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1. 以无穷远为计算边界的亥姆霍兹散射有限元方法
Finite Elements for Helmholtz Scattering with Infinity as a Computational Boundary
作者: Markus Wess, Anıl Zenginoğlu
类别: math.NA, gr-qc
发布日期: 2026-06-23
arXiv ID: 2606.25130v1
摘要翻译: 基于零无穷层构造，我们针对外亥姆霍兹方程建立了双曲型紧化的 H1 协调有限元格式。坐标变换将无穷远映射到有限外边界，并通过重新缩放移除主导振荡衰减。我们推导了变换后的方程以及具有有界系数的整体 sesquilinear 弱形式。紧化边界贡献了一个显式边界质量项，其迹在已知归一化下给出远场模式。我们将所得方法与使用完美匹配层 (PML) 的有限元离散化进行了比较，并报告了二维和三维的基准测试结果。数值实验包括单位圆盘的散射、捕获几何结构中的共振、三维的制造基准测试以及潜艇基准测试。
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链接: http://arxiv.org/abs/2606.25130v1
1. 研究对象和背景
本文研究有界障碍物在均匀介质中的时谐散射问题。设障碍物 $O\subset\mathbb{R}^d\ (d=2,3)$ 为有界 Lipschitz 域，其外部区域记为 $O^c=\mathbb{R}^d\setminus O$，物理坐标用 $\tilde{x}\in O^c$ 表示，$r=\|\tilde{x}\|$，$\hat{x}=\tilde{x}/r$。给定波数 $k>0$，物理散射场 $U^s:O^c\to\mathbb{C}$ 满足外域 Dirichlet 散射问题：
$$ (\Delta_{\tilde{x}}+k^2)U^s=0\quad\text{在 }O^c\text{ 内},\qquad U^s=-U^i\quad\text{在 }\partial O\text{ 上},\qquad \lim_{r\to\infty}r^{\frac{d-1}{2}}(\partial_r U^s-\mathrm{i}k U^s)=0\ \text{关于 }\hat{x}\text{ 一致}. $$求解该问题的数值难点主要来自两方面：计算域无界，且辐射解在任意大距离处保持振荡。传统处理方法包括人工截断并施加近似辐射条件，如 Engquist–Majda 或 Bayliss–Turkel 型局部吸收边界条件，以及高阶局部 Dirichlet-to-Neumann (DtN) 映射。然而，基于 Padé 逼近的局部 DtN 条件存在不随频率下降的相对误差下界。完全匹配层 (PML) 通过在物理区域外围添加复拉伸层来吸收出射波，截断误差可随频率指数衰减，但精度依赖于层宽和拉伸剖面。精确 DtN 映射或边界积分方程方法可以给出精确的非反射边界条件，但通常需要基本解或稠密边界算子，不易推广到变系数或非线性问题。
另一类思路是将无穷远映射为有限边界。纯空间紧致化（如 Chebyshev 谱方法）可将无界域变换到有限区间，但对振荡解会产生波长压缩，导致无限分辨率问题。无限元方法在外部区域采用编码出射渐近行为的特殊基函数，共轭无限元则将因子 $r^{-(d-1)/2}e^{\mathrm{i}kr}$ 纳入形函数，从而剩余振幅可数值逼近。这些方法表明，将出射渐近或复尺度衰减直接融入外部离散可得到高精度求解器。
超双曲紧致化源于 Penrose 对类光无穷远的共形处理，并在数值相对论中取得成功。其核心思想结合了空间紧致化与时间变换：通过径向坐标压缩将无穷远映射为有限边界 $\mathscr{I}$ (对应 $r\to\infty$)，同时对未知量施加重标度以消除代数衰减和主导振荡。重标度后的未知量在紧致化域上具有非振荡渐近行为，从而无需在边界上强制本质条件，且远场模式可直接从 $\mathscr{I}$ 上的迹读出（差一已知常数因子）。但此前超双曲紧致化多用于光滑、近球形的相对论模拟，而计算散射问题常涉及非光滑障碍物和非球形近场几何。本文借助 [2] 的思想，发展一种 $H^1$-协调有限元方法，在保留球形紧致化边界的同时允许内部任意障碍物几何，推导出全局有界系数的变分弱形式，并通过数值实验与径向 PML 离散比较，验证其精度和有效性。
2. 主要定理或主要结果
在假设障碍物 $O$ 有界 Lipschitz、径向紧致化映射如第 2.1.1 节所述（$Ω,S,R,L,G$ 满足 $Ω(S)=0,Ω'(S)<0, L>0$）且高度函数 $h$ 与 boost 函数 $H$ 满足定义 2.2（特别 $H(S)=1$, $(1-H^2)/G$ 有界）的条件下，本文证明了以下核心结论：
定理 3.1(一致性与有界性) 设 $U$ 是齐次 Helmholtz 方程 (1) 满足障碍物上 Dirichlet 数据 $U_0$ 与 Sommerfeld 辐射条件的解，且其经重标度后的拉回
$$ u(x) = Ω(\rho)^{\frac{d-1}{2}} e^{\mathrm{i}k h(r(\rho))} U(\tilde{x}(x)) $$属于 $H^1(D)$。则 $u$ 满足弱形式：求 $u\in V_1=\{v\in H^1(D):\operatorname{tr}|_{\partial O}v=U_0\}$，使得对所有 $v\in V_0=\{v\in H^1(D):\operatorname{tr}|_{\partial O}v=0\}$ 有 $a(u,v)=0$，其中 $$ a(u,v) := -\int_D \nabla u\cdot (G Q + L P)\nabla\bar{v}
- k^2\int_D \frac{1-H^2}{G}u\bar{v}
- \int_D \frac{1}{L}\Bigl(\frac{d-1}{2}Ω’\Bigr)^2 u\bar{v} -\mathrm{i}k\int_D H,n\cdot(u\nabla\bar{v}-\bar{v}\nabla u)
- \int_D \frac{Ω}{L}\Bigl(\frac{d-1}{2}Ω’\Bigr) n\cdot(u\nabla\bar{v}+\bar{v}\nabla u)
- \mathrm{i}k\int_{\mathscr{I}} u\bar{v}. $$ （这里 $n = x/\rho$ 为单位径向向量，$Q = nn^\top$, $P = I-Q$） 该弱形式的所有系数在 $D$ 上有界，且双线性型 $a$ 在 $H^1(D)\times H^1(D)$ 上有界：存在常数 $C_a(k,Ω,H,D)>0$ 使得 $$ |a(u,v)|\le C_a|u|{H^1(D)}|v|{H^1(D)},\quad\forall u,v\in H^1(D). $$
重要特征包括：
