5. 紧性与 $\Gamma$-收敛的传递 (命题 2, 4). 命题 4 是经典的非局部紧性结果: 若 $\{u_n\}$ 的加权非局部 Dirichlet 能量 (按 $\delta_n$ 缩放) 一致有界, 则其在 $L^p$ 中预紧, 且极限函数属于 $W^{1,p}$, 差商弱收敛到 $\nabla u$. 该结果结合 $\Gamma$-收敛 (定理 9) 保证了极小元的收敛性.

6. 数值方法. 本文采用的高效有限元离散并非本文核心理论创新, 而是利用 [19] 的方法, 通过将二维积分分解为一维积分的乘积来降低计算复杂度. 数值实验验证了理论收敛速度, 并展示了尖锐界面模型的实际可行性. 实验细节表明, 选取光滑核 (如高斯核) 与多立方基函数, 可以在规则网格上高效实现.

4. 与之前工作的比较

本文与已有非局部界面模型以及与局部 $p$-Laplacian 界面的研究相比, 有以下几个维度的推进与区别.

1. 与扩散界面方法的比较. 现有的非局部界面模型几乎全部采用扩散界面技术 (如 [3,9,10]): 将尖锐界面 $\Gamma$ 替换为宽度 $\delta$ 的带状区域 $\Gamma_\delta$, 并将界面条件转化为 $\Gamma_\delta$ 上的体积约束, 或者引入依赖于距离场的平滑过渡函数. 这类方法有两个主要缺点: (i) 需要将定义在界面上的函数 $g$ 延拓到整个 $\Gamma_\delta$ 或 $H^1(\Omega)$, 这不仅增加了实施难度, 而且对解的精度和理论分析带来额外复杂性; (ii) 模型的结构不够模块化, 难以推广到其他非线性界面条件. 本文的尖锐界面模型完全避免了体积延拓, 所有界面项直接定义在 $\Gamma$ 上, 非局部扩散仅在子区域内部进行, 界面惩罚项作为能量项的组成部分, 模型结构清晰. 这种简洁性使得推广到带有膜条件的 $p$-Laplacian 问题成为可能, 而扩散界面方法通常需针对每种非线性重新设计界面周围的平滑函数.

2. 与非局部 Dirichlet 边界惩罚方法的联系与区别. 本文的界面惩罚项形式受到 [11,12] 中处理非局部 Dirichlet 边界条件的启发, 但有本质不同: [11,12] 处理的是区域外边界, 而且通常需要在整个区域上定义非局部交互; 本文处理的是内部界面, 惩罚项同时涉及两个子区域的平均值, 且与扩散项共同保证了紧性. 具体地, 非局部 Poincaré 不等式 (引理 6) 将界面惩罚与子区域 Dirichlet 能量有机结合, 这是本文独立发展的技巧, 此前未有类似结果.

3. 对 $p$-Laplacian 界面问题的扩展. 文献中对于 $p$-Laplacian 界面问题的研究主要集中于局部模型, 参见 [17,18], 且往往假设某种穿过界面的正则性. 本文首次将 $p$-Laplacian 界面问题的非局部逼近建立在变分框架下, 并严格证明了 $\Gamma$-收敛. 特别地, 模型中引入的膜条件 ($\gamma|u_1-u_2|^{p-2}(u_1-u_2)$) 在局部情形对应于界面通量的跳跃线性依赖于解跳跃, 而在非局部逼近中, 这一项被自然地替换为 $K_{\delta,1}*u$ 与 $K_{\delta,2}*u$ 之差的 $p$ 次幂积分. 当 $p=2$ 且取 $\gamma=2/\delta$ 时, 模型退化为线性情况, 这提供了一致的处理框架.

4. 理论结果的完整性. 与已有非局部界面问题的工作相比, 本文不仅给出了形式模型, 还系统证明了: (i) 对任意固定 $\delta>0$ 极小元的存在唯一性 (定理 7, 8); (ii) 当 $\delta\to0$ 时的 $\Gamma$-收敛及极小元收敛 (定理 9 及其非线性版本); (iii) 提供了基于紧性的严格框架. 早先工作如 [9] 虽然也研究了能量泛函的逼近, 但未给出类似定理 7 和 8 的存在唯一性, 且其模型依赖于界面延拓. [3,10] 则主要关注数值计算, 缺乏变分收敛的完整理论.

5. 假设与适用范围. 本文的假设较为标准: $\Omega$ 为有界 Lipschitz 区域, 子域 $\Omega_i$ 光滑以保证迹定理和紧性; 核函数 $R, K$ 为非负、紧支撑、$C^1$ 且非增, 满足矩条件. 这些假设在非局部分析中常见. 代价方面: 由于使用惩罚项逼近界面连续性, 必须要求惩罚系数随 $\delta\to0$ 适当变化 (线性情况为 $1/\delta$, $p$-Laplacian 情况为常数 $\gamma$), 若参数选取不当, 可能会导致界面穿透或数值不稳定性; 但文中证明了所选参数在 $\Gamma$-收敛下给出正确极限. 此外, 由于非局部积分仅在子区域内进行, 紧性成立需要子区域具有一定正则性 (如 Lipschitz), 这与全空间非局部模型相比限制稍多. 然而, 相较于需要在整个 $\Omega$ 上取积分的扩散界面模型, 本文方法的计算效率可能更高, 因其核函数的支集可限制在子区域内部.

6. 数值方面的比较. 文中数值求解采用张量积分解的高效有限元法 [19], 相比直接计算 $2d$ 重积分可显著降低计算量. 这类加速技术并非本文原创, 但本文将其成功应用于界面问题, 并展示了线性收敛速度, 表明该数值方法与理论结果的一致性. 从文中未显式与其他数值格式 (如 [3,10]) 进行精度与效率的定量对比, 但作者强调了尖锐界面模型在实现上的便利性, 因为无需处理界面厚度参数和对应的自适应网格.

总体而言, 本文的核心贡献在于为非线性界面问题提供了一种结构简洁、理论完备的非局部逼近框架, 并通过 $\Gamma$-收敛严格建立了局部模型的收敛性. 与已有方法相比, 该模型避免了界面延拓, 更易于推广, 同时保证了极小元收敛这一重要的理论保证; 其技术核心——融合了界面惩罚的非局部 Poincaré 不等式——是处理此类问题的关键新工具.

4. 基于不定算子的谱粗空间：$H_k$-GenEO方法

Spectral coarse spaces based on indefinite operators: the $H_k$-GenEO method

作者: Théophile Chaumont-Frelet, Victorita Dolean, Mark Fry, Ivan G. Graham, Matthias Langer

类别: math.NA

发布日期: 2026-05-29

arXiv ID: 2605.31552v1

摘要翻译: GenEO（“重叠上的广义特征值问题”）是一种构造粗空间的方法，用于离散偏微分方程迭代求解器的预处理。该方法组合少量局部偏微分方程特征问题的模态，以获得全局粗空间。然后将粗求解与全局偏微分方程的局部求解相结合，得到预处理器。对于局部特征问题为半正定的情形，已发展了相当完整的GenEO理论。该理论主要应用于正定的全局偏微分方程，但最近也已推广至对流-扩散-反应问题，这些问题可能既非自伴也不正定。然而，当全局问题高度不定时，基于半正定局部特征问题构造的粗空间在实践中缺乏稳健性。本文考虑高度不定的全局偏微分方程问题，其特征为具有大参数$k$（且允许系数高度可变），我们发展了一种新的谱粗空间，其构造基于求解 extit{全局问题的局部副本}生成的特征值问题。我们对局部区域的直径不加任何约束，从而允许局部特征值问题是不定的。新方法（我们称之为$H_k$-GenEO）当$k$增大时，明显比基于半正定特征问题的方法更加稳健。我们针对预处理的GMRES迭代方法的稳健性提供了充分条件，这些条件用局部特征问题的容差和局部偏微分方程求解的子区域大小来表述。实践中观察到，在局部特征问题容差更弱的条件下，该方法关于$k$也是稳健的。实验还表明，该方法能够承受偏微分方程系数的大幅变化。

Comment: 无

链接: http://arxiv.org/abs/2605.31552v1

1. 研究对象和背景

本文研究用于求解高度不定偏微分方程有限元离散系统的两水平区域分解预条件迭代法。考虑如下带 Dirichlet 边界条件的二阶线性椭圆型方程

其中 $k>0$ 为波数，$A$ 是几乎处处正定的对称矩阵系数，$n$ 是几乎处处正的 $L^\infty$ 函数，$\Omega\subseteq\mathbb{R}^d$ ($d=2,3$) 为 Lipschitz 多边形或多面体。当 $k$ 很大时，方程呈高度不定性(非正定)，解具有高频振荡特征，给迭代求解带来本质困难。该问题源于薄膜振动等物理模型，虽非经典散射问题，但作为具有可变系数的高度不定模型问题，可用于检验新型迭代求解器的性能。

有限元离散后得到线性系统 $B\mathbf{u}=\mathbf{f}$，其中刚度矩阵$B$对应双线性形式 $b(u,v)=\int_\Omega (A\nabla u\cdot\nabla v - k^2 n uv)\,\mathrm{d}x$。当系统规模很大时，需采用良好的预条件子进行迭代求解。经典的一水平重叠型区域分解 Schwarz 方法通过在各重叠子域 $\Omega_j^\ell$ 上局部求解来构造预条件子，但随着子域数增多一般不具可扩展性，因此通常引入全局粗校正以增强可扩展性以及对系数非均匀性或高波数的鲁棒性。

近年来，基于谱信息的粗空间(spectral coarse space)受到广泛关注。最早的 GenEO (Generalised Eigenvalue problems on the Overlap) 方法由[34]提出，针对自伴正定问题，在每个粗子域上求解如下正半定广义特征值问题

并选取部分模态构成全局粗空间。该方法结合局部求解与粗求解，能在共轭梯度法下获得对子域数和系数跳跃的鲁棒性。但将其直接用于 $k>0$ 的不定问题(1.1)时，由于局部特征问题仅包含正半定部分，得到的粗空间缺乏捕捉振荡特征的能力，导致数值表现不鲁棒。

为处理不定问题，[5,7] 提出了 H-GenEO 粗空间，其局部特征问题修改为

在左端引入了不定算子，数值上显示出高频鲁棒性，但未提供理论分析。本文进一步提出 $H_k$-GenEO 粗空间，基于如下局部不定广义特征值问题

其左端仍为不定算子，右端则添加了一个正定项，使右端形式更接近分析中采用的加权范数。本文旨在为此方法建立严格的理论基础：在适当的网格尺寸、局部子域直径和特征值容忍度条件下，证明预处理 GMRES 的鲁棒收敛性，并将结果与现有相关工作进行对比。

2. 主要定理或主要结果

本文主要结果为定理 3.1 及其推论 3.2，给出了采用 $H_k$-GenEO 粗空间的两水平加性 Schwarz 预条件 GMRES 方法鲁棒收敛的充分条件。记局部子域的最大直径为 $H_\ell$，粗子域的最大直径为 $H_c$。设有限元网格拟一致且网格尺寸满足 $h\sim k^{-1}$，则当

时，预处理 GMRES 具有 $k$-鲁棒性。其中 $C_{\rm stab}>0$ 为问题 (2.1) 的稳定性常数，由假设 2.3 给出，即对任意 $f\in L^2(\Omega)$，弱解 $u$ 满足 $\|u\|_{1,k}\le C_{\rm stab}\|f\|$；$\tau = \min_{i=1}^N \lambda^i_{m_i+1}$ 为所有粗子域上选取特征模态后剩余的第一个正特征值的最小值，称为特征值容忍度。

具体地，定理 3.1 证明了：存在仅依赖于重叠次数的常数 $C$，使得若

且局部子域直径满足 $H_\ell \le h/(C_{\rm inv} k)$ 以保证局部问题的可解性，则预处理矩阵的场值距离原点有正下界，从而 GMRES 方法的残差在每一步以因子 $(1-1/\kappa)^{1/2}$ 减少，其中 $\kappa$ 依赖于 $a_{\max}, n_{\min}, C_{\rm stab}, \tau$ 等参数但独立于 $k$。

