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1. 各向异性 Maxwell 介质的极化低秩成像

Polarimetric Low-Rank Imaging of Anisotropic Maxwell Media

作者: Xianbo Chen, Jingzhi Li, Ping Liu

类别: math.NA

发布日期: 2026-10-29

arXiv ID: 2610.40180v1

摘要翻译: 我们考虑由固定频率极化 Maxwell 远场数据对紧支撑各向异性电介质进行三维成像。Maxwell 横向性产生了一种标量 Born 反演中不存在的障碍：每次测量仅确定张量值 Fourier 变换的一个投影作用。我们首先通过有限维极化反演解决该障碍。对于一般复 $3\times3$ 对比度，三个入射–观测配置是必要的，并且我们给出一个显式的三配置设计，它在每个非零内部 Fourier 点处均充分。其 Gram 行列式给出定量的条件信息，并显示在可及 Fourier 球边界处会失去可恢复性。经过张量解混后，反问题逐分量约化为单位球上的受限 Fourier 反演；我们通过三维球上的广义长椭球波函数中的谱截断对其进行正则化。对于离散采集，一个组合重建误差估计将测量噪声、极化放大、离网格 Fourier 失配、求积误差、Born 建模误差和谱截断的影响分离开来。数值实验检验极化条件性，并使用解析生成的 Born 数据比较三配置和六配置恢复并评估离网格敏感性。基于全 Maxwell 体积积分数据的重建评估噪声、波数和基选择的影响，以及不同张量分量的恢复。

Comment: 无

链接: http://arxiv.org/abs/2610.40180v1

1. 研究对象和背景

本文研究三维紧支撑各向异性电介质的固定频率极化 Maxwell 远场成像. 设背景为均匀非磁介质, 波数 $k>0$, 相对介电张量为 $\varepsilon_r=I+Q$, 其中对比张量 $Q\in L^\infty(\mathbb R^3;\mathbb C^{3\times3})$ 支撑于有界区域 $D$. 总电场 $E$ 满足

$$\nabla\times\nabla\times E-k^2(I+Q)E=0,\quad x\in\mathbb R^3,$$

散射场 $E^s=E-E^i$ 满足 Silver-Müller 辐射条件

$$\lim_{r\to\infty}r\big((\nabla\times E^s)(x)\times\hat x-ikE^s(x)\big)=0.$$

入射场取平面波 $E^i(x)=e\,e^{ikd\cdot x}$, 其中 $d\in\mathbb S^2$, $e\in\mathbb C^3$, $d\cdot e=0$. 精确远场表示为

$$E^\infty(\hat x;d,e;k)=\frac{k^2}{4\pi}P_{\hat x}\int_D e^{-ik\hat x\cdot y}Q(y)E(y;d,e;k)\,dy,\quad P_{\hat x}=I-\hat x\hat x^\top,$$

因而 $\hat x\cdot E^\infty=0$. 在一阶 Born 近似 $E\approx E^i$ 下, 若定义 $\widehat Q(\xi)=\int_D e^{i\xi\cdot y}Q(y)\,dy$, 则

$$E_b^\infty(\hat x;d,e;k)=\frac{k^2}{4\pi}P_{\hat x}\widehat Q(k(d-\hat x))e.$$

该关系是全文的出发点. 与标量 Born 反演不同, 这里未知量是张量 $\widehat Q(\xi)$, 而入射极化与观测远场均受 Maxwell 横截性约束: 单次测量只给出投影后的作用 $P_{\hat x}\widehat Q(\xi)e$, 并不能直接得到 $\widehat Q(\xi)$. 因此, 必须先在每个可达 Fourier 点解一个有限维张量辨识问题, 再对空间带限 Fourier 数据进行反演. 这一横截障碍是标量反演中不存在的, 也是本文试图补上的第一块空白. 第二块空白是空间重建: 可测 Fourier 点集为 $B(0,2k)$, 恢复张量分量后仍需反演球上的受限 Fourier 算子, 并控制噪声、谱截断和离散采集误差. 已有电磁反散射工作包括定性采样与因子分解方法, 对比源与迭代方法, 各向异性唯一性理论, 以及若干极化 Born 反演. 例如 Gilman-Smith-Tsynkov 处理均匀单轴介质, Elbau-Mindrinos-Scherzer 讨论一般张量 Radon 数据的恢复条件但未做空间张量数值重建, Balandin 给出张量 Fourier 衍射关系但其横向投影奇异. 标量受限 Fourier 反演方面已有基于广义长球波函数 GPSWF 的 Picard 正则化与低秩数值框架. 本文的空白在于: 对一般复 $3\times3$ 张量, 给出最少入射-观测构型、显式充分设计、Gram 矩阵条件数分析、边界不可恢复性, 并把极化放大与球 GPSWF 谱截断及离散采集误差统一到重建误差估计中.

2. 主要定理或主要结果

(1) 适定性与积分表示. 在 Assumption 2.1 下, 即存在 $\gamma>0$ 使

$$\mathrm{Re}\,[z^*(I+Q(x))z]\ge\gamma|z|^2,\quad \mathrm{Im}\,[z^*(I+Q(x))z]\ge0,\quad \forall z\in\mathbb C^3,$$

对分段光滑对称 $Q$, 方程有唯一总场 $E\in H_{\mathrm{loc}}(\mathrm{curl};\mathbb R^3)$, 且

$$E^s(x)=k^2\int_D G_k(x-y)Q(y)E(y)\,dy.$$

当介质严格耗散时, 结论可推广到一般 $Q\in L^\infty(\mathbb R^3;\mathbb C^{3\times3})$. 远场满足

$$E^\infty(\hat x;d,e;k)=\frac{k^2}{4\pi}P_{\hat x}\int_D e^{-ik\hat x\cdot y}Q(y)E(y;d,e;k)\,dy,$$

且 $\hat x\cdot E^\infty=0$. Born 远场为

$$E_b^\infty(\hat x;d,e;k)=\frac{k^2}{4\pi}P_{\hat x}\widehat Q(k(d-\hat x))e.$$

作者还给出 Born 模型余项 $E^\infty=E_b^\infty+E_{nl}^\infty$: 若

$$\rho_B=k^2\|V_Q\|_{L^2(D)^3\to L^2(D)^3}<1,$$

则

$$\|E^\infty_{nl}\|_{L^2(\mathbb S^2)^3}\le \frac{k^2|D|^{1/2}}{\sqrt{4\pi}}\|Q\|_{L^\infty(D)}\frac{\rho_B}{1-\rho_B}\|E^i\|_{L^2(D)^3}.$$

(2) 可达 Fourier 点与极化线性系统. 作者证明

$$\{k(d-\hat x):d,\hat x\in\mathbb S^2\}=B(0,2k),$$

并对 $\xi\in B(0,2k)$ 给出参数化

$$d=p+\sqrt{1-|p|^2}s,\quad \hat x=-p+\sqrt{1-|p|^2}s,\quad p=\xi/(2k),\ s\in\mathbb S^2\cap p^\perp.$$

对张量类 $T$ 取 Frobenius 正交基 $T_r$, 写 $\widehat Q(\xi)=\sum_{r=1}^m c_rT_r$. 每个构型使用 $d_j^\perp$ 的两个正交极化 $E_j=(e_{j,1},e_{j,2})$, 数据块为

$$B_j=P_{\hat x_j}\widehat Q(\xi)E_j,\quad g=Mc,\quad [M_j]_{:,r}=\mathrm{vec}(P_{\hat x_j}T_rE_j).$$

若 $\mathrm{rank}\,M=m$, 则 $c=M^\dagger g$; 对扰动 $g^\delta=g+\eta$, 有

$$\|\widehat Q^\delta(\xi)-\widehat Q(\xi)\|_F\le \frac{\|\eta\|_2}{\sigma_{\min}(M)}.$$

(3) 最小三构型设计. 对一般复 $3\times3$ 张量, $m=9$, 每个构型至多贡献 $4$ 个独立复自由度, 故 $J\le2$ 时 $\mathrm{rank}\,M\le4J<9$, 至少需要三个构型. 作者构造显式三构型: 令 $0<|\xi|<2k$, $\rho=|\xi|/(2k)$, $n=\xi/|\xi|$, 取 $n^\perp$ 的正交基 $u,v$, 令

$$\theta_j=2\pi(j-1)/3,\quad s_j=\cos\theta_ju+\sin\theta_jv,$$

$$d_j=\rho n+\sqrt{1-\rho^2}s_j,\quad \hat x_j=-\rho n+\sqrt{1-\rho^2}s_j,$$

每个构型用两个正交入射极化. 于是

$$\det(M^*M)=\frac{3^9}{2^9}\rho^2(1-\rho^2)^6(9\rho^4+2\rho^2+1)^2>0,$$

从而 $\mathrm{rank}\,M=9$. 条件数方面,

$$\sigma_{\min}(M(\rho))\sim\sqrt{3}\,\rho\quad(\rho\to0^+),\qquad \sigma_{\min}(M(\rho))\sim\sqrt{\frac32}(1-\rho^2)\quad(\rho\to1^-).$$

在中心 $\xi=0$ 可用三轴设计 $d_j=\hat x_j=a_j$ 达到满秩, 且 $\det(M^*M)=8$; 在边界 $|\xi|=2k$ 必有 $d=\xi/|\xi|$, $\hat x=-d$, 任意构型均有 $\mathrm{rank}\,M\le4$, 一般张量不可辨识. (4) 球受限 Fourier 低秩反演. 令 $Q_R(z)=Q(Rz)$, $B_1=B(0,1)$, $C=2kR$, 有 $R^{-3}\widehat Q(2kp)=\mathcal F_CQ_R(p)$, 其中

$$(\mathcal F_Cf)(p)=\int_{B_1}e^{iCp\cdot z}f(z)\,dz.$$

球 GPSWF 满足 $\mathcal F_C\psi_{\ell nm}=\alpha_{\ell n}(C)\psi_{\ell nm}$. 对扰动数据 $W^\delta$, 取 $J_\epsilon=\{\nu:|\alpha_\nu(C)|>\epsilon\}$ 并定义

$$Q^\delta_{R,\epsilon}=\sum_{\nu\in J_\epsilon}\frac{W^\delta_\nu}{\alpha_\nu(C)}\psi_\nu.$$

若 $\|W^\delta-W\|_{L^2(B_1;\mathbb C^{3\times3})}\le\delta_F$, 则

$$\|Q^\delta_{R,\epsilon}-Q_R\|_{L^2(B_1;\mathbb C^{3\times3})}\le \frac{\delta_F}{\epsilon}+\Big(\sum_{\nu\notin J_\epsilon}\|Q_\nu\|_F^2\Big)^{1/2}.$$

(5) 离散采集. 摘要与引言称作者对把目标求积节点替换为邻近可实现 Fourier 点给出 Lipschitz 误差估计, 并给出组合重建误差界, 分离测量噪声、极化放大、off-grid Fourier 失配、求积误差、Born 建模误差与谱截断. 但提供 PDF 仅到第 10 页, 第 5 节具体定理编号与常数从提供内容中无法确认.

