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1. 为什么凯洛格网格是径向的：一种隐藏的能量对称性
Why the Kellogg Mesh Is Radial: A Hidden Energy Symmetry
作者: Shun Zhang
类别: math.NA
发布日期: 2026-08-24
arXiv ID: 2608.22991v1
摘要翻译: 凯洛格（Kellogg）棋盘界面问题是鲁棒自适应有限元方法的一个经典基准。其成功的自适应网格是径向的：尽管存在大的对比度和非对称解，网格仍向界面交叉点强烈加密，却不显示任何角向结构。我们通过证明奇异解 $u(r,θ)=r^γμ(θ)$ 满足精确恒等式 $κ|\nabla u|^2=Λr^{2γ-2}$ 和 $κ|\nabla^2 u|_F^2=2(1-γ)^2Λr^{2γ-4}$ 来解释这一点，其中 $|\cdot|_F$ 是 Frobenius 范数，Hessian 在每个象限内分别计算，且 $Λ=γ^2\cos^2(πγ/4)$。关键之处在于消失的量：尽管 $κ$ 和 $u$ 也都依赖于角度，右端项却仅依赖于 $r$。这两个恒等式都跨越界面成立，且交叉点处的单元以相同方式平衡。结合等离散误差分布原则，这些恒等式表明目标单元密度是径向的，因此正确的网格除了向中心加密外不应显示任何其他结构。因此，熟悉的图像是鲁棒误差分析背后系数加权要求的空间结果，而非独立经验事实。这也使得该基准肉眼可读：一个非径向的凯洛格网格是计算未遵循系数加权局部困难的可见证据。这种解读特定于该基准；在加权困难非径向的两材料问题上，同样的鲁棒估计器会产生强材料偏置的网格。
Comment: 无
链接: http://arxiv.org/abs/2608.22991v1
1. 研究对象和背景
本文研究 Kellogg 棋盘式界面问题, 它是鲁棒自适应有限元方法的经典基准. 设 $\Omega=(-1,1)^2$, 扩散系数为
$$ \kappa(x)=\begin{cases} R, & x\in Q_1\cup Q_3,\\ 1, & x\in Q_2\cup Q_4, \end{cases} \qquad R>1, $$其中 $Q_1,\dots,Q_4$ 按逆时针排列. 模型问题为
$$ -\nabla\cdot(\kappa\nabla u)=0\quad\text{in }\Omega,\qquad u=g:=u_{\mathrm{exact}}\big|_{\partial\Omega}\quad\text{on }\partial\Omega. $$Kellogg 齐次奇异解具有形式 $u(r,\theta)=r^\gamma\mu(\theta)$, 其中 $\mu$ 在每个象限内是 $\cos(\gamma(\theta-\theta_i))$ 的适当组合, 参数 $\gamma,\rho,\sigma,R$ 满足匹配关系 (1.3). 经典取 $\gamma=0.1$ 时 $R\approx 161.4476387975881$, $\rho=\pi/4$, $\sigma\approx -14.92256510455152$. 该解在原点附近只有 $H^{1+\delta}$ 正则性 ($\delta<\gamma$), 且系数和角度分布都强烈不对称: 低系数象限中的振幅比高系数象限大约 $\sqrt{R}$ 倍. 因此从 $u$ 或 $|\nabla u|$ 本身完全看不出网格应具有径向对称性. 在自适应有限元文献中, Kellogg 问题的成功网格表现为向原点同心加密, 沿系数界面没有额外细化, 四个象限的元素数几乎相等; Chen 和 Dai 的数值比较清楚地展示了这一现象. 已有理论主要关注该基准上最优能量衰减 $(\#T)^{-1/2}$ 的恢复, 但最优收敛阶并不决定自由度的空间分配. 本文要填补的空白是: 为什么正确的 Kellogg 网格是径向的? 作者证明, 尽管 $\kappa$ 和 $u$ 都依赖角度, 但系数加权的局部误差量恰好只依赖半径; 再结合等分布原则, 就自然推出目标网格密度是径向的.
2. 主要定理或主要结果
本文的主要结果是如下精确恒等式. 定理 3.1 (隐藏能量对称性): 对 Kellogg 垂直两相棋盘构型及其齐次奇异模, 存在常数 $\Lambda>0$, 使得
$$ \kappa(x)|\nabla u(x)|^2=\Lambda r^{2\gamma-2}, $$$$ \kappa(x)|\nabla^2_P u(x)|_F^2=2(1-\gamma)^2\Lambda r^{2\gamma-4}, $$其中 $\nabla^2_P$ 是在四个象限内分别取二阶导数的破碎 Hessian, $|\cdot|_F$ 是 Frobenius 范数, 且
$$ \Lambda=\frac{\gamma^2 R}{1+R}=\gamma^2\cos^2\left(\frac{\pi\gamma}{4}\right). $$两个恒等式的右端都只依赖 $r$, 既不出现角度, 也不出现系数. 恒等式在每个开象限内成立; 由于振幅平衡 (3.19), 两个系数加权量的单侧迹跨界面相等, 因此可连续延拓到 $\Omega\setminus\{0\}$. 作为推论,
$$ \int_{B_h}\kappa|\nabla u|^2\,dx=\frac{\pi\Lambda}{\gamma}h^{2\gamma}, $$且每个象限各贡献四分之一. 推论 3.2 给出基准常数的闭式公式:
$$ R=\cot^2\left(\frac{\pi\gamma}{4}\right),\qquad \rho=\frac{\pi}{4},\qquad \sigma=\frac{\pi}{4}-\frac{\pi}{2\gamma}=-\frac{\arctan\sqrt{R}}{\gamma}, $$故 $\gamma=4\pi^{-1}\arctan(1/\sqrt{R})\in(0,1)$. 网格结论: 对不接触原点的界面拟合单元 $K$, 取 $s_K=1$, 局部误差尺度满足
$$ \varepsilon_K^2=2(1-\gamma)^2\Lambda h_K^2\int_K r^{2\gamma-4}\,dx \asymp h_K^4 r_K^{2\gamma-4}, $$即 $\varepsilon_K\asymp h_K^2 r_K^{\gamma-2}$. 等分布原则 $\varepsilon_K\asymp\varepsilon$ 给出 $h(r)\asymp\varepsilon^{1/2}r^{1-\gamma/2}$ 和局部密度 $n(r)\asymp\varepsilon^{-1}r^{\gamma-2}$. 对任意角区间 $I$ 和同心环域,
$$ N_I(r_1,r_2)=c_*\varepsilon^{-1}\frac{|I|}{\gamma}(r_2^\gamma-r_1^\gamma), $$所以等角扇区获得等量元素质量, 四象限平衡只是 $|I|=\pi/2$ 的特例. 对接触原点的单元, 固定 $s\in(0,\gamma)$, 局部尺度为
$$ \varepsilon_{K_{h_i,i}}=C_{s,K_1}h_i^\gamma,\qquad i=1,\dots,4, $$常数与象限无关, 等分布迫使 $h_i$ 可比. 因此目标网格密度是径向的, 只向中心细化, 界面本身不产生细化信号.
3. 研究方法、关键技术和核心工具
证明的核心障碍是: $\kappa$ 和 $u$ 各自都强烈依赖角度, 但它们组合成的系数加权局部量必须消去角度依赖. 作者不用全局正则性估计, 而是用元素级局部误差尺度
$$ \varepsilon_K=h_K^{s_K}\kappa_K^{1/2}|\nabla u|_{s_K,K}, $$它来自 Dupont-Scott 局部多项式逼近以及针对 Crouzeix-Raviart、间断 Galerkin、混合元等族的局部最优系数鲁棒先验估计. 证明分三个层次. 第一, 扇形内恒等式. 由于每个象限内 $\mu''+\gamma^2\mu=0$, 解为 $\mu_i=A_i\cos(\gamma(\theta-\theta_i))$; 角向振子守恒量 $\gamma^2\mu^2+(\mu')^2$ 为常数, 因此 $|\nabla u|^2=\gamma^2A_i^2r^{2\gamma-2}$. 对 Hessian, 将 $u$ 写成 $\operatorname{Re}(\lambda z^\gamma)$, 对全纯函数 $\Phi$ 有 $|\nabla^2(\operatorname{Re}\Phi)|_F^2=2|\Phi''|^2$, 从而 $|\nabla^2u|_F^2=2\gamma^2(1-\gamma)^2A_i^2r^{2\gamma-4}$. 第二, 也是真正的创新点: 用传输矩阵证明四象限的系数加权振幅平衡. 定义 $Y=(\mu,\tau)^T$, $\tau=\kappa\mu'$; 由 $u\in H^1(\Omega)$ 和 $\sigma=-\kappa\nabla u\in H(\mathrm{div};\Omega)$, $Y$ 的两个分量跨界面连续. 跨越宽度为 $\omega$、系数为 $\kappa$ 的扇区由矩阵
$$ M_\kappa(\omega)=\begin{pmatrix} \cos(\gamma\omega) & \frac{\sin(\gamma\omega)}{\gamma\kappa}\\ -\gamma\kappa\sin(\gamma\omega) & \cos(\gamma\omega) \end{pmatrix} $$传输, 其行列式为 1. 经过两个象限的矩阵 $N=M_1(\pi/2)M_R(\pi/2)$ 满足 $\operatorname{tr}N=2-\sin^2(\pi\gamma/2)(2+R+R^{-1})<2$. 单值性 $N^2Y(0)=Y(0)$ 结合 $\det N=1$ 与非零右上角元素排除 $N=\pm I$, 推出 $N$ 的两个特征值均为 $-1$, 从而 $\operatorname{tr}N=-2$, 得到 $\sin(\pi\gamma/2)=2\sqrt{R}/(1+R)$, 即 $R=\cot^2(\pi\gamma/4)$. 同时 $|\tau|=\gamma\sqrt{\kappa_i\kappa_j}|\mu|$ 在每个界面成立. 从界面两侧分别计算守恒量, 比较后得到 $\kappa_i\gamma^2A_i^2=\kappa_j\gamma^2A_j^2$, 因此四象限有公共常数 $\Lambda=\gamma^2R/(1+R)$. 第三, 将等分布原则转化为空间分配. 对远离原点的单元, 把定理 3.1 代入 $\varepsilon_K^2=h_K^2\int_K\kappa|\nabla^2u|_F^2dx$, 得 $\varepsilon_K\asymp h_K^2r_K^{\gamma-2}$, 于是 $n(r)\asymp\varepsilon^{-1}r^{\gamma-2}$. 对原点附近单元, 利用 Slobodeckij 半模的齐次缩放 $|\nabla u_i|_{s,K_{h_i,i}}=h_i^{\gamma-s}|\nabla u_i|_{s,K_{1,i}}$; 旋转不改变半模, 且 $\kappa_i^{1/2}|\nabla u_i|_{s,K_{1,i}}$ 与 $i$ 无关, 故 $\varepsilon_{K_{h_i,i}}=Ch_i^\gamma$, 等分布迫使四个原点单元尺寸可比. 整个推导不依赖具体有限元族、离散解和标记策略, 只依赖系数加权能量范数和等分布原则.
