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1. Helmholtz方程Cauchy问题的最小二乘弱Galerkin有限元格式
A Least-Squares Weak Galerkin Finite Element Scheme for Cauchy Problems in Helmholtz
作者: Chunmei Wang, Shangyou Zhang
类别: math.NA
发布日期: 2026-05-20
arXiv ID: 2605.21162v1
摘要翻译: 本文针对与Helmholtz方程相关的严重不适定Cauchy问题，提出并严格分析了一种最小二乘弱Galerkin（LS-WG）有限元方法。通过利用定义在间断函数空间上的弱Laplace算子，所提出的框架便于无缝处理复杂边界条件和内部交界面。我们强调了LS-WG格式在一般多边形和多面体剖分上的几何灵活性。进一步，我们证明了数值解的唯一性，并基于专门设计的离散能量范数导出了最优阶误差估计。大量的数值实验验证了理论收敛阶，并展示了该算法相对于传统Galerkin方法的稳健性和高效性。
Comment: 20 pages, 9 tables, 3 figures. arXiv admin note: substantial text overlap with arXiv:2605.14770
链接: http://arxiv.org/abs/2605.21162v1
1. 研究对象和背景
本文研究的是 Helmholtz 方程 Cauchy 问题, 即在有界多角形或多面体区域 $\Omega \subset \mathbb{R}^d$ ($d=2,3$) 上, 给定部分边界 $\Gamma_1$ (具正测度) 上的 Dirichlet 数据 $u = g_1$ 和 Neumann 数据 $\nabla u \cdot n = g_2$, 而在其余不可达边界 $\Gamma_2$ 上无任何信息, 求解函数 $u$ 满足方程
$$\Delta u + k^2 u = f \quad \text{在 } \Omega \text{ 内},$$其中 $k>0$ 为波数. 该问题在 Hadamard 意义下严重不适定, 边界数据的微小扰动可导致解的指数级放大. 它在逆散射、无损检测、声全息、热声成像、地球物理勘探等领域具有重要应用, 因而发展稳定高效的数值方法十分必要.
经典的数值处理方法包括拟可逆性方法[13]、交替迭代法[12]以及边界元结合 Tikhonov 正则化[9,25]等. 近十余年来, 弱伽辽金 (Weak Galerkin, WG) 有限元方法[46,47]因其允许完全不连续逼近函数并使用弱导数模拟微分算子而受到广泛关注. 该方法天然支持一般多边形与多面体网格, 能够处理悬挂节点和复杂界面. 其后续发展之一——原始‑对偶弱伽辽金 (PDWG) 方法[21,22]——通过引入对偶变量将非自伴问题转化为约束极小化, 但增加了整体自由度.
针对 Helmholtz 方程的 Cauchy 问题, 直接采用传统有限元离散会得到不定线性系统, 尤其在高波数下迭代求解极为困难, 且需 $C^0$ 连续基函数, 难以处理通用网格. 本文试图填补的空白是: 在 WG 框架下构造一个最小二乘弱伽辽金 (Least‑Squares Weak Galerkin, LS‑WG) 格式, 将原不适定 Cauchy 问题变换为对称正定 (SPD) 的离散系统, 通过引入弱拉普拉斯算子和适当设计的稳定子, 实现复杂边界条件下的无缝处理, 在此基础上严格证明离散解的唯一性并导出关于离散能量范数的最优阶误差估计, 最后通过数值实验验证收敛性与鲁棒性.
2. 主要定理或主要结果
设连续 Cauchy 问题 (1.1) 在 $\Omega$ 上存在唯一解 $u \in H^{m+1}(\Omega)$, $m \ge 1$ 为多项式次数. 记 $\mathcal{T}_h$ 为 $\Omega$ 的拟一致多边形或多面体剖分, 全局网格尺寸 $h = \max_{T\in\mathcal{T}_h} h_T$. 令 $W_h$ 为相应的弱伽辽金离散空间, $Q_h u = \{Q_0 u, Q_b u, Q_g (\nabla u)\}$ 是 $u$ 到 $W_h$ 的自然投影. 定义离散能量范数 $|||v||| = \sqrt{a(v,v)}$, 其中双线性形式 $a(\cdot,\cdot)$ 由 (3.1) 给出, 包含弱拉普拉斯项和稳定子. 本文的主要结论为:
唯一性 (Lemma 4.2): 若连续问题唯一可解, 则 LS‑WG 格式 (3.2) 存在唯一解 $u_h \in W_h$.
最优误差估计 (Theorem 6.2): 存在与 $h$ 和 $k$ 无关的常数 $C>0$, 使得
$$||| u_h - Q_h u ||| \le C k h^{m-1} \| u \|_{m+1,\Omega}.$$数值验证: 在 $k^2=10$ 和 $k^2=10^6$ 两种情形下, 于三角形网格和非凸多边形网格上使用 $P_2$, $P_3$, $P_4$ 元素进行测试, 结果均显示出约 $m-1$ 阶的收敛速率, 与理论分析吻合.
3. 研究方法、关键技术和核心工具
本文的技术核心在于如何在一个完全不连续的弱函数空间上, 将二阶椭圆算子与最小二乘思想结合, 并导出可证明最优误差估计的稳定化方案. 具体分析框架和关键工具有:
1. 弱函数与离散弱拉普拉斯算子
定义弱函数为三元组 $v = \{v_0, v_b, v_g\}$, 其中 $v_0 \in L^2(T)$, $v_b \in L^2(\partial T)$, $v_g \in [L^2(\partial T)]^d$, 三者独立, 不要求 $v_b$ 为 $v_0$ 的迹, $v_g$ 为 $\nabla v_0$ 的迹. 弱拉普拉斯 $\Delta_w v$ 通过分部积分定义: 对任意 $w \in H^2(T)$,
其离散版本 $\Delta_{w,r,T} v$ 是多项式空间 $P_r(T)$ 中唯一的函数满足上式对所有 $w \in P_r(T)$ 成立. 该算子使得在不连续空间上可逐元素计算二阶差分, 从而为最小二乘提供所需的局部残差 $\Delta_w u + k^2 u_0$.
2. 最小二乘双线性形式与稳定子设计
构造整体双线性形式
其中稳定子
$$s(u,v) = \sum_{T\in\mathcal{T}_h} \left[ k^2 h_T^{-3} \langle u_0 - u_b, v_0 - v_b \rangle_{\partial T} + k^2 h_T^{-1} \langle (\nabla u_0 - u_g) \cdot n, (\nabla v_0 - v_g) \cdot n \rangle_{\partial T} \right].$$稳定子的尺度 $h_T^{-3}$ 与 $h_T^{-1}$ 分别惩罚内部量与边界量及其法向导数的跳跃, 是达成最优 $h^{m-1}$ 阶的关键. 由于 $a(\cdot,\cdot)$ 对称正定, 离散系统为 SPD, 可直接用共轭梯度法高效求解.
3. 可交换性质与误差方程
一个决定性观察是 $L^2$ 投影与离散弱拉普拉斯的可交换性 (Lemma 4.1):
藉此可将连续方程投影到离散空间, 得到
$$a(Q_h u, v_h) = \sum_{T\in\mathcal{T}_h} (f, \Delta_w v_h + k^2 v_0)_T + s(Q_h u, v_h).$$与数值格式 (3.2) 相减, 即得误差方程
$$a(e_h, v_h) = -s(Q_h u, v_h), \qquad e_h = u_h - Q_h u.$$4. 能量估计与逼近性质
取测试函数 $v_h = e_h$, 由 Cauchy–Schwarz 不等式得
最后利用 $L^2$ 投影的标准逼近性质
$$\sum_{T} \| u - Q_0 u \|_{s,T}^2 \le C h^{2(m+1-s)} \|u\|_{m+1}^2, \; s=0,1,2,$$以及边界迹不等式
$$\|\phi\|_{\partial T}^2 \le C (h_T^{-1} \|\phi\|_T^2 + h_T \|\nabla\phi\|_T^2),$$细致估计稳定子中各项, 最终导出
$$s(Q_h u, Q_h u) \le C k^2 h^{2m-2} \|u\|_{m+1}^2,$$代入即得 Theorem 6.2 的误差界. 整个证明无需求解对偶问题, 完全基于能量范数的直接控制.
4. 与之前工作的比较
6. 与文献[50]中 LS‑WG 思想的关系
文中提及[50]可能初步引入了 LS‑WG 概念, 但所提供内容中未详细展开其理论与本文的具体区别. 从本文工作看, 其推进在于: a) 给出了完备的唯一性和误差分析; b) 明确揭示了稳定子中 $k^2 h^{-3}$ 与 $k^2 h^{-1}$ 的尺度对最优阶的必要性; c) 通过可交换性质大幅简化了分析. 若[50]仅提出格式而无理论, 则本文的贡献即为首次为该格式建立严格理论.