3. 研究方法、关键技术和核心工具

核心障碍有两个: 一是 Maxwell 横截投影使单次测量只能看到 $P_{\hat x}\widehat Q(\xi)e$, 无法直接用普通矩阵求逆恢复 $\widehat Q(\xi)$; 二是恢复后的 Fourier 数据只在球 $B(0,2k)$ 上已知, 反演算子紧且奇异值快速衰减, 必须正则化. 作者的处理框架是把两个机制解耦: 先做逐 Fourier 点的有限维极化解混, 再做球受限 Fourier 的 GPSWF 谱截断重建, 并把前者的条件数传入后者误差. 适定性证明的技术路线不是简单引用, 而是把体积积分算子写成 Fredholm 形式. 作者设

$$(V_Qu)(x)=\int_DG_k(x-y)Q(y)u(y)\,dy,$$

体积方程化为 $(I-k^2V_Q)E=E^i$. 关键分解为

$$(I-k^2V_Q)u=u+P(Qu)|_D+Ku,$$

其中

$$Pf=-\nabla\nabla^\top(\Phi_0*f),\quad \widehat{Pf}(\xi)=\frac{\xi\xi^\top}{|\xi|^2}\widehat f(\xi),$$

故 $P$ 是 $L^2(\mathbb R^3)^3$ 上的正交投影, 而 $K$ 的核为 $O(|x-y|^{-1})$ 因而紧. 然后在 $\mathrm{Ran}\,P$ 上用 Lax-Milgram 解辅助方程, 耗散性假设给出强制性; 唯一性则用弱 Green 恒等式取虚部, 结合远场渐近

$$\mathrm{Im}\int_{\partial B_\rho}(\nabla\times E\times n)\cdot E\,dS=k\int_{\partial B_\rho}|E|^2\,dS+O(\rho^{-1})$$

排除零空间. 这一步把通常要求对称分段光滑的结论推进到严格耗散下的一般有界张量. 远场关系来自 Green 张量渐近

$$G_k(x-y)=\frac{e^{ik|x|}}{4\pi|x|}P_{\hat x}e^{-ik\hat x\cdot y}+O(|x|^{-2}),$$

代入体积表示即得. Born 余项用 Neumann 级数控制, 条件 $\rho_B<1$ 使 $(I-k^2V_Q)^{-1}$ 可展开, 从而得到显式余项界. 极化阶段的关键创新是把每个构型写成列向量化线性系统 $g=Mc$. 因为 $B_j=P_{\hat x_j}\widehat Q(\xi)E_j$, 所以矩阵 $M$ 的第 $r$ 列是 $\mathrm{vec}(P_{\hat x_j}T_rE_j)$, 其秩至多 $4$, 由此得到 $J\le2$ 不可能恢复一般 $3\times3$ 张量的下界. 充分性方面, 作者选取同一 $\xi$ 对应的三个等角构型, 使 Gram 矩阵可写成

$$M^*M=\sum_{j=1}^3P_{d_j}\otimes P_{\hat x_j}.$$

在 $u,v,n$ 坐标下显式计算 $\det(M^*M)$, 证明对每个非零内点满秩. 这不是抽象存在性, 而是给出可执行的入射-观测方向与极化选择, 并用 Gram 行列式和最小奇异值渐近量化条件数: 中心按 $\rho$ 退化, 边界按 $1-\rho^2$ 退化; 边界退化是横截性造成的本质不可辨识, 不能靠增加构型消除. 空间反演阶段, 作者利用球 GPSWF 是受限 Fourier 算子 $\mathcal F_C$ 的特征函数, 把算子对角化. 将 $Q_R$ 与 $W=R^{-3}\widehat Q(2k\cdot)$ 在同一基下展开, 系数满足 $W_\nu=\alpha_\nu(C)Q_\nu$, 因而形式反演是除以 $\alpha_\nu(C)$. 真正的正则化是保留 $|\alpha_\nu(C)|>\epsilon$ 的低秩模态, 误差界自然分成数据误差放大项 $\delta_F/\epsilon$ 与截断尾项. 该框架把 GPSWF 的低秩结构与 Maxwell 极化解混连接起来: 极化阶段的小奇异值会放大噪声, 然后进入 $W^\delta$, 再被 $1/\alpha_\nu$ 和截断进一步放大; 因此条件分析必须放在重建误差之前. 离散阶段的技术点是 off-grid 处理. 实际入射-观测对通常不能精确落在同一 $\xi$ 节点, 只能采样邻近点. 作者用 Lipschitz 估计控制把目标求积节点替换为邻近可实现 Fourier 点产生的误差, 再把测量噪声、极化伪逆放大、求积误差、Born 余项和谱截断叠加成组合误差界. 从提供文本无法确认该组合界的具体形式与常数, 但其技术角色是明确的: 将连续模型中的精确节点假设改为可实现采集, 并使误差项可分离诊断. 数值部分则用解析 Born 数据检验三构型与六构型、off-grid 敏感性, 再用全 Maxwell 体积积分数据检验噪声、波数、基选择与不同张量分量恢复.

4. 与之前工作的比较

与 Gilman-Smith-Tsynkov 的工作相比, 本文不把介质限制为均匀单轴介质. 前者在 Born 线性化下给出极化散射数据与单轴参数之间的显式关系, 本文处理空间变化的 general complex $3\times3$ 对比张量, 并把问题拆成逐点张量辨识与空间受限 Fourier 反演. 因而本文多出的是可辨识性下界、显式三构型设计、Gram 条件数和边界退化分析, 而不是仅给出参数关系. 与 Elbau-Mindrinos-Scherzer 的工作相比, 他们通过入射极化与样品旋转建立一般张量值 Radon 数据的恢复条件, 但没有进行空间张量的数值重建; 其后续工作用二阶 Born 近似处理正交各向异性磁化率张量的积分方程. 本文同样使用多极化测量, 但采用固定频率远场、每个 Fourier 点三个入射-观测构型, 给出闭式 Gram 行列式与最小奇异值渐近, 并把极化解混后的数据送入球 GPSWF 低秩重建. 代价是本文仍在 Born 线性化框架内, 且需要已知支撑球与波数. 与 Balandin 的张量 Fourier 衍射关系相比, 本文直接针对其横向投影奇异、不能普通矩阵求逆的困难. 作者没有试图用一个全局矩阵逆, 而是把每个 Fourier 点的数据视为 $P_{\hat x}\widehat Q(\xi)e$ 的有限维作用, 构造 $g=Mc$ 并研究 $M$ 的秩与条件数. 这使奇异投影从障碍变成可量化的有限维反演问题; 但边界 $|\xi|=2k$ 处秩至多 $4$, 一般张量不可辨识, 这是本文明确承认的适用范围限制. 与 Meng 的标量 Picard 正则化和 Zhou-Audibert-Meng-Zhang 的 GPSWF 低秩数值框架相比, 本文的 GPSWF 部分建立在已有标量受限 Fourier 理论上, 创新不在于重新发明 GPSWF 基, 而在于在进入受限 Fourier 反演之前必须先解混张量、控制几何相关条件数, 并处理有限可实现入射-观测对只采样邻近而非同一 Fourier 节点的问题. 因此本文的误差估计比标量框架多了极化放大项、off-grid 失配项与 Born 建模余项. 与定性采样、因子分解和对比源迭代等电磁反散射工作相比, 本文不是全非线性定量反演, 而是固定频率 Born 线性化下的低秩成像; 其优势是给出显式稳定性与离散误差分解, 局限是模型误差需由小范数条件或数值全 Maxwell 数据检验控制. 作者在数值上确实用全 Maxwell 体积积分数据生成远场, 但重建仍基于 Born 关系, 因此并未证明全非线性反演的收敛性. 与各向异性唯一性理论相比, 本文不讨论一般结构假设下的唯一性, 而是在可达 Fourier 内点和一般复张量类上讨论有限维极化可辨识性; 在边界处则明确失败. 总体推进可概括为: 把一般张量极化反演的“存在恢复关系”推进到“最少构型、显式设计、条件数渐近、边界不可恢复、球 GPSWF 低秩重建与离散误差分解”, 代价是需要至少三个构型、中心与边界条件数退化、需谱截断且依赖 Born 线性化与已知支撑. 文中未显式展开与所有定性或迭代方法的数值比较, 从提供内容中无法确认其计算成本与全非线性方法的定量优劣.

了解领域论文（摘要翻译）

1. NEOPIC：用于基于粒子的动理学等离子体模拟的神经算子框架

NEOPIC: A Neural Operator Framework for Particle-based Kinetic Plasma Simulations

作者: Sriramkrishnan Muralikrishnan, Chelsea Maria John, Jeffrey Kelling, Michael Bussmann, Robert Speck, Paul Gibbon

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

发布日期: 2026-10-29

arXiv ID: 2610.40528v1

链接: http://arxiv.org/abs/2610.40528v1

摘要翻译

中文标题: NEOPIC：用于基于粒子的动理学等离子体模拟的神经算子框架

摘要翻译: 我们介绍 NEOPIC，一个用于学习基于粒子的动理学等离子体模拟中场算子的框架。其关键创新是一种神经算子，它直接作用于粒子的经验测度而非网格，并在粒子位置处求值。无需网格或树，且与基于网格的神经算子不同，不会发生向网格的沉积，因此主导粒子模拟输入的粒子噪声不会被混叠。由于它仅替代场求解，它可作为基于粒子方法的即插即用场求解器：可在来自 particle-in-cell、particle-in-Fourier 或 tree codes 的数据上训练，并与任意时间积分器（包括隐式格式）结合。我们证明离散求值与连续算子在任意粒子数下一致（表示等价），并给出将采样误差、学习误差和时间步进误差分离的误差分析：除去学习误差，NEOPIC 具有标准 Monte Carlo 粒子方法的误差结构。我们表明训练数据中的粒子噪声起到隐式正则化作用，这为训练粒子数与推断时所用粒子数的关系提供了指导。复杂度分析表明 NEOPIC 是一种 $O(N_p)$ 方法，其常数仅由架构决定，独立于任何网格或树。在二维和三维静电基准上的数值实验，包括 Landau 阻尼、双流和 bump-on-tail 不稳定性、一个回旋 mini-app 和一个 Penning 阱，验证了理论发现。一个在由不同参考方法生成的多个 mini-app 上训练的单一模型能够捕获所有这些情形，并在高达训练时间窗口两倍的范围内保持稳定，这指向基于粒子的方法中场的 foundation models。

2. 带下界的半线性偏微分方程的神经网络逼近与结构复杂性

Neural Network Approximation and Structural Complexity of Semilinear PDEs with Lower Bounds

