生成时间: 2026年10月27日 15:50:23
重点关注论文数量: 2
了解领域论文数量: 17
不相关论文数量: 39
重点关注论文（完整分析）
1. 三维含角区域上 Laplace 边界积分方程的奇性分析
Singularity Analysis of Laplace Boundary Integral Equations on Three-Dimensional Domains with Corners
作者: Wenmao Hua, Jun Lai, Baoling Xie, Lu Zhang
类别: math.NA
发布日期: 2026-10-26
arXiv ID: 2610.35664v1
摘要翻译: 三维非光滑区域上椭圆边值问题的解在棱边和顶点附近表现出奇性行为。利用层势理论，我们研究 Laplace 方程对应的边界积分方程，并刻画锥域上表面密度的主奇异指数。针对两类典型几何，我们给出了二面楔形区域上两族棱边奇异指数的显式公式，并通过连带 Legendre 函数及其导数的零点刻画了轴对称圆锥上的逐模态顶点奇异指数。基于这两个原型和伸缩不变性，我们为一般锥形区域建立相应的算子束，并发展了一个非线性特征值求解器。该求解器采用带分级网格的 Nyström 方法、对数修正梯形公式以及 Brent 方法，以准确高效地计算这些奇异指数。针对三面锥、正 m-面锥和液滴型锥的数值算例验证了方法的有效性，并展示了棱边奇异性和顶点奇异性之间的相互作用。
Comment: 无
链接: http://arxiv.org/abs/2610.35664v1
1. 研究对象和背景
本文研究三维含边和顶点的 Lipschitz 角域 $\Omega\subset\mathbb{R}^3$ 上 Laplace 方程的 Dirichlet 问题及其双层位势边界积分方程. 具体地, 考虑
$$\Delta u=0\quad \text{in }\Omega,\qquad u=f\quad \text{on }S=\partial\Omega,$$其中 $\Omega$ 可以为有界内部区域或无穷外部区域; 外部问题附加 $u(x)=o(1)$ 当 $|x|\to\infty$. 设
$$\Phi(x,y)=\frac{1}{4\pi |x-y|},\qquad D[\rho](x)=\mathrm{p.v.}\frac{1}{4\pi}\int_S \frac{(x-y)\cdot n_y}{|x-y|^3}\rho(y)\,ds_y,$$通过跳关系得到第二类边界积分方程
$$\left(\frac{\sigma}{2}I+D\right)\rho=f\quad \text{a.e. on }S,$$其中 $\sigma=\pm1$ 对应外部或内部问题. 对光滑边界, $D$ 紧且密度 $\rho$ 具有与边值数据相同的正则性; 但对分段光滑边界, 即使 $f$ 光滑, 密度也会在边和顶点附近出现分数幂、对数等奇异性, 从而破坏边界积分算子的紧性并限制高阶边界元方法的精度. 因此, 准确刻画密度 $\rho$ 的局部奇性指数是构造高阶数值格式、分析后验误差以及理解反散射唯一性的基础问题.
作者的几何框架是文中的 Lipschitz 三维角域: 对每个 $x\in S$, 存在半径 $r_x$ 和局部微分同胚 $\chi_x$, 使 $\chi_x(\Omega\cap B(x,r_x))$ 与切锥
$$\Gamma_x=\{r\vartheta: r>0,\ \vartheta\in G_x\}$$重合, 其中球面截面 $G_x\subset S^2$ 为 Lipschitz 区域. 按 $G_x$ 的拓扑, 边界分为面 $F$、边 $E$ 和顶点 $V$: 在 $F$ 处切锥为半空间; 在 $E$ 处切锥为二面楔 $\Gamma_x\simeq C_\alpha\times\mathbb{R}$, $C_\alpha\subset\mathbb{R}^2$ 是开角为 $\alpha\in(0,2\pi)$ 的平面扇形; 在 $V$ 处 $G_x$ 为球面多边形或球面曲线多边形. 本文不直接处理全局域, 而研究这些切锥上的局部奇性, 因为主奇异项由切锥决定. 作者聚焦于伸缩不变锥, 即 $x\in\Gamma\Rightarrow \lambda x\in\Gamma$ 对任意 $\lambda>0$ 成立; 该性质保持层位势核的齐次性, 使密度可写成分离变量形式, 并导出算子 pencil.
已有理论方面, 层位势理论对光滑边界有成熟 Fredholm 框架; 对多边形和多面体域, 椭圆边值问题解的渐近展开包含分数幂与对数项. 在位势理论层面, 文中所引 [18] 通过 Mellin 卷积刻画锥上层位势算子的可逆性, [19,20] 研究 Neumann–Poincaré 算子的本质谱. 计算方法上, 二维角域已有局部网格加密与递归压缩预条件、角点圆化、显式奇性展开、核分裂等; 三维方面, Li [26] 处理带孤立锥点的曲面, 其他高阶方法多针对光滑或轴对称曲面. 作者明确指出, 同时覆盖边奇性、顶点奇性及其在一般分段光滑锥上相互作用的系统处理仍然缺乏. 本文试图补上这一空白: 先对二面楔和轴对称圆锥两个可解析原型分别给出边与顶点奇异指数的显式或半显式刻画, 再基于伸缩不变性建立一般分段光滑锥的算子 pencil 和非线性特征值算法, 用数值例子展示三面锥、正 $m$ 面锥和液滴型锥上的边-顶奇性耦合.
2. 主要定理或主要结果
本文的主要严格结果集中在两类切锥模型上. 对开角 $\alpha\in(0,2\pi)$ 的无限二面楔 $S=S_1\cup S_2$, 作者先假设密度具有平移不变形式
$$\rho(y)=\begin{cases}r_y^\mu,&y\in S_1,\\ \eta r_y^\mu,&y\in S_2,\end{cases}\qquad \mu>-1,$$并在 Proposition 3.1 中算出双层位势算子作用后的显式表达式; 例如在 $S_1$ 上
$$D(\rho)(x)=-\frac{\eta\sin((\pi-\alpha)\mu)}{2\sin(\pi\mu)}r_x^\mu-\sum_{k=1}^{\infty}\frac{\eta\sin(k\alpha)}{2\pi(\mu-k)}r_x^k,$$在 $S_2$ 上有类似表达式. 该计算显示非整数 $\mu$ 产生 $r^\mu$ 主项, 而整数附近出现极点, 对应对数项.
在此基础上, Theorem 3.2 给出二面楔 Dirichlet 问题的截断密度展开. 假设 $\alpha$ 避开某个有限例外集, 且诱导的一维截面边值数据的偶部和奇部可由收敛 Taylor 级数表示, 则对每个正整数 $N$, 存在唯一系数 $b_0,\dots,b_N$ 和 $c_0,\dots,c_N$, 使截断密度
$$\rho^{(N)}(y)=\begin{cases}\sum_{k=0}^{N}b_k r_y^{k\pi/\alpha}+\sum_{k=0}^{N}c_k r_y^{k\pi/(2\pi-\alpha)},&y\in S_1,\\ \sum_{k=0}^{N}(-1)^{k+1}b_k r_y^{k\pi/\alpha}+\sum_{k=0}^{N}(-1)^k c_k r_y^{k\pi/(2\pi-\alpha)},&y\in S_2,\end{cases}$$满足边界积分方程至残差 $O(r^{N+1})$. 因此二面楔边附近密度的局部奇性由两族分数幂生成:
$$\mathcal{S}_{\mathrm{dihedral}}=\left\{\frac{k\pi}{\alpha}:k\in\mathbb{N}_0\right\}\cup\left\{\frac{k\pi}{2\pi-\alpha}:k\in\mathbb{N}_0\right\},$$主奇异指数为
$$\mu_{\mathrm{dihedral}}(\alpha)=\min\left\{\frac{\pi}{\alpha},\frac{\pi}{2\pi-\alpha}\right\}.$$这一结果与经典二维分析一致, 但以三维双层位势密度的形式给出.
对半顶角 $\alpha\in(0,\pi)$ 的轴对称圆锥, 取密度 ansatz
$$\rho(y)=r_y^\mu e^{im\varphi_y},\qquad \mu>-1,\ m\in\mathbb{Z}.$$Proposition 3.5 得到
$$D(\rho)(x)=\frac{\pi\sin^2\alpha}{4\sin(\pi\mu)}A_{\mu,m}(\alpha)r_x^\mu e^{im\varphi_x}+\frac{\sin^2\alpha}{4}\sum_{n=|m|}^{\infty}\frac{B_{n,m}(\alpha)}{n-\mu}r_x^n e^{im\varphi_x},$$其中
$$A_{\mu,m}(\alpha)=P_\mu^{-|m|}(-\cos\alpha)(P_\mu^{|m|})'(\cos\alpha)-(P_\mu^{-|m|})'(-\cos\alpha)P_\mu^{|m|}(\cos\alpha),$$$$B_{n,m}(\alpha)=\frac{2(n-|m|)!}{(n+|m|)!}P_n^{|m|}(\cos\alpha)(P_n^{|m|})'(\cos\alpha).$$对光滑边值数据, 主项须消失, 从而顶点奇异指数由 transcendental equation
$$V_{\mu,m}(\alpha)=\frac{\sigma}{2}+\frac{\pi\sin^2\alpha}{4\sin(\pi\mu)}A_{\mu,m}(\alpha)=0$$的根决定. Proposition 3.7 利用 Legendre 函数的 Wronskian 恒等式和 Euler 反射公式将其化简为: 内部问题
$$V_{\mu,m}(\alpha)=-\frac{\pi\sin^2\alpha}{2\sin(\pi\mu)}\kappa_{\mu,m}(P_\mu^{-|m|})'(-\cos\alpha)P_\mu^{-|m|}(\cos\alpha),$$外部问题
$$V_{\mu,m}(\alpha)=\frac{\pi\sin^2\alpha}{2\sin(\pi\mu)}\kappa_{\mu,m}P_\mu^{-|m|}(-\cos\alpha)(P_\mu^{-|m|})'(\cos\alpha),$$其中 $\kappa_{\mu,m}=(-1)^{|m|}\Gamma(\mu+|m|+1)/\Gamma(\mu-|m|+1)$. 由此定义
$$S_m^{(D)}(\alpha)=\{\mu\in\mathbb{R}_+\setminus\mathbb{Z}:P_\mu^m(\cos\alpha)=0\},$$$$S_m^{(N)}(\alpha)=\{\mu\in\mathbb{R}_+\setminus\mathbb{Z}:(P_\mu^m)'(\cos\alpha)=0\},$$则圆锥第 $m$ 模的奇异指数集合为
$$S_m(\alpha)=\begin{cases}S_m^{(D)}(\alpha)\cup S_m^{(N)}(\pi-\alpha),&\text{内部问题},\\ S_m^{(D)}(\pi-\alpha)\cup S_m^{(N)}(\alpha),&\text{外部问题},\end{cases}$$第 $m$ 模主指数 $\mu_{\mathrm{circular},m}(\alpha)=\min S_m(\alpha)$, 总主指数 $\mu_{\mathrm{circular}}(\alpha)=\min_{m\in\mathbb{N}_0}\mu_{\mathrm{circular},m}(\alpha)$.
作者还给出边值到系数的映射基础: Lemma 3.8 证明若 $g(r,\varphi)=\sum_m g_m(r)e^{im\varphi}$ 是顶点邻域内 $C^\infty$ 函数在圆锥上的迹, 则对每个 $m$ 和正整数 $N$, 存在 $d_{m,0},\dots,d_{m,N-1}$ 使
$$g_m(r)=\sum_{j=0}^{N-1}d_{m,j}r^{|m|+j}+O(r^{|m|+N}),\qquad r\to0,$$特别 $g_m(r)=O(r^{|m|})$. 于是取 $S_m(\alpha)$ 的前 $N$ 个非整数指数 $\mu_{m,1}<\cdots<\mu_{m,N}$, 可构造截断展开
$$\rho_m^{(N)}(r,\varphi)=\sum_{k=1}^{N}a_{m,k}r^{\mu_{m,k}}e^{im\varphi},$$并由 Proposition 3.5 得到
$$\left(\frac{\sigma}{2}I+D\right)\rho_m^{(N)}(x)=\sum_{n=|m|}^{\infty}\left(\sum_{k=1}^{N}\frac{\sin^2\alpha}{4}\frac{B_{n,m}(\alpha)}{n-\mu_{m,k}}a_{m,k}\right)r_x^n e^{im\varphi_x}.$$对一般分段光滑锥, 作者基于伸缩不变性构造 associated operator pencil, 并发展 Nyström 型非线性特征值求解器以计算奇异指数; 但就所提供 PDF 内容而言, 一般锥的完整定理陈述和收敛性证明未显示, 只能确认作者提出该框架并在三面锥、正 $m$ 面锥和液滴型锥上做了数值验证. 摘要还指出数值实验展示了边奇性与顶点奇性之间的相互作用, 但具体定量结果从提供内容中无法确认.
3. 研究方法、关键技术和核心工具
本文的核心技术障碍是: 双层位势算子 $D$ 在分段光滑边界上不再紧, 密度奇性同时受局部开角、球面截面拓扑和边界数据展开控制; 在一般分段光滑锥上边奇性与顶点奇性耦合, 且没有现成显式公式. 作者的处理不是简单套用能量估计, 而是把伸缩不变锥上的层位势作用写成 Mellin 型卷积/分离变量形式, 从而把偏微分方程的局部奇性转化为 Legendre 函数或一般球面截面上的非线性特征值问题.
第一项关键工具是 Mellin 变换与 Legendre 函数积分表示. 对二面楔, 作者沿 $z$ 方向积分消去平移不变性, 将二维核化为
$$\int_{-\infty}^{\infty}\frac{dz}{(A^2+z^2)^{3/2}}=\frac{2}{A^2},$$再对径向变量使用文中所引 Mellin 变换引理
$$\int_0^{\infty}\frac{t^\mu}{(1-2xt+t^2)^{1/2}}dt=-\pi\csc(\pi\mu)P_\mu(-x),\qquad -1第二项关键工具是 Legendre 函数的生成函数、加法定理与 Wronskian 恒等式. 对轴对称圆锥, 作者先利用法向和面积元将 $D(\rho)$ 写成
$$D(\rho)(x)=\frac{\sin\alpha}{8\pi}\frac{\partial}{\partial\alpha}\int_0^{2\pi}\int_0^1\frac{\rho(y)}{(r_x^2-2r_xr_y\chi(x,y,\alpha)+r_y^2)^{1/2}}dr_y d\varphi_y,$$其中 $\chi(x,y,\alpha)=\sin^2\alpha\cos(\varphi_x-\varphi_y)+\cos^2\alpha$. 径向积分用 Legendre 函数表示后, 角向积分借助球谐加法定理
$$P_\mu(-\chi)=P_\mu(-\cos\alpha)P_\mu(\cos\alpha)+2\sum_{k=1}^{\infty}P_\mu^{-k}(-\cos\alpha)P_\mu^k(\cos\alpha)\cos(k(\varphi_y-\varphi_x)),$$以及整数阶 Legendre 函数的类似展开, 再用 Fourier 模正交性把角向卷积对角化, 得到每个 $m$ 模独立的主项系数 $A_{\mu,m}(\alpha)$ 和解析尾项 $B_{n,m}(\alpha)/(n-\mu)$. 这一步把三维锥上的奇异指数问题化为单个 Legendre 方程. 随后作者用连接公式把 $A_{\mu,m}$ 化为 $P_\mu^{-|m|}(\pm\cos\alpha)$ 及其导数, 并用 Wronskian
$$W\{P_\mu^{-|m|}(x),P_\mu^{-|m|}(-x)\}=\frac{2}{\Gamma(|m|-\mu)\Gamma(\mu+|m|+1)(1-x^2)}$$和 Euler 反射公式消去繁琐组合, 最终把内部/外部问题的特征方程分别化为 $P_\mu^{-|m|}(\cos\alpha)=0$ 或 $(P_\mu^{-|m|})'(\cos\alpha)=0$ 型条件, 并解释 $\pi-\alpha$ 的互换来自内部/外部问题的对偶.