推论 3.2 进一步将该结果推广到一般网格，此时条件变为 $H_\ell \lesssim h/(C_{\rm inv} k)$ 且 $\tau \gtrsim (1+C_{\rm stab})^2 k^2$，常数依赖于逆不等式常数 $C_{\rm inv}$。值得注意的是，该理论对粗子域直径 $H_c$ 未施加显式上界，允许粗子域足够大以使得局部特征问题(1.4)不定，从而产生振荡模态以捕捉高频信息。数值实验甚至表明，仅需选择足够大但固定的常数 $\tau$ 即可实现 $k$-鲁棒收敛，且所需粗空间维数随 $k$ 增长非常温和。此外，与之前针对不定问题（基于(1.2)）的结果相比，本文将特征值容忍度条件从 $\tau \gtrsim (1+C_{\rm stab})^2 k^8$ 显著改进到 $k^2$，反映了新特征问题设计的优越性。

3. 研究方法、关键技术和核心工具

本文的核心技术障碍在于局部特征问题的不定性以及相应谱理论的缺失。作者的主要创新点在于：

1. 构造了新型的局部广义特征值问题 (1.4)，其右端形式选取为 $-\xi_i\nabla\cdot(A\nabla(\xi_i p)) + k^2\xi_i n(\xi_i p)$，这一形式在范数 $\|\cdot\|_{1,k}$ 下具有正定性，为后续分析中建立正交关系与投影估计奠定了基础。
2. 发展了抽象的、针对不定广义特征值问题的谱理论（第 2.2 节）。假设 $\tilde{V}$ 为有限维空间，$c$ 为对称正半定双线性形式，$\ker c$ 维数为 $s$，$b$ 为对称（可能不定）双线性形式。定理 2.9 证明了：若 $s\ge 1$ 且 $b$ 在 $\ker c$ 上正定，则存在一组基 $\{p_j\}$ 使得前 $\tilde{n}-s$ 个为特征向量，对应的有限特征值可排序为 $-\infty<\lambda_1\le\lambda_2\le\dots\le\lambda_{\tilde{n}-s}<\infty$，且满足广义正交性：$b(p_j,p_{j'})=\lambda_j\delta_{jj'}$，$c(p_j,p_{j'})=\delta_{jj'}$。该定理为在有限元子空间上分析 (1.4) 提供了框架，并允许 $\ker c$ 非零，对应无限特征值。
3. 在证明主定理时，采用“Elman 理论”来分析 GMRES 收敛性，即通过建立预处理矩阵范数的上界以及其场值到原点距离的下界来得到残差缩减因子。具体地，令 $M^{-1}$ 表示两水平加性 Schwarz 预条件子，则需估计 $\|M^{-1}B\|$ 和 $\inf_{v\neq 0} \frac{(M^{-1}B v, B v)}{\|B v\|^2}$。后者涉及 $M^{-1}$ 的对称正定部分，通过精细的局部化估计与分部积分技巧，将下界与 $\tau$ 及 $C_{\rm stab}$ 联系起来。
4. 证明过程中大量使用了加权范数 $\|\cdot\|_{1,k}$ 及其伴随的 $k$-加权内积，并充分利用了 Friedrichs 不等式(引理 2.7)和有限元逆估计(2.5)。此外，为了处理不定性，定理 2.9 中要求 $b$ 在 $\ker c$ 上正定，这通过假设问题(1.1)的适定性（$C_{\rm stab}<\infty$）和局部子域足够小来满足。
5. 在估计 $\tau$ 的条件时，作者引入了技术性常数 $C_0$，它来源于单形分解中单位分解函数与有限元函数的乘积估计，将局部特征值与全局稳定性常数联系起来。

整体证明框架为：先分析离散不定特征值问题的谱结构，证明正交投影的存在性及投影误差界；然后利用这些投影构造粗空间并分析两水平预条件子的场值；最后结合 Elman 理论导出收敛条件。

4. 与之前工作的比较

本文与已有工作在多个维度上有显著区别和推进：

1. 与经典 GenEO 方法（针对正定问题）的比较：经典的 GenEO [34] 使用正半定局部特征问题 (1.2)，对正定问题（$k=0$）可同时获得关于子域数和系数变异的鲁棒性。但当 $k$ 很大时，由于 (1.2) 不包含任何振荡信息，其粗空间无法有效处理不定性，数值上表现为不鲁棒。本文提出的 $H_k$-GenEO 由 (1.4) 构造粗空间，将不定算子置于左端，从而允许粗空间包含振荡模态。不仅数值上对 $k$ 鲁棒，更提供了完整的理论保证。

2. 与 H-GenEO [5,7] 的比较：H-GenEO 的局部特征问题为 (1.3)，其在数值上已表现出朴素的 $k$-鲁棒性，但缺乏理论分析。本文改用 (1.4) 的形式，关键在于右端添加的 $k^2\xi_i n(\xi_i p)$ 项：它使右端形式在分析范数 $\|\cdot\|_{1,k}$ 下正定，从而易于建立谱正交性和投影误差估计，而 (1.3) 的右端仅为 $-\xi_i\nabla\cdot(A\nabla(\xi_i p))$，没有相应的正定性，难以进行严格理论分析。因此本文在理论层面上实现了突破。

3. 与针对对流-扩散-反应方程的 GenEO 变体 [6,10] 的比较：文献 [6] 研究了将 $\Delta$-GenEO (基于 (1.2)) 应用于非自伴不定问题，其分析中得到的特征值容忍度条件为 $\tau \gtrsim (1+C_{\rm stab})^2 k^8$，非常苛刻。本文通过改用不定算子并调整右端形式，将条件大幅放松到 $k^2$，揭示了粗空间设计对理论条件的关键影响。后来 [10] 将估计改进至 $k^4$ 左右，但仍高于本文的 $k^2$。

4. 与其它 Helmholtz 问题粗空间方法的比较：

• [4,9] 等使用 Dirichlet-to-Neumann 映射构建粗空间，适用于无限域散射问题，但通常需预知传播方向等信息，不如谱方法通用。
• [27,28] 对 Helmholtz 问题提出了基于局部正半定 GEVPs 的粗空间，为保证局部问题正定，强制要求子域直径 $O(k^{-1})$，即局部 Helmholtz 问题接近强制的；理论证得预条件子为参数无关的压缩算子。本文不限制粗子域直径，允许其足够大以造成局部不定性，因而粗空间维数可更灵活。
• [23] 分析了常系数 Helmholtz 方程，在有吸收时子域可稍大，但无吸收时仍要求尺寸 $o(k^{-1})$ 以保证局部正定性。本文理论对粗子域尺寸无上界，且数值实验显示粗子域越大效果越好。
• [19,15] 等采用分段多项式粗空间，通过 hp-细化达到对 k 的鲁棒性，但不利用谱信息；其方法在系数剧烈变化时的表现未被分析。本文的谱粗空间自然能处理系数非均匀性，实验结果也表明其对高系数变化具有弹性。

5. 理论假设与代价：本文的充分条件要求局部子域直径满足 $H_\ell \lesssim k^{-1}$（或更一般的 $h/(C_{\rm inv} k)$），以保证局部 Neumann 型问题的可解性，这与多数 Helmholtz 子结构方法一致。粗子域可任意大，只需求解相应域上的不定广义特征值问题，其成本随粗子域增大而增加，但实践中适当选择粗子域尺寸可使特征值问题规模可控。特征值容忍度需满足 $\tau \gtrsim (1+C_{\rm stab})^2 k^2$，尽管数值上常数 $\tau$ 似乎已足够，但该理论估计仍可能是悲观的。

了解领域论文（摘要翻译）

1. 面向约束偏微分方程结构保持代理建模的胞腔层神经算子

Cellular Sheaf Neural Operators for Structure-Preserving Surrogate Modeling of Constrained PDEs

作者: Lennon J. Shikhman, Shane Gilbertie

类别: cs.LG, cs.CE, math.NA, physics.comp-ph, physics.plasm-ph

发布日期: 2026-05-31

arXiv ID: 2606.00937v1

链接: http://arxiv.org/abs/2606.00937v1

摘要翻译

中文标题: 面向约束偏微分方程结构保持代理建模的胞腔层神经算子

摘要翻译: 神经算子为偏微分方程模拟提供快速代理模型，但标准架构常将几何与离散化视为场数据的次要因素。物理状态通常被表示为网格-通道堆叠，即使不同量自然属于顶点、边、面、体元、边界或界面且必须满足相容性约束。我们提出胞腔层神经算子，一种面向结构保持神经偏微分方程代理的离散化感知框架。该方法在定向胞腔复形上表示偏微分方程状态，通过学习的限制映射耦合局部特征空间，并利用关联/Hodge信息传递来遵循计算几何。学习的更新头通过上边缘或通量映射进行，使得选定约束可以来源于胞腔复形结构，而不仅仅来自损失惩罚。对于磁流体动力学，这产生了由边电动势驱动的基于面的磁通量更新，以及由学习的面通量和体元源驱动的有限体积风格的流体更新。在湍流磁流体动力学和聚变平衡代理任务上，该方法改善了结构敏感的诊断指标，包括推演行为、散度控制、谱误差和平衡回归精度。这些结果表明，胞腔层结构是约束多物理系统中神经偏微分方程代理的一种有用的归纳偏置。

2. Alfeld 剖分上有限元 Stokes 复形的上同调

Cohomology of Finite Element Stokes Complexes on Alfeld Splits

作者: Pablo D. Brubeck, Yizhou Liang, Charles Parker

类别: math.NA

发布日期: 2026-05-29

arXiv ID: 2605.31348v1

链接: http://arxiv.org/abs/2605.31348v1

摘要翻译

中文标题: Alfeld 剖分上有限元 Stokes 复形的上同调

摘要翻译: 我们证明了，由 Fu、Guzmán 与 Neilan (2020, Math. Comp., 89, 1059–1091) 在 Alfeld 剖分网格上由分片多项式空间构成的有限元 Stokes 复形的上同调同构于连续 Stokes 与 de Rham 复形的上同调。我们还构造了新颖的“最小”协调有限元复形，其中 $H^1$ 协调空间为 Guzmán 与 Neilan (2018, SIAM J. Numer. Anal., 56, 2826–2844) 的最低阶空间，而 $L^2$ 协调空间为分片常数空间。这些最小复形也具有同构于连续 Stokes 与 de Rham 复形的上同调。我们进一步为这些最小复形构造了局部、有界的上链投影。所有结果对具有非平凡拓扑的强 Lipschitz 域以及在混合边界条件下均成立。

3. 高对比度谐振器非周期系统中亚波长导模的预解收敛与块逼近

Resolvent Convergence and Patch Approximation for Subwavelength Guided Modes in Non-Periodic Systems of High-Contrast Resonators

作者: Habib Ammari, Borui Miao, Jiayu Qiu

类别: math.SP, math.NA

发布日期: 2026-05-29

arXiv ID: 2605.30951v1

链接: http://arxiv.org/abs/2605.30951v1

摘要翻译

中文标题: 高对比度谐振器非周期系统中亚波长导模的预解收敛与块逼近

摘要翻译: 本文提出、分析并验证了一种快速算法，用于计算高对比度谐振器晶体中弯曲界面和非周期缺陷内的导模，其中 Floquet–Bloch 理论不适用。我们首先建立了控制连续算子到离散电容算子的预解收敛性。这一结果严格证明了将连续谱问题约化为离散特征值问题的合理性。然后，我们开发了一种离散算子的截断方案，称为块逼近，并针对块逼近推导了严格的误差估计。最后，我们通过各种算例验证了该方案的精度和效率。我们的框架为模拟高对比度谐振器非周期系统中的导模提供了一种通用、计算高效且理论严格的途径。