- 在无穷远边界 $\mathscr{I}$ 上不施加本质边界条件，唯一的边界项为附加的相位项 $\mathrm{i}k\int_{\mathscr{I}} u\bar{v}$。
- 当 $H=1+O(G)$ 时，入射波分量在 $H^1(D)$ 中不可容许（除非常数），从而变分形式自动选取出射分支（注记 1）。
- 远场模式通过 $u|_{\mathscr{I}}$ 与显式常数因子相联系：若物理远场定义为 $U(r,\hat{x}) = r^{-(d-1)/2}e^{\mathrm{i}kr}(U_\infty(\hat{x})+o(1))$，则取 $f=1$ 时 $u|_{\mathscr{I}}(\hat{x}) = S^{-(d-1)/2}e^{-\mathrm{i}kh_0}U_\infty(\hat{x})$。
数值实验部分验证了上述理论结果：在二维单位圆盘散射中，方法展现最优收敛率；与径向 PML 比较，超双曲紧致化在相同自由度下精度更高，且误差对波数 $k$ 的依赖性更弱；在三维球外人工基准和潜艇模型上亦取得预期精度。
3. 研究方法、关键技术和核心工具
本文的核心技术障碍在于如何将外域亥姆霍兹方程的奇性（无穷远与振荡衰减）通过坐标紧致化和未知量重标度完全吸收，并在此基础上构造一个既保持标准 $H^1$ 协调性、又使所有系数在紧致化边界处有界的变分弱形式。作者的主要创新在于非平凡的双线性型推导及对退化矩阵 $A = Ω^{d-1}M^{-1}$ 的精细分解。
1. 紧致化映射与系数分析
采用径向边界定义函数 $Ω(\rho)$，将物理径向坐标 $r$ 映射为 $\rho/Ω(\rho)$，使得 $\rho=S$ 对应 $r=\infty$。由此导出的雅可比矩阵及 $M^{-1}$ 含有因子 $Ω^{-(d-1)}$（公式 (11)）。引入重标度矩阵 $A=Ω^{d-1}M^{-1}$ 后，利用径向投影分解得到 $A = L P + G Q$（公式 (15)），其中 $G = Ω^2/L$ 在 $\mathscr{I}$ 上为零，导致径向余法向退化。这一退化是避免在无穷远处施加本质边界条件的几何根源。
2. 重标度与高度函数
为消除主振荡因子并保证边界正则性，物理解 $U$ 重标度为 $u = Ω^{-(d-1)/2} e^{-\mathrm{i}k h(r)}U$（一般形式为 $U = Ω^{(d-1)/2}e^{\mathrm{i}kh}u$）。引入实值径向高度函数 $h(r)$ 及其 boost $H(\rho)=dh/dr$，要求 $H(S)=1$ 且 $(1-H^2)/G$ 有界。这一条件等价于 $H=1+O(G)$ 在 $\mathscr{I}$ 邻域成立，从而将 $k^2$ 项中奇异的 $1/Ω^2$ 贡献抵消，留下 $k^2(1-H^2)/G$ 为有界函数。这本质上是将出射特征因子 $e^{\mathrm{i}kr}$ 分解出来，同时允许层内有灵活的高度函数选择（例如 $f=1$ 而非 $1/\rho$）。
3. 弱形式的系统推导
作者从物理弱形式出发，直接应用坐标变换与乘积法则，而不是先写出强形式 (23) 再去边界化。这样做的优点在于，得到的双线性型 $a(u,v)$ 保持振幅重标度因子以对称一阶项出现，且所有系数（如 $G,Q,L,P, H, Ω'$ 等）均受控。关键步骤是使用恒等式 (22) 及行列式变换，将含 $Ω^{-(d-1)}$ 的体积项转化为 $A$ 的散度形式，再结合 $A$ 的径向分解。最终形式 (33) 中的每一项系数都通过 $L$ 的正性、$G$ 的退化性以及 $H$ 的正则性保证了在 $D$ 上的 $L^\infty$ 有界性。
4. 有界性证明与远场提取
有界性论证基于各项被积函数的逐点估计：利用 $L\ge L_0>0$，$0\le G\le G_{\max}$，以及 $(1-H^2)/G\in L^\infty$，结合 H₁ 范数的 Cauchy–Schwarz 和迹不等式即得。$\mathscr{I}$ 上的边界项 $\mathrm{i}k\int_{\mathscr{I}} u\bar{v}$ 自然出现，无需额外近似。远场公式来自于高度函数的渐近展开 $h(r)=r+h_0+o(1)$ 与重标度定义，直接由 $u$ 在 $\mathscr{I}$ 上的点迹给出。
5. 数值实现与 PML 对比
离散采用标准 $H^1$-协调有限元空间。数值实验中，为与 PML 公平比较，使用相同阶数的有限元并在相同物理区域截断。超双曲方法仅需单层，无需调节 PML 吸收参数，且误差随网格加细以最优速率下降；对 $k$ 的依赖性明显弱于 PML（尤其在大波数时），这一现象与理论上系数的有界性及辐射条件的精确内含是一致的。
4. 与之前工作的比较
1. 与局部吸收边界条件和 PML 的比较
Engquist–Majda、Bayliss–Turkel 及后来的高阶局部 DtN 条件通过在人工边界上施加近似辐射条件截断区域，其相对误差存在与频率无关的下界 [13]。PML 通过在吸收层内引入复坐标拉伸实现指数衰减，但精度仍依赖于层厚、拉伸角度和截断处的边界条件 [15]。与这两类方法相比，本文的超双曲紧致化将真实的无穷远 $\mathscr{I}$ 作为计算边界精确包含，不引入模型误差；远场模式直接从 $u$ 在 $\mathscr{I}$ 上的迹获取，无需后处理或外推。数值实验显示，在相同自由度下，超双曲方法的 $L^2$ 误差显著低于 PML（例如圆盘散射实验中约 2–3 个数量级），且误差对波数 $k$ 的依赖更微弱（图 4.1—4.3）。代价是变换后的弱形式系数必须满足有界性条件（$(1-H^2)/G\in L^\infty$），但对高度函数的选取 (如 $H=1-G$) 易于满足，并未增加实施复杂度。
2. 与无限元方法的比较
经典无限元将标准有限元与外部近似空间耦合，外部形函数编码出射渐近行为。共轭无限元 [22,23] 通过因子 $r^{-(d-1)/2}e^{\mathrm{i}kr}$ 消除振幅和振荡，剩余幅值数值逼近。超双曲紧致化在精神上与其接近，同样提取了主导衰减与振荡。主要区别在于：（i）超双曲方法将外部区域紧致化到有限层，无穷远成为实际计算边界，从而全部使用标准 $H^1$ 有限元，不依赖特殊的无限单元库；（ii）通过高度函数 $h$ 和 boost $H$，层内变化可自由设计（仅需渐近匹配 $H=1$），相当于时间域中时间变换的自由度；（iii）远场模式直接由 $\mathscr{I}$ 上的点值读出，而不是由级数展开或插值得到。
3. 与纯空间紧致化谱方法的比较
Grosch–Orszag [29] 和 Boyd [30,31] 等发展的映射 Chebyshev 法将 $[0,\infty)$ 代数或指数映射到有限区间，适合衰减解，但对振荡解在映射边界附近剧烈压缩波长，导致无限分辨率要求。超双曲紧致化通过显式分解 $e^{\mathrm{i}kr}$ 避免了这一缺陷，使重标度后的解在 $\mathscr{I}$ 处正则，故有限元或谱离散均能有效逼近。与之相关的一种 PML 型方法 [34–36] 同样使用实坐标压缩并显式提取振荡因子，但它的外边界仍是人工截断，远场模式不能直接从边界迹获得；而本文的 $\mathscr{I}$ 是类光无穷远，具有辐射场的固有含义。
