生成时间: 2026年09月07日 15:40:12
重点关注论文数量: 1
了解领域论文数量: 2
不相关论文数量: 13
重点关注论文（完整分析）
1. 高度非均匀散射问题的多尺度有限元方法的频率显式收敛性分析
Frequency-explicit convergence analysis of a multiscale finite element method for highly heterogeneous scattering problems
作者: T. Chaumont-Frelet, Z. Kassali
类别: math.NA, math.AP
发布日期: 2026-09-04
arXiv ID: 2609.04930v1
摘要翻译: 我们分析了高度非均匀可穿透障碍物对时间谐波散射的数值逼近问题。在高频区域，这些问题的求解尤其具有挑战性，此时散射体尺寸 $L$ 远大于波长，即波数 $k$ 满足 $kL \gg 1$。这里我们进一步考虑散射体包含不同材料的情形，其特征尺寸为 $\varepsilon$，满足 $k\varepsilon \ll 1$。我们提出了一种高阶多尺度有限元方法，并给出了对 $k$ 和 $\varepsilon$ 均显式的误差分析。至关重要的是，我们的误差估计表明，使用高阶方法应能降低大频率下的计算成本，数值算例也印证了这一结论。
Comment: 无
链接: http://arxiv.org/abs/2609.04930v1
1. 研究对象和背景
本文研究高频时间谐波散射中强非均匀介质的数值逼近. 设 $D\subset\mathbb{R}^d$ ($d=2,3$) 为光滑有界且可穿透的散射体, 记其直径为 $\ell_D$; $\Omega$ 为包含 $\overline D$ 的光滑计算域, 记 $\Gamma=\partial D$, 并设 $d(\partial\Omega,D)\ge\ell_D$ 以简化若干几何常数. 在 $D$ 内材料具有小周期 $\varepsilon$ 的周期微结构: $\mu_\varepsilon=\hat\mu(x/\varepsilon)$, $A_\varepsilon=\hat A(x/\varepsilon)$, 在 $\Omega\setminus\overline D$ 中 $\mu_\varepsilon=1$, $A_\varepsilon=I$. 对波数 $k>0$ 与右端 $f\in L^2(\Omega)$, 模型问题为
$$-k^2\mu_\varepsilon u_\varepsilon-\nabla\cdot(A_\varepsilon\nabla u_\varepsilon)=f\ \text{in}\ \Omega,\qquad \nabla u_\varepsilon\cdot n-iku_\varepsilon=0\ \text{on}\ \partial\Omega,$$其弱形式为求 $u_\varepsilon\in H^1(\Omega)$ 使 $b_\varepsilon(u_\varepsilon,v)=(f,v)_\Omega$ 对所有 $v\in H^1(\Omega)$ 成立, 其中 $b_\varepsilon(\phi,v)=-k^2(\mu_\varepsilon\phi,v)_\Omega-ik(\phi,v)_{\partial\Omega}+(A_\varepsilon\nabla\phi,\nabla v)_\Omega$. 阻抗边界条件用于截断无界区域并近似 Sommerfeld 辐射条件. 本文关注所谓高频高对比区域: $k\ell_D\gg1$ 且 $k\varepsilon\ll1$, 即散射体远大于波长但波长远大于微结构尺寸. 直接离散要求网格同时分辨 $\varepsilon$ 与波长, 计算量随 $k^d\varepsilon^{-d}$ 增长; 多尺度有限元方法旨在仅用宏观网格尺寸 $H\gg\varepsilon$ 求解, 但误差分析必须对 $k$ 与 $\varepsilon$ 同时显式. 为此需要原问题稳定常数 $C^\varepsilon_{\mathrm{st}}:=k^2\sup_{\|f\|_{L^2}=1}\|u_\varepsilon\|_{L^2(\Omega)}$ 关于 $\varepsilon$ 的一致上界. 已有均质化理论, 如 [9] 的均质化结果一般只对单个右端 $f$ 给出; [14] 仅在系数不依赖某坐标、波速大于 $1$ 且满足几何符号条件 $x_d n_d\ge0$ 等特殊情形证明 $C^\varepsilon_{\mathrm{st}}\lesssim(k\ell_D)^3$; 一般情形的 $\varepsilon$ 一致稳定界此前缺乏. 本文旨在填补该空白, 并进一步获得频率显式的均质化误差与多尺度杂交混合 (MHM) 有限元误差界.