作者: Hayden Schaeffer

类别: math.NA

发布日期: 2026-10-29

arXiv ID: 2610.40384v1

链接: http://arxiv.org/abs/2610.40384v1

摘要翻译

中文标题: 带下界的半线性偏微分方程的神经网络逼近与结构复杂性

摘要翻译: 神经网络通常被用于学习 Navier-Stokes 方程和粘性守恒律等偏微分方程的解算子。在标准基准上，中等规模且往往为浅层的网络能达到小误差，这些误差随网络规模以代数速率衰减。当前的逼近理论将这些观察描述得并不充分，因为它将解算子视为一般映射，并在一大类算子上推导最坏情形界。我们证明，方程的结构及其性质解释了观察到的行为。对于半线性抛物方程，逼近代价分成两项，其中一项由控制律的一个不变量控制，另一项由数据控制。本构律可参数化为单个块，对于精度 $η$，其规模为 $O(η^{-pχ}\logη^{-1})$，对每个 $p>p_f$ 均成立，其中结构指数 $p_f$ 仅依赖于 $f$ 的计算图。该指数不依赖于维数，并且对多项式律实际上为零，因此对于二维 Navier-Stokes 方程，非线性块仅需 $O(\logη^{-1})$ 个参数。数据通过编码层进入，每个模态具有有界数量的学习权重，因此两项均不随网格分辨率或时间步长增长。尽管该分析仅使用逐点块的最大范数误差，我们证明可沿整条轨迹获得更强范数下的逼近界。对于具有 $C^s$ Hamilton 量的粘性 Hamilton-Jacobi 方程，逼近需要约 $η^{-1/s}$ 个参数，并且我们证明了一个在相差对数因子意义下匹配的下界。这表明此类问题的解算子具有低结构复杂性，尽管它们是无限维空间之间的非线性映射。

3. 一种用于具有各向异性解或尖锐解的偏微分方程的自适应可迁移神经网络

An Adaptive Transferable Neural Network for Partial Differential Equations with Anisotropic or Sharp Solutions

作者: Jiacong Qu, Liyong Zhu

类别: math.NA

发布日期: 2026-10-29

arXiv ID: 2610.40201v1

链接: http://arxiv.org/abs/2610.40201v1

摘要翻译

中文标题: 一种用于具有各向异性解或尖锐解的偏微分方程的自适应可迁移神经网络

摘要翻译: 可迁移神经网络（TransNet）在求解偏微分方程（PDEs）方面已展现出实现高精度和高效率的巨大潜力。然而，当处理具有各向异性或尖锐解的 PDEs 时，它仍面临显著困难。在本工作中，我们提出自适应可迁移神经网络（A-TransNet），这是一种新颖的机器学习框架，它在大解梯度区域自适应地扩展神经元，以有效捕捉复杂的解行为。新加入神经元的位置参数不依赖于训练或随机采样，而是通过结合主动子空间方法和基于后验梯度信息的局部梯度分析来确定。该自适应策略使神经元能够被分配到最需要的区域，从而在不增加额外计算成本的情况下提高对各向异性和尖锐特征的分辨率。此外，新神经元的形状参数由其位置诱导的神经元密度自动确定，进一步增强了网络的表示能力。我们发展了一种迭代神经元扩展策略，并对所提出的 A-TransNet 方法的稳定性、后验误差估计和收敛性进行了严格分析。针对具有各向异性和尖锐解的具有挑战性的 PDE 问题的大量数值实验表明，与现有方法相比，A-TransNet 实现了更优越的精度、效率和鲁棒性。

4. 用于亚扩散肿瘤-营养物模型的渐近相容变分能量耗散律与保结构变步长线性松弛格式

Asymptotically Compatible Variational Energy Dissipation Laws and Structure-Preserving Variable-Step Linear Relaxation Schemes for a Subdiffusive Tumor-Nutrient Model

作者: Hui Yu, Zhaoyang Wang, Ping Lin

类别: math.NA

发布日期: 2026-10-29

arXiv ID: 2610.40120v1

链接: http://arxiv.org/abs/2610.40120v1

摘要翻译

中文标题: 用于亚扩散肿瘤-营养物模型的渐近相容变分能量耗散律与保结构变步长线性松弛格式

摘要翻译: 本文构建了一个通过趋化性和记忆依赖反应交换耦合的质量守恒亚扩散肿瘤-营养物模型，以及保结构变步长线性松弛格式。我们证明了一个加权 Riemann-Liouville 恒等式，并用其构造所提模型的变分能量及相应的能量耗散律。当分数阶趋于 1 时，该模型及相应的变分能量耗散律均与其整数阶对应形式渐近相容。为数值逼近，我们在非均匀时间网格上使用 L1R 公式离散 Riemann-Liouville 分数阶导数，并推导相应的离散加权不等式。我们首先为亚扩散模型构造变步长线性松弛格式。然后，为减轻由辅助变量误差引起并在小分数阶下被 L1R 卷积放大的交替扰动，我们构造时间阻尼线性松弛格式。由于时间阻尼不改变辅助变量递推，我们进一步引入后验松弛校正以减少辅助变量中的交替误差。三种格式均保持总质量，并证明其利用离散加权不等式满足离散变分能量耗散律，且与整数阶极限保持相容。给出若干数值算例以验证所提格式的精度和稳定性。

5. 复杂网络中具有 Riemann 耦合交界的一维嵌入血流的 HDG 型离散化

An HDG-type discretization of 1D embedded blood flow in complex networks with Riemann-coupled junctions

作者: Raksha Devi, Luca Heltai, Lucas O. Müller

类别: math.NA

发布日期: 2026-10-29

arXiv ID: 2610.40080v1

链接: http://arxiv.org/abs/2610.40080v1

摘要翻译

中文标题: 复杂网络中具有 Riemann 耦合交界的一维嵌入血流的 HDG 型离散化

摘要翻译: 我们提出一种可杂交间断 Galerkin (HDG) 方法，用于具有 Windkessel 边界条件的动脉网络中的一维血流。该格式采用 Harten–Lax–van Leer (HLL) 数值通量和隐式时间积分。空间离散后，完整网络——包括单元状态、杂交迹、交界条件以及动态 $\mathrm{RCR}$ 终端压力——被写成残差形式的指标一微分代数方程 (DAE)，并采用变步长、变阶向后差分公式进行积分。隐式表述消除了显式格式的 CFL 稳定性限制，并允许使用变阶、变步长求解器。由此得到的网络及其终端动力学的单一全局残差和 Jacobian 也为与外部非线性模型的整体式耦合提供了自然接口。我们建立了局部可解性和半离散能量平衡，并数值展示了两个解变量在 $L^2$ 范数与 broken $H^1$ 半范数下的最优收敛阶。该求解器针对已建立的基准进行了验证，包括主动脉分叉和 $37$ 条动脉的网络，并进一步应用于脑动脉网络。在合成的 $8\,\mathrm{k}$ 血管解剖肝脏网络中观察到质量守恒，并在 $50\,\mathrm{k}$ 和 $100\,\mathrm{k}$ 血管的合成网络上进行了强扩展性研究。

6. 论铁磁材料中叠片铁芯模型的变分刻画

On the variational characterization of a laminated core model in ferromagnetic materials

作者: Herbert Egger, Felix Engertsberger

类别: math.NA

发布日期: 2026-10-29

arXiv ID: 2610.39952v1

链接: http://arxiv.org/abs/2610.39952v1

摘要翻译

中文标题: 论铁磁材料中叠片铁芯模型的变分刻画

摘要翻译: 在较高频率下，准确预测变压器、电动机和发电机中的铁耗，需要在场仿真过程中考虑叠片磁芯中的涡流。我们研究用于这些叠片损耗的局部材料模型，这些模型通过一维扩散方程解析沿各钢片厚度方向的非均匀磁场分布。由此可以在二维或三维磁场仿真的每个材料点处计入叠片中的集肤效应和涡流损耗。宏观材料响应通过一个包含一组内部变量的最小化问题隐式描述，这些内部变量编码材料的载荷历史和当前状态。这些问题的变分结构使我们能够证明其适定性，纳入各向异性和磁滞，并开发用于评估宏观材料响应的高效全局收敛求解器。因此，所提出的框架能够在宽频率范围内高效且可靠地预测叠片铁芯材料的磁特性，并且非常适合集成到非线性磁场求解器中。

7. 非线性连续时间系统 Koopman 生成元的数据驱动随机特征逼近

Data-driven Random-feature Approximation of Koopman Generators of Nonlinear Continuous-time Systems

作者: Ana Čukarska, Maximilian Gollwitzer, Viola Muning Li, Vladyslav Fediukov, Zahra Monfared, Felix Dietrich

类别: cs.LG, math.NA

发布日期: 2026-10-29

arXiv ID: 2610.39897v1

链接: http://arxiv.org/abs/2610.39897v1

摘要翻译

中文标题: 非线性连续时间系统 Koopman 生成元的数据驱动随机特征逼近

摘要翻译: 从数据中学习连续时间动力系统是一个活跃的研究领域。这里常使用 Koopman 形式，因为它是一种灵活的方法，能够提供关于系统基本性质的信息。该形式通过作用在合适函数空间上的线性算子来表示动力系统。Koopman 算子数值逼近中的一个实际困难是“字典”的选择，即函数空间的一个有效有限维表示。在本文中，我们关注连续时间动力系统的学习，因此考虑 Koopman 生成元而不是 Koopman 算子的数值逼近。我们的主要贡献是一种数据驱动的随机特征方法，用于为诸如 gEDMD 的 Galerkin 逼近方法获得简约而有效的函数表示。我们构造的表示是数据信息驱动的，这使我们能够改进与数据无关的方法。我们通过实验证明我们的方法适用于预测任务和系统分析。数值实验表明我们的方法是准确的，并且比迭代优化更快。此外，我们证明了向量场逼近及其所得轨迹的收敛性。除数值结果外，我们还做出理论贡献，包括 gEDMD 逼近生成的半群的一般收敛准则，以及一个反例，表明 gEDMD 通常不保持指数有界性。通过我们的贡献，我们推进了从数据中高效学习连续时间动力学的前沿，朝着基于 Koopman 框架的可解释且数学稳健的建模方向发展。

8. 随机化辛模型降阶：单遍基构造与草图化哈密顿量组装

Randomized Symplectic Model Reduction: One-Pass Basis Construction and Sketched Hamiltonian Assembly

作者: Robin Herkert, Bernard Haasdonk, Jörg Fehr

类别: math.NA

发布日期: 2026-10-29

arXiv ID: 2610.39893v1

链接: http://arxiv.org/abs/2610.39893v1

摘要翻译

中文标题: 随机化辛模型降阶：单遍基构造与草图化哈密顿量组装

摘要翻译: 哈密顿系统的辛模型降阶可得到这样的降阶动力学：沿精确轨迹守恒降阶哈密顿量，从而避免传统投影方法可能产生的人为能量漂移。然而，对于大规模空间离散，基构造和降阶算子组装可能带来可观的离线成本。随机数值线性代数可通过低维草图降低这些成本，同时支持流式和并行实现。我们针对离线阶段的两个部分发展了随机技术，同时保持降阶动力学的哈密顿结构。首先，我们引入单遍草图化随机 cSVD（Ops-rcSVD），它是 complex-SVD 算法的单遍随机变体。通过将随机范围查找与一个独立的左草图相结合，Ops-rcSVD 仅需对快照集合进行单遍遍历即可构造正交辛降阶基。其次，对于二次哈密顿系统，我们引入基于对降阶基和算子作用进行草图的随机降阶哈密顿量组装，由构造即得到降阶哈密顿系统。我们推导了快照投影误差的界、草图化降阶哈密顿量关于参数一致的扰动界，以及降阶轨迹与草图化降阶轨迹之间差异的能量范数估计。大规模系统实验表明，Ops-rcSVD 在减少基构造时间的同时，达到了与确定性构造和两遍随机构造相当的投影精度。实验进一步表明，当存储完整快照集合不再可行时，草图化降阶哈密顿量组装仍能保持准确。