4. 与之前工作的比较
与已有工作相比, 本文的推进是多方面的. 第一, Kellogg 原始工作 [18] 给出了相交界面椭圆问题的正则性分析和齐次奇异模, 但从未考虑该模的系数加权局部量具有径向对称性. 本文证明的恒等式 (3.21)–(3.22) 是全新的; 它解释了为什么数值上长期观察到的正确网格是同心圆式的. 第二, 在自适应有限元文献中, Morin-Nochetto-Siebert [22] 将 Kellogg 问题引入自适应迭代, Nochetto-Siebert-Veeser 的综述和 Bonito-Canuto-Nochetto-Veeser 的 Acta Numerica 综述都以恢复最优能量衰减 $(\#T)^{-1/2}$ 为主要成就. 本文明确指出, 最优收敛阶并不决定自由度空间分配; 从全局误差衰减无法推出网格的空间结构. 作者将鲁棒性要求推进到空间预测层面, 这是以往文献没有做的一步. 第三, 关于鲁棒估计器, 已有大量工作处理系数无关常数: Bernardi-Verf"urt 和 Petzoldt 的残差分析, 恢复型、均衡残差型以及 Ern-Vohral'ik 等的通量重构方法. 本文不提出新的估计器, 而是说明这些方法共同遵循的系数加权能量范数本身, 在 Kellogg 问题上会产生径向网格; 因此 Chen-Dai 图 1 中 H(div) 恢复型估计器得到四象限元素比 1.03 的网格, 而去掉系数权重的残差估计器得到 8.35 的强偏斜网格, 正是因为没有遵循系数加权局部难度. 第四, 本文还提供了基准参数的闭式公式, 免去了传统上数值求解非线性系统 (1.3) 的步骤; 对 $\gamma=0.1$, 闭式公式逐位复现了文献中抄来抄去的常数 (1.4). 第五, 作者明确区分了 Kellogg 棋盘的特殊性与一般两材料问题: 对另一个满足传输条件的分片二次多项式例子, $\kappa|\nabla^2w|_F^2$ 在界面两侧分别为 $8$ 和 $8q$, 等分布导致 $h_+\simeq q^{-1/4}h_-$, 网格应强烈偏向低系数侧. 这说明径向性是垂直棋盘的特殊全局匹配结果, 不是普适结论. 本文的代价和局限是: 它证明的是连续等分布模型下的目标密度, 并不证明任何具体自适应算法一定收敛到该密度; 常数可以依赖固定的 $\gamma$ 或 $R$, 不提供 $R\to\infty$ 的一致估计, 因为 $R$ 与 $\gamma$ 通过 $R=\cot^2(\pi\gamma/4)$ 绑定; 推导还假设网格界面拟合、形状正则且无 hanging node. 对更一般网格或 hp 方法, 作者未显式展开.
2. 极大规模时谐Maxwell方程的全矩阵自由三网格预条件子
A Fully Matrix-Free Three-Grid Preconditioner for the Time-Harmonic Maxwell Equations at Extreme Scale
作者: Shubin Fu
类别: math.NA
发布日期: 2026-08-24
arXiv ID: 2608.22903v1
摘要翻译: 三维时谐Maxwell模拟产生大规模复不定系统，其网格粗化严格受相位精度限制。尽管无矩阵有限元内核能高效利用GPU吞吐量，标准多重网格求解器最终受限于精确粗网格分解的内存和通信成本。我们提出一种用于旋度共形N{é}delec离散的全矩阵自由、无分解三网格预条件子，该离散包含完美匹配层(PML)和最优混合求积。该方法采用外层FGMRES求解未移位的细网格方程，而中间网格修正由以复移位$2h$–$4h$循环作为预条件的固定工作量FGMRES计算。这一策略将复移位限制在辅助预条件子中，从而保留物理Maxwell算子。局部傅里叶分析推导出混合Maxwell分支和相容的边传递，确定了稳健的移位和Jacobi阻尼参数。通过与解析Maxwell格林张量的对比验证，我们的方法展现出极强的可扩展性：使用单一的求解器配置，在仅64块NVIDIA A100 GPU上，约108.9亿个复边未知量的均匀和高度非均匀系统均在42.0–72.0秒内求解完成。
Comment: 无
链接: http://arxiv.org/abs/2608.22903v1
1. 研究对象和背景
三维时谐Maxwell方程在雷达散射、天线设计、光子学和电磁成像等领域有广泛应用。当计算区域跨越多个波长时，保持固定空间分辨率会使未知量个数随传播距离的三次方增长，构成极端规模的复不定线性系统。该问题在数学上具有特殊困难：curl-curl算子拥有巨大的梯度零空间，波动项使横向分支不定，PML截断引入复各向异性系数。为控制长路径传播的相位误差，本文采用最低阶Nédelec H(curl)协调有限元，并结合最优混合求积以抑制数值色散。然而，得到的离散系统极其庞大且复不定，求解器设计面临严重瓶颈。
具体地，设时间因子$\exp(-i\omega t)$，电场$\mathbf{E}$满足如下电场方程：
$$\nabla\times(\mu_r^{-1}\nabla\times \mathbf{E}) - k_0^2 \epsilon_r \mathbf{E} = f,\quad k_0=\omega\sqrt{\epsilon_0\mu_0},\ f=i\omega\mu_0 J.$$在矩形计算域$\Omega$上引入笛卡尔PML后，方程变为
$$\nabla\times(\tilde{\mu}_r^{-1}\nabla\times \mathbf{E}) - k_0^2 \tilde{\epsilon}_r \mathbf{E} = f,\quad \mathbf{n}\times\mathbf{E}=0\ \text{on}\ \partial\Omega,$$其中$\tilde{\epsilon}_r=\epsilon_r S$，$\tilde{\mu}_r=\mu_r S$，$S$为复拉伸张量。其弱形式是：求$\mathbf{E}\in H_0(\mathrm{curl};\Omega)$使得
$$(\tilde{\mu}_r^{-1}\nabla\times\mathbf{E},\nabla\times\mathbf{v})-k_0^2(\tilde{\epsilon}_r\mathbf{E},\mathbf{v})=(f,\mathbf{v}),\quad \forall\mathbf{v}\in H_0(\mathrm{curl};\Omega).$$用最低阶Nédelec元离散得到$A_h U_h=F_h$，其中$A_h=C_h-k_0^2 M_h$，$C_h$和$M_h$分别为离散curl-curl和质量算子。由于负质量项和PML张量，$A_h$复对称且不定。
最优混合求积是一种非标准两点规则，其混合参数$\tau=1/2$，可消除标准Gauss积分下$O(h^2|\xi|^4)$的色散误差，使色散阶变为四阶而不增加未知量。该设计对高波数问题至关重要，但同时也使离散算子不再满足Galerkin嵌套关系，粗网格算符必须重新离散，这给多水平方法带来了理论挑战。
求解此类系统的传统多水平方法通常在粗网格上执行精确因子分解，导致内存和通信开销超线性增长，严重阻碍可扩展性。本文旨在填补这一空白：设计一个完全无矩阵、无因子分解的三重网格预条件子，其所有层只涉及算子作用、对角Jacobi数据和几何传递，从而充分利用现代GPU的大规模并行吞吐。
2. 主要定理或主要结果
本文的主要结果分为算法设计、理论分析和数值验证三部分。
算法上，定义了物理层次$A_h,A_{2h}$和辅助移位层次$B_{2h},B_{4h}$，其中$B_\ell=C_\ell-(1+i\sigma)k_0^2 M_\ell$（$\ell=2h,4h$）。预条件子$M_h^{-1}$的每次应用为Algorithm 1：细网格预平滑（固定工作GMRES加阻尼Jacobi）、限制、中间网格校正（由内层FGMRES求解，右预条件子为复移位2h-4h V循环）、延长校正、细网格后平滑。该方法的关键是外层FGMRES作用于未移位方程，而移位只影响内层预条件子，因此物理算子保持不变。
理论方面，命题4.1证明了在平移不变均匀介质中，混合离散的符号满足$\hat{C}_h=h\hat{C}^*Q_f\hat{C}$，$\hat{M}_h=h^3Q_e$；广义特征值问题$(\hat{C}_h,\hat{M}_h)$的特征值为$0$和$\lambda_h(\theta)$（二重），其中
$$\lambda_h(\theta)=h^{-2}\sum_{j=1}^3\frac{4\sin^2(\theta_j/2)}{q_j(\theta_j)},\quad q_j(\theta_j)=(5+\cos\theta_j)/6.$$因而物理算子$\hat{A}_h$的特征值为$-k_0^2$（梯度分支）与$\lambda_h(\theta)-k_0^2$（两个横向分支）。命题4.2证明几何传递在24维边调和空间上闭合，延长符号为
$$p_r^{\alpha}(\theta)=c_r^{1-\alpha_r}s_r^{\alpha_r}\prod_{j\ne r}c_j^{2(1-\alpha_j)}s_j^{2\alpha_j},\quad c_j=\cos(\theta_j/2),\ s_j=\sin(\theta_j/2),$$且限制符号满足$\hat{R}_h=8\hat{P}_h^*$，从而保证了多水平校正的兼容性。
数值验证上，在64块NVIDIA A100-40G GPU上，对均匀介质、收敛透镜、周期和随机各向同性介质，采用同一求解器配置，求解了约$1.089\times10^{10}$个复边未知数的问题，总求解时间42.0–72.0秒，矩阵无关设置时间小于0.8秒，峰值内存低于37.7 GiB/GPU。以作者所知，这是迄今报告的最大规模三维时谐Maxwell curl-conforming有限元模拟。
3. 研究方法、关键技术和核心工具
本文的核心技术障碍来自两方面：首先，标准多重网格在粗网格上需要精确因子分解，在极端规模下不可承受；其次，广泛使用的复移位技法（如shifted Laplacian）以修改物理算子的谱为代价换取迭代稳定性，这不适用于需要保真求解时谐Maxwell问题的场合。作者的创新是引入“物理-辅助”解耦框架：外层FGMRES始终针对未移位方程$A_h U_h=F_h$，而中间网格校正通过内层FGMRES实现，其右预条件子是复移位$2h$–$4h$循环。这样复移位被严格限制在辅助级别，不改变物理算子。
在理论证明中，作者采用了局部Fourier分析（LFA），其关键步骤包括：
- 选择合适的坐标变换（3.2），将边平均坐标下的基函数无量纲化，使得质量符号和curl符号具有显式张量积结构。这避免了复杂积分，直接组装得到命题4.1的符号公式。
- 通过矩阵分解证明特征值。将质量符号平方根代入，得$\hat{M}_h^{-1/2}\hat{C}_h\hat{M}_h^{-1/2}=h^{-2}X^*X$，其中$X=Q_f^{1/2}\hat{C}Q_e^{-1/2}$恰为叉积矩阵，从而立即得到梯度零空间和两个横向分支。
- 建立粗细网格传递的调和闭包。由于粗算子通过重新离散得到而非Galerkin乘积，无法使用代数关系，必须对几何插值进行精确傅里叶分析。作者在边平均坐标下分解一维插值的alias系数，导出延长符号（4.7），并利用平移交换性与Fourier基正交性严格证明限制符号为延长符号的伴随（在适当的归一化下）。
- 参数选择基于符号的抽样有限工作分析，确定移位$\sigma$和Jacobi阻尼$\omega_J$的鲁棒区间。由于具体数值未在本文前半部分给出，无法在此详述，但该分析确保了算法在不熟悉介质上的稳健性。
整个证明框架是：先由单层符号获得谱信息，再由调和闭包获得多水平校正的相容性，最后辅以参数分析保障收敛。该方法完全避免矩阵组装和分解，所有算子只以逐元素局部形式作用，非常适合GPU并行。
4. 与之前工作的比较
本文与现有求解器相比，在多个维度上取得了推进。
（1）与稀疏直接法和层次压缩法相比：稀疏直接法在三维问题中内存填充和因子分解运算量超线性增长，层次压缩虽压缩了存储，但未消除全局因子分解这一瓶颈。本文通过完全无矩阵和无因子分解的迭代框架，将内存占用控制在每GPU 37.7 GiB以下，支撑了10.89亿未知数，远超直接法的实际能力。
（2）与区域分解和扫掠预条件子相比：扫掠预条件子沿方向顺序更新子域，其内在方向依赖性和子问题同步成本与GPU大规模并行架构不兼容。本文的多水平结构没有方向扫掠，只依赖全局粗网格通信，能够充分发挥A100 GPU的并行吞吐。
（3）与既有并行几何多重网格工作（文献[5]，据称达到1.3 billion未知数）相比：本文不仅将规模提升至约10.89 billion（大了一个数量级），而且关键是在最粗网格上也放弃了精确因子分解，改用固定工作FGMRES迭代，真正做到了“无因子分解”。文献[5]仍需要在最粗层装配和分解算子，成为可扩展性瓶颈。
（4）与常见的复移位预条件子（如用于Helmholtz方程的shifted Laplacian）相比：传统做法是用复移位算子替代原方程，这会改变解的谱，甚至引入非物理的耗散。本文通过外层FGMRES保留未移位物理方程，复移位只作为辅助循环中的预条件子出现，从而在理论上不损失解的物理保真性。这是概念上的本质区别。
（5）理论分析方面：针对非Galerkin嵌套粗网格算子的情况，本文给出了边调和闭包和传递符号的严格推导。这比仅依赖Galerkin三重网格的早期分析更普适，为采用重新离散粗算子的多种实用情形提供了理论基础。
需要指出的是，作者未显式展开与具体论文的定量比较，例如未报告与文献[5]在相同问题上的时间/内存对比。本文的代价可能包括：外层必须使用FGMRES而非普通GMRES或CG；固定工作迭代使预条件子非线性，收敛分析依赖数值实验；方法目前针对笛卡尔网格和PML，对非结构网格的推广尚不明确。此外，LFA中的参数鲁棒区间在本文中并未给出具体数值，这有待在完整论文中呈现。
3. 时域Maxwell方程组对称分裂格式的尖锐CFL稳定性与时间色散优化
Sharp CFL stability and temporal-dispersion optimization of symmetric splitting schemes for time-domain Maxwell equations
作者: Hui Duan, Hongliang Li, Lunzhong Guo
类别: math.NA
发布日期: 2026-08-23
arXiv ID: 2608.22315v1
摘要翻译: 我们分析时域Maxwell方程组的一族显式回文电-磁分裂中的系数设计。在四阶交错空间离散后，Fourier放大矩阵依赖于单个标量 $g_2=a(1-2a)/2$。我们证明 $a=1/4$ 是最大化谱CFL区间的唯一实系数，其阈值为 $s_*=12/(7\sqrt d)$。随后，我们识别出实系数在更高相位精度上的障碍：消除领先时间相位缺陷需要 $g_2=1/12$，而每个实成员都满足 $g_2\le 1/16$。由此产生的复共轭系数对每个固定的半离散Fourier模式给出四阶时间相位精度，其阈值为 $6\sqrt3/(7\sqrt d)$，而完整的场更新在时间上仍保持全局二阶。对于实Maxwell数据，物理输出是复轨迹的实投影；该投影与分支无关，并保持二阶误差界。我们进一步给出一个完全等价的加倍实算术实现，它阐明了辅助虚部的作用，而不改变数值方法。半离散收敛结果和数值实验证实了稳定性优化与相位优化之间的区别。
Comment: 无
链接: http://arxiv.org/abs/2608.22315v1
1. 研究对象和背景
本文研究时域Maxwell方程在无损、线性、各向同性介质中的传播问题。在能量尺度变换下，源-free Maxwell方程化为
$$ \partial_t \begin{pmatrix} h \\ e \end{pmatrix} = \nu \begin{pmatrix} 0 & -\nabla\times \\ \nabla\times & 0 \end{pmatrix} \begin{pmatrix} h \\ e \end{pmatrix} = A \begin{pmatrix} h \\ e \end{pmatrix}, $$其中 $h=\sqrt{\mu}H$, $e=\sqrt{\varepsilon}E$, $\nu=(\varepsilon\mu)^{-1/2}$。在周期或PEC边界条件下，系统满足能量守恒律。算符 $A$ 分裂为 $A=C+D$，其中
$$ C=\nu\begin{pmatrix}0&0\\ \nabla\times&0\end{pmatrix},\quad D=\nu\begin{pmatrix}0&-\nabla\times\\0&0\end{pmatrix}, $$且 $C^2=D^2=0$，因此子流为显式 $\mathrm{e}^{\tau C}=I+\tau C$, $\mathrm{e}^{\tau D}=I+\tau D$。本文关注五子流回文分裂族
$$ \Phi_{\Delta t}(a)=\mathrm{e}^{a\Delta t C}\mathrm{e}^{\frac12\Delta t D}\mathrm{e}^{(1-2a)\Delta t C}\mathrm{e}^{\frac12\Delta t D}\mathrm{e}^{a\Delta t C},\quad a\in\mathbb C, $$该族保持时间对称且至少二阶一致。空间离散采用四阶交错Yee格式，得到改进波数 $\eta_\xi$，并定义 $q=\nu\Delta t K_h$, $s=\nu\Delta t/\Delta x$，其中 $K_h^2=\Delta x^{-2}\sum_{\xi=1}^d \eta_\xi^2$，且 $0\le q^2\le \frac{49d}{9}s^2$。在此设定下，Fourier放大矩阵退化为仅依赖标量 $g_2(a)=a(1-2a)/2$，稳定性条件为 $|A_a(q)|\le1$。问题的重要性在于：长时间波传播同时受CFL时间步限制和数值色散影响，二者不一定能被同一时间积分器最优兼顾。本文旨在严格刻画该单参数族中稳定性优化与时间相位优化的关系，揭示实系数固有的障碍，并给出复数系数下的相位最优解及其物理解释。
2. 主要定理或主要结果
本文的主要结果如下。
(1) 公共放大矩阵。对每个固定的非零横向Fourier模式，能量尺度半离散Maxwell系统化为谐波振荡子，应用分裂(6)得到一步矩阵