4. 保持多不变量的离散梯度方法

A Multi-Invariant Preserving Discrete Gradient Methods

作者: Haodong Pu, Maohua Ran

类别: math.NA

发布日期: 2026-05-29

arXiv ID: 2605.30827v1

链接: http://arxiv.org/abs/2605.30827v1

摘要翻译

中文标题: 保持多不变量的离散梯度方法

摘要翻译: 本工作基于预估-校正策略，为保守系统引入了一种新颖的保结构方法。该框架将离散梯度校正应用于由显式单步或多步格式生成的预测值，在保持原预估器精度阶的同时守恒非线性不变量。该方法自然扩展到需要同时守恒多个不变量的情形。在温和条件下，建立了守恒性质、可解性、数值精度和稳定性。对 Lotka-Volterra 系统、正弦-Gordon 方程、刚体动力学和开普勒问题的长期数值模拟表明，与现有的投影和松弛方法相比，该方法具有更好的鲁棒性和守恒性质。

5. 用于非等温相场模型的熵稳定与能量守恒的全离散有限元方法

Entropy-stable and energy-conservative fully-discrete finite element method for non-isothermal phase-field models

作者: Aaron Brunk, Dennis Höhn

类别: math.NA

发布日期: 2026-05-28

arXiv ID: 2605.30559v1

链接: http://arxiv.org/abs/2605.30559v1

摘要翻译

中文标题: 用于非等温相场模型的熵稳定与能量守恒的全离散有限元方法

摘要翻译: 本工作提出了一种适用于耦合不可压 Navier-Stokes 方程的非等温相场系统的协调有限元格式。该数值格式通过使用熵作为主变量代替温度的变量变换，精确保持了熵产生和总能量守恒。进行了空间收敛性测试，并提供了代表性算例以展示格式的有效性。

不相关论文（基本信息）

1. 有限差分偏微分方程求解器中用于自适应网格细化的物理信息残差

Physics-Informed Residuals for Adaptive Mesh Refinement in Finite-Difference PDE Solvers

作者: Henry Kasumba, Ronald Katende

类别: math.NA, cs.CE, cs.LG

发布日期: 2026-06-01

arXiv ID: 2606.02475v1

链接: http://arxiv.org/abs/2606.02475v1

摘要: Classical finite-difference solvers remain reliable tools for partial differential equations, but their efficiency depends on where mesh resolution is placed. Uniform refinement can waste degrees of freedom when solution difficulty is localised near sharp gradients, fronts, oscillations, or constraint-sensitive regions. This paper studies a hybrid strategy in which a physics-informed neural network (PINN) is used not as the final solver, but as an off-grid residual probe for adaptive mesh refinement. The PINN residual is sampled over the domain, converted into cellwise indicators, and used to guide refinement before the final approximation is computed by a finite-difference solver. The method is evaluated on three benchmarks. The main full-solver validation uses the one-dimensional viscous Burgers equation with a nonuniform finite-difference solve on the adapted meshes. PINN-threshold refinement attains final relative $L^2$ error $0.021067$ with $60$ degrees of freedom, compared with $0.022617$ for uniform refinement with $192$ degrees of freedom. At matched mesh size, PINN-threshold reduces the error by about $67.5\%$. PINN-D"orfler refinement gives similar performance, with error $0.021264$ using $58$ degrees of freedom. A gradient indicator remains slightly more accurate, so the result supports usefulness rather than universal superiority. Manufactured 2D and 3D proxy tests, based on a nonlinear Schr"odinger equation and an incompressible Navier–Stokes system, show that PINN residuals can organise structured refinement and improve over random refinement, although they do not consistently outperform gradient or uniform baselines. The results support PINN-guided AMR as a residual-indicator strategy for transferring physics-informed diagnostic information into finite-difference mesh adaptation while preserving the classical solver as the final approximation engine.

2. 上下文算子网络的谱审查

Spectral Audit of In-Context Operator Networks

作者: Zhiwei Gao, Liu Yang, George Em Karniadakis

类别: math.NA, cs.LG

发布日期: 2026-06-01

arXiv ID: 2606.02427v1

链接: http://arxiv.org/abs/2606.02427v1

摘要: Existing evaluations of neural operators and in-context operator learning rely primarily on prediction error, but accurate output prediction does not guarantee the correct local dynamical structure. A model may match solutions while exhibiting incorrect sensitivities, distorted frequency response, spurious mode coupling, or unstable tangent behavior. We introduce a Jacobian-based spectral audit for in-context operator learning. For a fixed prompt, we differentiate the network output with respect to the query function and view the resulting Jacobian as a learned tangent operator. Projecting it onto Fourier modes, we obtain a local spectral characterization of the inferred operator, including frequency-dependent gains, phase structure, and cross-mode coupling. The audit complements standard prediction metrics by testing whether the model reproduces local mechanisms of the underlying PDE operator rather than only outputs. Across benchmarks, the audit reveals distinct operator-level phenomena, including phase transport, viscosity-dependent damping, nonlinear mode coupling, and reaction–diffusion stability structure. It also detects failures partially hidden by prediction-error metrics, including high-frequency degradation, incorrect phase recovery, and prompt–operator inconsistencies. Corrupted or internally inconsistent prompts lead to degraded tangent-operator structure even when pointwise predictions remain partially accurate. Our results suggest that prediction accuracy and local operator fidelity are distinct properties of learned neural operators. Our framework also provides a diagnostic for stability, sensitivity, and operator consistency.

3. 对称张量分布混合方法求解四阶椭圆奇异摄动问题

Symmetric-Tensor Distributional Mixed Method for Fourth-Order Elliptic Singular Perturbation Problem

作者: Xuehai Huang, Xinyue Zhao

类别: math.NA

发布日期: 2026-06-01

arXiv ID: 2606.02188v1

链接: http://arxiv.org/abs/2606.02188v1

摘要: A symmetric-tensor distributional mixed method for a fourth-order elliptic singular perturbation problem is developed in this paper. The moment variable is approximated by normal-normal continuous symmetric tensor elements, while the scalar variable is represented by an $H^1$-nonconforming virtual element space coupled with a polynomial multiplier on interior subsimplices of codimension two. Optimal parameter-uniform error estimates are derived, independent of the presence of boundary layers. A hybridized form of the method is also equivalent to stabilization-free weak Galerkin and $H^2$-nonconforming virtual element methods. In two dimensions, a close connection of the distributional mixed method to the classical Hellan-Herrmann-Johnson (HHJ) method is established, by naturally identifying the scalar virtual element-multiplier pair with the Lagrange finite element space. Thus the proposed method extends the two-dimensional HHJ method to arbitrary spatial dimensions. Three-dimensional numerical experiments support the theoretical convergence and robustness estimates.

4. 一致六面体网格上的迹保持hp插值与多项式提升

Trace-Preserving hp Interpolation and Polynomial Liftings on Conforming Hexahedral Meshes

作者: Situan Li, Weiying Zheng

类别: math.NA

发布日期: 2026-06-01

arXiv ID: 2606.02125v1

链接: http://arxiv.org/abs/2606.02125v1

摘要: Trace-compatible polynomial extensions are a recurring local ingredient in high-order finite element analysis on conforming hexahedral meshes. They are needed whenever prescribed edge and face traces must be preserved while a polynomial is extended into a neighboring cell or boundary patch. The main contribution of this paper is the construction of p-robust polynomial liftings on nonsingular conforming hexahedral boundary patches, with stable control of both the H^1 norm and the H^1-seminorm estimates needed for energy arguments. These liftings imply H^1-seminorm stable discrete harmonic extensions of polynomial Dirichlet traces. They also serve as boundary corrections for the conforming hp Clement interpolant, yielding trace-preserving interpolation operators for functions with only H^1 regularity. Under the uniform boundary-degree condition the constants are p-uniform; in the non-uniform case the stated logarithmic loss appears. We also treat meshes that may contain conforming singular boundary patches, where the loss remains polylogarithmic in the maximal local degree. Trace-preserving interpolation on reference cells and vertex-supported decompositions are developed as local tools for these patch and mesh-level constructions.

5. 对于四面体网格上离散调和延拓的 $p$-鲁棒迹提升与保边界 $hp$ 插值

$p$-Robust Trace Liftings for Discrete Harmonic Extensions and Boundary-Preserving $hp$ Interpolation on Tetrahedral Meshes

作者: Situan Li, Weiying Zheng

类别: math.NA

发布日期: 2026-06-01

arXiv ID: 2606.02086v1

链接: http://arxiv.org/abs/2606.02086v1

摘要: We construct p-robust polynomial trace liftings on three-dimensional tetrahedral meshes. The prescribed trace is a continuous piecewise polynomial function on a boundary face patch; the tetrahedra touching this patch have one common degree, while the interior degrees may be arbitrary. The lifting is degree-preserving, supported in the corresponding boundary layer, and satisfies both an H^1 estimate and a scaled boundary-layer L^2 estimate with constants independent of the mesh size and the polynomial degree. The construction is local and combines tetrahedral polynomial liftings, face-gluing arguments, and nonsingular vertex patches. As consequences of the construction, we obtain p-robust discrete harmonic extensions, including an H^1-seminorm-stable extension for the pure diffusion energy, and a boundary-preserving hp interpolation operator that keeps piecewise polynomial Dirichlet data exactly while retaining standard local approximation estimates.

6. 双曲面切片上线性波动方程的三维分部求和格式

3d Summation-by-Parts scheme for Linear Wave Equations on Hyperboloidal Slices

作者: Anuraag Reddy, Shalabh Gautam, Prayush Kumar

类别: gr-qc, math-ph, math.AP, math.NA

发布日期: 2026-06-01

arXiv ID: 2606.02051v1

链接: http://arxiv.org/abs/2606.02051v1

摘要: We derive a fully 3-dimensional Summation-By-Parts scheme for a class of linear wave equations on hyperboloidal slices that meet future null infinity on a Minkowski background. The scheme is derived in spherical polar coordinates, with a major strength being that it is provably stable and allows having grid points at the origin and on the $z$-axis, despite coordinate singularities, and at infinity, by introducing compactification followed by rescaling. Reducing it to the standard Cauchy problem, or on finite spacelike slices with an outer boundary, will follow a similar procedure. Interesting relations are obtained between the rescaling and compactification factors that simplify the equations, and the conditions on constraint addition terms are discovered to maintain symmetric hyperbolicity. Numerical implementation is achieved using finite-difference methods at second-order accuracy, which can be generalized to higher-order or spectral accuracies as well. Dissipation operators are given a more abstract treatment, which makes it possible to define them everywhere in the domain, including at the boundary points, in curvilinear coordinates, such that they satisfy the dissipative property (DP) in our energy norms. These generalizations reduce to the well-known Kreiss-Oliger dissipation operators whenever defined on a Cartesian grid in the bulk and satisfy the DP in the standard $L^2$-norms. We also propose new norm convergence tests that produce more accurate outputs. Promising results are obtained, giving hope for application to fully nonlinear systems, like the Einstein Field Equations, and extracting the resulting gravitational waves free of systematic errors or gauge ambiguities.

7. 用于瞬时频率和调频率估计的高阶同步压缩小波-线调频小波变换

High-order synchrosqueezed wavelet-chirplet transform for instantaneous frequency and chirprate estimation

作者: Shuixin Li, Jiecheng Chen, Qingtang Jiang, Gang Yu

类别: eess.SP, math.NA

发布日期: 2026-06-01

arXiv ID: 2606.01965v1

链接: http://arxiv.org/abs/2606.01965v1

摘要: The separation of multicomponent signals with crossing instantaneous frequency (IF) curves remains a fundamental challenge in time-frequency analysis. Although the synchrosqueezed wavelet-chirplet transform (SWCT) enhances time-frequency readability by introducing a chirprate variable, its effectiveness is constrained by the underlying assumption of local linear chirp. Consequently, this method does not perform well when analyzing signals characterized by strong frequency modulation. This paper extends the SWCT framework by relaxing the linear chirp assumption. We model signal components as having polynomial phase behavior over short intervals and derive compact expressions for high-order IF and chirprate reassignment operators. The proposed high-order synchrosqueezed wavelet-chirplet transform (HSWCT) enables accurate estimation of both IF and chirprate, and supports robust mode retrieval even with intersecting IF curves. Another key contribution is a rigorous mathematical analysis of the approximation errors of arbitrary-order reassignment operators for IF and chirprate estimation. When the chirprate vanishes, HSWCT simplifies to the traditional high-order synchrosqueezed wavelet transform. To our best knowledge, no theoretical analysis exists in the literature on the approximation of arbitrary-order SST IF reassignment operators to the IF. As a by-product of this work, our established theorem provides such an analysis, thereby filling a gap in the theoretical framework of high-order SSTs.