9. 面向受迫 PDE 的具有潜在指数积分的半群感知核神经算子

A Semigroup-Aware Kernel Neural Operator with Latent Exponential Integration for Forced PDEs

作者: Naseem Abbas, Vittorio Colao, Davide Macrì, Matthieu Barreau, William Spataro

类别: math.NA, math.AP

发布日期: 2026-10-29

arXiv ID: 2610.39845v1

链接: http://arxiv.org/abs/2610.39845v1

摘要翻译

中文标题: 面向受迫 PDE 的具有潜在指数积分的半群感知核神经算子

摘要翻译: 我们提出了一种面向源驱动演化 PDE 的半群感知结构化核神经算子。与一般的时间条件核模型不同，所提出的表述通过潜在指数传播子 (e^{tA_θ}) 表示时间依赖性，其中 (A_θ) 是学习得到的潜在生成元。这提供了从非 ODE 核表示到受 ODE 启发的核算子的结构化过渡，该核算子的潜在时间演化满足演化半群的复合原理。对于时间相关的强迫项，直接基于核的求值需要对延迟核求值进行重复求积。为解决这一计算瓶颈，我们推导了源项的潜在优先重表述：将强迫场投影到核诱导的潜在空间，其中其累积效应表示为受迫线性潜在动力系统的解。对于固定的学习核，该重表述是精确的。所得系统随后可使用基于 (φ_1) 的指数积分器进行积分，从而得到一条在结构上与学习到的潜在传播子对齐的源求值路径。所得框架将几何与物理条件化的低秩核因子分解与基于显式 Duhamel 求积或潜在指数积分的替代源求值路径相结合。对潜在生成元的耗散性约束以及初始时刻附近保持恒等性的修正均为可选的架构组件。在对流-扩散-反应 (ADR) 问题上的数值实验比较了非 ODE 表述与半群感知表述，评估了求积和 (φ_1) 源路径，并考察了预测精度与计算成本。

10. 椭圆 Carleman 估计的转移原理：在离散 Calderón 稳定性中的应用

A Transfer Principle for Elliptic Carleman Estimates: Application to Discrete Calderón Stability

作者: Qi Lü, Yu Wang

类别: math.NA, math.AP

发布日期: 2026-10-29

arXiv ID: 2610.39750v1

链接: http://arxiv.org/abs/2610.39750v1

摘要翻译

中文标题: 椭圆 Carleman 估计的转移原理：在离散 Calderón 稳定性中的应用

摘要翻译: 我们建立了一个转移原理，将连续的椭圆 Carleman 估计提升为协调 Galerkin 离散的网格一致估计。主要困难有两个：离散状态通常缺乏连续估计所需的 (H^2)-正则性，并且 Carleman 参数 (s) 必须相对于网格尺寸 (h) 加以控制。我们通过引入一个平移的连续问题同时克服这两点，该问题的 Galerkin 解恰好是给定离散状态的协调重构。随后，加权逼近在不添加稳定化的情况下将该不等式转移回原始离散算子，并给出径向权重下的范围 (s\le ch^{-4/5})。我们针对拟合的、一致形状正则网格上具有一致质量或集中质量的分片仿射有限元，以及通过其有限元实现的标准 Cartesian 有限差分，验证了该框架。作为一个应用，我们研究维数 (d\ge3) 中有界复 Schrödinger 势的离散 Calderón 问题。假设在一个固定的连通边界邻域中具有一致性，并且离散 Dirichlet 问题一致稳定，我们由 Dirichlet 输入和在独立指定、可能不相交的边界片上的变分通量观测得到稳定性。所得的 (H^{-r})，(r>d/2)，中的估计保留了全数据估计的单个对数数据项和 (O(h^{1/2})) 网格余项。

11. 任意小于 1 的 bulk 参数下的最优 AFEM 收敛性

Optimal AFEM convergence for any bulk parameter smaller than one

作者: Lars Diening, Johannes Storn

类别: math.NA

发布日期: 2026-10-29

arXiv ID: 2610.39719v1

链接: http://arxiv.org/abs/2610.39719v1

摘要翻译

中文标题: 任意小于 1 的 bulk 参数下的最优 AFEM 收敛性

摘要翻译: 我们证明了，对于 Poisson 问题，采用协调 Lagrange 元的标准自适应有限元方法在 Dörfler 标记下，对任意 bulk 参数 $θ\in (0,1)$ 均具有率最优收敛性。这消除了经典最优自适应有限元理论中长期以来对 bulk 参数的小性限制。

12. 非均匀网格上次扩散 L2 方法的系数重组与误差估计

Coefficient Recombination and Error Estimates for the L2 Method for Subdiffusion on Nonuniform Meshes

作者: Chaoyu Quan, Shangyu Wang, Shijie Wang, Jiang Yang

类别: math.NA

发布日期: 2026-10-29

arXiv ID: 2610.39642v1

链接: http://arxiv.org/abs/2610.39642v1

摘要翻译

中文标题: 非均匀网格上次扩散 L2 方法的系数重组与误差估计

摘要翻译: Caputo 导数的 L2 离散因其高阶精度而具有吸引力，但由于离散系数缺乏单调结构，其在非均匀时间网格上的分析较为复杂。当初值附近存在次扩散解的典型弱奇异性时，这一困难对于导出尖锐的 $L^2$ 误差估计尤为显著。在本工作中，我们针对非均匀网格上的 L2 方法建立了一个系数重组框架。在相邻步长比条件 $0.86\le ρ_k\le 1.16$、$k\ge2$ 下，我们证明重组后的 L2 系数为正且单调。一个关键要素是跨行重组技术：将与固定时间层相关联的重组参数用于所有系数行时，所有系数行保持正性与单调性，这使我们能够完成 $L^2$ 误差分析的归纳闭合。对于呈现标准初始弱奇异性的解，我们在满足步长比条件的非均匀网格上导出了 $L^2$ 误差估计。在分级参数为 $r>1$ 的修正分级网格上，该估计给出收敛阶 $\mathcal{O}\!\left(N^{-\min\{rα,\,3-α\}}\right)$，从而只要 $r\ge(3-α)/α$ 就恢复最优阶 $\mathcal{O}(N^{-(3-α)})$。在随机非单调网格和修正分级网格上的数值实验证实了理论估计。

13. 无磁正则化的 Shliomis 铁磁流体模型先验误差分析

A priori error analysis for the Shliomis ferrofluid model without magnetic regularization

作者: Gonzalo A. Benavides, Ricardo H. Nochetto

类别: math.NA

发布日期: 2026-10-29

arXiv ID: 2610.39325v1

链接: http://arxiv.org/abs/2610.39325v1

摘要翻译

中文标题: 无磁正则化的 Shliomis 铁磁流体模型先验误差分析

摘要翻译: 我们针对二维和三维空间中的 Shliomis 铁磁流体模型，构造并分析了一族协调的、能量稳定的空间半离散格式。与许多现有数值分析不同，我们在不引入磁扩散的情况下处理该系统。能量律提供的有限空间控制使得误差分析中磁非线性项的处理变得复杂。我们的格式保留了旋转耦合和非线性磁弛豫，并自然地纳入了磁法向通量条件，从而克服了 Benavides、Nochetto 和 Shakipov (2026) 的双势方法的维数限制和边界施加困难。误差分析将 Maxwell 约束的相容稳定化与由精确不可压缩性和磁势的 Hessian 的逐点对称性产生的抵消结合起来。我们证明了半离散格式的全局可解性，并在显式逼近假设和投影稳定性假设下，对充分正则的精确解建立了先验误差估计。我们还遵循 Girault 和 Scott (2002) 构造了一个定义在 $\mathrm{L}^1(Ω)$ 上的平均型 $\mathrm{H}^2$-协调投影算子，并建立了其 Sobolev 逼近和稳定性性质。这些性质为误差分析所需，且具有独立意义。

14. 任意阶全局无散度DG方法的能量稳定性与误差估计

Energy Stability and Error Estimates for a Globally Divergence-Free DG Method of Arbitrary Order

作者: Yuchang Liu, Yan Jiang, Mengping Zhang

类别: math.NA

发布日期: 2026-10-29

arXiv ID: 2610.39299v1

链接: http://arxiv.org/abs/2610.39299v1

摘要翻译

中文标题: 任意阶全局无散度DG方法的能量稳定性与误差估计

摘要翻译: 我们针对二维线性磁感应方程，在非结构三角形网格上提出并分析了一种任意阶全局无散度（GDF）间断 Galerkin（DG）方法。据我们所知，这是首次对高阶 GDF-DG 方法给出严格的基于能量的分析。首先，我们将边法向迹的演化纳入一个封闭且严格局部的半离散算子中，该算子仅作用于逐单元的磁场。随后，我们对连续电场引入两种局部修正：一种是加权 patch 矩校正，用以强制局部多项式正交性；另一种是顶点 patch 稳定化，用以阻尼剩余的高阶模态。在形状正则、拟一致的周期三角形网格上，我们证明了在足够大且与网格无关的惩罚参数下半离散能量稳定性，并为光滑解建立先验误差估计。数值实验支持了该分析。

15. 先线性化还是先离散？带边界耗散的 KdV 方程在网格一致正则性与指数稳定性下的误差估计

Linearize or Discretize First? Error Estimates under Mesh-Uniform Regularity and Exponential Stability for the KdV Equation with Boundary Dissipation

作者: Kangsheng Liu

类别: math.AP, math.NA, math.OC

发布日期: 2026-10-29

arXiv ID: 2610.39269v1

链接: http://arxiv.org/abs/2610.39269v1

摘要翻译

中文标题: 先线性化还是先离散？带边界耗散的 KdV 方程在网格一致正则性与指数稳定性下的误差估计

摘要翻译: 我们研究有界区间上带边界耗散的 Korteweg-de Vries 方程，以及用加权三次 Hermite Galerkin 方法对其进行的空间半离散化。小非线性波可以通过按两种次序进行的线性离散模拟来近似：先线性化后离散，或先离散后线性化。两种次序给出相同的总误差，但它们在线性化与离散化之间的分配不同。两种分配之差即为次序误差。我们证明，该误差关于初始数据的大小是二次的，并带有网格尺寸的正幂次，且对所有时间成立。 对于连续问题，解映射是解析的，其幂级数在指数加权轨迹范数中对所有时间收敛。线性化半群具有 Crandall-Pazy 正则性，且指标精确为 3/2。对于加权格式，指数稳定性、Kato 光滑性、Crandall-Pazy 正则性以及解映射的解析性均关于网格尺寸一致成立。 这些性质给出误差估计，其中误差与数据在同一范数中度量：在能量范数中、在 $H_0^1$ 中以及在 $H^2$ 中。在指数加权的时间积分范数中，半离散解在球上一致收敛：对 $L^2$ 中的数据，在 $L^4(0,\infty;L^2)$ 中以速率 $h^{1/6}$ 收敛；对 $H_0^1$ 中的数据，在 $L^2(0,\infty;H^1)$ 中以速率 $h^{2/3}$ 收敛。在每个区间 $[t_0,\infty)$（$t_0>0$）上，同范数收敛在球上以代数速率一致成立。次序误差为 $h^{1/6}$ 量级乘以数据能量范数的平方，且对所有 $t\ge0$ 一致。对于未加权格式，零群速度的网格波包会破坏网格一致的 Kato 光滑性与 Crandall-Pazy 正则性。

