to obtain high-order estimates of order $h^s$ as in the quasi-uniform setting, but for a different class of geometries. The key restrictions are the HT conditions and the bound $r\le Ch$, where $C<1$ is scale independent.
40. 用于 Stokes 流动的质量守恒混合应力公式的最优块预条件子
Optimal block preconditioners for a mass-conserving mixed stress formulation of Stokes flow
作者: Kaibo Hu, Jongho Park, Jindong Wang
类别: math.NA
发布日期: 2026-07-24
arXiv ID: 2607.22932v1
链接: http://arxiv.org/abs/2607.22932v1
摘要: We present optimal block diagonal and triangular preconditioners for a mass-conserving mixed stress formulation of Stokes flow. The algebraic formulation leads to a double saddle point system with unknowns corresponding to discrete stress, velocity, vorticity, and pressure. MINRES equipped with a block diagonal preconditioner for an augmented Lagrangian formulation of this system is analyzed and shown to be optimal, in the sense that the convergence rate is independent of key parameters such as mesh size and kinematic viscosity. GMRES equipped with a block triangular preconditioner is also analyzed using a field-of-values approach. Finally, we present numerical results for both two- and three-dimensional model problems to validate the parameter robustness of the proposed preconditioners.
41. 博弈动力学的参数化方法
Parameterized Methods for Game Dynamics
作者: Yijie Jin, Haomin Zhou
类别: math.NA
发布日期: 2026-07-24
arXiv ID: 2607.22899v1
链接: http://arxiv.org/abs/2607.22899v1
摘要: We introduce a parameterized computational framework for the evolution of strategic behavior in continuous games. We consider the collective dynamics of players through a time-dependent probability density over the strategy space, representing the likelihood of each strategy being chosen at any given time. Instead of directly solving the high-dimensional Fokker-Planck equation that governs this evolution, we represent the probability density as the pushforward of a reference distribution with parameterized pushforward maps and consider the evolution of the parameterized equations. The motivation for this work comes from a limitation of the parameterized Wasserstein gradient flow (PWGF) framework when it is applied to game dynamics. PWGF provides a parameterized approach for evolution equations of probability density that can be formulated as Wasserstein gradient flows. However, not all game dynamics admit such a gradient flow formulation. We generalize the parameterized pushforward map framework to the non-gradient flows and apply it to the game dynamics with provable error bound in Wasserstein metric. Numerical experiments with several non-gradient systems in economics are provided to demonstrate the effectiveness of this new framework.