第三项关键工具是伸缩不变性与算子 pencil. 对一般分段光滑锥, 密度按 $r^\mu$ 伸缩时, 双层位势算子作用后主项仍为 $r^\mu$ 乘以一个只依赖球面截面和 $\mu$ 的算子; 令边界方程的主项消失即得非线性特征值问题, 其未知量为奇异指数 $\mu$. 作者称其为 associated operator pencil. 由于球面截面 $\partial G$ 由有限条光滑弧组成, 且相邻弧以非退化角相交, 该 pencil 能同时编码边和顶点效应. 数值上, 作者采用 Nyström 方法并结合 graded meshes 处理端点奇性, 采用对数修正梯形公式处理对数奇异核, 再用 Brent 方法求非线性特征方程的根. 这些工具的组合使作者能够计算三面锥、正 $m$ 面锥和液滴型锥的奇异指数. 需要指出, 所提供文本只展示了方法框架和数值思路, 未给出一般锥 pencil 的完整适定性、谱离散误差或 Brent 求根收敛性的详细证明; 这些部分从提供内容中无法确认.
4. 与之前工作的比较
与二维角域边界积分方程的高阶方法相比, 本文的推进主要在三维双层位势密度层面. 文中明确提到 Serkh 和 Rokhlin [23] 的二维边奇性分析及其后续 [24]: [23] 对二面楔型二维问题给出显式奇性展开, [24] 通过允许非整数幂乘对数去掉了除有限多个开角外的限制. 本文 Proposition 3.1 和 Theorem 3.2 在三维二面楔上得到与二维分析完全一致的两族指数 $k\pi/\alpha$ 与 $k\pi/(2\pi-\alpha)$, 但对象是三维双层位势边界积分方程, 并给出截断密度满足方程至 $O(r^{N+1})$ 的残差估计. 因此, 本文不是简单重复二维结果, 而是把边奇性从二维边界积分方程推广到三维楔形切锥, 并为后续处理一般三维边提供可匹配的局部展开. 代价是二面楔定理要求 $\alpha$ 避开有限例外集, 且要求截面边值的偶部和奇部具有收敛 Taylor 展开; 这与 [24] 用对数项处理例外角度的思路相比, 本文在定理 3.2 中没有完全消除例外集, 虽然文中提到 [24] 的延拓可处理非整数幂乘对数.
与三维孤立锥点的工作相比, 文中特别引用 Li [26]. Li [26] 针对带孤立锥点的曲面构造高阶边界积分求解器, 采用单位分解离散并显式处理主密度奇性; 但其几何不覆盖边, 也不系统处理边与顶点相互作用. 本文则同时分析二面楔的边奇性和轴对称圆锥的顶点奇性, 并进一步提出一般分段光滑锥的 operator pencil, 目标是把边和顶点纳入同一框架. 在适用范围上, 本文从“孤立锥点”推进到“分段光滑锥, 球面截面由有限条光滑弧组成, 相邻弧以 $(0,2\pi)$ 内角相交”, 因而可覆盖三面锥、正 $m$ 面锥和液滴型锥; 在结果形态上, 本文对圆锥给出模-wise 指数由 $P_\mu^m(\cos\alpha)=0$ 或 $(P_\mu^m)'(\cos\alpha)=0$ 的零点刻画的半显式公式, 而 [26] 主要关注数值求解中的主奇性处理. 不过, 本文所提供内容没有给出一般锥上高阶边界元格式的完整误差分析, 因此与 [26] 的“高阶求解器”相比, 本文更偏向奇性指数分析与特征值计算框架, 尚未展示完整的高阶正问题求解器和收敛阶.
与锥上层位势算子理论工作 [18] 相比, [18] 通过 Mellin 卷积在合适加权 Sobolev 空间中建立边界积分算子的可逆性, 重点在适定性与函数空间; 本文则进一步追求奇异指数的显式/半显式刻画和可计算算法. 本文的 operator pencil 可视为把 [18] 的 Mellin 卷积观点具体化为非线性特征值问题, 并用 Nyström、graded meshes、对数修正梯形和 Brent 求根来计算指数. 与 Neumann–Poincaré 算子谱研究 [19,20] 相比, [19] 刻画旋转对称锥的本质谱, [20] 处理局部伸缩不变 Lipschitz 域; 这些工作关注算子谱和本质谱, 而本文关注双层位势密度在边和顶点附近的主奇异指数, 并把圆锥的模-wise 指数与 Legendre 函数零点直接联系起来. 本文的贡献在于把谱/算子理论语言转化为可数值求解的指数方程, 但代价是当前分析主要针对主奇异指数和局部切锥, 对完整谱的全局结构和高阶渐近项没有展开.
与二维高阶角域方法 [21–25] 相比, 本文的差异是维度与奇性类型. [21] 的局部网格加密加递归压缩预条件、[22] 的角点圆化、[23,24] 的显式奇性展开、[25] 的递归加密面板核分裂都主要处理二维角点; 本文处理三维边和顶点, 其中边奇性具有平移不变性, 顶点奇性具有球面模耦合. 本文对二面楔给出两族显式指数, 对圆锥给出 Legendre 零点刻画, 对一般锥给出数值 pencil, 这在高阶边界积分方法的设计上提供了局部奇性基函数和网格分级策略的理论依据. 但本文没有像 [21–25] 那样给出完整的预条件或核分裂算法, 也没有给出三维一般锥上的高阶离散误差定理; 因此其定位是奇性分析与非线性格林函数/指数求解器, 而不是完整的高阶正问题求解框架. 此外, 本文对外部问题假设右端满足与伴随核正交的条件, 这和经典双层位势外部问题处理一致; 对 Neumann 问题作者只说可由伴随性质类似处理, 未在提供内容中展开. 综合来看, 本文相对既有工作的主要推进是: 在三维双层位势框架下统一给出二面楔边指数的显式两族公式、轴对称圆锥顶点指数的 Legendre 零点公式, 以及一般分段光滑锥的 operator pencil 数值框架; 主要限制是局部切锥分析、主指数导向、一般锥严格收敛性未在摘录中展示, 以及二面楔定理的有限例外集和光滑数据假设.
2. 基于解分解的非线性 Poisson-Boltzmann 方程双网格有限元方法
A Two-Grid Finite Element Method for the Nonlinear Poisson-Boltzmann Equation Based on a Solution Decomposition
作者: Zhen Chao, Jingchun Chen, Cong He
类别: math.NA
发布日期: 2026-10-26
arXiv ID: 2610.34661v1
摘要翻译: 非线性 Poisson–Boltzmann 方程（PBE）是用于离子溶剂中生物分子静电学的广泛使用的隐式溶剂模型。其数值求解因点电荷、跨越分子表面的介电跳跃以及指数非线性而变得复杂。我们研究 PBE 的一种双网格有限元方法，该方法基于 Xie [J.~Comput.~Phys.~275 (2014) 294–309] 的解分解 $u=G+Ψ+\Phit$，其中 $G$ 解析地承载奇异性，$Ψ$ 求解一个线性界面问题，而 $\Phit$ 求解一个针对任意数量离子物种的半线性界面问题。$\Phit$ 的非线性问题仅在粗网格上求解，随后在细网格上作一次 Newton 修正，从而在细网格上只需求解对称正定的线性系统。分析采用指数函数的一个 $C^2$ 截断，该截断在超过水平 $τ$ 后按二次增长；当 $τ$ 高于某个显式界时，截断问题与原问题具有相同的解，且不需要离散最大值原理或网格角条件。在标准分片正则性假设下，我们证明 $\Phit$ 的有限元逼近的拟最优能量估计和 $L^2$ 估计，以及一个 $L^4$ 估计，并对双网格解证明三维情形下 $\enorm{u-\utg}\le C\big(h^{s-1}+H^{2(s-1)+t/2}\big)$，其中 $t=\min\{s,σ\}-1$。只要对偶问题至少与原问题一样正则（$σ\ge s$），粗网格就可选为 $H\sim h^{2/5}$ 而不损失精度。一个制造解测试验证了预测的收敛阶以及双网格修正对粗网格误差的二次依赖关系。
Comment: 27 pages, 1 figure
链接: http://arxiv.org/abs/2610.34661v1
1. 研究对象和背景
本文研究生物分子在离子溶剂中的静电势 $u$ 所满足的非线性 Poisson-Boltzmann 方程 (PBE). 设 $Ω\subset\mathbb{R}^3$ 为有界 Lipschitz 区域, 被分为溶质区域 $D_p$ 与溶剂区域 $D_s$, 界面为 $Γ=\partial D_p$. 溶质含 $n_p$ 个原子, 第 $j$ 个原子位于 $r_j$ 且电荷数为 $z_j$; 溶剂含 $n$ 种离子, 其电荷数与体浓度分别为 $Z_i$ 和 $\bar c_i$. 采用无量纲形式时, 模型为
$$-\epsilon_p\Delta u=\alpha\sum_{j=1}^{n_p}z_j\delta_{r_j}\ \text{in }D_p,\quad -\epsilon_s\Delta u+\rho(u)=0\ \text{in }D_s,$$并满足界面与边界条件
$$u(s^+)=u(s^-),\quad \epsilon_s\frac{\partial u(s^+)}{\partial n}=\epsilon_p\frac{\partial u(s^-)}{\partial n}\ \text{on }\u0393,\quad u=\hat g\ \text{on }\partial\u03a9.$$离子电荷密度为
$$\rho(\xi)=-\beta\sum_{i=1}^{n}Z_i\bar c_i e^{-Z_i\xi},$$并假设电中性条件 $\sum_{i=1}^{n}Z_i\bar c_i=0$ 与 $\bar c=\sum_{i=1}^{n}\bar c_i>0$ 成立, 因而 $\rho(0)=0$ 且 $\rho'(\xi)>0$. 该模型同时包含三类数值困难: 右端为原子中心处的 Dirac 测度, 使势在 $r_j$ 附近奇异; 介电系数在分子表面 $\u0393$ 上跳跃; 离子项关于势呈指数非线性. 直接对 $u$ 做有限元离散时, 点电荷奇异会污染误差估计, 界面跳跃要求界面拟合网格, 而指数非线性在三维中属于超临界增长, 使标准的全局 Lipschitz 或子临界假设失效.
为消除点电荷奇异, 本文采用 Xie 的解分解
$$u=G+\Psi+\widehat{\Phi},$$其中
$$G(r)=\frac{\alpha}{4\pi\epsilon_p}\sum_{j=1}^{n_p}\frac{z_j}{|r-r_j|}$$是溶质电荷的 Coulomb 势, 解析地承载全部点电荷奇异; $\Psi$ 求解一个线性界面问题
$$\Delta\Psi=0\ \text{in }D_p\cup D_s,\quad \Psi(s^+)=\Psi(s^-),\quad \epsilon_s\frac{\partial\Psi(s^+)}{\partial n}=\epsilon_p\frac{\partial\Psi(s^-)}{\partial n}+(\epsilon_p-\epsilon_s)\frac{\partial G(s)}{\partial n},$$并吸收边界条件与界面跳跃; 正则部分 $\widehat{\Phi}$ 求解半线性界面问题
$$\Delta\widehat{\Phi}=0\ \text{in }D_p,\quad -\epsilon_s\Delta\widehat{\Phi}+\rho(U+\widehat{\Phi})=0\ \text{in }D_s,$$在 $\u0393$ 与 $\partial\u03a9$ 上取齐次条件, 其中 $U=G+\Psi$ 是溶质在纯水中的势. 由于所有 $r_j$ 位于 $D_p$ 内部且与 $D_s$ 有正距离, $G\in C^\infty(D_s)$ 且 $\nabla G\in L^\infty(D_s)$, 因此 $\widehat{\Phi}$ 的问题具有有界数据, 非线性只出现在 $D_s$ 中. 其弱形式为求 $\widehat{\Phi}\in H$ 使得
$$a(\widehat{\Phi},v)+(\rho(U+\widehat{\Phi}),v)_{D_s}=0,\quad \forall v\in H_0^1(\u03a9),$$其中
$$a(v,w)=\epsilon_p\int_{D_p}\nabla v\cdot\nabla w\,dr+\epsilon_s\int_{D_s}\nabla v\cdot\nabla w\,dr,$$$$H=\{v\in H_0^1(\u03a9): e^{-Z_i v}\in L^2(D_s),\ i=1,\ldots,n\}.$$Xie 已在有限元框架下使用该分解, 并用修正 Newton 极小化方法求解; 但实际计算中绝大部分代数工作量集中在正则部分 $\widehat{\Phi}$ 的非线性系统上, 每次 Newton 迭代都需要在细网格上做大规模线性求解.
两网格方法可以降低这一成本: 只在粗网格 $H$ 上求解非线性问题, 再在细网格 $h\ll H$ 上做一次线性化 Newton 校正. 已有的两网格分析, 特别是 Holst, Szypowski 与 Zhu 针对半线性界面问题与正则化 PBE 的工作, 在三维全正则情形下给出 $H\sim h^{2/5}$ 即可保持精度的结论; 但对指数型超临界非线性, 其分析依赖离散最大值原理, 也就是刚度矩阵非对角元的符号条件或网格角条件. 对由分子表面生成的四面体网格, 这一条件是很强的限制. 此外, 在 Xie 分解中非线性项不是单纯只依赖 $\widehat{\Phi}$ 的 $\rho(\widehat{\Phi})$, 而是
$$\rho(U+\widehat{\Phi})\chi_{D_s},$$它通过 $U=G+\Psi$ 依赖空间位置, 在离散问题中 $U$ 又被 $U_h=G+\Psi_h$ 替代, 因此误差分析必须处理这一扰动. 本文试图补的空白正是: 在 Xie 解分解框架下, 对含任意离子种类、依赖 $U$ 且带指示函数的指数非线性半线性界面问题, 建立不依赖离散最大值原理与网格角条件的两网格有限元先验误差分析, 并给出粗网格尺度与细网格精度的显式关系.
