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1. $α$-极限问题：一个线性退化界面传输问题的收敛性
The $α$-Limit Problem: Convergence of a Linear Degenerate Interface Transmission Problem
作者: Toai Luong, Tadele Mengesha, Kerrek Stinson, Steven M. Wise, Ming Hei Wong
类别: math.AP, math.NA
发布日期: 2026-09-01
arXiv ID: 2609.01237v1
摘要翻译: 我们研究了一族线性退化界面传输问题的奇异极限，该问题源于新提出的双参数扩散域方法（DDM2p）中的正则化过程。对于 $α>0$，正则化问题在 $H^{1}(Ω)$ 上具有严格凸的变分形式。在极限 $α\to0$ 下，问题退化为具有非标准能量结构的弱耦合界面系统。为了刻画该极限，我们引入了一个闭的 Hilbert 子空间 $\mathcal{H}\subset H^{1}(Ω)$，该子空间通过环形子域 $Ω_2\subset Ω$ 上的辅助 Helmholtz 问题定义，并确定了 $\mathcal{H}$ 上的极限能量泛函 $\mathcal{E}_{0}$。我们利用标准框架证明了正则化能量 $\mathcal{E}_α$ 在强 $L^{2}(Ω)$ 拓扑下 $Γ$-收敛到 $\mathcal{E}_{0}$。因此，$\mathcal{E}_α$ 的极小元收敛到 $\mathcal{E}_{0}$ 的唯一极小元，而后者被证明等价于极限界面问题的解。我们进一步证明了在 $H^{1}(Ω)$ 中的强收敛性 $u_α\to u_{0}$，并建立了 $O(α)$ 的收敛速率。一维空间中的数值实验证实了预测的一阶收敛速率，并表明该速率是最优的。
Comment: 无
链接: http://arxiv.org/abs/2609.01237v1
1. 研究对象和背景
本文研究由 Two-Parameter Diffuse Domain Method (DDM2p) 的正则化过程诱导的一族线性退化界面透射问题的奇异极限. 设 $\Omega_1\subset \mathbb{R}^n$ 是具有 Lipschitz 边界 $\Sigma=\partial\Omega_1$ 的有界开集, $\gamma>0$, $\kappa\ge 0$, $q\in H^1(\Omega_1)'$, $g\in H^{-1/2}(\Sigma)$. 出发问题是单侧 Robin 边值问题: $-\Delta u_0^{(1)}+\gamma u_0^{(1)}=q$ in $\Omega_1$, $-n_1\cdot\nabla u_0^{(1)}=\kappa u_0^{(1)}+g$ on $\Sigma$. 它在 $H^1(\Omega_1)$ 中有唯一解. 由于边界 $\Sigma$ 可能很复杂, 为便于在矩形或分片多边形网格上计算, 将 $\Omega_1$ 紧嵌入更大的有界区域 $\Omega$, 记 $\Omega_2=\Omega\setminus\Omega_1$, 并引入外延解 $u_0^{(2)}$, 满足 $-\Delta u_0^{(2)}+\beta u_0^{(2)}=0$ in $\Omega_2$, $u_0^{(2)}=u_0^{(1)}$ on $\Sigma$, $n_2\cdot\nabla u_0^{(2)}=0$ on $\partial\Omega$. 把 Robin 问题与外延问题合起来记作问题 (P0), 其解为 $u_0=u_0^{(1)}\chi_1+u_0^{(2)}\chi_2$, 其中 $\chi_i$ 是 $\Omega_i$ 的特征函数. (P0) 是弱耦合的两侧界面问题, 但它在 $H^1(\Omega)$ 上没有自然的单个强制能量泛函.
为了让问题具有标准的变分结构, 对 $\alpha>0$ 引入退化正则化问题 (Pα): $-\Delta u_\alpha^{(1)}+\gamma u_\alpha^{(1)}=q$ in $\Omega_1$, $-\alpha\Delta u_\alpha^{(2)}+\alpha\beta u_\alpha^{(2)}=0$ in $\Omega_2$, $u_\alpha^{(1)}=u_\alpha^{(2)}$ on $\Sigma$, $-n_1\cdot\nabla(u_\alpha^{(1)}-\alpha u_\alpha^{(2)})=\kappa u_\alpha^{(1)}+g$ on $\Sigma$, $\alpha n_2\cdot\nabla u_\alpha^{(2)}=0$ on $\partial\Omega$. 它的能量泛函为 $E_\alpha[u]=\int_\Omega \frac12(D_\alpha|\nabla u|^2+c_\alpha u^2)dx+\int_\Sigma \frac12\kappa u^2dS-\langle q_0,u\rangle_\Omega+\langle g,u\rangle_\Sigma$, 其中 $D_\alpha=\chi_1+\alpha\chi_2$, $c_\alpha=\gamma\chi_1+\alpha\beta\chi_2$, $q_0=q\chi_1$. 对固定 $\alpha>0$, $E_\alpha$ 在 $H^1(\Omega)$ 上强制且严格凸, 因此存在唯一极小元 $u_\alpha$, 它正是 (Pα) 的解. 注意 (Pα) 在 $\Omega_2$ 内的方程等价于 $-\Delta u_\alpha^{(2)}+\beta u_\alpha^{(2)}=0$, 所以每个 $u_\alpha$ 都落在同一个由混合 Helmholtz 条件刻画的约束空间 $H$ 中. 随着 $\alpha\to 0$, $\Omega_2$ 上的系数 $\alpha$ 消失, 能量在 $H^1(\Omega)$ 上不再一致强制, 极限能量具有非标准的约束结构. 本文要填补的空白是: 从正则问题 (Pα) 到弱耦合极限问题 (P0) 的 $\Gamma$-收敛、极小元收敛以及定量收敛阶; 同时为 DDM2p 的双参数收敛分析提供目标能量 $E_0$.
2. 主要定理或主要结果
从摘要与正文摘录可确认以下严格结论.
(1) 辅助混合 Helmholtz 问题的适定性 (命题 3.1). 设 $U$ 是有界连通开集, $\partial U=\Gamma_1\cup\Gamma_2$, $\Gamma_1,\Gamma_2$ 都具有正 $(n-1)$ 维测度, $\Gamma_1$ 相对开且其相对闭包可用 Lipschitz 图局部表示. 对任意 $f\in H^{1/2}(\Gamma_1)$, 问题 $-\Delta u+\beta u=0$ in $U$, $u=f$ on $\Gamma_1$, $n\cdot\nabla u=0$ on $\Gamma_2$ 有唯一弱解 $u_f\in H^1(U)$, 并且存在只依赖于 $U$ 与 $\beta$ 的常数 $C_0>0$, 使得 $\|u_f\|_{H^1(U)}\le C_0\|f\|_{H^{1/2}(\Gamma_1)}$.
