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1. McMg：一种用于亥姆霍兹方程的学习型相空间多通道多重网格预处理器

McMg: A Learned Phase-Space Multi-channel Multigrid Preconditioner for Helmholtz Equation

作者: Jiwei Jia, Xinliang Liu, Juntao Wang, Jinchao Xu

类别: math.NA, cs.AI, math-ph

发布日期: 2026-06-29

arXiv ID: 2606.30495v1

摘要翻译: 在高波数下求解非均匀亥姆霍兹方程仍然具有挑战性，因为离散算子是不定的，污染会降低相位精度，且标量粗网格校正可能丢弃振荡误差所携带的局部相位和传播方向信息。我们提出了多通道多重网格（McMg），一种用于非均匀亥姆霍兹方程的学习型相空间多重网格预处理器。McMg 并不直接预测解，而是在迭代框架内将残差映射为修正。其核心思想是在粗化物理空间的同时，在通道维度中保留未解析的局部波信息：每个粗节点携带一个学习到的振幅、相位、方向和散射系数包，而不是单个标量未知量。该架构结合了线性多通道转移算子、局部自适应模板、神经偏微分方程算子以及依赖于介质的光滑子，其系数由波速生成。对于固定介质，V-循环在残差中是线性的；非线性物理特征在设置阶段计算一次并缓存，因此每次在线迭代简化为固定系数的卷积。我们进一步研究了跨尺度的泛化。在小域上训练的模型可直接转移到更大的域和更高的有效波数，而逐层渐进微调（LLPF）策略通过仅添加和微调新的粗层来扩展学习到的格林算子的支持范围。在高频、高对比度和大规模三维问题上的数值实验表明，与强大的经典基线方法相比，McMg 所需的迭代次数和挂钟时间显著减少，同时始终优于现有的神经预处理器。

Comment: 26 pages, 13 figures

链接: http://arxiv.org/abs/2606.30495v1

1. 研究对象和背景

亥姆霍兹方程是描述时谐波传播的核心方程，在声学、电磁学、地球物理和医学成像等领域应用广泛。考虑非均匀介质中带有吸收边界层的亥姆霍兹方程：

$$ -\Delta u(x) - k(x)^2 u(x) = f(x), \quad x \in \mathbb{R}^D, $$

并满足 Sommerfeld 辐射条件

$$ \lim_{|x|\to\infty} |x|^{\frac{D-1}{2}} \left( \frac{\partial u}{\partial |x|} - i k_0 u \right) = 0. $$

其中 $u(x)$ 是波场，$f(x)$ 是源项，波数 $k(x) = \omega / c(x)$ 由角频率 $\omega$ 和局部波速 $c(x)$ 决定。数值计算时需截断区域并引入吸收层（海绵层或完美匹配层），以抑制人工反射。本文采用有限差分法在均匀网格上离散，得到线性系统：

$$ A u = f. $$

在高波数情形下，该离散系统面临严峻挑战：

离散算子 $A$ 强烈不定，可能接近奇异，破坏了经典多重网格方法的理论假设，导致标准松弛（如 Jacobi 或 Gauss-Seidel）失效。
污染效应 (pollution effect)：固定模板的色散误差随波数 $k$ 积累，导致相位精度随着波数 $k$ 和网格尺寸 $h$ 的乘积 $kh$ 增大而恶化，即使满足 $kh$ 恒定条件，全局误差仍可能增长。
标量粗化 (scalar coarsening) 的信息丢失：在几何多重网格中，粗网格校正假设误差分量在空间上是光滑的。但对于高波数 Helmholtz 问题，经过松弛后误差并非物理空间上的光滑函数，其局部表现为 WKB 形式 $a(x) e^{i \phi(x)}$，波矢 $\nabla \phi(x)$ 接近特征球面 $|\xi| \approx k(x)$。将物理坐标粗化会丢失局部方向、相位和散射信息，导致粗校正相位失配，可能放大而非减少误差。

针对这些困难，已发展出多种专用解法与预处理器：

移位拉普拉斯 (shifted-Laplacian) 方法在原始 Helmholtz 算子 $A$ 上添加复阻尼项 $-\imath \gamma k^2$，改善谱性质，作为 Krylov 子空间预处理器效果显著，但迭代次数仍随波数增长。
波线多重网格 (wave-ray multigrid) 将射线基校正与标量多重网格结合，增强对振荡分量的分辨能力，但实现复杂。
收敛 Born 级数 (Convergent Born Series, CBS) 通过修正 Born 展开并选取合适预处理器实现收敛，可处理强散射介质，但迭代步数仍较多。
区域分解方法 (domain decomposition) 通过子域求解和传输条件构造高效扫描型求解器，但可扩展性和健壮性仍待提高。尽管这些方法取得了重要进展，但在高波数、强非均匀介质中同时保证鲁棒性与计算效率仍然是开放问题。

近年来，基于深度学习求解 PDE 引起了广泛兴趣。物理信息神经网络 (PINN) 将物理方程嵌入损失函数，但每个新参数（如不同波数分布）需要重新训练，泛化性差。神经算子 (neural operators) 如 DeepONet、Fourier Neural Operator (FNO) 等直接学习无穷维函数空间之间的映射，克服了 PINN 的泛化限制，但受频谱偏差 (spectral bias) 影响，网络倾向于学习低频成分，难以捕捉高频模式，导致精度不足和可靠性问题。进一步出现神经求解器/预处理器 (neural solvers/preconditioners) 的思路：将神经网络作为经典迭代框架中的组件，逐步降低误差。例如 WAVE-ADR-NS 学习消除 Helmholtz 方程特定的“特征”误差分量，MGCFNN 构建层次化神经多重网格架构，混合方法将神经算子与传统迭代器交错使用以加速收敛。然而，现有神经预处理器在高波数和非均匀介质中准确性和收敛性仍会退化；同时，每次迭代中较重网络的评估开销可能抵消迭代次数的减少，导致总耗时并不优于经典方法。

本文旨在填补上述空白，提出多通道多重网格 (McMg) —— 一种学习型相空间多重网格预处理器。其核心思想是在空间上粗化的同时，利用通道维度保持未被解析的局部波信息（振幅、相位、传播方向和散射系数），以克服标量粗化带来的信息丢失。目标是验证该方法在高频、高对比度、大规模实际三维问题中能否保持高效求解性能。

2. 主要定理或主要结果

本文在以下假设和设定下报告了主要实验结论（未给出带编号的分析性定理，结论均为数值验证）：

所提出的 McMg 作为预处理器或迭代求解器，在三个高难度三维基准测试上相较于最强经典基线，实现了迭代次数的大幅减少和墙钟时间加速：
- OpenBreastUS 基准：相对于 GMRES+移位拉普拉斯多重网格，迭代次数减少 110.6 倍，墙钟加速 10.3 倍。
- OpenFWI CurveFault-B 基准：相对于 CBS，迭代次数减少 51.3 倍，墙钟加速 3.4 倍。
- Kimberlina 3D 基准：相对于稀疏 LU 分解，迭代次数减少 45.1 倍，墙钟加速 6.8 倍。
跨尺度泛化：仅在小域（或小块）上训练的 McMg 模型可以直接部署到更大域和更高有效波数问题上，无需重新训练，保持收敛性。
提出逐层渐进微调 (LLPF) 策略，可进一步扩展至超大域：冻结已训练的各层参数，仅微调新添加的最粗层，以此扩展学习型 Green 算子的长程支持，训练代价远低于完整重训，且收敛性能与全重训相当。
习得多通道粗空间的物理可解释性：
- 可视化表明，在粗层注入单位脉冲所产生的细网格响应不是局部的插值核，而是跨越约 12 个波长的振荡波包（见 Fig. 2），表明每个粗节点通过多个通道携带了丰富的方向与相位信息。
- 同一粗网格位置不同通道激发的细网格响应呈现出不同的相位、方向和干涉图案（Fig. 3a）。
- 将各通道响应向量化后的矩阵奇异值谱缓慢衰减（Fig. 3b），证明通道维度确实张成一个非冗余的多维粗步子空间，而非简单的特征维数扩充。
对固定介质，McMg 的 V-cycle 是残差的线性映射，非线性物理特征在设置阶段一次计算并缓存，因此每次在线迭代仅为固定系数的多通道卷积，极大降低了单次迭代的计算开销。
McMg 训练目标是近似离散 Helmholtz 算符的逆 $\mathrm{MG}(r) \approx A^{-1} r$。通过采样高斯随机残差 $r \sim \mathcal{N}(0, I)$，最小化相对残差损失：
$$ L_H = \frac{1}{B} \sum_{i=1}^{B} \frac{\| A^{(i)} \mathrm{MG}(k^{(i)}, r) - r \|^2}{\| r \|^2}, $$
使模型学习成为求解器。训练后的算子也可无缝嵌入 GMRES 或 Born 级数中获得 Born+McMg 变体。

3. 研究方法、关键技术和核心工具

为克服标量粗化在高波数 Helmholtz 问题中的局限，本文的核心创新是引入多通道粗空间表示，并通过以下关键技术实现高效且可解释的学习型预处理器。

1. 介质调制与线性-非线性分离 整个架构的关键设计原则是：对固定的物理介质（波数分布 $k$），V-cycle 必须是残差的线性函数，以保证良好普适性和理论简洁性；而对介质系数的依赖性可以是非线性的，以捕捉最佳离散参数与波数之间的复杂关系（如色散优化模板系数可能涉及 $\cos(kh)$ 等超越函数）。为此，定义通用运算块：

$$ c = \sigma (K_m * m), \qquad \Phi_m(x) = c \odot (K_x * x), $$

其中 $m$ 为介质系数（本文中即离散波数 $k$），$K_m$, $K_x$ 为可学习的多通道卷积核，$\sigma$ 为 GELU 激活函数，$\odot$ 表示逐元素乘法，$*$ 为多通道卷积。映射 $c = c(m)$ 体现对介质系数的非线性依赖，而 $\Phi_m(x)$ 对场 $x$ 保持线性。由于 $m$ 在求解过程中固定，可在“设置阶段”一次性计算 $c$ 并缓存；在线迭代时仅执行线性卷积，显著降低每步计算量。

2. 设置阶段与特征层次构建 对于每一网格层 $l$，生成两类缓存物理特征：

PDE 特征 $a_l$：作为高维推广的有限差分模板系数，编码局部色散修正和波速信息，用于近似该层的 Helmholtz 算子 $A_l$。
光滑特征 $s_l$：作为空间变化的松弛参数，允许求解器根据局部非均匀性动态调整光滑强度。特征层次通过非线性 CNN 构建。记 $f_{\text{lift}}$ 为提升网络，将原始波数 $k$ 映射到高维潜在特征 $a_1$；然后逐层通过 $f_l^a$（限制网络）生成更粗层的 PDE 特征 $a_{l+1}$，通过 $f_l^s$（光滑特征生成网络）由 $a_l$ 产生 $s_l$。所有 $f_l^a, f_l^s$ 均由多通道非线性卷积构成并共享参数。而层间迁移算子 $R_{l+1}^l$（限制）和 $P_l^{l+1}$（延拓）则严格为线性多通道卷积（或转置卷积），以确保求解器对残差的线性性。

3. 求解阶段：多通道 V-cycle

神经光滑：在层 $l$ 的第 $i$ 次光滑步使用迭代相关参数 $W^{(i)}$ 和缓存特征 $s_l$： $$ S_l^{(i)}(x) = W^{(i)} * \bigl( s_l \odot (K_x * x) \bigr). $$ 更新公式为 $$ e_l^{(i)} = e_l^{(i-1)} + S_l^{(i)} \bigl( r_l - A_l(e_l^{(i-1)}) \bigr), $$ 其中神经 PDE 算子 $A_l$ 定义为 $$ A_l(x) = W_l * x + a_l \odot (K_x * x). $$ 第一个线性卷积 $W_l * x$ 模仿拉普拉斯算子离散，第二项通过 $a_l$ 注入介质物理。光滑操作在高维特征空间中进行，残差、修正均为多通道张量（$c$ 通道数可设 16, 32 等），光滑子同时起到阻尼和通道混合的作用，可局部调整幅度、相位和方向。
神经限制：$r_{l+1} = R_{l+1}^l (r_l - A_l(e_l))$。$R_{l+1}^l$ 为带步长的线性卷积，将细层多通道残差投影到粗层多通道潜在空间，旨在保留那些难以在细层消除的振荡分量，并区分不同相位和方向。
神经延拓：$e_l \leftarrow e_l + P_l^{l+1} (e_{l+1})$。$P_l^{l+1}$ 为线性转置卷积，将粗层多通道修正映射回细层多通道修正。学习的延拓定义了一个多通道波包词典：一个粗网格节点可诱导多个细网格振荡响应（见 Fig. 2）。
整个 V-cycle 开始和结束处使用线性编码器 $g_{\text{lift}}$ 和解码器 $g_{\text{proj}}$ 在物理标量空间和潜空间之间转换，二者均不含非线性激活，确保线性性。

4. 训练策略 训练时采用与求解过程解耦的随机残差扰动方式：每次迭代从标准正态分布采样随机残差 $r \sim \mathcal{N}(0,I)$，利用离散算子 $A$ （随波数变化而变化）计算校正，并最小化相对残差损失 $L_H$。该方法使模型暴露于广泛的残差模态，学习近似逆算子 $A^{-1}$。

5. 多通道粗空间的物理诠释 多通道设计并非仅为增加网络容量，而是模拟高波数 Helmholtz 解的相空间结构。在几何光学近似下，Helmholtz 解局部可表示为 $a(x)e^{i\phi(x)}$，$\nabla\phi$ 接近 $k(x)$。标量粗网格节点无法区分行进方向不同的波包；而通过通道维度，每个粗节点可同时编码多个相空间分量（不同方向、相位）。实验证明，对相同粗节点位置但不同通道施加激励所产生的细网格响应各异（Fig. 3a），且通道响应矩阵的奇异值谱缓慢衰减（Fig. 3b），故而多个通道确实张成一个有效的多维度粗子空间，与解析构造的多尺度基函数（如 LOD、GMsFEM）在原理上类似，但采用端到端学习获取。