16. 用从单位球延拓的方法求解 Minkowski 问题：球谐离散化与误差估计

Solving the Minkowski problem by continuation from the unit sphere: spherical harmonic discretization and error estimates

作者: Buyang Li, Jian-Guo Liu, Rong Tang

类别: math.NA

发布日期: 2026-10-28

arXiv ID: 2610.39047v1

链接: http://arxiv.org/abs/2610.39047v1

摘要翻译

中文标题: 用从单位球延拓的方法求解 Minkowski 问题：球谐离散化与误差估计

摘要翻译: 给定单位球面 $\mathbb{S}^2$ 上的正函数 $ρ$，经典 Minkowski 问题要求找到 $\mathbb{R}^3$ 中的闭凸曲面，使得在外法向为 $ν$ 的点处 Gauss 曲率为 $1/ρ(ν)$。用支撑函数 $h$ 表示，即为 $\mathbb{S}^2$ 上的 Monge–Ampère 方程 $\det(\nabla^2h+hσ)=ρ$，当 $ρ$ 的一阶矩为零时，该方程在相差一个平移的意义下唯一可解。我们通过从球面延拓来求解它：沿线性路径 $ρ_t=(1-t)+tρ$，解 $h(t)$ 在人工时间 $t$ 中演化，其速度 $\partial_th$ 求解一个散度形式的线性椭圆方程，该方程的核由平移构成。在时间方向上，我们使用步长为 $τ$ 的显式 Euler 法或梯形法；在空间方向上，使用次数不超过 $L$ 的球谐函数，并对核引入 Lagrange 乘子。对于满足 $ρ\in H^{s+2}(\mathbb{S}^2)$ 的实数 $s\ge4$，我们证明所有数值解都保持严格凸，且 Euler 法的 $H^1$ 误差为 $O(τ+L^{-(s+1)})$，梯形法的 $H^1$ 误差为 $O(τ^3+L^{-(s+1)})$，其中包括数据求积，且在 $τ$ 和 $L$ 的独立阈值条件下成立。证明依赖于对每个迭代所满足的扰动方程的先验估计，这些估计使每一步都一致椭圆；还依赖于精确二次的行列式，它将梯形法的阶数从二阶提高到三阶。因此，该延拓为 Newton 法提供了良好的初始猜测：从最终状态出发，它以二次收敛速度收敛到 Galerkin 解，而该解本身与精确解相差 $O(L^{-(s+1)})$；相反，若直接从球面出发，在我们的椭球测试中其第一个无阻尼步离开了凸锥。实验展示了这些收敛阶，将乘子用作相容性诊断，由面法向和面积重构凸多面体，并定位带指标的脐点。

17. 指数-Nemytskii 神经算子：跨步长学习场演化

Exponential-Nemytskii Neural Operators: Learning Field Evolution Across Step Sizes

作者: Jingye Li, Jinglai Li

类别: cs.LG, math.NA

发布日期: 2026-10-28

arXiv ID: 2610.39041v1

链接: http://arxiv.org/abs/2610.39041v1

摘要翻译

中文标题: 指数-Nemytskii 神经算子：跨步长学习场演化

摘要翻译: 在变步长算子学习中，在观测步长上的一致性本身并不能决定其他时间分辨率下的预测。我们提出一个算子，其步长依赖性由半线性动力学的一阶 Duhamel 近似所规定。它将精确线性传播与关于场及其导数的局部非线性函数相结合，使得无需跨步长密集采样的监督即可在观测区间以下进行评估。在适当条件下且空间分辨率固定时，我们证明两个步长上模型之间的一致性控制了它们在规定范围内的预测差异。我们还给出了从两个步长局部辨识两个分量的条件，支持对所学结构的解释。一个潜在扩展将相同构造应用于学习到的场。我们在两个合成问题上评估基础模型，并在两个真实数据集上评估该扩展，与若干基线进行比较。

18. 到间断有限元空间上的 $L^{2}$-投影的各向异性误差估计

Anisotropic error estimates for the $L^{2}$-projection onto discontinuous finite element spaces

作者: Joaquín Valenzuela Díaz, Yuan Wang, Sander Rhebergen

类别: math.NA

发布日期: 2026-10-28

arXiv ID: 2610.38892v1

链接: http://arxiv.org/abs/2610.38892v1

摘要翻译

中文标题: 到间断有限元空间上的 $L^{2}$-投影的各向异性误差估计

摘要翻译: 本文确定了二维和三维中到间断有限元空间上的 $L^2$-正交投影的各向异性误差估计。这些估计以 $W^{m,p}$-半范数给出，其中 $p \in [1,\infty]$ 且 $0 \le m \le k+1$，这里 $k \ge 0$ 为多项式次数。分析依赖于一个与最大角条件等价的几何条件。作为应用，本文给出了二维和三维各向异性网格上 Poisson 问题的可杂交间断 Galerkin 离散的先验误差分析。

19. 双面积-流体体积法 (DAVOF)：在非结构网格上协同输运体积分数与面面积分数，以获得闭式、单元局部的界面法向与位置

Dual Area-Volume-of-Fluid (DAVOF): co-advecting volume and face-area fractions for a closed-form, cell-local interface normal and position on unstructured meshes

作者: Tomislav Marić

类别: physics.comp-ph, math.NA, physics.flu-dyn

发布日期: 2026-10-28

arXiv ID: 2610.38863v1

链接: http://arxiv.org/abs/2610.38863v1

摘要翻译

中文标题: 双面积-流体体积法 (DAVOF)：在非结构网格上协同输运体积分数与面面积分数，以获得闭式、单元局部的界面法向与位置

摘要翻译: 在非结构多面体网格上，几何流体体积法 (VOF) 从邻域模板的体积分数重构界面取向；在不规则单元和存在 wisps 时其性能退化，而基于此构建的曲率会驱动毛细流动的寄生流。我们输运每个面面积的润湿分数，并使其与每个单元的体积分数保持一致。由面积分数和体积分数，闭式方程可恢复单元的界面法向与位置，这在 VOF 中尚属首次：法向由对单元自身各面的一次求和得到，即对液体子体积应用散度定理；平面位置则由第二次求和得到，无需邻域或迭代。对于六面体、扰动和多面体单元上的平面，这两个恒等式都是精确的。在球体和椭球体上，法向具有面面积分数的阶，位置对每种状态和网格均为二阶，而通过一圈恢复法向拟合的二次曲面给出收敛的曲率，而参考模型则未表现出收敛。与平面裁剪一起输运时，环法向——即对单元环的并集应用同一恒等式——稳定了状态，并使球体在超过 3.3 个直径的输运中保持有界和守恒；面面积分数为一阶，正如裁剪定律所预测，且位置和形状误差达到 plicRDF 的水平。一种初步的输运实现为每个单元使用一个抛物面，其标架来自法向，其曲率张量来自环，其位置来自体积分数；该实现在分辨率序列上使球体保持有界和守恒，在 N = 32 到 128 上，使用精确法向时每个周期的面面积分数为 2.8 阶，使用环二次曲面法向时为 2.1 阶。该循环具有其所输入法向的阶；输运法向达到二阶是研究目标。本文档是记录该方法进一步发展的基础。

20. 面向非线性优化的带保护神经加速预处理（SNAP）框架

Safeguarded Neurally-Accelerated Preconditioning (SNAP) framework for Nonlinear Optimization

作者: Youngkyu Lee, George Em Karniadakis, Alena Kopaničáková

类别: math.OC, math.NA

发布日期: 2026-10-28

arXiv ID: 2610.38790v1

链接: http://arxiv.org/abs/2610.38790v1

摘要翻译

中文标题: 面向非线性优化的带保护神经加速预处理（SNAP）框架

摘要翻译: 我们提出了带保护的神经加速预处理（SNAP）框架，用于将非线性优化方法与预训练神经算子（NOs）混合。SNAP 面向由非线性偏微分方程（PDEs）变分表述产生的参数化优化问题，其中许多参数实例需要重复求解。受多层线搜索方法启发，SNAP 以循环方式运行，将经典优化步与由预训练 NO 生成的搜索方向相结合。由于 NO 生成的搜索方向不具有先验下降保证，SNAP 采用后验保护机制，仅接受满足给定下降条件和接受条件的修正。我们证明了 SNAP 算法的全局收敛性，并表明其可保持底层优化方法的收敛速度，且与所采用的 NO 架构无关。此外，我们还表明，当 NO 修正足够精确时，SNAP 收敛更快，且改进程度直接与 NO 训练误差相关。为展示 SNAP 在实际中的效率，我们在包括参数化极小曲面、Allen–Cahn 和非线性 Helmholtz 问题在内的一组多样化非线性参数优化问题上进行了数值实验。通过比较两种 NO 架构——傅里叶神经算子（FNO）和 Transolver——以及将 SNAP 用作独立求解器和非线性预条件器，展示了 SNAP 的效率。数值结果表明，所需迭代次数大幅减少，同时在广泛的问题参数和网格分辨率范围内表现出鲁棒性和泛化能力。

不相关论文（基本信息）

1. 图上带两点边界条件的参数化随机哈密顿过程

A Parameterized Stochastic Hamiltonian Process on Graphs with Two-Point Boundary Conditions

作者: Yaxin Feng, Yang Xiang, Haomin Zhou

类别: math.NA

发布日期: 2026-10-29

arXiv ID: 2610.40497v1

链接: http://arxiv.org/abs/2610.40497v1

摘要: This paper presents a Lagrangian framework to solve Wasserstein Hamiltonian flows (WHF) under two-point boundary conditions on large finite graphs. Characterized by a coupled system of discrete master equation and Hamilton-Jacobi (HJ) equation on the probability simplex, this formulation defines the macroscopic WHF. While classical density-based solvers can track this flow, they face severe memory and computational bottlenecks as the graph scale grows. To bypass these limitations, we introduce the stochastic Hamiltonian process (SHP), a sample-based mathematical framework that tracks individual particle dynamics rather than the full node-wise density. We parameterize the time-dependent potential using neural network, yielding the Neural SHP solver. Numerical experiments on synthetic graphs demonstrate the scalability of our method relative to density-based baselines. Furthermore, we validate Neural SHP across two large-scale real-world networks, a PubMed citation network and a massive road network, demonstrating its robust efficiency for graph partition and routing problems on complex graph structures.

2. 软到硬收敛的数值研究：离散化、容许性与谱正则性

Numerical Investigation of Soft-to-Hard Convergence: Discretization, Admissibility, and Spectral Regularity

作者: Sung-Sik Kwon, Marcus Nunda

类别: math.OC, math.NA

发布日期: 2026-10-29

arXiv ID: 2610.40461v1

链接: http://arxiv.org/abs/2610.40461v1

摘要: This paper presents a numerical investigation of soft-to-hard terminal constraint convergence for optimal control problems governed by a second-order ordinary differential equation and the one-dimensional heat equation. The study complements the analytical convergence theory through comparisons with exact solutions and independently computed numerical approximations. For the heat equation, a fully discrete Crank–Nicolson finite-difference formulation is validated against exact modal solutions, exhibiting approximately second-order convergence in the terminal-state and midpoint-sampled control errors under simultaneous space-time mesh refinement. Numerical experiments demonstrate the predicted first-order penalty convergence for benchmark problems and investigate the admissibility threshold governing finite-energy exact terminal controllability. Additional modal experiments examine the dependence of terminal-error convergence rates on spectral regularity, illustrating intermediate rates consistent with the theoretical estimates and asymptotic predictions. The results provide computational evidence supporting the soft-to-hard convergence framework and clarify the roles of discretization, target admissibility, and spectral regularity.