2. 主要定理或主要结果
设均质化系数如下: $\mu_0$ 在 $D$ 内等于 $\mu_H=\langle\hat\mu\rangle_Y$, 在 $\Omega\setminus\overline D$ 内等于 $1$; 对 $j=1,\dots,d$, 令 $\hat\chi_j\in H^1_\sharp(Y)$ 为单元问题
$$-\nabla_y\cdot(\hat A\nabla_y\hat\chi_j)=-\sum_{\ell=1}^d\partial_{y_\ell}\hat A_{j\ell}$$的零均值解, 并记 $\hat\chi=(\hat\chi_1,\dots,\hat\chi_d)$. 若 $\hat C_{j\ell}=\partial_{y_j}\hat\chi_\ell$, 则 $A_0$ 在 $D$ 内取 $A_H=\langle\hat A(I-\hat C)\rangle_Y$, 在外部取 $I$. 定理3.3证明稳定常数关于 $\varepsilon$ 一致有界: $C_{\mathrm{st}}:=\sup_{0<\varepsilon\le\ell_D}C^\varepsilon_{\mathrm{st}}<+\infty$, 即对所有 $f\in L^2(\Omega)$ 与 $0<\varepsilon\le\ell_D$ 一致地有
$$k^2\|u_\varepsilon\|_{L^2(\Omega)}\le C_{\mathrm{st}}\|f\|_{L^2(\Omega)}.$$这里 $C_{\mathrm{st}}$ 可依赖 $k$, 但不依赖 $\varepsilon$; 文中先以引理3.1建立 $C^\varepsilon_{\mathrm{st}}$ 对 $\varepsilon$ 的连续性. 定理4.1给出频率显式均质化误差估计: 令 $\chi_\varepsilon$ 在 $D$ 内为 $\hat\chi(x/\varepsilon)$, 在 $\Omega\setminus\overline D$ 内为零, 则
$$k\|u_\varepsilon-u_0-\varepsilon\chi_\varepsilon\cdot\nabla u_0\|_{H^1_k(\Omega\setminus\Gamma)}\lesssim C_{\mathrm{st}}\big(k\|u_0\|_{H^1_k(\Omega)}+|u_0|_{H^2(D)}\big)\lesssim C_{\mathrm{st}}C^{\mathrm{H}}_{\mathrm{st}}(k\varepsilon)^{1/2}\|f\|_{L^2(\Omega)},$$其中 $u_0$ 是对应均质化问题的解, $C^{\mathrm{H}}_{\mathrm{st}}$ 是均质化问题的稳定性常数. 系4.2进一步得到 $C^{\mathrm{H}}_{\mathrm{st}}\lesssim C_{\mathrm{st}}\lesssim(1+C_{\mathrm{st}}(k\varepsilon)^{1/2})C^{\mathrm{H}}_{\mathrm{st}}$. 对多尺度杂交混合有限元, 取单元尺寸为 $H$, 多项式界面次数为 $\ell\ge0$, 且假定单元上用于构造基函数的 Neumann 型局部问题被精确求解. 若 $\varepsilon\ll H$, $C_{\mathrm{st}}^2(k\varepsilon)^{1/2}\ll1$, $C_{\mathrm{st}}(kH)^{\ell+1}\ll1$, 则离散解 $u_H$ 满足
$$k\|u_\varepsilon-u_H\|_{L^2(\Omega)}+\|\nabla(u_\varepsilon-u_H)\|_{L^2(\Omega)}\lesssim kH+C_{\mathrm{st}}(kH)^{\ell+1}+C_{\mathrm{st}}^2(k\varepsilon)^{1/2}.$$该估计推广了无振荡系数时标准有限元的误差界 [8,13,26]; 当 $\varepsilon\sim H$ 时, 经典理论要求增加含 $\varepsilon/H$ 的共振项, 而本文的估计在尺度分离条件 $\varepsilon\ll H$ 下不含该项. 误差估计中 $C_{\mathrm{st}}(kH)^{\ell+1}$ 随 $\ell$ 增大快速衰减, 故高阶单元可补偿 $C_{\mathrm{st}}$ 随频率增长带来的影响; 数值实验支持这一结论. 需要强调的是, 理论结果假设基函数精确计算, 实际计算中第二层网格取 $h\sim\varepsilon$, 但其离散误差未被纳入定理分析.