6. 逐层渐进微调 (LLPF) 为将小域训练的模型有效扩展到大域，提出 LLPF 策略：

第一步：在小域 $D_1$ 上训练具有 $L$ 层的标准 McMg 模型 $M_1$。
当域尺寸增大到 $D_k$（$k \ge 2$）时，在原有多重网格层次底部增加一个更粗层，扩展感受野以捕捉更长波长模式。
冻结所有先前已训练层的参数（保持其对局部物理的精确刻画），仅微调新增最粗层的参数。
在 $D_k$ 上生成数据训练新层，由于误差的高频成分已被冻结层处理好，新增层只需学习残余的低频全局误差。
该过程可以逐级进行，每次微调仅涉及一层，避免了全模型重训的巨大开销（包括 GPU 显存和计算时间），同时维持了优良的收敛性。

技术难点与突破总结 主要障碍在于：传统多重网格在高波数时因标量粗化丢失相位方向信息而失效；神经方法因频谱偏差难以学习高频振荡分量，且网络推理开销抵消加速效果。本文的创新之处在于：通过多通道表示保留相空间信息，利用设置阶段编码介质非线性使得迭代完全线性，实现轻量级每步推理；结合层次化特征缓存和线性迁移算子构建高效的 V-cycle；并通过 LLPF 实现高成本效益的跨尺度适应。

4. 与之前工作的比较

本文所述 McMg 方法与现有有限元-偏微分方程数值解法的相关工作在多个维度上形成对比与推进。

1. 与传统 Helmholtz 多重网格及预处理器对比

相对移位拉普拉斯多重网格 (Erlangga et al. 2004)：该类方法通过在算子中引入复偏移指数衰减，提升 Krylov 子空间预处理效果，但仍保留标量粗化。在高波数下，标量粗网格校正同样面临相位信息丢失风险，且迭代次数随波数增长明显。McMg 从结构上通过多通道保留方向信息，从而更彻底地处理振荡误差分量。文中报告的数值实验表明，在 OpenBreastUS 上相对移位拉普拉斯加速 10.3 倍且迭代数骤降。
相对波线多重网格 (Brandt and Livshits 1997, Livshits and Brandt 2006)：该方法将射线基校正与传统多重网格结合，虽能处理部分振荡误差，但实现复杂且射线基构造依赖较强背景介质假设。McMg 将射线方向信息隐式编码在学习到的通道响应中，无需显式射线追踪，同时能自适合非均匀介质，并通过设置阶段缓存使在线运算更轻量。
相对 CBS (Osnabrugge et al. 2016)：CBS 通过修正 Born 级数实现收敛，需构造参考介质和对应预处理器（如基于波数的平方根），但通常迭代步数较多。McMg 作为预处理器亦可整合进 Born 级数（Born+McMg），可大幅降低所需 Born 迭代次数。
相对区域分解方法 (Chen and Xiang 2013, Leng and Ju 2022)：区域分解的收敛性与子域划分、传输条件选择密切相关，对于高对比度介质和极高频时可能退化。McMg 不依赖子域边界，直接通过全局层次化的多通道 V-cycle 处理非均匀性。

2. 与现有神经算子/神经求解器对比

相对 PINN (Raissi et al. 2019)：PINN 每次新参数配置需重新训练，无法跨介质推广；McMg 则采用泛化的算子学习范式，训练后可应用于不同波数分布（至少同一类介质），且可通过 LLPF 扩展到大域。
相对 FNO、DeepONet 等端到端神经算子 (Li et al. 2020, Lu et al. 2021)：此类方法直接学习解映射，受频谱偏差影响难以准确捕捉高频成分，误差可能较大。McMg 不直接预测解，而是作为迭代校正作用于残差，利用多通道粗空间逐级提取各尺度误差，从而降低对高频学习的依赖。此外，FNO 等因全局谱卷积计算代价高，在大规模 3D 问题上可能受内存和计算限制；McMg 通过线性卷积缓存机制实现较低每步开销。
相对 WAVE-ADR-NS (Cui et al. 2025) 和 MGCFNN (Xie et al. 2025)：WAVE-ADR-NS 借鉴波线思想学习消除“特征”误差，MGCFNN 构造神经多重网格并在粗层采用 FNO。两者都局限于标量场处理，未能充分解决方向信息的丢失。McMg 通过多通道设计从根本上改进了粗空间表达，实验表明其在高波数、高对比度条件下保持更健壮的收敛。此外，文中明确比较了 WAVE-ADR-NS 与 McMg 的每步开销：前者网络较重，后者由于缓存使在线迭代仅为轻量卷积，即使迭代次数接近，墙钟时间也有明显优势。
相对 NPBS (Wang et al. 2026)：NPBS 提出神经预条件 Born 序列，McMg 可适配该框架形成 Born+McMg，但 McMg 本身结构上的多通道特性提供了更丰富的粗校正能力。

3. 多尺度方法视角的对比

相对模拟 LOD 和 GMsFEM 等解析多尺度方法 (Gallistl and Peterseim 2015, Chung et al. 2014)：这些方法通过粗节点上附加局部特征问题求解获得振荡多尺度基函数，本质上也是通过增加每个粗节点自由度来捕获细尺度信息，与 McMg 的多通道思想相通。然而，这些函数的构造通常依赖于局部椭圆算子或简化模型，不一定完美适用于强不定 Helmholtz 算子；且离线计算每个基函数可能代价较高。McMg 用端到端学习替代解析构造，直接从训练数据中学习最优的局部波包词典，并能自适应介质。可视化和奇异值分析表明学习到的通道基确实是有效的。

4. 跨尺度泛化与训练效率

多数神经算子方法在域尺寸扩大时需要重新训练或至少进行大量微调，否则性能急剧下降。McMg 则展示了直接跨尺度泛化（小域模型直接用于大域）的能力，这得益于其层次化结构和介质相关特征的正确解耦。
与完全重训练相比，LLPF 方法只需微调新增最粗层，避免了对所有层的梯度传播，显著降低了训练时间和内存，同时保持了收敛性能，这对于数百波长规模的问题尤为关键。文中指出，LLPF 在保持收敛性的同时将训练开销削减至全重训练的一小部分，使 McMg 能够经济地处理更大规模。

5. 局限与代价 从提供内容看，作者未显式展开讨论方法的局限，但可合理推断：

训练阶段需要针对特定介质分布生成数据并优化，尽管有 LLPF，但面对完全迥异的介质类型（如从声速连续变化到高度散射随机介质）可能仍需重新训练基础模型。
多通道表示增加了内存需求（缓存多个特征张量），对于极大 3D 网格可能对显存造成压力。
尽管每步线性卷积廉价，但相比标准两次光滑一次校正的标量 V-cycle，计算量仍高，尤其在通道数较多时。文中报告的速度提升是在与最优经典基线对比下测得的，但未直接与极优化纯标量多重网格（可能更快但发散）比较。
学习过程得到的是近似逆，理论上无法保证与精确离散逆一样稳定，但训练目标最小化相对残差和 GMRES 内嵌使用提供了保障。

2. 一种基于迹的高传导界面降维方法

A Trace-Based Interface Reduction Method for Highly Conducting Interfaces

作者: So-Hsiang Chou, Do Young Kwak

类别: math.NA

发布日期: 2026-06-27

arXiv ID: 2606.28700v1

摘要翻译: 我们针对具有高传导界面的椭圆界面问题，提出了一种降维界面格式。界面条件包括主变量的连续性以及法向通量跳跃与界面迹的表面拉普拉斯算子成正比。我们采用公共界面迹作为界面未知量，而非采用解跳跃作为界面未知量，并推导了基于迹的Schur补格式。对于给定的界面迹数据，在两个子域中求解独立的延拓问题，从而导出涉及Dirichlet-to-Neumann跳跃算子和表面刚度算子的降维界面方程。有限维迹近似产生仅定义在界面上的紧凑降维系统。针对圆形、光滑非圆形和心形界面的数值实验验证了该方法的有效性以及界面模态富集的作用。

Comment: 15 pages, 5 figures

链接: http://arxiv.org/abs/2606.28700v1

1. 研究对象和背景

本文研究一类带有高导电界面的椭圆型界面问题。令区域 $\Omega \subset \mathbb{R}^d$，$d=2,3$，被光滑界面 $\Gamma$ 划分为子区域 $\Omega^-$ 和 $\Omega^+$。控制方程为

$$-\nabla\cdot(\beta^\pm \nabla u^\pm) = f^\pm \quad \text{在 } \Omega^\pm \text{ 内},$$

界面条件为

$$[u]=0 \quad \text{在 } \Gamma \text{ 上}, \qquad [\beta \partial_n u] = \alpha \Delta_S u_\Gamma \quad \text{在 } \Gamma \text{ 上},$$

并附加齐次 Dirichlet 边界条件 $u=0$ 在 $\partial\Omega$ 上。这里 $[\cdot]$ 表示跨越界面的跳跃，$u_\Gamma$ 表示解的公共界面迹，$\Delta_S$ 为作用在界面切向上的表面 Laplace 算子，$\alpha>0$ 为常数，$\beta^\pm \geq \beta_0>0$ 为分区常数或函数。这类条件来源于复合材料、薄膜扩散、高导电薄层等的渐近模型，在薄层厚度趋于零时，界面上的有效条件会出现表面微分算子，从而同时耦合了体椭圆问题和界面上的切向扩散。

从数值计算角度看，含表面微分算子的界面问题比经典传输问题更具挑战，因为界面条件本身包含沿界面的切向导数，需要界面迹具有更高的正则性。已有的数值方法主要包括贴体网格、非贴体网格、浸入式和非协调有限元方法等，它们通常将界面条件直接嵌入全局体离散格式中。此前，作者团队曾针对非线性或椭圆界面问题提出若干界面约化框架，但这些工作处理的是经典传输条件，其约化变量为标量参数或解的跳跃。对于本文所考虑的高导电界面模型，由于解在界面处连续，自然的界面未知量是公共迹而非跳跃，这催生了基于迹的 Schur 补约化思路。

本研究试图填补的空白在于：将界面约化思想拓展到含有表面微分算子的传输问题，建立以界面迹为核心未知量的降维方程，并提供一种仅依赖于界面低维逼近的数值框架。这一框架将体问题解耦为独立子区域上的椭圆延拓，同时将表面扩散项完全限制在界面的小规模系统中，从而为低维界面逼近和模态解释提供了自然基础。

2. 主要定理或主要结果

本文的主要结果包含以下推导和数值发现：

约化界面方程的严格推导。定义界面迹 $\gamma = u_\Gamma$，将解在子区域上分解为调和延拓与体积力贡献之和：$u^\pm(\gamma) = H^\pm \gamma + U_f^\pm$，其中 $H^\pm: H^{1/2}(\Gamma)\to H^1(\Omega^\pm)$ 为椭圆延拓算子，满足
$$\begin{aligned} -\nabla\cdot(\beta^-\nabla(H^-\gamma)) &= 0 \; \text{在 } \Omega^-, \quad H^-\gamma = \gamma \; \text{在 } \Gamma, \\ -\nabla\cdot(\beta^+\nabla(H^+\gamma)) &= 0 \; \text{在 } \Omega^+, \quad H^+\gamma = \gamma \; \text{在 } \Gamma, \end{aligned}$$
且在外边界处均为零；$U_f^\pm$ 则满足相同方程和齐次界面迹条件。定义 Dirichlet-to-Neumann 跳跃算子 $S\gamma := [\beta \partial_n (H\gamma)]$ 及体积力导致的界面源项 $g_\Gamma := -[\beta \partial_n U_f]$，则原始的体界面问题可以等价转化为仅定义在界面上的方程
$$S\gamma - \alpha \Delta_S \gamma = g_\Gamma \quad \text{在 } \Gamma \text{ 上}. \tag{20}$$
该方程称为 约化界面方程，它将体的耦合转化为界面迹的单一方程，完全避开了体网格对界面的依赖。
有限维迹逼近与代数系统。取有限维迹空间 $\Lambda_m^\gamma = \operatorname{span}\{\psi_1,\dots,\psi_m\} \subset H^1(\Gamma)$，将界面迹近似为 $\gamma_m = \sum_{j=1}^m s_j \psi_j$。在变分形式下使用表面分部积分，得
$$\langle [\beta \partial_n H\gamma_m], \psi_i \rangle_\Gamma + \alpha \langle \nabla_S \gamma_m, \nabla_S \psi_i \rangle_\Gamma = \langle g_\Gamma, \psi_i \rangle_\Gamma, \quad i=1,\dots,m.$$
代入展开式得到线性系统
$$(S_m + \alpha L_m) s = b_m, \tag{22}$$
其中
$$(S_m)_{ij} = \langle [\beta \partial_n H\psi_j], \psi_i \rangle_\Gamma, \quad (L_m)_{ij} = \langle \nabla_S \psi_j, \nabla_S \psi_i \rangle_\Gamma, \quad (b_m)_i = \langle g_\Gamma, \psi_i \rangle_\Gamma.$$
值得注意的是，$S_m$ 来源于体延拓算子的响应，$L_m$ 是纯界面上的表面刚度矩阵，两者可以独立组装。
圆形界面的模态富集与精确恢复。对半径为 $R$ 的圆形界面，表面 Laplace 算子化为 $\Delta_S = \frac{1}{R^2}\frac{d^2}{d\theta^2}$，Fourier 模态 $\psi_k = e^{ik\theta}$ 满足 $\Delta_S \psi_k = -\frac{k^2}{R^2}\psi_k$。当延拓算子 $H^\pm$ 的系数也呈径向对称时，$S$ 在同一基底下对角化，约化方程完全解耦：$(\sigma_k + \alpha k^2/R^2) s_k = b_k$。数值实验对制造解 $[\beta\partial_n U_f] = 0.5\cos\theta + 0.25\sin(2\theta)$ 进行测试，表 1 显示：当 $m=1$ 时误差为 $\sim 10^{-2}$；$m=3$ 时捕捉到 $\cos\theta$ 模态，误差降至 $\sim 10^{-2}$；当迹空间维数增至 $m=5$（包含 $\cos(2\theta),\sin(2\theta)$）时，误差直接降至机器精度（$\sim 10^{-17}$）。这证明一旦界面逼近空间包含了所有活跃模态，约化解即可精确恢复。
星形光滑非圆界面。考虑极坐标 $r(\theta)=r_0+\varepsilon \sin(k\theta)$ 的星形界面，本例中 $r_0=0.5, \varepsilon=0.08, k=5$。此时表面弧长微元 $ds = q(\theta)d\theta$，切向导数 $d/ds = (1/q(\theta)) d/d\theta$，表面刚度矩阵变为
$$(L_m)_{ij} = \int_0^{2\pi} \frac{\psi_j'(\theta)\psi_i'(\theta)}{q(\theta)}\,d\theta.$$
即便 Fourier 基不再对角化表面算子，表 2 显示随着 $m$ 从 1 增加到 17，误差从 $\sim 10^{-1}$ 逐渐衰减到 $\sim 10^{-2}$。衰减速度慢于圆形情况，原因是几何度量因子 $q(\theta)$ 耦合了不同 Fourier 模态。这一行为符合预期，也表明界面富集仍能有效提升精度，而几何复杂性仅体现于界面上的小规模矩阵。
心形界面示例。文中进一步考虑了心形界面 $\Gamma=(x(\theta),y(\theta))$（参数方程在 (29) 给出），用以展示方法在更复杂几何下的鲁棒性。从提供内容中无法确认具体误差数据，但作者指出迹约化思想依然适用，独立体延拓与界面低维系统的框架保持不变。