2. 一维亥姆霍兹方程的一种伯努利相位拟合有限差分方法与波数显式分析
A Bernoulli phase-fitted finite difference method and wavenumber-explicit analysis for the one-dimensional Helmholtz equation
作者: Ansgar Jüngel, Panchi Li, Zhiwei Sun, Zhiwen Zhang
类别: math.NA
发布日期: 2026-05-20
arXiv ID: 2605.21144v1
摘要翻译: 我们针对区间 $(0, L)$ 上带有阻抗边界条件的亥姆霍兹方程，提出了一种伯努利相位拟合（BPF）有限差分方法。该格式源于亥姆霍兹算子单向分解的复化沙尔费特—古梅尔离散化，既给出了一种相位拟合的内部离散，也给出了精确的离散阻抗边界封闭。对于齐次问题，该方法对平面波是精确的，因此格式既不引入内部数值色散，也不产生边界处的人工反射。对于非齐次问题，我们证明了适定性，导出了波数显式的稳定性估计，并建立了对所有 $kh\notinπ\mathbb Z$ 成立的二阶一致性和收敛性，其中 $k$ 为波数，$h$ 为网格尺寸。特别地，在固定分辨率条件 $kh\le s_0$（其中 $0 Comment: 无 链接: http://arxiv.org/abs/2605.21144v1 本文研究一维Helmholtz方程在区间$(0,L)$上带阻抗边界条件的数值解法。研究对象为如下边值问题: 其中$k>0$为波数, $f$为源项, $g_0,g_L\in\mathbb{C}$为给定入射波数据. 该模型描述时谐声波、电磁波等传播现象. 当$k$很大时, 解具有高频振荡特性, 其数值逼近的核心困难在于所谓的 污染效应: 对标准离散方法(如通常的三点有限差分), 数值误差无法仅通过每波长保持固定自由度来控制, 为达到给定精度, 总自由度需以比自然尺度$O(k)$更快的速度增长. 因此, 设计具有色散校正或相位拟合的格式以减轻或消除污染效应是有穷元与有限差分领域经久不衰的课题. 已有若干途径: 在有限元框架下, 高阶$hp$方法可显著减少污染; 间断Galerkin、边界元及多尺度方法被用于提升高波数鲁棒性; Trefftz类方法直接在逼近空间引入局部精确解以捕捉振荡. 在有限差分框架中, 常见思路是构造可调系数的优化紧致格式或色散最小化格式, 或采用移位波数思想迫使离散相速度更好匹配真解. 文献[7]提出渐近色散校正, 为一般有限差分算子提供系统改善精度的手段. 对于一维情形, 已有结果更强: 齐次方程可构造精确三点差分格式[31], 非齐次方程的无污染格式也已在[30]中提出. 但其分析依赖Taylor展开, 仅在小$kh$假设下有效($h$为网格步长). 近期基于Fourier分析的工作[16]对带Dirichlet边界条件的标准三点格式给出了尖锐的波数显式误差估计, 但未能推广到阻抗边界, 且针对的是未校正的格式. 本文试图填补的空白是: 在包含全Nyquist区间$0 它将波分解为左行和右行分量, 对应算子分别零化平面波$\mathrm{e}^{\mathrm{i}kx}$与$\mathrm{e}^{-\mathrm{i}kx}$. 由此, 我们通过复化经典Scharfetter–Gummel通量离散直接逼近一阶因子, 再复合得到一种 Bernoulli 相位拟合 有限差分格式. 该格式在齐次时精确重现平面波, 因此内部无数值色散, 边界无人工反射. 本文的核心分析任务在于: 对该格式建立适用的波数显式稳定性、一致性与收敛性估计, 并在适当分辨率条件下证明无污染意义下的误差界. 本文对上述一维Helmholtz阻抗边值问题提出了一种Bernoulli相位拟合(BPF)有限差分格式(6), 并建立了完整的波数显式分析. 主要理论结果包含以下定理. 精确性(命题2): 若$f\equiv 0$, 则连续问题的平面波解$u(x)=\alpha\mathrm{e}^{\mathrm{i}kx}+\beta\mathrm{e}^{-\mathrm{i}kx}$的节点值$u_i=u(x_i)$精确满足BPF离散系统, 即 其中$m(s)=\mathrm{e}^{-\mathrm{i}s/2}\cos(s/2)$. 从而格式在内部无数值色散, 在边界无伪反射. 适定性与波数显式稳定性(定理3): 对任意$kh\notin\pi\mathbb{Z}$, 源$f$与边界数据$g_0,g_L$, BPF离散问题存在唯一解$u^h=\{u_i\}_{i=0}^n$. 解满足以下先验估计: 其中$\Theta(s)=|B(\mathrm{i}s)|^2 = s^2/(4\sin^2(s/2))$, 且 特别地, 若$kh\le s_0<\pi$且$kL\ge\pi$, 则$A_0(kh,kL)$关于$k$一致有界. 一致性(定理12, 文中给出): 当解充分光滑时, BPF格式的局部截断误差满足二阶一致性, 即对$i=1,\dots,n-1$有 且边界截断误差也为$O(h^2)$, 其中常数$C$依赖于解的高阶导数但与$k$的关系被显式追踪. 收敛性(定理13): 在$kh\notin\pi\mathbb{Z}$的条件下, 误差$e_i = u(x_i)-u_i$满足 常数$C_0$显式依赖于$kh$和$kL$, 且在固定分辨率$kh\le s_0<\pi$与$kL\ge\pi$之下与波数$k$无关, 即实现了在Nyquist主带内的无污染二阶收敛. 该误差界也可解读为关于$k$的$O(k^{-2})$衰减. 本文的证明框架围绕三个技术支柱展开: 复化Scharfetter–Gummel离散的结构性质、平面波精确提升技巧、以及离散能量与通量恒等式的联合运用. 格式构造与核心代数结构
格式的核心源于对一阶算子$\partial_x\pm\mathrm{i}k$的Scharfetter–Gummel逼近. 回忆经典SG通量: 其中$B(z)=z/(\mathrm{e}^z-1)$为Bernoulli函数, 具有性质$B(-z)=\mathrm{e}^z B(z)$, $B(-z)-B(z)=z$. 将实漂移速度$v$复化为$\mathrm{i}k$, 即得离散单向算子 它们精确零化右行与左行平面波. 复合得整体离散系统(6), 边界项引入因子$m(kh)$以保证平面波精确性. 经过代数演算(附录命题22), 内部算子可写为等价的三点形式: 其中$\Theta(s)=|B(\mathrm{i}s)|^2$. 这表明该格式实质上是以移位波数$\tilde k = k/\sqrt{\Theta(kh)}$的色散校正格式, 但源项被因子$1/\Theta$缩放, 从而保证了相位一致性与后续分析的便利. 平面波提升与稳定性分析
稳定性分析的主要障碍有二: 一是阻抗边界带来的非齐次边界条件, 二是离散能量范数在$kh\ge\pi/2$时无法直接控制(因$\cos(kh)$变号). 作者通过如下步骤克服. 本文与既有有限差分/有限元Helmholtz求解器在多维度上形成对比, 尤其体现在波数显式分析、适用范围与无污染性质方面. 与标准三点有限差分格式及色散校正格式的比较
经典三点格式$(-\Delta_h - k^2)u_i = f_i$的数值色散导致误差界为$O(k^4 h^2)$并隐含因子$k L$, 造成污染效应. 为了达到给定精度, 需$k^3 h^2$充分小. 文献[7]提出的渐近色散校正格式采用$(-\Delta_h - \tilde k^2)u_i = f_i$, 其中$\tilde k = k/\sqrt{\Theta(kh)}$, 通过移位波数消除了首阶色散. 这种格式在齐次时退化为精确三点格式[31], 但对于非齐次源项, 它并未调整右端项. 本文BPF格式等价于 可见左端算子的波数调整以因子$\Theta$整体缩放离散Laplace算子, 同时源项自然地保留原始$f_i$而未经$\Theta$缩放. 这种处理使得格式在齐次时也精确, 且非齐次时具有更好的相位一致性. 在分析层面, [7]的误差估计基于Taylor展开, 典型结果为$O(k^2 h^2)$, 仅当$kh$小量时有效; 而本文获得的是$O(h^2)$形式的误差界, 且常数在$kh\le s_0<\pi$内独立于$k$, 首次将波数显式无污染收敛性扩展到全Nyquist主带. 此外, [30]的格式亦采用移位波数但源项未缩放, 且其无污染证明仍限于小$kh$假设. 本文的进步在于同时克服了阻抗边界带来的技术困难并给出了大范围波数显式界. 与精确/无污染三点格式的对比
[31]针对齐次方程构造了精确三点格式, 其系数依赖于$kh$的正切函数, 从而节点处完全消除误差. 本文的BPF格式在齐次情形下与之等价, 但额外提供了对非齐次项的处理以及完整的稳定性与收敛性分析. [31]未涉及源项存在时的理论保证. 因此, 本文一方面推广了格式的使用范围, 另一方面将其置于系统的波数显式分析框架下. 与Fourier分析方法的比较
近期工作[16]对标准三点格式在Dirichlet边界条件下进行了尖锐的Fourier-symbol误差分析, 得到如$k\|e\|_{0,h} + |e|_{1,h} \le C k^2 h^2$的估计, 其中$C$与$k$无关当$kh$固定. 此结果虽为波数显式, 但仅适用于未校正格式且边界是简单的Dirichlet类型. 本文借鉴了Fourier-symbol的思路, 但需处理两点差异: 阻抗边界条件带来的非齐次项和算子修正. 作者通过平面波精确提升将误差方程转化为齐次离散辐射条件, 再借助正弦展开和符号分析得到类似形式的界(定理13). 此外, 由于格式的相位拟合性质, 最终误差阶提升为$O(h^2)$, 而标准格式只能达到$O(k^2 h^2)$的误差界. 这是格式和分析双重推进的结果. 与Trefftz及波基方法的比较
这类方法(如[4,18,19,20])在近似空间中包含平面波或一般局部解, 可直接消除色散和污染, 但通常涉及数值积分或系统矩阵的病态问题, 且在一维情形下可能过于繁重. 本文提供的BPF格式是简单的三点有限差分, 计算成本低, 且严格证明了在Nyquist主带内的无污染收敛, 对一维问题更具实用优势. 局限与适用条件
本文结果明显的限制是空间维度(仅一维)和格式对特定波数的精确性仅限于齐次部分. 条件$kh\notin\pi\mathbb{Z}$排除了网格与半波长整数倍重合的退化情形; 均匀误差界要求在Nyquist主带内, 即$kh\le s_0<\pi$, 同时$kL\ge\pi$保证至少半波长. 当$kh$接近$\pi$时, 稳定性常数$A_0$中$\sec(s/2)$将增大, 但仍保持有界若$s$以某个$s_0<\pi$为上界. 因此该方法适用于每波长至少多于2个网格点的配置. 与高维推广可能相关的困难是因子分解的形式在高维不能直接成立, 但本文的启发式思路可能为构造其它相位拟合格式提供借鉴. A Priori Error Analysis of a High-Order Selective Discontinuous Galerkin Method for Elliptic Interface Problems 作者: Fang Liu, Haroun Meghaichi, Xu Zhang 类别: math.NA 发布日期: 2026-05-18 arXiv ID: 2605.19039v1 摘要翻译: 本文针对与界面不贴合的笛卡尔网格上的椭圆界面问题，发展了一种高阶选择间断伽辽金（SDG）方法。该方法在界面单元上使用间断伽辽金（DG）格式，在其余区域使用连续伽辽金（CG）格式。相应地，我们基于文献[1]的高阶Frenet IFE基函数，构造了一个新的、局部协调的混合浸入有限元（HIFE）空间。与DG方法相比，该SDG方法的计算开销显著降低，且与CG方法的计算开销保持在同一量级。我们证明了新的HIFE空间在 $h$-加密下达到最优逼近，并建立了SDG格式的适定性。在能量范数和 $L^2$ 范数下导出了先验误差估计。给出了数值算例以验证理论分析。 Comment: 无 链接: http://arxiv.org/abs/2605.19039v1 本文研究二维有界矩形区域 $\Omega\subset\mathbb{R}^2$ 上的椭圆界面问题。区域 $\Omega$ 被一条光滑界面曲线 $\Gamma$ 分割为两个子区域 $\Omega^-$ 和 $\Omega^+$，满足 $\Gamma = \partial\Omega^-\cap\partial\Omega^+$。在该区域上考虑如下扩散方程： 其中扩散系数 $\beta$ 是分段常数： 在界面 $\Gamma$ 上，解满足物理跳跃条件（位移连续、法向通量连续）： 其中 $[v]_\Gamma = v^+|_\Gamma - v^-|_\Gamma$，$n$ 为 $\Gamma$ 的单位法向（从 $\Omega^-$ 指向 $\Omega^+$）。当采用高阶近似（多项式次数 $m\ge 2$）时，为保证有限元空间能够获得最优逼近阶，进一步要求解的正则性并引入拉普拉斯扩展跳跃条件： 这类界面问题在材料科学、流体力学、电磁场等许多领域中出现，其典型特征是扩散系数在界面处的间断导致解的整体正则性降低：尽管在每一子区域内部解可以很光滑，但在界面附近产生局部低正则性。标准的有限元方法通常需要网格贴合界面（body-fitted mesh），每次界面移动或演化时都需要重新剖分网格，计算成本高。为克服这一困难，浸入有限元（Immersed Finite Element, IFE）方法被提出，它允许网格不依赖于界面（interface-unfitted mesh），而在基函数中直接构造满足界面跳跃条件的形状函数，从而避免重新剖分。 自从 Li (1998) 首次针对一维问题构造线性 IFE 空间以来，IFE 方法已经取得了长足进展。在二维情形，线性 IFE 和双线性 IFE 空间相继被提出，其逼近性质也得到证明。为获得更高精度，Adjerid 与 Lin (2015) 构造了满足扩展跳跃条件的一维高阶 IFE；Adjerid 等 (2019) 通过最小二乘弱施加条件的方式构造了高维 IFE；Guo 与 Lin (2019) 基于 Cauchy 扩张提出了一种弱施加界面条件的高阶 IFE。最近，Adjerid 等 (2020, 2022) 引入了基于 Frenet-Serret 变换的高阶几何一致（GC）IFE 空间。