3. 研究方法、关键技术和核心工具
证明的第一个关键是建立 $C^\varepsilon_{\mathrm{st}}$ 的一致上界. 由于无法对任意 $\varepsilon$ 直接做紧性论证, 作者先证明连续依赖性: 对相邻尺度 $\varepsilon,\varepsilon'$, 利用周期系数的 Lipschitz 连续性比较 $u_\varepsilon$ 与 $u_{\varepsilon'}$, 得到
$$b_\varepsilon(u_\varepsilon-u_{\varepsilon'},v)\lesssim\ell_D\frac{|\varepsilon-\varepsilon'|}{\varepsilon\varepsilon'}\|u_{\varepsilon'}\|_{H^1_k(\Omega)}\|v\|_{H^1_k(\Omega)},$$再结合命题2.4的对偶范数估计推出 $|C^\varepsilon_{\mathrm{st}}-C^{\varepsilon'}_{\mathrm{st}}|\lesssim\ell_D(C^\varepsilon_{\mathrm{st}})^2|\varepsilon-\varepsilon'|/(\varepsilon\varepsilon')$, 从而得到引理3.1的连续性. 随后用反证法: 若上确界无界, 则存在 $\varepsilon_n\to0$ 和归一化右端 $f_n$ 使 $k^2\|u_{\varepsilon_n}\|_{L^2}\ge n$. 将解和右端同时除以 $\|u_{\varepsilon_n}\|_{L^2}$ 得到归一化解 $\hat u_n$ 与右端 $\hat f_n$, 满足 $\hat f_n\to0$ 于 $L^2$. 从能量等式可得 $\hat u_n$ 在 $H^1(\Omega)$ 中有界, 因此可选子列使 $\hat u_n$ 在 $L^2$ 强收敛于 $u_0$, 且 $\nabla\hat u_n$ 双尺度收敛到 $\nabla u_0+\nabla_y u_1$. 利用双尺度收敛和均质化理论, 可证 $b_{\varepsilon_n}(\hat u_n,v)\to b_0(u_0,v)$ 对所有 $v$; 另一方面又有 $b_{\varepsilon_n}(\hat u_n,v)=(\hat f_n,v)_{\Omega}\to0$, 故 $u_0$ 满足齐次均质化问题. 由均质化问题的唯一性(及唯一延拓原理)得 $u_0=0$, 这与 $\|u_0\|_{L^2}=1$ 矛盾. 这一方法继承了 [9] 的紧性思路, 但把结果从单个右端 $f$ 提升为对所有 $L^2$ 右端一致, 这正是后续有限元误差分析所需要的. 均质化误差定理4.1的证明沿用 [14] 的校正子框架: 令辅助函数 $W_\varepsilon$ 满足 [14, Eq. (4.28)] 型估计 $\|W_\varepsilon\|_{H^1_k}\lesssim C_{\mathrm{st}}\|\psi\|_{(H^1_k)'}$, 从而把原文中依赖局部半径和层厚度的常数 $C_{\mathrm{layer}}(kR,kR_0,b_{\min})$ 全部替换为不依赖 $\varepsilon$ 的整体稳定常数 $C_{\mathrm{st}}$; [14] 的若干引理可直接平移, 只有涉及 $u_0$ 的正则性项被替换为 $|u_0|_{H^2(D)}$. 对 MHM 方法的分析, 文中采用与散射体界面 $\Gamma$ 拟合的曲网格, 在每个单元上通过带 $\ell$ 次多项式 Neumann 数据局部问题构造基函数, 从而把单元尺度 $H\gg\varepsilon$ 内的振荡信息吸收进离散空间. 在基函数精确求解的假设下, 整体误差被分离为宏观均质化误差和局部离散误差; 附录 A 专门建立用于单元局部 Neumann 问题的辅助均质化估计, 这是控制 $H$-单元内 $\varepsilon$ 尺度振荡的关键. 需要说明的是, 摘录文本没有展开第六节的 inf-sup 稳定性和局部逼近引理, 因此无法从提供内容重建每一个精细常数; 但引言明确说明其证明是把(1.1)的稳定常数纳入标准有限元工具, 并与定理4.1耦合. 作者强调第二个创新点是首次在多尺度有限元误差界中显式显示高阶单元能够抑制 $C_{\mathrm{st}}$ 随频率增长的影响, 而非仅仅沿用传统有限元的色散误差直觉.