2. 主要定理或主要结果
本文的核心严格结果可分为三部分. 第一部分是非线性项的 $C^2$ 截断及其与原始问题的等价性. 作者定义
$$E_\tau(y)=\begin{cases}e^y,& y\le\tau,\\ e^\tau\left(1+(y-\tau)+\frac12(y-\tau)^2\right),& y>\tau,\end{cases}$$并令
$$\rho_\tau(\xi)=-\beta\sum_{i=1}^{n}Z_i\bar c_i E_\tau(-Z_i\xi),\quad P_\tau(\xi)=\int_0^\xi\rho_\tau(\eta)\,d\eta.$$引理 3.1 与 3.2 证明 $E_\tau\in C^2(\mathbb{R})$, $E_\tau'>0$, $0 当 $Z_{\max}|\xi|\le\tau$ 时, $\rho_\tau(\xi)=\rho(\xi)$ 且相应导数相等. 定理 3.4 证明截断问题 存在唯一解 $\Phi_\tau\in H_0^1(\u03a9)$, 它是严格凸泛函 在 $H_0^1(\u03a9)$ 上的唯一极小元, 并给出先验能量估计 其中 $\|v\|_a=a(v,v)^{1/2}$, $M_U=\|U\|_{0,\infty,D_s}$. 定理 3.5 进一步证明对截断解与原始问题的任意解 $\widehat{\Phi}\in H$ 都有 定理 3.6 给出关键的等价阈值: 若 则截断解 $\Phi_\tau$ 属于 $H$, 并且就是原始问题 (15) 的唯一解; 特别地原始解 $\widehat{\Phi}$ 不依赖 $\tau\ge\tau_0$, 且 这一定理同时给出原始半线性问题存在唯一性的一个新证明, 且该证明在 $H_0^1(\u03a9)$ 中工作, 而不需要 Xie 原文中的集合 $H$ 作为解空间. 第二部分是正则部分有限元近似的先验误差估计. 作者在假设 4.1 下工作: 存在 $1 的解满足 $w\in H^\sigma(D_p\cup D_s)$ 且 其中 $C_R$ 不依赖 $c$ 与 $f$. 记 采用拟合界面的、形状正则且拟一致的四面体网格与连续分段线性有限元空间, 对以扰动数据 $U_h=G+\Psi_h$ 计算得到的离散解 $\Phi_h$, 摘要中报告定理 4.6 与 4.8 以及推论 4.9 证明了能量范数与 $L^2$ 范数的准最优误差估计, 并证明了 $L^4$ 误差估计; 其中扰动只通过 $\|\Psi-\Psi_h\|_{0,2,D_s}$ 进入. 由于提供的 PDF 摘录在第 10 页处结束, 定理 4.6 与 4.8 的完整右端常数、逐项误差阶表达式以及 $L^4$ 估计的精确形式未能从摘录中逐项核对; 但可确认其结论类型与上述扰动结构. 第三部分是两网格方法的误差定理. 算法仅在粗网格 $H$ 上求解非线性截断问题, 然后在细网格 $h$ 上做一次 Newton 校正, 细网格上只需求解对称正定线性系统. 对两网格解 定理 5.2 证明在三维中 因此取 即可保持细网格非线性解的精度. 特别地, 当对偶问题至少与原始问题同样正则, 即 $\sigma\ge s$ 时, $t=s-1$, 上述指数化为 $2/5$, 因而可取 $H\sim h^{2/5}$ 而不损失精度; 若 $\sigma 本文真正的技术障碍不是单一的能量估计, 而是三重耦合: 指数非线性属于超临界增长, Galerkin 解先验上只控制在 $H^1$ 中, 没有显然的 $L^\infty$ 界; 非线性项通过 $U=G+\Psi$ 依赖空间位置, 并在离散问题中被 $U_h=G+\Psi_h$ 扰动; 界面介电跳跃使原始问题与对偶问题都只有分片 $H^s$ 或 $H^\sigma$ 正则性, 其中 $s,\sigma$ 可能小于 2. 作者没有沿用 Holst, Szypowski 与 Zhu 依赖离散最大值原理的路线, 而是把证明拆成截断, 先验 $L^\infty$ 界, 截断问题误差分析与两网格校正四层. 第一层关键工具是 $C^2$ 二次增长截断. 把 $e^y$ 替换为 $E_\tau(y)$ 后, 截断非线性 $\rho_\tau$ 严格递增, 导数有正下界与线性上界, 二阶导数有界, 且 $\rho_\tau$ 至多二次增长. 这使截断问题的泛函 $J_\tau$ 严格凸且强制, 并在 $H_0^1(\u03a9)$ 上连续: 因为 $\Phi\in L^6(\u03a9)$ 而 $P_\tau(U+\Phi)$ 至多三次增长, 所以 $v\mapsto P_\tau(U+v)$ 作为 Nemytskii 算子从 $L^3(D_s)$ 连续到 $L^1(D_s)$, 从而可用凸分析的弱下半连续性与 Gâteaux 微分得到唯一极小元与唯一弱解. 这一步的重要性在于, 误差分析可以完全在截断问题上进行, 不再需要离散解的逐点界, 也不需要离散最大值原理或网格角条件; 代价是常数中会出现 $e^\tau$, 但作者强调对 $\tau\ge\tau_0=2Z_{\max}M_U$ 截断与原问题同解, 因而 $e^\tau$ 反映的是实际离子密度尺度, 不是人为放大. 第二层关键工具是先验 $L^\infty$ 界. 作者对截断解或原始解 $\Phi$ 取测试函数 由于 $\phi,\psi\in H_0^1(\u03a9)$ 且 $a(\Phi,\phi)=a(\phi,\phi)$, $a(\Phi,\psi)=a(\psi,\psi)$, 在集合 $\{\phi>0\}\cap D_s$ 上有 $U+\Phi\ge0$, 由单调性与 $\rho_\tau(0)=0$ 得 $(\rho_\tau(U+\Phi),\phi)_{D_s}\ge0$, 于是 $a(\phi,\phi)\le0$; 下界同理. 该论证只用到 $\rho_\tau$ 的单调性与 $\rho_\tau(0)=0$, 不要求 $\u0393$ 超出 Lipschitz 正则性. 由此得到 $-M_U\le\Phi\le M_U$, 再结合 $Z_{\max}|U+\Phi|\le2Z_{\max}M_U=\tau_0$ 与 $\rho_\tau$ 在 $|Z_i\xi|\le\tau$ 时等于 $\rho$ 的性质, 证明截断问题与原始问题同解. 这是本文避免离散最大值原理的核心. 第三层是有限元误差分析. 对截断问题, 非线性项具有至多二次增长与有界二阶导数, 因此在有界能量集合上提供受控的 Lipschitz 性质; 同时线性化算子 因 $\rho_\tau'>0$ 而保持强制性与对称正定性. 能量范数误差通过相减弱形式并取误差为测试函数得到, 其中非线性差项用单调性处理. $L^2$ 估计通过对偶问题 (30) 得到, 对偶系数取 $c=\rho_\tau'(U+\widehat{\Phi})$, 它满足 $0 将 $L^2$ 与能量估计组合起来, 以控制二次增长非线性在误差方程中的高阶项. 这些估计不要求网格满足 Delaunay 型角条件, 也不要求离散解满足最大值原理. 第四层是两网格校正的误差传播. 粗网格上求解非线性截断问题得到 $\Phi_H$, 细网格上只做一次 Newton 校正. 由于截断非线性是 $C^2$ 且二阶导数有界, Newton 校正后的误差关于粗网格误差是二次的; 作者把粗网格误差分为正则性插值误差 $H^{s-1}$ 与对偶正则性带来的 $H^t$ 型项, 再与细网格线性部分误差 $h^{s-1}$ 平衡, 得到 细网格线性系统为对称正定, 因为其双线性形式由 $a(\cdot,\cdot)$ 与乘子 $\rho_\tau'(U_h+\Phi_H)>0$ 组成, 所以可用共轭梯度法. 作者还说明粗网格非线性问题既可使用细网格数据 $U_h$, 也可使用粗网格数据 $U_H$; 后者使整个非线性计算留在粗网格, 但误差估计形式不变. 需要说明的是, 提供的 PDF 摘录没有给出第 5 节中细网格 Newton 校正的逐项离散公式与定理 5.2 的完整证明细节, 上述关于校正二次性的描述依据摘要与第 1 节的说明, 无法从摘录中逐行核对. 与 Xie 的原始工作相比, 本文首先不是提出一个新的分解, 而是在同一分解 $u=G+\Psi+\widehat{\Phi}$ 上补足严格的先验误差分析与两网格加速理论. Xie 在有限元求解器中已经使用该分解, 并用修正 Newton 极小化方法处理正则部分的非线性, 同时为了数值防溢出把指数在某个水平截断. 本文定理 3.6 把这种实践中的截断提升为严格等价定理: 只要 $\tau\ge2Z_{\max}M_U$, 截断问题与原始问题具有同一个唯一解, 且 $|\widehat{\Phi}|_\infty\le M_U$, $\|U+\widehat{\Phi}\|_{\infty,D_s}\le2M_U$. 这不仅解释了实际截断的无害性, 还给出原始半线性问题存在唯一性的新证明, 并且证明在 $H_0^1(\u03a9)$ 中完成, 而不必把解空间限制在 Xie 原文的集合 $H$ 上. 代价是本文的分析需要 $M_U=\|U\|_{0,\infty,D_s}$ 有限, 常数中会出现 $e^\tau$; 但作者指出 $\tau$ 可取在显式阈值 $2Z_{\max}M_U$ 附近, 因此常数反映真实离子密度, 不是额外的人为放大. 在计算成本上, Xie 的求解器每次 Newton 迭代都需要在细网格上做大规模线性求解; 本文把非线性求解限制在粗网格, 细网格只保留两次对称正定线性求解, 一次用于 $\Psi_h$, 一次用于 Newton 校正. 但本文没有在提供内容中给出与 Xie 实际三维生物分子算例的逐项 CPU 时间比较, 这部分无法确认. 与 Holst, Szypowski 与 Zhu 的两网格半线性界面问题分析相比, 本文的推进主要在假设与适用范围. 他们的分析以正则化 PBE 为主要例子, 在三维全正则情形下同样得到粗网格 $H\sim h^{2/5}$ 即可保持精度; 但对指数型超临界非线性, 其证明依赖离散最大值原理, 也就是要求刚度矩阵非对角元满足符号条件, 对由分子表面生成的四面体网格而言这是很强的网格角条件. 本文用 $C^2$ 二次增长截断替代离散最大值原理: 先证明截断问题适定, 再证明截断解满足不依赖网格的 $L^\infty$ 界, 最后证明阈值以上的截断问题与原问题同解. 因此本文的误差分析不需要离散最大值原理, 也不需要网格角条件. 另一个差异是本文的非线性项为 $\rho(U+\widehat{\Phi})\chi_{D_s}$, 通过 $U=G+\Psi$ 依赖空间位置, 并在离散问题中由 $U_h=G+\Psi_h$ 替代; 这比只含 $\rho(\widehat{\Phi})$ 的半线性界面问题多出一层数据扰动. 本文把该扰动最终压缩为 $\|\Psi-\Psi_h\|_{0,2,D_s}$ 项, 从而在误差定理中显式追踪其影响. 此外, 本文允许任意数目离子种类, 而对称 1:1 电解质只是其中特例; 在对称 1:1 情形, 截断非线性为 $\kappa^2\sinh_\tau$, 它是 Xie 原文修正双曲正弦的 $C^2$ 版本, 且 $\cosh_\tau=\sinh_\tau'$, 这比原文仅在 $\pm\tau$ 处连续但不可微的修正函数更适合误差分析. 与经典两网格有限元方法相比, 本文处理的是超临界指数非线性而非全局 Lipschitz 或子临界非线性. 经典两网格分析通常要求非线性项满足全局 Lipschitz 条件, 或至少满足可在能量空间中获得先验界的增长条件; 指数非线性在 $H^1$ 中不满足这些条件. 本文的截断把非线性变成至多二次增长且二阶导数有界的 $C^2$ 函数, 使凸分析, Nemytskii 算子连续性与对偶估计可用; 随后再证明截断不改变解. 这一路线放宽了非线性增长假设, 但代价是引入了截断参数 $\tau$ 与先验界 $M_U$ 的要求. 在误差阶方面, 本文给出的是依赖分片正则性指标 $s$ 与对偶正则性指标 $\sigma$ 的一般估计 当 $\sigma\ge s$ 时可取 $H\sim h^{2/5}$, 与全正则情形的最优粗网格尺度一致; 当 $\sigma 本文的局限与代价也需要明确. 分析采用拟合界面的多面体区域, 连续分段线性有限元与拟一致网格; 拟一致性只在第 5 节用于把网格尺寸与未知量数目联系起来. 作者在摘录中说明, 对曲线界面若网格以 $O(h^2)$ 分辨率逼近界面, 结果仍可保持, 但本文不展开该推广; 因此不能把本文结论直接视为任意曲线界面与任意网格的完整理论. 误差常数依赖 $e^\tau$ 与 $M_U$, 且要求 $U\in L^\infty(D_s)$; 虽然这是线性子问题 $\Psi$ 的性质, 但在极低正则界面情形仍需单独验证. 两网格方法仍要在粗网格上求解非线性问题, 只是把细网格非线性求解替换为一次 Newton 校正; 当对偶正则性差于原始正则性时, 粗网格必须取得更细, 粗网格节省会下降. 此外, 从提供内容中无法确认本文是否对三维实际生物分子算例给出了与 APBS, DelPhi 或 MIBPB 等软件的数值成本比较; 作者在第 6 节只报告了 manufactured-solution 测试, 用于验证收敛阶与两网格校正关于粗网格误差的二次依赖. Quantitative approximation theorem of neural operators for the Dirichlet-to-Neumann map 作者: Pu-Yun Kow, Andrea Aspri, Elisa Francini, Jenn-Nan Wang 类别: math.AP, math.NA 发布日期: 2026-10-26 arXiv ID: 2610.35404v1 链接: http://arxiv.org/abs/2610.35404v1 中文标题: 神经算子对 Dirichlet-to-Neumann 映射的定量逼近定理 摘要翻译: 本文研究用于 Dirichlet-to-Neumann 映射（DN 映射）的神经算子，该映射出现在著名的 Calderón 问题中。在许多用于由 DN 映射的知识重构未知电导率的数值方法中，需要在给定电导率函数处评估相应的 DN 映射，这涉及求解电导率方程的边值问题并取 Neumann 数据。这样的过程耗时且不有效。因此，构造一个代理模型来替代将电导率函数映射到相应 DN 映射的前向映射是一种有利策略。本文的主旨是推导这些神经算子的参数复杂度的定量界。特别地，在适当假设下，它们的深度最多随 $\varepsilon^{-1}$ 对数增长，而相关谱秩和投影秩按代数速率增长。需要注意的是，一般神经算子会遭受“参数维数灾难”。本工作展示了针对 DN 映射的神经算子的构造，其不受这种灾难的影响。 An unconditionally convergent BDF2-type finite element integrator for the inertial Landau-Lifshitz-Gilbert equation 作者: Michele Aldé, Hywel Normington, Dirk Praetorius, Michele Ruggeri 类别: math.NA 发布日期: 2026-10-26 arXiv ID: 2610.35382v1 链接: http://arxiv.org/abs/2610.35382v1 中文标题: 惯性Landau-Lifshitz-Gilbert方程的无条件收敛BDF2型有限元积分器 摘要翻译: 我们提出并分析了一种全离散数值格式，用于逼近惯性Landau-Lifshitz-Gilbert（iLLG）方程的解，该方程是描述亚皮秒时间尺度下磁化强度动力学的最新模型。该方法采用一阶有限元进行空间离散，并采用两步后向差分公式（BDF2）进行时间离散，其中惯性项由BDF2与后向Euler方法的凸组合处理。我们证明了格式的适定性与无条件稳定性，并得到了磁化强度所满足的iLLG固有单位长度约束在实现中的误差的显式界。此外，我们推导了近似解所满足的离散能量律，其形式类似于该模型固有的耗散能量律。最后，我们采用构造性方法证明了近似解无条件收敛到问题的弱解。数值实验评估了格式的性能并验证了理论结果。 A Robust ADI-Based Direct Discontinuous Galerkin Method for Two-Dimensional Singularly Perturbed Parabolic PDEs 作者: Vijay Kumar, Gautam Singh 类别: math.NA 发布日期: 2026-10-26 arXiv ID: 2610.35343v1 链接: http://arxiv.org/abs/2610.35343v1 中文标题: 面向二维奇异摄动抛物型偏微分方程的一种稳健的基于 ADI 的直接间断 Galerkin 方法 摘要翻译: 本文提出二维单参数奇异摄动抛物问题，并针对其数值求解发展了一种算子分裂直接间断 Galerkin（DDG）方法。为提高计算效率，采用交替方向隐式（ADI）分裂方法将原始的二维问题分解为两个独立的一维子问题。空间离散在分片均匀 Shishkin 网格上采用 DDG 方法进行，从而能够精确分辨奇异摄动问题（SPPs）中出现的陡峭边界层。时间离散则采用向后 Euler 格式，以保证稳健性与稳定性。理论分析建立了空间离散的最优收敛阶为 $k$ 阶，其中 $k$ 表示多项式基函数的次数，并在时间维上具有 $(Δt)$ 阶收敛性。我们证明了所得的全离散格式是相容的、稳定的且收敛的。数值实验验证了理论结果，并表明了所提方法的有效性与稳健性。 Towards Exact Multidimensional Fluxes: A Structure-Preserving Data-Informed Framework for Riemann Solvers 作者: Yucheng Zhang, Clint Dawson 类别: physics.comp-ph, math.NA 发布日期: 2026-10-26 arXiv ID: 2610.35265v1 链接: http://arxiv.org/abs/2610.35265v1 中文标题: 面向精确多维通量：Riemann 求解器的保结构数据信息驱动框架 摘要翻译: 有限体积和间断 Galerkin 方法利用其面上的时空通量积分来更新单元，而该积分通常依赖于局部 Riemann 问题的解。在多维 Riemann 问题情形下，由于复杂的波相互作用，该积分难以精确计算。本文表明，精确的时空通量积分是关于时间步长和面尺寸的多项式。这启发了一种基于学习的方法，因为只需学习从初始状态到多项式系数的映射。我们提出一个保结构、数据信息驱动的 Riemann 求解器框架，该框架使用在高分辨率模拟数据上训练的神经网络来学习这一映射。包括相容性和对称性在内的关键结构性质由构造保证。只要高分辨率模拟收敛到精确通量积分，且网络能够准确预测系数，该框架就能给出近乎精确的通量积分以用于数值模拟。
我们利用二维 Euler 方程的经典内爆和爆炸问题来演示该框架。在非分裂 MUSCL-Hancock 格式中，学习得到的 Riemann 求解器在 CFL 数为 0.9 时保持稳定，而使用精确一维 Riemann 求解器的同一格式则失败。与后者在 CFL=0.45 时相比，学习格式略更精确，并且所需计算时间约少 30%。 Regularized Sequential Reconstructions of Dynamic Nonsmooth Sources from Sparse Boundary Flux Data 作者: Davide Bianchi, Tianrui Dai, Hongqin Zhang, Shupeng Zhang, Zhidong Zhang 类别: math.NA 发布日期: 2026-10-26 arXiv ID: 2610.35173v1 链接: http://arxiv.org/abs/2610.35173v1 中文标题: 稀疏边界通量数据下动态非光滑源的正则化序贯重构 摘要翻译: 我们考虑单位圆盘 $Ω\subset\mathbb R^2$ 上热方程的反源问题，其通量测量在有限多个边界点处给出。源在预先给定的区间上关于时间分段常数，并且在每个区间上是具有 Lipschitz 边界的未知区域的特征函数。每个区间上的测量分解为当前源的贡献和来自更早源的历史项；减去后者后，留下一个关于前向算子 $F_τ$ 的线性反问题，该算子在 $L^2(Ω)$ 上稠密定义且无界。我们证明 $F_τ$ 不是可闭的，因此关于闭无界算子的经典正则化理论不适用。我们改用一种使其紧化的 Hilbert 尺度惩罚进行正则化。所得的 Galerkin–Tikhonov 重构是序贯的：每个源由截至其区间末端观测到的通量以及此前重构出的源来恢复。我们证明其收敛性和唯一性，表明较早的重构误差以时间上的指数衰减传播到较晚的数据中，并建立关于传感器数量的稳定性估计。一个含噪测量的数值实验表明了序贯逼近的可行性，并通过 Fourier 逼近和神经网络边界校正加以改进。 Structure-preserving discretisations for the quasi-incompressible, thermal Maxwell-Stefan equations 作者: Boris D. Andrews, Aaron Baier-Reinio, Aaron Brunk, Patrick E. Farrell, Mária Lukáčová-Medviďová 类别: math.NA 发布日期: 2026-10-26 arXiv ID: 2610.35157v1 链接: http://arxiv.org/abs/2610.35157v1 中文标题: 准不可压缩热 Maxwell-Stefan 方程的保结构离散化 摘要翻译: 我们研究一个在准不可压缩约束下模拟互溶混合物的热 Maxwell-Stefan-Navier-Stokes 系统。该系统由质量和动量守恒律以及熵平衡律组成；相较于能量平衡律，采用熵平衡律是因为它便于发展两种保结构离散方法。第一种方法在时间上采用半隐式 Euler 格式，并结合适当的空间有限元，保证熵产生，并在逐点意义下强制准不可压缩约束，但在数值上耗散总能量。第二种方法在时间上采用连续 Petrov-Galerkin 离散，同样产生熵并逐点满足约束，此外还守恒总能量，并允许高阶时间近似。一系列数值实验展示了两种数值方法的保结构性质和行为。 An explicit third-order low-regularity integrator for the cubic nonlinear Schrödinger equation 作者: Shengxin Luo, Bin Wang, Yan Wang 类别: math.NA 发布日期: 2026-10-26 arXiv ID: 2610.34918v1 链接: http://arxiv.org/abs/2610.34918v1 中文标题: 三次非线性薛定谔方程的显式三阶低正则性积分器 摘要翻译: 本文针对一维环面 $\mathbb{T}$ 上的三次非线性薛定谔方程，提出了一种显式三阶低正则性积分器。通过在 Fourier 空间中对积分进行精细的数值逼近，我们构造了一个三阶格式，并给出了严格的全局误差分析。更准确地说，我们证明：对于初值属于 $H^{r+4}(\mathbb{T})$ 且 $r>1/2$ 的情形，该格式在 $H^{r}(\mathbb{T})$ 中达到三阶收敛。此外，对于 $H^{4}(\mathbb{T})$ 中的初值，建立了在 $L^{2}(\mathbb{T})$ 中的三阶收敛性。数值实验验证了理论收敛结果。 One-Sided $p$-Explicit Stability for High-Order Cut Finite Element Nitsche Methods 作者: Erik Burman, Peter Hansbo, Mats G. Larson, Karl Larsson 类别: math.NA 发布日期: 2026-10-26 arXiv ID: 2610.34853v1 链接: http://arxiv.org/abs/2610.34853v1 中文标题: 高阶切割有限元 Nitsche 方法的单侧 $p$-显式稳定性 摘要翻译: 我们针对具有对称 Nitsche 边界条件的高阶切割有限元给出了一个单侧稳定性论证。在每个边界片上，一个 $H^1$-投影将离散函数分解为一个单一多项式和稳定化残差。内向片段控制物理域内的多项式；完整单元估计仅控制残差。对于光滑外部边界，并在依赖于几何的网格限制下，这给出了具有因子 $p^2/h$ 的显式法向通量估计，以及具有相同缩放的一个充分 Nitsche 罚参数。常数独立于切割位置和多项式次数。一个相邻区间估计量化了多项式延拓的指数代价，并说明在稳定性证明中避免这种做法的动机。我们得到 $hp$-误差估计：相对于最佳逼近，在能量范数中损失多项式次数的半次幂，在 $L^2$ 中损失其次数的一次幂；并在充分细的网格上，对解析延拓得到关于多项式次数的指数收敛。平移圆盘实验显示，在所测试的多项式次数和切割位置下，归一化通量常数和罚参数阈值有界，且分辨良好的收敛曲线符合预测阶数。物理稳定化误差和依赖于延拓的稳定化误差分别报告。 Numerical Approximation of Maxwell Transmission Eigenvalue Problem in Dispersive Media via Holomorphic Operator Function 作者: Chao Wang 类别: math.NA 发布日期: 2026-10-26 arXiv ID: 2610.34804v1 链接: http://arxiv.org/abs/2610.34804v1 中文标题: 色散介质中 Maxwell 传输特征值问题的全纯算子函数数值逼近 摘要翻译: 本文针对色散介质中的 Maxwell 传输特征值问题，提出并分析了一种数值方法。