$$ M_a(q)=\begin{pmatrix}A_a(q)&B_a(q)\\ C_a(q)&A_a(q)\end{pmatrix},\quad \det M_a(q)=1, $$其中
$$ A_a(q)=1-\frac{q^2}{2}+\frac{g_2(a)}{2}q^4,\quad g_2(a)=\frac{a(1-2a)}{2}, $$$$ B_a(q)=-q+\frac{1-2a}{4}q^3,\quad C_a(q)=q+a(a-1)q^3+\frac{a^2(1-2a)}{4}q^5. $$特征多项式为 $\lambda^2-2A_a(q)\lambda+1=0$。因此CFL与色散问题归结于判定 $|A_a(q)|\le1$ 的区间。
(2) 实系数CFL最优唯一性（定理3.2）。在实系数族内，唯一使包含 $q=0$ 的连续谱稳定区间最大的系数是 $a=1/4$，此时 $g_2=1/16$，且 $|A_{1/4}(q)|\le1$ 当 $0\le q^2\le16$。结合四阶交错stencil的界限，对应的 $d$ 维CFL阈值为
$$ s_{\rm stab}^{*}=\frac{12}{7\sqrt d}. $$实际稳健步长要求 $s (3) 实系数相位障碍与复数相位优化（定理3.3）。数值相位 $\theta_a(q)$ 定义为 $\cos\theta_a(q)=A_a(q)$，$\theta_a(q)\sim q$ $(q\to0)$。其展开为 消去前导时间相位缺陷要求 $g_2=1/12$。但任意实系数满足 $g_2\le1/16<1/12$，因此没有实系数能实现四阶相位精度。在复数域中，$g_2=1/12$ 给出共轭对 对这两个系数有 即对固定的半离散波数 $K_h$，频率误差为 (4) 相位优化格式的谱CFL阈值（命题3.4）。当 $a=a_\pm$ 时，$A_a(q)=1-\frac{q^2}{2}+\frac{q^4}{24}$，其满足 $|A_a(q)|<1$ 当 $0 稳健步长要求 $s (5) 半离散二阶收敛（命题3.5）。在周期边界、常系数、固定交错网格下，设定所有非零Fourier模式满足 $0\le q\le q_0 这里 $\|\cdot\|_h$ 为复数离散 $\ell^2$ 范数（对 $\Phi_{\rm disp}$）。该命题只针对固定网格，未声称对连续Maxwell解的 $O(\Delta t^2+h^4)$ 一致估计。 (6) 复数轨迹的实投影与等价实现（推论3.7）。从相同实初值出发，$a_+$ 与 $a_-$ 生成的复数轨迹互为共轭，因此实部投影相同，且这个投影不损失二阶误差界。文中还给出了精确等价的 $2N$ 维实数实现，表明复数数据类型并非数学本质，只是实现上需处理辅助虚部。 本文的核心贡献并非引入新的空间离散或高阶组合，而是在固定五子流回文分裂框架下，对系数设计问题给出完整的数学分析。主要技术障碍与处理如下。 第一，将多维Maxwell问题约化为标量系数问题。通过Fourier变换，在能量尺度变量下，半离散Maxwell算子的每个横向Fourier模式只对应一个 $2\times2$ 谐波振荡子，放大矩阵具有对称形式。这一约化依赖于四阶交错空间离散的代数结构，使得稳定性条件与相位条件都只通过 $g_2(a)=a(1-2a)/2$ 这一标量进入。作者严格推导了放大矩阵的四个分量，并给出了特征多项式的形式，为后续优化提供了单一变量。 第二，实系数CFL最大化的证明。此处真正的困难在于证明最优性的唯一性。作者利用 $g_2(a)$ 的恒等式 $g_2=\frac1{16}-(a-\frac14)^2$，立即得到上界。然后证明若 $a\ne\frac14$，则在 $q^2=8$ 处 $A_a(8)=1-4+32g_2<-1$，从而稳定区间不可能覆盖 $[0,16]$。对于 $a=1/4$，$A_{1/4}(q)=-1+(q^2-8)^2/32$，所以稳定区间恰为 $0\le q^2\le16$。这个证明非常简洁，但给出了尖锐阈值。 第三，相位匹配与实系数障碍。数值相位 $\theta_a$ 的展开由 $A_a$ 确定。作者计算出 $\theta_a=q+(\frac1{24}-\frac{g_2}{2})q^3+O(q^5)$，因此消除 $q^3$ 项需要 $g_2=1/12$。由于 $g_2\le1/16$，所以实系数不可能。这个障碍是定理3.3的核心，它揭示了一个基本的不相容性：在本族中，实系数无法同时达到最优稳定域（$g_2=1/16$）和相位匹配（$g_2=1/12$）。复数系数的引入是对该障碍的解析延拓。 第四，整体二阶性与半离散收敛证明。复数相位优化虽然提高了固定Fourier模式的相位精度，但完整一步映射的局部截断误差仍是 $O(\Delta t^3)$，这从 $B_a(q)$ 中 $q^3$ 的系数与 $-\sin q$ 的展开不匹配可以看出。为了得到功率有界性，作者使用Cayley-Hamilton公式将 $M_a(q)^n$ 表示为 $\cos(n\theta)I+\frac{\sin(n\theta)}{\sin\theta}(M_a(q)-A_a(q)I)$。对于 $\Phi_{\rm stab}$，$q^2=8$ 处 $A=-1$ 且 $M=-I$，但副对角元线性消失，所以该点是可穿越的；对于 $\Phi_{\rm disp}$，则没有这种内点退化。将这些逐模式估计与离散Fourier变换结合，再用标准telescoping技巧即可得到二阶误差界。值得注意的是，该证明假设 $q_0$ 严格低于阈值，因此回避了端点缺陷矩阵的幂界问题。 第五，复数轨迹的实投影与等价实化。作者注意到复数系数产生的轨迹在复数化相空间演化。因为初值是实的，所以 $a_+$ 与 $a_-$ 的轨迹互为共轭，取实部后得到唯一物理输出。这里证明了投影分支独立性，并给出了一个等价的实数维系统，将实部和虚部同时推进。这个实现并没有删去虚部动力学，因此不改变数值格式，只是为实际计算提供了回避复数运算的方式。这一处理使得复数系数方法在工程实现中更具可行性。 整体上，本文的独创性在于对低阶段显式分裂给出了可调参数空间的精确边界，并区分了稳定性优化与相位优化这两个相互竞争的目标。它没有试图提高全局收敛阶，而是以牺牲部分CFL阈值来获得更好的色散特性，并用数学定理保证了这种权衡的必然性。 本文的比较对象涵盖经典FDTD、交替方向方法、辛FDTD、算子分裂、能量守恒分裂以及复数系数积分器。分几个方面比较。 与经典Yee FDTD及优化解的关系。Yee格式在交错网格上具有二阶空间和时间精度，本文考虑的四阶空间离散配合五子流分裂，在系数选择上拥有更多自由度。与Kusaf等[23]的优化symplectic-FDTD系数相比，本文在单参数回文族内证明了 $a=1/4$ 是唯一达到最大稳定区间的实系数，并给出了精确阈值 $s_{\rm stab}^*=12/(7\sqrt d)$。文献[23]的系数在实践中表现良好，但可能未经严格的最优性证明；本文补上了这一缺口，并指出该系数并非相位最优。 与高阶分裂方法比较。最近的高阶保结构分裂（如六阶方法[20]、结合边界闭包和无条件稳定的方法[22]）追求更高的全局阶或额外的结构性质。本文有意限制在五子流结构，不增加子流数，因此全局阶保持二阶。它以相位色散优化为目标，即对每个固定半离散模式给出四阶时间相位精度，但不提高完整场的收敛阶。这种取舍是本文与高阶级方法的本质区别：本文的优化是在低阶段代价下的色散改善，而不是全局误差阶提升。 与复数系数积分器的比较。复数系数在几何积分中已有理论基础[24-26]，通常用于提高阶数或实现某些稳定性性质。本文引入复数系数并非为了提升全局阶（实际上仍是二阶），而是用来满足相位匹配条件 $g_2=1/12$。这可以看作解析延拓的技巧：实系数无法达到该条件，但复数系数可以。文中特别指明，该方法不声称更严格的能量守恒，也不改变二阶误差界。此外，与一般复数系数方法不同，本文明确推导了等价的实数实现，说明复数不是必须的，只是数学上的简洁表示。这为实际计算提供了便利。 与ADI和LOD方法的比较。ADI和LOD方法旨在放松CFL限制，常常表现为无条件稳定。本文没有追求无条件稳定，而是在显式回文分裂的框架下优化可达到的CFL区间。本文给出的阈值是尖锐的，并且代价是相位优化时阈值下降约13.4%（从 $12/(7\sqrt d)$ 到 $6\sqrt3/(7\sqrt d)$）。这说明了稳定性与色散在显式低阶段格式中的固有张力。 局限与未覆盖之处。作者明确说明，本文的分析限于常系数、周期或PEC边界、均匀介质和固定网格。变系数、吸收层、边界闭包等没有纳入Fourier分析。半离散收敛结果不包含空间离散误差，因此没有给出对连续Maxwell解的一致 $O(\Delta t^2+h^4)$ 估计。这些是未来工作可能的延伸点。 Inertial Manifold Neural Operator for Dissipative Time-Dependent Partial Differential Equations 作者: Xiaoyang Xie, Clarence W. Rowley 类别: math.NA, cs.LG, math.DS 发布日期: 2026-08-24 arXiv ID: 2608.23546v1 链接: http://arxiv.org/abs/2608.23546v1 中文标题: 惯性流形神经算子用于耗散时间相关偏微分方程 摘要翻译: 在本文中，我们引入了惯性流形神经算子（IMNO）来求解耗散时间相关偏微分方程（PDE）。由于耗散，此类系统的长时间动力学通常表现出有效的低维结构。与标准神经算子架构（如傅里叶神经算子（FNO））不同，IMNO 显式利用低维结构，在非线性耗散 PDE 的长时程自回归训练和预测中实现更好的物理可解释性、准确性和稳定性。对于平移等变 PDE，我们进一步提出了所提神经算子的平移等变变体（IMNO-SE），确保输入中的空间平移在输出中引起相同的空间平移。这种保持对称性的归纳偏置显著提高了其在平移等变 PDE 中的性能。我们进行了广泛的基准实验，以数值评估 IMNO 的性能。 A variational framework for bond-based peridynamics with spatially varying horizons and its asynchronous time integration 作者: Prateek Prateek, Giuseppe Capobianco, Kai Partmann, Kestin Weinberg, Michael Ortiz, Sigrid Leyendecker 类别: cs.CE, math.NA, physics.comp-ph 发布日期: 2026-08-24 arXiv ID: 2608.23121v1 链接: http://arxiv.org/abs/2608.23121v1 中文标题: 具有空间变化视界的键基近场动力学的变分框架及其异步时间积分 摘要翻译: 键基近场动力学为断裂模拟提供了一个非局部框架，该框架不需要位移场的空间导数。然而，当空间变化的视界与非均匀离散化同时使用时，经典的单视界键基近场动力学公式会导致物质点之间的非对称相互作用。这些非对称相互作用违反平衡定律，并可能引入鬼力和虚假波反射等非物理伪影。在本工作中，我们针对具有空间变化视界的键基近场动力学建立了一个变分公式。从拉格朗日-达朗贝尔原理出发，我们推导了运动控制方程，并证明双视界近场动力学公式自然地产生于内能的变分。 在此变分结构的基础上，我们构造了异步变分积分器，允许在求解域的不同区域使用不同的时间步长。这对于具有局部细化的动态断裂模拟尤其有用，因为此类模拟仅在需要高分辨率或预期裂纹扩展的区域附近才需要小时间步长。涉及波传播、受拉预裂纹板和Kalthoff-Winkler冲击实验的数值算例表明，所提出的框架消除了由非均匀视界引起的虚假反射，保留了物理上一致的断裂模式，并获得了与均匀细化模拟相当的结果。同时，与标准速度-Verlet方法相比，异步变分积分器减少了内力评估的次数。因此，所提出的方法为具有空间变化视界的键基近场动力学模拟提供了一致的变分基础和高效的时间积分策略。 A Massively Parallel Three-Grid Preconditioner for the High-Frequency Helmholtz Equation 作者: Shubin Fu, Yitong Wang, Zixiao Zhao 类别: math.NA 发布日期: 2026-08-24 arXiv ID: 2608.22946v1 链接: http://arxiv.org/abs/2608.22946v1 中文标题: 高频亥姆霍兹方程的大规模并行三重网格预条件子 摘要翻译: 对三维时谐波在多个波长上的传播进行精确模拟，需要控制相位误差并高效求解大型不定系统。我们开发了一种基于紧凑27点插值优化有限差分（IOFD）离散格式的三重网格求解器。其波数相关模板支持每个最短波长六个点的细网格分辨率，并在 (2h) 网格上以每个最短波长仅三个点进行无偏移物理校正。该方法在 (h) 和 (2h) 网格上保持无偏移IOFD算子，而一个复偏移的 (2h)–(4h) 辅助循环为粗网格逆的免分解迭代近似提供预条件。将偏移限制在该辅助循环中保留了粗网格校正的传播特性。与出射格林函数的比较验证了在高达 (6144^3) 的网格序列上的相位和相对振幅精度，最大问题在每个坐标方向上跨越约1024个波长。相同的固定求解器配置在光滑、间断、高对比度和地球物理速度模型上均保持稳健收敛，并展现出可扩展的并行性能。特别地，一个在每个坐标方向上跨越约340个波长的问题，仅使用64块NVIDIA A100 GPU便在18.1秒内求解完成。 A Jin–Xin Relaxation Gradual Convergence Method for Conservation-Law PINNs 作者: Emmanuel Lorin, Yuchong Tang, Xu Yang, Yi Zhu 类别: math.NA 发布日期: 2026-08-23 arXiv ID: 2608.22493v1 链接: http://arxiv.org/abs/2608.22493v1 中文标题: 守恒律PINNs的Jin–Xin松弛逐步收敛方法 摘要翻译: 非线性双曲守恒律的Jin–Xin松弛引入了一个松弛参数，该参数控制分辨激波的内部层宽度；极限守恒律的不连续性仅在奇异极限中随着该宽度消失而出现。物理信息神经网络（PINNs）使用光滑网络近似，因此不太适合这种极限，而具有固定参数的松弛PINNs仅能分辨单一尺度，无法跟随向极限解的多尺度过渡。我们提出了Jin–Xin松弛逐步收敛方法（JXRGCM），该方法将松弛参数视为延拓变量，按计划将其退火至零，并从先前阶段热启动每个阶段，从而使得近似通过逐渐变尖锐的尺度跟随松弛剖面。在亚特征条件下，我们建立了一个稳定性估计，其常数与松弛参数无关；结合松弛极限，该估计为标量守恒律提供了关于熵解的$L^2$收敛速率$\mathcal{O}(\varepsilon^{1/4})$。对Burgers方程、浅水溃坝问题和Sod激波管的数值实验表明，与固定参数松弛PINNs和其他物理信息方法相比，JXRGCM提高了激波和稀疏波的分辨率。 On the extreme eigenvalues of the Gram Matrix in Physics-Informed Neural Networks for the Poisson Equation 作者: Bangti Jin, Longjun Wu 类别: math.NA 发布日期: 2026-08-23 arXiv ID: 2608.22426v1 链接: http://arxiv.org/abs/2608.22426v1 中文标题: 关于泊松方程物理信息神经网络中Gram矩阵的极值特征值 摘要翻译: 由微分神经正切核（DNTK）诱导的Gram矩阵的最小和最大特征值在基于梯度类算法训练的过参数化PINN的分析中起着关键作用。然而，由于多个微分算子的存在所带来的挑战，极值特征值的理论分析仍然完全缺失。本文针对无偏置项的两层RePU神经网络，对带有Dirichlet边界条件的Poisson方程的无穷DNTK矩阵的极值特征值给出了显式的下界和上界。该设置在采样点和输入维度(d)方面相当一般：在推导最小特征值的下界时，要求采样点为(\delta)-分离的，并且(d\geq 3)。这些结果推广了神经正切核的结果，并且据我们所知，是首次关于DNTK谱的结果。 A Hybrid Physics-Informed Neural Network Framework for Computing Dispersion Relations of SH Waves in Generalized Hetrogeneous Layered Media with Applications 作者: Subhajyoti Sarkar, Santimoy Kundu 类别: math.DS, math.NA 发布日期: 2026-08-23 arXiv ID: 2608.22353v1 链接: http://arxiv.org/abs/2608.22353v1 中文标题: 一种用于计算广义非均匀分层介质中SH波频散关系的混合物理信息神经网络框架及其应用 摘要翻译: 本文提出了一个数学和计算框架，用于计算连续变化非均匀分层结构中剪切水平（SH）波的频散关系。该方法将非均匀层的贡献从完整频散关系中分离出来，利用物理信息神经网络（PINN）学习该贡献，随后结合训练好的模型来确定完整的频散关系。研究了离散化问题的数学性质，包括有限差分系统的奇异性和层解的振荡行为，同时为PINN近似建立了泛化误差估计。该框架首先在地震学构型上进行了测试，该构型由覆盖在花岗岩半空间上的非均匀砂岩层组成，其中考虑了剪切模量和密度具有独立变化率的指数非均匀性。所提出的方法在特殊情况下通过解析解进行了验证，而对于一般构型，Haskell矩阵方法表明，随着均匀子层数量的增加，其收敛于PINN预测的连续变化频散关系。参数研究进一步证实了与基本物理的一致性。该方法的一个重要特征是，非均匀层方程可以独立训练并在多种设置中重复使用。为证明这一点，将同一训练好的网络与受根本不同物理定律控制的压电和压磁基底耦合，凸显了PINN框架作为非均匀分层介质频散分析可复用计算模块的潜力。 