8. GLT 序列与 GLH 序列的块雅可比/高斯-赛德尔预处理

Block Jacobi/Gauss-Seidel preconditioning for GLT sequences, and GLH sequences

作者: Carlo Garoni, Stefano Serra-Capizzano

类别: math.NA

发布日期: 2026-06-01

arXiv ID: 2606.01888v1

链接: http://arxiv.org/abs/2606.01888v1

摘要: The theory of generalized locally Toeplitz (GLT) sequences is an apparatus for computing the spectral and singular value distribution of sequences of matrices that possess a (possibly hidden) Toeplitz-like structure. These sequences, which are known as GLT sequences, arise in several applications, including the discretization of differential equations. Associated with any GLT sequence is a special function called symbol. In this paper, we prove that, if $\{A_n\}_n$ is a GLT sequence with symbol $κ$ and $P_n$ is any block Jacobi or block Gauss-Seidel preconditioner for $A_n$ with a fixed number of blocks independent of $n$, then $\{P_n\}_n$ is a GLT sequence with symbol $κ$, just like $\{A_n\}_n$. This result allows us to predict a remarkable efficiency of block Jacobi/Gauss-Seidel preconditioning for GLT sequences, which is in fact illustrated through numerical experiments. It also allows us to extend the Fasino-Tilli theorem on the zero distribution of Hankel matrix sequences generated by $L^1$ functions to a larger class of matrix sequences called generalized locally Hankel (GLH) sequences.

9. 用于选主元Cholesky分解的变分自由能选主元策略

Variational Free Energy Pivot Selection for Pivoted Cholesky

作者: Louise Schaub, Peter Zaspel

类别: math.NA

发布日期: 2026-06-01

arXiv ID: 2606.01821v1

链接: http://arxiv.org/abs/2606.01821v1

摘要: Pivoted Cholesky factorizations construct low-rank approximations of symmetric positive definite matrices by sequentially selecting pivots from the residual diagonal. Classical greedy and randomized rules, such as randomly pivoted Cholesky, target the algebraic trace-norm error of the residual. In many applications, however, the matrix enters a nonlinear matrix functional whose value, not the trace-norm error, determines solution quality, and residual-based rules ignore this structure. We derive a pivot rule that maximizes the exact one-step change of such a functional under Cholesky-consistent rank-1 updates, for a functional combining log-determinant, quadratic, and trace terms. This functional arises as the variational free energy in Gaussian process regression, where the matrix is a kernel matrix. The resulting per-step gain admits a closed-form additive decomposition into complexity, data-fit, and trace contributions, and is used directly as a pivot-selection criterion. We refer to the resulting method as $Δ$-VFE pivoted Cholesky. At each iteration, the criterion is evaluated on a batch of $s$ candidate pivots sampled proportionally to the residual diagonal via incremental Woodbury updates, at a total cost of $\mathcal{O}(snr^2)$ for an $n\times n$ matrix and target rank $r$. This matches the asymptotic complexity of randomly pivoted Cholesky up to the batch factor $s$. Cholesky-consistent rank-1 updates yield monotonically non-decreasing functional values, and the proposed rule maximizes the per-step gain among them. Numerical experiments show improved objective values and predictive accuracy at low to moderate ranks compared to classical and randomly pivoted Cholesky, while preserving trace-norm approximation quality.

10. 面向量子线性系统求解器的Pauli结构预处理

Pauli-structured preconditioning for quantum linear system solvers

作者: Hantao Nie, Zhijian Lai, Dong An

类别: quant-ph, math-ph, math.NA

发布日期: 2026-06-01

arXiv ID: 2606.01733v1

链接: http://arxiv.org/abs/2606.01733v1

摘要: Preconditioning is a fundamental technique for accelerating classical linear system solvers, and understanding when its benefits persist in quantum linear system (QLS) solvers is important for assessing the practical resource requirements of quantum linear algebra. In QLS algorithms, however, the potential advantage of preconditioning may be offset by the normalization overhead incurred by composing separate block-encodings of the system matrix and the preconditioner, as observed in recent work. This limitation leaves open whether additional algebraic structure can make preconditioning effective in quantum access models. Motivated by this question, we show that Pauli-structured representations of both the system matrix and the preconditioner allow the preconditioned operator to be accessed through regrouped Pauli expansions. In this setting, algebraic regrouping of Pauli products can reduce the Pauli coefficient weight of the preconditioned operator, thereby altering the normalization parameters relevant to quantum algorithms. We derive explicit size and coefficient-weight bounds for the regrouped Pauli representations, and we trace their consequences for both direct block-encoding constructions and randomized Pauli linear system solvers. These results identify when Pauli-structured preconditioning can reduce the effective complexity parameters of QLS algorithms, rather than merely improving the classical condition number. Numerical experiments on a finite-dimensional synthetic benchmark show reductions in norm-aware direct block-encoding diagnostics and in the randomized QLS per-sample depth proxy.

11. 带点奇性的薛定谔方程傅里叶谱方法的渐近恢复

Asymptotic Recovery in Fourier Spectral Methods for the Schrödinger Equation with Point Singularities

作者: Yanjie Li, Sihong Shao

类别: math.NA, math.SP, quant-ph

发布日期: 2026-06-01

arXiv ID: 2606.01718v1

链接: http://arxiv.org/abs/2606.01718v1

摘要: This paper studies the Fourier spectral method (FSM) for the Schrödinger equation with singular potentials $V \in H^{s}$, where $s > \max\{d/2-2,-1\}$ and $d$ denotes the spatial dimension. This setting includes a broad class of singular potentials, such as the 3D Coulomb potential and the 1D Dirac-delta potential. First, we combine the Feshbach-Schur map with a refined perturbation argument to derive sharp convergence orders for FSM, yielding order $2s+2$ for eigenvalues and order $s+1$ for eigenfunctions in the $H^1$ norm. More importantly, the $H^1$ error with respect to the projected eigenfunction converges with a higher order $s+1+b$, where $b=\min\{s+2-d/2-\varepsilon,\; s+1,\; 2\}>0$ for arbitrarily small $\varepsilon>0$, revealing a super-convergence phenomenon. Second, in the presence of potentials with isolated point singularities, we develop an asymptotic-recovery (AR) technique to post-process the FSM solutions. The resulting method, dubbed AR-FSM, fully exploits the super-convergence property and achieves convergence orders $2s+2+2b$ for eigenvalues and $s+1+b$ for eigenfunctions in the $H^1$ norm, while the AR post-processing requires only a computational cost that is linear in the number of FSM degrees of freedom. The analysis introduces a rigorous definition of point singularities and develops a foundational framework for their study. It further establishes an asymptotic expansion of eigenfunctions consisting of a regular component in $H^{s+4}$ together with $d+1$ asymptotic functions associated with each singular point. Numerical experiments confirm the sharpness of these theoretical bounds.

12. 一种求解椭圆反问题的交替学习型配点法

An alternating learning-based collocation method for solving inverse elliptic problems

作者: Zhizhong Kong, Jerry Zhijian Yang, Cheng Yuan

类别: math.NA, physics.comp-ph

发布日期: 2026-06-01

arXiv ID: 2606.01622v1

链接: http://arxiv.org/abs/2606.01622v1

摘要: We propose the Alternating Learning-Based Collocation (ALBC) method for solving inverse elliptic problems. Our approach employs sinusoidal shallow networks as adaptive basis generators. By alternately updating the state variable and the unknown parameter, we decompose the original nonconvex joint optimization problem into a sequence of tractable linear subproblems. This strategy effectively overcomes the fixed-basis limitations of classical collocation methods while avoiding the slow convergence typically encountered in deep learning approaches. Theoretically, we establish stability estimates and prove the convergence of the proposed algorithm. Numerical experiments on five benchmark problems demonstrate the efficacy of ALBC, which consistently outperforms the standard collocation method in accuracy. Furthermore, it achieves performance comparable to or better than that of physics-informed neural networks at a substantially lower computational cost. Finally, the method remains robust under noise levels of up to twenty percent.

13. 基于有限元插值的免稳定虚拟元方法

Stabilization-free virtual element methods based on finite element interpolation

作者: Jikun Zhao, Wenhao Zhu, Bei Zhang, Shipeng Mao

类别: math.NA

发布日期: 2026-06-01

arXiv ID: 2606.01614v1

链接: http://arxiv.org/abs/2606.01614v1

摘要: In this paper, we introduce a new framework for designing stabilization-free virtual element methods (VEMs) based on an finite element interpolation-based strategy, where we can simultaneously eliminate the stabilization terms in the discretizations of diffusion and reaction terms. The core idea is to construct a computable, polynomial-preserving, and norm-equivalent interpolation operator from the virtual element space to a (local) finite element space. Leveraging the properties of this operator, we design two types of stabilization-free schemes. The first scheme requires the interpolation to preserve the polynomial consistency related to the bilinear forms, thereby maintaining both consistency and stability as in the standard VEM. The second scheme relaxes this consistency requirement. While it may not satisfy the standard polynomial consistency, the second scheme retains optimal convergence with simpler construction, fewer degrees of freedom and, more importantly, applicable to more complex problems such as those involving nonlinearities or variable coefficients. We construct concrete interpolation operators for both conforming and nonconforming virtual elements in two and three dimensions. These operators are then employed to realize stabilization-free schemes for conforming and nonconforming VEMs. Numerical experiments confirm the optimal convergence rates of the proposed methods. The presented framework can be extended to design stabilization-free schemes for other polytopal discretization methods, such as the hybrid high-order method and the weak Galerkin method.

14. 通过二阶几何监督学习混沌动力学

Learning Chaotic Dynamics through Second-Order Geometric Supervision

作者: Shinhoo Kang, Hai V. Nguyen, Tan Bui-Thanh

类别: math.NA, cs.LG

发布日期: 2026-06-01

arXiv ID: 2606.01596v1

链接: http://arxiv.org/abs/2606.01596v1

摘要: Learning chaotic dynamical systems from data requires more than short-term predictive accuracy: the learned model must preserve the attractor geometry and its invariant statistics. Trajectory (zero-order) and Jacobian (first-order) matching supervise the values and tangent structure of the vector field, but neither constrains how the field bends away from its tangent plane. A model can thus match values and tangents at the supervised states yet curve differently from the truth, remaining locally accurate while drifting toward spurious attractors and distorting long-time statistics. We show that enforcing second-order consistency mitigates these failures, but forming the full Hessian is prohibitive in high dimensions. We propose model-constrained randomized Jacobian matching, which compares the Jacobians of the true and learned vector fields at randomly perturbed inputs. A Taylor expansion shows that the expected randomized Jacobian loss decomposes into the nominal Jacobian mismatch plus a Hessian mismatch scaled by the noise variance, implicitly enforcing second-order consistency at $\mathcal{O}(d^2)$ cost without forming the $\mathcal{O}(d^3)$ Hessian tensor. Using only Jacobian evaluations, the method scales to high dimensions where explicit Hessian matching does not. Numerical experiments confirm that second-order methods are robust. For Lorenz63, first-order methods produce catastrophic Lyapunov-exponent outliers under minimal temporal supervision, which second-order methods eliminate while recovering the correct attractor. For coupled Lorenz96, an out-of-distribution forcing sweep separates the methods: all agree up to $F=16$, but beyond $F=18$ only second-order methods preserve the invariant measure and Lyapunov spectrum. On both systems, randomized Jacobian matching performs comparably to explicit Hessian matching at much lower cost.