该问题被表述为指标为零的全纯 Fredholm 算子函数的特征值问题。在离散化方面，我们采用 Nedelec 元。该格式的收敛性通过 Karma 的抽象逼近理论建立。为计算由此产生的非线性矩阵特征值问题，我们采用一种并行多步围道积分方法。数值实验展示了所提方法的可行性与精度。 A High-Order Geometric-Compatible DGSEM with Subcell Limiting for the GPR System 作者: Jielin Yang 类别: math.NA 发布日期: 2026-10-26 arXiv ID: 2610.34755v1 链接: http://arxiv.org/abs/2610.34755v1 中文标题: 用于GPR系统的高阶几何相容DGSEM及其子单元限制 摘要翻译: 针对固定Euler网格上的约化Godunov-Peshkov-Romenski系统，我们构造了一个与状态方程无关的几何相容有限体积（GC-FV）方法，以及一个高阶节点间断Galerkin谱元推广（GC-DG）。密度由 $\rho=\rho_0\det(\mathbf{A})$ 恢复，因此质量守恒是畸变方程的一个隐含伴随守恒律。以直线路径的Dal Maso-LeFloch-Murat涨落对（fluctuation pair）为出发点，最小改变余因子投影给出单值的密度通量，并在不涉及跳跃商（jump quotients）的情况下给出精确的诱导局部Lax-Friedrichs耗散。分部求和性质将几何两态恒等式提升至高阶，并给出GC-DG的等价子单元表示。随后，我们在相同的Legendre-Gauss-Lobatto子单元上将GC-DG与相容的GC-FV算子相混合。由于两个算子诱导出相同的边界密度通量，全局子单元混合在抑制激波产生振荡的同时，保持了精确的半离散质量平衡。一个互补的模态/面跳跃指示子提供逐单元的混合系数。数值实验验证了高阶收敛性、全离散质量漂移的预期时间缩放，以及对流体、非线性弹性、热波和弹塑性问题（包括真正多维的构型）的鲁棒激波捕捉能力。 A Discontinuous Galerkin Method for Biharmonic Dirichlet Control Problems With Patch Reconstruction 作者: Wei Gong, Ruo Li, Jun Yin 类别: math.NA, math.OC 发布日期: 2026-10-26 arXiv ID: 2610.34669v1 链接: http://arxiv.org/abs/2610.34669v1 中文标题: 基于贴片重构的双调和 Dirichlet 控制问题间断 Galerkin 方法 摘要翻译: 本文提出并分析一种内罚间断 Galerkin 方法，用于由双调和方程支配、通过法向导数施加控制的 Dirichlet 边界控制问题。该方法基于重构间断逼近，该逼近具有任意高阶的逼近空间且每个单元只有一个未知量。我们构造了一种在该空间中逼近状态和伴随状态的离散格式，并在能量范数和 $L^2$ 范数下推导了先验误差估计。分析借助 medius argument 在多边形区域上可获得的有限正则性条件下进行。数值实验验证了所预测的收敛阶。 Lie Symmetry-Guided Adaptive Sampling for Physics-Informed Neural Networks 作者: Runze Zhao 类别: math.NA 发布日期: 2026-10-25 arXiv ID: 2610.34345v1 链接: http://arxiv.org/abs/2610.34345v1 中文标题: 物理信息神经网络的李对称引导自适应采样 摘要翻译: 物理信息神经网络（PINNs）已成为求解偏微分方程（PDEs）的一种有前景的无网格框架，在物理、工程和医学中具有广泛应用。然而，其性能对训练点的采样策略高度敏感。传统的均匀采样往往无法充分分辨具有复杂解特征的区域，导致效率低且精度受限。为应对这些挑战，我们提出了一种用于 PINNs 的李对称驱动自适应采样框架，称为 Lie-AS-ISC。对于允许李对称群的 PDEs，利用对称变换导出解与其变量之间的内在函数关系，从而能够构造具有精确解且张成解域的基础点。在此基础上，通过联合考虑绝对误差和不变曲面条件，发展了一种自适应采样策略，同时引入双指标损失以增强训练约束。所提方法显著提高了预测精度。实验结果表明，该方法在精度上始终优于现有方法。 ESPC-Transolver: An Equivalent-Surface Physics-Guided Coupling-Aware Transolver for 3D Multiple Scattering 作者: Guoqing Zhao, Liang Li, Xing Li, Hao Kuang 类别: math.NA 发布日期: 2026-10-25 arXiv ID: 2610.34246v1 链接: http://arxiv.org/abs/2610.34246v1 中文标题: ESPC-Transolver：一种面向三维多重散射的等效表面物理引导耦合感知 Transolver 摘要翻译: 跨几何泛化尤其具有挑战性，因为以几何为条件的多重散射代理模型必须从几何特征推断未见对象的散射特性，同时准确捕捉由重复波交换产生的复杂且相位敏感的相互作用。我们提出 ESPC-Transolver，一种等效表面物理引导、耦合感知的 Transolver，它将局部散射响应表征与对象间相互作用建模分离。对于每个声硬散射体，在一组共享的球面波激励下构造规范的散射声压响应。局部 Dirichlet-to-Neumann（DtN）映射和模态散射转换将这些响应表示在包围球面上，从而为不同几何提供物理一致的输入通道。共享的对象局部 Transolver 块编码这些响应和入射场迹，而相对位置感知的切片注意力建模对象间耦合。从局部算子恢复的预测界面压力和法向通量定义等效源，用于重建外部近场和远场散射。我们在固定频率下的三维声学多重散射问题上评估该方法，并将其与几何条件预测器和替代神经网络骨干进行比较。ESPC-Transolver 在熟悉几何族的留出实例上实现更低的散射场预测误差，并有效泛化到此前未见的 Peanut 和 Torus 几何。外部场重建再现多重散射干涉图案，将跨几何界面预测与近场压力和远场散射恢复联系起来。 Convergence Analysis of Variable-Separation-Based Model Reduction for Linear Parametric Partial Differential Equations 作者: Liang Chen, Qiuqi Li, Yusong Ye, Pingwen Zhang 类别: math.NA 发布日期: 2026-10-25 arXiv ID: 2610.34139v1 链接: http://arxiv.org/abs/2610.34139v1 中文标题: 线性参数偏微分方程基于变量分离的模型降阶的收敛性分析 摘要翻译: 本文分析变量分离（VS）模型降阶方法的收敛性；该方法通过在采样参数值处求解残差问题，依次构造线性参数偏微分方程（PDEs）解的分离逼近。我们证明，在误差族满足有限维逼近条件，且控制双线性形式满足对称性、一致强制性和一致有界性的假设下，所得的 VS 逼近在参数平均能量范数下几乎必然收敛到精确参数解。对于算子与右端项具有有限项分离表示的参数偏微分方程，我们为误差族构造有限维逼近空间，并给出其维数与逼近误差的界，从而验证这些逼近条件，并得到经过 $N$ 次迭代后期望参数平均误差能量的 $O((\log N)^{-1})$ 上界。当解族包含于一个固定有限维空间时，我们得到期望误差能量更尖锐的 $O(N^{-1})$ 界；此外，若算子还与参数无关，则在几乎必然有限的迭代次数后对几乎每个参数都可精确恢复。数值实验展示了 VS 逼近的误差衰减，以及参数无关情形下的有限步恢复。 An optimization-based property-preserving limiter for two-phase flow in porous media on non-matching meshes 作者: Sanghyun Lee, Seungmin Lee, Chen Liu 类别: math.NA 发布日期: 2026-10-24 arXiv ID: 2610.33940v1 链接: http://arxiv.org/abs/2610.33940v1 中文标题: 基于优化的、用于非匹配网格上多孔介质两相流的保性质限制器 摘要翻译: 多孔介质中的两相流在工程与科学的众多应用中均有出现。其数值模拟的一个基本挑战是：饱和度输运方程的近似可能违反最大值原理，并且即使解仍位于物理界内，也可能出现伪振荡。本文针对非匹配网格上两相流系统的间断 Galerkin 离散，提出一种保界反振荡后处理限制器。基于优化的单元平均限制器通过最小化与原始单元平均的 $L^2$ 距离，在保持全局质量的同时对单元平均施加给定界。为了抑制残余振荡，将单元平均限制器与 minmod 限制器结合；minmod 限制器仅在由问题单元检测器识别出的单元中激活，以避免不必要的精度损失；随后，Zhang-Shu 缩放限制器对点值施加界。此外，我们证明该单元平均限制器在非匹配网格上保持固定时刻数值解的精度。在均质、非均质和裂缝型多孔介质上的数值实验验证了所提方法的精度与鲁棒性。 Optimization-based Cell Average Limiters as Metric Projections: Constrained Evolution and Convergence Analysis for Convection-diffusion Equations 作者: Seungmin Lee, Sanghyun Lee, Chen Liu 类别: math.NA 发布日期: 2026-10-24 arXiv ID: 2610.33449v1 链接: http://arxiv.org/abs/2610.33449v1 中文标题: 基于优化的单元平均限制器作为度量投影：对流扩散方程的约束演化与收敛性分析 摘要翻译: 基于优化的单元平均限制器强制实现全局守恒性和界，并已成功应用于从相场模型到可压缩 Navier-Stokes 方程等各种非线性系统的间断 Galerkin (DG) 离散。本文以热方程和对流扩散方程为原型，分析由单元平均限制产生的格式的收敛性。将守恒性和容许性编码为微分包含，可将该限制器识别为约束演化 Lie 分裂中的后向 Euler 邻近步。该邻近步是到凸容许集上的度量投影，我们证明它在 $L^2$ 范数下是非扩张且稳定的，并且不增加单元平均的全变差。结合标准误差分析，非扩张性在逆 CFL 条件下，对于多项式次数为 $k$ 的 DG 离散的隐式 Euler 时间步进，给出了 $O(τ+h^{k+1})$ 估计，以及一个无条件的 $O(τ+h^{k})$ 估计。数值实验证实了预测的收敛速率和界的强制执行。 Hybridizable Discontinuous Galerkin methods for the Keller-Segel model 作者: Silvia Bertoluzza, Gabriella Bretti, Micol Pennacchio, Christophe Prud’homme 类别: math.NA 发布日期: 2026-10-23 arXiv ID: 2610.33260v1 链接: http://arxiv.org/abs/2610.33260v1 中文标题: Keller-Segel 模型的可杂交间断 Galerkin 方法 摘要翻译: 我们提出一种可杂交间断 Galerkin (HDG) 有限元方法，用于近似时间依赖的 Keller-Segel 模型的解；该模型是描述趋化细胞运动的漂移-扩散系统。我们发展并分析了相应的 HDG 格式，并针对所提出的离散化建立了理论收敛性结果。最后，我们给出数值实验以验证理论发现。 Operator limits of random matrices: Edge universality and sharp approximations 作者: Hyung-Gil Kwak 类别: math.PR, math-ph, math.NA 发布日期: 2026-10-26 arXiv ID: 2610.35808v1 链接: http://arxiv.org/abs/2610.35808v1 摘要: We establish operator universality at the spectral edge of general Wigner matrices by showing that their suitably rescaled Lanczos tridiagonalizations, constructed from deterministic initial vectors, converge to the stochastic Airy operator with Dirichlet boundary condition.
The result requires control of the entire tridiagonal structure on the scale relevant to the limiting differential operator. The Lanczos coefficients for non-Gaussian Wigner matrices are strongly dependent. We analyze their fluctuations by combining multi-resolvent local laws with quantitative control of Krylov subspaces and a comparison of the resulting Jacobi coefficients with those of the Gaussian ensembles. This approach identifies the limiting Brownian potential directly from the original matrix entries, without invoking Dyson Brownian motion in the step of Green function comparison. The resolvent limit yields joint convergence of edge eigenvalues, embedded eigenfunctions, and spectral measures.
For Gaussian, Laguerre and circular $β$-ensembles we construct sharp couplings to the stochastic Airy, Bessel and Sine operators, preserving the finite-matrix laws. For fixed $β> 0$, their squared Hilbert-Schmidt inverse errors have almost-sure orders $n^{-1/3}, n^{-1}$ and $n^{-1}$, respectively, with $a > 0$ in the hard-edge upper bound. We also identify the positive leading order coefficient for the circular ensemble, whose second derivative jumps at $β= 2$. Persistence Pays: Square-root condition-number dependence for Metropolis-adjusted kinetic Langevin Monte Carlo with a non-reversibly updated accept/reject uniform 作者: Nawaf Bou-Rabee 类别: math.PR, math.NA, stat.CO 发布日期: 2026-10-26 arXiv ID: 2610.35781v1 链接: http://arxiv.org/abs/2610.35781v1 摘要: We prove accelerated mixing for a Metropolis-adjusted kinetic Langevin sampler: the OBABO splitting with friction $2\sqrt m$, adjusted with Neal’s non-reversibly updated uniform, which is redrawn after every epoch of $E$ steps. For targets $e^{-U}$ on $\mathbb{R}^d$ with $mI\preceq\nabla^2U\preceq LI$ and an $L_H$-Lipschitz Hessian, started from a Gaussian at an approximate minimizer, the chain reaches total variation accuracy $\varepsilon$ after $\tilde O\big(\sqrt{κd}+\sqrt{L_H}(d/m)^{3/4}\big)$ gradient evaluations, where $κ=L/m$ and $\tilde O$ hides logarithmic factors. The $\sqrtκ$ term is necessary, already on Gaussian targets. The proof discretizes Lu’s entropic hypocoercivity by means of an entropy inequality for free transport, so that the relative entropy of the unadjusted chain decays at a rate of order $\sqrt m$ up to a discretization bias. With the non-reversibly updated uniform, the leading energy errors of velocity Verlet telescope along a path. A coupling argument combines the two. On Bayesian logistic regression posteriors for two standard data sets, which satisfy these assumptions, the gradient cost of the chain with the non-reversibly updated uniform grows roughly like the square root of the condition number at the posterior mode, and that of standard adjusted OBABO roughly like its $3/4$ power. Multi-Fidelity Particle Flow Filters based on Log Homotopy 作者: Abhinab Bhattacharjee, Andrey A. Popov, Amit N. Subrahmanya, Adrian Sandu 类别: cs.CE, math.NA 发布日期: 2026-10-26 arXiv ID: 2610.35712v1 链接: http://arxiv.org/abs/2610.35712v1 摘要: This work proposes a new Multi-Fidelity Particle Flow Filter (MFPFF) based on a linear control variate framework and multilevel Monte Carlo methods. Conventional particle filters often suffer from weight collapse and particle degeneracy in high-dimensional spaces. Particle flow filters address this by smoothly transporting prior particles toward the posterior distribution, thereby avoiding the need for resampling. This work constructs multi-fidelity particle flow filters by augmenting a small number of expensive high-fidelity particles (full-order model runs) with a large number of inexpensive low-resolution particles (reduced-order surrogates constructed via intrusive Galerkin projection). These surrogates act as control variates to enhance the statistical estimate of the full-order posterior, yielding enhanced analysis accuracy at comparable computational cost. The work focuses on two particle flows based on log homotopy, the stochastic Gromov flow and deterministic Exact flow. Numerical experiments with the Korteweg-De Vries (KdV) and Quasi-Geostrophic (QG) test problems demonstrate that the proposed MFPFF achieves accuracy comparable to high-fidelity simulations while significantly reducing computational expense. Computing Data-Aware Diagonal Frame Decompositions 作者: Andrea Ebner, Dirk Lorenz, Mathias Staudigl 类别: math.NA, math.OC 发布日期: 2026-10-26 arXiv ID: 2610.35691v1 链接: http://arxiv.org/abs/2610.35691v1 摘要: We present a method for computing a specific type of matrix decomposition that combines a structured framework with data-driven adaptation. This framework allows for an efficient and stable representation of signals or images using frames by identifying sets of vectors that naturally capture the essential features of the given data. The resulting decomposition enables the development of regularization techniques that enhance the quality of reconstruction by preserving important signal characteristics while suppressing noise. We develop an optimization-based algorithm to compute the decomposition and establish convergence properties of the resulting iterates. Numerical experiments on synthetic data and tomographic image reconstruction illustrate the behavior of the proposed method and its performance compared to conventional spectral regularization methods. Graph-Convolutional Reinforcement Learning for Finite-Time Stabilization of a Chaotic Chua Circuit via Unified Flexible Terminal Sliding-Mode Control 作者: Hooman Fatoorehchi, Majid Roohi, Rasmus Kold, Matthias Ehrhardt, Andreas Basse-O’Connor, Sebastian