4. 与数值相对论中紧致化应用的比较
超双曲紧致化在数值相对论中广泛用于引力波波形提取、黑洞微扰和准正规模计算 [44–51]。那些应用大多依赖光滑的球形或近球形几何，且往往使用谱方法。本文首次将其发展为适用于任意 Lipschitz 障碍物几何的 $H^1$-协调有限元框架，并对 Helmholtz 方程给出了完整的变分理论，填补了该方法在频域散射有限元模拟中的空白。数值实验涵盖了非光滑障碍物（如潜艇基准）和三维情形，证明了方法的通用性。
5. 局限性与代价
从提供的材料看，作者未显式讨论方法的限制，但从构造可知：（i）紧致化中心必须置于障碍物内部，且障碍物不能与紧致化边界相交；（ii）$\mathcal{I}$ 固定为球面，对高度各向异性的散射体可能需要更复杂的紧致化几何（文中提及可推广至一般光滑闭曲面但未实现）；（iii）与 PML 类似，引入对称双线性型的非自伴低阶项，但不影响有限元收敛性。此外，方法要求解在 $\mathscr{I}$ 邻域具有精确的出射球面波展开，对非均匀介质或非线性问题，重标度因子可能需要重新设计。但这些均在标准散射问题假设之内，与大部分截断技术相同。
了解领域论文（摘要翻译）
1. 闭合曲面Willmore流的$A$-稳定向后差分全离散格式的误差估计
Error estimates for $A$-stable backward difference full discretizations of Willmore flow of closed surfaces
作者: Nils Bullerjahn, Balázs Kovács
类别: math.NA
发布日期: 2026-06-24
arXiv ID: 2606.25934v1
链接: http://arxiv.org/abs/2606.25934v1
摘要翻译
中文标题: 闭合曲面Willmore流的$A$-稳定向后差分全离散格式的误差估计
摘要翻译: 针对闭合二维曲面的Willmore流，给出了$A$-稳定向后差分全离散格式（1阶和2阶）的最优阶$H^1$-范数误差估计的证明。该数值方法对耦合演化方程组进行离散，空间上采用多项式次数至少为二的演化曲面有限元，时间上采用1阶或2阶向后差分方法。收敛性分析基于稳定性分析，而稳定性分析则利用二阶系统的反对称结构进行能量估计，并结合Dahlquist的$G$-稳定性以及Nevanlinna和Odeh的乘子技术，同时给出了一个体现Dahlquist精神的新上界。数值实验展示并补充了理论结果。
2. 基于数据驱动多尺度空间的算子学习用于随机非均匀多孔介质中的非线性流动
Operator Learning on the Data-Driven Multiscale Space for Nonlinear Flow in Random Heterogeneous Porous Media
作者: Maria Vasilyeva, Raphael Pangilinan
类别: math.NA
发布日期: 2026-06-24
arXiv ID: 2606.25820v1
链接: http://arxiv.org/abs/2606.25820v1
摘要翻译
中文标题: 基于数据驱动多尺度空间的算子学习用于随机非均匀多孔介质中的非线性流动
摘要翻译: 我们提出一种基于粗尺度数据驱动多尺度空间的算子学习框架，用于随机非均匀多孔介质中的非线性流动。该多尺度空间由局部代表性细尺度解快照构造而成，从而得到解流形的精确低维表示。该多尺度基函数用作神经算子的主干，而分支网络从输入渗透率场预测对应的降阶系数。与Galerkin投影方法不同，该神经算子学习从渗透率场到解系数的全局非线性映射，提供了更大的灵活性、更高的精度，并且无需在线非线性粗网格求解和系数评估。数值结果表明，对于高对比度非均匀介质中的非线性流动，所提方法相比基于投影的方法获得了良好的精度且计算成本显著降低。
3. 针对向列型Helmholtz–Korteweg方程的混合$C^{0}$内部惩罚方法
A hybrid $C^{0}$-interior penalty method for the nematic Helmholtz–Korteweg equation
作者: Tim van Beeck
类别: math.NA
发布日期: 2026-06-24
arXiv ID: 2606.25642v1
链接: http://arxiv.org/abs/2606.25642v1
摘要翻译
中文标题: 针对向列型Helmholtz–Korteweg方程的混合$C^{0}$内部惩罚方法
摘要翻译: 向列型Helmholtz–Korteweg方程模拟了时间谐波声波在向列型Korteweg流体（如向列相液晶）中的传播。该偏微分方程在经典Helmholtz方程的基础上增加了两个额外的四阶项，其中一项在向列场方向上是各向异性的。我们利用Cordes条件改进了Farrell等人(2025)之前的连续分析，并提出了一种$C^0$-杂交内部惩罚离散化方法。所提出的离散化方法比$C^1$-协调方法具有更大的灵活性，并且非常适合三维空间和曲面区域上的应用。我们证明了该方法对于任何大于等于2的多项式次数是稳定的，与空间维度无关，前提是各向异性足够小。此外，我们在最小正则性假设下证明了离散解序列收敛到连续解，并在连续解具有额外正则性时导出了收敛速率。最后，我们通过数值算例展示了该方法的能力。
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1. Tensorion: Muon优化器的张量感知推广
Tensorion: A Tensor-Aware Generalization of the Muon Optimizer
作者: Vladimir Bogachev, Vladimir Aletov, Alexander Molozhavenko, Sergei Kudriashov, Maxim Rakhuba
类别: cs.LG, cs.CV, math.NA
发布日期: 2026-06-24
arXiv ID: 2606.25975v1
链接: http://arxiv.org/abs/2606.25975v1