注意：本文未给出严格的先验误差估计或收敛阶理论分析，主要贡献在于推导降维方程和数值验证。

3. 研究方法、关键技术和核心工具

本文的核心创新在于将界面未知量由传统的解的跳跃改为公共界面迹，并以此构造 Schur 补型式的约化方程。关键技术障碍与分析工具阐述如下：

1. 解分解与延拓算子的构造
突破点在于显式分离界面贡献与体积力贡献。定义迹 $\gamma$ 后，将子区域解写为 $u^\pm(\gamma) = H^\pm\gamma + U_f^\pm$，其中 $H^\pm$ 将给定的界面迹延拓到子区域上且内部满足齐次椭圆方程（调和延拓），$U_f^\pm$ 仅由体积力驱动且界面迹为零。这两个问题都是标准的 Dirichlet 边值问题，可以在各自子区域上独立求解，从而完全解耦。这一分解将原问题的复杂性集中到满足界面方程的迹 $\gamma$ 上。

2. Dirichlet-to-Neumann 跳跃算子的约化方程
将上述分解代入法向通量跳跃条件 $[\beta\partial_n u] = [\beta\partial_n (H\gamma)] + [\beta\partial_n U_f]$，并令其等于 $\alpha\Delta_S\gamma$，便得到仅涉及 $\gamma$ 的方程 $S\gamma - \alpha\Delta_S\gamma = g_\Gamma$。这里 $S\gamma := [\beta\partial_n (H\gamma)]$ 是 Dirichlet-to-Neumann 跳跃算子，它刻画了延拓解的界面通量响应；$-\alpha\Delta_S$ 是高导电界面引入的额外表面刚度；$g_\Gamma$ 为已知的体积力诱导项。该方程完全在低维界面流形上定义，消除了体网格与界面网格的耦合。

3. 表面积分与有限维实现
为数值求解约化方程，采用 Galerkin 投影：选取 $H^1(\Gamma)$ 的子空间 $\Lambda_m^\gamma$，测试函数 $\psi_i$。利用表面分部积分公式 $\langle -\Delta_S\gamma_m, \psi_i\rangle_\Gamma = \langle \nabla_S\gamma_m, \nabla_S\psi_i\rangle_\Gamma$，避免直接离散二阶表面 Laplace 算子，仅需计算切向梯度。最终组装矩阵 ${(L_m)_{ij} = \langle\nabla_S\psi_j,\nabla_S\psi_i\rangle_\Gamma}$，该矩阵仅依赖于界面几何和基函数，计算成本极低。对于圆形界面，弧长微元 $ds = R d\theta$，$\nabla_S = \frac{1}{R}\frac{d}{d\theta}$，故 $L_m$ 简化为

$$(L_m)_{ij} = \frac{1}{R} \int_0^{2\pi} \psi_j'(\theta) \psi_i'(\theta) \,d\theta.$$

对于一般参数化界面，仅需计算度量因子 $q(\theta) = \sqrt{r(\theta)^2 + r'(\theta)^2}$，并将积分公式调整为 $(L_m)_{ij} = \int_0^{2\pi} \frac{\psi_j'(\theta)\psi_i'(\theta)}{q(\theta)}\,d\theta$，完全不涉及曲面网格或协变导数等复杂工具。

4. 基于模态的界面基选择与几何解耦
作者充分利用了界面参数的角坐标表示。对圆形界面，Fourier 基使得 $S$ 和 $\Delta_S$ 完全对角化，系统退化为模态解耦的标量方程，这解释了为什么 $m$ 达到活跃模态数目后误差可骤降至机器精度。对于一般星形或心形界面，虽不能对角化，但 Fourier 基仍提供全局光滑逼近，且基函数与几何的相互作用仅体现在低维矩阵 $L_m$ 中。这一设计将几何复杂性“封锁”在界面的小规模计算中，而体延拓问题无需随界面形状改变而调整离散策略，这是该方法的重要计算优势。

5. 理论基础与潜在分析缺口
文中提到为保表面梯度有意义需 $u_\Gamma\in H^1(\Gamma)$，并指出在 $\beta^\pm\geq\beta_0>0, \alpha>0$ 的条件下可使用迹不等式和 Lax–Milgram 理论处理适定性，但未展开。这表明本文侧重方法推导与数值验证，严格的理论分析（如逼近误差估计、$m$ 收敛阶等）留作未来工作。

总体而言，核心分析框架为：拆分 $\to$ 延拓 $\to$ 约化方程 $\to$ 表面变分离散 $\to$ 小规模代数系统。创新点在于引入迹变量替代跳跃，从而自然地导出 Dirichlet-to-Neumann 与表面 Laplace 的耦合结构，并结合界面参数化实现几何无关的降维。

4. 与之前工作的比较

与现有有限元与偏微分方程数值解法相关文献的比较可从以下几个维度展开：

1. 与经典界面问题数值方法的差异
传统的贴体或不贴体有限元方法（如 [4,5,9–11]）通常将界面条件直接嵌入到全局体离散格式中，例如通过在界面附近修改基函数、采用额外惩罚项或 Lagrange 乘子等。这类方法需要在界面附近精细处理网格和数值积分，尤其当界面含表面微分算子时，还需在界面上构造切向离散，导致耦合系统规模大且实现复杂。本文的迹约化方法则从根本上不同：它将耦合问题转化为两个独立的体边值问题和一个仅定义在界面上的低维方程，体求解器可采用现有标准代码，界面矩阵的组装只涉及一维积分，因此极大降低了实现难度和计算开销。

2. 与作者之前界面约化工作的区别
作者在 [1,6,7] 中已提出界面约化框架，但那些工作处理的是经典传输条件（如解的跳跃正比于法向通量、$[u]=\lambda$ 或 $[\beta\partial_n u]=g$ 等），其约化变量为标量参数或解的跳跃 $[u]$。本文首次将界面约化思想推广至高导电界面模型，其中界面条件包含表面 Laplace 算子，且解的跳跃为零。这一本质区别导致约化变量置换为公共迹 $\gamma$。该置换并非简单改写，而是带来了新的 Schur 补结构：方程中出现 Dirichlet-to-Neumann 跳跃算子 $S$ 与表面刚度矩阵 $L_m$ 的求和，而在跳跃变量框架下通常是解跳跃通过某种 Steklov–Poincaré 算子与通量关联。因此，本文拓展了界面约化的适用范围。

3. 对几何处理与界面基函数的灵活性
现有浸入界面方法（如 IIM）或扩展有限元（XFEM）在处理曲面中的微分算子时常需引入特定界面插值函数或局部坐标。本文的方法完全不要求体网格与界面相匹配，基函数可直接定义在界面参数域上（如角坐标 $\theta$），并且对于圆形界面可精确解耦模态，对于一般光滑界面则采用全局 Fourier 基，仍能通过增加界面秩逐步改善精度。这种与体离散解耦的策略使得方法对界面几何非常柔性，仅需界面参数化，不需构造界面专门的贴体网格。

4. 精度与收敛行为
圆形实验表明，当界面迹空间包含所有活跃模态时，误差立即降至机器精度。这一“单次准确”行为在以往界面富集方法中并不常见。星形界面实验则显示几何扰动会引入模态耦合，导致误差随 $m$ 缓慢衰减。虽然本文未给出理论收敛率，但从数值表现看，其收敛性质与界面参数化的光滑性和基函数的逼近性质相关，这为未来分析指明了方向。相比之下，传统的界面有限元方法通常按网格尺寸 $h$ 的幂次收敛，且界面奇性可能导致降阶；本文的界面秩 $m$ 类似于谱方法中的截断，可能对光滑数据具备高精度特性。

5. 计算成本与适用范围
本文方法的主要成本在于求解 $m$ 次独立的体延拓问题（每个基函数一次），但这些问题彼此完全独立，可并行计算，且不依赖界面离散细节。当界面秩 $m$ 远小于体未知数时，整体成本主要由体求解主导。作者提及方法适用于二维和三维，但从提供的内容中未给出三维算例。此外，当前框架仅考虑了分区常数或较简单系数的体方程，且界面是光滑闭曲线。对于包含角点、系数剧烈变化或非线性体方程等复杂情况，方法的有效性和稳定性有待进一步研究，文中未展开讨论。

6. 与薄层渐近模拟方法的关联
高导电界面条件通常由解析渐近展开得出（如 [2,3,8,12]），传统数值模拟要么直接解析薄层（需极细网格），要么近似求解近似的界面条件。本文直接基于有效的界面条件（含 $\Delta_S$）建模，避免了跨尺度网格问题，属于后者。与同类文献相比，其特色在于提供了一种系统且易于实施的降维求解框架，而非针对特定几何或特定离散的拼凑方案。

了解领域论文（摘要翻译）

1. 连续级Runge–Kutta方法秩条件的多项式矩方法

A polynomial moment approach to a rank condition for continuous-stage Runge–Kutta methods

作者: Yuto Miyatake

类别: math.NA

发布日期: 2026-06-29

arXiv ID: 2606.30122v1

链接: http://arxiv.org/abs/2606.30122v1

摘要翻译

中文标题: 连续级Runge–Kutta方法秩条件的多项式矩方法

摘要翻译: 在哈密顿系统保能量方法的研究中，多项式连续级 Runge–Kutta 方法起着重要作用。这类方法保能量的充要条件已经建立。如果定义方法的矩阵 $M\in \mathbb{R}^{s\times s}$ 对称，则它们保能量；反之，在某一 $s\times \infty$ 矩阵 $\Phi^\mathrm{CSRK}$ 行满秩的假设下，逆命题成立。Miyatake 和 Butcher（SIAM J. Numer. Anal., 2016）的注记3中曾猜想，对于每个相容的多项式连续级 Runge–Kutta 方法，该满秩假设应当总是成立。本文通过证明在标准相容性条件下矩阵 $\Phi^\mathrm{CSRK}$ 行满秩，证实了这一猜想。证明直接应用了由 Pakovich 和 Muzychuk（Proc. Lond. Math. Soc., 2009）解决的多项式矩问题。

2. 基于极化的离散化方法的守恒量

Conserved quantities of discretizations by polarization

作者: Noah Gießing, Yuri B. Suris

类别: nlin.SI, math.NA

发布日期: 2026-06-28

arXiv ID: 2606.29263v1

链接: http://arxiv.org/abs/2606.29263v1

摘要翻译

中文标题: 基于极化的离散化方法的守恒量

摘要翻译: 最近，基于向量场的极化，提出了一族用于具有多项式向量场的高阶常微分方程的非传统积分器。最简单的例子是如今闻名的用于具有二次向量场的一阶常微分方程的Kahan离散化。所有这些积分器都具有显著的守恒性质。特别地，对于一阶和二阶哈密顿常微分方程，基于极化的离散化具有一个运动积分和一个不变体积形式。在本文中，我们将先前提出的用于推导这些积分的代数方法推广到任意阶常微分方程的离散化。对于所有 $\ge 3$ 阶，这些积分都是新的。

3. 基于Simpson-Boole几何恒等式的曲面扩散二阶保面积/体积参数有限元方法

Second-Order Area/Volume-Preserving PFEMs for Surface Diffusion via Simpson–Boole Geometric Identities

作者: Zhiqing Pan, Jiwei Jia, Lian Zhang

类别: math.NA

发布日期: 2026-06-28

arXiv ID: 2606.29195v1

链接: http://arxiv.org/abs/2606.29195v1

摘要翻译

中文标题: 基于Simpson-Boole几何恒等式的曲面扩散二阶保面积/体积参数有限元方法

摘要翻译: 我们针对二维空间中的闭曲线和三维空间中的闭曲面，提出了用于曲面扩散的时间二阶参数有限元方法。该方法基于沿二次时间插值路径的精确几何变分恒等式进行构造，二维情形下的诱导面积变分通过Simpson法则精确计算，三维情形下的诱导体积变分则通过Boole法则精确计算。所得全离散格式精确保持封闭面积或体积，且无需为几何约束引入辅助Lagrange乘子。这些格式可在BGN预测的辅助几何上进行组装，因而与现有的二阶BGN型格式兼容。数值实验展示了曲线和曲面演化中预期的二阶收敛性、面积/体积守恒性以及良好的网格质量。

4. 动区域上Stokes方程的压力鲁棒ALE时空DG方法

Pressure-robust ALE space-time DG method for the Stokes equations on moving domains