15. 四层重叠Schwarz方法作为多重网格粗化求解器求解复杂几何中不可压缩非牛顿流动

Four-Level Overlapping Schwarz as Multigrid Coarse Solver for Incompressible Non-Newtonian Flow in Complex Geometries

作者: Stephan Köhler, Oliver Rheinbach

类别: math.NA

发布日期: 2026-05-31

arXiv ID: 2606.01433v1

链接: http://arxiv.org/abs/2606.01433v1

摘要: For complex geometries, the coarse problem of geometric multigrid can be too large to be solved by a direct solver. Here, we report on the use of domain decomposition applied to the multigrid coarse problem. Additive overlapping Schwarz methods are domain decomposition methods for the iterative solution of partial differential equations whose numerical and parallel scalability can be improved by the addition of coarse levels. A successful coarse space for such methods, inspired by iterative substructuring, is the generalized Dryja–Smith–Widlund (GDSW) space. A monolithic two-level overlapping Schwarz preconditioner based on a GDSW coarse space has been introduced for the solution of saddle-point problems arising from incompressible fluid problems, and has subsequently been extended to a three-level method. In the present work, for the first time, we consider a monolithic four-level overlapping Schwarz preconditioner, obtained by applying the two-level monolithic GDSW construction recursively three times, so that the second- and third-level coarse problems are themselves treated by overlapping Schwarz and only the smallest fourth-level coarse problem is solved by a sparse direct method. Numerical results are presented for a three-dimensional incompressible stationary Stokes problem with a Carreau-type non-Newtonian viscosity model posed on the complex geometry of an extrusion die, on up to $4\,000$,MPI ranks, comparing the four-level preconditioner with its two-level and three-level counterparts in both roles. This work is part of the StroemungsRaum project, funded by the German Bundesministerium für Forschung, Technologie und Raumfahrt (BMFTR, formerly BMBF) as part of the SCALEXA program on new methods and technologies for exascale computing.

16. 铁磁流体：建模与逼近

Ferrofluids: Modeling and Approximation

作者: Gonzalo A. Benavides, Ricardo H. Nochetto, Mansur Shakipov

类别: math.NA, math-ph

发布日期: 2026-05-31

arXiv ID: 2606.01392v1

链接: http://arxiv.org/abs/2606.01392v1

摘要: Starting from Maxwell’s and linear momentum balance equations, we derive a ferrofluid model using the generalized Onsager’s principle. Guided by a discrete perturbation estimate, we design and analyze families of Galerkin schemes that converge to sufficiently regular solutions and derive error estimates. Finally, we numerically explore the model with our proposed method.

17. 免转置线性代数

Transpose-free linear algebra

作者: Diana Halikias, Michiel E. Hochstenbach, Alex Townsend

类别: math.NA

发布日期: 2026-05-31

arXiv ID: 2606.01335v1

链接: http://arxiv.org/abs/2606.01335v1

摘要: We study the limitations of matrix-free algorithms that access a matrix $A$ only through forward matrix-vector products (matvecs) $x \mapsto Ax$, without access to the transpose $A^\top$ or its action. This setting arises naturally in operator learning, inverse problems, and matrix-free PDE solvers, where adjoint evaluations may be unavailable or prohibitively expensive. We show that the lack of transpose access creates severe and sometimes insurmountable theoretical barriers. For Krylov methods, we prove that the sequence of projected operator norms produced by Arnoldi iteration can follow any prescribed nondecreasing curve, showing that forward matvecs alone provide essentially no reliable information about the spectral norm. For several core problems, including least squares, norm estimation, column subset selection, and local maximum volume, we establish non-identifiability results; distinct matrices can generate identical forward-query transcripts while having fundamentally different solutions. We also prove quantitative lower bounds on the number of forward matvecs required for approximation tasks. In particular, any algorithm that computes a near-optimal rank-$k$ approximation must use at least $n$ queries, and estimating the Frobenius norm to relative accuracy $\eps$ requires $Ω(\eps^{-2})$ queries when $n$ is sufficiently large, matching the complexity of Hutchinson-type estimators up to constants. Although some problems remain solvable without transpose access, the transpose-free setting is fundamentally more limited in both identifiability and efficiency.

18. 基于变分空间的编码器–解码器神经算子的高效逼近

Efficient Approximation for Encoder–Decoder Neural Operators via Variation Spaces

作者: Jia-Qi Yang, Lei Shi

类别: stat.ML, cs.LG, math.FA, math.NA, math.ST

发布日期: 2026-05-31

arXiv ID: 2606.01244v1

链接: http://arxiv.org/abs/2606.01244v1

摘要: We study operator learning using encoder–decoder neural networks. Inspired by the function-space theory of neural networks, we introduce a variation space as an infinite-dimensional structural class for nonlinear operators. This space is defined through vector-valued measures directly on the input and output spaces. For operators in this space, we establish approximation bounds for encoder–decoder two-layer networks in the Bochner $L^q$ norm. The resulting error bound decomposes into the input encoding error, the output encoding error, and a finite-width approximation term of order $N^{-1/2}$, with a constant independent of the input and output encoding dimensions. When the input and output encoding errors decay polynomially in the encoding dimensions, these estimates yield algebraic approximation and learning rates. The results provide an theoretical guarantees for efficient neural operator learning beyond general Lipschitz or Fréchet differentiable operator classes.

19. 残差加权随机化雅可比: 通过残差集中性与异步扩展得到的锐化界

Residual-Weighted Randomized Jacobi: Sharpened Bounds via Residual Concentration and Asynchronous Extension

作者: Evan Coleman

类别: math.NA, cs.DC

发布日期: 2026-05-31

arXiv ID: 2606.01232v1

链接: http://arxiv.org/abs/2606.01232v1

摘要: We study randomized stationary methods for symmetric positive definite linear systems in which component $j$ is selected with probability proportional to $|r_j|^\ell$. This power-weighted family interpolates continuously between uniform randomized Jacobi as $\ell \to 0$ and Gauss–Southwell greedy relaxation as $\ell \to \infty$. For the central case $\ell = 2$, we sharpen the standard one-step convergence analysis using the inverse participation ratio (IPR) $ν^2(r) = n\|r\|_4^4/\|r\|_2^4$, which equals $1$ when the residual is uniform and grows toward $n$ as it concentrates. The resulting bound amplifies the expected per-step progress by exactly $ν^2$ over the uniform-sampling baseline. The IPR can be computed online at $O(n)$ cost and doubles as a per-iteration diagnostic. We extend the analysis to asynchronous power-weighted Jacobi via the Avron–Druinsky–Gupta framework, obtaining an epoch-based convergence theorem in which the IPR controls both the progress coefficient and the allowed-delay window. Numerical experiments on shared-memory hardware support the sharpened bound and show the IPR trajectory is essentially concurrency-insensitive. Unexpectedly, consistent-reads execution, the easier case for the ADG analysis, destabilizes power-weighted sampling at high concurrency while inconsistent reads remain stable; the same IPR that amplifies progress amplifies a thread-collision rate that inconsistent reads appear to absorb. We propose a feedback-damping mechanism and verify two predictions about its dependence on problem size.

20. 具有弱奇异激波的非线性浅水方程的高阶正则化及其有限体积法逼近

A high-order regularization of the non-linear shallow water equations with weakly singular shock waves and its approximation by finite volume methods

作者: Rezwana Razzaque Angana, Dimitrios Mitsotakis

类别: math.AP, math.NA

发布日期: 2026-05-31

arXiv ID: 2606.01200v1

链接: http://arxiv.org/abs/2606.01200v1

摘要: Considered herein is a high-order regularization of the nonlinear shallow water equations within the framework of water wave theory. The regularized system is Galilean invariant and its solutions maintain an energy level that closely matches that of the nonlinear shallow water equations. However, in contrast to the classical nonlinear shallow water system, which admits discontinuous shock waves, the regularized formulation gives rise to weakly singular shock waves, which have continuous spatial profiles with unbounded spatial derivatives at isolated points. Using dynamical systems techniques, we establish the existence of such waves. Although weakly singular traveling waves remain continuous over their entire domain, their numerical approximation via finite element or pseudospectral schemes is affected by the emergence of spurious oscillations. To address this issue, we explore several finite volume methods for the accurate numerical approximation of these solutions. Our results demonstrate that the regularized system effectively reproduces the dynamics of the nonlinear shallow water equations in several scenarios. Moreover, our computations indicate that weakly singular shock waves are dynamically stable and can arise from general initial conditions connecting two asymptotic states. In contrast, other weakly singular structures, such as cusped solitons, appear to be structurally unstable, as we were unable to generate them from generic initial data.

21. 半盲解卷积的分片常值水平集方法：在条形码解码中的应用

A piecewise constant levelset approach for semi-blind deconvolution: Application to barcode decoding

作者: Adriano De Cezaro, Eduardo Hafemann, Antonio Leitão

类别: math.NA

发布日期: 2026-05-31

arXiv ID: 2606.01156v1

链接: http://arxiv.org/abs/2606.01156v1

摘要: We consider a semi-blind deconvolution problem arising in the decoding of blurred linear barcodes. Building on the Piecewise Constant Level Set (PCLS) framework introduced in [De,Cezaro et al., Inv.,Probl., 29 (2013), 015003], we propose and analyze a solution method based on augmented Lagrangians to obtain stable approximate solutions to the corresponding inverse problem with respect to noisy measurements. We establish the existence of generalized multipliers for the augmented Lagrangian functional under consideration, as well as the absence of duality gaps. These results provide the theoretical foundation required to prove regularization properties of the approximate solutions produced by the proposed strategy. Furthermore, we present an associated ADMM-type iterative scheme for the explicit computation of approximate barcodes. Numerical experiments are carried out for various variance values (responsible for the blurred effect) and several levels of noise, validating the effectiveness of the proposed method.

22. 面向近晶液晶的张量梯度流与准熵及其保持耦合物理约束的离散化

Tensor gradient flow with quasi-entropy for smectic liquid crystals and discretizations keeping coupled physical constraints

作者: Jie Xu, Xiaomei Yao

类别: math.NA, cond-mat.soft, physics.comp-ph

发布日期: 2026-05-30

arXiv ID: 2606.00659v1

链接: http://arxiv.org/abs/2606.00659v1

摘要: A gradient flow for the concentration and a $2\times 2$ tensor is constructed to describe smectic liquid crystals. The free energy consists of the entropy term and interaction term involving squared second order spatial derivatives. The entropy term incorporates the concentration in the quasi-entropy originally proposed for the tensor only, which is a strictly convex and lower semicontinuous function imposing coupled constraints between the concentration and the tensor. An evolution equation for the boundary normal derivative of the concentration is proposed in addition to the equations for the concentration and the tensor, giving an energy dissipation system. Numerical schemes are designed with emphases on using the entropy term to keep the coupled constraints, and the discretization of the boundary normal derivatives satisfying summation by parts. Existence, uniqueness, energy dissipation and error estimates are established. Numerical results indicate the efficiency and robustness of the scheme. Configurations of defects different from other layer structures are observed.