(2) 极限空间 $H$ 的定义与良定性. 定义 $H=\{v\in H^1(\Omega): v^{(2)}=v|_{\Omega_2}\ \text{满足}\ -\Delta v^{(2)}+\beta v^{(2)}=0\ \text{in}\ \Omega_2,\ n_2\cdot\nabla v^{(2)}=0\ \text{on}\ \partial\Omega\}$. 由命题 3.1 可知 $H$ 是 $H^1(\Omega)$ 的闭 Hilbert 子空间. 对任意 $v^{(1)}\in H^1(\Omega_1)$, 存在唯一的 $v\in H$ 使得 $v|_{\Omega_1}=v^{(1)}$; 反之若 $v\in H$, 则 $v|_{\Omega_1}\in H^1(\Omega_1)$. 换言之, $H$ 中的函数由其在 $\Omega_1$ 上的部分完全决定, $\Omega_2$ 上的部分是混合 Helmholtz 外延.
(3) 极限能量. 在 $H$ 上定义 $E_0[u]=\frac12\int_{\Omega_1}(|\nabla u|^2+\gamma u^2)dx+\frac12\int_\Sigma\kappa u^2dS-\langle q,u\rangle_{\Omega_1}+\langle g,u\rangle_\Sigma$, 其中将 $u|_{\Omega_1}$ 仍记为 $u$. 这里 $u_0$ 是 $E_0$ 在 $H$ 上的唯一极小元, 并且它等价于原耦合问题 (P0) 的解. 对 $u\in H$, 系数定义直接给出恒等式 $E_\alpha[u]=E_0[u]+\frac{\alpha}{2}\int_{\Omega_2}(|\nabla u|^2+\beta u^2)dx$.
(4) $\Gamma$-收敛. 将 $E_\alpha|_H$ 按 $+\infty$ 延拓到 $L^2(\Omega)$ 得到 $F_\alpha$, 同样得到 $F_0$. 则当 $\alpha\to 0$ 时, $F_\alpha$ 在强 $L^2(\Omega)$ 拓扑下 $\Gamma$-收敛到 $F_0$. 因此 $E_\alpha$ 在 $H$ 上的唯一极小元 $u_\alpha$ 收敛到 $E_0$ 的唯一极小元 $u_0$.
(5) 强 $H^1$ 收敛与误差阶. 对问题 (Pα) 的解 $u_\alpha$ 与极限解 $u_0$, 除 $L^2$ 收敛外, 作者证明更强的强收敛 $u_\alpha\to u_0$ in $H^1(\Omega)$, 并建立一阶误差估计 $\|u_\alpha-u_0\|_{H^1(\Omega)}=O(\alpha)$ ($\alpha\to 0$).
(6) 数值实验. 一维数值实验验证了预测的一阶收敛率, 并提示该 $O(\alpha)$ 阶在一般意义下是尖锐的. 从提供的内容无法确认定理编号或常数是否显式给出; 上述是摘录中明确出现的数学结论.
3. 研究方法、关键技术和核心工具
证明的核心困难是: 当 $\alpha\to 0$ 时 $\Omega_2$ 上的二次项以 $\alpha$ 加权消失, 所以 $E_\alpha$ 不能在固定 $H^1(\Omega)$ 上一致强制; 直接对极小元列取极限会丢失 $\Omega_2$ 部分的信息. 作者的真正创新点是不把极限问题当作一个局部 PDE 系数趋于零的平凡极限, 而是把约束集 $H$ 写入变分框架.
(1) 公共约束空间. 对 $\alpha>0$, (Pα) 在 $\Omega_2$ 内的方程虽带因子 $\alpha$, 但除以 $\alpha$ 后正是 $-\Delta u_\alpha^{(2)}+\beta u_\alpha^{(2)}=0$; 外边界条件也化为零 Neumann 条件. 因此所有 $u_\alpha$ 都属于 $H$. 于是问题被约化为在固定 Hilbert 空间 $H$ 上极小化 $E_\alpha$. 这一约化避免了 $\alpha\to 0$ 时 $H^1(\Omega_2)$ 范数被 $\alpha$ 加权所抹掉的问题.
(2) 构造 $H$ 的辅助 Helmholtz 问题. $H$ 不是简单的局部 Dirichlet 空间, 而是通过混合边值 Helmholtz 问题定义的像空间. 命题 3.1 的证明使用提升函数 $\tilde f\in H^1(U)$, 令 $w=u-\tilde f$ 属于 $V=\{v\in H^1(U): v=0\ \text{on}\ \Gamma_1\}$, 在 $V$ 上用 Lax-Milgram 定理求解. 关键在于 $\Gamma_1$ 具有正测度, 因而 $V$ 上成立 Poincaré 不等式, 使双线性形式 $A(v_1,v_2)=\int_U(\nabla v_1\cdot\nabla v_2+\beta v_1v_2)dx$ 在 $V$ 上强制. 得到的先验估计 $\|u_f\|_{H^1(U)}\le C_0\|f\|_{H^{1/2}(\Gamma_1)}$ 随后用于控制 $H$ 中函数在 $\Omega_2$ 上的 $H^1$ 范数.
(3) 能量差恒等式. 对每个 $u\in H$, 有 $E_\alpha[u]-E_0[u]=\frac{\alpha}{2}\|u^{(2)}\|_{H^1_\beta(\Omega_2)}^2$, 其中 $\|u^{(2)}\|_{H^1_\beta(\Omega_2)}^2=\int_{\Omega_2}(|\nabla u^{(2)}|^2+\beta|u^{(2)}|^2)dx$. 这个恒等式把奇异参数 $\alpha$ 的全部影响集中为 $\Omega_2$ 上的一个小加权能量项. 它在 $H$ 上给出逐点收敛 $E_\alpha[u]\to E_0[u]$, 从而为 $\Gamma$-收敛的 limsup 不等式提供了常数恢复序列.
(4) $\Gamma$-收敛与紧性. 作者采用标准框架: 把 $E_\alpha|_H$ 和 $E_0$ 通过 $+\infty$ 延拓为 $L^2(\Omega)$ 上的 $F_\alpha$, $F_0$, 再证明强 $L^2$ 拓扑下的 liminf 不等式与恢复序列. liminf 的困难在于 $u_\alpha$ 可能在 $\Omega_2$ 上缺乏与 $\alpha$ 无关的 $H^1$ 一致界. 这里命题 3.1 起到关键作用: 若 $u_\alpha$ 都在 $H$ 中且 $E_\alpha(u_\alpha)$ 有界, 则 $\Omega_1$ 部分的能量给出 $H^1(\Omega_1)$ 一致界; $u_\alpha^{(2)}$ 由其边界迹唯一确定, 迹定理与命题 3.1 再把 $H^1(\Omega_2)$ 范数控制到迹范数, 从而得到整个 $H^1(\Omega)$ 的一致界. 由此可以取出弱 $H^1$ 收敛子列并证明下半连续性.