Accurately computing quasiperiodic parabolic equations within finite-size domains via modeling quasiperiodic boundary conditions 作者: Xiaofang Han, Kai Jiang, Meng Li 类别: math.NA 发布日期: 2026-08-23 arXiv ID: 2608.22241v1 链接: http://arxiv.org/abs/2608.22241v1 中文标题: 在有限尺寸域内通过建模准周期边界条件精确计算准周期抛物型方程 摘要翻译: 准周期系统表现出无衰减的长程有序，并且自然地定义在全空间上。然而，在实际应用中，计算在有限尺寸域上进行，这使得选择保持全局准周期结构的边界条件成为一个关键的建模挑战。特别是，传统边界条件不包含计算域之外准周期场的任何信息。传统的周期边界条件（PBCs）由于对无理数的有理逼近而遭受丢番图误差，限制了其精度。受此启发，我们针对准周期问题提出一类准周期边界条件（QBCs），该类边界条件避免了传统丢番图误差带来的局限性。通过利用低维物理域与高维环面之间的同态，QBCs 有效地捕获了边界处的长程结构。为验证所提方法，我们将 QBCs 应用于求解有限尺寸域内的准周期抛物型方程（QPEs），并建立了严格的收敛性结果。数值实验表明，QBCs 显著降低了丢番图误差的影响。当用于建模有限尺寸 QPEs 并与合适的数值离散格式相结合时，它们能够对高正则性和低正则性情形进行精确且高效的计算，同时表现出较 PBCs 改进的收敛性。 Weighted Laplacian Flow: A Deterministic Particle Flow with Provable Convergence 作者: Weiye Gan, Tangjun Wang, Zuoqiang Shi 类别: math.NA, math.AP 发布日期: 2026-08-22 arXiv ID: 2608.21831v1 链接: http://arxiv.org/abs/2608.21831v1 中文标题: 加权拉普拉斯流：一种具有可证明收敛性的确定性粒子流 摘要翻译: 从目标概率密度中采样是统计学、机器学习和科学计算中的一项基本任务。我们引入了加权拉普拉斯流，这是一种确定性粒子流方法，可将样本从易于处理的初始密度传输到归一化未知的目标密度。该方法演化目标分布与传输分布之间的对数密度比，并通过求解与目标密度相关的加权泊松方程来构造粒子速度。这种设计避免了选择核函数的必要性，并允许沿流动直接控制粒子权重。我们建立了所提出的偏微分方程系统的全局适定性，并证明了传输密度在$L^\infty$距离和Kullback-Leibler散度下均收敛到目标密度。在次线性强迫条件下，该方法在有限时间内实现精确收敛。针对多模态、重尾和十维目标的数值实验表明，加权拉普拉斯流能够执行长距离质量传输，克服能量势垒。 Eigenvalue asymptotics and uniform eigenfunction bounds for the fractional Laplacian in the interval 作者: Cheng Zhang 类别: math.CA, math-ph, math.NA, math.SP 发布日期: 2026-08-24 arXiv ID: 2608.23457v1 链接: http://arxiv.org/abs/2608.23457v1 摘要: We prove a three-term asymptotic formula for the eigenvalues of the fractional Laplacian in the bounded interval. This improves the eigenvalue asymptotics of Kulczycki–Kwaśnicki–Małecki–Stós and Kwaśnicki, and confirms the conjectural $O_α(n^{-2})$ remainder by the numerical simulations of Kaleta–Kwaśnicki–Małecki. We also prove that the normalized eigenfunctions are bounded uniformly in the eigenvalue index $n$ and the fractional order $α$. This settles the conjecture proposed by Kwaśnicki through numerical experiments. Transform-Based Multilinear Algebra via Tensor Decomposition 作者: Yidan Mei, Shenghan Mei, Ziqin He, Can Chen 类别: math.NA 发布日期: 2026-08-24 arXiv ID: 2608.23366v1 链接: http://arxiv.org/abs/2608.23366v1 摘要: Transform-based tensor products, including the T-product and its more general form, namely the higher-order tensor-tensor product, have become fundamental tools for multilinear data analysis in applications such as image processing, signal reconstruction, and robotics. While invertible transforms enable tensor computations to be carried out via matrix operations in the transform domain, the resulting storage and computational costs remain prohibitive for high-dimensional, higher-order tensors. To address this challenge, we integrate low-rank tensor decomposition techniques, specifically tensor train decomposition (TTD) and hierarchical Tucker decomposition (HTD), into transform-based multilinear algebra to improve computational and memory efficiency. In particular, we develop TTD- and HTD-based formulations for the T-product and its associated key algebra, such as block diagonalization and tensor singular value decomposition, by operating directly on the factor matrices or tensors of the decompositions. The framework is further generalized to the higher-order tensor-tensor product and applied to multilinear model order reduction problems. We demonstrate the effectiveness and efficiency of our framework with numerical examples. Least-Squares and Low-Rank Approximation for Linear Relations Using a Diagrammatic Language 作者: Júlia de Araújo Mota, Iago Leal de Freitas, Lucas Rufino, João Paixão 类别: cs.SC, math.NA, math.RA 发布日期: 2026-08-24 arXiv ID: 2608.23324v1 链接: http://arxiv.org/abs/2608.23324v1 摘要: We employ the machinery of linear relations to the study of optimization problems in linear algebra. We first show that the relational version of the pseudo-inverse can be realized through a generalization of the least-squares problem. This allows one to prove that the pseudo-inverse realizes the solution of certain relational optimization problems. Our main result is showing that a certain truncation of this pseudo-inverse defines a solution to a relational version of the classical low-rank approximation problem which recovers both the Eckart-Young Theorem and several optimization problems involving pairs of matrices and vector spaces. The longest-edge bisection algorithm may produce degenerating tetrahedra 作者: Sergey Korotov 类别: math.NA, math.CO 发布日期: 2026-08-24 arXiv ID: 2608.23139v1 链接: http://arxiv.org/abs/2608.23139v1 摘要: An explicit sequence of tetrahedra generated by the longest-edge bisection algorithm is shown to degenerate. The example violates shape regularity and both the minimum- and maximum-angle conditions, demonstrating that arbitrary tie-breaking among longest edges does not guarantee nondegeneration. A particle method for the Boltzmann equation via amortized sampling from Green’s function of the lifted linear operator 作者: Zichang Ju, Lei Li, Yang Yu 类别: math.NA, physics.comp-ph 发布日期: 2026-08-24 arXiv ID: 2608.22880v1 链接: http://arxiv.org/abs/2608.22880v1 摘要: The collision operator for the Boltzmann equation is a nonlinear nonlocal operator. When lifted in the extended 2-particle space, it is viewed as the projection of a collisional linear operator. In this paper, we propose a particle method that samples the post-collision relative velocity directly from the Green’s function of this operator (the transition probability of the generated time-continuous Markov chain). The normalizing flow amortized sampling is then proposed to reduce the sampling complexity. The resulted method takes $O(N)$ each time where $N$ is the particle number, and conserves momentum and energy exactly. This method does not require the boundedness of the kernel and, more importantly, it allows learning the Green’s function directly from the scattering data without selecting the kernel in a specified family. Mirror descent algorithms with logarithmic barriers 作者: Alberto De Marchi, Yura Malitsky, Adrien B. Taylor 类别: math.OC, cs.LG, math.NA 发布日期: 2026-08-24 arXiv ID: 2608.22834v1 链接: http://arxiv.org/abs/2608.22834v1 摘要: This work derives convergence guarantees for mirror descent and proximal mirror descent algorithms when a logarithmic barrier is used as a distance-generating function. Standard approaches cannot be applied when the solution lies on the boundary, where the Bregman divergence blows up. We show that, in a specific setting, both methods enjoy an $O(\log k / k)$ rate, which is also tight. In addition, our contributions include: (i) a new technique for handling the blow-up; (ii) a resolution of a gap in the theory of relative smoothness; and (iii) a comparison of the proposed approach with interior-point methods. Symmetric Taylor–Hood elements for the linear stress gradient problem 作者: Jun Hu, Ting Lin, Shudan Tian, Min Zhang 类别: math.NA 发布日期: 2026-08-24 arXiv ID: 2608.22730v1 链接: http://arxiv.org/abs/2608.22730v1 摘要: We develop mixed finite element methods for the linear stress-gradient elasticity problem based on symmetric Taylor–Hood elements. We establish the stability of the symmetric Taylor–Hood pair on simplicial meshes in both two and three dimensions, thereby resolving the stability left open by Brezzi, Fortin, and Marini in 1993. Combing the Nitsche’s method, we further construct finite element schemes for linear stress problem in general boundary condition. Numerical experiments confirm the theoretical results. Diagonal Frog meets ADI: trading matrix exponentials for rational maps in the Fokker–Planck equation 作者: Andrey Itkin, Rakhymzhan Kazbek 类别: math.NA, physics.comp-ph, q-fin.CP, q-fin.PR 发布日期: 2026-08-24 arXiv ID: 2608.22703v1 链接: http://arxiv.org/abs/2608.22703v1 摘要: A companion paper \cite{ItkinDF2026} introduced the Diagonal Frog (DF) positivity-preserving schemes for anisotropic Fokker–Planck equations, advancing each directional substep by a Krylov-computed matrix exponential, which dominates the cost. Replacing that exponential by a rational map $r(γL)$ reduces the substep to a banded