Nis Bay Villadsen, Philip L. Fosbøl 类别: math.DS, math.NA 发布日期: 2026-10-26 arXiv ID: 2610.35613v1 链接: http://arxiv.org/abs/2610.35613v1 摘要: This paper introduces a graph convolutional reinforcement learning framework for the finite-time stabilization of chaotic dynamics in a physical Chua circuit. The circuit is modeled directly in terms of capacitor voltages, inductor current, and physical component parameters. This preserves the connection between the mathematical model and the underlying electrical system. A unified, flexible terminal sliding-mode controller is developed to achieve stabilization within a finite time in the presence of uncertainties in the lumped model, external disturbances, and bounded control inputs. The controller incorporates state-dependent nonlinear exponents to provide flexible convergence characteristics while avoiding singular terms near equilibrium. To minimize dependence on manual controller tuning, the admissible controller parameters are optimized using an actor-critic reinforcement learning scheme with a graph convolutional network that leverages the circuit dynamics’ coupling structure. The learned parameters are constrained to the ranges required by the theoretical stability conditions. Numerical experiments demonstrate the rapid suppression of chaotic oscillations and the finite-time convergence of the state and sliding variables while respecting the prescribed input-saturation limits. Compared to a reference control method, the proposed approach reduces the IAE, ISE, ITAE, and ITSE by 56.27,%, 56.40,%, 61.69,%, and 64.70,%, respectively. These results suggest that graph-based reinforcement learning can effectively tune parameters while retaining the robustness and analytical structure of the underlying sliding-mode controller. Computing ${}_{q+1}{F}_q$ via rational conformal maps 作者: John Lopez, Richard Mikael Slevinsky 类别: math.NA 发布日期: 2026-10-26 arXiv ID: 2610.35577v1 链接: http://arxiv.org/abs/2610.35577v1 摘要: It is well-known that certain Möbius (linear fractional) transformations provide analytic continuation of the Gauss hypergeometric function. These transformations work by exchanging the roles of the three regular singular points and $0$, $1$, and $\infty$ of the Gauss hypergeometric differential equation. Not one of these transformations, however, brings the points $e^{\pm{\rm i}π/3}$ closer to the origin, and convergence in a neighbourhood of these is also troublesome. In this work, we produce a Maclaurin series in optimal complexity after rational conformal mapping that induces pointwise convergence to any ${}_{q+1}F_q$ throughout the cut complex plane $\mathbb{C}\setminus[1,\infty)$. We show that for the optimal parameter values this series converges faster than an approximate mapped Chebyshev series obtained by the ultraspherical spectral method. ADACT: PDE-Constrained Thermal Hardware/Software Co-Design Adjoint-Driven Optimal Control via Finite Difference Discretization 作者: Don Li 类别: math.OC, cs.AR, eess.SY, math.NA 发布日期: 2026-10-26 arXiv ID: 2610.35569v1 链接: http://arxiv.org/abs/2610.35569v1 摘要: By virtue of the sheer compute demands of large-scale artificial intelligence (AI) and high-performance computing (HPC), modern multi-core microprocessors suffer from severe power density limits and thermal throttling, entailing an urgent need for principled methods for predictive, physics-informed dynamical thermal management (DTM). Here, we introduce a hardware/software co-design physics-informed PDE-constrained DTM throttling optimal control algorithm, ADACT (Adjoint-Driven Adaptive Co-Design Thermal Control), that minimizes thermal risk and cooling energy overhead while maximizing core execution throughput. The underlying physical temperature field is governed by a two-dimensional nonhomogeneous parabolic reaction-diffusion equation with spatially localized, time-varying power source terms and variable convective cooling subject to nonhomogeneous Dirichlet boundary conditions. We employ a discretization of the continuous state PDE via a Crank-Nicolson finite difference scheme with a 5-point discrete Laplacian operator on a uniform spatial mesh, which preserves unconditional numerical stability and $O(Δx^2 + Δy^2 + Δt^2)$ second-order temporal and spatial accuracy. To efficiently compute exact gradients, we construct and implement a continuous backward-in-time adjoint state method with L-BFGS as our optimization algorithm of choice. The accuracy and second-order convergence rates of both forward state and backward adjoint solvers are quantitatively verified via the Method of Manufactured Solutions (MMS), and gradient accuracy is validated using directional derivative Taylor tests. Numerical experiments demonstrate that ADACT dynamically throttles software power workloads and modulates hardware cooling efforts to strictly uphold physical thermal thresholds while maintaining operational throughput. Model reduction in time: Learning flow maps for reducing stochastic dynamical systems in time 作者: Shreya Jha, Benjamin Peherstorfer 类别: cs.LG, math.NA 发布日期: 2026-10-26 arXiv ID: 2610.35504v1 链接: http://arxiv.org/abs/2610.35504v1 摘要: Predictions with dynamical systems can be computationally expensive—even with reduced models—because reaching future prediction times can require many sequential time integration steps. We propose Flow-Map First-Order Trajectory Matching (FM-FTM), a nonintrusive model reduction approach that accelerates ensemble predictions of stochastic dynamical systems by learning flow maps that replace multiple time-integration steps. Our method uses first-order trajectory matching (FTM) to learn a deterministic representation of the stochastic probability-current dynamics and then reduces its evolution over time into a flow map whose evaluation directly yields predictions over long time horizons. Numerical experiments on evolution problems with stochastic forcing demonstrate that FM-FTM accelerates ensemble predictions by up to two orders of magnitude while accurately predicting time marginals and ensemble statistics. Acceleration of the BSS Algorithm 作者: Tizian Sommerfeld 类别: math.NA 发布日期: 2026-10-26 arXiv ID: 2610.35474v1 链接: http://arxiv.org/abs/2610.35474v1 摘要: Bartel, Schäfer and Ullrich investigated the BSS subsampling algorithm, which is based on the work by Batson, Spielman and Srivastava. While this algorithm works well, the bottleneck is its complexity of $\mathcal{O}(σ^2nm^3)$ for a frame of $n$ vectors of dimension $m$ and oversampling $σ^2$. We continue their work by showing that the algorithm can be implemented in $\mathcal{O}(σ^2nm^{2.5})$. Moreover, for some special cases, an even faster implementation is possible. In particular, BSS subsampling of Fourier-type frames on a full frequency cube can be implemented in essentially $\mathcal{O}(σ^2m^2(n+m^{1.5}))$.} A framework for trackable and modular rounding error analysis 作者: Erin Carson, Yuxin Ma, Meiyue Shao 类别: math.NA 发布日期: 2026-10-26 arXiv ID: 2610.35420v1 链接: http://arxiv.org/abs/2610.35420v1 摘要: Robust and meaningful rounding error analysis is becoming increasingly challenging as modern numerical computations involve a rapidly evolving variety of arithmetic formats, precisions, and computational kernels. We propose a trackable and modular framework for rounding error analysis, together with a new notation for bounding accumulated rounding errors. The analysis is trackable in the sense that the origin and propagation of individual error terms remain explicit in the final result, and modular in the sense that local error bounds can be replaced according to the arithmetic, precision, or computational kernel without repeating the structural analysis. We also introduce the notation $γ_n=(1-{\bm u})^{-n}-1$, which provides an alternative to the classical $γ_n^{\mathrm h}=n{\bm u}/(1-n{\bm u})$ notation. A key feature is that our notation does not require the restriction $n{\bm u}<1$ and conveniently handles errors arising from division. Using block LU factorization as a representative example, we show that the framework distinguishes the error sources of different algorithmic variants and permits improved kernels, such as compensated summation and accurate matrix multiplication, to be incorporated directly into the resulting error bounds. Measurement-dependent reduction of open dynamics in the quantum phase estimation algorithm: the visible subspace and the damped Fejér kernel 作者: Marina O. Gordeichuk, Oleg M. Kiselev 类别: math-ph, math.NA, quant-ph 发布日期: 2026-10-26 arXiv ID: 2610.35332v1 链接: http://arxiv.org/abs/2610.35332v1 摘要: For the quantum phase estimation (QPE) algorithm we study the minimal space of observables that is sufficient to predict the entire output distribution under open dynamics of the counting register. Building on the general Krylov theory of reduction, we compute the structure of this space explicitly for the full Fourier measurement. The outcome probabilities are determined by cyclic sums of the state matrix. If the adjoint generator is diagonal on the matrix units, the spectral classes inside the residue classes of the index difference modulo the register dimension $M$ give a constructive basis, an exact dimension formula, and attainable bounds: the dimension of the visible space lies between $M$ and $M^2-M+1$. The class includes independent qubit dephasing, and an analytically solvable reference model of Gorini–Kossakowski–Lindblad–Sudarshan (GKLS) type yields a damped Fejér kernel; two random-unitary channels with the same Ramsey contrast show that higher harmonics are distinguishable in the QPE distribution. For approximate reduction we use a finite-time Gramian and an a posteriori residual estimate, with a quadratic error order for a reducing unperturbed subspace and a compatible initial state. Numerical calculations check the consistency of the representations and the behaviour of the error under non-Toeplitz perturbations. Self-consistent flow maps based on commutator-free exponential integrators 作者: Xiaofeng Cai, Linghui Kong 类别: math.NA, physics.plasm-ph 发布日期: 2026-10-26 arXiv ID: 2610.35327v1 链接: http://arxiv.org/abs/2610.35327v1 摘要: We develop a flow-map framework for nonlinear transport equations, comprising a self-consistent exact frozen flow and its numerical realization. An exponential–pullback identity represents each frozen transport exponential in a commutator-free exponential integrator as the pullback induced by its exact backward characteristic map. Composing these maps defines the self-consistent exact frozen flow, a time-discrete construction with autonomous frozen fields formed from linear combinations of transport fields evaluated at the stage approximations. A local freezing-error identity and a one-step estimate for the Lie–Euler Vlasov–Poisson update separate field variation, field approximation, and characteristic integration errors; these estimates do not assert global PDE convergence. We specialize this construction to the Vlasov–Poisson system and obtain the self-consistent numerical flow through velocity-space quadrature, a discrete Poisson solve, field interpolation, and numerical integration of the frozen characteristic equations. Under suitable regularity assumptions, we