(5) 强 $H^1$ 收敛与 $O(\alpha)$ 误差. 只用 $\Gamma$-收敛只能得到 $L^2$ 收敛. 为了得到一阶 $H^1$ 率, 作者利用变分等式而非仅仅紧性. 令 $e=u_\alpha-u_0$. 定义双线性型 $B_0(u,v)=\int_{\Omega_1}(\nabla u\cdot\nabla v+\gamma uv)dx+\int_\Sigma\kappa uvdS$, $B_2(u,v)=\int_{\Omega_2}(\nabla u\cdot\nabla v+\beta uv)dx$. 则极小条件分别为 $B_0(u_\alpha,v)+\alpha B_2(u_\alpha,v)=L(v)$, $B_0(u_0,v)=L(v)$ 对所有 $v\in H$ 成立. 相减并取 $v=e$, 得 $B_0(e,e)+\alpha B_2(e,e)=-\alpha B_2(u_0,e)$. 再利用 $e\in H$, 所以 $e^{(2)}$ 是 $e|_\Sigma$ 的 Helmholtz 外延, 因而 $\|e^{(2)}\|_{H^1(\Omega_2)}\le C\|e|_\Sigma\|_{H^{1/2}(\Sigma)}\le CB_0(e,e)^{1/2}$. 于是右端被 $\alpha C\|u_0\|_{H^1(\Omega_2)}B_0(e,e)^{1/2}$ 控制, 得到 $B_0(e,e)^{1/2}\le C\alpha$, 再由外延估计得到 $\|e\|_{H^1(\Omega)}=O(\alpha)$. 这一误差机制比普通能量差给出的 $O(\alpha^{1/2})$ 更强, 是本文强收敛结果的定量核心.
(6) 推广性. 文中指出, 虽然系数取常数 $D_\alpha=\chi_1+\alpha\chi_2$, 结果可自然推广到椭圆变系数 $D_\alpha=A\chi_1+\alpha B\chi_2$, 其中 $A,B\in L^\infty(\Omega;\mathbb{R}^{d\times d})$ 对称正定且最小特征值一致有正下界. 摘录中未给出常数依赖与变系数情形的完整证明.
4. 与之前工作的比较
本文最直接的对比对象是作者前两篇关于 DDM2p 的工作. 文献 [23,24] 已在固定 $\alpha>0$ 的情形证明 $E_{\varepsilon,\alpha}$ 随 $\varepsilon\to 0$ $\Gamma$-收敛到 $E_{0,\alpha}$, 并证明解 $u_{\varepsilon,\alpha}$ 在 $H^1(\Omega)$ 中收敛到 $u_{0,\alpha}$; 但那两篇没有给出理论收敛阶, 只给出 $\varepsilon$ 一阶渐近并由数值支持. 本文补上的是另一个方向的极限: 在去掉扩散界面近似后, 让正则化参数 $\alpha\to 0$, 证明 $E_{0,\alpha}\xrightarrow{\Gamma}E_{0,0}$ 与 $u_{0,\alpha}\to u_{0,0}$ in $H^1$, 并给出 $O(\alpha)$ 理论速率. 因此本文为图 2.1 中的 $(\varepsilon,\alpha)$ 双参数收敛程序提供了关键的中间目标能量 $E_{0,0}$.
与许多单参数 DDM 工作相比, 本文的极限问题不是直接在原区域 $\Omega_1$ 上写出的一个局部 Robin 方程. 传统 DDM 通常用相场近似 $\chi_1$ 和 $\delta_\Sigma$, 再证明 $\varepsilon\to 0$ 时恢复 $\Omega_1$ 上的 Dirichlet/Neumann/Robin 问题; 极限边界条件只出现在 $\Sigma$ 上. 在 DDM2p 中, 由于显式引入第二个参数 $\alpha$, 极限问题还包含 $\Omega_2$ 上的混合 Helmholtz 外延, 因此极限能量必须在非局部约束空间 $H$ 上定义. 这种非标准能量结构是该方法与已有单参数 DDM 分析的主要区别.
从计算的角度看, DDM2p 中的 $\alpha$ 并不是可有可无的辅助参数: 作者明确指出, 在有限元或有限差分离散中, 加入 $\alpha$ 能保证整体刚度矩阵对角元有下界 $\alpha$; 如果不做这种正则化, $\Omega_2$ 内可能产生指数小的对角元, 导致条件数恶化. 已有 DDM 文献虽然常在实际数值中使用类似截断或修正, 但理论框架往往不显式分析 $\alpha$ 的作用. 本文把 $\alpha$ 视作一个真正的奇异摄动参数, 并证明其极限不损失 $\Omega_2$ 外延结构, 这比单纯令 $\varepsilon\to 0$ 的分析更接近可计算格式.
在假设方面, 本文只要求 $\Omega_1$ 有 Lipschitz 边界且紧含于 $\Omega$, $\partial\Omega$ 与 $\Sigma$ 不相交; 对 $\alpha$-极限本身不要求 $\Sigma$ 光滑. 作为对比, 相场 DDM 的 $\varepsilon$ 近似通常需要界面附近有符号距离函数, 这在实际构造中往往要求 $\Sigma$ 足够光滑. 作者还说明主要结果可推广到对称正定变系数 $A\chi_1+\alpha B\chi_2$, 这比仅处理标量常系数更宽. 代价是论文中的数值验证只在一维进行; 高维与变系数情形的一阶率和尖锐性并未由数值实验直接验证.
本文尚未回答双参数 DDM 的终极收敛问题. 文中收敛图显示, $(\varepsilon,\alpha)\to(0,0)$ 的联合 $\Gamma$-收敛、沿路径 $\alpha=\alpha(\varepsilon)$ 例如 $\alpha=\varepsilon^2$ 的收敛, 以及 $u_{\varepsilon,\varepsilon^p}\to u_{0,0}$ 的收敛阶, 都留作未来工作. 从当前摘录看, 作者也未显式展开与 [1,10,11,15,20,22,27] 等单参数 DDM 理论在误差常数或正则性假设上的详细定量比较; 本文的价值主要在于第一次严格刻画 $\alpha$-极限及其一阶率.