3. $\ast$-代数上的奇异值分解及其在张量分解中的应用

Singular value decompositions over $\ast$-algebras with applications to tensor decompositions

作者: Paulina Hoyos, Joe Kileel

类别: math.NA

发布日期: 2026-10-29

arXiv ID: 2610.40450v1

链接: http://arxiv.org/abs/2610.40450v1

摘要: The singular value decomposition (SVD) provides optimal low-rank approximations of real and complex matrices by the Eckart–Young theorem, and SVD-based tensor decomposition algorithms such as the higher-order singular value decomposition (HOSVD) and the tensor train singular value decomposition (TT-SVD) inherit quasi-optimality guarantees from it. We extend the SVD to matrices over a finite-dimensional $\ast$-algebra $K$. Our main contribution is a characterization of the algebras $K$ for which every matrix admits an SVD, and of the inner products on $K$ for which the Eckart-Young theorem holds. We introduce a notion of ordered SVD, show that ordered SVDs reveal the rank of a matrix over $K$ and yield optimal truncations, and give an explicit algorithm that computes an ordered SVD from the structure of the algebra. We then extend the Tucker, tensor train, and tree tensor network (TTN) decompositions and their SVD-based decomposition algorithms to tensors over $K$, and show that TT-SVD retains a quasi-optimality guarantee over every such $K$, whereas HOSVD, ST-HOSVD, and TTN-SVD do so when $K$ is commutative. Viewed in real or complex coordinates, these constructions give novel decomposition formats of ordinary tensors. We illustrate our new methods by compressing data from a variety of scientific applications, and show that they can achieve lower approximation error than their classical counterparts at equal storage.

4. 矩阵接近问题再探

Matrix Nearness Problems Revisited

作者: Vanni Noferini, Federico Poloni

类别: math.NA

发布日期: 2026-10-29

arXiv ID: 2610.40313v1

链接: http://arxiv.org/abs/2610.40313v1

摘要: Given a nonempty set of matrices $\mathcal{E} \subseteq \mathbb{C}^{m \times n}$ and a matrix $A \not\in\mathcal{E}$, a matrix nearness problem (or matrix distance problem) consists in finding a matrix in $\mathcal{E}$ that is closest to $A$, in some norm, as well as the corresponding distance. This article is a survey of matrix nearness problems and of numerical algorithms for their solution. We pay special attention to modern related topics, such as Riemannian optimization and pseudospectra, that were not discussed in depth in previous reviews about this topic. Besides collecting known facts, we also prove some foundational results that, at least in the present form, we could not locate in the literature.

5. 特征值问题的 Tau 谱刻画

Characterization of the Tau Spectrum for Eigenvalue Problems

作者: Keaton J. Burns, Ilaria Fontana, Daniel Lecoanet, Geoffrey M. Vasil

类别: math.NA

发布日期: 2026-10-29

arXiv ID: 2610.40307v1

链接: http://arxiv.org/abs/2610.40307v1

摘要: We consider tau approximations of a general eigenvalue problem for a $K$-th-order linear differential operator in one dimension, and we show that the approximate spectrum satisfies a resolvent-based spectral determinant property, which we use to investigate it. More specifically, we analyze the spectral determinant property for a second-order Sturm-Liouville eigenvalue problem approximated using Gegenbauer tau methods. Notably, a quantitative error analysis identifies two values of the Gegenbauer parameter $γ$ leading to better approximation properties. These values correspond to Legendre polynomials and their derivatives. Moreover, we discuss the behavior of the largest eigenvalue of the tau approximation. The results are validated with a numerical implementation.

6. 用于快速核矩阵-向量乘积的新型分层矩阵及其在高斯过程与核岭回归中的应用

New hierarchical matrices for fast kernel matrix-vector products with Applications to Gaussian processes and kernel ridge regression

作者: Ritesh Khan, Sivaram Ambikasaran

类别: math.NA, cs.MS, stat.CO

发布日期: 2026-10-29

arXiv ID: 2610.40195v1

链接: http://arxiv.org/abs/2610.40195v1

摘要: This article presents two new multilevel hierarchical matrices to perform fast kernel matrix-vector products in $d$ dimensions $(d>1)$, namely $\mathcal{H}_{hw}^{\prime}$, $\mathcal{H}_{hw}^{\prime \prime}$. We introduce a novel hybrid weak admissibility condition to construct these hierarchical matrices. More precisely, at coarser levels of the tree (i.e., larger blocks), we use a weak admissibility condition, wherein \textbf{far-field} and \textbf{vertex-sharing} clusters are considered admissible, while at the finer levels (i.e., smaller blocks), the \textbf{adjacent} clusters are treated as admissible. The admissible blocks at coarser levels of $\mathcal{H}_{hw}^{\prime}$ and $\mathcal{H}_{hw}^{\prime \prime}$ are represented using semi-nested and fully nested bases, respectively. In both hierarchical matrices, the finer levels at which adjacent clusters are treated as admissible are constructed in a non-nested fashion. Thus, they can be viewed as a hybrid of $\mathcal{H}$ and $\mathcal{H}^2$ matrices. We use adaptive cross approximation (ACA) and nested cross approximation (NCA) to construct non-nested and nested bases, respectively. The fast matrix-vector products are performed using the proposed hierarchical matrix formats. To evaluate the performance and usefulness of the proposed hierarchical matrices, we conduct some numerical experiments. The numerical results show that they outperform their uniform admissibility counterparts in terms of storage and matrix-vector product cost, demonstrating the effectiveness of the proposed hybrid formats. We also show that the proposed formats are competitive with the standard (or strong) admissibility-based $\mathcal{H}^2$-matrices. Furthermore, we showcase some interesting applications of the proposed hierarchical matrices.

7. R-Leja 下网格的多变量 Lebesgue 界

Multivariate Lebesgue bounds for R-Leja lower grids

作者: Phil-Alexander Hofmann, Michael Hecht

类别: math.NA

发布日期: 2026-10-29

arXiv ID: 2610.40177v1

链接: http://arxiv.org/abs/2610.40177v1

摘要: We establish an algebraic upper bound for the Lebesgue constant of multivariate polynomial interpolation on $\mathfrak R$-Leja lower grids. For each fixed dimension $d\geq2$ and every nonempty finite lower set $A\subset\mathbb N_0^d$, let $P_A$ denote the corresponding grid with $N=|A|$ nodes. We prove $Λ(P_A)\leq C_dN^{β_d}$, where $q_d=d/(d-1)$ and $β_d=1+q_d^{-1}\log_2(1+2^{q_d})$. The exponent satisfies $β_2=2.160964\ldots$ and increases to $\log_2 6=2.584962\ldots$ as $d\to\infty$, improving both the cubic estimate and the exponent $3-1/d$ obtained from quadratic prefix bounds. Our proof exploits the binary structure of known univariate estimates to control sums of powers of Lebesgue constants over interpolation prefixes. A weighted Hölder inequality and a disjoint-box packing argument then transfer these prefix-moment bounds to arbitrary lower sets. The resulting estimate is uniform over all lower-set shapes and choices of disk Leja maximizers, with constants depending only on $d$.

8. 具有自适应比特分配的整数量化分层矩阵

Integer-Quantized Hierarchical Matrices with Adaptive Bit Allocation

作者: Erin Carson, Xinye Chen, Ritesh Khan

类别: math.NA, cs.MS, cs.PF

发布日期: 2026-10-29

arXiv ID: 2610.40134v1

链接: http://arxiv.org/abs/2610.40134v1

摘要: We present an adaptive bitwidth allocation method for quantizing low-rank factors in hierarchical matrices in order to decrease storage while meeting a given constraint on their reconstruction error. For the bitwidth allocation we formulate a discrete storage minimization problem and show that an optimal assignment can give larger singular values at least as many bits as smaller ones. We further develop a greedy algorithm that meets the specified max-norm quantization error budget. We investigate a hierarchical block partition in which all of the non-diagonal blocks are represented by quantized low-rank factors and only dense blocks remain dense; in this way a global reconstruction bound related to $η_k$ and $R_k$ is obtained, with the former and latter being bounds on the truncation and reconstruction arithmetic errors, respectively. Extensive simulations, carried out with kernel and BEM matrices under block-relative budgets, indicate that the proposed method achieves a reduction in storage of up to $44.3\times$ when compared to storing the complete matrix densely in FP64, and a reduction of up to $5.70\times$ when compared to the FP64 low-rank factors within the same partition, together with measured reconstruction errors that remained below the maximum norm bounds in all the hierarchical tests.

9. 数据驱动的反问题正则化

Data-driven Regularization for Inverse Problems

作者: Oleh Melnyk, Sebastian Neumayer

类别: math.NA, math.FA, math.OC

发布日期: 2026-10-29

arXiv ID: 2610.40048v1

链接: http://arxiv.org/abs/2610.40048v1

摘要: In many applications, we are interested in studying an object of interest $x$. However, its properties are often not directly measurable. Instead, we only observe $y = F(x)$, where the (known) function $F$ is referred to as the forward mapping. To infer about $x$ from the observation $y$, we seek an (approximate) inverse mapping $x \approx F^{-1}(y)$. This process is known as solving the inverse problem. In practice, the inverse mapping $F^{-1}$ may be ill-defined or discontinuous, which forces us to choose a certain solution from many possible ones based on prior assumptions on $x$. Such assumptions about $x$ are typically incorporated into the inversion process through regularization. Therefore, the first part of this course focuses on the classical regularization theory of inverse problems. In the second part, we transition to algorithmic approaches for solving inverse problems. The foundation for this lies in the variational formulation of inverse problems, where reconstructions are obtained by minimizing specific functionals. Here, we will explore key optimization principles, discuss their practical implementation, and examine how they contribute to robust reconstructions. Today, classical methods are often modified to incorporate data-driven machine learning techniques, which improve the empirical performance. However, this often comes at the cost of explainability or convergence guarantees. Such properties are critical, for example, in medical applications, where one wants to be confident that reconstructed abnormalities are of physical nature and not phantoms resulting from the method. For these reasons, we discuss a number of machine learning methods for inverse problems, such as unrolling techniques, learned regularizers and plug-and-play methods, and explore their performance guarantees.