23. 通过有噪费曼-卡茨监督驯服PINNs的损失景观：算子预处理与非渐近误差界

Taming the Loss Landscape of PINNs with Noisy Feynman-Kac Supervision: Operator Preconditioning and Non-Asymptotic Error Bounds

作者: Nathanael Tepakbong, Hanyu Hu, Chengyu Liu, Xiang Zhou

类别: stat.ML, cs.LG, math.NA, math.OC, math.ST

发布日期: 2026-05-30

arXiv ID: 2606.00643v1

链接: http://arxiv.org/abs/2606.00643v1

摘要: Physics-Informed Neural Networks (PINNs) often train slowly or fail to converge on challenging partial differential equations (PDEs), a behavior recently linked to severely ill-conditioned loss landscapes inherited from the underlying differential operator. We study PINNs augmented with a pointwise data-fidelity term, added at a few points in the domain to the standard residual and boundary losses. We show that this supervision term acts as an operator-level preconditioner: for suitable weights, our comparison bounds guarantee a substantially smaller condition number than under the standard PINN loss, independently of how the pointwise labels are obtained. For a broad class of PDEs admitting a Feynman-Kac (FK) representation, we generate such labels by Monte Carlo averages of the FK functional, resulting in what we call ``FK-PINNs", and using the excess risk decomposition approach, we derive non-asymptotic $L^2(Ω)$-error bounds for FK-PINNs with $\tanh$ activation trained by finitely many steps of gradient descent. Along the way, we establish pseudo-dimension bounds for first- and second-order derivatives of $\tanh$ neural networks, which are of independent interest and, to the best of our knowledge, new. Numerical experiments on Poisson, Schrödinger, mean exit time, and committor problems corroborate the theory, and show that FK-PINNs can successfully solve PDEs for which standard PINNs exhibit severe failure modes.

24. 深度神经网络为不适定反问题提供正则化

Deep neural network yields regularization for ill-posed inverse problems

作者: Qiao Zhu, Lan Wang, Ye Zhang

类别: math.NA

发布日期: 2026-05-30

arXiv ID: 2606.00517v1

链接: http://arxiv.org/abs/2606.00517v1

摘要: This paper studies the regularization of ill-posed inverse problems by deep neural networks (DNNs). We extend architecture-based regularization from shallow networks to deep models by developing a deterministic framework in which the admissible network class is enlarged adaptively and the resulting architecture complexity acts as the regularization mechanism. We propose two discrepancy-principle-driven expanding DNN algorithms to treat the cases where an explicit parameter-radius bound is available and unavailable, respectively. For both algorithms, we prove the finite termination of the adaptive expansion procedure and the convergence of the regularized solutions as the noise level vanishes. In addition, we derive explicit asymptotic bounds on the terminal network architecture, thereby quantifying how the required network complexity scales with the noise level. Numerical experiments on several representative linear and non-linear inverse problems support the theoretical findings and illustrate the practical usefulness of the proposed framework.

25. 基于学习代理动力学的连续数据同化

Continuous Data Assimilation with Learned Surrogate Dynamics

作者: Wenwen Li, Daniel Sanz-Alonso

类别: math.DS, math.NA, math.OC, stat.ML

发布日期: 2026-05-30

arXiv ID: 2606.00480v1

链接: http://arxiv.org/abs/2606.00480v1

摘要: Continuous data assimilation seeks to estimate the state of a dynamical system from partial observations. In many applications, however, the state dynamics are unknown or prohibitively expensive to simulate at the required resolution, leading to model error. Motivated by this challenge and the increasing adoption of machine learning surrogates in data assimilation, this paper develops a unified finite-dimensional analysis of nudging algorithms that employ learned surrogate models of the dynamics. We first establish general conditions on the dynamics and observations that guarantee accurate tracking for nudging with the true dynamics model, both in the noise-free and noisy settings. We then show that nudging algorithms that employ surrogate models retain exponential convergence up to an explicit error floor that quantifies the effects of surrogate approximation error and observation noise. Finally, we analyze surrogate models obtained by learning either the vector field or the short-time solution map of the system, and quantify the amount of training data needed to ensure accurate nudging in the noise-free setting. Numerical experiments support the theory.

26. 基于提升的接口降维框架用于非线性传输和特征值问题

A Lifting-Based Interface Reduction Framework for Nonlinear Transmission and Eigenvalue Problems

作者: So-Hsiang Chou

类别: math.NA

发布日期: 2026-05-30

arXiv ID: 2606.00468v1

链接: http://arxiv.org/abs/2606.00468v1

摘要: We present a lifting-based interface reduction framework for nonlinear transmission and eigenvalue problems. The method represents the solution as a sum of a bulk component and a lifting component that carries the interface jump, thereby reducing the original problem to a nonlinear system posed on the interface. A low-dimensional approximation is obtained by restricting the interface unknown to a finite-dimensional subspace. The corresponding lifting modes are precomputed and reused, leading to a formulation in which the bulk operator remains fixed and the essential behavior is governed by a small number of interface degrees of freedom. For eigenvalue problems, the same framework yields a reduced system in which eigenvalues are identified through the near-singularity of a parameter-dependent interface matrix. The associated eigenvectors reveal dominant interface modes, providing a direct interpretation of the spectral structure. Numerical experiments show that both approximation accuracy and spectral behavior are determined primarily by the interface representation rather than by the bulk discretization. In particular, enriching the interface space rapidly improves accuracy and reveals additional eigenmodes, while mesh refinement alone has limited effect. These results indicate that transmission and eigenvalue problems are effectively governed by a small number of interface modes, offering a simple and computationally efficient perspective on model reduction.

27. 基于厄米块嵌入与广义量子信号处理的二维Black-Scholes方程求解

Solving 2D Black Scholes Equation via Hermitian Block Embedding and Generalised Quantum Signal Processing

作者: James W. Greenwell, Jingbo Wang, Des Hill

类别: quant-ph, math.NA

发布日期: 2026-05-30

arXiv ID: 2606.00458v1

链接: http://arxiv.org/abs/2606.00458v1

摘要: The Black Scholes equation provides a fundamental model for the no arbitrage pricing of financial derivatives. After finite difference discretisation, the pricing problem can be formulated as a finite dimensional linear algebra problem involving the inverse of a non Hermitian time step matrix. Recent advances in quantum linear algebra algorithms, particularly the generalised quantum signal processing (GQSP)algorithm, enable matrix functions to be implemented through polynomial transformations of a suitable unitary or Hermitian form. In this paper, we develop a Hermitian block embedding method that enables GQSP to be applied to the two dimensional Black Scholes equation. Numerical simulations for two asset European call options are performed to evaluate the proposed approach. GQSP based solutions are benchmarked against the classical polynomial approximation with backward Euler finite difference method, showing close agreement. This indicates that the Hermitian block embedding construction accurately captures the dynamics of the original non Hermitian operator. These results demonstrate the feasibility of combining Hermitian block embeddings with GQSP for multidimensional Black Scholes problems and provide a proof of principle for applying modern quantum linear algebra techniques to option pricing.

28. 数据驱动的谱预测用于加速大规模电子结构计算

Data-Driven Spectral Prediction for Accelerating Large-Scale Electronic Structure Calculations

作者: Abhiram Badrinarayanan, Davor Davidovic, Edoardo Di Napoli, Jurica Novak, Luigi Genovese, Gustavo Ramirez-Hidalgo, Xinzhe Wu

类别: physics.comp-ph, cond-mat.mtrl-sci, cs.LG, math.NA

发布日期: 2026-05-29

arXiv ID: 2606.00401v1

链接: http://arxiv.org/abs/2606.00401v1

摘要: Simulating large molecular systems comprising thousands of atoms requires highly scalable methodologies. While modern Density Functional Theory (DFT) codes exhibit linear scaling, solving the associated large, sparse generalized eigenproblems remains a critical computational bottleneck on exascale architectures. In the context of the LimitX project, we propose a data-driven framework to accelerate these calculations. By shifting the machine learning target from discrete eigenvalues to the coefficients of an interpolating Chebyshev polynomial, and by comparing both all-atom and fragment-based structural representations, we successfully overcome the dimensionality constraints of large-scale spectral prediction. We investigate three machine learning models (Kernel Ridge Regression, Graph Neural Networks, and Random Forests) trained on a novel 2 TB dataset of protein dimers. The predicted spectra provide initial guesses that effectively bypass early Self-Consistent Field (SCF) iterations in BigDFT. Ultimately, these spectral predictors will be deployed to dynamically optimize upcoming rational filter-based eigensolvers, such as FrASE, which is currently in initial development.

29. $L_p$范数的经验逼近

Empirical Approximation of $L_p$ Norms

作者: Feng Dai, Egor Kosov, Noel Murasko

类别: math.FA, math.CA, math.NA, math.PR

发布日期: 2026-05-29

arXiv ID: 2606.00347v1

链接: http://arxiv.org/abs/2606.00347v1

摘要: We study empirical $L_p$ moments of a random vector $\pmb\varphi$ based on its i.i.d.\ copies $\pmb\varphi^1,\ldots,\pmb\varphi^m$, that is, $\frac1m\sum_{j=1}^m |\langle \pmb\varphi^j,y\rangle|^p$. Our main result is a new estimate for the expected uniform deviation [ \mathbb{E}\sup_{y\in D}\biggl| \frac1m\sum_{j=1}^m |\langle \pmb\varphi^j,y\rangle|^p -\mathbb{E}|\langle \pmb\varphi,y\rangle|^p \biggr| ] over an arbitrary index set $D$. The proof is based on a new bound for Talagrand’s $γ$-functional, sharper than the standard Dudley-type entropy estimate. We then apply this estimate to the following two problems. First, for $p>2$, we study Marcinkiewicz-type discretization of $L_p$ norms on an $N$-dimensional subspace $X_N\subset B(Ω)$ of bounded functions on a probability space $(Ω,μ)$. We obtain bounds in terms of the norm of the embedding $ (X_N,\|\cdot\|_{L_p(μ)})\hookrightarrow B(Ω). $ In particular, we prove that when this norm is of order $N^{1/p}$ and [ m \ge C(p), N\log N,(\log\log N)^{p-1}, ] then $m$ random samples suffice to approximate the $L_p(μ)$ norm uniformly on $X_N$ by the sampled discrete $L_p$ norm. This substantially improves the previously known bound in this setting $ m \ge C(p)\, N(\log N)^{\min\{p,3\}}, $ and is optimal up to the factor $(\log\log N)^{p-1}$ in the random-sampling setting. Second, for $1\le p<2$, we obtain an $L_p$ analogue of the restricted isometry property via random sampling for bounded orthogonal systems and, more generally, for $N$-element systems $\mathcal D_N$ satisfying a Riesz-type condition. We prove that when [ m \ge C(p), s\log N,(\log s)^2,\log\log s, ] then $m$ random samples suffice to guarantee an $L_p$ restricted isometry-type property uniformly over the class of all $s$-sparse functions generated by $\mathcal D_N$.

30. 随机舍入增大小奇异值

Stochastic Rounding Increases Small Singular Values

作者: Linkai Ma, Tingzhou Yu, Petros Drineas

类别: math.NA, cs.LG

发布日期: 2026-05-29

arXiv ID: 2606.00312v1

链接: http://arxiv.org/abs/2606.00312v1

摘要: Over the past half-dozen years, stochastic rounding (SR) has regained significant attention as a quantization scheme for low-precision floating-point arithmetic, with applications spanning numerical analysis and modern machine learning systems. Recent work has shown that SR acts as an implicit regularizer by increasing the smallest singular value of extremely tall-and-thin (or, symmetrically, short-and-fat) matrices. In this work, we substantially sharpen and extend this understanding in two directions. First, we show that the regularization effect of SR is not restricted to extreme aspect ratio regimes: it persists for matrices with constant aspect ratio. Second, we demonstrate that SR does not merely regularize the smallest singular value, but instead lifts entire clusters of singular values at the tail of the spectrum. Together, these results provide a more general characterization of stochastic rounding as a spectral regularizer, revealing that its effects extend beyond extremal aspect ratios and act on a broader portion of the singular value spectrum.

31. 基于对称 Hermite 求积的平衡截断法从导数数据学习线性动力系统

Symmetric Hermite quadrature-based balanced truncation for learning linear dynamical systems from derivative data

作者: Sean Reiter, Steffen W. R. Werner

类别: math.NA, cs.LG, eess.SY, math.DS, math.OC

发布日期: 2026-05-29

arXiv ID: 2606.00298v1

链接: http://arxiv.org/abs/2606.00298v1

摘要: Data-driven reduced-order modeling is an essential component in the computer-aided design of control systems. In this work, we present a novel symmetric Hermite formulation of the quadrature-based balanced truncation algorithm that constructs linear reduced-order models from evaluations of the full-order system’s transfer function and its derivative. Significantly, the Hermite formulation preserves desirable qualitative properties of the system used to generate the data, such as state-space Hermiticity and, consequently, asymptotic stability.