该方法利用界面曲线的 Frenet 局部坐标将弯曲界面映射为直线，并在映射后的虚构矩形上构造多项式函数，再通过弱形式强制拉普拉斯扩展跳跃条件，从而得到一组精确满足界面条件的形状函数。这些基函数不仅具有最优逼近能力，且其局部空间构成 $H^1$-协调的，这是许多其他 IFE 方法所不具备的优点。 另一方面，有限元格式的连续性处理直接影响自由度数量。连续伽辽金（CG）法强制全局连续性，在界面问题中为满足界面条件需要特殊的处理（如部分惩罚的 PPIFE 方法）；而间断伽辽金（DG）法则通过增加惩罚项处理单元间的不连续性，全局自由度显著增加。2014 年，Zhang 等提出了选择间断伽辽金（Selective DG, SDG）方法，其思想是仅在需要局部加密的区域使用 DG 格式，其他区域使用标准 CG 格式，从而在保留 DG 灵活性的同时大幅减少自由度。 本文的工作是将高阶 Frenet IFE 基函数与 SDG 框架结合，构造一种面向界面问题的混合浸入有限元空间（Hybrid IFE, HIFE），并对其误差展开严格分析。新方法的要点是： 与完全 DG-IFE 方法 [22] 相比，本文的 SDG 方案将惩罚边集从全部内部边缩减到仅由界面单元的所有边构成的集合 $\breve{E}^s_h$；混合 IFE 空间是 [22] 中 DG-IFE 空间的真子空间，因此其逼近性质和相应的先验误差估计不能直接沿用 [22] 的结果，而需要单独建立。此外，惩罚边集的缩减使得适定性分析需要新的论证。本文从理论层面证明了该 HIFE 空间在 $h$-细化下达到最优逼近，证明了 SDG 格式的适定性，并给出了能量范数和 $L^2$ 范数下的最优先验误差估计。数值实验验证了理论分析。 在假设网格满足界面-单元交截条件 H1 (界面至多与每条边交于一点) 和 H2 (界面至多与每个单元交于两条边) 以及解具有足够正则性 ($u\in H^{m+1}(\Omega,\Gamma;\beta)$) 的前提下，本文证明了以下结论： 适定性 (定理 1)：对于 SDG 双线性形式 $a_h(\cdot,\cdot)$，当对称情形 $\epsilon=1$ 时无条件强制；当 $\epsilon=0$ 或 $\epsilon=-1$ 时，只要惩罚参数 $\sigma_0^B$ 充分大 (具体地，$\sigma_0^B \ge C\frac{(\beta^+)^2}{\beta^-}$，其中常数 $C$ 仅依赖于单元几何和逼近次数 $m$，且 $\alpha\ge 1$)，双线性形式在能量范数 下满足强制性 同时，存在无关于 $h$ 和 $\beta^\pm$ 的常数 $C$ 使得 $a_h(\cdot,\cdot)$ 连续： 由 Lax-Milgram 定理即得 SDG 解的存在唯一性。 投影最优逼近 (引理 6 和 7)：定义投影算子 $\Pi_h: H^{m+1}(\Omega,\Gamma;\beta)\to S_h^m(\Omega)$，其在非界面单元为标准拉格朗日插值，在界面单元为 $L^2$ 投影，满足 以及能量范数估计 先验误差估计 (第 4 节, 具体定理未在当前摘录，但摘要明确)：当解 $u$ 充分光滑时，SDG 近似解 $u_h$ 满足 其中常数 $C$ 与 $h$ 和界面位置无关，但可能依赖于 $\beta^\pm$ 的比值。 本文的证明面临三个核心障碍：(i) 混合 IFE 空间仅在界面单元上含有非多项式基函数，且全局连续性结构复杂，经典投影插值估计不易直接推广；(ii) 惩罚边集 $\breve{\mathcal{E}}_h^s$ 是界面元素的所有边，而非全部内部边，导致相容项估计中仅涉及部分边，且强制性常数依赖于惩罚参数选择；(iii) 扩散系数的间断性迫使所有估计必须显式地控制 $\beta^+/\beta^-$ 比值的影响。 作者的主要创新点及关键工具如下： 混合 IFE 空间的构造与局部协调性：在非界面单元上采用标准 $Q_m$ 多项式，在界面单元上采用 Frenet IFE 基函数，自然构成 $H^m(\mathcal{T}_h,\Gamma;\beta)$ 的子空间。Frenet IFE 基函数通过弱形式 (2.6)-(2.7) 精确满足界面跳跃条件，因此空间具有局部 $H^1$-协调性 (即在界面单元内部 $\beta$ 乘以梯度在界面处法向连续)。这避免了在非界面单元上引入任何与界面相关的近似，也使得连续性仅需在非界面单元间的边上保持，从而显著缩小惩罚区域。 惩罚边集缩减下的适定性分析：由于 $\breve{\mathcal{E}}_h^s$ 比 $\breve{\mathcal{E}}_h$ 小得多，直接移植标准 DG 强制性证明会因相容项积分区域过小而失败。作者在引理 3 中，针对 $\epsilon=0$ 和 $\epsilon=-1$ 的非对称情形，通过 Cauchy-Schwarz 不等式与 Young 不等式的精细搭配，将相容项 与跳量部分和梯度部分分别联系起来，最终要求 $\sigma_0^B$ 不低于 $C(\beta^+)^2/\beta^-$ 来吸收梯度项，从而恢复强制性。引理 2 中的跟踪不等式 是控制边值的关键，该结果依赖于 [22] 中对 Frenet IFE 空间的精细分析。 最优逼近性质的证明：投影 $\Pi_h$ 分别在两类单元上定义，其整体逼近性质不能由单个单元的直接估计相加得到，因为界面上估计需要将误差映射回 Frenet 虚构矩形 $\hat{K}_F$。引理 5 表明每个 $\hat{K}_F$ 与最多 49 个原始网格单元相交，因此通过拼接估计仍可得到 $\|u-\Pi_h u\|_{H^i(\Omega)}$ 的整体误差界，且保持最优阶。在能量范数证明 (引理 7) 中，跳量项 $[u-\Pi_h u]$ 的 $L^2(B)$ 界通过逆估计和迹不等式转化为体积误差，再结合引理 6 的 $H^1$ 半范估计即得。 误差分析框架：虽然当前摘录未完整展现最终误差定理的证明过程，但标准做法是借助 Lax-Milgram 定理和引理 7 的投影误差界，利用 Strang 或 Céa 引理获得能量范数误差 $\|u-u_h\|_h \le C \inf_{v_h\in S_h^m} \|u-v_h\|_h$，进一步结合对偶论证可得 $L^2$ 估计。文中通过引入合适的算子实现了这一框架。 本文与已有文献的比较主要体现在方法构造、分析难度和计算效率三个层面。 与完全 DG-IFE 方法 [22] 的比较： 与 CG 方法 (PPIFE) 的比较： 与其他高阶 IFE 方法的比较： 理论贡献： Composite B-Spline Current Deposition and Interpolation Operators for Thin-Wire Finite-Difference Time-Domain Simulations 作者: Cole Gruninger, Boyce E. Griffith 类别: math.NA 发布日期: 2026-05-20 arXiv ID: 2605.21450v1 链接: http://arxiv.org/abs/2605.21450v1 中文标题: 用于细导线时域有限差分模拟的复合B样条电流沉积与插值算子 摘要翻译: 针对斜向闭合环天线的Holland-Simpson细导线时域有限差分（FDTD）模拟会表现出持续的低频寄生电流，因为电流沉积算子未能保持电荷守恒。与一个沿导线采样切向电场的插值算子一起，该沉积算子可实现为分布关于一个支在导线上的正则化δ函数的正则化。我们证明，当导线承载恒定电流时，电荷守恒要求沉积电流在离散意义下无散，并且我们引入了一族能满足该条件且达到机器精度的复合B样条正则化。由于B样条核是分片多项式且分段点事先已知，故可精确计算耦合线积分，从而允许在每个网格平面交叉处设置子区间分段点的复合高斯-勒让德求积。将插值算子取为沉积算子的离散伴随可保持反对称性，并确保离散无旋电场在闭合回路中不产生净电动势。对中心馈电偶极子以及圆形和方形环天线的数值实验表明，所提出的正则化方法能产生与方向无关且与已知特性一致的阻抗值，而简单的三线性正则化则会在闭合回路中产生非物理的寄生低频电流。 Error analysis of a finite element scheme for parametric mean curvature flow based on the DeTurck trick 作者: Klaus Deckelnick, Vanessa Styles 类别: math.NA, math.AP 发布日期: 2026-05-20 arXiv ID: 2605.21445v1 链接: http://arxiv.org/abs/2605.21445v1 中文标题: 基于DeTurck技巧的参数化平均曲率流有限元格式的误差分析 摘要翻译: 本文研究参数化平均曲率流近似的数值格式的误差分析。所研究的格式基于利用DeTurck技巧的重参数化，由Elliott和Fritz在[15]中提出。在半离散情形下，对于阶为 $k \geq 2$ 的有限元空间离散，我们证明了位置向量的最优 $H^1$-误差估计。数值实验确认了这一误差界，并表明该格式关于网格点分布具有良好的性质，正如[15]中所观察到的。 Achieving Material Robustness via Symmetric Stress Finite Element Discretizations 作者: Pablo Brubeck, Charles Parker, Umberto Zerbinati 类别: math.NA 发布日期: 2026-05-20 arXiv ID: 2605.21425v1 链接: http://arxiv.org/abs/2605.21425v1 中文标题: 通过对称应力有限元离散实现材料鲁棒性 摘要翻译: 当离散连续介质力学中变分问题中的对称应力张量时，必须选择如何施加应力张量的对称性：(i) 强施加，要求离散张量逐点对称，或 (ii) 弱施加，通过引入拉格朗日乘子。对于Hellinger–Reissner弹性力学和不可压缩流动的速度–应力公式的$H(\mathrm{div})$协调有限元离散，其中Cauchy应力张量的对称性与角动量守恒相关，我们证明这一选择可能会显著影响数值格式的精度。通过一系列受纤维增强材料、液晶聚合物网络和极性流体启发的各向异性本构律的基准问题，我们表明弱施加对称性的格式可能会产生任意差的应力近似——即使对于零应力构型也是如此。然而，强施加对称性的格式能够独立于本构律提供精确的应力近似，我们称这一性质为材料鲁棒性。我们提出了一个统一的理论，严格解释了这一行为。 Addition Theorems for Real Vector Spherical Harmonics and Explicit Matrix Representations of the Quasi-Periodic Elastic Single Layer Potential 作者: Xin Feng 类别: math.AP, math-ph, math.NA 发布日期: 2026-05-20 arXiv ID: 2605.20736v1 链接: http://arxiv.org/abs/2605.20736v1 中文标题: 实向量球谐函数的加法定理与准周期弹性单层位势的显式矩阵表示 摘要翻译: 本文针对一维周期阵列中与Kelvin张量相关的准周期弹性单层位势$\mathcal{S}_D^{α,0}$发展了一种多极展开方法。该方法的关键步骤是推导实向量球谐函数$V_{lm}$, $W_{lm}$, $X_{lm}$的平移加法定理。这些加法定理使得能够以闭合形式精确计算$\mathcal{S}_D^{α,0}$的所有矩阵元素。通过完全在球谐基下工作，所提出的分析方法克服了因弱奇异核的直接曲面离散化而普遍存在的收敛性差和网格依赖性问题。此外，涉及的无穷级数利用多重对数函数精确求值，从而无需截断级数。作为一个应用，积分方程$\mathcal{S}_D^{α,0}[f]=\varphi$被简化为一个线性系统。该框架进一步推广到每个单胞中含两个不相交球的二聚体几何构型中，其中非对角矩阵通过Lerch超越函数显式地给出。 Arbitrary-order structure-preserving discretizations for geometric curvature flows 作者: Ganghui Zhang, Boris D. Andrews, Patrick E. Farrell 类别: math.NA 发布日期: 2026-05-19 arXiv ID: 2605.20371v1 链接: http://arxiv.org/abs/2605.20371v1 中文标题: 几何曲率流的任意阶保结构离散格式 摘要翻译: 几何流，即浸入流形随时间按其自身几何演化，展现出重要的结构性。例如，表面扩散在保持体积的同时耗散表面积；希望在离散化时保持这些性质。这促进了大量关于这些流的保结构离散化研究，尽管时间上只是低阶的。在本文中，我们首次提出了几何曲率流（曲线缩短流/平均曲率流以及曲线/表面扩散）的离散化，该离散化在空间和时间上任意阶地保持了面积和体积的演化。关键思想是以特殊方式引入辅助变量，使得经过时间连续 Petrov–Galerkin 离散后，面积耗散律的推导可以重现。这些辅助变量由时间上保结构的一般策略所指明，该策略适用于许多其他问题。所提出的格式同样以最小变形率策略的方式保持网格质量。我们在几个基准测试中展示了其保结构性质和高阶收敛性。 A Penalty-Free Asymmetric Nitsche’s Method for Edge Elements 作者: Tianwei Yu 类别: math.NA 发布日期: 2026-05-19 arXiv ID: 2605.20367v1 链接: http://arxiv.org/abs/2605.20367v1 中文标题: 一种用于棱边元的无惩罚非对称Nitsche方法 摘要翻译: 我们证明了使用Nédélec棱边元的无惩罚非对称Nitsche方法在弱施加切向Dirichlet边界条件求解curl-curl型问题时的稳定性。主要结果是在四面体网格的孤立片条件下，非对称双线性形式的inf-sup稳定性估计。讨论了该方法在curl-椭圆问题和一个磁对流扩散问题中的应用。 