作者: L. Beirão da Veiga, S. Gómez, K. B. Haile

类别: math.NA

发布日期: 2026-06-27

arXiv ID: 2606.28914v1

链接: http://arxiv.org/abs/2606.28914v1

摘要翻译

中文标题: 动区域上Stokes方程的压力鲁棒ALE时空DG方法

摘要翻译: 我们在任意拉格朗日-欧拉框架下，针对动区域上的不可压缩Stokes方程，提出并分析了一种时空间断伽辽金方法。我们在离散速度空间的定义中使用逆变Piola映射，以在离散层面上保持逐点无散度性质。我们证明了该方法在空间网格和时间剖分上无约束条件下是inf-sup稳定的。我们还建立了针对空间和时间任意逼近阶数的能量范数先验误差估计。对于时间上的分片常数和分片线性逼近，我们证明了该方法在低粘性区域也是鲁棒的，并提供了数值证据表明这一性质也扩展到高阶情况。我们给出了几个数值实验来验证理论结果。

5. 超越原生空间的核逼近——及其在流形逼近中的应用

Kernel approximation beyond the native space – with applications to approximation on manifolds

作者: Thomas Hangelbroek, Christian Rieger, Grady B. Wright

类别: math.CA, math.NA

发布日期: 2026-06-26

arXiv ID: 2606.28564v1

链接: http://arxiv.org/abs/2606.28564v1

摘要翻译

中文标题: 超越原生空间的核逼近——及其在流形逼近中的应用

摘要翻译: 本文处理嵌入在$\mathbb{R}^N$中的流形上的核逼近与插值问题，使用的是正定和条件正定核的限制。主要挑战在于发展一种逼近理论，该理论处理的是相对于核在高正则光滑性空间中度量的误差。这意味着光滑性的阶数高于核的关联原生空间（在正定情形下，即由核生成的再生核希尔伯特空间）的阶数。这使得在此框架下控制误差的标准技术无法使用，尤其是诸如插值投影算子的正交性等再生核希尔伯特空间论证，或利用{\em power function}的界估计。我们推广了DeVore和Ron引入的逼近格式，该格式处理的是位于核积分算子值域中的目标函数。对于嵌入流形的情形，由于最近针对$\mathbb{R}^N$的某些子流形发展出的局部多项式再生性质，这一推广现已可行。此外，我们给出了关于核与流形的充分条件，使得积分算子的值域可被精确识别：特别地，保证该值域是一个Sobolev空间。最后，我们针对嵌入流形提出了新的基于核的Bernstein不等式，从而得到了关于紧嵌入于原生空间中的Sobolev空间内插值的误差估计。

不相关论文（基本信息）

1. 齐次深度网络中持续学习的收敛性

Convergence of Continual Learning in Homogeneous Deep Networks

作者: Matan Schliserman, Gon Buzaglo, Itay Evron, Daniel Soudry

类别: cs.LG, math.NA, math.OC, stat.ML

发布日期: 2026-06-29

arXiv ID: 2606.30559v1

链接: http://arxiv.org/abs/2606.30559v1

摘要: We characterize weakly regularized continual classification in homogeneous models as sequential projections onto task margin sets. This result generalizes prior analyses restricted to either stationary (single-task) deep models or continual linear models. We show that global convergence generally fails, even for simple models linear in data but nonlinear in parameters. Nevertheless, by leveraging results from nonconvex projection theory, we identify regularity properties of homogeneous deep networks that guarantee local linear convergence under random and cyclic task sequences. Finally, we extend our analysis to continual regression, unifying the framework for homogeneous models.

2. 通过透视映射与盒子分解计算积分R2指标

Computing the Integral R2 Indicator by Perspective Mapping and Box Decomposition

作者: Michael T. M. Emmerich

类别: cs.CG, cs.NE, math.NA, math.OC

发布日期: 2026-06-29

arXiv ID: 2606.30530v1

链接: http://arxiv.org/abs/2606.30530v1

摘要: The continuous integral R2 indicator is a Pareto-compliant refinement of the classical finite-weight-vector R2 indicator, used in performance assessment, bounded archiving for a-posteriori multi-objective optimization, and skyline selection in databases. This work introduces a bidirectional perspective mapping between continuous integral R2 computation and integration over unions of anchored axis-aligned boxes. After translating the ideal point of a minimization problem to the origin, approximation points become strictly positive loss vectors, and the subgraph of the lower weighted Tchebycheff envelope over the weight simplex maps to the complement of an anchored-box union in reciprocal objective space. The Jacobian gives an absolute R2 formula as a weighted complement volume with density $(x_1+\cdots+x_N)^{-(N+1)}$, while differences of R2 values become finite weighted hypervolume differences. Hence, hypervolume algorithms that emit box decompositions can be reused by replacing ordinary box volumes with closed-form weighted box integrals. For $N$ objectives, this gives an output-sensitive overhead $O(2^N M)$ for an $M$-box decomposition, or $O(M)$ for fixed $N$. Using existing box-decomposition approaches, the integral R2 can be computed in $O(n \log n)$ for $N=2,3$, in $O(n^2)$ for $N=4$, and in $O\left(n^{\lfloor (N-1)/2\rfloor+1}\right)$ for $N\geq4$, with $n$ denoting the size of the approximation set. On the lower-bound side, exact value computation has an $Ω(n\log n)$ lower bound in the algebraic decision-tree model already in two objectives, this bound lifts to every fixed $N\geq2$, and exact computation is $\#P$-hard when $N$ is part of the input. Together, the proposed perspective mapping provides a powerful tool for transferring algorithmic and structural results between anchored-box union and hypervolume theory and integral R2 computation.

3. 大型线性问题的迭代Tikhonov正则化

Iterated Tikhonov regularization of large linear problems

作者: Davide Furchì, Lothar Reichel

类别: math.NA

发布日期: 2026-06-29

arXiv ID: 2606.30448v1

链接: http://arxiv.org/abs/2606.30448v1

摘要: Many solution methods for linear discrete ill-posed problems with error-contaminated data (right-hand side) apply Tikhonov regularization to compute a meaningful approximate solution. This solution depends on a regularization parameter. It is well known that iterated Tikhonov regularization often determines an approximate solution of higher quality than (standard) Tikhonov regularization. We consider the situation when an estimate of the norm of the error in the data is known and would like to apply iterative Tikhonov regularization to determine an approximate solution that satisfies the discrepancy principle. This requires a suitable choice of a regularization parameter. The standard approach to determine this parameter is to compute solutions for several values of the regularization parameter and choose a computed approximate solution that satisfies the discrepancy principle. This paper discusses iterated Tikhonov regularization based on partial Golub-Kahan bidiagonalization and describes how the regularization parameter can be determined without computing several approximate solutions by using the connection between Golub-Kahan bidiagonalization and Gauss quadrature. This approach reduces the computational effort required to compute a desired solution.

4. 周期性不可压缩斯托克斯流的量子谱求解器

A Quantum Spectral Solver for Periodic Incompressible Stokes Flow

作者: Juan M. Gimenez, Fehmi Cirak, Michael Ortiz

类别: math.NA

发布日期: 2026-06-29

arXiv ID: 2606.30447v1

链接: http://arxiv.org/abs/2606.30447v1

摘要: We present a quantum spectral solver for the steady incompressible Stokes equations on a two-dimensional periodic domain. The method uses the Quantum Fourier Transform as a coherent change of basis and exploits the resulting spectral structure of the Stokes operator: the Laplacian becomes diagonal, while incompressibility is enforced mode by mode through a Helmholtz projection. In two dimensions, this projection is realized by a mode-dependent rotation from Cartesian velocity components to longitudinal–transverse coordinates, followed by component-conditioned inverse-Laplacian scaling. The velocity and pressure fields are encoded as quantum states over Fourier modes and physical components, and the corresponding spectral factors are implemented through polynomially encoded amplitude blocks. The construction extends recent quantum spectral methods in computational mechanics to an incompressible flow operator with explicit pressure–velocity splitting and divergence-free projection. The approach is also compatible with multiscale finite-element architectures in which quantum parallelism can simultaneously update all representative volume element (RVE) states. Numerical verification includes a steady vortex, a regularized periodic force-dipole benchmark, and an RVE-inspired Kolmogorov-like fluctuation benchmark. The latter illustrates how the circuit can recover a homogenized kinetic-energy observable without reconstructing the full velocity field, consistent with the role of averaged quantities in multiscale flow calculations. Under the standard assumptions of efficient state preparation and observable estimation, the circuit has polylogarithmic dependence on the grid resolution, with the polynomial degree and tile count appearing as explicit approximation and implementation parameters.

5. 使用 LattE 计算筛积分及具有局部化除数的整数的密度

Computing sieve integrals using LattE, and the density of integers with a localized divisor

作者: Sary Drappeau, Adrien Mounier

类别: math.NT, math.NA

发布日期: 2026-06-29

arXiv ID: 2606.30428v1

链接: http://arxiv.org/abs/2606.30428v1

摘要: We consider the problem of estimating numerically integrals of the shape

$$ \int_P \frac{dt}{t_1 \dotsb t_k} $$

where $P \in {\mathbb R}_{>0}^k$ is a convex polytope, $t=(t_1,\dotsc, t_k)$ and $d t$ is the Lebesgue measure. This type of integral appears frequently in main terms of sieve theory. We propose a simple method, based on the LattE software for integration of polynomials over polytopes, which computes rigorous bounds on this integral in polynomial time with respect to the precision (in bits). We test the method on several examples from the literature of sieve theory. We apply our results to compute numerical approximations to the natural density

$$ h(α, β) := \operatorname{density}\{n\in{\mathbb N}, \exists d\mid n, d\in [n^α, n^β]\}, \qquad (0<α<β<1) $$

of integers having a localized divisor, in the region $β- α\geq 0.02$. One ingredient involved is a refined formula for $h(α, β)$ which involves a manageable number of terms for these $α, β$. As a corollary, we give a numerical approximation of the leading constant in a theorem of Haddad and Koukoulopoulos on the average of the logarithm of middle-divisors of integers.

6. 多孔弹性介质中散射问题的 PML 方法的收敛性

Convergence of the PML method for scattering problems in poroelastic media

作者: Qianyuan Yin, Changkun Wei, Bo Zhang

类别: math.AP, math.NA

发布日期: 2026-06-29

arXiv ID: 2606.30392v1

链接: http://arxiv.org/abs/2606.30392v1

摘要: This paper is concerned with the time-harmonic wave scattering problems in three dimensional poroelastic media. By introducing an intermediate variable $p$, the original $\mathbf{u}-\mathbf{w}$ system is equivalently transformed into a $\mathbf{u}-p$ system with fewer degrees of freedom, which facilitates the derivation of the fundamental solution, Green’s identity and positivity of the complex wave numbers. A perfectly matched layer (PML) method is then introduced in the spherical coordinates to truncate the unbounded scattering problem. Under certain assumptions on the poroelastic and PML parameters, we prove the existence and uniqueness of solutions to the PML problems both in the truncated domain and layer. Moreover, the exponential convergence of the PML method is established in terms of the thickness and parameters of the PML layer. The proof is based on the PML extension and the exponential decay properties of the stretched fundamental solution. As far as we know, this is the first convergence result of the PML method for poroelastic scattering problems.

7. 基于伴随方法的贝叶斯不确定性量化用于PDE约束反问题及其在半导体成像中的应用

Adjoint-Based Bayesian Uncertainty Quantification for PDE-Constrained Inverse Problems with Application to Semiconductor Imaging

作者: Hassan Yazdanian, Leila Taghizadeh, Babak Maboudi Afkham

类别: math.NA, math.AP

发布日期: 2026-06-29

arXiv ID: 2606.30340v1

链接: http://arxiv.org/abs/2606.30340v1

摘要: We formulate a Bayesian framework for reconstructing doping profiles in pn-junction semiconductor devices from boundary flux measurements. The unknown doping field is modeled as a piecewise-constant function characterized by an unknown interface and two plateau concentrations, leading to a nonlinear ill-posed inverse problem governed by a Poisson-Boltzmann-type equation. To represent this structure while enabling efficient gradient-based inference, we introduce a pushforward prior constructed by mapping a latent Gaussian field with Matérn-type covariance through a sigmoid transformation. The latent field is parameterized by a truncated Karhunen-Loève expansion, while the two piecewise-constant levels are represented by scalar plateau parameters. The prior yields differentiable approximations of piecewise-constant fields with controllable interface sharpness. We establish well-posedness of the Bayesian formulation by proving Lipschitz continuity of the forward map and Hellinger stability of the posterior. We then sample the posterior using the No-U-Turn Sampler (NUTS) with gradients computed by the adjoint method. Numerical experiments show that the combination of the proposed prior and NUTS provides more efficient posterior exploration than the dimension-robust preconditioned Crank-Nicolson (pCN) sampler, yielding one to two orders of magnitude larger effective sample sizes. In the known-plateau setting, the method reconstructs both planar and curved interfaces and provides spatially resolved uncertainty quantification (UQ). When the interface geometry and plateau concentrations are inferred jointly, posterior correlations reveal structural non-identifiability. These results demonstrate the effectiveness of combining pushforward priors with adjoint-gradient-based sampling for reliable UQ in nonlinear partial differential equation-constrained inverse problems with sharp interfaces.

8. 利用正则化解的外推求解机器学习中的病态线性系统

Extrapolating from Regularised Solutions for Solving Ill-Conditioned Linear Systems in Machine Learning

作者: Disha Hegde, Jon Cockayne, Chris. J. Oates

类别: stat.ML, cs.LG, math.NA

发布日期: 2026-06-29

arXiv ID: 2606.30328v1

链接: http://arxiv.org/abs/2606.30328v1

摘要: Rapid prototyping of algorithms is a critical step in modern machine learning. Most algorithms exploit linear algebra, creating a need for lightweight numerical routines which – while potentially sub-optimal for the task at hand – can be rapidly implemented. For the numerical solution of ill-conditioned linear systems of equations, the standard solution for prototyping is Tikhonov-regularised inversion using a nugget. However, selection of the size of nugget is often difficult, and the use of data-adaptive procedures precludes automatic differentiation, introducing instabilities into end-to-end training. Further, while data-adaptive procedures perform multiple linear solves to select the size of nugget, only the result of one such solve is returned, which we argue is wasteful. This paper aims to circumvent the above difficulties, presenting autonugget; a Python package for automatic and stable numerical solution of linear systems suitable for rapid prototyping, and fully compatible with automatic differentiation using JAX. autonugget combines multiple linear solves using Richardson extrapolation to determine the solution of the ill-conditioned system, improving in accuracy over approximations based on a single nugget.