摘要: Common first-order optimizers, such as Adam, implicitly treat each parameter block as an unstructured vector, which disregards the multilinear weight structure present in many modern machine learning models. Recent work has shown that exploiting matrix structure can improve optimization dynamics. A notable example is Muon, which performs steepest descent under the spectral norm constraint. We take the next step and introduce Tensorion, a tensor-aware optimizer that extends Muon’s constrained optimization perspective from matrices to higher-order tensors. Tensorion is built around a linear minimization oracle (LMO) over a tensor norm ball. The norm is carefully chosen to balance two objectives: tightly bounding the tensor spectral norm, while still keeping the LMO tractable. This LMO becomes computable because it reduces to operations on adaptively selected unfolding matrices. Notably, when restricted to order-2 tensors (i.e., matrices), Tensorion recovers Muon exactly. Experiments on tensor-based computer vision problems suggest that Tensorion can offer improved convergence behavior and more stable gradient updates compared with Adam-based and existing tensor-aware baselines in the evaluated settings.
2. Dirichlet-to-Neumann 映射神经算子学习的主部分解
Principal-Part Decomposition for Neural Operator Learning of Dirichlet-to-Neumann Maps
作者: Shuo Ling, Wenjun Ying, Han Zhou
类别: math.NA
发布日期: 2026-06-24
arXiv ID: 2606.25952v1
链接: http://arxiv.org/abs/2606.25952v1
摘要: Dirichlet-to-Neumann (DtN) maps send boundary values of a partial differential equation (PDE) solution to its normal derivative on the boundary. Learning such maps across varying domains is important for boundary-value problems, but a black-box neural operator must model both the operator’s non-smoothing principal behavior and its dependence on boundary geometry. We use the boundary integral representation of the DtN map to obtain, for smooth planar boundaries, a useful principal-part decomposition: a geometry-independent leading operator can be written as a universal Fourier multiplier, while the remaining geometry-dependent correction is smoother. We propose Principal-Part Decomposed Neural Operators (PPDNO), a hybrid analytic-neural framework that turns this decomposition into a geometry-conditioned operator learning model. PPDNO computes the principal part by FFT and trains a low-rank Deep Operator Network (DeepONet)-type architecture to approximate only the residual correction across families of boundary geometries. This design keeps the exact linear action on the boundary data, exposes the sampled boundary as an input to the model, and turns the learned target into a smoother operator family. We justify the decomposition theoretically by proving smoothing and separated-approximation properties of the residual, and we derive finite-node and training-error bounds for the reconstructed full map. Experiments on interior Laplace problems over elliptical and Fourier-parameterized domains, and on exterior Helmholtz problems over rose curves, show that PPDNO improves accuracy over direct neural operator baselines while adding little inference overhead and generalizing to unseen boundary data. These results suggest that analytic operator structure and geometry-conditioned learning can be combined effectively for boundary solution maps.