solve, but the positivity argument no longer applies. We prove that for eventually exponentially positive generators the entrywise sign of $r(γL)$ at large steps is decided by a single number, the value $r(\infty)$ taken on infinitely stiff modes. Nonnegativity holds above a computable threshold when $0\le r(\infty)<1$, and at most on a bounded interval, empty or vanishingly narrow in all our tests, when $r(\infty)<0$. The criterion rejects the Crank–Nicolson (trapezoidal) method, where $r(\infty)=-1$, and selects the subdiagonal Padé$(0,2)$ method, which is second order, L-stable and provably positive above an explicit threshold. The resulting DF-ADI scheme costs $O(N)$ per step, keeps the implicit factorized mixed derivative unchanged, is second order in space and time, and conserves discrete mass exactly. In the strong cross-diffusion regime, however, the directional factors demand a step larger than the mixed derivative permits, so the composite second-order scheme is only empirically positive there, and the criterion serves to discriminate the well-behaved multiplicative factors from the stabilizing-correction schemes rather than to guarantee positivity. Against the Krylov exponential it runs ten to thirty-two times faster at matched accuracy in our tests with the gain growing with the mesh. We extend the construction to the backward Kolmogorov equation and to jump-diffusion models. A general-purpose sensitivity method for multiple simultaneous parameter perturbations in stochastic reaction networks 作者: David F. Anderson, Jingyi Ma 类别: stat.ME, math.NA, math.PR, q-bio.QM 发布日期: 2026-08-23 arXiv ID: 2608.22627v1 链接: http://arxiv.org/abs/2608.22627v1 摘要: Stochastic reaction networks are continuous-time Markov chain models for interacting populations, with applications in biochemistry, epidemiology, ecology, and related areas. We study finite-difference sensitivity estimation when a single estimator requires several nearby parameterized paths. Existing variance-reducing couplings are typically pairwise, so that repeated use is either inefficient or requires application-specific choices in multi-path settings. We introduce the multi-path stacked coupling (MSC), a space-time Poisson construction that jointly generates any finite collection of parameterized paths. Each pairwise marginal of MSC has the same law as the corresponding split coupling pair, allowing existing variance bounds to transfer directly; in finite-state settings, we also obtain first-order expansions for the mean and second moment of finite-difference numerators. We apply MSC in three settings of practical importance: estimating many first derivatives simultaneously, estimating a single first derivative using a wider finite-difference stencil, and estimating higher-order derivatives. Numerical experiments on a processive phosphorylation network demonstrate strong performance in each of the three application areas considered, consistent with the theoretical advantages of MSC: across all three applications, MSC achieves the smallest root mean square error (RMSE) among the methods considered over the tested computational budgets. On the Accuracy of Gradient Random Walk Methods for the Heat, FitzHugh-Nagumo, and Burgers’ Equations 作者: Stephen Abkin, Prabir Daripa 类别: math.NA 发布日期: 2026-08-23 arXiv ID: 2608.22592v1 链接: http://arxiv.org/abs/2608.22592v1 摘要: Gradient Random Walk (GRW) methods represent the spatial derivative of a solution with weighted particles and recover the solution by cumulative summation. Measured accuracy depends not only on the particle count but also on where the reconstruction is evaluated, how the boundary data are incorporated, and how the physical solution is recovered from the computed field. We separate these contributions for the heat equation, a scalar FitzHugh-Nagumo traveling front, and Burgers’ equation treated through the Cole-Hopf transformation, using multi-seed ensembles, paired reconstructions of identical trajectories, and deterministic controls that distinguish stochastic from systematic error. For the heat equation, an apparent error plateau at fixed bin count is traced to a half-bin mismatch between the cumulative sum and its comparison points, and realigning the comparison removes it. For Burgers’ equation, the accuracy of the recovered solution is set by the boundary data for the transformed variable, and exact transformed data remove this limit. For the FitzHugh-Nagumo front, errors in the profile, front location, and speed decrease under particle refinement, before and after the translational component is removed, verifying the deterministic reaction-weight formulation. With the evaluation and boundary conventions held fixed, the stochastic error decreases in the particle count $N$, consistent with the Monte Carlo convergence rate $O(N^{-1/2})$. These findings identify the operations that govern measured GRW accuracy and show how to improve it. Two-level domain-decomposition AdaGrad method for scalable training of graph neural networks 作者: Laurynas Varnas, Julien Herrmann, Alexander Heinlein, Serge Gratton, Alena Kopaničáková 类别: math.NA, cs.LG 发布日期: 2026-08-23 arXiv ID: 2608.22575v1 链接: http://arxiv.org/abs/2608.22575v1 摘要: Graph neural networks (GNNs) have emerged as a powerful framework for learning from graph-structured data. However, their efficient training remains challenging, particularly in distributed computing environments. This challenge arises from the use of message passing, which couples all graph nodes, leading to expensive optimization steps, high memory requirements, and substantial communication overhead. To alleviate these limitations, we propose a novel domain-decomposition (DD) variant of AG2m, an AdaGrad method enhanced with second-order curvature information and momentum, denoted by DD-AG2m. The proposed DD-AG2m alternates between AG2m optimization on the original (global) graph and AG2m optimization on the partitioned graphs. To incorporate global information at reduced cost, we further introduce a two-level variant (2DD-AG2m) that performs global optimization steps on a coarse graph obtained by randomly subsampling nodes within each subdomain. Numerical experiments spanning graph classification, node-level regression, and spatiotemporal forecasting tasks demonstrate that the proposed DD methods reduce the computational cost required to achieve the same predictive performance by a factor of 4-8. Moreover, for the fixed computational cost, they improve the predictive performance of GNNs by up to 22% compared with the baseline AG2m. Optimal compression of kernel matrices by interpolets with application to high-dimensional approximation 作者: Helmut Harbrecht, Lucio Antonio Rosi 类别: math.NA 发布日期: 2026-08-23 arXiv ID: 2608.22560v1 链接: http://arxiv.org/abs/2608.22560v1 摘要: We consider the compression of kernel matrices on the unit interval $[0,1]$ by interpolets that have sufficiently many vanishing moments. We define a compression rule which discards most matrix coefficients without compromising the accuracy offered by the underlying discretization. Since interpolets can be scaled such that the compressed kernel matrices are well conditioned, we derive a fully discrete scheme that solves a kernel interpolation problem under consideration in linear overall complexity. We finally generalize this approach to the unit $n$-cube $[0,1]^n$ by means of the sparse grid combination technique. Numerical experiments are carried out to validate the theoretical findings. Non-asymptotic Analysis of Matérn Regression: The Roles of Target and Kernel Lengthscales 作者: Daniel Sanz-Alonso 类别: math.ST, math.NA, stat.CO 发布日期: 2026-08-23 arXiv ID: 2608.22553v1 链接: http://arxiv.org/abs/2608.22553v1 摘要: Theoretical guarantees for kernel regression are typically formulated in terms of smoothness, but practical accuracy depends critically on how the design resolution compares with the target and kernel lengthscales. We develop a finite-sample theory for Matérn regression