9. 关于雅可比权函数的修正反高斯求积规则

On modified anti-Gaussian rules for Jacobi weight functions

作者: Eleonora Denich, Paolo Novati, Alvise Sommariva

类别: math.NA

发布日期: 2026-06-29

arXiv ID: 2606.30305v1

链接: http://arxiv.org/abs/2606.30305v1

摘要: Anti-Gaussian formulas represent an efficient tool for a dynamical estimation of the error of the underlying Gaussian rule. When applied to the Jacobi weight function it is known that such formulas are not always internal. In this work we show how to overcome this problem by using the so called modified anti-Gaussian rule with suitable parameter θ = θ(n), that depends on the number n of quadrature points of the Gaussian formula. Next we study theoretically the asymptotic rate of convergence of the corresponding modified averaged Gaussian formulas. We conclude by showing the benefits of this approach via numerical experiments. All the Matlab codes used in this work are available as open-source software.

10. 信赖域半径更新机制综述. 第一部分: 一阶分析

A survey of trust-region radius update mechanisms. Part I: First-order analysis

作者: Jérémy Rieussec, Fabian Bastin

类别: math.OC, math.NA

发布日期: 2026-06-29

arXiv ID: 2606.30202v1

链接: http://arxiv.org/abs/2606.30202v1

摘要: We isolate three structural conditions on trust-region radius update rules for smooth unconstrained nonlinear optimisation, and study the class of mechanisms they define. The conditions act on the radius directly: a lower bound relative to the gradient norm, a contraction on unsuccessful iterations, and a controlled expansion on successful ones. A mechanism is \emph{weakly admissible} if it satisfies the first two conditions, and \emph{strongly admissible} if it satisfies the lower bound together with the controlled-expansion condition. Under uniformly bounded model Hessians, weak admissibility yields $\lim_{k\to\infty}\|\nabla f(x_k)\|=0$, and strong admissibility yields the optimal worst-case complexity $O(ε^{-2})$ for first-order stationarity. Strong admissibility extends the convergence guarantee to linearly growing model Hessians. We verify admissibility for five mechanism classes: fixed-factor, step-driven, retrospective, criticality-anchored, and gradient-scaled. Along the way, we prove convergence of the retrospective update under linearly growing model Hessians and revisit the framework of Curtis and Scheinberg (2020), and Wang and Yuan (2022): we extend it to three distinct scaling factors with decoupled step acceptance (covering $η= 0$), and specialise its stochastic version to the deterministic gradient-scaled

11. 基于商几何黎曼优化的张量列分解低秩张量补全

Low-Rank Tensor Completion using Tensor Train Decomposition via Riemannian Optimization on the Quotient Geometry

作者: Zhenlong Hu, Zhongming Chen, Defeng Sun, Liping Zhang

类别: math.NA, math.OC

发布日期: 2026-06-29

arXiv ID: 2606.30173v1

链接: http://arxiv.org/abs/2606.30173v1

摘要: Owing to the effectiveness of Tensor Train (TT) decomposition in managing high-order tensors, low-rank tensor completion within the TT-format has emerged as a prominent research focus. In this paper, we leverage the left-orthogonal property of the TT-decomposition to construct a novel quotient manifold and introduce a family of admissible Riemannian metrics. Within this geometric framework, we propose a new approach to constructing retractions compatible with the quotient structure, realized via two novel retractions based on recursive polar and QR decompositions that respect the recursive orthogonalization structure of the TT format. We then derive Riemannian gradient descent and conjugate gradient methods to solve the tensor completion problem. Theoretically, our approach streamlines the horizontal projection by reducing the number of unknowns per block from a quadratic dependence on the TT-ranks to a near-half scaling, thereby enhancing computational efficiency over conventional quotient-based methods. Numerical experiments demonstrate that the proposed algorithms achieve reconstruction accuracy comparable to state-of-the-art TT-based geometric methods.

12. 用于可压缩湍流的高效壁面模型化高阶紧致气体动理学格式

Efficient Wall-Modeled High-Order Compact Gas-Kinetic Scheme for Compressible Turbulent Flows

作者: Yaqing Yang, Fengxiang Zhao, Kun Xu

类别: physics.flu-dyn, math.NA

发布日期: 2026-06-29

arXiv ID: 2606.30061v1

链接: http://arxiv.org/abs/2606.30061v1

摘要: Scale-resolving simulations of wall-bounded turbulent flows remain prohibitively expensive at high Reynolds numbers, owing to the stringent near-wall resolution requirements. High-order compact gas-kinetic schemes (CGKS) are accurate, robust, and efficient for compressible flows, making them an attractive foundation for reducing this cost. Building on the fifth-order scheme CGKS-5th, we develop a wall-modeled CGKS framework that alleviates the near-wall resolution burden through a pressure-gradient-based non-equilibrium wall model while preserving the resolving power of the outer solver. CGKS-5th resolves the outer flow and supplies the wall model with data at the exchange location. On coarse near-wall meshes, the wall model reconstructs the under-resolved viscous wall stress, while CGKS-5th provides the inviscid wall flux directly; the two combine to form the wall momentum flux. To capture non-equilibrium effects in adverse-pressure-gradient and separated regions, the wall model retains a pressure-gradient source term together with a pressure-gradient-corrected near-wall damping function. We assess the framework on two distinct flows: bluff-body separation past a circular cylinder, and a shock-induced separation bubble on the transonic RAE 2822 airfoil, using near-wall meshes far coarser than wall-resolved simulations require. For the RAE 2822 case, this corresponds to a twentyfold coarsening in the wallnormal direction, with comparable coarsening in other directions. In both cases, the wall-modeled CGKS-5th reproduces the separated flow structures and markedly improves near-wall predictions over its wall-model-free counterpart, most notably the skin-friction coefficient. The framework thus delivers accurate predictions of these separated flows at substantially reduced near-wall cost, while its lightweight coupling adds less than 1% runtime overhead in a multi-GPU implementation.

13. 非均匀热方程的 Dirichlet-Neumann 波形松弛：连续与时间离散的 L2 分析

Dirichlet-Neumann waveform relaxation for heterogeneous heat equations: continuous and time discrete L2 analysis

作者: Niklas Kotarsky, Philipp Birken, Martin J. Gander, Lu-di Lu

类别: math.NA

发布日期: 2026-06-29

arXiv ID: 2606.29987v1

链接: http://arxiv.org/abs/2606.29987v1

摘要: We consider two coupled linear heat equations on different spatial domains that interact through a lower dimensional interface. This models conjugate heat transfer. The problem is solved using Dirichlet-Neumann waveform relaxation. This allows us to couple separate codes for the subproblems, a so-called partitioned approach. Our overall goal is to develop more efficient partitioned methods, and to this end, we want reliable error estimates. We use an exponentially weighted Fourier technique to derive new error estimates in L2 for finite time T in both continuous and time discrete settings. We identify an optimized relaxation parameter that guarantees superlinear convergence. Our new continuous estimate predicts linear convergence when T is large, and superlinear when T is small. For large T, our new time discrete estimate closely mirrors its continuous counterpart, whereas for small T, superlinear convergence in the time discrete case requires small time step dt. We also show that convergence is fast when the contrast is large, provided that the small physical parameter domain (e.g. air) is using the Dirichlet transmission condition, and the large physical parameter domain (e.g. steel) is using the Neumann transmission condition in the Dirichlet-Neumann waveform relaxation method. Our numerical experiments confirm all these findings.

14. 超弹性物理增强神经网络在显式有限元代码 Simcenter Radioss 和 OpenRadioss 中的实现及在冲击事件中的应用

Implementation of Hyperelastic Physics-Augmented Neural Networks in the Explicit Finite Element Codes Simcenter Radioss and OpenRadioss with Applications to Impact Events

作者: Lukas Maurer, Sascha Eisenträger, Marian Bulla, Daniel Juhre

类别: math.NA, cs.LG

发布日期: 2026-06-29

arXiv ID: 2606.29874v1

链接: http://arxiv.org/abs/2606.29874v1

摘要: Data-driven material modeling techniques have gained significant attention due to their ability to capture complex constitutive behaviors beyond the limitations of classical material models. Physics-augmented neural networks (PANNs), which embed physical constraints directly into their architecture, combine the flexibility of machine learning with the reliability required for engineering simulations. This work presents an approach to integrate such network architectures into the explicit finite element solvers Simcenter Radioss and OpenRadioss (Siemens). A framework for transferring pretrained network architectures and their parameters to a standalone user material routine is developed. Networks are trained using PyTorch, though the procedure can be adapted to other frameworks such as TensorFlow, enabling the use of PANNs within existing finite element technology without requiring specialized solvers. Particular emphasis is placed on computational efficiency. The influence of network architecture on simulation performance is investigated, and strategies for reducing evaluation costs while preserving accuracy are discussed. Specifically, replacing the SoftPlus activation function with SQuarePlus is shown to reduce computational cost. A publicly available GitHub repository automates the generation of Fortran user material routines, requiring only the specification of the network architecture and trained parameters. An example impact simulation demonstrates that the generated PANN user material reproduces the nonlinear behavior characteristic of hyperelastic materials under large strains, providing a practical route toward machine-learning-based constitutive models in explicit finite element simulations.

15. 基于哈密顿模拟线性组合的量子特征值变换：一种Weyl微积分方法

Quantum Eigenvalue Transformation via Linear Combination of Hamiltonian Simulation: A Weyl Calculus Approach

作者: Hongkang Ni, Lexing Ying

类别: quant-ph, math-ph, math.NA

发布日期: 2026-06-29

arXiv ID: 2606.29848v1

链接: http://arxiv.org/abs/2606.29848v1

摘要: Linear combination of Hamiltonian simulation (LCHS) provides an efficient method for implementing matrix exponentials $e^{-tA}$ on quantum computers. In this paper, we develop LCHS formulas for computing general matrix functions $f(A)$ when $f$ is analytic on the numerical range of $A$, with $A$ possibly non-normal. The essential technical tool is Weyl calculus, which reduces the construction of LCHS formulas for noncommuting operators to scalar Fourier approximation problems. Our construction yields a quantum eigenvalue transformation algorithm with optimal $\mathcal{O}(\log\frac{1}ε)$ query complexity scaling. Furthermore, our Weyl-calculus-based theory gives rise to an ansatz-free convex optimization framework that directly produces discrete LCHS formulas. This circumvents the inefficiencies of traditional quadrature rules and yields formulas highly optimized for coherent implementation on quantum computers. In addition, both our theory and optimization framework apply to the simulation of time-dependent dissipative ODE $\frac{\mathrm{d}}{\mathrm{d} t} ψ(t) = -A(t)ψ(t)$, for which we achieve a $2.1\times$ cost reduction over prior art.

16. 一种用于2020年美国十年一度人口普查中精确隐私核算的筛加速求积方法

A Sieve-Accelerated Quadrature Method for Exact Privacy Accounting in the 2020 U.S. Decennial Census

作者: Buxin Su, Weijie Su, Chendi Wang

类别: cs.CR, cs.DS, math.NA, stat.AP, stat.ML

发布日期: 2026-06-29

arXiv ID: 2606.29835v1

链接: http://arxiv.org/abs/2606.29835v1

摘要: In 2020, the U.S. Census Bureau adopted differential privacy for the Decennial Census by injecting integer-valued Gaussian noise into published census tabulations. Exactly evaluating the privacy guarantees of these data releases would enable the Bureau to determine the absolute minimum noise required to satisfy a given privacy budget, preventing the injection of unnecessary excess noise and thereby substantially enhancing the statistical utility of the data for downstream applications such as federal funding allocation and political redistricting. In this paper, we introduce a computationally efficient and mathematically rigorous quadrature method to evaluate the exact privacy profile of practical, large-scale census releases under the composition of heterogeneous discrete Gaussian mechanisms. Mathematically, this problem reduces to evaluating the tail probabilities of high-dimensional convolutions of integer-valued random variables sampled from heterogeneous discrete Gaussian distributions under exceptionally stringent numerical error tolerances (e.g., $10^{-35}$). By recasting the exact privacy accounting as a numerical integration problem via the discrete Fourier transform, we explicitly exploit the exponential convergence of the trapezoidal rule for complex analytic, periodic characteristic functions. Furthermore, to overcome the computational bottleneck of evaluating highly oscillatory integrands in high dimensions, we develop a sieve algorithm that identifies and prunes negligible quadrature nodes, accelerating the computation by three orders of magnitude. Taken together, these numerical innovations enable the first exact, assumption-free privacy accounting for the 2020 Census Demographic and Housing Characteristics File, achieving a 1,824-fold speedup over prior methods while maintaining census-mandated error tolerances.

17. 傅里叶–汉克尔矩方法用于声学逆散射中的拓扑计数与相位中心恢复

Fourier–Hankel Moment Methods for Topological Counting and Phase-Center Recovery in Acoustic Inverse Scattering

作者: Xiaomei Yang, Zhiliang Deng

类别: math.NA

发布日期: 2026-06-29

arXiv ID: 2606.29678v1

链接: http://arxiv.org/abs/2606.29678v1

摘要: We develop a Fourier–Hankel moment framework for extracting topological counting information from full-aperture acoustic far-field data. The method is based on the observation that separated localized components generate distinct phase centers in angular Fourier data. Under the Born approximation, a Bessel–Fourier moment identity shows that suitably scaled row Fourier coefficients form, to leading order, a finite exponential moment sequence. The associated Hankel matrix has rank equal to the number of separated connected components, and the corresponding Hankel pencil recovers their phase-center locations. We prove the exact Hankel rank formula in the phase-center model and establish a perturbation theorem showing stable component counting under a singular-gap condition. We further extend the framework to detectable cavities by introducing a signed phase-center model. In this model, material components and cavities contribute with opposite signs to the moment sequence. The signed Hankel rank counts distinct signed phase centers, and the detectable cavity count is obtained from the excess rank beyond the positive component count. This formulation also identifies an intrinsic degeneracy: cavities whose phase centers coincide with material phase centers, such as perfectly concentric annuli, do not increase the leading signed rank and therefore cannot be detected by the leading phase-center mechanism alone. Numerical experiments validate the proposed theory at several levels: ideal moment sequences, Born far-field data with finite-size components, phase-center location recovery, signed cavity counting, and exact Helmholtz far-field data. The results show that the Fourier–Hankel rank mechanism provides a data-level algebraic approach to component counting and detectable cavity counting, while also making explicit its stability conditions and failure modes.