3. 基于提升和张量列车近似的齐次玻尔兹曼方程快速格式
A fast scheme for the homogeneous Boltzmann equation based on lifting and tensor train approximation
作者: Kun Huang, Yingda Cheng, Irene M. Gamba
类别: math.NA, physics.comp-ph
发布日期: 2026-06-24
arXiv ID: 2606.25914v1
链接: http://arxiv.org/abs/2606.25914v1
摘要: We propose a fast deterministic scheme for the space-homogeneous Boltzmann equation that exploits the low-rank structure of the velocity distribution. This paper consists of two independent contributions. The first is a \emph{lifting-projection (LP) scheme}, inspired by the approach in the recent theoretical breakthroughs \cite{guillen2025landau, imbert2026monotonicity, guillen2025landau2} on the well-posedness of the Landau and Boltzmann equations. In particular, the approach lifts the nonlinear 3D Boltzmann equation to the 6D linear Kac master equation, advanced over a single time step, and projected back to its marginal in 3D. The second contribution is a \emph{low-rank tensor method} for evaluating the collision operator, in which the lifted solution is represented in tensor train (TT) format and computed via a TT cross approximation algorithm with interpolation, complemented by a TT-friendly conservation correction that enforces conservation of mass, momentum, and energy. When the solution is low-rank in velocity, the method scales linearly in $n$ when cubic interpolation is used (and quadratic in $n$ when spectral interpolation is used), where $n$ is the number of grid points in each velocity direction. Therefore, our methods offer significant computational savings over existing deterministic solvers in such cases. Numerical experiments on 2D and 3D benchmarks, including the BKW exact solution and anisotropic initial data, confirm the computational scaling, the expected order of accuracy and verify the effectiveness of the conservation correction.
4. 面向微分代数方程的谱延迟校正方法内并行化
Spectral deferred corrections parallelized across the method for differential-algebraic equations
作者: Matthias Bolten, Lisa Wimmer
类别: math.NA
发布日期: 2026-06-24
arXiv ID: 2606.25908v1
链接: http://arxiv.org/abs/2606.25908v1
摘要: In this work, the performance of different spectral deferred corrections (SDC) methods applied to initial value problems for differential-algebraic equations (DAEs) of index one is analyzed. The SDC method solves a series of correction equations, and after each iteration, the numerical solution is corrected by adding the obtained approximation of the error. The formulation of the SDC method allows for a parallelization across the method to obtain small-scale parallelism, for which a number of processes equal to the number of collocation nodes can be used. Recently, an analytical approach to finding optimal diagonal coefficients for parallel SDC was proposed. So far, no analysis with the new coefficients for parallel SDC applied to DAEs was done. We demonstrate that parallel SDC methods solve DAE problems with high accuracy faster than the associated sequential SDC schemes, obtaining speedup in a small-scale parallel regime.
5. 用于长期碎裂云传播的物理信息统计学习模型
A Physics-Informed Statistical Learning Model for Long-Term Fragmentation Cloud Propagation
作者: Yema Paul, Diogo Sousa, Albert Servitje, Rodrigo Pariente, Emmanuel Delande, Christophe Taillan, Joan Pau Sanchez
类别: math.NA
发布日期: 2026-06-24
arXiv ID: 2606.25893v1
链接: http://arxiv.org/abs/2606.25893v1
摘要: This paper introduced a Hierarchical Generative Density Model (HGDM) for the long-term propagation of orbital fragmentation clouds. Validation against high-fidelity Monte Carlo simulations showed that the proposed surrogate accurately reproduces the dominant multidimensional structures of propagated clouds while consistently outperforming classical band-formation approximations based on independent angular variables. Accurate cloud reconstructions were obtained using only a few hundred to a few thousand propagated fragments, yielding reductions exceeding two orders of magnitude in computational cost and three orders of magnitude in storage requirements; future work will investigate its application to large-scale debris-environment evolution and collision-cascade simulations associated with the Kessler syndrome.
6. 针对具有不确定接触结构的流行病动力学动理学模型的多保真度方法
Multi-fidelity methods for kinetic models of epidemic dynamics with uncertain contact structure
作者: Liu Liu, Andrea Medaglia, Hao Xie, Mattia Zanella
类别: math.NA
发布日期: 2026-06-24
arXiv ID: 2606.25835v1
链接: http://arxiv.org/abs/2606.25835v1
摘要: In this work, we develop a multi-fidelity strategy for kinetic models in epidemiology with uncertain contact dynamics. Assessing and controlling the population-level effects of contact dynamics requires the development of models for understanding observable effects of heterogeneous contact structures, whose formation depends on complex social phenomena. These can be captured taking into account high-dimensional uncertain quantities. The proposed approach combines high-fidelity kinetic solvers with a hierarchy of low-fidelity surrogates, including reduced macroscopic models and coarse kinetic descriptions, remaining applicable even in regimes where a macroscopic closure is unavailable. This hierarchical framework identifies representative parameter samples and reconstructs full solutions via projection-based techniques, enabling efficient uncertainty propagation while drastically reducing computational cost. Numerical experiments in high-dimensional stochastic settings demonstrate that accurate statistical estimates of epidemic observables can be obtained with significantly reduced computational costs compared to standard approaches.
7. RNA二级结构吸引盆的动力学度量与能量景观超度量性分析
Kinetic metric for basins of attraction of RNA secondary structures and analysis of the ultrametricity of the energy landscape
作者: A. P. Zubarev
类别: cond-mat.dis-nn, cond-mat.stat-mech, math.NA
发布日期: 2026-06-24
arXiv ID: 2606.25824v1
链接: http://arxiv.org/abs/2606.25824v1
摘要: A method is proposed for testing the hypothesis of ultrametric organization of the energy landscape of RNA secondary structures, based on the analysis of transition kinetics between basins of attraction. The method relies on a kinetic metric constructed from the spectral decomposition of the symmetrized Kramers transition rate matrix and the Mahalanobis distance, which is not an ultrametric by construction. A complete computational framework is developed, including automatic filtering of noisy eigenmodes and a procedure for analyzing disconnected structure graphs arising from stochastic sampling. The method is demonstrated on a set of reference and random RNA sequences. It is shown that, for a fixed nucleotide composition, the degree of nontrivial ultrametricity of the energy landscape is determined by the order of nucleotides.