4. 与之前工作的比较
与均质化文献 [9] 相比, 本文的主要推进是把均质化结果从单个固定右端 $f$ 提升为对任意 $f\in L^2(\Omega)$ 一致成立的稳定估计(1.1). 这一右端无关性对 Galerkin 误差分析至关重要, 因为离散误差方程中出现的是与离散残差相关的任意右端, 而非仅仅原始数据 $f$. 与 [15] 相比, 本文在需要显式刻画 $C_{\mathrm{st}}$ 对 $k$ 的依赖时沿用 [15] 的技术, 因而所谓 fully explicit 的 $C_{\mathrm{st}}$ 上界只是在关于系数的合适额外假设下成立, 并非无条件显式; 摘录没有给出这些额外假设的具体形式. 与 [14] 相比, 本文对 $C^\varepsilon_{\mathrm{st}}$ 的一致界去掉了 [14] 中的特殊几何与材料假设, 包括系数不依赖某空间坐标、波速大于 $1$、以及 $x_d n_d\ge0$ 的符号条件; 代价是 [14] 能给出显式 $C^\varepsilon_{\mathrm{st}}\lesssim(k\ell_D)^3$, 而本文的一般性定理通常只给出有限常数 $C_{\mathrm{st}}(k)$, 不直接给出它的 $k$ 增长阶. 此外, 本文要求周期系数具有 $W^{1,\infty}$ 正则性, 比 [14] 允许的分片常值模式更强; 作者在注记3.2中认为该正则性假设可放松, 但未给出严格证明. 与标准有限元误差估计 [8,13,26] 相比, 在没有系数振荡的形式极限(可令 $k\varepsilon\to0$)下, 新估计(1.2)回到经典形式; 新的贡献是显式保留了 $C_{\mathrm{st}}$ 和 $(k\varepsilon)^{1/2}$ 项, 并且表明在固定 $H$ 时增大多项式次数 $\ell$ 可使 $C_{\mathrm{st}}(kH)^{\ell+1}$ 迅速变小. 对多尺度方法本身, 文中定位了与局部正交分解 (LOD) 类方法 [10,21,30,31] 的差异: LOD 方法在 $kH\ll1$ 下对高频鲁棒, 且当 $\varepsilon\sim H$ 时不受共振效应影响, 但其局部问题需要在由多个单元组成的补丁上求解, 计算成本更高; 本文的 MHM 方法只做单元素上的 Neumann 局部求解, 在尺度分离 $\varepsilon\ll H$ 下可能更高效. 类似评论也适用于 [27] 的 MsGFEM. 相对于经典 MsFEM [3,24,33], 本文在 $\varepsilon\ll H$ 时不出现 $\varepsilon/H$ 共振项; 但若 $\varepsilon\sim H$, 则仍须像 [24] 那样加入共振项, 而 LOD 方法在该情形更具优势. 代价方面, 文中所有误差估计都假设多尺度基函数被精确求解; 数值实现中采用的第二层网格 $h\sim\varepsilon$ 的误差未进入理论, 因此完整全离散多尺度方法的严格误差分析仍是空白. 同样, 本文假设网格精确拟合散射体边界, 没有讨论等参有限元或变分犯罪; 如注记5.2所述, 作者预期可仿照 [15] 引入 Strang 型引理来推广, 但未在本文件出. 数值例子部分在摘录中只有概述, 未包含具体表格和图, 故无法对其精度和计算量做进一步比较.
了解领域论文（摘要翻译）
1. 一个用于模拟奇异摄动问题解的简单浅层神经网络
A simple shallow neural network for emulating the solution to singularly perturbed problems
作者: Christos Xenophontos, Aayushman Raina
类别: math.NA
发布日期: 2026-09-04
arXiv ID: 2609.05180v1
链接: http://arxiv.org/abs/2609.05180v1
摘要翻译
中文标题: 一个用于模拟奇异摄动问题解的简单浅层神经网络
摘要翻译: 我们考虑（前馈）神经网络（NN）用于模拟具有两个小参数的奇异摄动二阶边值问题的解。我们描述了一个浅层NN，它利用了可用的解的渐近展开。这些光滑部分和层部分的可加分解允许导数估计，这些估计在微分阶数和奇异摄动参数方面是显式的 \cite{melenk, Irene, SX}。利用这种分解，我们提出了一个简单的NN，使用$\tanh$激活函数以及不同的训练目标（如残差或能量最小化）来模拟此类问题的解。关键思想是用适当的指数函数扩充逼近空间，类似于有限元方法中的富化空间，例如~\cite{Kellogg}。一维和二维的数值算例，包括一个光滑的非张量积域，说明了在测试的摄动范围内所产生的参数鲁棒行为。
2. 一类满足无穷多个熵条件的高阶间断伽辽金方法，具有可证明的误差估计和一般非线性守恒律的强收敛性
A class of high-order discontinuous-Galerkin methods satisfying infinitely many entropy conditions with provable error estimates and strong convergence for general nonlinear conservation laws
作者: Yuanzhe Wei, Chi-Wang Shu
类别: math.NA, physics.comp-ph
发布日期: 2026-09-04
arXiv ID: 2609.04687v1
链接: http://arxiv.org/abs/2609.04687v1
摘要翻译
中文标题: 一类满足无穷多个熵条件的高阶间断伽辽金方法，具有可证明的误差估计和一般非线性守恒律的强收敛性
摘要翻译: 我们提出了一种新的框架，利用算子半群为标量守恒律推导半离散间断伽辽金（DG）方法，然后应用该框架构造了一类在非均匀网格上满足具有一般E-通量的无穷多个局部熵不等式的高阶OFDG型格式[13]。此类格式通过使用文[1]意义下的熵稳定数值通量，进一步推广到任意空间维数下的守恒律方程组。最后，我们证明了非线性标量守恒律光滑解的最优误差估计，并借助补偿紧性方法证明了严格凸守恒律间断解的强收敛性。
不相关论文（基本信息）
1. 浅层神经网络在混合Sobolev空间中的逼近
Shallow neural network approximation in mixed Sobolev spaces
作者: Yuwen Li, Guozhi Zhang
类别: math.NA, cs.LG
发布日期: 2026-09-04
arXiv ID: 2609.05263v1
链接: http://arxiv.org/abs/2609.05263v1
摘要: We investigate the best $L_2$ approximation of mixed Sobolev spaces by shallow neural networks with $n$ neurons and general activation functions. We first establish an activation-independent Fourier-block principle: if an activation has univariate approximation order $ρ$ in the sense of the Fourier-block property, then the global approximation rate has algebraic order $\min\{α,ρ\}$ for target functions of mixed smoothness $α$, up to explicit logarithmic factors. To verify this property for concrete activations, we introduce a structured univariate approximation condition that implies the Fourier-block property with explicit parameters. For $\mathrm{ReLU}^k$, a matching algebraic lower bound identifies $\min\{α,k+1\}$ as the optimal algebraic approximation exponent in any dimension, up to logarithmic factors in the upper bound. The framework also yields the exponent $\min\{α,k+1\}$ for cardinal B-splines and soft-$\mathrm{ReLU}^k$, and the full mixed-smoothness exponent $α$ for ELU and cosine activations, again up to logarithmic~factors.