Scalable parallel 3-D TEM inversion via rational approximation of the matrix exponential 作者: Ralph-Uwe Börner, Stefan Güttel, Thomas Günther 类别: math.NA, physics.geo-ph 发布日期: 2026-05-19 arXiv ID: 2605.19951v1 链接: http://arxiv.org/abs/2605.19951v1 中文标题: 可扩展并行三维瞬变电磁反演：基于矩阵指数的有理逼近 摘要翻译: 我们提出了一种新颖的大规模三维电磁反演并行实现方法，该方法基于高斯-牛顿框架，并结合了用于瞬变模拟的矩阵指数有理近最佳逼近。方法采用并行直接求解器处理由有理逼近的部分分式表示导出的移位线性方程组，并利用MPI在共享内存架构上实现高效的并行执行。该方法的一个关键特性是时间依赖性完全包含在所采用有理函数的残差中，使得正演响应和灵敏度的计算实际上独立于所需的观测时间数。模型正则化通过光滑性约束实现，并使用Raviart-Thomas元进行表述。线性化反问题通过LSQR求解，其中使用隐式并行雅可比算子。数值实验展示了成功恢复一个具有约70万自由度的合成三维电导率结构。研究进一步讨论了由同时存储多个稀疏矩阵分解引起的内存消耗和共享内存可扩展性方面的计算瓶颈。文中概述了基于预条件迭代求解器和分布式高性能计算架构的可能改进。Julia语言实现已作为开源软件发布，以支持可重复研究并促进地球物理反演社区的进一步发展。 A second-order product-type implicit-explicit Runge-Kutta method preserving unit length and energy dissipation structures for gradient flows of vector fields 作者: Jianan Li, Shuang Liu, Tao Tang, Jiang Yang 类别: math.NA 发布日期: 2026-05-19 arXiv ID: 2605.19942v1 链接: http://arxiv.org/abs/2605.19942v1 中文标题: 一种保持单位长度与能量耗散结构的二阶乘积型隐式-显式龙格-库塔方法用于向量场梯度流 摘要翻译: 单位向量场的梯度流出现在众多物理模型中，例如调和映射热流、向列液晶及磁化动力学。设计能同时保持单位长度约束并耗散能量的数值格式对此类系统的可靠模拟至关重要。虽然投影方法可有效强制执行单位长度约束，但在投影下保证能量耗散，尤其在高阶格式中，仍具挑战性。有别于传统的隐式-显式龙格-库塔（IMEX-RK）方法，本文提出了一种构造乘积型IMEX-RK格式的通用方法，它更适配各种模型，旨在设计保结构数值格式。针对具有Dirichlet能量的单位向量场梯度流，我们利用乘积型IMEX-RK方法和投影技术，设计了一种线性的二阶数值格式，该格式同时保持能量耗散和单位长度约束。数值实验验证了该格式的精度、稳定性与保结构性质。据我们所知，这是第一个能够对调和映射热流同时保持单位长度与原有Dirichlet能量的二阶线性格式。 A dual-field structure-preserving mixed finite element discretization for incompressible Hall MHD equations 作者: Yi Zhang 类别: math.NA, math-ph 发布日期: 2026-05-19 arXiv ID: 2605.19920v1 链接: http://arxiv.org/abs/2605.19920v1 中文标题: 面向不可压缩Hall MHD方程的一种双场结构保持混合有限元离散 摘要翻译: 本文针对不可压缩Hall MHD方程，提出一种新颖的双场结构保持混合有限元离散。该离散满足点态质量守恒、磁高斯定律以及电流密度守恒，并且遵循离散能量律，该能量律在耗散情形下精确刻画能量耗散，在理想情形下则退化为能量守恒。数值实验验证了该方法在时间和空间上的精度以及结构保持性质。 When can a neural operator replace a coarse solve? Architectural principles for two-level preconditioning 作者: Hugo Melchers, Michael Abdelmalik, Victorita Dolean 类别: math.NA 发布日期: 2026-05-19 arXiv ID: 2605.19867v1 链接: http://arxiv.org/abs/2605.19867v1 中文标题: 神经网络算子何时能替代粗求解？二层预条件子的架构原则 摘要翻译: 神经网络算子越来越多地被用作经典数值方法中的即插即用加速器，但人们很少清楚哪些架构要素对哪种角色至关重要。本文针对一个关键角色回答了这个问题：作为离散线性偏微分方程二层预条件子内部的粗空间校正。通过沿着两个设计轴——输入离散化（采样与对基函数积分）和源项线性性——系统地改变四种类DeepONet架构，我们表明，这个2$\times$2设计中最有利的角落由单一架构占据，即神经格林算子（NGO），而偏离它则会产生可预见的失效模式：结构非对称的预条件谱、在自伴问题上预条件共轭梯度的崩溃，以及在非自伴问题上的停滞。作为粗空间校正，NGO在扩散和对流-扩散问题上达到了与精确粗求解相同的迭代次数。我们还刻画了固定尺寸的学习粗空间在高Helmholtz波数下的失效，并将其归结为基函数的性质而非架构的性质。这一原则可推广：将输入针对用于输出的基函数进行积分，正是使得神经网络算子能够充当Galerkin型粗空间校正的原因。 A Unified Transmissibility-Based Interior Penalty DG Method for Heterogeneous and Anisotropic Diffusion 作者: Gregory Etangsale, Vincent Fontaine, Anis Younes 类别: math.NA 发布日期: 2026-05-19 arXiv ID: 2605.19454v1 链接: http://arxiv.org/abs/2605.19454v1 中文标题: 基于传递率的统一内罚 DG 方法用于非均质各向异性扩散 摘要翻译: 我们针对非均质各向异性扩散问题，通过在一个紧致杂交内罚（H-IP）方法中对骨架未知量进行精确代数消去，导出了一个原始间断伽辽金（DG）格式。所得的统一内罚 DG（UIP-DG）格式包含从杂交格式继承的基于传递率的权系数，以及分别作用于原始跳跃和法向扩散通量跳跃的两个稳定化项。这些惩罚项分别与逐面传递率的调和平均及算术平均的倒数成比例。这一构造为先前独立引入的若干内罚方法提供了统一的视角，同时产生了一种稳定性不依赖于扩散对比度和各向异性的稳健方法。我们证明了该格式的相容性、强制性和有界性，并对所有变体导出了拟最优能量范数先验误差估计。数值实验验证了理论结论。 DG = FEM + flat elements, Part I: Diffusion 作者: Jiří Szotkowski, Václav Kučera, Chi-Wang Shu, Antoine Quiriny, Jonathan Lambrechts, Nicolas Moës, Jean-François Remacle 类别: math.NA 发布日期: 2026-05-18 arXiv ID: 2605.19037v1 链接: http://arxiv.org/abs/2605.19037v1 中文标题: DG = FEM + 平坦单元，第一部分：扩散 摘要翻译: 我们为 Poisson 问题建立了经典连续有限元方法（FEM）与间断伽辽金方法（DG）之间简单、严格且易于实现的联系。其核心思想是在单元界面处插入一层厚度趋于零的“虚拟”单元。通过修改这些单元上的扩散系数使其与厚度成正比，我们证明了 FEM 格式收敛到具有梯形边缘求积的 Babuška-Zlámal DG。该格式易于实现，只需 (i) 在网格编辑中引入退化界面单元，以及 (ii) 在原本未修改的 FEM 代码中设置一个 Jacobian 阈值，即可通过调和有限元方法（TFEM）框架处理退化单元。我们给出了所得 TFEM-DG 格式的严格推导，证明了最优 $H^1$ 和 $L^2$ 误差估计，并展示了二维和三维的数值实验。该方法允许在 FEM 代码中轻松实现 DG，甚至只需极少的编码工作即可在单元级别上自适应地在 FEM 和 DG 之间切换。该框架易于扩展，我们将在另一篇专门讨论演化非线性一阶双曲系统的姊妹篇中展示这一点。 On the Regularity and Generalization of One-Step Wasserstein-guided Generative Models for PDE-Induced Measures 作者: Likun Lin, Zhongjian Wang, Jack Xin, Zhiwen Zhang 类别: cs.LG, cs.AI, math.NA, stat.ML 发布日期: 2026-05-20 arXiv ID: 2605.21388v1 链接: http://arxiv.org/abs/2605.21388v1 摘要: Despite the remarkable empirical success of generative models, the available theory on their statistical accuracy in scientific computing remains largely pessimistic. This paper develops a theoretical framework for understanding the regularity of transport maps and the generalization properties of one-step Wasserstein-guided generative models for PDE-induced probability measures. We consider normalized target densities associated with linear elliptic and parabolic equations on bounded domains, as well as diffusion and Fokker–Planck equations on the torus. Under standard structural assumptions, we prove that these target measures satisfy doubling conditions. By combining this fact with regularity theory for optimal transport between doubling measures, we show that the optimal transport map from a uniform source measure to the target measure is Hölder continuous. This regularity yields an approximation-theoretic justification for one-step generative models that learn PDE-induced distributions via a single pushforward map. As a representative instance, we study DeepParticle and derive excess-risk bounds characterizing the discrepancy between the learned map and the population-optimal map. We also establish a robustness estimate under target shift and illustrate the theory with experiments which support the derived rates. Data-Efficient Neural Operator Training via Physics-Based Active Learning 作者: Alicja Polanska, Lorenzo Zanisi, Vignesh Gopakumar, Stanislas Pamela 类别: cs.LG, cs.AI, math.NA, physics.comp-ph 发布日期: 2026-05-20 arXiv ID: 2605.21348v1 链接: http://arxiv.org/abs/2605.21348v1 摘要: Solving partial differential equations with neural operators significantly reduces computational costs but remains bottlenecked by high training data requirements. Active learning offers a natural framework to mitigate this by selectively acquiring the most informative samples in an iterative manner. We introduce physics-based acquisition - a novel physics-informed active learning algorithm that leverages the partial differential equation residual to guide data selection. We validate the method by presenting numerical experiments for the 1D Burgers equation and the 2D compressible Navier-Stokes equations. We show that, in our experiments, physics-based acquisition consistently outperforms random