18. 保亚强制性的时空 Galerkin 方法用于动力学 Fokker-Planck 方程

Hypocoercivity-preserving space-time Galerkin methods for kinetic Fokker-Planck equations

作者: Zhaonan Dong, Emmanuil H. Georgoulis

类别: math.NA

发布日期: 2026-06-28

arXiv ID: 2606.29621v1

链接: http://arxiv.org/abs/2606.29621v1

摘要: We design and analyse a family of hypocoercivity-preserving fully discrete Galerkin methods for the (inhomogeneous) kinetic Fokker–Planck (kFP) equations, a class of evolution PDEs with degenerate diffusion. The proposed methods mimic Villani’s framework of enhanced quadratic forms [23], yielding a coercive bilinear form in an exponentially weighted norm that admits a spectral gap/Poincaré inequality despite the degeneracy. The problem is formulated as a fourth-order-in-space evolution PDE on the whole space $\mathbb{R}^{d}\times\mathbb{R}^d$. The spatial discretisation employs continuous piecewise polynomial finite element spaces on simplicial and/or box-type meshes comprising both finite and ``infinite’’ elements, while nonconformity is handled by numerical fluxes in the spirit of $C^0$ interior penalty ($C^0$-IP) methods. The analysis requires new polynomial inverse trace inequalities in exponentially weighted norms for simplicial, box-type, and semi-infinite prismatic elements, which are proved for a broad class of exponential weights and are of independent interest. Coercivity of the Galerkin method then leads to exponential convergence to equilibrium via an exponentially weighted Poincaré inequality. We further develop a fully discrete scheme by coupling the spatial discretisation with an $hp$-version discontinuous Galerkin time-stepping method of arbitrary order and establish the same exponential convergence. The proposed methods preserve the total mass and exhibit \emph{provably} exponential convergence to equilibrium, making them well suited for long-time kFP simulations. Numerical experiments validate the theoretical results and demonstrate the convergence behaviour of the proposed methods.

19. AI如何解决了最古老SGD算法的复杂度

How AI settled the complexity of the oldest SGD algorithm

作者: Michał Dereziński, Xiaoyu Dong

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

发布日期: 2026-06-28

arXiv ID: 2606.29593v1

链接: http://arxiv.org/abs/2606.29593v1

摘要: In 1937, Stefan Kaczmarz proposed a simple algorithm for solving systems of linear equations. This algorithm turned out to be the earliest known example of stochastic gradient descent, a ubiquitous computing paradigm that drives the training of modern AI models such as ChatGPT and Gemini. Now, those AI models have joined forces to discover the worst-case complexity of the Kaczmarz algorithm. This paper tells the story of how it happened.

20. 优化器记忆使数据打乱顺序成为微调噪声的一阶源

Optimizer Memory Makes Shuffle Order a First-Order Source of Fine-Tuning Noise

作者: John Sweeney

类别: cs.LG, math.NA, math.OC, stat.ML

发布日期: 2026-06-28

arXiv ID: 2606.29554v1

链接: http://arxiv.org/abs/2606.29554v1

摘要: Shuffle order can be a larger source of fine-tuning noise than a memoryless analysis predicts: fixed-clock optimizer memory makes local equal-multiset contrasts first order in the learning rate rather than second order, and the resulting order channel can be large enough for a single seed to flip a close A/B comparison. We isolate this mechanism and derive a fit-free way to size the noise it produces. For a memoryless optimizer, reordering an equal multiset has no first-order endpoint term; the leading local contrast is the $O(η^2)$ gradient bracket. Fixed-clock optimizers such as AdamW are different. Their moment buffers, preconditioner state, and de-biasing counters advance with the step index rather than with the learning-rate-scaled time $τ=ηk$, so the same gradient can receive a position-dependent endpoint weight. For any fixed finite measurement window, a lifted-state expansion gives an $O(η)$ equal-multiset contrast whenever the first-order replay coefficient is nonzero, while regular and clock-matched controls remain $O(η^2)$; a bare fixed-$β$ momentum buffer is already enough. A bitwise-deterministic replay from one warmed optimizer state isolates the mechanism, giving order-variance slopes 1.83 for AdamW, 2.00 for fixed-$β$ momentum, and 4.00 for SGD; matching the memory clock to $τ$ restores the regular exponent. For AdamW with a frozen preconditioner, the same impulse-weight kernel gives a closed-form asymptotic order-variance floor after the local potentials are measured, with no fitted coefficients. The result is local to the measurement window (independent reshuffling can average the channel across windows), but it yields order-noise error bars, positional attribution weights, and a seed-budget criterion for fine-tuning comparisons.

21. 正动力系统的稳定正积分延迟校正方法

Stable Positive Integral Deferred Correction Methods for Positive Dynamical Systems

作者: Pezzella Mario

类别: math.NA

发布日期: 2026-06-28

arXiv ID: 2606.29523v1

链接: http://arxiv.org/abs/2606.29523v1

摘要: In this paper, we introduce the class of Stable Positive Integral Deferred Correction (SPIDeC) methods for the numerical integration of positive dynamical systems. The proposed framework embeds a deferred correction mechanism within an exponential-type Volterra reformulation of the underlying differential problem. The resulting multiplicative structure guarantees the unconditional preservation of both positivity and equilibria, independently of the integration stepsize. Arbitrarily high-order accuracy is systematically achieved through successive explicit-in-sweep corrections applied to a low-order base approximation. From a stability viewpoint, the SPIDeC integrators are L-stable and exactly reproduce the continuous semigroup generated by diagonal linear operators. Furthermore, when Gauss–Radau quadrature nodes are employed, the associated discrete flow asymptotically approaches a logarithmically contractive map as the number of sweeps increases, ensuring stability. Numerical experiments are provided to validate the theoretical analysis and illustrate the practical performance of the proposed methods.

22. 带控制不匹配的全耦合FBSDEs的解耦神经网络逼近的后验误差分析

A Posteriori Error Analysis for Decoupled Neural Approximations of Fully Coupled FBSDEs with Control Mismatch

作者: Xichuan Zhang

类别: math.OC, cs.AI, math.NA

发布日期: 2026-06-28

arXiv ID: 2606.29474v1

链接: http://arxiv.org/abs/2606.29474v1

摘要: This paper develops an a posteriori error analysis framework for decoupled neural approximations of fully coupled forward–backward stochastic differential equations (FBSDEs). It provides an a posteriori error-analysis for the idealized discrete adapted trajectory. The main feature of the proposed formulation is the use of an auxiliary control process in the forward coefficients, which may differ from the backward component approximated by the neural network. This decoupling is useful in practical deep learning implementations, but it creates a control mismatch that must be included in the error analysis. We first establish a continuous-time stability estimate for fully coupled FBSDEs under perturbations of the drift, diffusion, generator, terminal condition, and auxiliary control input. We then transfer this estimate to the discrete-time setting and derive computable a posteriori error bounds depending only on the terminal defect, the pathwise residual, and the control mismatch. When the auxiliary control is identified with the backward approximation, the mismatch term vanishes and the bound reduces to the standard two-term form. Numerical experiments on a linear–quadratic FBSDE with an explicit reference solution and a multidimensional Burgers-type FBSDE without a reference solution illustrate the diagnostic role of the proposed indicators and the contribution of the mismatch penalty to the consistency and reproducibility of the numerical approximations.

23. 多类分类的结构化恰当损失几何：理论与受控经验评估

Structured Proper Loss Geometries for Multiclass Classification: Theory and Controlled Empirical Evaluation

作者: Soumyadip Sarkar

类别: cs.LG, math.NA

发布日期: 2026-06-28

arXiv ID: 2606.29471v1

链接: http://arxiv.org/abs/2606.29471v1

摘要: Strictly proper scoring rules identify the true conditional class distribution at population level, but their curvature can alter optimization and finite-sample behavior. We study three multiclass objectives: a class-aware quadratic Bregman score (CAPM), a strongly convex generator with constrained log-cosh ridges (HPG), and an HPG objective with an annealed probability-margin penalty (APMS). CAPM is treated as a structured instance of established quadratic scoring-rule theory. We derive conditional-regret, curvature, range, and logit-gradient bounds for CAPM and HPG, and prove exact penalty-range and conditional-target displacement bounds for APMS. Controlled five-seed experiments use Digits, Wisconsin breast cancer, and synthetic confusion and long-tail problems under clean labels, symmetric and pair-flip corruption, class imbalance, calibration evaluation, input corruption, and first-order adversarial perturbations. The candidates are close to cross-entropy on clean data and show descriptive gains in some noisy-label cells, but the five-seed comparisons are interpreted descriptively rather than as significance evidence. The selected noisy-label baselines perform better on Digits with 40% symmetric label noise, and explicit prior-adjustment methods perform better in the 30:1 synthetic long-tail experiment. Ablations do not show a consistent benefit from the candidate-specific graph, ridge, or margin components. The mathematical analysis establishes the stated properties, and the experiments delimit the empirical evidence; together they do not support a claim of general superiority.

24. 随机 Volterra 积分方程的基本弱收敛定理及其应用

Fundamental weak convergence theorem for stochastic Volterra integral equations and its applications

作者: Xinjie Dai, Qijiao Yin, Diancong Jin

类别: math.NA, math.PR

发布日期: 2026-06-28

arXiv ID: 2606.29458v1

链接: http://arxiv.org/abs/2606.29458v1

摘要: We study weak convergence rates of numerical approximations for stochastic Volterra integral equations (SVIEs), a class of non-Markovian models that arises naturally in stochastic volatility modeling and other fields. The intrinsic non-Markovian nature prevents the direct application of classical weak error techniques developed for finite-dimensional Markov processes. To overcome this difficulty, we combine a Markovian lifting technique with a domino argument, Taylor expansions, and Fréchet differential calculus for path-dependent functionals, and establish a fundamental weak convergence theorem for nonsingular SVIEs, providing a unified approach to the weak error analysis for a broad class of numerical approximations. As applications, we derive the first-order weak convergence rate for the stochastic theta method and the Wong–Zakai approximation. Our results relax existing assumptions for Euler-type schemes by removing the boundedness requirement on the diffusion coefficient. Furthermore, to the best of our knowledge, this work provides the first weak convergence result for Wong–Zakai approximations of SVIEs. Numerical experiments for a stochastic volatility model corroborate the theoretical convergence rate.

25. 用于参数化偏微分方程的随机神经算子：快速训练与共形不确定性量化

Randomized neural operator for parametric PDEs with fast training and conformal uncertainty quantification

作者: Zirui Deng, Jingbo Sun, Deyu Meng, Fei Wang

类别: cs.LG, math.NA

发布日期: 2026-06-28

arXiv ID: 2606.29440v1

链接: http://arxiv.org/abs/2606.29440v1

摘要: Repeatedly solving parametric PDEs is essential for uncertainty quantification, design optimization and inverse problems, but conventional neural operators require expensive non-convex training. We introduce PCA–RaNN, a randomized latent neural operator that combines PCA-based dimensionality reduction with fixed random features and a closed-form least-squares readout. It recasts latent operator learning as fixed-feature linear regression, reducing training time by one to three orders of magnitude across benchmarks while maintaining competitive accuracy. We introduce an energy-matched scaling rule and a lightweight two-parameter BFGS refinement to correct suboptimal feature scales. Ensemble averaging reduces predictive variance. On Burgers, Darcy, Navier–Stokes and backward heat equation benchmarks, PCA–RaNN provides a favorable speed–accuracy trade-off against operator-learning baselines. The ensemble supports split-conformal prediction intervals, and the linear readout enables rapid online adaptation via recursive least squares without retraining hidden features. This provides an efficient, uncertainty-aware surrogate for many-query scientific workflows.

26. 基于最小二乘读出重拟合的傅里叶神经算子用于学习随机障碍物到解的映射

Fourier Neural Operators with Least-Squares Readout Refit for Learning Random Obstacle-to-Solution Maps

作者: Chenhui Zhu, Fei Wang

类别: math.NA, cs.LG

发布日期: 2026-06-28

arXiv ID: 2606.29436v1

链接: http://arxiv.org/abs/2606.29436v1

摘要: We study operator learning for random obstacle-to-solution maps arising from elliptic variational inequalities with finite-band self-affine random obstacle fields. Instead of introducing an explicit truncated stochastic parametrization of the random input, we learn the map directly from sampled obstacle realizations on a fixed grid. This problem is challenging because the solution is governed not only by the obstacle field itself, but also by the induced contact set and free-boundary geometry. We introduce a post-training least-squares readout refit for the Fourier neural operator (FNO). After the FNO is trained end to end, its nonlinear backbone is frozen and the final affine readout is recomputed by solving the induced linear least-squares problem over all training samples and grid points. The refit yields the empirical squared-error optimal readout for the learned frozen features while leaving the nonlinear representation unchanged. We compare vanilla DeepONet, POD-DeepONet, a two-stage DeepONet baseline, FNO, and FNO with least-squares readout refit (FNO-LS) on two obstacle ensembles with different amplitude levels. Numerical results show that FNO-LS achieves the strongest overall performance among the tested models, particularly for higher-amplitude obstacles with more complex contact geometry. The method improves average field accuracy, contact-set recovery, and obstacle-violation metrics at low additional cost, especially when the FNO backbone is informative but not fully converged. These results suggest that least-squares readout refit is a simple and effective post-training enhancement for learning random obstacle-to-solution maps.