on periodic domains with quasi-uniform designs, covering noiseless interpolation and noisy kernel ridge regression. Our minimax result shows that accurate recovery requires a design dense enough to resolve the target lengthscale and sufficient information at that scale to overcome noise. We prove that Matérn interpolation additionally requires the design to resolve the kernel lengthscale: if the kernel is too short relative to the point spacing, an additional error remains even when the target is well resolved. For noisy Matérn regression, we derive a three-term fixed-ridge risk characterization consisting of target-scale bias, kernel-scale bias, and variance, and show that optimizing over the ridge parameter yields four distinct contributions. When the target is no more than twice as smooth as the kernel, choosing a kernel lengthscale longer than the target does not worsen the oracle risk, whereas choosing one too short can. Thus these resolution conditions determine when accurate recovery becomes possible, while smoothness determines how rapidly the error decreases thereafter. Quasi-optimality of adaptive FEM for optimal control problems involving Dirac measures governed by biharmonic equation 作者: Asha K. Dond, Neela Nataraj, Subham Nayak 类别: math.NA 发布日期: 2026-08-23 arXiv ID: 2608.22528v1 链接: http://arxiv.org/abs/2608.22528v1 摘要: This article establishes the quasi-optimality of adaptive nonconforming finite element methods for a class of optimal control problems involving Dirac measures governed by the biharmonic equation. The nonconforming Morley finite elements are employed for discretising both the state and adjoint variables. A modified right-hand side through a companion operator that maps Morley finite elements toaconformingspacehelpstoovercomethechallengeinhandlingpointsourcesontheright-hand side. A priori and a posteriori error estimates for the optimal control problems are derived. Further,optimal convergence rates for adaptive finite element methods are established using an axiomatic framework: by proving key properties such as stability, reduction, discrete reliability, and quasi-orthogonality. Numerical experiments for three types of optimal control problems are discussed extensively and they validate the theoretical results. Stability analysis of consistent splitting implicit-explicit multistep methods up to ninth-order accuracy for incompressible flows 作者: Yuanyuan Kang, Hong-lin Liao, Guidong Liu 类别: math.NA 发布日期: 2026-08-23 arXiv ID: 2608.22481v1 链接: http://arxiv.org/abs/2608.22481v1 摘要: This work presents a concise, unified stability theory of high-order decoupled \lan{implicit-explicit linear multistep (IELM)} methods based on the well-known consistent splitting technique for the incompressible Navier-Stokes equation. With the help of the recent semi-generating function approach and the global discrete energy analysis, one can establish the unconditional stability of a consistent splitting IELM method with respect to the $\ell^{\infty}(H^1)\cap \ell^{2}(H^2)$ norm if the associated implicit-explicit controllability intensity is larger than $\sqrt{2}/2$, a constant determined by the Stokes pressure estimate. It is shown that the $β$-parameterized GBDF-$\rmk$ ($2\le \rmk\le5$) schemes and $γ$-parameterized SIELM-$\rmk$ ($2\le \rmk\le9$) schemes can fulfill this requirement of implicit-explicit controllability intensity by choosing proper parameters so that they can theoretically maintain the unconditional stability of the associated consistent splitting IELM methods. Numerical experiments are also included to support our theory. An Efficient IMEX-SDIRK2 mr-ccSAV Scheme for the Forced Navier–Stokes Equations with Uniform-in-Time Enstrophy Bounds 作者: Honglin Liao, Haifeng Wang, Xiaoming Wang 类别: math.NA 发布日期: 2026-08-23 arXiv ID: 2608.22445v1 链接: http://arxiv.org/abs/2608.22445v1 摘要: We propose and analyze an IMEX-SDIRK2 mean-reverting concurrent-correction scalar auxiliary variable (mr-ccSAV) method for the forced two-dimensional periodic Navier–Stokes equations in vorticity form. The viscous term is treated by Alexander’s SDIRK2 method and advection explicitly. Each stage requires two elliptic solves with the same shifted Laplacian and the solution of either a cubic or a linear scalar algebraic equation. For initial vorticity in $\dot L^s(Ω)$, $s>2$, a stage solution exists for every positive time step; uniqueness is established separately under an explicit small-step condition. The principal result is a direct, unconditional uniform-in-time enstrophy bound for arbitrary positive time steps. For persistently bounded forcing, this estimate is absorbing: the influence of the initial data decays, and the forcing contribution does not accumulate in time. Under additional regularity, uniformly bounded step sizes, and bounded neighboring step ratios, we also establish uniform-in-time $H^1$ and $H^2$ vorticity bounds without a small-step condition. For smooth solutions, the method converges optimally at second order. Numerical experiments confirm its accuracy, long-time robustness, and effectiveness of a companion embedded time-step selector. Upper Hölderian with Explicit Exponent of Solution Mapping with Applications to Ball Constrained Least Squares Problems 作者: Yu Wang, Shenglong Hu 类别: math.OC, math.NA 发布日期: 2026-08-23 arXiv ID: 2608.22436v1 链接: http://arxiv.org/abs/2608.22436v1 摘要: In this paper, we propose an extension of the well-known Robinson implicit function theorem for generalized equations from the upper Lipschitzian case to the upper Hölderian case. Explicit exponents dependence between the generalized equation and its linearization is determined. Applications to ball constrained least squares problems, including linear least squares and separable nonlinear least squares, are studied. In particular, we establish that the solution mapping of ball constrained linear least squares under linear perturbation is locally upper Hölder continuous with exponent $1/3$, which is of independent interest. Ultilizing the upper Hölderian version of the implicit function theorem, we show the local upper Hölderian of the solution mapping of parametric ball constrained linear least squares. Optimal Condition Numbers in Low-Rank Positive Semidefinite Matrix Sensing 作者: Mingxuan Sun, Zhiqiang Xu 类别: cs.IT, math.FA, math.NA 发布日期: 2026-08-23 arXiv ID: 2608.22418v1 链接: http://arxiv.org/abs/2608.22418v1 摘要: In this paper we focus on the stability of positive semidefinite matrix sensing maps $Φ_{\mathcal{A}}(X)=(\langle A_i,X\rangle)_{i=1}^m$ where $A_i\succeq 0$, $X\succeq0$ and $\operatorname{rank}(X)\le r$. We introduce the bi-Lipschitz constants of $Φ_{\mathcal{A}}(X)$ and define the global condition numbers as the ratio of upper and lower Lipschitz constants. We give deterministic universal lower bounds for these condition numbers, that depend only on the rank $r$ and on the underlying field. We then investigate the random rank-one Gaussian measurements and show that our lower bounds on condition numbers are asymptotically sharp, and therefore the random rank-one Gaussian measurements are asymptotically optimal. As an application, we derive the stability guarantees for an $\ell_1$-residual PhaseLift-type estimator at the optimal sampling scale. A New Generalized Low-Rank Cholesky Factor ADI Algorithm for Large-Scale Stein Equations 作者: Umair Zulfiqar 类别: math.NA, eess.SY 发布日期: 2026-08-23 arXiv ID: 2608.22406v1 链接: http://arxiv.org/abs/2608.22406v1 摘要: The low-rank alternating direction implicit (ADI) method is an efficient solver for large-scale Stein equations with low-rank solutions. This paper shows that, as in the continuous-time Lyapunov equation case, the low-rank Cholesky factor ADI (LRCF-ADI) method for Stein equations implicitly performs $\mathcal{H}_2$-pseudo-optimal model order reduction for discrete-time systems. This observation leads to an automatic shift-generation strategy, allowing LRCF-ADI to select subsequent shifts without user intervention.