2. 麦克斯韦电场积分算子的形状全纯性与稀疏逼近
Shape Holomorphy and Sparse Approximation of the Maxwell Electric Field Integral Operator
作者: Paul Escapil-Inchauspé, Carlos Jerez-Hanckes
类别: math.NA, math.AP
发布日期: 2026-08-31
arXiv ID: 2609.00466v1
摘要翻译: 不确定形状障碍物的时谐麦克斯韦散射的不确定性量化所需要的不仅仅是散射场的全纯依赖性：对于边界元方法而言，必须是边界积分算子族本身关于形状参数全纯地依赖。存在两个障碍。电场积分方程的自然能量空间 $\boldsymbol H^{-1/2}_{\mathrm{div}_Γ}(Γ)$ 依赖于几何，而现有的针对弱奇异核的算子值形状全纯性理论是在 $L^2$ 中建立的，无法触及该空间。我们消除了这两个障碍。一种曲面逆变 Piola 变换将依赖于几何的麦克斯韦迹空间与一个固定的参考空间等同起来，并且在拉回变分表述中，曲面雅可比行列式精确抵消。主要的分析成分是一个一致分数阶映射定理 $H^{-1/2}\to H^{1/2}$，该定理适用于一致 $C^{1,1}$ 曲面上的复形变标量单层族，它通过将拉普拉斯主部实现为固定环境空间上复系数 Newton 问题的迹而得到。因此，拉回算子对 $\bm b\in\ell^p(\N)$, $0 Comment: 无 链接: http://arxiv.org/abs/2609.00466v1 研究对象是时谐麦克斯韦散射中理想导电(PEC)障碍物的外问题与其边界积分方程. 令 $\hat D\subset\mathbb R^3$ 为有界 Lipschitz 区域, 参考边界 $\hat\Gamma=\partial\hat D$ 是紧致可定向的 $C^{1,1}$ 曲面. 给定满足 $\nabla\times\nabla\times E_{\mathrm{inc}}-\kappa^2 E_{\mathrm{inc}}=0$ 的整个入射场, 散射场 $E_{\mathrm{sc}}$ 在外部区域满足 $\nabla\times\nabla\times E_{\mathrm{sc}}-\kappa^2E_{\mathrm{sc}}=0$, Silver-Müller 辐射条件, 以及边界条件 $\gamma_D(E_{\mathrm{sc}}+E_{\mathrm{inc}})=0$. 利用外向 Helmholtz 基本解 $G_{\kappa}(x,y)=e^{i\kappa|x-y|}/(4\pi|x-y|)$, 对表面电流 $j$ 定义电场矢势 EFIE 由迹算子 $T_{\kappa,\Gamma}:=\gamma_D E_{\kappa}$ 给出; 再经对偶同构 $\mathcal I_\Gamma:Y(\Gamma)\to X(\Gamma)'$ 写成 $A_{\kappa,\Gamma}:=\mathcal I_\Gamma T_{\kappa,\Gamma}\in\mathcal L(X(\Gamma),X(\Gamma)')$. 其自然能量空间为 范数为 $\|u\|_X^2=\|u\|_{H^{-1/2}_\parallel}^2+\|\mathrm{div}_\Gamma u\|_{H^{-1/2}}^2$. 这里是 Maxwell 与标量积分方程的主要区别: 能量空间不仅含切向分量的低正则性, 还含表面散度的低正则性, 且 $X(\Gamma_y)$ 随参数几何变化. 在不确定性量化、逆散射、鲁棒设计与降阶模型中, 边界本身可能依赖大量或可数无穷多个参数. 本文将固定参考边界 $\hat\Gamma$ 按照 变形, 记 $b_j=\|\psi_j\|_{W^{2,\infty}(\hat\Gamma;\mathbb R^3)}$. 经典的参数化偏微分方程理论表明: 当 $b=(b_j)\in\ell^p(\mathbb N)$, $0 已有的两块结果均不直接覆盖该问题: Dölz-Henríquez [17] 对弱奇异标量核证明了参数到边界积分算子的形状全纯, 但其范围是 $L^2$, 依赖 $|\hat x-\hat y|^{-\nu}$, $\nu\in(0,2)$, 不提供 $H^{-1/2}\to H^{1/2}$ 的分数阶光滑; Jerez-Hanckes, Schwab 与 Zech [26] 对 Maxwell 体域变分问题建立了参数到解的复全纯, 但这是解级而非算子级, 也不能直接告诉我们在 $H^{-1/2}_{\mathrm{div}_\Gamma}$ 中算子族的逼近性质. 本文要填补的空白正是: 让形状全纯性成立在 EFIE 的自然迹空间 $X(\Gamma)$ 中, 并由此得到算子族、表面电流与远场的稀疏逼近. 主要结果在以下整体假设下成立: $\hat\Gamma$ 满足 Assumption 2.1, 即紧致、可定向、$C^{1,1}$, 带有限一致图册; 变形场 $\psi_j\in W^{2,\infty}(\hat\Gamma;\mathbb R^3)$, $b_j=\|\psi_j\|_{W^{2,\infty}}$, $b=(b_j)\in\ell^p(\mathbb N)$, $0 (1) Piola 同构与 Jacobian 消去. 对 $y\in U$, 定义表面逆变 Piola 映射 $P_y:\hat X=X(\hat\Gamma)\to X(\Gamma_y)$ 为 $(P_y\hat u)\circ r_y=J_y^{-1}F_y\hat u$, 其中 $F_y=D_{\hat\Gamma}r_y$ 是切向微分, $J_y$ 是曲面 Jacobian. Piola 同构定理证明 $P_y$ 是一致同构, 且与表面散度满足 因而可把依赖于几何的空间 $X(\Gamma_y)$ 拉到固定的 $\hat X$ 上. 在拉回弱形式中, 两个双重面积分里的 $J_y^{-1}$ 分别与相应表面测度的 $J_y$ 逐项抵消; 固定参考变分形式中不出现法向场、逆表面度量或显式 Jacobian, 几何效应只通过 $F_y$ 和变形后的 Helmholtz 核进入. 对复参数 $z$, 核为 $G_{\kappa,z}(\hat x,\hat y)=e^{i\kappa\sqrt{q_z}}/(4\pi\sqrt{q_z})$, 其中 $q_z=(r_z(\hat x)-r_z(\hat y))\cdot(r_z(\hat x)-r_z(\hat y))$, $\cdot$ 取复双线性欧氏积. (2) 一致分数阶映射定理 (Theorem 5.1). 在 admissible 复管 $O_\rho$ 上, 对所有 $z\in O_\rho$, 标量单层算子 $V_z f(\hat x)=\int_{\hat\Gamma}G_{\kappa,z}(\hat x,\hat y)f(\hat y)ds_{\hat y}$ 满足 这里 admissible 半径指 $\rho_j>1$, $\sum_j(\rho_j-1)b_j\le\varepsilon_0$; 常数仅依赖 $\kappa$ 与 Assumptions 2.1/2.2 的一致几何常数. 这个 $H^{-1/2}\to H^{1/2}$ 估计是全文的分析核心, 它补上了 $L^2$ 理论所缺的一阶平滑. (3) EFIE 算子族的 $(b,p,\varepsilon)$-全纯性与一致可逆性. 