2. 一种针对非线性端口哈密顿微分代数方程的保结构高效积分的广义标量辅助变量方法
A Generalized Scalar Auxiliary Variable Method for Structure-Preserving and Efficient Integration of Nonlinear port-Hamiltonian DAEs
作者: Aashutosh Sharma, Andreas Bartel, Manuel Schaller
类别: math.NA
发布日期: 2026-09-04
arXiv ID: 2609.05246v1
链接: http://arxiv.org/abs/2609.05246v1
摘要: We develop an energy-optimal generalized scalar auxiliary variable (EOP-GSAV) framework for nonlinear index-one port-Hamiltonian differential-algebraic equations (pH-DAEs). Exploiting the port-Hamiltonian structure, we separate the nonlinear effort, interconnection, and dissipation terms from a constant implicit core. The resulting BDF-1 and BDF-2 schemes require one linear solve per time step with a reusable factorization, while retaining discrete passivity and accurate tracking of the Hamiltonian. The schemes are compared with the implicit midpoint method equipped with full, modified, and frozen-Jacobian Newton iterations. Numerical experiments ranging from a strongly state-dependent nonlinear stress test to large-scale benchmarks demonstrate robust and competitive performance, with substantial efficiency gains in matched-accuracy regimes. A comparison with SUNDIALS IDA shows comparable work-precision behavior at equal order despite a non-specialized Python/SciPy implementation, while unrestricted variable-order adaptive IDA is faster in the high-accuracy regime.
3. PRICE：比特币价格预测中大语言模型适配选择的系统性研究
PRICE: A Systematic Study of LLM Adaptation Choices for Bitcoin Price Forecasting
作者: Maryam Fakhari, Mehran Safayani
类别: cs.LG, cs.AI, math.NA
发布日期: 2026-09-04
arXiv ID: 2609.05235v1
链接: http://arxiv.org/abs/2609.05235v1
摘要: Cryptocurrency markets exhibit extreme volatility and non-stationary dynamics that challenge conventional forecasting methods. Although Large Language Models (LLMs) have shown promise for time series forecasting, the combined effects of adaptation choices remain largely unexplored in financial settings. This study introduces PRICE, a structured approach for adapting LLMs to short-term Bitcoin price forecasting. Built on a 4-bit quantized LLaMA-3 8B model, PRICE investigates how fine-tuning, numerical representation, prompting, inference, and decoding jointly influence forecasting performance. PRICE integrates Parameter-efficient fine-tuning with Low-Rank Adaptation (LoRA), Recursive multi-step inference, Integer-rounded numerical representation, Context-Task-Format (CTF) prompting, and Exact zero-temperature decoding. Ablation studies show that each component contributes to forecasting accuracy and reliability. LoRA enables efficient training on limited hardware, recursive inference improves accuracy, integer-rounded values reduce errors, CTF prompting outperforms Chain-of-Thought, Implicit Chain-of-Thought (iCoT), and few-shot prompting, and zero-temperature decoding improves stability during recursive forecasting. Comparative evaluation against eight transformer-based and time-series foundation models shows that PRICE achieves the lowest forecasting errors on both validation and test sets while maintaining robust performance across evaluation periods. Despite being based on a model primarily pretrained on text rather than time-series data, PRICE achieves competitive or superior performance relative to specialized foundation models. These findings demonstrate that adaptation choices critically determine the accuracy and robustness of LLMs for numerical time-series forecasting.