8. 间隙感知精确非负矩阵分解：一个双侧SVD规范和一个三区域W-秩分类
Gap-Aware Exact Nonnegative Matrix Factorization: A Two-Sided SVD Gauge and a Three-Regime W-Rank Taxonomy
作者: Mithil Ramteke
类别: math.NA, math.ST
发布日期: 2026-06-24
arXiv ID: 2606.25715v1
链接: http://arxiv.org/abs/2606.25715v1
摘要: We extend the cone-ray exact-NMF pipeline of Ramteke (arXiv:2606.22451) from the uniform-support regime r_+ = r to the gap regime r_+ > r, and classify recoverable nonnegative factorisations by the rank of the W-factor into a three-regime taxonomy. Regime A (rank(W) = r_+, full column rank): a two-sided SVD-gauge cone-ray pipeline W = U_{r+}(G) Q, H = P V_{r+}(K)^T with G, K on Stiefel manifolds and square consistency Q P = diag(S_r, 0). On 10x10 dense random gap matrices it gives 100/100 recovery at r_+ = 5 and 6. We explain this via two geometric facts: slack enclosure (the data cone has codimension r_+ - r in the outer cone) and NRF-variety thickness (valid gauges form a positive-measure set, so the blind SVD lands on one with probability one). Regime B (rank(W) = r, W a column subset of M): a rank-deficient-W branch enumerating r_+-subsets of M’s columns with per-column LP tests. On the block-diagonal family diag(C, J_k), where additivity of nonnegative rank collapses the valid gauges to a single point and the blind SVD pipeline fails, the column-subset branch restores recovery in milliseconds. Regime C (r < rank(W) < r_+, W not a column subset): exposed by the regular octagon’s slack matrix. An exact size-6 NRF exists and is reachable by the symmetric formulation at an oracle gauge derived from a known factorisation (residual 1.5e-10), but the blind problem is open: 50 Haar random restarts and Riemannian gradient descent on the Stiefel/Grassmann gauge all stall, because the alt-LP residual is piecewise constant on cells of gauge-space, so local descent cannot cross cell walls. A combined toolkit (Regime B then A) covers regimes A and B with no regression on dense draws; Regime C remains open, with the regular octagon as the cleanest unsolved test case.
9. 均匀多面体上球面的二次$G^1$样条逼近
A Quadratic $G^1$ Spline Approximation of the Sphere over Uniform Polyhedra
作者: Ema Češek, Aleš Vavpetič
类别: math.NA
发布日期: 2026-06-24
arXiv ID: 2606.25697v1
链接: http://arxiv.org/abs/2606.25697v1
摘要: In this paper, we study geometrically continuous quadratic splines over triangulations. While a rich variety of $C^1$ quadratic splines is available over planar domains, and such splines can also be constructed on the torus, the problem becomes significantly more challenging on more general surfaces. We first construct a $G^1$ spline over a regular spherical $n$-gon, subdivided into $3n$ triangles. Based on this construction, we obtain a quadratic $G^1$ spline approximation of the sphere induced by an arbitrary uniform polyhedron, where each $n$-gonal face is subdivided into $3n$ triangles. The construction uses only quadratic triangular patches and yields explicit control points depending on the geometry of the underlying polyhedron and one free parameter. We also analyze the resulting approximation quality and curvature behavior, and illustrate the construction on Platonic and Archimedean solids.
10. 一种用于求解高维椭圆线性偏微分问题的经济型瀑布式张量多重网格方法
An economic cascadic tensor multigrid method for solving high dimensional elliptic linear partial differential problems
作者: Jingyu Huang, Chenliang Li
类别: math.NA
发布日期: 2026-06-24
arXiv ID: 2606.25643v1
链接: http://arxiv.org/abs/2606.25643v1
摘要: In this paper, based on the tensor form, we propose a class of economic cascadic tensor multigrid method(ECTMG) for solving high dimensional elliptic linear partial differential problems. Compared with traditional methods, the new method not only reduces the storage space but also lowers the computational complexity from $\mathcal{O}(n^{3})$ to $\mathcal{O}(n^{2})$. We analyze the convergence rate of the conjugate gradient method which is based on the tensor form($\mathrm{CG}\_\mathrm{BTF}$), and then provide the convergence analysis for the new method. Finally, the effectiveness of the new method is verified through numerical examples.
11. 参数化椭圆重构与Robin边界条件中小随机性的抛物型偏微分方程的后验误差估计
Parametric elliptic reconstructions and a posteriori error estimates for parabolic partial differential equations with small randomness in a Robin boundary condition
作者: Nakidi Shravani, Gujji Murali Mohan Reddy, Amiya Kumar Pani, Stig Larsson
类别: math.NA
发布日期: 2026-06-24
arXiv ID: 2606.25640v1
链接: http://arxiv.org/abs/2606.25640v1
摘要: We obtain reliable a posteriori residual-based error estimates for parabolic partial differential equations with small randomness in a Robin-type boundary condition. The uncertainty is addressed via a perturbation approach, transforming the problem with small random input data into a sequence of deterministic problems. Finite element approximations, combined with the backward Euler time discretization, are employed for the resulting problems. To ensure optimal spatial accuracy, the elliptic reconstruction framework is suitably adapted to this setting. This is achieved by introducing a parametric elliptic reconstruction operator that unifies the a posteriori analysis of deterministic parabolic problems with that of parabolic problems with small uncertainties. The obtained a posteriori error estimator is robust with respect to the parameter that describes the amount of uncertainty, in the sense that the constants appearing in the bounds are independent of the parameter, as well as of the mesh size and the time-step. In addition, numerical experiments are presented to validate the theoretical findings and illustrate the robustness of the proposed estimators.