通过把拉回 EFIE 弱形式写成由 $V_z$ 和 $F_z$ 组成的复合, 对每个 admissible 半径序列, pulled-back 算子族 $z\mapsto\hat A_z$ 在 $O_\rho$ 上逐坐标全纯且算子范数一致有界, 即实参数族 $y\mapsto\hat A_y$ 是 $(b,p,\varepsilon)$-全纯的, 值在 $\mathcal L(\hat X,\hat X')$. 若波动数平方 $\kappa^2$ 不属于任何实几何 $\Gamma_y$($y\in U$)的内电特征值, 则先由点态非共振性、$y\mapsto\hat A_y$ 的算子范数连续性以及 $U$ 的紧性导出一致可逆界; 再经复扰动, 逆算子在较小的 admissible 复管上存在且全纯. 从而对满足同样全纯性的入射数据, 表面电流 $y\mapsto\hat j_y\in\hat X$ 以及远场模式都是 $(b,p,\varepsilon)$-全纯的. (4) Legendre 系数的 $\ell^p$ 可加性与稀疏逼近. 在 $(b,p,\varepsilon)$-全纯性与 $b\in\ell^p$ 的前提下, 将参数限制到 Bernstein 椭圆族 $E_\rho$ 时, 算子族的 Legendre 系数属于 $\ell^p$; 因此算子族本身、表面电流与远场都允许维数无关的最佳 $N$ 项稀疏多项式逼近. 更高阶混合导数满足各向异性 Cauchy 型界. 固定同一参考网格、同一有限元空间与同一代数指标, 参数到 Galerkin 刚度矩阵的映射继承同样的全纯性, 且逼近常数不依赖空间离散维数. 证明可以分成几何层、解析层和代数量层, 每一层都有非平凡的障碍. 几何层: 处理依赖几何的能量空间. EFIE 的自然拓扑是 $X(\Gamma)=\{u\in H^{-1/2}_\parallel:\operatorname{div}_\Gamma u\in H^{-1/2}\}$, 它与 $\Gamma$ 本性绑定; 简单与 $r_y$ 复合不会把 $\operatorname{div}_{\Gamma_y}$ 变成 $\operatorname{div}_{\hat\Gamma}$ 的低阶扰动. 作者采用的工具是表面逆变 Piola 变换: $P_y$ 在拉回坐标下乘 $J_y^{-1}F_y$. 关键性质是 $\operatorname{div}$ 的换位关系, 由 $\nabla_{\Gamma_y}\varphi\circ r_y=F_y^{-\top}\nabla_{\hat\Gamma}(\varphi\circ r_y)$ 从定义直接推出. 为了证明该映射确实是 $\hat X$ 到 $X(\Gamma_y)$ 的同构且常数一致, 要用到 $C^{1,1}$ 假设: $H^s_\parallel$ 等于切向投影 $\Pi_\Gamma$ 作用在 $H^s(\Gamma;\mathbb R^3)$ 上的值域; $F_y,J_y^{-1}$ 都属于 $W^{1,\infty}$ 且一致有界, 所以乘法和复合在这些 Sobolev 空间上一致有界. 若只假设 Lipschitz, 则必须如 [7] 用内蕴定义, 该事实不成立. 几何层的另一个要点是 Jacobian 消去是精确的而非渐近的: 在 EFIE 的两个项中分别代入 $P_y$, 得到 $J_y^{-1}$ 乘以测度 $J_y ds$, 所有显式 Jacobian 消失. 这使固定参考形式不含法向、逆度量, 从而所有几何信息都集中在 $F_z$ 和核内. 该形式的拉回还使一个参考边界元空间可用于所有参数. 复化与复分离引理. 为了让实参数 $y$ 获得全纯延拓, 需要在复参数 $z\in\mathbb C^{\mathbb N}$ 下定义几何. 作者规定 admissible 半径条件 $\sum_j(\rho_j-1)b_j\le\varepsilon$, 并在乘积管 $O_\rho$ 上定义 $r_z$. 此时没有物理切平面; 关键引理证明 $q_z=(r_z(\hat x)-r_z(\hat y))\cdot(r_z(\hat x)-r_z(\hat y))$ 的实部一致正定: $\Re q_z\ge c_0|\hat x-\hat y|^2$. 证明中先将每个 $z_j$ 逐坐标投影到最近的实点 $y_j$, 利用 $\ell^p\subset\ell^1$ 和 admissibility budget 控制 $\sum|z_j-y_j|b_j$; 这里只用到管域 $O_\rho$ 而非常 Bernstein 椭圆, 这是引理选择管域的原因. 由此可以得到一致弱奇异界 $|G_{\kappa,z}|\le C_\kappa/|\hat x-\hat y|$ 与逐坐标全纯性. 真正的分析核心: Theorem 5.1 的证明. 在实曲面上, 经典论证把 $V$ 视为 Newton 势的迹: $V$ 是从 $H^{-1/2}$ 到 $H^{1/2}$ 的算子, 因为 $\Delta^{-1}$ 把 $H^{-1/2}$ 变为 $H^{3/2}_{loc}$, 取迹后到 $H^{1/2}$. 复形变后 $G_{\kappa,z}$ 不再是某个实系数椭圆方程的基本解, 该论证立即失效. 作者的新思路是: 尽管 $r_z(\hat\Gamma)$ 不是物理曲面, 但 admissible 复几何始终是一致正则实几何的小扰动; 于是可以把 $V_z$ 的 Laplace 主部写成固定环境空间上一个复系数 Newton 问题的迹. 更具体地, 他们在环境空间构造一个扇形且一致强制的复系数椭圆算子, 使得其 Newton 位势在边界上的迹正是主部核算子, 再对边界数据用迹定理; $C^{1,1}$ 假设在此提供所要求的定量环境实现: 切向投影的可交换性、法向场的正则性、双Lipschitz常数和曲面 Jacobian 的一致界. 这个证明的设定是, 只把复几何作为振幅扰动而不是把复曲面当作真实物理域; 因而避免了复曲面上的微分几何. 作者还说明, 为保持复可微, 边界积分与对偶均取复双线性配对; 而辅助环境问题为强制可以取 sesquilinear 能量内积, 最后通过复共轭回到线性对偶. 算子值全纯的传递与一致可逆性. 当 Theorem 5.1 可用, 固定参考弱形式中的每项都被 $V_z$ 控制. 复合后逐坐标全纯来自 $F_z$ 和核的逐坐标全纯; 一致有界来自 uniform fractional mapping theorem. 一致可逆性不是额外假设: 利用共振条件只排除实参数集上的内电特征值, 与 $z\mapsto\hat A_z$ 的连续性和 $U$ 的紧性结合, 得到 $\sup_{y\in U}\|\hat A_y^{-1}\|<\infty$; 对每个实点的逆再在复管中做 Neumann 级数, 得到更小 admissible 管上的一致可逆全纯延拓. 逼近结果: $(b,p,\varepsilon)$-全纯性再与经典逼近理论结合. 从 Legendre 系数在 polyellipse 上的 Cauchy 积分公式可得到各向异性衰减界; 由于 $b\in\ell^p$, $p<1$, 系数排序后的尾部可按 $N$ 项控制, 因此得到 best $N$-term 且维数无关的逼近率. 在 Galerkin 层, 因为 Piola 变换把所有参数问题放在同一个参考网格和索引上, 参数只影响核与 $F_z$, 参数到矩阵映射也全纯, 且离散空间维数不影响参数逼近阶. 