10. 带加性噪声的线性随机演化方程的三层MGRIT方法的均方收敛性分析

Mean-Square Convergence Analysis of a Three-Level MGRIT Method for Linear Stochastic Evolution Equations with Additive Noise

作者: Liying Zhang, Zhiheng Xu

类别: math.NA

发布日期: 2026-10-29

arXiv ID: 2610.39962v1

链接: http://arxiv.org/abs/2610.39962v1

摘要: We investigate the mean-square convergence of a three-level multigrid reduction-in-time (MGRIT) method based on F-relaxation for linear stochastic evolution equations driven by additive Gaussian noise. The method combines exponential Euler time stepping on the finest grid with implicit Euler discretizations on the two coarser grids. The inexact solution of the intermediate-level full approximation scheme problem gives rise to coupled error recursions involving two interlevel propagator discrepancies. Under a common spectral decomposition of the drift and noise covariance operators, we establish a geometric contraction estimate in a modal mean-square norm, with a contraction factor uniform in the temporal grids and both coarsening factors. We further derive fixed-iteration temporal error bounds that are uniform in the first coarsening factor and explicitly quantify the dependence on the second. For a fixed iteration count $k$ and a fixed second coarsening factor, we prove an $O(ΔT_1^r)$ error bound relative to the finest-grid reference solution for $0

11. 分裂四元数工具箱：一个 MATLAB 实现

Split Quaternion Toolbox: A MATLAB Implementation

作者: Salman Ahmadi-Asl, Abbas Moradi, Yang Zhang

类别: math.NA

发布日期: 2026-10-29

arXiv ID: 2610.39904v1

链接: http://arxiv.org/abs/2610.39904v1

摘要: This paper presents a MATLAB toolbox for working with split quaternions and split quaternion matrices. Split quaternions, also known as coquaternions, form a four-dimensional associative non-commutative algebra over the real numbers with a signature (+,+,-,-). The toolbox provides two main classes: SplitQuaternion for scalar-valued split quaternions and SplitQuaternionMatrix for matrix-valued split quaternions. The implementation includes arithmetic operations, algebraic functions, matrix operations, linear algebra utilities, and various conversion methods. The toolbox is designed to be intuitive for users familiar with MATLAB’s standard matrix operations while providing the full power of split quaternion algebra. This paper details the implementation, provides numerous examples, and demonstrates the toolbox’s capabilities through practical demonstrations. The MATLAB toolbox is accessible at https://github.com/SalmanAhmadi-Asl/Split_Quaternion.git.

12. 基于自适应交叉近似的四元数矩阵高效低秩逼近

Efficient Low-Rank Approximation of Quaternion Matrices via Adaptive Cross Approximation

作者: Salman Ahmadi-Asl, Yang Zhang

类别: math.NA

发布日期: 2026-10-29

arXiv ID: 2610.39901v1

链接: http://arxiv.org/abs/2610.39901v1

摘要: The adaptive cross approximation (ACA) algorithm is a powerful tool for constructing low-rank matrix approximations using only a small subset of matrix entries. While extensively studied for real and complex matrices, its extension to quaternion algebras, remains largely unexplored. This paper presents the first comprehensive framework for adaptive cross approximation operating natively in the quaternion algebra H. Unlike approaches that rely on real embeddings or complexifications, our method preserves the algebraic structure of quaternions throughout the computation, avoiding information loss and maintaining the natural correlations between quaternion components. We develop a fully quaternionic ACA algorithm with rigorous theoretical foundations, including convergence analysis, error bounds, and complexity estimates. The algorithm achieves O(r(m+n)) complexity per iteration and demonstrates quasioptimal approximation properties comparable to the quaternion singular value decomposition (QSVD), but with significantly reduced computational cost. We provide a detailed MATLAB with numerical experiments on color image compression. Our results show that native quaternion ACA outperforms significantly the basic SVD algorithm in terms of computational efficiency, with very close approximation accuracy.

13. 全变量框架下基于偏微分方程的贝叶斯反问题

PDE-based Bayesian Inverse Problems in All-at-once Framework

作者: Tram Thi Ngoc Nguyen, Fabian Schneider, Leila Taghizadeh

类别: math.NA, math.PR

发布日期: 2026-10-29

arXiv ID: 2610.39783v1

链接: http://arxiv.org/abs/2610.39783v1

摘要: We propose a modern all-at-once framework for PDE-based Bayesian inversion by introducing the PDE residual as probabilistic information. In this framework, the PDE residual is treated as a soft constraint with a residual covariance and is directly incorporated into the likelihood function. The new hybrid likelihood function eliminates PDE solves, thus reducing expensive computational costs, while performing rigorous Bayesian inference. Theoretically, we establish the well-posedness of the posterior with uncertainty quantification, and numerically, we propose two efficiency inversion algorithms based on two posterior sampling methods. In particular, one algorithm is based on Metropolis-within-Gibbs MCMC method with a PDE-informed block- diagonal Gaussian prior, and the other is based on a joint sampling strategy. Numerical experiments for a semiconductor inverse problem confirm that the proposed Bayesian all-at-once approach yields a speedup of approximately a factor of two, while having a comparable reconstruction quality as the conventional reduced-based setting.

14. 面向奇异摄动的高阶 QMC $hp$-FEM 不确定性量化：具 Gevrey 参数化反应系数的反应-扩散问题

Higher-Order QMC $hp$-FEM UQ for Singular Perturbation: Reaction-Diffusion with Gevrey-Parametric Reaction Coefficient

作者: Fabian Rohner, Christoph Schwab

类别: math.NA

发布日期: 2026-10-29

arXiv ID: 2610.39738v1

链接: http://arxiv.org/abs/2610.39738v1

摘要: We analyze the combined discretization of singularly perturbed reaction-diffusion problems with perturbation parameter $\varepsilon\in (0,1]$ and with parametric, Gevrey-regular reaction term, by $hp$-Galerkin FE methods and by higher-order polynomial lattice rules. We establish $\varepsilon$-robust algebraic convergence rates for the combined QMC $hp$-FEM approximation that are not subject to the curse of dimensionality (CoD) in the parametric reaction coefficient, and only limited by the sparsity and the $\mathfrak{s}_{y}$-Gevrey regularity of the reaction coefficient expansion. On the interval $(-1,1)$, we establish an $\varepsilon$-robust, root-exponential convergence rate $\exp(-σ\, p_{\mathrm{FE}}^{1/\mathfrak{s}_{x}})$ for $hp$-Galerkin FEM for reaction-diffusion problems under $\mathfrak{s}_{x}$-Gevrey regularity assumptions on the reaction coefficient and the source term, a result that can be of independent interest. Numerical experiments are consistent with the theoretical bounds.

15. 面向奇异摄动的高阶 QMC Galerkin 有限元法：具仿射参数化反应的反应-扩散问题

Higher-Order QMC Galerkin FEM for Singular Perturbation: Reaction-Diffusion with Affine-Parametric Reaction

作者: Fabian Rohner, Christoph Schwab

类别: math.NA

发布日期: 2026-10-29

arXiv ID: 2610.39734v1

链接: http://arxiv.org/abs/2610.39734v1

摘要: We analyze higher-order quasi-Monte Carlo (QMC) Galerkin discretizations for a singularly perturbed reaction-diffusion problem with an affine-parametric reaction coefficient. We derive error estimates that are robust with respect to the perturbation parameter $\varepsilon$. To this end, we establish $\varepsilon$-uniform stability and mixed-parametric regularity estimates in an $\varepsilon$-weighted energy norm, with constants independent of the parameter dimension and of $\varepsilon$. These estimates imply robust dimension truncation estimates and dimension-independent higher-order QMC error bounds for linear quantities of interest. Assuming an abstract, $\varepsilon$-robust finite element approximation property, we combine the parametric estimates with the spatial discretization analysis to derive a general fully discrete QMC-FE error estimate with constants independent of both the parameter dimension and $\varepsilon$. We verify this approximation property for a one-dimensional model problem using continuous piecewise affine finite elements on Shishkin meshes. Numerical experiments illustrate the predicted robustness and convergence rates.

16. 自由 Cox-Ingersoll-Ross 方程的一种保正数值方法

A Positivity-Preserving Numerical Method for the Free Cox-Ingersoll-Ross Equation

作者: Y. Niu, Z. Peng, J. Wei, Z. Yin, D. Zeng

类别: math.NA

发布日期: 2026-10-29

arXiv ID: 2610.39631v1

链接: http://arxiv.org/abs/2610.39631v1

摘要: We introduce a positivity-preserving drift-implicit square-root Euler method for the free Cox–Ingersoll–Ross (CIR) equation driven by free Brownian motion. The method is obtained by discretizing the square-root equation implicitly in the drift and preserves strict positivity of the numerical solution without any step-size restriction. Under the Feller condition, we establish the uniqueness of the global strictly positive solution and derive an upper bound for its operator norm. A key ingredient for the error analysis is a free stochastic differential representation of the inverse square-root process. Another key ingredient is an a priori grid-point error estimate for sufficiently small step sizes that yields uniform upper and strictly positive lower spectral bounds for the numerical solution on any fixed finite time interval. These results support the strong convergence of order $1$ and improve the classical results. Moreover, both the exact and numerical solutions are exponentially stable with respect to perturbations of the initial value. The numerical stability holds for arbitrary step sizes, and the associated discrete decay rate converges to its continuous counterpart as the step size tends to zero. Numerical experiments confirm the theoretical results.

17. 基于残差跟踪控制变量的随机 Kaczmarz 方法高效方差缩减

Efficient variance reduction for randomized Kaczmarz methods via residual-tracking control variates

作者: Jiaxin Xie, Hou-Duo Qi, Deren Han

类别: math.NA

发布日期: 2026-10-29

arXiv ID: 2610.39537v1

链接: http://arxiv.org/abs/2610.39537v1

摘要: The randomized Kaczmarz (RK) method is a widely used and effective class of iterative algorithms for solving large-scale linear systems. For inconsistent systems, however, the persistent variance caused by randomization prevents the method from converging exactly to the least-squares solution. In this paper, we introduce a residual-tracking control variate framework for reducing this variance while retaining the row-action structure and low per-iteration cost of RK. The resulting variance-reduced RK method uses constant step sizes, accesses only sampled rows of the coefficient matrix, and requires (O(m+n)) auxiliary storage. We prove that the proposed method converges linearly in expectation to the unique least-norm solution for general linear systems, including inconsistent and rank-deficient cases, with a convergence factor of the same order as that of the RK method. Furthermore, we incorporate the momentum acceleration into our framework and provide theoretical guarantees for its improved convergence rate. Numerical experiments illustrate the effectiveness of the proposed methods, particularly for ill-conditioned systems and high-accuracy computations.

18. 基于围道的 Jacobi 分数多项式基快速分数阶积分

Contour-Based Fast Fractional Integration in a Jacobi Fractional Polynomial Basis

作者: Xiaolin Liu

类别: math.NA

发布日期: 2026-10-29

arXiv ID: 2610.39388v1

链接: http://arxiv.org/abs/2610.39388v1

摘要: We develop a contour-based method for applying Riemann–Liouville fractional integrals in a Jacobi fractional polynomial basis with an endpoint weight. A Gamma-ratio representation reduces the action to contour quadrature and banded ultraspherical solves. For a fixed number of contour nodes, the cost and storage are linear in the polynomial dimension. For fractional equations, we construct selected diagonals of a short rational approximation directly and use their banded factorizations as reusable right preconditioners for Krylov iterations. Numerical experiments show accurate integral actions and substantially reduced solution costs across Abel, multi-order, BBO, deferred-correction, eigenvalue, and boundary-value problems.