32. 基于似然信息的降阶模型近似后验的误差界

Error bounds for approximate posteriors from likelihood-informed reduced-order models

作者: Han Cheng Lie, Jakob Scheffels, Elisabeth Ullmann

类别: math.NA

发布日期: 2026-05-29

arXiv ID: 2606.00274v1

链接: http://arxiv.org/abs/2606.00274v1

摘要: In the design of computational methods for Bayesian inverse problems, costly forward model evaluations make it difficult to sample from or compute the posterior. This motivates the need for approximate forward models that are cheaper to evaluate. We consider reduced-order forward models which exploit the lower-dimensional structure in the Bayesian inverse problem by projecting to the “likelihood-informed subspace” of the parameter space where the prior-to-posterior update is significant. However, the theoretical properties of these reduced-order forward models and their impact on the solution of the Baysian inverse problem are not always well-understood. In this work we consider linear Gaussian inverse problems with a possibly singular prior covariance matrix. We analyse a recently proposed reduced-order model which uses a Petrov-Galerkin projection to likelihood-informed subspaces that arise in optimal low-rank approximations of the posterior covariance matrix. We bound the error in the resulting approximation of the root prior-preconditioned Hessian of the data misfit. Based on this we also bound the errors of the approximate posterior covariance and mean. Our analysis shows that this reduced-order model recovers the exact posterior when the rank of the reduced-order model is equal to the “intrinsic dimension” of the inverse problem, i.e. the rank of the prior-preconditioned Hessian. Two numerical experiments from structural engineering illustrate the performance of our bounds.

33. 连续时间随机游走到具有无界系数的时间分数阶扩散的收敛速率

Convergence Rates of Continuous-Time Random Walks to Time-Fractional Diffusions with Unbounded Coefficients

作者: Artur Sidorenko, Vasilii Kolokoltsov

类别: math.PR, math.NA

发布日期: 2026-05-29

arXiv ID: 2605.31471v1

链接: http://arxiv.org/abs/2605.31471v1

摘要: We investigate uniform weak convergence rates for probabilistic numerical methods applied to backward time-fractional diffusion equations whose dynamics are driven by diffusions with possibly unbounded coefficients, such as the Geometric Brownian Motion. The fractional structure is represented through a random time-change by the inverse of a stable subordinator. To approximate the underlying fractional dynamics, we combine discrete Markov chain schemes for the diffusion component with heavy-tailed random walk approximations of the time change. Our analysis builds on Feller semigroup techniques and a high-order sensitivity framework for diffusion semigroups based on the Kunita stochastic flows and tensor fields. We derive uniform bounds for all orders of sensitivities, establish a quasi-contraction property for the associated semigroup, and transfer these estimates to the fractional setting via the convolution representation with the inverse subordinator. As a result, under killing conditions which dominate at least the base-space semigroup growth, we obtain weak convergence rates for the combined continuous-time-random-walk scheme to the time-fractional diffusion, with a logarithmic regime before the discount dominates the stronger smooth-space growth.

34. 正规矩阵的谱密度估计

Spectral density estimation for normal matrices

作者: Cameron Musco, Christopher Musco, Rikhav Shah, John Urschel, Nicholas West

类别: math.NA

发布日期: 2026-05-29

arXiv ID: 2605.31430v1

链接: http://arxiv.org/abs/2605.31430v1

摘要: The spectral density estimation problem asks for an algorithm that, given an $n\times n$ matrix $A$, outputs a probability measure that is a good approximation to the uniform distribution on the eigenvalues of $A$, called the spectral density of $A$. This paper considers the setting where $A$ is a large normal matrix that is accessible only through matrix-vector product queries. We provide an algorithm that makes just $m$ matrix-vector queries to $A$ and returns, with high probability, a measure within earth mover’s distance $O(1/m+\log m/{\sqrt n})$ of the true spectral density of $A$. We provide a complementary lower bound that any algorithm producing an $\varepsilon$-approximation to the true spectral density for large matrices must make $Ω(1/\varepsilon)$ matrix-vector queries. The lower bound holds even for the more restricted case of real symmetric input matrices. In combination with our upper bound, it shows that spectral density estimation is essentially no harder for complex normal matrices than for real symmetric matrices.

35. 基于多项式混沌的随机分岔分析：分支逼近解的一致性与收敛性

Stochastic bifurcation analysis via polynomial chaos: consistency and convergence of branch-approximating solutions

作者: Giacomo Venier, Isabella Carla Gonnella, Federico Pichi, Gianluigi Rozza

类别: math.NA, math.DS

发布日期: 2026-05-29

arXiv ID: 2605.31288v1

链接: http://arxiv.org/abs/2605.31288v1

摘要: Parameter-dependent dynamical systems that exhibit bifurcations pose significant computational challenges, as traditional continuation methods require repeated, costly simulations across large ranges of parameter values to capture sudden qualitative changes in the solution. In this work, we propose a systematic approach to reconstruct the branches of the entire bifurcation diagram in a single numerical solver leveraging generalized Polynomial Chaos (PC) expansion. By treating the parameter as a random variable, we cast the deterministic parameter-dependent model in a weak stochastic form, and then use a Galerkin projection to recover bifurcation branches globally across the parameter domain without iterative pointwise continuation. We show that the resulting Galerkin system, in the non-uniqueness regime, produces many discrete algebraic roots that naturally split into two classes: highly oscillatory solutions and branch-approximating ones. We develop a rigorous theoretical framework that establishes consistency, proves convergence of the branch-approximating solutions to the true steady states, and guarantees uniqueness of the Galerkin solution under suitable assumptions. Finally, we confirm these theoretical results with numerical experiments on several parameter-dependent ordinary differential equations (ODEs), demonstrating the accuracy and computational efficiency of our single-run framework in capturing complex bifurcation diagrams for both scalar and vector-valued systems.

36. 三维调和势边值问题的全纯神经网络框架

A holomorphic neural network framework for 3D boundary value problems governed by harmonic potentials

作者: Enrico Ballini, Allan Peter Engsig-Karup, Tito Andriollo

类别: math.NA, cs.LG

发布日期: 2026-05-29

arXiv ID: 2605.31231v1

链接: http://arxiv.org/abs/2605.31231v1

摘要: We present a neural-network-based framework for the solution of three-dimensional boundary value problems where the solution is expressible in terms of harmonic potentials. The approach leverages the Whittaker integral formula, which allows representing the solution through functions that are holomorphic with respect to a suitable complex variable. These functions are subsequently approximated using holomorphic neural networks, which guaranty fulfillment of the holomorphicity requirement. A key feature of the proposed formulation is that the governing partial differential equations (PDEs) are satisfied exactly by construction. Therefore, in contrast to standard physics-informed neural networks, no residual minimization of PDEs is required in the interior of the domain, and training is based exclusively on boundary collocation points. The method is validated against three-dimensional Laplace and linear elasticity problems, where, in the latter case, displacement and stress fields are expressed via the Papkovich-Neuber potentials. The numerical results show an accurate approximation of both scalar and vector fields, with errors remaining controlled throughout the domain. Overall, the work demonstrates that the incorporation of analytical structures into neural network architectures provides a natural and effective framework for the meshless approximation of three-dimensional boundary value problems while preserving the underlying properties of the governing equations.

37. 深度ReLU神经网络对各向异性与混合光滑函数的逼近与学习

Approximation and learning of anisotropic and mixed smooth functions by deep ReLU neural networks

作者: Yunfei Yang, Jun Fan

类别: stat.ML, cs.LG, math.NA

发布日期: 2026-05-29

arXiv ID: 2605.31152v1

链接: http://arxiv.org/abs/2605.31152v1

摘要: This paper studies how efficiently deep ReLU neural networks can approximate and learn smooth functions. When the error is measured in $L^p([0,1]^d)$ norm and the approximator is a network with width $W$ and depth $L$, recent works have proven the supper approximation rate $\mathcal{O}((WL)^{-2s/d})$ for Besov space $\mathcal{B}^s_{q,r}([0,1]^d)$ under the Sobolev embedding condition $s/d>1/q-1/p$. In order to overcome the curse of dimensionality in this rate, we extent this result to anisotropic and mixed smooth function classes. We establish the approximation rate $\mathcal{O}((WL)^{-2\tilde{s}})$ for anisotropic Besov space $\mathcal{B}^{\boldsymbol{s}}_{q,r}([0,1]^d)$ with anisotropic smoothness $\boldsymbol{s}=(s_1,\dots,s_d)$ under the embedding condition $\tilde{s} > 1/q-1/p$, where the mean smoothness $\tilde{s} = (\sum_{i=1}^d s_i^{-1})^{-1}$. For mixed smooth Besov space $\mathcal{MB}^s_{q,r}([0,1]^d)$ with mixed smoothness $s>1/q-1/p$, we show that the approximation rate $\mathcal{O}((WL)^{-2s})$ holds up to logarithmic factors. Using these results, we also derive approximation bounds for the composition of anisotropic Besov functions. As an application, it is shown that deep ReLU neural networks can achieve minimax optimal rates up to logarithmic factors for a wide range of smooth function classes.

38. 非线性守恒律的可扩展贝叶斯推断

Scalable Bayesian Inference for Nonlinear Conservation Laws

作者: Tim Weiland, Philipp Hennig

类别: cs.LG, math.NA

发布日期: 2026-05-29

arXiv ID: 2605.31127v1

链接: http://arxiv.org/abs/2605.31127v1

摘要: Nonlinear conservation laws are at the heart of many of the most important dynamical systems in science and engineering. In practical applications, such systems are often subject to various sources of uncertainty, e.g. due to sparse or noisy measurements. Inferring physical quantities and fields of interest then becomes an ill-posed problem which both classical numerical methods and modern deep learning-based methods struggle to treat appropriately. Recent work has framed classical numerical methods as Bayesian inference under Gaussian process priors, resulting in a physics-aware treatment of uncertainties. Following this line of work, we develop a novel numerically conservative method for uncertainty-aware simulations of nonlinear conservation laws. We use recent sparse approximation techniques to scale up to large-scale forward and inverse problems. For forward simulation, we inherit the accuracy of classical solvers while providing structured uncertainty quantification. On inverse problems, we recover posteriors over nonparametric source fields in seconds – outperforming neural baselines that take minutes to produce a less accurate point estimate.

39. 利用t-积中的半张量积求解耦合张量方程 $\mathcal{A} \ltimes \mathcal{X} =\mathcal{B}, \ \mathcal{X}\ltimes \mathcal{C}=\mathcal{D}$

Solving Coupled Tensor Equation $\mathcal{A} \ltimes \mathcal{X} =\mathcal{B}, \ \mathcal{X}\ltimes \mathcal{C}=\mathcal{D}$ using Semi-Tensor Products in the t-product

作者: Mansi Wankhede, Ranjan Kumar Das

类别: math.NA

发布日期: 2026-05-29

arXiv ID: 2605.31122v1

链接: http://arxiv.org/abs/2605.31122v1

摘要: This paper investigates the solution of coupled third-order tensor equation $\mathcal{A} \ltimes \mathcal{X} = \mathcal{B},\ \mathcal{X} \ltimes \mathcal{C} = \mathcal{D},$ of arbitrary dimensions by incorporating semi-tensor product (STP) within t-product framework, where the unknown $\mathcal{X}$ can take form of vector, matrix, or tensor. For the unknown $\mathcal{X}$, we establish a necessary and sufficient condition that provides an equivalence criterion for the existence of solutions. Moreover, the explicit structure (Toeplitz, Circulant) of $\mathcal{C}$ and $\mathcal{D}$ is characterized. Theoretical results are supported by several illustrative examples.