acquisition and matches the state of the art in data efficiency. At the same time, it has the unique advantage of injecting a physics inductive bias into the training process, ensuring that simulation cost is spent where the model’s physical understanding is weakest. Experimental detection of inclusions for the time-harmonic elastic wave equation 作者: Sarah Eberle-Blick, Jochen Moll 类别: math.AP, math.NA 发布日期: 2026-05-20 arXiv ID: 2605.21158v1 链接: http://arxiv.org/abs/2605.21158v1 摘要: We are concerned with the reconstruction of inclusions in elastic bodies based on measurements from a laboratory experiment. In doing so, we solve the inverse problem of the time-harmonic elastic wave equation, in contrast to the stationary wave equation and the corresponding lab experiment proposed earlier in Eberle and Moll (2021). The investigation of the harmonic problem leads to a better reconstruction compared to the stationary one. Since we deal with real measurement data, we have to take into account, that those measurements always include measurement errors, so that we have to handle noisy data. Thus, we consider the linearized monotonicity method for noisy data and introduce a modified version of this method. Based on this, we reconstruct the inclusions numerically. Stochastic Galerkin and Monte-Carlo methods for parabolic problems: Numerical performance of variational matrix-free approximations 作者: Moataz Dawor, Nils Margenberg, Markus Bause 类别: math.NA 发布日期: 2026-05-20 arXiv ID: 2605.21046v1 链接: http://arxiv.org/abs/2605.21046v1 摘要: Stochastic Galerkin methods offer unexplored potential for the numerical simulation of parabolic problems with random variables, in particular if they are combined with variational discretizations of the space and time variables. Due to the high dimensionality, the solution of the arising algebraic systems do not become feasible without efficient solvers, preconditioners, and software architectures. A stochastic Galerkin discretization with an embedded slabwise finite element approximation of the space and time variables is proposed and analyzed numerically. For solving the linear systems, GMRES iterations are block-preconditioned by a geometric multigrid (GMG) technique using a local Vanka smoother for the space-time subsystems. Monte-Carlo methods are also used for solving random parabolic problems and studied here for the purpose of comparison. The Monte-Carlo approach is built on the space-time finite element formulation together with the GMRES-GMG solver technology. All algorithms have been implemented in a unified matrix-free framework based on the deal.II software library. Comparative numerical evaluations illustrate the performance properties of both approaches, including convergence of the discretizations and statistics of the algebraic solver. Superiority of the stochastic Galerkin approach is observed. Rectangular Multispectral Perturbation Theory 作者: Christof Vermeersch, Sarthak De, Bart De Moor 类别: math.NA 发布日期: 2026-05-20 arXiv ID: 2605.21013v1 链接: http://arxiv.org/abs/2605.21013v1 摘要: We provide a first systematic treatment of so-called rectangular multispectral perturbation theory. With their paper from 2003, Hochstenbach and Plestenjak [“Backward Error, Condition Numbers, and Pseudospectra for the Multiparameter Eigenvalue Problem” in Linear Algebra and its Applications] extended perturbation theory from one-parameter eigenvalue problems to multiple spectral parameters. After two decades, we take it one step further and consider a different manifestation of the multiparameter eigenvalue problem that consists of one matrix equation with rectangular coefficient matrices. We perform a norm-wise backward error analysis, define condition numbers for both eigenvalues and eigenvectors, and introduce the pseudospectrum while also considering the computational implications of working with multiple spectral parameters. The rectangular shape hampers a direct application of the existing definitions and properties. For example, the left null space at a given eigenvalue is non-trivial and the dimensions of the left and right eigenvectors are different. Through numerical examples, we illustrate and link the different concepts from the perturbation theory. A system identification application seem to suggest that, in optimization-driven problems for which multiparameter reformulations exist, the globally optimal solutions tend to coincide with the best-conditioned eigenvalues. Conditioning and backward errors for nonlinear eigenvalue problems with eigenvector nonlinearities 作者: Vilhelm Peterson Lithell, Victor Janssens, Elias Jarlebring, Karl Meerbergen, Wim Michiels 类别: math.NA 发布日期: 2026-05-20 arXiv ID: 2605.20933v1 链接: http://arxiv.org/abs/2605.20933v1 摘要: We consider eigenvalue condition numbers and backward errors for a class of symmetric nonlinear eigenvalue problems with eigenvector nonlinearities. For both of these quantities, we derive explicit and computable expressions that can be evaluated with little computational effort for a given eigenpair, assuming the matrix perturbations are measured by the spectral or Frobenius norm. We also show how symmetric perturbations can be exploited in the analysis. By means of two numerical experiments we demonstrate that problems incorporating eigenvector nonlinearities potentially need to be treated with additional care, when compared to the linear or eigenvalue-nonlinear theory. Precise Asymptotics and Exact Formulas for Tensor Product Energies of Fibonacci Lattices 作者: Melia Haase, Nicolas Nagel 类别: math.NT, math.MG, math.NA 发布日期: 2026-05-20 arXiv ID: 2605.20895v1 链接: http://arxiv.org/abs/2605.20895v1 摘要: We consider the asymptotics of sums of the form where $(F_n)_{n \in \mathbb N} = (1, 1, 2, 3, 5, 8, 13, \dots)$ are the Fibonacci numbers. Such sums appear, for example, in the context of discrepancy theory and numerical integration methods reformulated as energy minimization problems.
We show that for parameters $σ> 1$ and a large class of functions $f$ the above sum behaves asymptotically like for some constants $C$ and $D$. These constants can be given via infinite series connected to the Dedekind zeta function over the algebraic number field $\mathbb Q(\sqrt5)$.