与 Dölz–Henríquez [17] 的推进与差异: 他们的框架在固定边界上用弱奇异标量核, 证明参数到核算子在 $\mathcal L(L^2(\hat\Gamma))$ 中的形状全纯, 靠的是 kernel bound $|r_z(\hat x)-r_z(\hat y)|^{-\nu}$, $\nu\in(0,2)$; 这只适合 $L^2$ 的乘积估计, 不能给出 $H^{-1/2}\to H^{1/2}$ 的分数阶平滑. Maxwell EFIE 若只按分量作用在 $L^2$ 上, 无法控制能量空间; 因为 EFIE 的形式需要 $V_z$ 把 $H^{-1/2}$ 数据提高到 $H^{1/2}$ 才能与对偶函数配对. 本文没有重复 [17] 的 $L^2$ 论证, 而是额外证明了 Theorem 5.1, 使对偶配对的双方均可定义; 这是框架层面的推进. 代价是: 振幅序列 $b_j$ 现在取 $W^{2,\infty}$ 范数, 而 [17] 只取 $C^{0,1}$; 对 $p<1$ 的可加性要求更强的量仍属于 $\ell^p$, 因此 admissible deformation fields 更窄. 与 Jerez-Hanckes, Schwab 和 Zech [26] 的差别: [26] 对形状参数化的 Maxwell 变分问题证明了参数到散射解的全纯, 在仿射参数情形要求 (1.2), 即 $\sum_j\|D T_j\|_{L^\infty}<1$ 且 $\|T_j\|_{W^{1,\infty}}\in\ell^p$. 本文证明的是参数到 EFIE 算子族在自然迹拓扑下的全纯, 比参数到解高一层: 解代理必须对每个入射场重新构建; 算子代理一次构建后可用于任意入射场、右端项与线性观测. 在结构上, [26] 走体域变换路线, 把几何信息放进变换后微分算子的行列式与逆 Jacobian, 这会破坏常数系数 Helmholtz 基本解的显式结构; 本文的边界 Piola 路线保留 $G_\kappa$ 的显式形式, 仅在两点差中代入变形映射. 从假设看, [26] 的仿射变形只要求 $W^{1,\infty}$, 而这里为 $W^{2,\infty}$; 这与要求 $C^{1,1}$ 参考面一致. 与边界元形状导数路线 [25,18,19] 的差别: 这些工作对小幅扰动 $\Gamma_\delta=\{x+\delta\psi(x)\}$ 做一阶 Taylor 展开, 要求标称边界 $C^{2,1}$, 因为形状导数问题的数据包含平均曲率 $\operatorname{div}_\Gamma n\in W^{1,\infty}$; 一阶稀疏边界元框架 [18] 及其三维 EFIE 实现 [19] 也在此框架内. 本文不做振幅截断: 不形成形状导数, 而是把整个变形族复化, 因此参考面只需 $C^{1,1}$; 沿指定变形方向的第一参数导数从全纯算子映射的导数恢复, 高阶混合导数由各向异性 Cauchy 型界控制, 而不是由形状导数的递推公式控制. 这是参数尺度上的推进, 也把适用边界正则性从 $C^{2,1}$ 降到 $C^{1,1}$. 其代价是必须假设无穷维参数序列的振幅满足 $\ell^p$ 可加且几何保持一致 bi-Lipschitz; 而一阶方法对单一扰动幅度没有这样的全局几何条件. 与经典形状微分 [14,15] 的对比: Costabel-Le Louër 的形状演算基于伪齐次核展开并假设光滑边界, 本文的 $C^{1,1}$ 设定不适合; 作者明确说不采用 [14,15]. 其他限制与开放问题: 从提供内容看, 作者没有在开形状空间上建立 Fréchet 全纯, 而只讨论固定仿射参数族 $r_y$; 这比某些抽象 shape holomorphy 弱, 但对边界元逼近已经够用. 全局注入性和一致下 bi-Lipschitz bound (2.4) 是独立于导数小量的 co-Lipschitz 条件, 并不由 $\sum_j\|D\psi_j\|_{L^\infty}<1$ 自动推出, 这对变形场是实质限制. 共振排除是对每个实几何都要求 $\kappa^2$ 非内电特征值, 并不检查是否能一致避开; uniform invertibility follows from compactness. 作者还声明 $C^{1,1}$ 假设不一定最优并留到 Section 9. 这些都应纳入成本考虑. Solving the Incompressible Navier-Stokes Equations on Oriented Curved Surfaces Discretized by Point Clouds 作者: Alejandra Foggia, Ivo F. Sbalzarini 类别: math.NA, cs.CE, physics.comp-ph, physics.flu-dyn 发布日期: 2026-08-31 arXiv ID: 2609.00216v1 链接: http://arxiv.org/abs/2609.00216v1 中文标题: 点云离散的定向曲面上不可压缩Navier-Stokes方程的求解 摘要翻译: 我们提出了一种无网格数值求解器，用于求解由表面点云表示的定向曲面上不可压缩Navier-Stokes方程。在曲面上，与刚性和压力-速度耦合相关的数值挑战更加严峻。此外，曲面上的向量微积分与欧几里得情形不同。所提出的方法在欧拉参考系中直接作用于表面点云，无需计算网格。它在空间和时间上实现了一致的高阶精度近似；我们展示了高达六阶的精度。不可压缩约束被局部地施加为弱人工可压缩近似，避免了全局矩阵求逆。我们证明该方法提供了曲面向量场和微分算子的一致且收敛的近似。我们研究了误差、空间分辨率和人工马赫数之间的关系，并刻画了人工振荡的频谱。我们给出了在对称曲面（如球面和环面）以及参数化和非参数化非对称曲面上的不可压缩Navier-Stokes方程的数值解。由于所提出的方法直接作用于非结构化表面点云，它为图像导出几何上的模拟提供了一种有前景的方法，例如来自显微镜视频的生物形态发生模拟。 Families of relative periodic orbits in the planar three-body problem via consecutive alignments 作者: Joan Gimeno, Marc Jorba-Cuscó, Begoña Nicolás 类别: math.DS, astro-ph.EP, math-ph, math.NA, physics.class-ph 发布日期: 2026-09-01 arXiv ID: 2609.01585v1 链接: http://arxiv.org/abs/2609.01585v1 摘要: Relative periodic orbits (RPOs) are solutions of the three-body problem that are periodic in a uniformly rotating reference frame and, in general, quasi-periodic in inertial coordinates. We present a numerical procedure for computing and continuing one-parameter families of RPOs of the planar Newtonian three-body problem.