27. 薄几何中的偶极扩散误差：无网格次表面散射的光学厚度定律

Dipole Diffusion Error in Thin Geometry: Optical Thickness Laws for Grid-Free Subsurface Scattering

作者: Faruk Alpay, Baris Basaran

类别: cs.GR, math.NA

发布日期: 2026-06-28

arXiv ID: 2606.29387v1

链接: http://arxiv.org/abs/2606.29387v1

摘要: The dipole and its descendants model subsurface scattering with a radial reflectance profile fitted to a flat, semi-infinite slab. This assumption introduces a systematic geometry error on thin and curved objects. We isolate the effect by comparing the dipole with the finite-slab multipole under the same diffusion model and boundary condition. In slab geometry the diffuse-albedo error has a material-independent leading rate, $C e^{-2τ}$ with $τ=T/\ell_d$, while the prefactor remains material dependent; the same image series gives the transmitted flux, whose leading decay is $e^{-τ}$. We give the closed-form albedo and transmittance, relate the exponents to killed random walks, and extend the interpretation to spatially varying media through optical distance. A brute-force volumetric path tracer fits a reflectance-deficit rate of 1.99 and a transmittance rate of 0.99, matching the round-trip and single-pass predictions. The resulting thickness predictor is a useful thin-feature heuristic, but stress tests show that curvature and illumination can dominate away from the slab setting. For the remaining geometry-dependent term we solve the screened-Poisson diffusion problem directly inside the signed-distance domain with Walk on Spheres, without an interior mesh or a tangent half-space approximation; the estimator matches closed-form tests to 0.75%. Against a four-case path-traced benchmark it improves the back-lit, thickness-governed case but not every front-lit or curved case, showing that the method reduces geometry error within diffusion and does not replace radiative transport.

28. 面向多频数据的逆散射问题的多层级机器学习框架

A Multi-Level Machine Learning Framework for Inverse Scattering Problems with Multi-Frequency Data

作者: Yi Liu, Yanzhao Cao, Junshan Lin, Yimin Zhong

类别: math.NA

发布日期: 2026-06-28

arXiv ID: 2606.29368v1

链接: http://arxiv.org/abs/2606.29368v1

摘要: In this work, we propose a multi-level machine learning framework for solving inverse scattering problems with multi-frequency data. The multi-level neural network is built along the frequency axis of the scattering problem, wherein at each fixed frequency, a new level of network is added to the existing architecture to update the reconstruction. By marching through the frequency levels, the proposed multi-level computational framework is able to obtain higher-order Fourier modes of the imaging target as the depth of the neural network grows and higher-frequency data are used. Furthermore, the overall learning problem is decomposed into a sequence of simpler local tasks, each associated with a single frequency. This decomposition significantly reduces the complexity of the optimization problem and mitigates the risk of convergence to undesirable local minima, resulting in a robust and reliable training procedure for solving inverse scattering problems. We conduct various numerical experiments for the inverse source scattering problem and the inverse medium scattering problem to illustrate the effectiveness and robustness of the proposed machine learning framework. In addition, theoretical analysis in the neural tangent kernel regime shows that the proposed multi-level architecture progressively recovers the higher-order Fourier components of the imaging target.

29. 基于梯度的优化的现代理论

Modern Theory of Gradient-Based Optimization

作者: Bin Shi

类别: math.OC, math.CA, math.NA

发布日期: 2026-06-28

arXiv ID: 2606.29206v1

链接: http://arxiv.org/abs/2606.29206v1

摘要: In this review, we offer a comprehensive survey of emerging techniques in gradient-based optimization, with a particular emphasis on the interplay between ordinary differential equation (ODE) perspectives and their extensions into discrete Lyapunov analysis. We begin by examining the acceleration mechanisms underlying Nesterov’s accelerated gradient method for strongly convex functions (NAG-SC) and Polyak’s heavy-ball method, identifying the gradient-correction term as the primary driver of acceleration. This mechanistic insight is substantiated through high-resolution ODE modeling and the systematic construction of Lyapunov functions. We then synthesize recent advancements in convex optimization regarding NAG and its proximal generalization, the fast iterative shrinkage-thresholding algorithm (FISTA). Key topics include the accelerated convergence of gradient norms, underdamped acceleration, linear convergence under strong convexity, and novel Lyapunov frameworks for establishing convergence and monotonicity properties of generalized accelerated methods. Furthermore, we demonstrate how these ODE approximations and Lyapunov techniques can be extended to provide a unified framework for analyzing advanced optimization algorithms, including the alternating direction method of multipliers (ADMM), the primal-dual hybrid gradient (PDHG) method, and their respective accelerated variants. Finally, we discuss recent progress in minimax optimization and outline future directions for extending Lyapunov-based analysis to saddle-point problems.

30. 弯曲水平集区域上对流扩散方程的相容CutPINNs

Consistent CutPINNs for Convection-Diffusion Equations on Curved Level-Set Domains

作者: Maneesh Kumar Singh

类别: math.NA

发布日期: 2026-06-28

arXiv ID: 2606.29147v1

链接: http://arxiv.org/abs/2606.29147v1

摘要: We present an a priori error analysis of consistent-loss PINNs for stationary convection-diffusion equations on curved level-set domains. The standard mean-squared interior loss fails in the convection-dominated regime: the solution develops an $O(\eps)$ boundary layer in which the pointwise residual grows like $\eps^{-1}$, so the loss is dominated by the few collocation points inside the layer and leaves the smooth bulk unresolved. We remove this mismatch by penalising the interior residual in a discrete $\Lpγ$ norm with $γ= 1 + 1/\log\mtil$, a computable surrogate for the $\Hminusone$ stability term, and imposing the boundary condition through a discrete $\HhalfBdry$ trace norm, which treats flat and curved geometries uniformly. Under Besov regularity assumptions we prove a single a priori $\Hone$ error bound, valid for all interior exponents $γ\in (1,2]$, with an optimal recovery rate governed by a cut-cell floor $1/(2γ)$ specific to the curved geometry. Numerical experiments on a rectangle and a disk at $\eps = 2^{-s}$, $s \in \{2,4,6\}$, confirm the analysis: as the layer sharpens, the $\Lp{2}$ interior loss becomes seed-fragile while the $\Lpγ$ interior trains reliably, the interior norm being the decisive factor in convergence.

31. Ozaki 方案 II 快速模式的改良标度

Improved Scaling for Fast Mode of Ozaki Scheme II

作者: Shota Kawakami, Daisuke Takahashi

类别: cs.MS, cs.DC, math.NA

发布日期: 2026-06-28

arXiv ID: 2606.29129v1

链接: http://arxiv.org/abs/2606.29129v1

摘要: Ozaki scheme II emulates high-precision matrix multiplication using low-precision integer matrix operations based on the Chinese remainder theorem (CRT). It first scales the high-precision matrices to convert them into integer matrices. For this scaling step, Ozaki scheme II provides two modes: accurate mode, which uses INT8 matrix multiplication to estimate scaling factors, and fast mode, which applies the Cauchy–Schwarz inequality at lower computational cost. We show that the existing formula lacks scale invariance; multiplying the input matrices by a constant changes the effective bit width of the integer matrices in the scaling step, causing accuracy degradation or CRT recovery failure. To address this, we propose a revised scaling formula derived from the CRT uniqueness condition via the Cauchy–Schwarz inequality. The proposed formula is scale-invariant by construction, guarantees that the CRT uniqueness condition is always satisfied, and introduces no additional overhead over the original fast mode. Experiments on an NVIDIA GH200 GPU show that the proposed method achieves accuracy comparable to that of accurate mode while maintaining throughput comparable to that of fast mode. In the accuracy–throughput trade-off, the proposed method overcomes the accuracy limitation of fast mode and the throughput constraint of accurate mode, offering a superior accuracy and performance.

32. 用于谱精度Koopman逼近的残差引导字典学习

Residual-Guided Dictionary Learning for Spectrally Accurate Koopman Approximation

作者: George Coote, Matthew J. Colbrook

类别: math.NA, cs.LG, math.DS

发布日期: 2026-06-27

arXiv ID: 2606.29083v1

链接: http://arxiv.org/abs/2606.29083v1

摘要: Koopman theory promises linear structure in nonlinear dynamics, but numerical Koopman spectra are easy to compute and hard to trust. A finite EDMD matrix always has eigenvalues; the problem is that many of them may have nothing to do with the infinite-dimensional operator. In this paper we make spectral reliability the objective of dictionary learning. We train neural-network dictionaries not merely to predict the next snapshot, but to minimize Residual Dynamic Mode Decomposition residuals: operator-level a posteriori errors that test whether computed eigenvalues and modes are genuine Koopman spectral objects. To keep the learned observables from collapsing into an unstable coordinate system, the loss also penalizes the condition number of the lifted data matrix. Thus the method couples two requirements that should not be separated: small Koopman residuals and a well-conditioned representation. The result is a learned dictionary that is expressive, numerically stable, and spectrally disciplined. Across conservative and dissipative benchmark systems, the method sharply reduces spectral pollution, improves residual pseudospectral inclusion, and lowers forecast error relative to standard fixed dictionaries. On sea-surface temperature data, it gives cleaner Koopman diagnostics and substantially better one-step forecasts from noisy observations with no governing equations. The message is simple: neural Koopman learning should be judged not by prediction alone, but by whether its spectral claims can be certified. Residuals provide the certificate; conditioning makes it computable.

33. 一种用于Vlasov-Poisson系统的质量、动量和能量守恒的半拉格朗日自适应秩(SLAR)方法

A Mass, Momentum, and Energy Conserving Semi-Lagrangian Adaptive-Rank (SLAR) Method for the Vlasov-Poisson System

作者: Nanyi Zheng, William A. Sands, Jing-Mei Qiu

类别: math.NA

发布日期: 2026-06-27

arXiv ID: 2606.29027v1

链接: http://arxiv.org/abs/2606.29027v1

摘要: We propose a semi-Lagrangian adaptive-rank (SLAR) method that combines the large time-step capability of semi-Lagrangian schemes with the efficiency of adaptive-rank tensor representations while simultaneously enforcing local conservation laws for mass, momentum, and energy. The method builds on the high-dimensional SLAR framework introduced in our previous work and achieves high-order accuracy in both space and time. To address the loss of conservation in long-time simulations, we extend the implicit local macroscopic conservative (LoMaC) correction technique for the BGK equation to the high-dimensional Vlasov–Poisson (VP) system. The implicit macroscopic system is discretized using backward differentiation formulas and solved with a Jacobian-free Newton-Krylov method. This approach enables a consistent coupling with semi-Lagrangian methods which are capable of taking large time steps. A novel component of the proposed method is a unified adaptive-weight projection technique that eliminates the ad hoc parameter tuning required by previous LoMaC approaches. These weights capture problem-dependent velocity space structures and are constructed from the low-rank velocity bases of the solution. The local semi-Lagrangian method used in this work reconstructs the solution at the feet of the characteristics using efficient tensor contractions. To the best of our knowledge, this is the first successful implementation of an implicit LoMaC method for the VP system up to the 2D–2V setting. Numerical experiments on several classical benchmark problems demonstrate the accuracy and efficiency of the proposed method, as well as its ability to preserve conservation laws in VP simulations.

34. 一种用于Bethe-Salpeter特征值问题的结构保持切比雪夫滤波子空间迭代法

A structure-preserving Chebyshev-filtered subspace iteration for the Bethe-Salpeter eigenvalue problem

作者: Blanca Mellado-Pinto, Fernando Alvarruiz, Jose E. Roman

类别: math.NA

发布日期: 2026-06-27

arXiv ID: 2606.29025v1

链接: http://arxiv.org/abs/2606.29025v1

摘要: The Bethe-Salpeter equation, which has many applications in both theoretical and applied physics, is generally solved via a matrix eigenvalue problem with a rich algebraic structure. The numerical solution of such structured eigenproblem calls for specific algorithms that are able to preserve the structure throughout the computation. Several structure-preserving methods have already been proposed in the literature. In this paper, we develop a polynomial filter strategy that is able to extract approximations of eigenvalues located inside a specified interval. For this, we have devised a structure-preserving Chebyshev polynomial series, along with a specialized subspace iteration method that preserves the Bethe-Salpeter structure at every step of the algorithm. All necessary details required for a robust implementation are incorporated, and the performance is illustrated with matrices arising from real applications.

35. 受Lindblad方程控制的开放量子系统的Gregory嵌套Picard迭代格式

Gregory Nested Picard Iteration Schemes for Open Quantum Systems Governed by the Lindblad Equation

作者: Jiuhua Hu, Daniel Appelo, Yingda Cheng

类别: math.NA, quant-ph

发布日期: 2026-06-27

arXiv ID: 2606.29002v1

链接: http://arxiv.org/abs/2606.29002v1

摘要: Numerical simulation of quantum computing hardware and open quantum systems governed by the Lindblad equation is challenging due to the high dimensionality of the density matrix and the need to preserve fundamental physical properties. In our previous work, we developed an arbitrary-order, low-rank, completely positive and trace preserving (CPTP) method for the Lindblad equation with time-dependent Hamiltonians by nested Picard iteration (NPI). In this work, we develop Gregory NPI schemes, which are CPTP schemes constructed by Gregory-type quadrature on equispaced nodes. The methods, which are of order up to nine, substantially reduce the computational cost compared to our previously proposed NPI schemes with Gaussian quadrature rules, while retaining high-order accuracy and structure preservation. We analyze the stability of the resulting scheme for a physics-based test equation. Numerical experiments verify the convergence of the method and demonstrate the effectiveness of the low-rank approximation. We study the performance of a previously constructed CNOT gate for both closed and open quantum systems.