In special cases we even observe simple closed-form expressions for such sums as above, explicitly proving that Solving Multivariate Polynomial Systems and Rectangular Multiparameter Eigenvalue Problems with MacaulayLab 作者: Christof Vermeersch, Bart De Moor 类别: cs.MS, math.NA 发布日期: 2026-05-20 arXiv ID: 2605.20884v1 链接: http://arxiv.org/abs/2605.20884v1 摘要: We present the Matlab toolbox MacaulayLab, which implements numerical linear algebra algorithms for solving multivariate polynomial systems and rectangular multiparameter eigenvalue problems. Its structure and functionality are the result of several years of research and algorithmic development. We demonstrate how the software works and compare its performance with other software packages, such as PNLA, PHCpack, and MultiParEig. Some core features of MacaulayLab are the fact that it solves two key problems via one common approach, works independently of the chosen polynomial basis and monomial order, and is capable of dealing with positive-dimensional solution sets at infinity. The toolbox (including its future updates) and a large collection of test problems are freely available online. Efficient and simple fourth-order compact finite difference methods for convection-diffusion-reaction equations on arbitrary curved domains 作者: Qiwei Feng, Bin Han, Peter Minev 类别: math.NA 发布日期: 2026-05-20 arXiv ID: 2605.20842v1 链接: http://arxiv.org/abs/2605.20842v1 摘要: In this paper, we discuss the 2D convection-diffusion-reaction equation with variable smooth coefficients and the Dirichlet boundary condition on a complicated, thin, and curved domain. We propose the fourth-order compact FDM at every grid point with the uniform Cartesian mesh. For the regular stencil center, we utilize the fourth-order compact 9-point FDM to approximate the solution. According to the preliminary analysis, we use vertical and horizontal transformations to derive fourth-order compact FDMs in 10 cases for all irregular stencil centers. To obtain the left-hand side of the stencil of the fourth-order FDM in each case, we only need to solve an at most $6 \times 24$ linear system which is presented with the explicit formula. The right-hand side of the FDM is constructed in explicit expression for any irregular stencil centers too. To achieve the fourth-order consistency, up to second-order partial derivatives of convection, diffusion, reaction, and source terms are used for the FDM at the regular stencil center, and the FDM at an irregular stencil center only requires first-order partial derivatives of convection, diffusion, reaction, and source terms, and up to third-order derivatives of the Dirichlet boundary function and the parametric expression of the boundary curve. We test challenging domains with 100-leaf, high-curvature, high-frequency, sharply varying, and nearly overlapping boundary curves, the proposed FDM produces the high accuracy and the stable fourth-order convergence rate in $l_2$ and $l_{\infty}$ norms. All stencils of our FDMs have a simple desired structure by only keeping grid points inside $Ω$ in the standard compact 9-point stencil for both regular stencils and boundary stencils, but without assuming any information outside the domain $Ω$. Hamiltonian and Symplectic Tensors in the T-product Algebra 作者: Susana Lopez-Moreno, Taehyeong Kim 类别: math.NA 发布日期: 2026-05-20 arXiv ID: 2605.20829v1 链接: http://arxiv.org/abs/2605.20829v1 摘要: We study Hamiltonian and symplectic tensor structures in the T-product algebra. We define T-Hamiltonian and T-symplectic tensors and characterize them through their Fourier-domain slices. For T-Hamiltonian tensors we establish the standard block form and the spectral symmetry of T-eigenvalues, while for T-symplectic tensors we derive the inverse and exponential-map properties. Our main result is a constructive T-Williamson normal form for tensors whose Fourier-domain slices are real symmetric positive-definite matrices. We also show that, under the Hermitian symplectic convention adopted here, this decomposition does not extend directly to arbitrary Hermitian positive-definite Fourier-domain slices, and we derive a real-valued recovery criterion under Fourier conjugate symmetry. Numerical experiments verify the construction, exhibit runtime trends consistent with the slice-wise complexity $O(pn^3)$, and illustrate the framework on a Fourier-domain encoding of covariance-matrix families arising in continuous-variable quantum dynamics. Data-informed posterior approximation for Bayesian linear inverse problems 作者: Haibo Li 类别: math.NA 发布日期: 2026-05-20 arXiv ID: 2605.20770v1 链接: http://arxiv.org/abs/2605.20770v1 摘要: Computing posterior distributions in large-scale Bayesian linear inverse problems is challenging due to the high dimensionality of the parameter space. In this work, we develop a data-informed framework that shifts the computational focus from the parameter space to the data space. We rigorously characterize an intrinsically low-dimensional data space, establish its isometric embedding into the parameter space, and show that the prior-to-posterior update is confined to a data-informed subspace. This perspective allows posterior inference to be carried out in a reduced data-informed subspace. Based on this formulation, we propose a quotient-space Golub–Kahan bidiagonalization method to construct data-informed Krylov subspaces, and integrate empirical Bayesian inference into the iterative framework, enabling simultaneous hyperparameter estimation and posterior approximation in a matrix-free manner. Numerical experiments on representative problems support the theoretical framework and demonstrate the effectiveness of the resulting method. Persistent-Homology-Guided Topology Scanning of Qualitative Indicators for Acoustic Inverse Scattering 作者: Xiaomei Yang, Jiaying Jia, Zhiliang Deng 类别: math.NA, math-ph 发布日期: 2026-05-20 arXiv ID: 2605.20673v1 链接: http://arxiv.org/abs/2605.20673v1 摘要: Qualitative methods such as the linear sampling method and the factorization method reconstruct acoustic scatterers through sampling indicators. In practice, these indicators are gray-scale fields on a prescribed sampling window and a binary obstacle shape is obtained only after thresholding. The choice of threshold is usually empirical and may be unstable when the indicator contains noise-induced artifacts or when the scatterer has nontrivial topology, such as multiple components or holes. This paper proposes a topology-aware postprocessing framework based on persistent homology. Given any normalized qualitative indicator, we scan the persistent homology of its superlevel sets and use the resulting zero- and one-dimensional persistent features to estimate or impose the topology of the unknown scatterer. A topology-guided threshold is then selected by minimizing a Betti-number discrepancy together with mild geometric penalties. The method is indicator-agnostic: it can be applied to the linear sampling indicator, the factorization-method indicator, or a normalized fusion of indicators. The main formulation is single-frequency and therefore remains close to the classical qualitative inverse scattering setting. We present the mathematical construction, an automatic topology detection rule based on persistence lifetimes and lifetime gaps, and a detailed algorithmic protocol for numerical implementation. Numerical tests verify that the proposed method is effective. Multilevel Isogeometric Projection Stabilization via Quasi-Interpolation for Advection-Diffusion-Reaction Equations 作者: Zakaria El Hasnaoui, Ahmed Ratnani 类别: math.NA 发布日期: 2026-05-20 arXiv ID: 2605.20583v1 链接: http://arxiv.org/abs/2605.20583v1 摘要: This paper presents a novel multilevel projection-based stabilization method for advection-dominated advection–diffusion–reaction problems within the framework of Isogeometric Analysis (IGA). The proposed approach extracts and penalizes fine-scale fluctuations using continuous B-spline quasi-interpolants, avoiding both the highly sensitive parameter tuning typical of SUPG methods and the discontinuous auxiliary spaces required by classical Local Projection Stabilization. Stabilization is applied hierarchically across nested levels of the discrete space via explicit mesh-dependent weights. We establish the mathematical soundness of the method by proving a priori error estimates, supplemented by a discrete inf-sup condition established for the one-dimensional setting with constant advection under a numerically validated stability hypothesis that ensures robust streamline derivative control. Numerical experiments on stringent benchmarks demonstrate the method’s ability to effectively suppress spurious oscillations across a variety of regimes, including the limiting cases of pure advection and advection–reaction. Notably, despite being a fully linear formulation, the method achieves significant reduction of undershoots near sharp layers, delivering performance comparable to complex nonlinear shock-capturing schemes. Furthermore, by utilizing a robust global parameter scaling, the proposed approach significantly alleviates the parameter sensitivity that typically affects residual-based alternatives, reducing the strong dependence on problem-dependent tuning. Adaptive Multi-Fidelity Structural Optimization under Fluid-Structure Interaction 作者: Aditya Narkhede, Erick Rivas, Kevin Wang 类别: physics.comp-ph, math.NA, math.OC 发布日期: 2026-05-19 arXiv ID: 2605.20501v1 链接: http://arxiv.org/abs/2605.20501v1 摘要: The design of structures and vehicles subject to fluid-structure interaction (FSI) often requires high-fidelity coupled analysis. While the design variables pertain to the structure, the computational cost is dominated by the fluid solver, making iterative optimization prohibitively expensive. This paper presents an adaptive multi-fidelity optimization method combining high-fidelity FSI analysis with a lightweight surrogate for fluid-induced loads and a decision model that selects between surrogate and high-fidelity fluid evaluations. During optimization, completed FSI analyses incrementally update a non-intrusive surrogate model based on nearest-neighbor search and radial interpolation. A hybrid Lagrangian-Eulerian mapping function is developed to transfer fluid loads between structural designs. The evolution of surface orientation is handled by decomposing the traction vectors into local orthonormal bases. An adaptive Gaussian process regression model is employed to predict surrogate error and quantify uncertainty, allowing risk-aware selection of when coupled analysis is required. As design evaluations cluster near the optimum, the accuracy of the surrogate model naturally improves, thereby reducing the reliance on the fluid solver. It requires no offline training, preserves the high-fidelity structural model in all design evaluations, and ensures that the final design is evaluated by high-fidelity FSI analysis. The fundamental idea is justified theoretically using a simplified model problem, which shows that the leading-order error is a monotonically increasing, concave, and bounded function of the fluid added mass. The framework is demonstrated on two benchmark problems. For shape optimization of a flexible panel under shock loading, results show an $80\%$ reduction in computational cost while maintaining accuracy within $2.3\%$ of fully high-fidelity FSI optimization. A Simple GPU-Accelerated Solver for the Schrödinger Operator with Applications to Ground States and Hamiltonian Simulation 作者: Xinyu Liu, Xiangxiong Zhang 类别: math.NA, physics.comp-ph, quant-ph 发布日期: 2026-05-19 arXiv ID: 2605.20491v1 链接: http://arxiv.org/abs/2605.20491v1 摘要: We extend the tensor-product direct solver from the Laplacian to the Schrödinger operator $-Δ+ V$. When the potential $V_1$ is separable, the operator $-Δ+ V_1$ is inverted or exponentiated at cost $O(N^{1+1/d})$ in $d$ dimensions via per-axis eigendecomposition. On a single NVIDIA A100 GPU, this costs less than one second for $10^9$ degrees of freedom in 3D. For non-separable potentials $V = V_1 + V_2$, the same solver provides a preconditioner $(-Δ+ V_1)^{-1}$ for the preconditioned conjugate gradient (PCG) method and a propagator for operator-splitting time integrators. For bounded $V_2$, we prove that the preconditioned operator has a bounded condition number and a clustered spectrum with at most finitely many outlier eigenvalues, independently of the mesh size, and also independently of the domain size when $V_1$ is a confining potential. This explains the mesh- and domain-independent PCG iteration counts observed in practice. We apply this method to ground state computation via inverse iteration for linear problems and via the $a_u$ gradient flow for Gross–Pitaevskii energy in 3D, and also Hamiltonian simulation via the approximated qHOP and Magnus-2 splitting methods from 3D to 9D on a single NVIDIA GH200 GPU. Entropy-stable discretizations for the compressible Euler equations using simple adaptive averages 作者: Carlo De Michele, Ayaboe K. Edoh 类别: physics.flu-dyn, math.NA 发布日期: 2026-05-19 arXiv ID: 2605.20472v1 链接: http://arxiv.org/abs/2605.20472v1 摘要: Entropy stabilization of the compressible Euler system is achieved by adapting the averages that are applied to the density and internal energy variables. The approach achieves non-linear robustness despite the use of simplified symmetric means (e.g., arithmetic, geometric, or harmonic evaluations), including their related expansions for asymptotic entropy conservation. The proposed formulation works via centralized convective terms and can naturally