prove that the self-consistent exact frozen flow for this system is a volume-preserving diffeomorphism. Temporal self-convergence tests on Vlasov–Poisson and guiding-center benchmarks yield rates consistent with the design orders and illustrate conservation behavior and the evolution of solution structures in long-time simulations with large time steps. A theoretically exact filtered back-propagationformula for single-time wave inversion 作者: Markus Haltmeier 类别: math.AP, math.NA 发布日期: 2026-10-26 arXiv ID: 2610.35302v1 链接: http://arxiv.org/abs/2610.35302v1 摘要: We consider single-time wave inversion, that is, recovering the initial pressure $f$ from the wave field $\mathcal{W}_T f = u(\cdot,T)$ at a single time $T$, where $u$ solves the wave equation with sound speed $c$ and initial data $(f,0)$. In two dimensions this is the main reconstruction step of full-field photoacoustic projection imaging, where each camera image is a two-dimensional wave field at a single time instant. An efficient direct method is back-propagation by the Fourier multiplier $2\cos(cT|ξ|)$. It yields exact inversion in odd dimensions but not in two. We give its error in closed form, with bounds, and derive an exact explicit filtered back-propagation $f = 2\mathcal{W}_T \mathcal{M}_K \mathcal{W}_T f$ in analogy to filtered backprojection for the Radon transform. A Numerical Method for the Fractional Semilinear Calderón Problem 作者: Mukul Dwivedi, Jesse Railo, Andreas Rupp 类别: math.NA, math.AP 发布日期: 2026-10-26 arXiv ID: 2610.35299v1 链接: http://arxiv.org/abs/2610.35299v1 摘要: We present a finite element method for reconstructing a coefficient in a fractional semilinear equation from a single exterior measurement. Global $\mathcal{O}\left(k^{-2}\right)$ Convergence of Multilevel Gradient-Regularized Newton Methods 作者: Nick Tsipinakis, Matthias Voigt 类别: math.OC, cs.LG, math.NA 发布日期: 2026-10-26 arXiv ID: 2610.35195v1 链接: http://arxiv.org/abs/2610.35195v1 摘要: We develop and analyze multilevel gradient-regularized Newton methods for unconstrained convex optimization with Lipschitz-continuous Hessians. By computing Newton corrections in lower-dimensional subspaces, the proposed methods aim to reduce computational costs while preserving the global $\mathcal{O}\left(k^{-2}\right)$ convergence rate in function values of the full-space method. To establish this rate for multilevel methods, we introduce a class of row-nested restriction operators that ensures monotonicity of the reduced-gradient norms across levels. Numerical experiments with row-nested operators demonstrate convergence behavior comparable to full-space gradient-regularized Newton, while offering substantial computational savings on large-scale instances. Inverse heat source problem from terminal data on a set of low Hausdorff dimension 作者: Kai Yu, Zhiyuan Li, Zhidong Zhang 类别: math.AP, math.NA 发布日期: 2026-10-26 arXiv ID: 2610.35014v1 链接: http://arxiv.org/abs/2610.35014v1 摘要: We study the recovery of a spatial source in the heat equation from terminal observations on a compact interior set $E$ with positive $β$ dimensional Hausdorff content for some $β>d-1$. The prescribed temporal forcing vanishes on a nontrivial interval immediately preceding the observation time. Under a spectral Sobolev prior and a quantitative noncancellation condition on the terminal spectral multipliers, we establish conditional logarithmic stability. For scattered observations on such sets, we derive a sampling inequality under a quasi uniformity assumption. Under explicit sampling and independent noise assumptions, we prove a logarithmic convergence rate in probability for the corresponding Tikhonov reconstructions with a data driven parameter choice. Two numerical experiments using a finite candidate approximation of a fixed Hausdorff set assess discrete spectral sampling stability and illustrate the effects of the waiting interval and measurement noise on reconstruction. Randomized Krylov Subspace Methods for Shifted Linear Systems 作者: Xiang Yuan, Gang Wu 类别: math.NA 发布日期: 2026-10-26 arXiv ID: 2610.34964v1 链接: http://arxiv.org/abs/2610.34964v1 摘要: We propose several novel Krylov subspace algorithms termed shifted randomized GMRES (RandGMRES-Sh), shifted randomized FOM (RandFOM-Sh), shifted randomized GMRES with deflated restarting (RandGMRES-Sh-DR), and shifted randomized FOM with deflated restarting (RandFOM-Sh-DR), for solving shifted linear systems. These methods are designed to significantly reduce the computational overhead of orthogonal basis generation while preserving orthogonality under specific conditions. Besides, this unified framework makes it possible to solve all shifted linear systems at the same time by using a single shared basis. These new approaches combine multiple advanced techniques together effectively, such as randomized sketching, the control of collinear residual directions, and deflated restarting methods. Compared with the standard versions of traditional algorithms, the new algorithms show a steady but moderate improvement in convergence speed. Furthermore, we theoretically prove that although GMRES and sketched GMRES have different algorithmic solution steps, their iterates are all confined to and together span the same basic Krylov subspace. We also work out the theoretical relationship between the solutions produced by RandGMRES-Sh and RandFOM-Sh at the same iteration step before the first restart. Moreover, we strictly prove how the solutions of RandGMRES-Sh and RandFOM-Sh are related to those of their traditional versions, GMRES-Sh and FOM-Sh, respectively. A Variational Geometric Approach to Exact Recovery by $\ell_1/\ell_2$ Minimization 作者: Yulan Liu, Rongrong Lin 类别: math.NA 发布日期: 2026-10-26 arXiv ID: 2610.34842v1 链接: http://arxiv.org/abs/2610.34842v1 摘要: For affine-constrained $\ell_1/\ell_2$ minimization with nonzero data, a variational geometric approach is developed to characterize exact recovery. Necessary and sufficient conditions for local and global exact recovery are established in terms of directional derivatives and cones associated with sublevel sets of the ratio objective. The local analysis further reveals that local minimality, first-order growth, and tilt stability are equivalent. These pointwise characterizations lead to improved sufficient conditions for uniform recovery of sparse signals. The sufficient condition for uniform local recovery has coefficient $c_s=\max\{1,(\sqrt{s}-1)/2\}$ for $s\ge2$. For each fixed $s$, the coefficient $c_s$ is optimal within the considered family of local null-space conditions. Projection estimates yield a scalar bound for uniform global recovery with coefficient two, improving the earlier coefficient three. The coefficient two is optimal within the considered family of global null-space bounds when both $s$ and $n$ may vary. A positive margin in the local null-space condition provides explicit growth moduli, localization radii, and bounds on linear objective perturbations that leave the localized minimizer fixed. Examples show that uniform local recovery does not imply uniform global recovery. A discrete fractional Grönwall inequality for dissipative systems via resolvent kernel estimate 作者: Lei Li, Yunxiao Liu, Jiwei Zhang 类别: math.NA 发布日期: 2026-10-26 arXiv ID: 2610.34827v1 链接: http://arxiv.org/abs/2610.34827v1 摘要: We establish a discrete fractional Gronwall inequality for dissipative systems with time-dependent forcing. For completely positive discretizations, we bound individual resolvent entries by integrals of the continuous response kernel, under controlled perturbation to the L1 weights and a stepsize restriction. The proof constructs a supersolution for each resolvent column. The resulting Mittag-Leffler weights retain both the decay of initial data and the historical contributions of forcing. The framework covers L1 and several standard schemes on uniform meshes. For linear subdiffusion, it yields decay-preserving L1 error estimates separating first order and $(2-α)$ order contributions, with stronger long-time decay under additional solution regularity. Active Flux and Polygons: why a stabilisation and how to implement boundary conditions 作者: Remi Abgrall, Yongle Liu 类别: math.NA 发布日期: 2026-10-26 arXiv ID: 2610.34761v1 链接: http://arxiv.org/abs/2610.34761v1 摘要: In a recent paper, the authors developed an Active Flux-type algorithm for polygonal meshes (see \cite{BP_Pampa_VEM}). The main difference between Active Flux methods on polygonal meshes and those on Cartesian or simplex meshes is that the basis functions are not explicitly known. These functions describe the solution using point-value approximations on the polygon boundaries and the average value within the polygons. In \cite{BP_Pampa_VEM}, it was shown that directly transposing the algorithm from simplexes to polygons leads to a mild instability, and a remedy was proposed without a rigorous explanation. In the present paper, we clarify this issue and demonstrate that the solution presented in \cite{BP_Pampa_VEM} is indeed correct. Furthermore, another challenge-the implementation of boundary conditions-was previously solved heuristically. Building on \cite{pampa_dg}, we show how to properly implement these boundary conditions. Finally, the method is illustrated through several examples with hybrid meshes. An efficient coupled block preconditioner for the Marker-and-Cell discretization of the Stokes-Darcy equations 作者: Zhen Chao, Tianwen Li 类别: math.NA 发布日期: 2026-10-26 arXiv ID: 2610.34692v1 链接: http://arxiv.org/abs/2610.34692v1 摘要: We consider the iterative solution of the large, sparse and nonsymmetric double saddle-point systems arising from the stationary Stokes-Darcy equations in two dimensions, discretized by the Marker-and- Cell (MAC) finite difference method. Existing block triangular preconditioners rest on approximations of two Schur complements, the second of which is hard to approximate uniformly in the physical parameters. We propose instead a coupled preconditioner that keeps the interface coupling and regularizes the constraint block by a small multiple of the identity, so that the second Schur complement never has to be approximated. We analyze the spectrum of the preconditioned matrix, both in the ideal case and for the inexact version used in computations, and derive from it a rule for scaling the regularization parameter with the physical parameters. Numerical results show mesh-independent convergence of preconditioned GMRES and robustness with respect to the viscosity and the permeability. Dynamics-Aware Sampling and Recovery for Tensor-Valued Dynamical Systems 作者: Seok-Young Chung, Longxiu Huang, Yisen Wang 类别: math.NA, cs.IT 发布日期: 2026-10-25 arXiv ID: 2610.34621v1 链接: http://arxiv.org/abs/2610.34621v1 摘要: We study sampling and recovery for tensor-valued dynamical systems evolving under the t-product, using partial spatial observations over time. Although the t-product Fourier transform yields slicewise matrix dynamics, recovery does not generally decouple across slices. Eigenvalues shared across Fourier-domain frontal slices couple the corresponding eigenspaces into aggregated spectral blocks, which determine the tensor-specific recovery geometry and space–time tradeoff.