The method exploits consecutive syzygies, understood here as configurations in which the three bodies are aligned and their velocities satisfy the corresponding symmetry conditions. Matching the positions and momenta at two consecutive alignments reduces the computation of RPOs to a low-dimensional nonlinear problem. Its solutions are then numerically continued, and linear stability is determined from the nontrivial eigenvalues of the rotated monodromy matrix after removing the neutral directions associated with conserved quantities and continuous symmetries.
The procedure is applied to several mass distributions and initial configurations, producing families of Poincaré, Hill, and binary-type solutions. These families exhibit transitions from nearly circular to highly eccentric motion, changes of stability near resonances and turning points, and absolute periodic solutions when the rotation angle is a rational multiple of 2π. In the Hill families, the continuation connects satellite configurations with circumstellar motion as the smallest body loses its gravitational binding to the intermediate body. Circumbinary and circumstellar configurations are also obtained in the binary regime.
The results illustrate the dynamical diversity of RPOs and provide coherent three-body motions that can be used as prescribed trajectories in restricted four-body models. Greedy sampling designs via reduced basis methods: optimal recovery in the uniform norm 作者: Sebastian Neumayer, Kateryna Pozharska, Tino Ullrich 类别: math.NA, math.FA 发布日期: 2026-09-01 arXiv ID: 2609.01578v1 链接: http://arxiv.org/abs/2609.01578v1 摘要: We study optimal sampling recovery in reproducing kernel Hilbert spaces (RKHS) in the uniform norm. For every RKHS with bounded kernel, we establish new comparisons between linear sampling widths and Gelfand widths that overcome the known square-root gap, without requiring a measure or a Christoffel-type condition. Our bounds rely on nested sampling designs obtained by kernel interpolation at (weak) P-greedy points. Under additional (polynomial) decay assumptions the decay rate of the Gelfand widths directly transfers to the sampling widths. With either a logarithmic oversampling or passing to the square root of the Gelfand widths we obtain a direct comparison (requiring no decay assumption) between them. This is particularly effective for super-polynomial decay, such as in Paley-Wiener spaces. Our results follow from representations of both widths in terms of kernel translates and yield, in the opposite direction, a new existence result for a sharp reduced basis selection. Numerical experiments for Legendre, mixed-Sobolev, and Paley-Wiener kernels illustrate our findings. Score-Based Generative Data Assimilation for Integrating Aggregated Surveillance Data into Agent-Based Models in Epidemic Tracking 作者: Siming Liang, Jacob Hauck, Minglei Yang, Adam Spannaus, Heidi Hanson, Guannan Zhang 类别: math.NA, q-bio.QM 发布日期: 2026-09-01 arXiv ID: 2609.01434v1 链接: http://arxiv.org/abs/2609.01434v1 摘要: Reliable epidemic monitoring often requires inferring regional infection burden and transmission heterogeneity from noisy, spatially aggregated, and potentially sparse surveillance data. Agent-based models (ABMs) are attractive for this task because they represent individual behavior, contact heterogeneity, and localized interventions, but these same features make them difficult to calibrate online. We develop a generative AI-based data-assimilation (GenDA) framework for partially observed epidemic ABMs that estimates both the epidemic state and a heterogeneous parameter field while respecting the gap between observable macrostates and latent agent-level microstates. GenDA combines a training-free, score-based generative update for macrostate correction with a direct parameter update based on macrostate discrepancies, followed by a macro-micro reassignment step that restores consistency with the ABM. In controlled and geographically explicit synthetic experiments, the framework recovers regional epidemic burden, dominant hotspot structures, and effective transmission heterogeneity from aggregated observations, while improving post-assimilation forecasts relative to state-only assimilation. Accelerated primal–dual dynamics and algorithms for convex optimization with nonlinear inequality constraints 作者: Xin He 类别: math.OC, math.NA 发布日期: 2026-09-01 arXiv ID: 2609.01415v1 链接: http://arxiv.org/abs/2609.01415v1 摘要: We consider convex optimization with nonlinear inequality constraints and develop a primal–dual multiplier framework that is consistent in continuous and discrete time. We first propose continuous-time dynamics with Nesterov-type vanishing damping $α/t$, together with suitable extrapolations of the dual variable and the nonlinear constraint mapping. Under convexity assumptions and $α\geq3$, we establish $\mathcal O(t^{-2})$ convergence rates for both nonlinear feasibility and the objective residual. We then derive an inexact accelerated primal–dual algorithm through a compatible discretization of a perturbed version of the dynamics. For composite convex objectives, a weighted summability condition on the primal inexactness yields the $\mathcal O(k^{-2})$ rates for feasibility and the objective residual, thereby matching the accelerated rates of their continuous-time counterparts. To the best of our knowledge, this is the first Nesterov-type primal–dual multiplier framework for convex optimization with nonlinear inequality constraints. Accelerating the Improved Arrow–Hurwicz Iteration via the Anderson Algorithm for Steady-State Navier–Stokes Equations 作者: Sinan Ergen, Mustafa Ağgül, Mustafa Türkyılmazoğlu 类别: math.NA 发布日期: 2026-09-01 arXiv ID: 2609.01288v1 链接: http://arxiv.org/abs/2609.01288v1 摘要: We apply Anderson acceleration to the improved Arrow–Hurwicz (IAH) method for the finite element solution of the steady-state incompressible Navier–Stokes equations. The IAH scheme avoids saddle-point solves by decoupling the velocity and pressure updates, but can require prohibitively many iterations, particularly at high Reynolds numbers. To place the acceleration on a rigorous footing, we reformulate the IAH iteration as a nonlinear fixed-point operator $G$ for the grad-div augmented discrete formulation induced by the scheme and establish its well-definedness, Lipschitz continuity, and Fréchet differentiability, thereby verifying the required smoothness conditions locally near the fixed point. Numerical experiments on problems with known analytical solutions, lid-driven cavity flow up to $Re = 15{,}000$, and channel flow over a full step demonstrate that the resulting Anderson-accelerated improved Arrow–Hurwicz algorithm substantially reduces iteration counts and CPU time while retaining the reported manufactured-solution convergence rates and centerline-velocity agreement. Projection-based low-rank assembly in IgA 作者: Tom-Christian Riemer, Martin Stoll 类别: math.NA, math.AP 发布日期: 2026-09-01 arXiv ID: 2609.01218v1 链接: http://arxiv.org/abs/2609.01218v1 摘要: Isogeometric Analysis (IgA) uses the same spline functions to represent the computational domain and to approximate the solution. This allows exact geometry descriptions, but the resulting mass and stiffness matrices are expensive to assemble and to store, especially in three dimensions. We present a projection-based low-rank approach for assembling the mass and stiffness tensors of orientation-preserving