40. 具有退化系数的椭圆型微分方程的一种新的切比雪夫配置法

A novel Chebyshev collocation method for elliptic -type differential equations with degenerate coefficient

作者: Enze Yuan, Huantian Xie, Ziwu Jiang, Jianwei Zhou

类别: math.NA

发布日期: 2026-05-29

arXiv ID: 2605.31055v1

链接: http://arxiv.org/abs/2605.31055v1

摘要: A novel collocation scheme is presented for elliptic-type differential equations with degenerate coefficients and homogeneous Dirichlet boundary conditions. The use of weighted orthogonal Chebyshev polynomials for the basis functions leads to stiffness matrices with sparse structure, enabling efficient direct calculations. By an orthogonal projection, rigorous analyses are devoted to deriving a-priori error estimates of spectral accuracy in two norms. Furthermore, ample numerical experiments are conducted and compared with error data, convergence rates, condition numbers and $N$-$\log$ curves to confirm the theoretical analyses results. Our proposed method achieves spectral accuracy and handles boundary singularities efficiently, as demonstrated by theoretical analyses and numerical experiments.

41. 论杂交混合域分解方法在永磁同步电机中的应用

On the Application of Hybrid Mixed Domain Decomposition Methods to Permanent Magnet Synchronous Machines

作者: Timon Seibel, Sebastian Schöps, Kersten Schmidt

类别: cs.CE, math.NA

发布日期: 2026-05-29

arXiv ID: 2605.31032v1

链接: http://arxiv.org/abs/2605.31032v1

摘要: In this work, we study the application of a hybrid mixed domain decomposition (HMDD) method for the rotor-stator coupling of a permanent magnet synchronous machine. For this, we derive a variational formulation on the electric machine inspired by hybridized discontinuous Galerkin methods using a mixed magnetostatics problem, an affine material law and boundary conditions respecting the symmetry of the motor. We are then able to locate the resulting finite element method within the HMDD framework presented in arXiv:2604.22543. This enables us naturally to transfer the well-posedness results and error estimates for the HMDD method to the finite element method considered in this work. Lastly, as a proof of concept, we consider an academic example and compare the resulting magnetic flux density and potential lines to their counterparts obtained by a well-established in-house code using iso-geometric analysis.

42. 一种面向指标感知学习的非侵入式方法

A non-intrusive approach to index-aware learning

作者: Peter Förster, Idoia Cortes Garcia, Wil Schilders, Sebastian Schöps

类别: cs.CE, math.NA

发布日期: 2026-05-29

arXiv ID: 2605.30955v1

链接: http://arxiv.org/abs/2605.30955v1

摘要: We present a non-intrusive version of the index-aware learning framework introduced in arXiv:2309.00958. Index-aware learning itself is an approach for learning the time and parameter dependent solutions of differential-algebraic equations (DAEs), in particular those of electrical circuits. A key feature of the approach is that it ensures the learned solutions to remain physics-consistent, i.e.\ inherent constraints of the solution, such as e.g.\ Kirchhoff’s laws, remain fulfilled. In general, this is achieved by leveraging a decoupling of the DAE into its differential and algebraic parts, while the non-intrusive version of the approach additionally relies on results from arXiv:2604.20475 and arXiv:2107.07755. We illustrate the overall workflow and compare the non-intrusive and intrusive versions using a buck converter as an example.

43. 基于结构信息的分块结构矩阵Kronecker秩的界

Structure-Informed Bounds on the Kronecker Rank of Block-Structured Matrices

作者: Allison Fuller, Malena Español, Misha Kilmer

类别: math.NA

发布日期: 2026-05-29

arXiv ID: 2605.30908v1

链接: http://arxiv.org/abs/2605.30908v1

摘要: We derive theoretical bounds on the Kronecker rank of block-structured matrices that possess both inner and outer structure. Building on the matrix-to-tensor and tensor-to-matrix framework of Kilmer and Saibaba ($\textit{SIAM J. Matrix Anal. Appl.}$, 2022), we show that the Kronecker rank of a matrix $\mathbf{A}$ equals the dimension of the span of its distinct blocks (the $\textit{inner blockspan}$), and equals as well the dimension of the corresponding span for a permuted matrix $\mathbf{B}$ whose inner and outer structures are interchanged. We give two proofs of this equality: one via a direct dimension argument, and one via an explicit isomorphism between the outer blockspan of $\mathbf{A}$ and the column space of the second-mode unfolding of the associated tensor. These results yield nested containments that translate structural information, such as Toeplitz, Hankel, banded, or sparse patterns, into computable upper bounds on the Kronecker rank. We also establish a duality between element-level sparsity in $\mathbf{A}$ and block-level sparsity in $\mathbf{B}$. Numerical experiments confirm the theory across several classes of structured matrices and, for two sparse matrices drawn from the SuiteSparse collection, provide a structural explanation for observed singular value decay.

44. 基于分数阶布洛赫方程的磁共振动力学：一种混合计算框架

Magnetic Resonance Dynamics via Fractional Bloch Equation: a Hybrid Computational Framework

作者: Neetu Garg, Diptiranjan Biswal, Varsha R

类别: math.NA

发布日期: 2026-05-29

arXiv ID: 2605.30856v1

链接: http://arxiv.org/abs/2605.30856v1

摘要: Bloch equations are a powerful tool in describing the dynamics of nuclear magnetization in magnetic resonance phenomena. The fractional generalization of the Bloch equation effectively captures the anomalous relaxation and diffusion in porous, heterogeneous, and complex media. These equations describe how nuclear magnetization evolves under the influence of magnetic fields and relaxation processes. This work effectively employs a hybrid approach, the Laplace residual power series method, to investigate and analyze the fractional Bloch equation. A series solution is derived as the approximate solution for magnetization components. The influence of fractional order on each magnetization component in magnetization dynamics is analyzed and illustrated graphically. We conduct an error analysis to demonstrate the reliability and effectiveness of the proposed approach. The superiority of the suggested approach is shown using a comparative study with existing methods. The findings indicate the potential of the suggested approach as a reliable tool in understanding fractional magnetic resonance systems arising in applications such as NMR spectroscopy, MRI, MRF, and other complex heterogeneous materials.

45. 多孔弹性方程约束的分布最优控制问题

Distributed optimal control problems governed by poroelasticity equations

作者: Arbaz Khan, Jeonghun J. Lee, Harpal Singh

类别: math.OC, math.NA

发布日期: 2026-05-29

arXiv ID: 2605.30839v1

链接: http://arxiv.org/abs/2605.30839v1

摘要: In this paper, we propose and analyze a novel two-field symmetric formulation with solid displacement and fluid pressure as main unknowns for the Biot’s consolidation model in poroelasticity. Firstly, we prove the well-posedness of the new formulation and then show the existence and uniqueness of optimal control where the fluid sources in the model act as a control variable. We prove a priori error estimates for the fully discrete scheme with backward Euler time discretization and a variational approximation of the control variable. A numerical example is presented to validate the performance of the proposed novel scheme.

46. 基于分层代理和的可扩展无Ewald求和的周期性Stokes流边界积分方程框架

A scalable Ewald-free BIE framework for periodic Stokes flow via hierarchical proxy sums

作者: Tianyue Li, Dhairya Malhotra, Shravan Veerapaneni

类别: physics.flu-dyn, math.NA

发布日期: 2026-05-29

arXiv ID: 2605.30805v1

链接: http://arxiv.org/abs/2605.30805v1

摘要: Particulate Stokes flow in confined, periodic geometries underlies a broad class of problems in biophysics, microfluidics, and the rheology of complex fluids. Boundary integral equation (BIE) methods are a natural tool for such problems, but existing periodization schemes rely either on periodic Green’s functions, which are restrictive for complex confining geometries, or on free-space schemes that solve auxiliary proxy strengths alongside the surface densities in an extended linear system whose cost scales unfavorably in three dimensions. We present a BIE framework for three-dimensional particulate Stokes flow in periodic pipes with circular cross-sections, wall-bounded doubly-periodic, and triply-periodic geometries that uses only the free-space Green’s function and avoids both Ewald summation and the extended linear system. Proxy sources placed on equivalent surfaces of the kernel-independent FMM (KIFMM) form the auxiliary basis, and contributions from far image boxes are captured by a hierarchical proxy sum made absolutely convergent by a net-force-zero compatibility condition. The resulting periodization precomputation depends only on the periodic-box geometry, independent of the kernel and of the surfaces inside the box, and is reused verbatim across the Stokeslet, stresslet, and rotlet. Combined with high-order adaptive surface discretizations, the method achieves high-order accuracy at $\mathcal{O}(N)$ cost with a single layer of image boxes in the near field. Numerical examples on dense polydisperse suspensions with thousands of particles and on flow through complex periodic channels, together with strong and weak scaling studies, demonstrate efficient performance on systems with millions of degrees of freedom on distributed-memory architectures.

47. 闪电加多项式逼近：角域中奇异函数的最优根指数收敛

Lightning Plus Polynomial Approximation: Optimal Root-Exponential Convergence for Singular Functions in Corner Domains

作者: Shuhuang Xiang, Jun Xiang, Shunfeng Yang, Yuee Zhong

类别: math.NA

发布日期: 2026-05-29

arXiv ID: 2605.30796v1

链接: http://arxiv.org/abs/2605.30796v1

摘要: This work presents a rigorous convergence analysis for the lightning plus polynomial approximation scheme, which employs rational approximations constructed with tapered, exponentially clustered poles. This pole placement strategy was originally introduced by Trefethen and his collaborators for the resolution of corner singularities. We establish optimal root-exponential convergence for the class of prototype functions of the form $g(z)z^α$ or $g(z)z^α\log z$, where $g$ is analytic on a neighborhood of the solution domain. The results obtained in this work confirm the validity of Conjectures 3.1 and 5.3 stated in [SIAM J. Numer. Anal., 61:2580-2600, 2023], and demonstrate that the choice $σ_{\mathrm{opt}} =\frac{\sqrt{2(2 -β)}π}{\sqrtα}$ achieves the theoretically optimal convergence rate $\mathcal{O}\left(e^{-\sqrt{2(2 - β)Nα}π}\right)$. In particular, for the specific case $β= 0$, the proposed scheme achieves the same optimal convergence rate as the best rational approximation to $x^α$ on $[0,1]$ established by Stahl. Furthermore, working within the decomposition framework for corner domains proposed by Gopal and Trefethen, this paper provides a rigorous proof of optimal root-exponential convergence for lightning plus polynomial approximation problems, and explicitly derives the optimal pole clustering parameter.

48. 由拟熵闭合近似分子模型导出的棒状液晶张量梯度流

Tensor gradient flow for rod-like liquid crystals from molecular model with closure approximation by quasi-entropy

作者: Yongyong Cai, Jie Xu, Haixin Zhang

类别: cond-mat.soft, math.NA, physics.comp-ph

发布日期: 2026-05-29

arXiv ID: 2605.30735v1

链接: http://arxiv.org/abs/2605.30735v1

摘要: In tensor dynamics for liquid crystals derived from molecular models, a common problem is closure approximation. For rod-like molecules, the Bingham closure has proved to outperform other methods because it inherits the gradient flow structure of the molecular model, but is difficult to achieve efficient computations maintaining the gradient flow structure. We propose a closure approximation by the quasi-entropy that has been successfully applied to the free energy, based on which we construct the tensor gradient flow. The quasi-entropy closure has the same symmetry properties as the Bingham closure. The resulting tensor gradient flow is able to constrain the eigenvalues of the tensor within the physical range, guaranteeing the positive definiteness of the dissipation operator given by the higher-order tensors. The quasi-entropy closure is easy to implement since it can be reduced to minimizing an elementary function of three variables. As a result, we construct a numerical scheme preserving the eigenvalue constraints and energy dissipation, with the closure approximation decoupled from solving the scheme. Numerical simulations are carried out for the interface between the isotropic and the uniaxial nematic phase, as well as the defect evolutions, where the higher-order tensors indeed make a difference.