12. 带有乘性随机力的Allen-Cahn型多值p-拉普拉斯方程的纽曼问题
The Neumann problem for a multivalued p-Laplace equation of Allen-Cahn type with a multiplicative stochastic force
作者: C. Bauzet, F. Lebon, K. Schmitz, C. Sultan, A. Zimmermann
类别: math.AP, math.NA
发布日期: 2026-06-24
arXiv ID: 2606.25615v1
链接: http://arxiv.org/abs/2606.25615v1
摘要: In this paper, we consider a parabolic problem with constraint written as a differential inclusion, driven by a multiplicative colored noise and involving a p-Laplace operator (for $p \geq 2$), nonlinear random source terms and subject to Neumann boundary conditions on a bounded Lipschitz domain of $R^d$ with $d \geq 1$. This contribution aims at proving existence and uniqueness of a solution for such a multivalued problem. On one hand, the existence result is proved by the analysis of a semi-implicit time discretization scheme constructed on a smoother version of our problem, itself obtained by a regularization “à la Moreau-Yosida” of the subdifferential term. The key point of our approach consists in finding a clever relation between the time step denoted $τ$ and the Moreau-Yosida regularization parameter denoted $ε$ in view to pass simultaneously to the limit with respect to $τ$ and $ε$. On the other hand, the uniqueness of the solution is proved by standard arguments.
13. 关于带二重等距节点的抛物与三次样条函数空间的高斯型求积公式的误差界
On the error bounds of the Gauss-type quadrature formulae associated with spaces of parabolic and cubic spline functions with double equidistant knots
作者: Geno Nikolov, Petar B. Nikolov
类别: math.NA
发布日期: 2026-06-24
arXiv ID: 2606.25586v1
链接: http://arxiv.org/abs/2606.25586v1
摘要: In two papers from 1995 P. Köhler and G. Nikolov showed that Gauss–type quadrature formulae associated with spaces of spline functions with equidistant knots are asymptotically optimal in certain Sobolev classes of functions. In particular, Gauss–type quadratures associated with the spaces of spline functions of degree $r-1$ with double equispaced knots are asymptotically optimal definite quadrature formulae of order $r$ when $r$ is even, and it is conjectured that the asymptotical optimality property persists also in the case of odd $r$. For $r=3,\,4$, these quadrature formulae have been constructed by G. Nikolov, who also proved estimates for their error constants. The aim of this note is to refine the estimates for the error constant in the case $r=3$, and to point out to some error estimates in both cases $r=3$ and $r=4$, which are easier to evaluate and could be sharper than those which involve the uniform norm of the $r$-th derivative of the integrand. \medskip \noindent \textbf{Keywords and Phrases:} Spline functions, monosplines, Peano representation of linear functionals, definite quadrature formulae, error estimation of quadratures, Bernoulli polynomials.\medskip \noindent \textbf{Mathematics Subject Classification 2020:} 41A55, 65D30, 65D32.
14. 二维时空分数阶扩散方程逆源问题的高效Krylov子空间求解器
Efficient Krylov solvers for inverse source problem in 2D space-time fractional diffusion equation
作者: Asim Ilyas, Stefano Serra-Capizzano
类别: math.NA
发布日期: 2026-06-24
arXiv ID: 2606.25520v1
链接: http://arxiv.org/abs/2606.25520v1
摘要: In this work, we consider a two-dimensional time-space fractional diffusion equation with a variable coefficient and investigate the inverse source problem of reconstructing the source term f(x,y) , after regularizing the problem using the quasi-boundary value method to mitigate ill-posedness. A finite difference discretization results in a large-scale linear system with a multilevel Toeplitz-like block structure. We perform a spectral analysis of the associated matrix sequences, employing tools from Generalized Locally Toeplitz (GLT) theory, and construct efficient preconditioners based on the GLT analysis. The proposed preconditioners preserve the multilevel structure of the discretization matrices and leads to a general eigenvalue clustering around one for the preconditioned sequence. Numerical experiments validate the theoretical findings and demonstrate that the proposed approach significantly accelerates the convergence of the GMRES method in reconstructing the source term in the two-dimensional space-time fractional diffusion equation.
15. 面向分析的几何去特征化用于起始电压预测
Towards analysis-aware geometry defeaturing for inception voltage predictions
作者: Ondine Chanon
类别: math.NA, math.AP
发布日期: 2026-06-24
arXiv ID: 2606.25482v1
链接: http://arxiv.org/abs/2606.25482v1
摘要: Defeaturing is routinely employed in numerical simulations of medium- and high-voltage equipment, not only to reduce computational costs but also to make meshing possible. But the errors that this practice introduces in the predicted breakdown/inception voltages are currently neglected. This work presents a goal-oriented \textit{a posteriori} error estimation framework that aims at assessing such defeaturing errors in inception voltage computations. The methodology combines a first-order approximation of the inception voltage error with dual-weighted residual estimators, providing an error bound that quantifies the impact of defeaturing on the simulation results. The approach builds upon the streamer integral model for inception voltage prediction and uses the recently proposed certified goal-oriented analysis-aware defeaturing estimator of~[1] for elliptic PDEs. The first-order approximation of the inception voltage is a linear functional J of the background electric field whose definition involves a line integral along the critical field line, and is therefore only of low regularity. To meet the abstract regularity assumption of the goal-oriented theory, we introduce a Gaussian mollification of J. The methodology is illustrated on a pin-plate benchmark with a small protrusion of varying size and shape, using a single shared adaptive mesh and an adaptive coupling of the mollification width to the local mesh size in order to ensure that the defeaturing error dominates over the discretization and the regularization errors.
16. 基于 Lean 的线搜索方法的形式化
Formalization of Line Search Methods by Lean
作者: Yiyang Zhang, Kenneth W. Shum
类别: math.OC, cs.MS, math.NA
发布日期: 2026-06-24
arXiv ID: 2606.25412v1
链接: http://arxiv.org/abs/2606.25412v1
摘要: This paper presents a formalization of line search methods in the Lean 4 theorem prover. Our goal is to advance machine verification of nonlinear optimization theory by translating standard textbook definitions and convergence arguments into rigorous Lean code. We formalize fundamental notions related to gradient descent and descent directions, adaptive step-size selection via backtracking line search, and several classical line search criteria, including the Armijo, Goldstein, and Wolfe conditions, as well as nonmonotone variants. We further formalize a key convergence result, namely the Zoutendijk theorem, which plays a central role in the global convergence analysis of gradient-based iterative methods. By providing machine-checkable definitions and proofs for line search theory, this work complements existing formalizations of first-order optimization methods and establishes a foundation for the verified development of more advanced algorithms in nonlinear programming.