We establish deterministic identifiability conditions for the initial tensor state. In the diagonalizable case, these are frame conditions on the aggregated blocks. We extend them to non-diagonalizable systems through generalized eigenspaces and Jordan structure. A temporal–spectral factorization of the finite-horizon stacked recovery system underlying Tensor Dynamical Sampling (DS) yields a decoupled two-stage formulation once the temporal threshold is reached. Under independent spatial subsampling, we prove high-probability recovery guarantees using empirical Gram matrices on the aggregated blocks. The bounds depend on aggregated multiplicities, incoherence, and population conditioning. Stability estimates separate the spatial frame factor from the temporal Vandermonde factor and show that increasing temporal depth may worsen numerical conditioning even after identifiability is ensured.
We implement Tensor DS and evaluate it on synthetic and real data. Phase transitions track the predicted blockwise sampling thresholds, and stability experiments illustrate Vandermonde conditioning. Experiments on sea surface temperature and turbulence velocity data show that a learned t-product system operator generally yields more accurate and structurally faithful reconstructions than vectorized low-rank, Tucker-type, graph-regularized, graph-spectral, and vectorized dynamical sampling baselines. Energy stability of explicit–implicit–null time discretizations for nonlinear degenerate parabolic equations with state-dependent coefficients 作者: Yue Wu, Chi-Wang Shu 类别: math.NA 发布日期: 2026-10-25 arXiv ID: 2610.34539v1 链接: http://arxiv.org/abs/2610.34539v1 摘要: We study nonlinear degenerate parabolic equations with state-dependent coefficients admitting a scalar energy potential. The explicit–implicit–null (EIN) method avoids severe time step restrictions of explicit time stepping and costly Newton–Raphson iterations required by a fully implicit treatment of the nonlinearity. Using orthogonal projection of the Kirchhoff transform, we prove nonlinear energy stability for first- to third-order EIN time stepping coupled with a wide range of spatial discretizations that admit self-adjoint positive semidefinite discrete operators. Under an upper bound for the diffusion coefficient on relevant stage segments, two IMEX coefficient conditions yield sufficient stabilization bounds without any time step restriction. The estimates dissipate the original convex energy and allow degenerate diffusion. We also extend the analysis to diffusion systems with a scalar energy potential in a fixed positive definite metric, and to consistent positive-weight quadrature discretizations. Applications include porous-medium flow, nonlinear heat transfer in plasma, and multicomponent diffusion. Symbolic Direct Construction of Augmented Systems for Levenberg-Marquardt Power Flow 作者: Tianshi Cheng, Venkata Dinavahi 类别: eess.SY, cs.PF, math.NA 发布日期: 2026-10-25 arXiv ID: 2610.34509v1 链接: http://arxiv.org/abs/2610.34509v1 摘要: Levenberg-Marquardt power flow (LMPF) addresses ill-conditioned equations and provides least-squares stationary points for infeasible operating requests. Conventional implementations form damped normal equations, where J^T J introduces fill, squares the spectral condition number of the undamped part, and adds sparse-product cost. We propose an augmented LMPF formulation with a symbolic-permutation-based Jacobian operator that fills the augmented blocks directly. We prove that the matrix is symmetric quasi-definite for positive damping, enabling fixed-order, pivot-free QDLDL factorization. This property also motivates the solver choice: KLU value-only refactorization can reuse an unsuitable pivot order, forcing fresh full factorizations on difficult LMPF trajectories. We further introduce one-probe adaptive trust-region initialization for the radius and damping. Direct assembly is 7.5-11.9x faster than the fastest coordinate-list path and 18.0-58.8x faster than cached normal-equation sparse-product formation; QDLDL reduces numerical refactorization and triangular-solve time by up to 3.40x on identical matrices. Adaptive initialization solves flat-started RTE cases in 19-25 factorizations, 7.5-11.6x faster than reproduced three-step LM. On case2383wp, our 84-iteration run performs 84 factorizations and 168 resolves in 237 ms, versus published times of 200.780-441.72 s with fewer outer iterations, corresponding to speedups of 847-1860x. These results require complete factorization and resolve accounting and reveal initialization and legacy-parameter effects. DC-initialized NR exhibits fast quadratic convergence throughout every tested solvable loading range, motivating clearer, reproducible definitions of ill-conditioned power-flow benchmarks. Second-Order Steps, Spectral Accuracy: Discrete Parallel Transport of Invariant Subspaces 作者: Hanwen Zhang 类别: math.NA 发布日期: 2026-10-25 arXiv ID: 2610.34494v1 链接: http://arxiv.org/abs/2610.34494v1 摘要: Eigensolvers return orthonormal bases of invariant subspaces, but each basis is determined only up to a unitary factor, so bases computed at different parameter values cannot, in general, be interpolated directly. Discrete parallel transport resolves this ambiguity by aligning consecutive bases through the polar factors of their overlaps, but approximates the continuous parallel-transported frame only to second order. We show that the resulting frames can nevertheless be interpolated with spectral accuracy. For analytic families of invariant subspaces sampled at $N+1$ Chebyshev nodes, the computed frames are exact samples of an analytic frame related to the parallel-transported frame by a gauge that remains within $O(N^{-2})$ of the identity throughout a fixed complex neighborhood, uniformly in $N$. Consequently, Chebyshev interpolation approximates this analytic frame with geometric convergence. Moreover, the continuous parallel-transported frame can be recovered to spectral accuracy by solving a small matrix differential equation, without additional eigensolves or derivatives of the underlying matrix family. The results do not require the family to be Hermitian and remain valid through eigenvalue crossings and branch points, provided the selected invariant subspace depends analytically on the parameter. Numerical examples confirm the predicted rates. Constellations for Lyapunov exponents 作者: Nima Alibabaei 类别: math.DS, math.NA 发布日期: 2026-10-25 arXiv ID: 2610.34306v1 链接: http://arxiv.org/abs/2610.34306v1 摘要: We study the maximal Lyapunov exponent of random products of positive matrices, and prove that it can be calculated to arbitrary precision with arithmetic work polynomial in the number of requested digits. In nondegenerate settings, the complexity of our method depends on the intrinsic complexity of the dynamics—which we characterize—and not on the Euclidean dimension of the ambient space in which the dynamics live. As numerical applications, we determine the Hausdorff dimension of random intersections of Sierpiński gaskets, and Viswanath’s constant—the growth rate of the random Fibonacci sequence.
We first establish a general inequality for nonnegative matrix families showing that, if a signed measure with finite support is sufficiently close to being stationary, then it predicts the Lyapunov exponent accurately. This framework also encompasses Pollicott’s Fredholm determinant approximation. We then show that, just like drawing a constellation by picking bright stars, one can construct a sequence of signed measures approximating stationarity by picking informative orbits of the projective dynamics. Furthermore, we prove that our construction uses an asymptotically optimal number of atoms, which is governed by an intrinsic quantity we call constellation dimension. Recovering fractal boundary dimensions from Robin eigenvalues 作者: Habib Ammari, Yat Tin Chow, Fuqun Han 类别: math.NA 发布日期: 2026-10-25 arXiv ID: 2610.34284v1 链接: http://arxiv.org/abs/2610.34284v1 摘要: We study Robin problems in which a small local boundary coefficient competes with the increasing surface measure of a finite-depth fractal approximation. For a fixed smooth domain, layer potentials give Green function and eigenpair expansions; contour residues and a finite-rank reduction yield a quantitative spectral-gap-dependent eigenvalue remainder. For varying Lipschitz domains, a uniform Poincaré inequality and a trace estimate with respect to normalized surface measure give a uniform ground-state expansion and a rank-one resolvent singularity as the total Robin boundary mass tends to zero. Under the boundary-resolution hypothesis and the condition on the approximation scales, ground-state measurements with known Robin coefficients recover the lower and upper Minkowski dimensions for sequences with bounded total Robin boundary mass. We describe Neumann, generalized Robin, and Dirichlet-type operator limits under the corresponding compact Mosco convergence hypotheses. Koch snowflake computations illustrate the reconstruction formulas, and an exact finite-stage area correction improves the adjacent-stage dimension estimator. Numerical Construction of Quasi-Periodic Solutions for Nonlinear PDEs: II. Unbounded Perturbation 作者: Mingwei Fu, Bin Shi 类别: math.NA 发布日期: 2026-10-25 arXiv ID: 2610.34194v1 链接: http://arxiv.org/abs/2610.34194v1 摘要: Many nonlinear partial differential equations (PDEs) arising in physical models involve unbounded perturbations, where nonlinear terms contain spatial derivatives. Such perturbations can be expressed as compositions of unbounded operators with bounded nonlinear maps. In this paper, we propose alternating numerical schemes for constructing quasi-periodic solutions to this class of PDEs, including the nonlinear acoustic wave (NLAW) equation and the Korteweg-de Vries (KdV) equation. To render the nonlinear perturbations bounded, we apply inverse differential operators as a preconditioning; this, however, also rescales the linear part by polynomial factors in the spatial mode $n$. Two main difficulties arise. The first is to control the measure of the resulting An efficient method for reconstructing the potential in the fractional Calderón problem with a single measurement 作者: Yun-Long Liu, Jin Wen, Jun-Liang Fu 类别: math.NA 发布日期: 2026-10-25 arXiv ID: 2610.34175v1 链接: http://arxiv.org/abs/2610.34175v1 摘要: In this paper, we develop a numerical method for determining the potential function in one-dimensional and two-dimensional fractional-order Calderón problems through a single measurement. The fractional Laplacian is discretized using the finite difference method, and the POD-RBF method is used to construct the forward problem surrogate model that can provide the predicted value and its exact gradient in time. The inverse problem is solved using Tikhonov regularization and the conjugate gradient method. The efficiency and effectiveness of the developed method are demonstrated through numerical experiments, and the theoretical results are verified. The Calderón problem for the multi-fractional Laplacian Schrödinger equation with drift terms 作者: Jin Wen, Yun-Long Liu, Jun-Liang Fu 类别: math.NA 发布日期: 2026-10-25 arXiv ID: 2610.34077v1 链接: http://arxiv.org/abs/2610.34077v1 摘要: This paper studies the Calderón inverse problem for the multi-fractional Laplacian Schrödinger equation with drift terms, whose leading operator is a sum of fractional Laplacians of different orders, perturbed by a drift vector field and a potential. We consider two inverse settings: in the homogeneous source case (f=0), we aim to simultaneously recover the drift coefficient (b) and the potential (q) from the associated Dirichlet-to-Neumann map; in the non-homogeneous case (f\neq 0), we aim to simultaneously determine (b), (q), and the source term (f). Under suitable assumptions, we establish the well-posedness of the forward problem, derive a strong unique continuation principle, prove Runge-type approximation properties in fractional Sobolev spaces and weighted (C^k) spaces, and obtain generalized Alessandrini identities. These tools yield global uniqueness results for the inverse problems. In addition, we develop a numerical reconstruction framework combining finite difference discretization, radial basis function surrogate modeling, and hierarchical Bayesian inversion. Numerical examples in one dimension for smooth and discontinuous coefficients and sources illustrate the effectiveness of the proposed method and support the theoretical uniqueness results. Greedy Orthogonal Polynomial Manifolds with Lipschitz Control for Dimensionality Reduction and Reduced Modeling 作者: Tomoki Koike, Benjamin Peherstorfer, Elizabeth Qian 类别: cs.LG, math.NA 发布日期: 2026-10-24 arXiv ID: 2610.33784v1 链接: http://arxiv.org/abs/2610.33784v1 摘要: Polynomial manifolds provide nonlinear low-dimensional representations through polynomial decoders. We propose to represent the polynomial terms of the decoder using features that are orthonormal on the training data. This empirical orthogonality enables a greedy algorithm, Greedy-OPM, which jointly selects encoder modes and polynomial features while minimizing the reconstruction error under a prescribed bound on the decoder’s Lipschitz constant. Numerical experiments show that this Lipschitz constraint improves generalization away from the training data while retaining high reconstruction accuracy. The Exact Dihedral Angle Sum Range for Circumcentral-Cevian Jaw Tetrahedra 作者: Sergey Korotov 类别: math.NA 发布日期: 2026-10-24 arXiv ID: 2610.33783v1 链接: http://arxiv.org/abs/2610.33783v1 摘要: Hajja, Hammoudeh, Hayajneh and Martini showed that a tetrahedron whose cevians to the circumcenters of the opposite faces are concurrent is either a kite or a special jaw. We determine the exact dihedral-angle-sum range on the jaw component. The lower endpoint is attained only by the regular tetrahedron, while the upper endpoint is approached only in the degenerate limit. The proof reduces the family to one scale-invariant parameter and an explicit monotonicity argument. On the stability of second-order methods for DDEs and PDDEs 作者: Javier Rodríguez-Gutiérrez, Jesús Martín-Vaquero 类别: math.NA 发布日期: 2026-10-24 arXiv ID: 2610.33780v1 链接: http://arxiv.org/abs/2610.33780v1 摘要: In this paper, a one-parameter family of second-order implicit–explicit (IMEX) multistep methods is studied for delay differential equations (DDEs). The analysis is based on the matrix problem $y'(t)=A y(t) + B y(t-τ(t))$, where $A$ is Hermitian negative definite but $B$ is any matrix. For constant, grid-aligned delays, a field-of-values argument yields unconditional stability whenever a suitable generalized field of values $F_p(A,B)$ is contained in the open unit disk. For a constant non-grid-aligned delay, we show that quadratic interpolation does not reduce the corresponding stability disk. For genuinely variable delays, the method is formulated as a nonautonomous augmented system, and product-level sufficient conditions are obtained from a common quadratic Lyapunov inequality and from a quantitative slow-variation hypothesis.