4. 重新审视动态离子存在下的动理学静电电子非线性(KEEN)波
Revisiting kinetic electrostatic electron non-linear (KEEN) waves in the presence of dynamical ions
作者: Rostislav-Paul Wilhelm, Philipp Krah, Kai Schneider, Fabio Bacchini, Virginie Grandgirard
类别: physics.plasm-ph, math.NA
发布日期: 2026-09-04
arXiv ID: 2609.04941v1
链接: http://arxiv.org/abs/2609.04941v1
摘要: We revisit the kinetic electrostatic electron nonlinear (KEEN) waves studied by Afeyan et. al in 2014 using a hybrid flow-mapping strategy that combines the characteristic mapping method (CMM) with numerical flow iteration (NuFi). The study extends the classical setup to dynamical ions, comparing their impact on the long-time KEEN dynamics with the static-ion case. To this end, we extend the CMM-NuFI framework with a multi-map strategy, assigning one map to the ion and one to the electron characteristic flow. The resulting problem exhibits a wide separation of spatial and temporal scales, driven by the fine structures generated by the ponderomotive force and by the large ion-to-electron mass ratio, which renders it computationally prohibitive for conventional grid-based methods. Our multi-map CMM-NuFI method efficiently resolves these disparate scales, enabling long-time simulations of KEEN dynamics with fully dynamical ions.
5. 保持迹的Fortin算子的构造
Construction of trace-preserving Fortin operators
作者: Franziska Eickmann, Tabea Tscherpel
类别: math.NA
发布日期: 2026-09-04
arXiv ID: 2609.04926v1
链接: http://arxiv.org/abs/2609.04926v1
摘要: We present a unifying framework to construct local Fortin operators for conforming mixed finite element pairs for the Stokes equations. The operators are constructed to satisfy the divergence-preservation property, local stability and approximation properties, and certain trace-preservation properties. For the latter, some of the finite element pairs require an enrichment of the velocity space. We present the construction for the $P_d-P_0$ element, the Bernardi-Raugel element, the conforming Crouzeix-Raviart element, a modified MINI element and generalised Taylor-Hood elements. Furthermore, we discuss implications of the existence of such a Fortin operator beyond inf-sup stability. These include applications to non-Newtonian fluid flow, problems with inhomogeneous Dirichlet boundary conditions, as well as uniform inf-sup stability relevant for domain approximation and moving domains.
6. 基于Flash Attention的快速高斯和
Fast Gauss Sums via Flash Attention
作者: Nicolaj Rux, Sebastian Neumayer
类别: cs.LG, math.NA
发布日期: 2026-09-04
arXiv ID: 2609.04910v1
链接: http://arxiv.org/abs/2609.04910v1
摘要: Gaussian kernel sums are the computational core of maximum mean discrepancies (MMDs), kernel gradient flows, Stein variational gradient descent (SVGD), and many other kernel methods. At the same time, softmax attention has received an extraordinary amount of hardware-aware code engineering, culminating in flash attention. We show that Gauss kernel sums with arbitrary, signed weights can be evaluated via flash attention: two small input augmentations turn the normalized softmax reduction into the unnormalized Gauss sum, without writing a single line of custom GPU code. For feature dimension D>8 in fp16, this approach beats compiled PyTorch code as well as PyKeOps kernels (often significantly) in speed, memory-overhead and accuracy. Indeed, its memory scaling remains linear.