17. 一种基于矩阵对角化的全解耦任意高阶时间步进格式，用于各向异性相场枝晶生长模型
A fully-decoupled arbitrarily high-order time-stepping scheme based on matrix diagonalization for the anisotropic phase-field dendritic crystal growth model
作者: Weiwen Wang, Shaoqin Zheng
类别: math.NA
发布日期: 2026-06-24
arXiv ID: 2606.25345v1
链接: http://arxiv.org/abs/2606.25345v1
摘要: We propose a fully-decoupled arbitrarily high-order time-stepping scheme for the anisotropic phase-field dendritic crystal growth model. The scheme combines an auxiliary-variable formulation with algebraically stable Runge-Kutta methods and satisfies a discrete energy dissipation law. To address the computational bottleneck arising from the coupled linear system in existing high-order schemes, a matrix diagonalization technique is introduced to transform the coupled linear elliptic system into a set of independent constant-coefficient elliptic equations. The resulting equations can be solved separately and in parallel, thereby improving computational efficiency. Numerical experiments in both two and three dimensions are presented to verify the convergence, energy stability, and efficiency of the proposed scheme. Comparisons with the original coupled formulation demonstrate the effectiveness of the matrix diagonalization strategy, while additional tests illustrate the advantages of high-order temporal discretizations. Simulations under different anisotropy coefficients, latent heat parameters, rotation angles, and initial nucleus shapes are also presented to investigate their effects on dendritic morphology.
18. 无三维矩阵：笛卡尔偏微分方程求解器的统一张量积视角
No 3D Matrices: A Unified Tensor-Product View of Matrix-Free Cartesian PDE Solvers
作者: Yong Yi Bay, Kathleen A. Yearick
类别: math.NA, cs.CE, physics.comp-ph, physics.flu-dyn
发布日期: 2026-06-23
arXiv ID: 2606.25148v1
链接: http://arxiv.org/abs/2606.25148v1
摘要: Every Cartesian three-dimensional PDE solver hides a structural secret that production CFD codes have used for half a century and that graduate-level textbooks rarely state plainly. The derivative matrices, the compact Padé line solves, the Galerkin mass inversions, the alternating-direction-implicit substeps, and even the fast Poisson and Helmholtz diagonalization transforms all factor along the coordinate axes and collapse into repeated one-dimensional banded kernels executed along the grid lines. The three-dimensional operator exists only on paper; it is never assembled, factored, or stored. This paper is the manual for that collapse. We derive the Kronecker-product algebra that makes it exact, carry it cleanly through central differences, compact schemes, tensor-product Galerkin, B-spline and isogeometric methods, collocation, ADI time stepping, and direct Poisson and Helmholtz solves, and bring into the open the three production tricks that turn the reduction into hardware-conscious floating-point throughput on real machines: the multi-right-hand-side reshape that exposes a sweep as one batched line kernel (a dense BLAS-3 GEMM when the line factor is dense or element-local, a banded or stencil kernel when it is not), the sum factorization that rescues high-order Galerkin from the $O(p^{2d})$ quadrature trap, and the pencil decomposition that keeps every direction contiguous across an MPI cluster. For fixed stencil width or fixed polynomial degree, the compute cost stays $O(N)$ in the total number of unknowns $N = N_x N_y N_z$; the operator storage drops to $O(N_x + N_y + N_z)$ up to bandwidth constants; direct separable Poisson and Helmholtz solvers add the expected transform cost; the line kernels are embarrassingly parallel. These facts are familiar to practitioners but rarely assembled in one place; this paper collects them and shows how to use them.
19. 求解器精确性，学习灵活性：Stokes流的等变边界校正算子
Solver Exactness, Learned Flexibility: Equivariant Boundary-Correction Operators for Stokes Flow
作者: Denis Gueyffier
类别: physics.flu-dyn, math.NA
发布日期: 2026-06-23
arXiv ID: 2606.25075v1
链接: http://arxiv.org/abs/2606.25075v1
摘要: The drag and mobility of bodies in viscous (Stokes) flow govern problems in shape design, suspensions, or microorganism swimming. Classical solvers compute them accurately but expensively; purely learned surrogates are fast but unreliable off their training data. We combine both: a solver’s exactness for the part of the solution operator known in closed form, and learning for the part that is not. For incompressible Stokes flow the elliptic-core kernel is already known: in free space the Leray projector is a single rotation-equivariant Stokeslet with no free parameters, and the boundary-integral solver built on it is exact to machine precision at $O(N)$. The one object with no closed form is the boundary correction. We split the operator: fix the core exactly and equivariantly, and learn only that correction, as a well-conditioned second-kind operator. On a Stokes testbed where the exact solve is ground truth, the split gives a working solver ($2 \times 10^{-3}$ end-to-end, $5$-$16\times$ more data-efficient than a black-box DeepONet) and overturns three expectations. (i) Conditioning is not the bottleneck: a $10^{16}$-conditioned first-kind and a bounded second-kind operator give the same error. (ii) Cross-shape generalization is governed by the descriptor’s equivariance, not capacity: a noninvariant descriptor degrades by $>10^5\times$ under rotation, while canonicalization restores near-machine transfer. (iii) Coverage, not expressivity, is the lever; a local equivariant kernel removes the heavy out-of-distribution tail, cutting worst-case interior error from $O(10)$ to $\sim 10^{-7}$. We then open the central exterior problem in 3D: a completed double layer, made exact by quadrature by expansion, is second-kind well-conditioned and $SO(3)$-equivariant, reproduces the analytic drag of spheres and ellipsoids, and composes across bodies.