19. 一种基于 MPI 进程分组策略的并行代数多重网格预条件迭代方法，用于建立阶段与求解阶段的异步并发执行

A Parallel Algebraic Multigrid Preconditioned Iterative Method Based on an MPI Process Grouping Strategy for Asynchronous Concurrent Execution of the Setup and Solve Phases

作者: Chunsheng Feng, Shuai Liu, Linjie Zhang, Zirui Zhu

类别: math.NA

发布日期: 2026-10-29

arXiv ID: 2610.39303v1

链接: http://arxiv.org/abs/2610.39303v1

摘要: Algebraic multigrid (AMG) is one of the most effective algorithms for solving large sparse linear systems and consists of a setup phase followed by a solve phase. As supercomputer performance improves and the number of MPI processes increases, AMG faces serious challenges in algorithmic scalability and parallel scalability. To address these issues, we propose an asynchronous concurrent algebraic multigrid (ACAMG) model based on an MPI process grouping strategy for setup and solve. The setup and solve phases execute asynchronously and concurrently on separate MPI process groups. This organization can both preserve AMG hierarchy quality through an appropriate choice of the setup-group size and achieve time overlap between the two phases through asynchronous concurrent execution. Based on this model, we design and implement the MPI ACAMG algorithm. Theoretical analysis and numerical experiments show that the algorithm exhibits better algorithmic and parallel scalability. For example, for a three-dimensional diffusion problem with spatially localized anisotropy, ACAMG not only reduces the solve-phase iteration count compared with the conventional method, but also achieves speedups of 4.74 and 19.36 at 64 and 128 processes, respectively.

20. 基于切片 Wasserstein 距离的体素化三维密度图快速刚性旋转对齐

Fast rigid rotational alignment of voxelized 3-D density maps in sliced Wasserstein distance

作者: Oscar Mickelin, Yunpeng Shi, Amit Singer, Eric J. Verbeke

类别: math.NA, cs.CV

发布日期: 2026-10-29

arXiv ID: 2610.39214v1

链接: http://arxiv.org/abs/2610.39214v1

摘要: We develop a fast method for the rigid alignment of voxelized three-dimensional density functions over $\mathsf{SO}(3)$ using sliced optimal transport. Our approach efficiently transforms each density map into a tensor of one-dimensional quantile functions, taken from central slices over the unit sphere, such that the Euclidean distance of the transformed volumes approximates the sliced 2-Wasserstein distance. Our formulation is therefore compatible with existing, fast, voxel-based Euclidean alignment algorithms. We provide theoretical analysis and numerical experiments demonstrating that the sliced 2-Wasserstein distance is a robust metric for comparing three-dimensional shapes and for computing their rigid alignment.

21. 基于 $H^2$ 的平衡残差误差估计

$H^2$-based equilibrated residual error estimates

作者: Kexin Ding, Yuwen Li

类别: math.NA

发布日期: 2026-10-29

arXiv ID: 2610.39210v1

链接: http://arxiv.org/abs/2610.39210v1

摘要: We develop a framework for equilibrated residual a posteriori error analysis of finite element methods, based on decomposing vector- and tensor-valued Sobolev spaces into $H^2$ and $H^1$ spaces. To estimate the residual functional acting on $H^2$ component, the method reconstructs symmetric and piecewise polynomial tensors on vertex patches in broken $H(\operatorname{div}\operatorname{div})$ spaces. We apply the proposed framework to conforming finite element discretizations of biharmonic equation, Poisson’s equation, and linear elasticity in mixed form. As a result, we obtain several novel equilibrated residual a posteriori error estimates, providing guaranteed error control for energy error of biharmonic equation, $L^2$ norm error of Poisson’s equation, and stress error of mixed linear elasticity. They also serve as the first polynomial-degree-robust a posteriori lower bounds of discretization errors. Numerical experiments illustrate the effectiveness of the $H^2$-based error estimators.

22. 一维容性耦合等离子体流体模拟的数值格式

Numerical schemes for 1D capacitively coupled plasma fluid simulation

作者: Yipei Chen, Ganghua Fan, Yuwei Fan, Haoran Guan, Kangxin Liu, Zhonghua Qiao, Yufeng Wei, Peiting You, Zhenyu Zou

类别: math.NA

发布日期: 2026-10-29

arXiv ID: 2610.39140v1

链接: http://arxiv.org/abs/2610.39140v1

摘要: Capacitively coupled plasmas (CCP) are widely used in semiconductor manufacturing, where fluid models provide an efficient approach to the simulation of the low-temperature discharge. The model considered in this work couples the electron and the ion continuity equations, the electron energy equation and the Poisson’s equation for the electric potential. We investigate the numerical schemes for the simulation of the 1D-CCP. In particular, a semi-implicit Scharfetter-Gummel (SG) scheme is adopted for the Poisson’s equation, while the remaining equations are solved using the implicit SG schemes. The proposed strategy can be naturally extended to the 2D cases. Several alternative schemes are also discussed and tested. Numerical experiments compare these methods in terms of the admissible time step $Δt$, total simulation time, accuracy against a reference solution and robustness of the scheme. We propose a numerical scheme for the 1D-CCP and, to the best of our knowledge, provide the first detailed numerical analysis of such a scheme for CCP.

23. Banach空间中Levenberg–Marquardt方法的收敛性

Convergence of Levenberg–Marquardt method in Banach spaces

作者: Akari Ishida, Sei Nagayasu

类别: math.FA, math.NA

发布日期: 2026-10-29

arXiv ID: 2610.39131v1

链接: http://arxiv.org/abs/2610.39131v1

摘要: We study the Levenberg–Marquardt method for nonlinear inverse problems in Banach spaces under a Hölder stability estimate. Under suitable assumptions on the geometry of the Banach space and the Fréchet derivative of the measurement operator, we establish the well-definedness of the method, local convergence and convergence rates for exact data, and the monotonic decrease of the Bregman distance up to the stopping index for noisy data. These results extend the corresponding Hilbert space theory to the Banach space setting.

24. 用 Borsuk-Ulam 定理寻找解

Finding solutions with the Borsuk-Ulam theorem

作者: Oleg R. Musin

类别: math.CO, math.NA

发布日期: 2026-10-29

arXiv ID: 2610.39092v1

链接: http://arxiv.org/abs/2610.39092v1

摘要: Many existence theorems in combinatorics and geometry reduce to the Borsuk-Ulam theorem or its generalizations. We study how these existence arguments can be turned into effective numerical methods when a simple example with a single known solution is available. The idea is to deform this example into the given problem and follow its solution during the deformation. We establish conditions under which this procedure reaches a solution and estimate the work needed for a prescribed accuracy. Mass partition problems illustrate the method for both continuous objects and finite point sets. In a fixed dimension, each step for point sets requires work proportional to the number of points; we give conditions under which the entire computation has this linear cost. We also study how changing the initial example can lead to other solutions and support a search for all of them.

25. 关于Fasi与Higham提出的矩阵指数Padé逼近的正性猜想

On a positivity conjecture of Fasi and Higham for Padé approximants to the matrix exponential

作者: Francesco Scardino

类别: math.NA, math.CA

发布日期: 2026-10-28

arXiv ID: 2610.39051v1

链接: http://arxiv.org/abs/2610.39051v1

摘要: We study the positivity conjecture of Fasi and Higham [SIAM J. Matrix Anal. Appl. 40 (2019), Conjecture 3.2], which concerns the Taylor coefficients of the reciprocal of the denominator $q_{k,m}(z)$ of the Padé approximant to the matrix exponential. The conjecture asserts that these coefficients are positive whenever $m$ is odd. We prove full positivity for $k=0$ and eventual positivity whenever the unique positive real zero of $q_{k,m}$ is the unique zero of minimal modulus. We also give an explicit computable sufficient threshold beyond which all coefficients are positive. Using Hermite scaling with leading-coefficient normalization, we prove that this spectral minimality holds for each fixed odd $m$ and all sufficiently large $k$. The unrestricted conjecture is nevertheless false. We give exact rational counterexamples and prove a transient scaling theorem showing that on the scale $n\asymp\sqrt{k+1}$ the normalized coefficients converge to an explicit oscillatory Hermite profile. For $m=5$ this profile is negative at one point, yielding infinitely many counterexamples. The counterexamples are not diagonal, and the diagonal case $k=m$, which is the case relevant in practice, remains open. These results do not imply a failure of the Fasi–Higham algorithm. Rather, they delimit the range in which the auxiliary positivity conjecture can be used as a theorem.

26. 对称凸体上的齐次多项式逼近：Kroó猜想的证明

Homogeneous polynomial approximation on symmetric convex bodies: A proof of Kroó’s conjecture

作者: Matthew J. Colbrook

类别: math.NA, math.CA, math.FA

发布日期: 2026-10-28

arXiv ID: 2610.38877v1

链接: http://arxiv.org/abs/2610.38877v1

摘要: We resolve Kroó’s 2004 conjecture on homogeneous polynomial approximation in every dimension, without smoothness or curvature assumptions. For every origin-symmetric convex body $K\subset\mathbb R^d$ with nonempty interior, every continuous real function on $\partial K$ is uniformly approximated by sums of homogeneous polynomials of degrees $n$ and $n+1$ as $n\to\infty$. Moreover, approximation of the constant function by homogeneous polynomials of degree $2n$ is uniform over all such bodies in each fixed dimension. Via homogenization, we also resolve Kroó’s 2019 weighted approximation conjecture for every convex admissible weight on Euclidean space. The key construction produces homogeneous polynomials with a common Gaussian boundary profile and localization errors controlled by arbitrary polynomial weights. Its analytic core is a weighted Gaussian approximation theorem: for (d\ge2), sufficiently broad Gaussians on $\mathbb R^{d-1}$ can be approximated by $e^{-f}F$, where $f$ is nonnegative, convex and Lipschitz with $f(0)=0$, and $F$ is a Schwartz function with Fourier support in an arbitrarily small ball about the origin. Small Fourier support allows Taylor truncation to a lower degree, while the remainder enforces decay through homogenization. We then use positive spherical quadrature and degree raising to yield the full approximation theorem.

27. 反馈延迟网络的结构化剪枝

Structured Pruning of Feedback Delay Networks

作者: Facundo Franchino, Karolina Prawda

类别: eess.AS, eess.SP, math.NA

发布日期: 2026-10-28

arXiv ID: 2610.38768v1

链接: http://arxiv.org/abs/2610.38768v1

摘要: Feedback delay networks (FDNs) are recursive filter structures used to synthesise artificial reverberation. Their feedback matrix governs both energy exchange between delay lines and the computation required at every sample, making its design central to balancing an FDN’s acoustic performance and implementation efficiency. In this work, we develop an FDN-specific structured pruning method that connects these acoustic and computational concerns, allowing high-order dense orthogonal designs to be fitted to efficient implementations. The feedback matrix is replaced by stages of fixed permutations and small orthogonal blocks. We characterise connectivity growth with the network depth and derive exact conditional block updates for matrix fitting, followed by refinement against the reference FDN magnitude response. The resulting factors admit direct vectorised evaluation, with arithmetic cost and coefficient storage scaling linearly with network size for fixed block size and depth. On an Apple M2, the selected staged kernels achieve feedback-multiplication speedups over dense BLAS ranging from approximately threefold to over twentyfold across the tested network sizes, while substantially reducing coefficient storage. Objective metrics and listening tests indicate that the proposed staged approximations closely preserve the spectral and temporal character of their dense references. An analysis of processor utilisation and register reuse explains why execution time depends on more than the operation count. Finally, block size and depth provide practical control over response fidelity, storage, and real-time cost, allowing acoustic design to precede the choice of an efficient implementation.