The standard LRCF-ADI method requires shifts outside the unit circle. We generalize the method to allow shifts anywhere in the complex plane, including on the unit circle. This extension enables numerical integration for frequency-limited Stein equations by interpolating the integrand at points on the unit circle. It also enables non-intrusive, data-driven balanced truncation and frequency-limited balanced truncation using experimentally measurable transfer function samples on the unit circle, without requiring access to a state-space realization. Numerical results for large-scale models demonstrate the effectiveness of the proposed methods as low-rank Stein equation solvers and data-driven model order reduction methods. Gaussian process learning with flow map refinement for parameter estimation in dynamical systems 作者: Yue Hao, Dongwei Ye 类别: cs.LG, cs.CE, math.NA 发布日期: 2026-08-23 arXiv ID: 2608.22324v1 链接: http://arxiv.org/abs/2608.22324v1 摘要: Parameter estimation is a central task in data-driven learning of dynamical systems. It aims to recover the underlying physical parameters from observed time-series data, thereby providing interpretable insights into the physical mechanisms governing the system. Gradient/derivative matching methods based on Gaussian process provide an efficient way to perform parameter estimation. Those methods avoid repeated numerical integration and enforce local derivative consistency. However, such local matching may result in global inconsistency with the governing flow map, particularly under scarce and noisy observations. To address this limitation, we propose a framework based on Gaussian process learning with flow map refinement (GPL-FMR), a two-stage parameter estimation framework. The first stage is based on Gaussian process learning algorithm and the posterior obtained from which is transferred as an informative prior to the second stage based on flow-map refinement. The second stage further improves the parameter estimation via optimisation based on global dynamical constraints. We demonstrate and analyse its performance on multiple numerical examples, including the Van der Pol oscillator, the Lotka-Volterra model, and the Lorenz-63 system. The results show that the proposed framework consistently improves parameter estimation accuracy, particularly under scarce and noisy observations. Natural superconvergence points and asymptotic expansions for spline finite elements in one dimension 作者: Peng Yang, Zhimin Zhang 类别: math.NA 发布日期: 2026-08-23 arXiv ID: 2608.22304v1 链接: http://arxiv.org/abs/2608.22304v1 摘要: We study the natural superconvergence points and asymptotic expansions of one-dimensional spline finite element approximations. For a spline space of degree $k$ and any smoothness $0\leμ\le k-1$, we prove that the $s$-th derivative of the error exhibits enhanced convergence of order $O(h^{k+2-s})$ at points where $k-s$ is even, provided the mesh is symmetric within a region of size $Ch|\ln h|$ around the point. This condition is known to be optimal for the cases of low derivative order $s=0,1$; the present analysis shows that the same local condition is sufficient for all admissible $s$. Moreover, by expanding the error in Legendre polynomials, a closure theorem determines the leading-order Legendre coefficients (the asymptotic expansion of the error) by combining the Galerkin orthogonality with the superconvergence conditions. For $μ=k-1$ (B-splines) and $μ=k-2$, the Galerkin orthogonality conditions vanish and the coefficients are determined solely by the superconvergence conditions. The asymptotic expansion can be expressed through a simple antiderivative recurrence on Legendre polynomials. The resulting polynomial’s zeros encode the complete set of superconvergence points for all derivative orders. Numerical experiments for selected $(k,μ)$ pairs confirm the theoretical predictions. On Convergence Behavior of Randomized Kaczmarz-type Methods for Solving Doubly Noisy Linear Systems 作者: Yudan Gan, Gang Wu 类别: math.NA 发布日期: 2026-08-23 arXiv ID: 2608.22259v1 链接: http://arxiv.org/abs/2608.22259v1 摘要: The randomized Kaczmarz (RK) method is an efficient iterative projection algorithm with low computational complexity for solving consistent linear systems. However, noise is inevitable in real-world applications, and both the coefficient matrix and the right-hand side vector may be contaminated by noise. The convergence analysis of RK-type methods for doubly noisy linear systems, where both the system matrix and the measurement vector are perturbed, remains relatively limited. In this paper, we investigate the limiting behavior of the RK algorithm for solving doubly noisy inconsistent linear systems without imposing any additional initial assumptions. Furthermore, to the best of our knowledge, this work provides the first convergence analysis of the randomized extended Kaczmarz (REK), randomized block Kaczmarz (RBK), and randomized double block Kaczmarz (RDBK) algorithms for doubly noisy linear systems. We prove that these algorithms converge to a neighborhood of the least-squares solution of the underlying noiseless system. Compared with existing theoretical estimates, the proposed bounds effectively characterize the convergence behavior of these algorithms when applied to doubly noisy linear systems. Finally, numerical experiments are conducted to validate the theoretical results. Sharp Barron Regularity Results for Coulombic Many-Electron Wave Functions 作者: Pingbing Ming, Hao Yu 类别: math.AP, cs.LG, math.NA 发布日期: 2026-08-23 arXiv ID: 2608.22252v1 链接: http://arxiv.org/abs/2608.22252v1 摘要: We establish sharp Barron regularity for Coulombic many-electron wave functions after extraction of the universal cut-off Jastrow factors. Following the factorization of Fournais et al. An hp-version time stepping spectral Monte Carlo method for semi-linear parabolic equations 作者: Jiaying Feng, Zhiyuan Hui, Changtao Sheng, Chenglong Xu 类别: math.NA 发布日期: 2026-08-23 arXiv ID: 2608.22190v1 链接: http://arxiv.org/abs/2608.22190v1 摘要: In this paper, we present an $hp$-version time-stepping spectral Monte Carlo method for solving semi-linear parabolic equations. The key innovation lies in constructing an exponentially accurate stochastic algorithm that integrates a residual iteration scheme on Gauss-type nodes in both temporal and spatial directions with a reconstruction strategy rooted in spectral methods. To address the long-time simulations and initial singularities that are often challenging for traditional stochastic algorithms (e.g., walk-on-spheres method), we further develop an $hp$-version time-stepping framework that employs multiple time steps and, respectively, geometric time partitions with linearly increasing polynomial degrees to handle these difficulties. Notably, the proposed algorithm bypasses the need to solve linear systems required by traditional spectral methods and remarkably supports parallel computation at both temporal and spatial grid points. We rigorously establish exponential convergence rates for the multistep method within a finite number of iterations. Extensive numerical experiments are conducted to demonstrate the spectral accuracy and computational efficiency of the proposed method in long-time simulations, problems with initial singularities, and a five-dimensional problem, thereby validating the theoretical results. A posteriori error estimates for the two-step explicit exponential Adams method for parabolic equations 作者: Xianfa Hu, Wansheng Wang 类别: math.NA 发布日期: 2026-08-23 arXiv ID: 2608.22162v1 链接: http://arxiv.org/abs/2608.22162v1 摘要: In this paper, we derive optimal order a posteriori error estimates for the variable step-size explicit two-step exponential Adams (E-Adams2) method for parabolic problems. We begin by introducing an E-Adams2 approximation, deffned by the piecewise linear approximate solutions, which leads to suboptimal error estimates. To recover optimal order error estimates, we introducean appropriate reconstruction of the approximation with the second order residual for the explicit E-Adams2 method, which plays key roles in deriving optimal order a posteriori error estimates for the proposed explicit method for linear and semilinear parabolic equations. Various numerical experiments are carried out to verify the correct convergence rates of the a posteriori quantities, and the high efffciency of the adaptive algorithm. Loss Landscape Features That Make Adam Stall: Definitions, Estimators, and the Preconditioned Hessian View 作者: Rodion Podorozhny 类别: cs.LG, math.NA 发布日期: 2026-08-23 arXiv ID: 2608.22145v1 链接: http://arxiv.org/abs/2608.22145v1 摘要: Across implicit-neural-representation (INR) architectures and analytic benchmarks we observe that a thoroughly tuned Adam (especially its learning rate (lr), e.g. in a hyperparameter sweep from $lr = 0.05$ to $10^{-8}$) can potentially reach a very low loss even on ill-conditioned loss landscape or converge at a plateau far above the loss attained by second-order methods. This report defines the measured metrics that help determine if Adam can mitigate the ill-conditioning on a given loss landscape. We provide the indicators by which each outcome is determined, that are: the condition number of the Hessian and of the Adam-preconditioned Hessian $D^{-1/2}HD^{-1/2}$ (with the derivation from Adam’s update rule), the diagonal mass $ρ$ that distinguishes axis-aligned from cross-coupled ill-conditioning, the negative spectral mass estimated by stochastic Lanczos quadrature, and the gradient energy fractions over curvature bands, including the flat fraction that indicates the Adam stall. A worked out $2\times 2$ example and an illustration show the reasons why a diagonal preconditioning by Adam can remove axis-aligned ill-conditioning by rescaling and why it cannot do the same if the ill-conditioning is cross coupled. In addition, we present a case study of FINER image fitting architecture that goes over the whole loss landscape analysis framework: the fitting architecture description, reasons due to which its landscape stalls Adam at saddles, the measured PSNR values through our tuned baselines to the $120$–$134$,dB results of the blockwise second order methods, the error maps behind those numbers, and description of the benefits such image fitting accuracy gives in practice. Blockwise Stabilized Adaptive Cubic Regularization with Subsolvers via Recurrence 作者: Rodion Podorozhny 类别: cs.LG, math.NA 发布日期: 2026-08-22 arXiv ID: 2608.22129v1 链接: http://arxiv.org/abs/2608.22129v1 摘要: Cubic-regularized Newton methods have the optimal $\mathcal{O}(\varepsilon^{-3/2})$ global rate and an automatic saddle-escape mechanism, but their subproblem is most often solved by a full eigendecomposition, limiting feasible model size. We introduce a blockwise optimizer that partitions parameters by tensor, minimizes an independent cubic model with an adaptive cubic constant $M_b$ per block, and accepts or rejects each block step against a monotone guard on the full loss. The subproblem solver is chosen by block size: small blocks use lazy exact cubic steps from explicitly formed per-block Hessians; arbitrarily large tensors use a matrix-free Chebyshev-bounded Krylov subspace built by the Lanczos process. The cubic shift bounds the required polynomial degree whenever the gradient-driven shift dominates negative curvature, renders the shifted operator positive semidefinite before any polynomial is applied, and preserves the $\mathcal{O}(\varepsilon^{-3/2})$ rate under inexact subproblem solves. We prove these claims, and the blockwise scheme carries a monotone per-block descent guarantee. Experiments cover FINER INRs (about 199k parameters) and a 91.4M-parameter ViSIR INR, where the blockwise cubic step remains exact in the cubic-model sense on every block, including the 88.5M-parameter decoder tensor (97% of the model). Run to full convergence on FINER, the ARC-$\varphi_1$ optimizer reaches 133.5 dB PSNR while tuned Adam plateaus at 78.2 dB at the same extended budget; in the roughly 70 minutes Adam takes to reach its peak, ARC-$\varphi_1$ reaches 95.6 dB. A companion report isolates the loss-landscape features responsible for Adam’s behavior. Virtual element approximations for distributed or Neumann boundary optimal control problems governed by poroelasticity equation 作者: Thirupathi Gudi, David Mora, Nitesh Verma 类别: math.OC, math.NA 发布日期: 2026-08-22 arXiv ID: 2608.22107v1 链接: http://arxiv.org/abs/2608.22107v1 摘要: This paper investigate the conforming virtual element method for the optimal control problem governed by the linear poroelasticity equation. Both distributed control and Neumann boundary controls are considered with the optimize-then-discretize approach. We establish the well-posedness of the discrete state and adjoint problems in three-field formulation by deriving the first-order optimality condition. The optimal a priori error estimates are developed through a novel poroelastic projection for both control problems, and these estimates are uniform with respect to the relevant physical parameters of the model problem. Moreover, the numerical experiments are carried out using the primal dual active set strategy, and demonstrate the support for the derived theoretical results. On Convergence of an Accelerated Modified Newton Method for Nonlinear Equations 作者: Sanwar Ahmad, Joy Watson, Mohammad Tabanjeh 类别: math.NA 发布日期: 2026-08-22 arXiv ID: 2608.22060v1 链接: http://arxiv.org/abs/2608.22060v1 摘要: Newton’s iteration is a fundamental tool for root-finding and numerical solutions of systems of equations. The iteration rapidly refines the initial approximation to the exact root, and in general the convergence is quadratic. Since the method requires finding the function value and its derivative at each iteration, in some cases, it may not converge. This is because the value of the derivative gets close to zero. In this paper, we introduce a modified and stable algorithm of Newton’s iteration method that addresses this issue, reduces computational cost, and improves efficiency. In addition. we analyze the convergence properties of the modified method to demonstrate its effectiveness. Data-driven reduced-order models for the radiative transfer equation 作者: Yinxi Pan, Kui Ren, Shanyin Tong 类别: math.NA 发布日期: 2026-08-22 arXiv ID: 2608.22041v1 链接: http://arxiv.org/abs/2608.22041v1 摘要: We present a data-driven reduced-order modeling (ROM) framework for the zeroth angular moment of the solution to the radiative transfer equation (RTE) rather than the full phase-space solution. Our construction is based on the Peierls integral formulation of the angularly averaged density. For media with isotropic scattering, the density satisfies a closed second-kind Fredholm equation with a globally attenuated, weakly singular kernel. We project this equation directly. For media with anisotropic scattering, we utilize the average-fluctuation decomposition to derive a closed system for a projection-based ROM. Numerical simulations are presented to