adhere to additional structures of the flow equations such as kinetic-energy- and pressure-equilibrium-preservation. Synchronous and Asynchronous Parallelism Approaches for Generalized Canonical Polyadic Tensor Decomposition with GenTen 作者: Jeremy M. Myers, Eric T. Phipps 类别: math.NA 发布日期: 2026-05-19 arXiv ID: 2605.20353v1 链接: http://arxiv.org/abs/2605.20353v1 摘要: The Canonical Polyadic (CP) tensor decomposition is a well-known method for interpretable analysis of high-dimensional data. Recently, the Generalized CP method (GCP) was introduced by Hong and Kolda to allow for flexible choice of the loss function in the optimization problem defining the CP model, enabling more interpretable decompositions of strongly non-Gaussian data such as count or binary data. Furthermore, Kolda and Hong introduced a version of GCP that leverages randomization and stochastic optimization to address scalability to large, sparse data sets. In this work, we take these ideas a step further and consider synchronous and asynchronous algorithms for parallel GCP tensor decomposition through the GenTen software package, exploiting both shared and distributed memory parallelism. We build on shared memory parallel CP decomposition algorithms utilizing Kokkos for portability across CPU and GPU architectures to support the random sampling and stochastic optimization methods required by GCP. We then couple this approach to the well-known medium-grained distributed memory parallelism scheme developed for traditional CP decompositions through MPI, providing a synchronous, hybrid MPI+Kokkos, parallel GCP decomposition capability. Finally, we propose an asynchronous distributed parallelism approach building on related techniques for federated learning to achieve even better scalability to large data sets. We study the effectiveness of the proposed synchronous and asynchronous approaches vis-a-vis computational cost and accuracy on synthetic and publicly-available real-world datasets of varying sizes, dimensions, and sparsity patterns using several loss functions. A Novel Stochastic Particle-Field Algorithm for a Reaction-Diffusion-Advection Cancer Invasion Model 作者: Jingyuan Hu, Zhongjian Wang, Jack Xin, Zhiwen Zhang 类别: math.NA 发布日期: 2026-05-19 arXiv ID: 2605.20140v1 链接: http://arxiv.org/abs/2605.20140v1 摘要: In this paper, we present a novel numerical framework for solving a specific biological reaction-diffusion-advection system of cancer growth in three dimensions (3D) using particles of variable mass. We adopt empirical particle measures to represent cell density and dynamically construct the concentration fields of multiple related chemical species throughout the 3D domain. Efficient interaction between the particles and the spatial grid is achieved through a Particle-in-Cell (PIC) algorithm, while diffusion in space is solved rapidly using a spectral method. We demonstrate that for this particular system, the rate of change of particle mass remains bounded over finite time intervals. Furthermore, in addition to the inherent positivity preservation of cell density guaranteed by the empirical particle measures, the concentrations constructed by the algorithm are also unconditionally positivity-preserving on the spatial grid. Moreover, we present a rigorous error analysis for the proposed method, and numerical experiments confirm the theoretical convergence rates. To the best of our knowledge, this is the first numerical work to solve this system in three dimensions, wherein a rapid spread of cells driven by haptotactic flux is observed, similar to the behavior documented in the two-dimensional case. Reliable sampling-based RKHS norm estimation via superconvergence 作者: Tizian Wenzel, Abdullah Tokmak, Christian Fiedler 类别: math.NA 发布日期: 2026-05-19 arXiv ID: 2605.20091v1 链接: http://arxiv.org/abs/2605.20091v1 摘要: Kernel methods are one of the cornerstones of learning-based control, modern system identification, surrogate modelling, and related fields. A key advantage of this class of learning and function approximation methods is the availability of quantitative error bounds, which in turn play a key role in guaranteeing the safety of learned controllers and related learning-based algorithms. However, these error bounds rely on a particular property of the target function – its reproducing kernel Hilbert space (RKHS) norm – which is usually impossible to obtain in practice. Motivated by this severe shortcoming, we present a novel sampling-based RKHS norm estimation approach with a solid theoretical foundation, leveraging very recent advances in the theory of superconvergence in kernel methods. Our method is applicable to a broad range of practically relevant function classes and requires only reasonable prior knowledge about the target function. Extensive numerical experiments demonstrate the efficacy and practical applicability of the proposed method. By providing a reliable RKHS norm estimation approach, we remove a major obstacle to the practical deployment of learning-based control algorithms. Quasi-optimal complexity of iterative Galerkin methods driven by an elliptic reconstruction error estimator 作者: Maximilian Brunner, Gregor Gantner, Christoph Lietz, Dirk Praetorius 类别: math.NA 发布日期: 2026-05-19 arXiv ID: 2605.20057v1 链接: http://arxiv.org/abs/2605.20057v1 摘要: We study an iterative Galerkin method for quasilinear elliptic problems in the Browder-Minty setting. The resulting discrete nonlinear systems are solved by linearization via a (damped) Zarantonello iteration. Unlike prior work, adaptive mesh refinement is driven by an elliptic reconstruction error estimator, which is natural in the sense that the a posteriori bounds for the linearization and discretization errors are well separated. For this setting, we present the first comprehensive convergence analysis of the corresponding algorithm. We prove unconditional full R-linear convergence of a suitable quasi-error that combines linearization and discretization errors. For sufficiently small adaptivity parameters, we further establish optimal convergence rates with respect to the number of degrees of freedom and quasi-optimal complexity, i.e., optimal convergence rates with respect to the overall computational cost. Numerical experiments underpin the theoretical findings. Error Bounds for Importance Sampling with Estimated Proposal Distributions 作者: Cathrine Aeckerle-Willems, Ilja Klebanov, Simon Weissmann 类别: math.ST, math.NA 发布日期: 2026-05-19 arXiv ID: 2605.19989v1 链接: http://arxiv.org/abs/2605.19989v1 摘要: Importance sampling with data-driven proposal distributions is widely used in practice. A common workflow first generates an auxiliary sample of size $N$ from an approximation of the target distribution, constructs a density estimate $\hat q$ such as a kernel density estimator (KDE), and then draws $n$ importance samples from this learned proposal. Despite its practical relevance, the theoretical properties of this hierarchical procedure remain poorly understood, since classical importance sampling theory assumes a fixed proposal.
We address this gap by deriving non-asymptotic error bounds for standard, defensive, and self-normalized importance sampling estimators with random proposals. Our results separate the Monte Carlo error, scaling as $n^{-1/2}$, from the proposal approximation error measured through the mean integrated absolute and squared errors (MIAE and MISE) of $\hat q$. To obtain explicit convergence rates in $(N,n)$, we establish MIAE and MISE bounds for KDEs constructed from geometrically ergodic Markov chains in stationary and non-stationary regimes. Combining these results yields quantitative guarantees for importance sampling with KDE-based proposals. Our theory provides practical guidance for selecting defensive mixture weights in a nonparametric importance sampling framework. Revisiting recursive methods for Dyson and Keldysh in NEGF: Part I 作者: Edoardo Di Napoli, Alessandro Pecchia, Gustavo Ramirez-Hidalgo 类别: math.NA, cs.CE, physics.comp-ph 发布日期: 2026-05-19 arXiv ID: 2605.19910v1 链接: http://arxiv.org/abs/2605.19910v1 摘要: The simulation of quantum transport in nanodevices requires the solution of the Dyson and Keldysh equations, a task dominated by the inversion of massive, block-tridiagonal matrices. While the Recursive Green’s Function (RGF) method has long been the standard $O(N)$ solver for quasi-1D systems, its formulation has typically been restricted to sequential execution and nearest-neighbor interactions. In this work, we carefully reformulate RGF through the lens of Domain Decomposition and Schur Complement theory. This allows us to extend the recursive formalism to block $n$-diagonal systems (handling higher-order stencils) and to derive a parallel algorithm, Domain-Decomposition based RGF (DDRGF), which stitches macroscopic domains via reduced interface systems. We explore data dependencies in DDRGF in detail, by means of block-sparse structures and tracing back to the desired output as a block tridiagonal approximation, giving a clear, reproducible and extensible formulation. We validate these algorithms using \texttt{LibNEGF.jl}, a Julia-based implementation, demonstrating that the structural insights of domain decomposition provide a robust pathway for high-performance quantum transport simulations on modern multi-core clusters. The theory presented here lays down the base for tackling the Keldysh problem, to be similarly handled in future stages of our work. Although the target here is the acceleration of kernels in the non-equilibrium Green’s function method, the algorithms and the implementations presented can be immediately used in any application involving block $n$-diagonal systems. Adaptive Reduced-Basis Trust-Region Methods for Defect Identification in Elastic Materials 作者: Benedikt Klein, Mario Ohlberger, Thomas Schuster 类别: math.NA, math.OC 发布日期: 2026-05-19 arXiv ID: 2605.19896v1 链接: http://arxiv.org/abs/2605.19896v1 摘要: Monitoring the integrity of elastic structures using ultrasonic waves requires the efficient identification of material parameters from measured surface displacements. The displacement field is governed by Cauchy’s equation of motion, i.e., an elastic wave equation. Consequently, defect localization leads to a high-dimensional spatial parameter identification problem for a hyperbolic system with given initial and boundary conditions. Stable parameter reconstructions typically rely on regularization techniques such as the iteratively regularized Gauss–Newton method (IRGNM). However, its practical application is computationally demanding due to the high-dimensional nature of the problem.
To address this bottleneck, we propose a reduced-order modeling approach that simultaneously reduces the state and parameter spaces using adaptively constructed reduced-basis spaces. This yields online-efficient surrogate models for both the forward and adjoint evaluations required in derivative-based optimization. To ensure reliability, the IRGNM iteration is embedded into an adaptive, trust-region framework that provides accuracy of the reduced-order approximations. The approach extends our recent contributions, which focus on elliptic and parabolic problems, to the hyperbolic setting. We demonstrate the reliability and effectiveness of the method for defect detection through numerical experiments. Graph Neural Networks for Community Detection in Graph Signal Analysis 作者: Roberto Cavoretto, Alessandra De Rossi, Enrico Montini 类别: math.NA, cs.LG 发布日期: 2026-05-19 arXiv ID: 2605.19733v1 链接: http://arxiv.org/abs/2605.19733v1 摘要: Community detection is a central problem in graph analysis, with applications ranging from network science to graph signal processing. In recent years, Graph Neural Networks (GNNs) have emerged as effective tools for learning low-dimensional representations of graph-structured data and have shown strong performance in clustering tasks, particularly on large and high-dimensional graphs. This paper investigates the use of GNN-based community detection within a graph signal interpolation framework. After reviewing the main classes of GNN architectures for community detection according to a standard taxonomy, we integrate the resulting graph communities into a Partition of Unity Method (PUM) for interpolation with Graph Basis Functions (GBFs). In this approach, GNN-derived communities are used to construct local subdomains on which GBF interpolants are computed and subsequently combined into a global approximation. Numerical experiments on benchmark %graph datasets, including geometric and urban network examples demonstrate that the proposed combination of GNN-based clustering and GBF-PUM interpolation yields accurate signal reconstructions. The results indicate that deep learning-based community detection can provide effective graph partitions for localized interpolation schemes, supporting its use in scalable graph signal analysis. Diffusion Graph Posterior Sampling for Nonlinear Inverse Problems with Application to Electrical Impedance Tomography 作者: Giovanni S. Alberti, Damiana Lazzaro, Serena Morigi, Matteo Santacesaria, Shibo Wang 类别: eess.IV, cs.LG, math.NA 发布日期: 2026-05-19 arXiv ID: 2605.19621v1 链接: http://arxiv.org/abs/2605.19621v1 摘要: Deep generative models have emerged as state-of-the-art for solving inverse problems, but applying them to inverse problems for PDEs, like electrical impedance tomography (EIT) remains challenging. Because physical domains are naturally discretized as unstructured meshes rather than regular grids, standard convolutional architectures are often inadequate. In