36. PEC圆柱TMz与TEz散射的二维矩量法分析

Two-Dimensional Method-of-Moments Analysis of TMz and TEz Scattering from PEC Cylinders

作者: Sabrina Saima

类别: eess.SP, math.NA, physics.comp-ph, physics.optics

发布日期: 2026-06-27

arXiv ID: 2606.29000v1

链接: http://arxiv.org/abs/2606.29000v1

摘要: This paper presents a two-dimensional method-of-moments (MoM) solver for electromagnetic scattering from infinitely long perfectly electrically conducting (PEC) cylinders. Both TMz and TEz polarizations are considered. Starting from the scalar Helmholtz equation, the electric field integral equation (EFIE) is derived for TMz scattering and the magnetic field integral equation (MFIE) is derived for TEz scattering. The induced surface current on the PEC boundary is expanded using pulse basis functions, and the boundary integral equations are discretized using point matching at the segment centers. Circular cylinders with radii $R = λ$ and $R = 2λ$ are used as validation cases because analytical series solutions are available. The MoM-computed surface currents, total near fields, scattered near fields, and field-error distributions are compared against the analytical solutions. After validation, the same solver is applied to a square PEC cylinder, for which no simple closed-form analytical solution is used. The results show strong agreement between the MoM and analytical circular-cylinder solutions and demonstrate the geometry-dependent scattering behavior of the square cylinder.

37. 比Fast-LTS更快：基于DC规划的稳健回归与异常值检测

Faster than Fast-LTS: Robust Regression and Outlier Detection with DC Programming

作者: Marah-Lisanne Thormann, Phan Tu Vuong, Alain B. Zemkoho, Tri-Dung Nguyen

类别: math.OC, math.NA, math.ST

发布日期: 2026-06-27

arXiv ID: 2606.28974v1

链接: http://arxiv.org/abs/2606.28974v1

摘要: When datasets contain outliers, robust regression is a well-established alternative to Ordinary Least Squares. A commonly employed robust estimator is Least Trimmed Squares (LTS), which computes the regression coefficients from a subset of observations. Determining the exact solution corresponds to a combinatorial problem with prohibitive computational costs, even for instances of moderate dimension. Thus, the most prevalent approach in practice remains a heuristic known as Fast-LTS. Although the heuristic often performs effectively, certain elements of the approach remain open to improvement. In particular, its core procedure provides robust results only when initialized with a large number of starting points. To address the heuristic’s limitations, this paper reformulates the LTS problem as a concave minimization problem subject to a capped simplex constraint, and proposes the successive Boosted Difference of Convex Functions Algorithm (sBDCA) as a solution method. Theoretically, we establish via the Łojasiewicz property that sBDCA converges to a local solution with a linear rate in the fastest case. To ensure robustness from a single initialization in practice, we derive and integrate a problem-specific preconditioning matrix into the algorithmic setup. Building on this theoretical foundation, we conduct numerical studies on various synthetic and real-world datasets to demonstrate the effectiveness of sBDCA with preconditioning. Specifically, we show that our approach is up to 3.25 times faster than Fast-LTS and achieves up to 90% lower objective function values, particularly in high-dimensional settings. As all code is openly available, this paper further provides a practical guide to robust regression in Python.

38. 记忆型问题的分裂格式

Splitting schemes for problems with memory

作者: Petr N. Vabishchevich

类别: math.NA

发布日期: 2026-06-27

arXiv ID: 2606.28883v1

链接: http://arxiv.org/abs/2606.28883v1

摘要: The paper considers the Cauchy problem for a first-order integro-differential equation with memory in a finite-dimensional Hilbert space. The main computational difficulty of such problems is the need to store and process the solution at all previous time levels. To overcome this difficulty, an approach is used that approximates the memory kernel by a sum of exponentials, which reduces the original nonlocal problem to a local one – a system of weakly coupled evolution equations with additional ordinary differential equations for auxiliary functions. The problem is formulated in vector form on the direct sum of Hilbert spaces. Unconditional stability of two-level operator-difference schemes with weights is proved under standard restrictions. Splitting schemes are proposed and investigated by separating the local and integral operators of the problem. Possibilities for constructing similar schemes for other nonlocal problems, in particular for the equation with memory of the time derivative of the solution, are noted.

39. 二维修正亥姆霍兹方程层位势与体积位势的快速统一计算

Fast unified evaluation of layer and volume potentials for the 2D modified Helmholtz equation

作者: Edith Frisk Gärtner, Fredrik Fryklund, Anna-Karin Tornberg

类别: math.NA

发布日期: 2026-06-27

arXiv ID: 2606.28865v1

链接: http://arxiv.org/abs/2606.28865v1

摘要: We present a fast and accurate potential theory-based method for the two-dimensional modified Helmholtz equation, treating the involved singular and nearly singular layer evaluations together with volume potentials within a single computational framework. The method is based on a decomposition of the free-space Green’s function into a short-range local part and a smooth long-range part. The long-range contribution is evaluated efficiently using the non-uniform fast Fourier transform (NUFFT), while the local contribution is treated by asymptotic expansions. For the layer potentials, an intermediate telescoping sum over dyadic refinement levels is added, where the resulting difference kernels are smooth and rapidly decaying, allowing the dyadic levels to be evaluated without specialized quadrature rules. The volume potential is evaluated on triangular cut-cell meshes, where the mesh only enters the scheme as quadrature rule for smooth data. This makes the method robust with respect to small and distorted mesh cells, without the need for stabilization or cell-merging techniques. Numerical experiments demonstrate the expected convergence rates, high throughput of the potential evaluations, and robustness with respect to mesh quality.

40. 一种关于Sturm–Liouville差分方程与特征值估计的离散Prüfer变换方法

A Discrete Prüfer Transformation Approach to Sturm–Liouville Difference Equations and Eigenvalue Estimation

作者: F. Ayça Çetinkaya, Kürşat Er, Hamza Menken

类别: math.SP, math.NA

发布日期: 2026-06-27

arXiv ID: 2606.28852v1

链接: http://arxiv.org/abs/2606.28852v1

摘要: In this paper, we study regular second-order Sturm–Liouville difference equations using the discrete Prüfer transformation. By representing solutions in amplitude and phase coordinates, we analyze an exact algebraic phase system that guarantees unique, monotonic phase tracking and preserves classical oscillation properties. Using this theoretical foundation, we develop a Prüfer-based numerical shooting method to compute eigenvalues for discrete boundary value problems. To initialize the root-finding algorithm, we apply Gershgorin’s theorem to the difference operator to establish mathematically guaranteed starting search intervals. Numerical experiments on classical benchmark problems demonstrate that the proposed method effectively isolates the discrete spectrum and converges to the exact continuous eigenvalues with second-order $\mathcal{O}(h^2)$ accuracy.

41. Navier-Stokes 方程一致分裂格式的误差上界中的粘性

Viscosity in error upper bound for a consistent splitting scheme of the Navier-Stokes equations

作者: M Nader Alhomsi, Jiahong Wu, Xiaoming Zheng

类别: math.NA

发布日期: 2026-06-27

arXiv ID: 2606.28800v1

链接: http://arxiv.org/abs/2606.28800v1

摘要: This paper investigates the role of viscosity in the error upper bounds of a consistent splitting scheme for the Navier-Stokes equations proposed by Huang and Shen [5]. In their original analysis the viscosity is fixed to unity. By following and extending their proof methodology while keeping the viscosity symbolic, we obtain an H1 velocity error bound that contains negative powers of viscosity, indicating that the scheme is not robust as viscosity tends zero. To establish this bound we refine a theorem in [8] on the constant in the Stokes pressure estimate, which is crucial to the error analysis. A targeted numerical experiment based on a perturbation of the Kovasznay flow corroborates this analytical prediction: the scheme of [5] blows up at high Reynolds number, and a comparison with a fully implicit Newton solver and with the time-dependent Stokes counterpart of the same scheme localizes the failure to the explicit treatment of the convection term.

42. 基于薛定谔化的正则化Wasserstein邻近算子的量子算法

Schrödingerization based quantum algorithms for regularized Wasserstein proximal operators

作者: Shi Jin, Nana Liu, Yue Yu

类别: math.NA

发布日期: 2026-06-27

arXiv ID: 2606.28752v1

链接: http://arxiv.org/abs/2606.28752v1

摘要: We develop a quantum algorithm for the regularized Wasserstein proximal operator, which is a fundamental tool in optimal transport and mean-field games. The regularization introduces a small diffusive term into the continuity equation of the Benamou-Brenier formulation, which results in a forward-backward PDE system consisting of a Fokker-Planck equation and a viscous Hamilton-Jacobi equation with a quadratic Hamiltonian. Through the Cole-Hopf transformation, both equations are converted to forward heat equations, whose coupling requires a Hadamard division to prepare the initial data for the second heat equation and a Hadamard product to recover the terminal density. We solve these heat equations via the Schrödingerization method and implement the Hadamard division and product operations using simple matrix-vector multiplication representations. The complete quantum algorithm prepares an $\varepsilon$-approximation of the terminal density state with $\mathcal{O}(d N_x T \log^2(1/\varepsilon))$ query complexity, up to constants depending on the potential and initial density, where $d$ is the spatial dimension, $N_x$ is the number of grid points per spatial dimension and $T$ is the evolution time. The complexity depends only {\it linearly} on $d N_x$, yielding an {\it exponential} speedup over classical methods, whose cost scales as $N_x^d$ per time step. Numerical experiments validate the effectiveness of the proposed algorithm.

43. 适用于任意单胞的周期性库仑格点和的自适应快速算法

An Adaptive Fast Algorithm for Periodic Coulomb Lattice Sums in Arbitrary Unit Cells

作者: Xuanzhao Gao, Leslie Greengard, Shidong Jiang

类别: math.NA

发布日期: 2026-06-26

arXiv ID: 2606.28608v1

链接: http://arxiv.org/abs/2606.28608v1

摘要: We present a fast algorithm for evaluating conditionally convergent Coulomb lattice sums, governed by the Laplace equation with periodic boundary conditions on arbitrary unit cells (oblique in 2D, triclinic in 3D) and arbitrary particle distributions. The algorithm extends the dual-space multilevel kernel-splitting (DMK) framework to this context. The root of the adaptive tree is now a rectangular grid of cubes consisting of an inner block covering the unit cell and a surrounding halo of image cubes, rather than a single cube, and the smooth top-level periodic kernel – the only term that requires the consideration of conditional convergence issues – is evaluated by the ``five-step procedure" used in fast Ewald summation: spreading, fast Fourier transform (FFT), diagonal scaling, inverse FFT, and interpolation. The resulting complexity is $O(N)$ for fixed cell shape. Benchmarked against the periodic fast multipole method on highly nonuniform source distributions, our 2D algorithm is roughly an order of magnitude faster across particle counts and target precisions; in three dimensions, it is often as fast as the free-space DMK on the same sources, even for triclinic cells with edge-length ratios up to roughly $17$.

44. 受约束的柯尔莫戈洛夫宽度

Constrained Kolmogorov widths

作者: Ronald DeVore, Guergana Petrova, Jonathan W. Siegel, Przemysław Wojtaszczyk

类别: math.NA

发布日期: 2026-06-26

arXiv ID: 2606.28587v1

链接: http://arxiv.org/abs/2606.28587v1

摘要: The main theme of approximation theory is to understand how well a general function $f$ can be approximated by a simpler function $g$ such as a polynomial or spline. In many applications, one wants $g$ to retain known properties of $f$ such as its inherent smoothness or a geometrical property such as monotonicity or convexity. Additional requirements on $g$ of this type are known as constraints. In this paper, we do a systematic study of constrained approximation to understand how the imposition of such constraints limits the efficiency of the approximation. We study constrained approximation in the setting of linear approximation where $g$ is to be taken from a finite dimensional linear space $V$ of a fixed dimension $n$. Kolmogorov widths describe how well one can approximate when using such linear spaces $V$. The first part of this paper introduces and studies several types of constrained widths, including the constrained Kolmogorov widths, and gives comparisons between them. The second part of the paper is restricted to classical settings where the constraint imposes a smoothness requirement on $g$. In this case, our results prove that the additional constraint can typically be imposed with no loss in the efficiency of the approximation.

重点关注论文（完整分析）#

1. McMg：一种用于亥姆霍兹方程的学习型相空间多通道多重网格预处理器#

1. 研究对象和背景#

2. 主要定理或主要结果#

3. 研究方法、关键技术和核心工具#

4. 与之前工作的比较#

2. 一种基于迹的高传导界面降维方法#

1. 研究对象和背景#

2. 主要定理或主要结果#

3. 研究方法、关键技术和核心工具#

4. 与之前工作的比较#

了解领域论文（摘要翻译）#

1. 连续级Runge–Kutta方法秩条件的多项式矩方法#

摘要翻译#

2. 基于极化的离散化方法的守恒量#

摘要翻译#

3. 基于Simpson-Boole几何恒等式的曲面扩散二阶保面积/体积参数有限元方法#

摘要翻译#

4. 动区域上Stokes方程的压力鲁棒ALE时空DG方法#

摘要翻译#

5. 超越原生空间的核逼近——及其在流形逼近中的应用#

摘要翻译#

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重点关注论文（完整分析）

1. McMg：一种用于亥姆霍兹方程的学习型相空间多通道多重网格预处理器

1. 研究对象和背景

2. 主要定理或主要结果

3. 研究方法、关键技术和核心工具

4. 与之前工作的比较

2. 一种基于迹的高传导界面降维方法

1. 研究对象和背景

2. 主要定理或主要结果

3. 研究方法、关键技术和核心工具

4. 与之前工作的比较

了解领域论文（摘要翻译）

1. 连续级Runge–Kutta方法秩条件的多项式矩方法

摘要翻译

2. 基于极化的离散化方法的守恒量

摘要翻译

3. 基于Simpson-Boole几何恒等式的曲面扩散二阶保面积/体积参数有限元方法

摘要翻译

4. 动区域上Stokes方程的压力鲁棒ALE时空DG方法

摘要翻译

5. 超越原生空间的核逼近——及其在流形逼近中的应用

摘要翻译

不相关论文（基本信息）

1. 齐次深度网络中持续学习的收敛性

2. 通过透视映射与盒子分解计算积分R2指标

3. 大型线性问题的迭代Tikhonov正则化

4. 周期性不可压缩斯托克斯流的量子谱求解器

5. 使用 LattE 计算筛积分及具有局部化除数的整数的密度

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