7. 耦合port-Hamiltonian常微分方程的能量一致分裂与分解方法
Energy-Consistent Splitting and Decomposition Approaches for Coupled port-Hamiltonian ODEs
作者: Marius Mönch, Nicole Marheineke, Andreas Bartel, Kevin Schäfers, Michael Günther
类别: math.NA, math.DS
发布日期: 2026-09-04
arXiv ID: 2609.04884v1
链接: http://arxiv.org/abs/2609.04884v1
摘要: Operator splitting provides an attractive approach for the numerical integration of (coupled) port-Hamiltonian systems, as it allows the underlying system structure to be exploited at the level of the individual subproblems. However, the choice of the decomposition is not unique and may strongly affect both the computational efficiency and the preservation of the energy behavior of the original system. In this work, we investigate this interplay systematically and introduce energy consistency as a criterion for assessing splitting methods for port-Hamiltonian ordinary differential equations. We derive sufficient conditions under which a splitting based on a given decomposition inherits the energy behavior of the continuous system and use these conditions to analyze several decomposition strategies for coupled port-Hamiltonian systems. In particular, we compare decompositions that preserve the structure with approaches that exploit lower-dimensional subsystem dynamics or separated time scales. The analysis is complemented by numerical experiments using Strang splitting and its multiple-time-stepping extension. A scalable electro-thermal benchmark with fast electrical and slow thermal dynamics is employed to assess accuracy, energy behavior, and computational efficiency. The results demonstrate that preserving the port-Hamiltonian structure of the subflows is essential for energy-consistent splitting, whereas decompositions that exploit subsystem structure or time-scale separation can provide substantial computational advantages. In particular, the time-scale decomposition yields significant efficiency gains for systems with pronounced multirate characteristics, while structure-destroying decompositions may lead to undesirable energy behavior.
8. 高斯圆问题中哈代猜想的数值实验
Numerical experiments on the Hardy conjecture for the Gauss circle problem
作者: Satoshi Yamaguchi, Shoichi Fujima, Shigehiko Kuratsubo, Eiichi Nakai, Tsuyoshi Yoneda
类别: math.NT, math.NA
发布日期: 2026-09-04
arXiv ID: 2609.04725v1
链接: http://arxiv.org/abs/2609.04725v1
摘要: The classical unsolved Gauss circle problem concerns estimating the error between the number of lattice points inside a circle and the area of the circle as its radius tends to infinity. About a century ago, Hardy proposed a conjecture concerning this problem. In this paper, we attempt to provide numerical evidence in support of the Hardy conjecture through large-scale numerical computations.
9. 关于重启共轭梯度法的Forsythe猜想的完全解决
A Complete Resolution of Forsythe’s Conjecture for Restarted Conjugate Gradients
作者: Matthew J. Colbrook, George Stepaniants, Alex Townsend
类别: math.NA, math.DS, math.OC
发布日期: 2026-09-04
arXiv ID: 2609.04659v1
链接: http://arxiv.org/abs/2609.04659v1
摘要: Forsythe’s conjecture, published in 1968, asserts that for each restart length $s$, every exact-arithmetic restarted conjugate-gradient iteration on a real symmetric positive definite problem either terminates or has normalised residuals that converge separately along the even and odd restart subsequences. Apart from the classical steepest-descent case, this asymptotic question remained unresolved in full generality for nearly six decades. We give a complete classification by restart length in the original finite-dimensional setting and identify a sharp threshold. For $s=2$ and $s=3$, every problem either terminates or has convergent even and odd residual directions. For every $s\ge4$, there is a diagonal positive definite counterexample of dimension $s+4$ which never terminates and whose even residual directions do not converge. Together with Akaike’s theorem for $s=1$, this shows that the conjectured universal conclusion is true precisely for $s\in\{1,2,3\}$ and false for every $s\ge4$. The positive results follow from a degree-independent double-orthogonality identity and an analysis of the low-degree fixed-point sets. At restart length four, rational interval arithmetic and Sturm sequences certify a transverse Hopf point of the leading vector field of the rescaled squared-weight map. Analytic periodic-orbit and shadowing arguments yield the counterexample at restart length four, and degree elevation extends the construction to every larger restart length.
10. GPU上的自适应多层间断伽辽金方法
Adaptive Multilevel Discontinuous Galerkin Methods on GPUs
作者: Michał Wichrowski
类别: math.NA
发布日期: 2026-09-03
arXiv ID: 2609.04534v1
链接: http://arxiv.org/abs/2609.04534v1
摘要: I present a matrix-free symmetric interior penalty discontinuous Galerkin method for adaptively refined Cartesian meshes on GPUs. At non-matching interfaces, auxiliary \emph{shadow cells} represent the adjacent coarse polynomial on the fine level. This approach converts non-matching interfaces into matching faces, permitting uniform face evaluation throughout the mesh. I prove that this construction is equivalent to the standard non-matching formulation. Furthermore, this representation yields a local geometric multigrid method in which frozen shadows provide inter-level boundary data and carry residual contributions to coarser levels. A primal–dual pairing eliminates shadow assembly from the Krylov iteration. Numerical experiments with cubic elements ($p=3$) on an NVIDIA A100 show stable multigrid convergence under increasing refinement depth and efficient GPU execution.