this paper, we propose a novel framework that extends diffusion posterior sampling (DPS) to graph-structured data. We develop an unconditional score-based diffusion model directly on a 2D triangular mesh to learn an accurate prior over the physical solution space. Furthermore, we introduce a regularized variant, RDPS, which incorporates explicit regularization terms, such as total variation and generalized Tikhonov, to complement the implicit diffusion prior and mitigate severe ill-posedness. Extensive experiments on synthetic and real 2D EIT datasets demonstrate that RDPS produces stable, physically plausible reconstructions. Our approach generalizes well to out-of-distribution inclusion geometries, is highly robust to measurement noise, and outperforms current state-of-the-art solvers (e.g., GPnP-BM3D, DP-SGS) in reconstruction accuracy and artifact reduction. A Spline-based Physics-Informed Numerical Scheme: Accurate Smooth Solutions for Differential Equations 作者: Ayman Mourad, Fatima Mroue 类别: math.NA, math.AP 发布日期: 2026-05-19 arXiv ID: 2605.19564v1 链接: http://arxiv.org/abs/2605.19564v1 摘要: The rise of Physics-Informed Neural Networks (PINNs) has popularized the concept of solving differential equations via residual minimization. However, neural networks are often viewed as Assessing Finite Element Choice in Structural Topology Optimization and A Posteriori Error Estimation 作者: Jyotiranjan nayak, Shafeequdheen P, Vijayakrishna Rowthu 类别: math.NA 发布日期: 2026-05-19 arXiv ID: 2605.19441v1 链接: http://arxiv.org/abs/2605.19441v1 摘要: This study investigates the impact of finite element selection on structural topology optimization using the SIMP (Solid Isotropic Material with Penalization) method. Specifically, it compares linear (P1) and quadratic (P2) triangular elements with the conventional bi-linear quadrilateral (Q1) elements. Numerical experiments performed on benchmark problems including a cantilever beam, a bridge structure, and a beveled beam reveal notable differences in both the final optimized objective value (compliance) and the accuracy of the finite element solutions. The accuracy is evaluated using an a posteriori error estimator, highlighting the influence of element type on solution quality and optimization performance. Evaluation of External Magnetic Flux Density in Piezo-Flexomagnetic Nanobeams Using a Hybrid 1D-2D Finite Element Framework 作者: Lala Samprit Ray, Bishweshwar Babu 类别: cond-mat.mtrl-sci, math.NA 发布日期: 2026-05-19 arXiv ID: 2605.19422v1 链接: http://arxiv.org/abs/2605.19422v1 摘要: This study numerically evaluates the external magnetic flux density generated in air by the bending of a piezo-flexomagnetic nanobeam. In several classes of non-contact sensors, the magnetic field induced in the surrounding medium is often more useful than the internal magnetic response. However, most theoretical studies on piezo-flexomagnetic nanostructures neglect the external magnetic domain. The proposed framework employs a coupled hybrid finite element formulation combining a 1D Timoshenko beam model with a 2D magnetostatic problem encompassing both the beam body and the surrounding air domain. The formulation is verified against analytical solutions of magnetically isolated piezo-flexomagnetic beams. The results demonstrate the presence of a significant external magnetic flux distribution in free-standing structures, even in the absence of piezomagnetic coupling. A systematic sensitivity analysis further identifies the material parameters most strongly influencing the external transverse magnetic flux density. These findings provide insight into the design of nanoscale non-contact magnetoelastic sensing systems. Measurement Selection Strategies for Position Estimation in Indoor Environments 作者: Neetu R. R, Shrihari Vasudevan, Ranjani H. G 类别: eess.SP, cs.ET, math.GN, math.NA 发布日期: 2026-05-19 arXiv ID: 2605.19406v1 链接: http://arxiv.org/abs/2605.19406v1 摘要: Time-based indoor positioning techniques rely on multiple access points (APs) and measurements between the user equipment (UE) and the APs. In dense indoor environments, occlusion-induced non-line-of-sight (NLoS) propagation introduces significant delays in these measurements, thereby degrading position estimation accuracy. To address this challenge, this paper proposes measurement selection strategies to improve position estimation accuracy. A ray-tracing (RT) simulator is employed to characterize the propagation environment and derive AP neighborhood information, which is subsequently used to design and evaluate different measurement selection strategies. The approaches explored include AP neighborhood-based cardinality selection, intersection and union of measurements from AP neighborhoods, and fixed measurement selection. Experiments demonstrate the efficacy of the proposed measurement selection strategies in environments under significant NLoS conditions. Fast algorithms for interpolation with clamped $L$-splines of order four 作者: O. Kounchev, H. Render, G. Simeonov, Ts. Tsachev 类别: math.NA 发布日期: 2026-05-19 arXiv ID: 2605.20283v1 链接: http://arxiv.org/abs/2605.20283v1 摘要: Interpolation and smoothing using cubic and generalized splines are fundamental tools in data analysis and statistical modeling. Recently, fast computational algorithms were developed for natural $L$-splines of order four, which arise as piecewise solutions to the differential operator $L_ξ^2 = (\frac{d^2}{dt^2} - ξ^2)^2$. In this paper, we extend this mathematical framework to the important case of clamped (or complete) boundary conditions, where the first derivatives at the interval endpoints are prescribed. We explicitly construct the governing linear system for the interpolation problem and mathematically prove that the resulting tridiagonal matrix is strictly row diagonally dominant, thereby guaranteeing its invertibility and the numerical stability of the fast algorithm. The proposed method is implemented in MATLAB. Furthermore, the developed clamped $L$-splines provide a foundation for constructing multivariate clamped polysplines, which serve as a promising alternative to Physics-Informed Neural Networks (PINNs) for solving partial differential equations in Mathematical Physics. From Simple to Complex: Curriculum-Guided Physics-Informed Neural Networks via Gaussian Mixture Models 作者: Jianan Yang, Yiran Wang, Shuai Li, Fujun Cao, Xuefei Yan, Junmin Liu 类别: cs.LG, math.NA 发布日期: 2026-05-19 arXiv ID: 2605.19263v1 链接: http://arxiv.org/abs/2605.19263v1 摘要: Physics-informed neural networks (PINNs) offer a mesh-free framework for solving partial differential equations (PDEs), yet training often suffers from gradient pathologies, spectral bias, and poor convergence, especially for problems with strong nonlinearity, sharp gradients, or multiscale features. We propose the Curriculum-Guided Gaussian Mixture Physics-Informed Neural Network (CGMPINN), which integrates Gaussian mixture modeling with dynamic curriculum learning. Specifically, a GMM is periodically fitted to the PDE residual distribution to quantify spatially varying learning difficulty. A smooth curriculum schedule progressively shifts training focus from easy to harder regions, while precision-based variance modulation suppresses unreliable clusters during early optimization. This dual curriculum is governed by a shared curriculum parameter and can be combined with self-adaptive loss balancing. We further establish theoretical guarantees, including sublinear convergence of the gradient norm for the induced time-varying loss, uniform equivalence between the curriculum-weighted and standard PDE losses, and a generalization bound with an explicit weighting-induced bias characterization. Experiments on six benchmark PDEs spanning elliptic, parabolic, hyperbolic, advection-dominated, and nonlinear reaction-diffusion types show that CGMPINN consistently achieves the lowest relative $L_2$ and maximum absolute errors among all compared methods, reducing relative $L_2$ error by up to 97.8% over the standard PINN at comparable cost. Our code is publicly available at https://github.com/Mathematics-Yang/CGMPINN. Multiresolution analysis on tessellation graphs for inertial particle dynamics 作者: Keigo Matsuda, Thibault Maurel-Oujia, Kai Schneider 类别: physics.flu-dyn, math.NA, physics.comp-ph 发布日期: 2026-05-19 arXiv ID: 2605.19244v1 链接: http://arxiv.org/abs/2605.19244v1 摘要: A multiresolution technique on tessellation graphs for particle dynamics is proposed. This allows to split spatial field data given on millions of discrete particle positions into scale-dependent contributions. The Delaunay tessellation is used to define the graph, and Voronoi cell volumes are used to satisfy volume conservation. Our approach enables computation of the scale-dependent statistics of particle dynamics by leveraging a wavelet transformation of Lagrangian point particle data and is useful for characterizing particle clustering in turbulent flows. The technique is systematically verified by using synthetic data of randomly distributed particles in a two-dimensional plane. Then the applicability of the technique is demonstrated by extracting the scale-dependent particle velocity divergence of inertial particles in homogeneous isotropic turbulence from direct numerical simulation data. The result is verified by comparing the energy spectrum of the divergence with that obtained by a Fourier-based approach. Finally, the wavelet-based filtering to the particle velocity divergence is demonstrated to extract the effect of caustics in inertial particle clustering. Sharp analysis of sketched least squares and randomized low-rank approximation 作者: Ethan N. Epperly, Robert J. Webber 类别: math.NA, stat.CO 发布日期: 2026-05-18 arXiv ID: 2605.19096v1 链接: http://arxiv.org/abs/2605.19096v1 摘要: Two widely used randomized algorithms are the sketch-and-solve method for least-squares regression and the randomized SVD for low-rank approximation. These algorithms apply a random embedding to compress a target matrix, and they perform computations on the compressed matrix to save computational cost. This paper asks, what is the optimal random embedding in these algorithms? Also, what is the sharpest possible error bound for the optimal embedding? The paper proves that a random orthonormal matrix is minimax optimal for the sketch-and-solve algorithm while any rotation-invariant embedding is minimax optimal for the randomized SVD. Following these results, the paper obtains the best possible error bounds for sketched least-squares and the randomized SVD. Last, empirical experiments provide evidence of universality phenomena, in which several random embeddings lead to similar accuracy to the optimal embeddings in practice. Recovering Complex Unitary Eigenspaces from Real-Valued Embeddings 作者: Stefanie Günther, N. Anders Petersson 类别: math.NA 发布日期: 2026-05-18 arXiv ID: 2605.19041v1 链接: http://arxiv.org/abs/2605.19041v1 摘要: We consider the problem of recovering a unitary eigendecomposition of a complex unitary matrix from that of its embedded real-valued formulation. Such formulations arise naturally in scientific computing workflows that employ real-arithmetic solvers by representing complex matrices in term of their real and imaginary parts. While the reconstruction is trivial when the spectrum of the real-valued embedding is simple, degenerate and/or complex conjugated eigenvalues introduce ambiguities because each eigenspace may include contributions from both the unitary matrix and its complex conjugate. We prove that this ambiguity can always be resolved by applying a structured projection to the eigenspaces of the real-valued embedding, followed by a rank-revealing orthonormalization. The resulting procedure recovers the eigenvalues and an unitary eigenbasis for the original unitary matrix, with correct multiplicities of degenerate eigenvalues.1. 研究对象和背景
2. 主要定理或主要结果
3. 研究方法、关键技术和核心工具
4. 与之前工作的比较
3. 椭圆界面问题的高阶选择间断伽辽金方法的先验误差分析
1. 研究对象和背景
2. 主要定理或主要结果
3. 研究方法、关键技术和核心工具
4. 与之前工作的比较
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11. 基于传递率的统一内罚 DG 方法用于非均质各向异性扩散
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12. DG = FEM + 平坦单元，第一部分：扩散
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black boxes" requiring significant computational overhead and stochastic optimization. Moreover, PINNs typically treat boundary conditions (BCs) as soft constraints" within the loss function and this makes the optimization process struggling to enforce the BCs properly. This paper introduces the \textbf{Spline-based Physics-Informed Numerical Scheme (SPINS)}, a numerical framework designed to solve both initial and boundary value problems of ordinary differential equations (ODEs). By replacing the neural network architecture of traditional PINNs with a structured spline basis, SPINS achieves high accuracy and interpretability with a minimal parameter set. In addition, the BCs are automatically satisfied from the choice of the splines architecture. Therefore, SPINS provides smooth numerical solutions for ODEs allowing analytical differentiation. Moreover, SPINS benefits from the automatic differentiation where computing the gradient of the physics-informed loss function is an easy task making the optimization process very fast using gradient-based optimizers such as the L-BFGS-B algorithm. We demonstrate the efficacy of SPINS on nonlinear second order ODEs with several choices of BCs using cubic and quintic interpolating splines and present its natural extension to high order ODEs.27. 评估结构拓扑优化中的有限元选择与后验误差估计
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