The approach developed herein is illustrated through several examples in which the discussed theory is applied not only to DDEs, but also to semidiscrete parabolic problems given by partial delay differential equations with a diffusion term. Inverse problems with fractional operators: an analytical study and its numerical linear algebra treatment via the GLT theory 作者: Asim Ilyas 类别: math.NA 发布日期: 2026-10-24 arXiv ID: 2610.33763v1 链接: http://arxiv.org/abs/2610.33763v1 摘要: Fractional derivatives model memory effects and nonlocal interactions beyond classical integer-order models. Fractional differential equations (FDEs) thus describe anomalous diffusion, transport in heterogeneous media, and hereditary dynamics. Their nonlocality, however, makes inverse problems challenging: unknown sources, coefficients, or initial conditions must be recovered from indirect and noisy data. Inverse problems for FDEs are usually ill-posed in the Hadamard sense, so regularization, such as the quasi-boundary value method, is required for stable reconstruction. Discretization produces large linear systems whose matrices inherit the nonlocal structure of fractional operators. Understanding the asymptotic spectral properties of these matrix sequences is essential for efficient solvers and preconditioners. Generalized Locally Toeplitz (GLT) theory provides a unified framework for the spectral distribution of large structured matrices and for preconditioner design. This thesis uses GLT theory to analyze matrix sequences from inverse problems for FDEs, construct structure-preserving preconditioners, and study nonlinear operations in the GLT algebra. In particular, it proves maximal distribution results for geometric means of GLT sequences: when the symbols commute, the geometric mean sequence remains GLT without symbol invertibility. This resolves a long-standing conjecture and shows maximality in the noncommuting degenerate case. Overall, the thesis integrates regularization, spectral analysis, and preconditioning into a coherent computational framework for inverse problems in nonlocal fractional dynamics, while enriching structured matrix analysis. On the power of adaption and randomization 2: Non-symmetric convex sets 作者: David Krieg, Erich Novak, Mario Ullrich 类别: math.FA, math.NA 发布日期: 2026-10-24 arXiv ID: 2610.33733v1 链接: http://arxiv.org/abs/2610.33733v1 摘要: Let $S:X\to Y$ be a bounded linear operator between real Banach spaces, and let $F\subset X$ be nonempty, bounded and convex. We prove Here $a_k$ is the minimal worst-case error of continuous affine algorithms of rank at most $k$, and $e_n^{\rm ran}$ and $e_n^{\rm ada}$ are the minimal errors of adaptive randomized and deterministic algorithms using at most $n$ continuous linear measurements. This shows that the gain of adaption and randomization, and even nonlinearity, is limited by a factor $27n$ for approximation in convex sets. We discuss the optimality of the result by examples. An improved fast iterative moment method for the simulation of near-continuum gas flows with Shakhov model 作者: Zhenning Cai, Zhicheng Hu, Guanghan Li, Chunwu Wang, Xiaowen Wang 类别: math.NA 发布日期: 2026-10-24 arXiv ID: 2610.33537v1 链接: http://arxiv.org/abs/2610.33537v1 摘要: The fast iterative moment (FIM) method developed in [G. Li, C. Wang, and Z. Hu, Comput. Fluids, 311 (2026), 107031] achieves fast convergence for near-continuum gas flows modeled by the Boltzmann-BGK equation through alternating iterations between the high-order moment system and the consistent hydrodynamic equations. In this paper, we extend the FIM solver to the Shakhov model and improve it in the following aspects. A finite volume method with linear reconstruction is employed for high-resolution spatial discretization of both systems. Following the original FIM method, an implicit-explicit scheme treating the Shakhov collision term implicitly is adopted for the high-order moment system. A local Fourier analysis of a linearized one-dimensional BGK model reveals the need for more hydrodynamic iterations at smaller Knudsen numbers and their slower convergence on refined grids. Motivated by these findings, a Newton-multigrid method with a new regularization strategy is introduced for the hydrodynamic equations, generally providing an effective hydrodynamic correction with one step per alternating iteration, thereby reducing the need for empirical tuning. Numerical experiments for planar Couette flow, lid-driven cavity flow, and heat transfer in a bottom-heated cavity are carried out to investigate the performance of the improved FIM solver. The results demonstrate improved efficiency and robustness over the original FIM method, particularly at small Knudsen numbers, and favorable performance with both first- and second-order spatial discretizations. Robust augmented Block Preconditioning for two-phase Incompressible Inductionless MHD 作者: Bijie Song, Lingxiao Li, Jie Zhang 类别: math.NA 发布日期: 2026-10-24 arXiv ID: 2610.33385v1 链接: http://arxiv.org/abs/2610.33385v1 摘要: In this paper, we develop a robust augmented block preconditioner for three-dimensional, two-phase, incompressible inductionless MHD equations. We employ a high-order monolithic and charge-conservative arbitrary Lagrangian–Eulerian (ALE) finite element method as base scheme, for which the solution variables are current density, velocity, electric potential and pressure. Grad-div stabilization is added in both the current density equation and discrete momentum equation. Backward Euler method is used for the time discretization. At each time step, fully coupled solution strategy is adopted. This leads to large-scale, ill-conditioned, nonsymmetric, and indefinite dual saddle-point system. To solve this system efficiently, we construct an augmented block preconditioner within the constraint preconditioning framework. The ultimate preconditioner combines the augmented Lagrangian technique with approximate block factorization for current density sub-system and fluid sub-system. The fully coupled four-field system is then solved using preconditioned FGMRES iteration. Numerical experiments confirm the high-order accuracy and discrete charge conservation of the ALE-FEM. Drop of heavy fluid in light fluid is taken as benchmark to demonstrate the robustness of the preconditioner with respect to the discontinuity of physical parameters and the mesh size. Derivative-Informed Dimension Reduction of Stochastic Differential Equations: A Path-Space Sensitivity Approach 作者: Joanna Zou, Han Cheng Lie, Youssef Marzouk 类别: math.NA, math.PR, stat.CO 发布日期: 2026-10-23 arXiv ID: 2610.33210v1 链接: http://arxiv.org/abs/2610.33210v1 摘要: Uncertainty in the drift of a stochastic differential equation (SDE) induces uncertainty in statistical observables defined as expected functionals on path space. We propose a derivative-informed method for identifying low-dimensional structure in the drift-to-observable map, with the aim of improving the tractability of uncertainty quantification. This approach can be viewed as a generalization of the active subspace method to infinite-dimensional input spaces and to path-space observables of SDEs. Representing uncertainty in the drift by random perturbations in a Hilbert space, we construct a path-space sensitivity operator using first-order Fréchet derivatives of the observable with respect to the perturbation. We establish conditions under which the leading eigenfunctions of this operator correspond to directions of maximum mean squared sensitivity of the observable, thus enabling low-dimensional approximation with certified error bounds. We further introduce a martingale control variate of the Fréchet derivative that reduces error in sample-based estimators of the operator. Using a Galerkin scheme for numerical discretization, we show that reduced representations of the drift from the derivative-informed basis yield more accurate approximations of mean first hitting times of SDEs than the Karhunen-Loève expansion of the same truncation order. A Gaussian Process State Space Formulation for Materials Constitutive Modeling: Application to Quasi-brittle Materials 作者: Houlin Xu, Chenyang Li, K. T. Ramesh, Michael D. Shields 类别: cs.CE, cs.LG, math.NA, math.PR 发布日期: 2026-10-23 arXiv ID: 2610.33175v1 链接: http://arxiv.org/abs/2610.33175v1 摘要: Constitutive models for history-dependent, path-dependent materials typically encode this dependence in prescribed functional forms through internal variables such as damage or plastic strain that are phenomenological, model-dependent, and not directly observable. These rigid forms lack flexibility to match observations, make calibration potentially non-unique, and complicate uncertainty quantification. This paper develops a Gaussian-process state-space model (GP-SSM) that replaces the fixed form of the constitutive model with a flexible GP regression, and the prescribed internal variables with latent state variables learned directly from observable stress-strain histories. As a GP, the model is naturally endowed with a measure of uncertainty and can be constrained to satisfy physical principles. We demonstrate this framework for quasi-brittle materials under triaxial compression, where the latent variable evolution is driven by the current-state stress invariants and the increment of equivalent strain to reflect irreversible, pressure-sensitive history. A sparse multi-output GP then maps the latent state and current strain tensor to the six-component stress response, trained by variational sparse inference. The trained model is exported into Abaqus/Explicit through a user material subroutine. Four verification cases are reported: (1) offline rollout on validation and test histories; (2) a single element under changing confinement; (3) a frozen-memory unloading-reloading extension; and (4) a multi-element structural model compared against the Karagozian and Case Concrete (KCC) model. These results show that the learned state-space update reproduces held-out RVE-scale histories in closed-loop rollout, tracks evolving confinement, supports unloading without violating the irreversibility of the learned history state, and reproduces structural-level behavior consistent with KCC.3. 研究方法、关键技术和核心工具
4. 与之前工作的比较
了解领域论文（摘要翻译）
1. 神经算子对 Dirichlet-to-Neumann 映射的定量逼近定理
摘要翻译
2. 惯性Landau-Lifshitz-Gilbert方程的无条件收敛BDF2型有限元积分器
摘要翻译
3. 面向二维奇异摄动抛物型偏微分方程的一种稳健的基于 ADI 的直接间断 Galerkin 方法
摘要翻译
4. 面向精确多维通量：Riemann 求解器的保结构数据信息驱动框架
摘要翻译
5. 稀疏边界通量数据下动态非光滑源的正则化序贯重构
摘要翻译
6. 准不可压缩热 Maxwell-Stefan 方程的保结构离散化
摘要翻译
7. 三次非线性薛定谔方程的显式三阶低正则性积分器
摘要翻译
8. 高阶切割有限元 Nitsche 方法的单侧 $p$-显式稳定性
摘要翻译
9. 色散介质中 Maxwell 传输特征值问题的全纯算子函数数值逼近
摘要翻译
10. 用于GPR系统的高阶几何相容DGSEM及其子单元限制
摘要翻译
11. 基于贴片重构的双调和 Dirichlet 控制问题间断 Galerkin 方法
摘要翻译
12. 物理信息神经网络的李对称引导自适应采样
摘要翻译
13. ESPC-Transolver：一种面向三维多重散射的等效表面物理引导耦合感知 Transolver
摘要翻译
14. 线性参数偏微分方程基于变量分离的模型降阶的收敛性分析
摘要翻译
15. 基于优化的、用于非匹配网格上多孔介质两相流的保性质限制器
摘要翻译
16. 基于优化的单元平均限制器作为度量投影：对流扩散方程的约束演化与收敛性分析
摘要翻译
17. Keller-Segel 模型的可杂交间断 Galerkin 方法
摘要翻译
不相关论文（基本信息）
1. 随机矩阵的算子极限：边缘普适性与精确逼近
2. 坚持终有回报：具有非可逆更新接受/拒绝均匀变量的 Metropolis 调整动能 Langevin 蒙特卡洛的条件数平方根依赖
3. 基于对数同伦的多保真度粒子流滤波器
4. 计算数据感知的对角框架分解
5. 基于统一柔性终端滑模控制的混沌蔡氏电路有限时间镇定的图卷积强化学习
6. 通过有理共形映射计算 ${}_{q+1}{F}_q$
7. ADACT：PDE约束的热硬件/软件协同设计——基于有限差分离散的伴随驱动最优控制
8. 时间上的模型降阶：学习流映射用于随机动力系统的时间降阶
9. BSS 算法的加速
10. 一个可追踪且模块化的舍入误差分析框架
11. 量子相位估计算法中开放动力学的测量依赖约化：可见子空间与阻尼 Fejér 核
12. 基于无对易子指数积分器的自洽流映射
13. 单时刻波反演的一个理论上精确的滤波反传播公式
14. 分数阶半线性 Calderón 问题的一种数值方法
15. 多层梯度正则化牛顿法的全局 $\mathcal{O}\left(k^{-2}\right)$ 收敛性
16. 低 Hausdorff 维数集合上基于终端数据的逆热源问题
17. 位移线性系统的随机Krylov子空间方法
18. 通过 $\ell_1/\ell_2$ 最小化实现精确恢复的变分几何方法
19. 通过预解核估计的耗散系统离散分数阶 Grönwall 不等式
20. Active Flux 与多边形：为何需要稳定化以及如何实现边界条件
21. 面向 Stokes-Darcy 方程 Marker-and-Cell 离散的一种高效耦合块预条件子
22. 张量值动力系统的动力学感知采样与恢复
23. 带状态依赖系数的非线性退化抛物方程的显式–隐式–零时间离散的能量稳定性
24. Levenberg-Marquardt 潮流增广系统的符号直接构造
25. 二阶步进，谱精度：不变子空间的离散平行输运
26. 李雅普诺夫指数的星座
27. 由 Robin 特征值反演分形边界维数
28. 非线性偏微分方程拟周期解的数值构造：II. 无界扰动
bad'' frequencies, since the bounds on the measures of the excluded frequency sets are enlarged by polynomial factors in $n$; we show that this measure nevertheless remains small. The second concerns resonances involving the perturbed eigenvalues of the linearized operator at large scales. Our key observation is that, when the dispersive frequencies grow polynomially in $n$ with degree greater than one, such resonances cannot occur if the magnitude of the temporal index lies below a threshold proportional to $|n|$ raised to one less than that degree. It therefore suffices to consider the complementary regime, in which the growth induced by the unbounded perturbation can be controlled in terms of the temporal index. At large scales, an additional complication arises: for a given spatial index $n$, all temporal indices in a box may be bad’’. We show, however, that at each scale there is at most one such spatial index. Consequently, the resulting counting loss is only polynomial and can be absorbed into the exponential-logarithmic threshold, allowing us once again to retain control over the measures of the excluded frequency sets.29. 单次测量下分数阶 Calderón 问题中势函数重构的一种高效方法
30. 带漂移项的多分数阶 Laplacian Schrödinger 方程的 Calderón 问题
31. 用于降维与降阶建模的具有Lipschitz控制的贪婪正交多项式流形
32. Circumcentral-Cevian Jaw 四面体的精确二面角和范围
33. 关于DDEs和PDDEs二阶方法的稳定性
34. 带分数阶算子的反问题：一项解析研究及其基于 GLT 理论的数值线性代数处理
35. 论自适应与随机化的能力 2：非对称凸集
36. 用于Shakhov模型近连续气体流动模拟的改进快速迭代矩方法
37. 两相不可压缩无感应MHD的鲁棒增广块预条件
38. 导数信息驱动的随机微分方程降维：一种路径空间灵敏度方法
39. 材料本构建模的高斯过程状态空间表述：在准脆性材料中的应用