illustrate the effectiveness of the ROMs we implemented. DySCo: Dynamically consistent data-driven downscaling of extremes in climate projections 作者: S. Stamatelopoulos, M. Wang, I. Lopez-Gomez, L. Zepeda-Nunez, Z. Y. Wan, R. Carver, F. Sha, T. P. Sapsis 类别: cs.LG, math.NA, physics.ao-ph 发布日期: 2026-08-22 arXiv ID: 2608.21998v1 链接: http://arxiv.org/abs/2608.21998v1 摘要: Regional climate risk assessment is critical for applications such as infrastructure design, disaster forecasting, and insurance resource allocation. However, estimating regional (i.e., high-spatial-resolution) risk with global climate models (GCMs) remains computationally prohibitive, which has driven the development of downscaling methods for coarse GCM outputs. Downscaling is vital for rare events, since quantifying their extreme properties requires high spatial resolution and very long GCM simulations. These methods non-intrusively increase GCM resolution while correcting statistical biases from unresolved fine-scale processes, thereby improving the accuracy of extreme event statistics with long return periods. A key challenge is preserving dynamical consistency, as freely evolving GCM trajectories are not expected to track the observational dataset used for training the correction operator. This is critical for causal extreme event analyses, where storyline-based risk assessment, i.e., extreme event catalogs, is necessary for effective planning. We address this challenge by introducing Dynamically and Statistically Consistent downscaling (DySCo), a non-intrusive framework yielding high-resolution climate projections consistent with coarse GCM dynamics. DySCo relies on a data-driven reformulation of nudging to create dynamically paired training trajectories without intrusive GCM modifications. Using these paired trajectories, we train a dynamically and statistically consistent, two-stage operator. We evaluate the method by downscaling the Community Earth System Model v2 Large Ensemble (LENS2) in time and space towards historical reanalysis. Results show DySCo achieves superior dynamical consistency with the coarse GCM trajectories, essentially applying a minimal, causal correction to the GCM, preserving top statistical performance comparable to state-of-the-art unsupervised models. The Smallest Singular Value of Nonuniform Fourier Matrices 作者: Liang Chen, Rongrong Lin, Haizhang Zhang 类别: math.NA 发布日期: 2026-08-22 arXiv ID: 2608.21960v1 链接: http://arxiv.org/abs/2608.21960v1 摘要: We study the smallest singular value of nonuniform Fourier matrices in two settings: clustered nodes and perturbations of an equispaced grid. By reducing the problem to spectral norm estimates for periodic nonuniform interpolation matrices, we obtain nearly optimal bounds in both cases. For clustered nodes, we derive a local separation condition in which each required gap depends only on the sizes of the two neighboring clusters. For perturbations with the bound (1/4\leq L<1/2), our result confirms the conjecture of Austin and Trefethen on the (2)-norm Lebesgue constant up to a logarithmic factor. A diffusion time-changed stochastic SIS epidemic model: well-posedness, long-time behavior, and numerical approximation 作者: Xiaotong Li, Huaqian Zhou, Ruchun Zuo 类别: math.NA, math.PR 发布日期: 2026-08-22 arXiv ID: 2608.21930v1 链接: http://arxiv.org/abs/2608.21930v1 摘要: In this paper, we propose and analyze a diffusion time-changed susceptible-infected-susceptible (SIS) epidemic model driven by time-changed Brownian motion. We prove that the proposed model admits a unique global positive solution for any initial value in $(0,N)$. The extinction and persistence of the disease are then investigated. To approximate the diffusion time-changed SIS model, we construct a positivity-preserving logarithmic Euler-Maruyama (LEM) method. Assuming that the time-changed is given by the inverse of a standard $α$-stable subordinator with $α\in(0,1)$, we prove that the numerical solution converges strongly to the exact solution with order $α$. Finally, numerical experiments are provided to confirm the predicted convergence rates and illustrate the positivity-preserving property of the proposed method. Rapid Earthquake-to-Tsunami Waveform Generation via Large-Scale Multi-GPU FFT Convolution Applied to the Cascadia Subduction Zone 作者: Bowen Shi, Sreeram Venkat, Stefan Henneking, Omar Ghattas 类别: cs.DC, math.NA, physics.geo-ph 发布日期: 2026-08-22 arXiv ID: 2608.21763v1 链接: http://arxiv.org/abs/2608.21763v1 摘要: Data-driven methods for earthquake and tsunami early warning rely on large ensembles of rupture scenarios and their resulting waveforms, but generating such datasets with repeated high-fidelity seismic and tsunami simulations is prohibitively expensive. We exploit the linear time-invariant structure of both dynamics to precompute elastic Green’s functions and acoustic-gravity adjoint responses, reducing the source-to-waveform map to two consecutive convolution operators. We evaluate these convolutions with a distributed, FFT-accelerated GPU pipeline that partitions the large seafloor grid across GPUs and directly generates the final observation waveforms. We demonstrate the scalability of this pipeline for the Cascadia Subduction Zone with 963 subfaults, 2,416,530 seafloor grid points, 64 observation locations, and 256 timesteps, requiring 9.45 TiB of aggregate GPU memory. On 64 GB200 GPUs within one NVL72 domain, the pipeline generates waveforms in 24 ms per rupture once the response operators are resident, enabling large rupture ensembles to be evaluated within minutes. Wasserstein Moment Nudging for Vlasov-Poisson Data Assimilation 作者: Liyao Lyu, Xinyue Yu, David Schneidinger, Hayden Schaeffer 类别: math.NA, physics.comp-ph, physics.plasm-ph 发布日期: 2026-08-22 arXiv ID: 2608.21725v1 链接: http://arxiv.org/abs/2608.21725v1 摘要: We introduce a continuous data assimilation method for particle-in-cell simulations of the Vlasov-Poisson equation when only hydrodynamic moments are observed. The forecast state is an empirical measure on phase space, whereas the observed fields (density, bulk velocity, and temperature) constrain only a few velocity moments and leave the velocity-space shape of the distribution undetermined. We construct the moment feedback as a Wasserstein gradient flow of a moment-mismatch functional over phase-space measures. The resulting drift acts directly on particle positions and velocities, couples the density, momentum, and energy residuals through a single variational structure, and vanishes on the entire moment-compatible set. Under the standard Wasserstein metric, the energy residual produces a position correction that grows quadratically with the particle speed, and the particle system falls outside standard well-posedness theory. Our primary formulation pairs the quadratic moment mismatch with a velocity-weighted Wasserstein metric that penalizes spatial transport at large peculiar velocity relative to the observed bulk flow, which removes this growth. A direction-split variant retains the plain metric instead. Under the same weighted metric, an alternative moment-relative-entropy functional yields an affine, shape-preserving velocity correction and explicit global moment balances for the space-inhomogeneous system. We prove that the finite-particle scheme with linear Lenard-Bernstein collisions is globally well posed. In 1D1V and 2D2V experiments with several collision models, the nudged formulations reduce bulk-velocity and temperature errors by up to two orders of magnitude relative to an unassimilated run. Tensor greedy double block extended Kaczmarz method for solving inconsistent tensor linear systems under t-product 作者: Jérémie Mabiala, Lionel Tondji 类别: math.NA 发布日期: 2026-08-22 arXiv ID: 2608.21705v1 链接: http://arxiv.org/abs/2608.21705v1 摘要: The randomized extended Kaczmarz method is an effective iterative framework for solving large-scale inconsistent linear systems. In this paper, we extend this framework to third-order inconsistent tensor linear systems under the t-product and propose the Tensor Greedy Double Block Extended Kaczmarz (TGDBEK) method. At each iteration, TGDBEK dynamically constructs active blocks of row and column slices via a residual-based greedy selection strategy, prioritizing the slices associated with the largest residual norms. Unlike existing tensor block Kaczmarz variants that rely on static, predefined partitions – the tensor randomized extended block Kaczmarz (TREBK) method, its greedy counterpart (TREGBK), and the tensor randomized extended average block Kaczmarz (TREABK) method – TGDBEK adapts the active block sizes dynamically at each step using a single intuitive threshold parameter $η$. We establish the theoretical linear convergence of TGDBEK to the unique minimum-norm least-squares solution $\mathcal{A}^\dagger * \mathcal{B}$. Extensive numerical benchmarks on synthetic dense and sparse tensor systems, as well as multidimensional multichannel color and 3D volumetric MRI image deblurring problems, demonstrate that TGDBEK substantially outperforms state-of-the-art tensor Kaczmarz solvers in both iteration count and CPU running time. Data-Driven Dynamic Algorithm Dispatch with Large Language Models 作者: Rushil Shah, Emmanuel Lujan, Rabab Alomairy, Alan Edelman 类别: cs.AI, cs.CE, math.NA 发布日期: 2026-08-21 arXiv ID: 2608.21584v1 链接: http://arxiv.org/abs/2608.21584v1 摘要: We introduce a large language model (LLM)-driven approach for generating dynamic algorithmic dispatch heuristics in high-performance linear algebra. By combining prompt engineering with LLaMA 3 and a curated performance database, the model learns to synthesize selection heuristics that exploit structural patterns to identify fast algorithmic choices. A case study on LU factorization demonstrates the model’s ability to replicate expert-designed strategies. This work, developed as part of the DARPA-MIT SmartSolve project, highlights the promise of LLMs for algorithmic discovery and the development of more adaptive, fast linear algebra software. A Data-Driven Computational Framework for Incompressible Flow in Hydraulic Networks 作者: Pedro B. Bazon, Cristian G. Gebhardt, Roberto F. Ausas 类别: math.NA 发布日期: 2026-08-21 arXiv ID: 2608.21575v1 链接: http://arxiv.org/abs/2608.21575v1 摘要: The classical procedure for solving hydraulic networks relies on the assumption of constitutive equations, which state the relationship between the pressure gradient and fluxes along an edge. In this paper, we propose a data-driven framework that bypasses these constitutive models, formulating the incompressible flow problem directly on the graph topology. By assigning discrete measured data points to network edges, the problem is cast as a mixed-integer quadratic optimization over nodal pressures, edgewise states, and data assignments, accommodating both laminar (convex) and turbulent (non-convex) regimes. To solve this, we evaluate three algorithms: a GPU-accelerated Brute Force method, the Alternating Direction Method (ADM), and Deterministic Annealing (DA). The Brute Force method certifies global optima for small networks, establishing a good baseline for the iterative solvers. We demonstrate that ADM is highly sensitive to its initialization, requiring a faithful surrogate model to avoid local minima. In contrast, DA eliminates this dependence through unsupervised clustering. By annealing the data assignment from the centroid to strict nearest-neighbor projections, DA consistently reaches the global optimum without prior manifold reconstruction. Furthermore, numerical experiments reveal that DA is robust to noisy data, remains thermodynamically admissible on all but the coarsest and noisiest datasets, and sustains its convergence rate on larger networks where Brute Force is intractable and ADM degrades. Finally, the framework is successfully validated on complex configurations, including mixed-component networks and a $958$-edge arteriovenous bed featuring a non-Newtonian Carreau–Yasuda model, demonstrating its scalability and practical applicability. A Memory-Efficient Adjoint State Optimization Method Based on Time-Reversible Dynamical Low-Rank Approximation 作者: Lukas Einkemmer, Julian Mangott 类别: math.NA, math.OC, physics.comp-ph, physics.plasm-ph 发布日期: 2026-08-21 arXiv ID: 2608.21545v1 链接: http://arxiv.org/abs/2608.21545v1 摘要: The primary challenge of conducting PDE-constrained optimization for high-dimensional problems, such as kinetic equations, is the often prohibitive memory cost. Computing gradients using the adjoint state method would require the storage of the entire time history of the forward solution. For such problems, where the memory cost for storing a single instance of the forward solution can already be a limiting factor, this is clearly not feasible. In this paper, we propose a memory-efficient adjoint state method that compresses the forward and adjoint solution with a dynamical low-rank approximation (a model order reduction technique) and bypasses the need to store the entire forward solution by employing a time-reversible low-rank integrator. The dynamical low-rank approach introduces a number of challenges: reversibility can fail in the rank-deficient case and the low-rank trajectories can show chaotic behavior. In particular, the latter has a number of important consequences for the optimization problem. We address those challenges and show that our method can drastically reduce the memory requirement for gradient-based optimization of kinetic equations. In particular, we consider two examples from kinetic plasma physics: optimizing beam profiles to suppress a bump-on-tail instability and shaping a particle beam using external electric fields.3. 研究方法、关键技术和核心工具
4. 与之前工作的比较
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\cite[Definition1.4]{FournaisEtAl2005}, for a Coulombic eigenfunction $ψ$ we define the successive quotients by [ φ=e^{-F_{2,\mathrm{cut}}}ψ\quad\text{and}\quad φ_3=e^{-F_{3,\mathrm{cut}}}φ=e^{-(F_{2,\mathrm{cut}}+F_{3,\mathrm{cut}})}ψ. ] Then [ φ,φ_3\in\mathcal{B}^s(\mathbb{R}^{3N}) \qquad\text{for every }s<2. ] This range is optimal among universal factorizations. No factor depending only on the particle number and the nuclear data, but not on the eigenfunction or its eigenvalue, can make every corresponding quotient belong to $\mathcal{B}^2$.
We also determine the exact endpoint growth. Writing $\varepsilon=2-s$, we prove that, for either $u=φ$ or $u=φ_3$, there is a computable constant $M$ independent of $\varepsilon$ such that [ \left|u\right|{\mathcal{B}^{2-\varepsilon}}\leq\frac{M}{\varepsilon^2}\left|u\right|{\mathcal{B}^1}. ] For the unperturbed two-electron atom we prove, with a constant independent of $\varepsilon$, [ \left|\left|φ_3\right|_{\mathcal{B}^{2-\varepsilon}}-\frac{32πZ\lvertφ_3(0,0)\rvert}{\varepsilon^2}\right|\leq\frac{C}{\varepsilon}. ] Hence the quadratic rate in the upper bound is sharp whenever $\lvertφ_3(0,0)\rvert\neq0$, as is the case for the ground state.24. 半线性抛物方程的 hp-版本时间步进谱蒙特卡洛方法
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