11. 一种基于能量的守恒-耗散潜在神经演化算子用于磁化动力学
An Energy-Based Conservative-Dissipative Latent Neural Evolution Operator for Magnetization Dynamics
作者: Sebastian Schaffer, Lukas Exl
类别: cs.LG, math.NA, physics.comp-ph
发布日期: 2026-09-03
arXiv ID: 2609.04530v1
链接: http://arxiv.org/abs/2609.04530v1
摘要: We develop an energy-based reduced-order model for micromagnetic magnetization dynamics that couples a convolutional autoencoder to a structured latent neural ordinary differential equation. Motivated by the precessional-dissipative structure of the Landau-Lifshitz-Gilbert equation, the latent vector field is generated from the gradient of a learned scalar potential through an antisymmetric operator and a symmetric positive-semidefinite dissipative operator. This potential is learned in nonunique latent coordinates and is not identified with the Gibbs free energy, but decreases monotonically along autonomous continuous-time solutions, while the antisymmetric component permits motion along its level sets. The encoder, decoder, latent energy, and operators are trained jointly on short trajectory windows using latent and decoded-rollout losses alone, without time-derivative supervision, physical-energy labels, or dissipation penalties. At inference, an initial state is encoded once, evolved in latent space, and decoded only at the requested output times, enabling substantially cheaper trajectory prediction than the micromagnetic solver used to generate the training data. We compare quadratic, deep, and additive deep-quadratic latent energies on two datasets parameterized by field amplitude and generated for the two applied-field directions of the NIST $μ$MAG Standard Problem 4. Dissipative-only and antisymmetric-dissipative models achieve comparable accuracy on short training-style windows but differ substantially on uninterrupted rollouts, for which the antisymmetric-dissipative models provide markedly more accurate trajectory predictions. The deep-quadratic energy gives the best overall accuracy for both field directions and exhibits slower error growth when rollouts are extended to twice the training horizon.
12. 改进用于逼近逆矩阵的超幂方法的效率
Improving the efficiency of a Hyperpower method for approximating inverse matrices
作者: Alicia Cordero, Juan R. Torregrosa
类别: math.NA
发布日期: 2026-09-03
arXiv ID: 2609.04433v1
链接: http://arxiv.org/abs/2609.04433v1
摘要: In this manuscript, scalar norm-type accelerators are used for first time in order to increase the order of convergence of known iterative methods for solving the matrix equation $X^{-1}-A=0$, while decreasing the number of matrix-matrix products. Specifically, our proposed scheme upgrades the order of convergence in a $50\%$, achieving the sixth-order of convergence with the computational cost of fourth-order Hyperpower method, in terms of matrix-matrix multiplications. In the main results, not only the convergence and its order is proven, but also its efficiency and stability. This new technique provides good numerical results, even compared with existing fourth-, sixth- and eighth-order schemes in large matrices of up to $10^6$ entries, but also promises the opening of a new kind of procedures with good performance and scalability. Finally, an application to digital image restoration is made to show the behavior of the method, compared with existing ones.
13. 通过不动点迭代高精度计算Lewis权重
Computing Lewis Weights to High Precision by Fixed-Point Iteration
作者: Swati Padmanabhan
类别: cs.DS, math.NA
发布日期: 2026-09-03
arXiv ID: 2609.04338v1
链接: http://arxiv.org/abs/2609.04338v1
摘要: The $\ell_p$-Lewis weights of a matrix are defined by a fixed-point equation. For $p<4$, Cohen and Peng [CP15] showed that iterating an equivalent rearrangement of this equation computes Lewis weights to high precision; for $p\geq4$, prior high-precision methods instead use optimization-based approaches. We show that the direct Lewis fixed-point iteration, appearing in the thesis of Lee [Lee16], computes Lewis weights to high precision for every $p>2$. For a matrix $\mathbf{A}\in\mathbb{R}^{m\times n}$ partitioned into row blocks $\mathbf{A}_{[1]},\ldots,\mathbf{A}_{[k]}$, we compute, for $p>2$, coordinatewise $\varepsilon$-approximate $\ell_p$ block Lewis weights in $O\left(p\log\frac{p\sqrt{\sum_{i=1}^k\operatorname{rank}(\mathbf{A}_{[i]})}}{\varepsilon}\right)$ rounds of exact leverage-score-vector computations. For ordinary Lewis weights, this becomes $O\left(p\log\frac{p\sqrt{m}}{\varepsilon}\right)$, improving the $O\left(p^2\log(m/\varepsilon)\right)$ bound of Gribling, Sidford, and Zhang [GSZ26] for $p\geq4$. Our main observation is that each direct Lewis update contracts the KL divergence to the true weights by a factor of $1-\frac{2}{p}$. We also give an alternate explanation of this contraction through volume sampling and entropic independence. Synthetic experiments closely match our predicted local contraction rates and iteration counts grow approximately linearly with $p$; real-data experiments illustrate the information-concentration tradeoff of finite-$p$ block Lewis designs.