tensor-product B-spline geometries. For the mass tensor, we exploit the polynomial structure of the determinant of the Jacobian of the geometry map and represent it in reduced spline product spaces by univariate coefficient transfer operators; with exact quadrature and without truncation, the resulting low-rank tensor is an exact reformulation of the standard Galerkin mass tensor. For the stiffness tensor, we split the rational weight function into a polynomial numerator, again represented in reduced spline product spaces, and the reciprocal determinant, which is in general not a spline function and is therefore approximated by an $L^2$-projection onto a tensor-product spline space. Both constructions are carried out entirely in the tensor-train (TT) format, with the projection system solved by the alternating minimal energy (AMEn) method, so that full high-order coefficient tensors are never formed and the multidimensional integrals reduce to univariate integrals and contracted products. The method is implemented in MATLAB using GeoPDEs and the TT-Toolbox. Numerical experiments show that it is competitive with full assembly and with the interpolation-based low-rank method, and that it applies in two situations in which interpolation is problematic: a singular interpolation system and nearly singular geometries. The construction is restricted to orientation-preserving tensor-product B-spline geometries and does not cover NURBS. On the Gram matrix of standard inner products of asymmetrically-weighted Hermite functions 作者: Ruiyang Dai 类别: math.NA, math-ph 发布日期: 2026-09-01 arXiv ID: 2609.01144v1 链接: http://arxiv.org/abs/2609.01144v1 摘要: Let A denote the infinite Gram matrix associated with the standard L2 inner product of asymmetrically-weighted (AW) Hermite functions. We derive an explicit representation of its entries and its Cholesky factorization. We further show that this factorization admits a natural interpretation on a scaled Bargmann-Fock basis. An explicit formula for the inverse of A is also obtained. We then consider the corresponding finite Gram matrix and analyze its asymptotic property, as well as that of its Schur complement. The analysis is motivated by numerical methods for plasma physics, in particular Galerkin spectral methods applied to the Vlasov-Poisson (VP) system. As an application, we demonstrate how the derived Gram matrix formulas and asymptotic results can be exploited in the analysis and implementation of a Galerkin spectral method for the VP system. 32-point DFT Approximations Based on Minimal Frobenius Error and DFT Symmetries 作者: L. Andrade-Silva, W. A. S. Aleixo, R. J. Cintra 类别: eess.SP, math.NA 发布日期: 2026-09-01 arXiv ID: 2609.01115v1 链接: http://arxiv.org/abs/2609.01115v1 摘要: This work introduces low-complexity, multiplierless approximations for the 32-point discrete Fourier transform. The proposed methods are obtained by minimizing the Frobenius error compared against the DFT matrix over a set of trivial multipliers. A row-wise, symmetry-constrained parameterization is employed to reduce the search space size, rendering the task computationally tractable. The resulting approximations could outperform the reference method in the literature according to energy-based error measurements. A sparse matrix factorization is provided for efficient computation; the arithmetic costs are 152 real additions and 34 bit-shifts only. Symmetry Inheritance and Symmetry-Reduced Finite Element Analysis for Second-Order Linear Elliptic Problems with Robin Boundary Conditions 作者: Wei Jiang, Xianlong Pan 类别: math.NA 发布日期: 2026-09-01 arXiv ID: 2609.01107v1 链接: http://arxiv.org/abs/2609.01107v1 摘要: A symmetry framework is developed for second-order linear elliptic equations subject to Robin boundary conditions on bounded domains. Orthogonal transformations of the domain are represented through left group actions on scalar, vector, and second-order tensor fields. The corresponding transformation rules for the gradient, divergence, diffusion flux, and conormal boundary term are derived in detail. Based on these relations, symmetry groups are introduced for the principal coefficient tensor, the first-order and zeroth-order coefficients, the differential operator, the Robin coefficient, and the boundary operator. The symmetry properties of the volume and boundary source terms are then incorporated into a common symmetry group for the complete boundary value problem. Under the assumption of unique solvability, it is proved that every element of this common group is also a symmetry of the solution. For reflection symmetries, homogeneous generalized Neumann conditions are obtained on artificial symmetry boundaries, leading to an exact reduction of the computational domain. A corresponding finite element formulation is presented, and numerical results are provided to verify the theoretical symmetry properties and the validity of the resulting domain-reduction strategy. Three examples illustrate radial reduction from multiple dimensions to one dimension, reflection-based domain reduction, and a variable-coefficient problem in which all coefficients and source terms are nonzero. Randomized inexact block triangular preconditioners for double saddle-point systems in PDE-constrained optimization 作者: Siqi Liang, Na Huang 类别: math.NA 发布日期: 2026-09-01 arXiv ID: 2609.01031v1 链接: http://arxiv.org/abs/2609.01031v1 摘要: We develop a new class of inexact block triangular preconditioners for double saddle-point systems arising from PDE-constrained optimization. The proposed preconditioners are constructed through matrix factorization techniques while preserving the inherent block structure of the original systems. A comprehensive spectral analysis of the preconditioned matrices is provided, yielding explicit bounds for both real and nonreal eigenvalues. To enable efficient construction of the inexact preconditioners, randomized strategies are introduced to select the required subblocks. We establish high-probability bounds for the expected approximation error, with the error estimates explicitly characterized in terms of the eigenvalues of the associated matrices. Numerical experiments demonstrate the effectiveness, robustness, and scalability of the proposed preconditioners, and validate the efficiency of the randomized construction strategies. OptiXDE: A fast optical-inspired solver for differential equations 作者: Yang Yang, Mingjiao Yan, Zongliang Zhang 类别: math.NA 发布日期: 2026-09-01 arXiv ID: 2609.01009v1 链接: http://arxiv.org/abs/2609.01009v1 摘要: OptiXDE is a matrix-free spectral operator framework for differential equations on uniform grids and embedded domains. Inspired by angular-spectrum propagation in Fourier optics, it maps transform-diagonal spatial operators to analytical modal multipliers and composes them with physical-space operators for nonlinearities, geometry and boundary enforcement. A common transform–operator–inverse-transform backbone is demonstrated across transient diffusion, periodic and embedded-domain Poisson problems, the cubic nonlinear Schr"odinger equation, viscous Burgers dynamics, the two-dimensional Allen–Cahn equation and incompressible flows from the Taylor–Green vortex to embedded-cylinder vortex shedding. Transform-compatible linear problems are recovered near the floating-point limit, whereas errors on the singular L-shaped domain remain localized near the re-entrant corner and regularized interface. Nonlinear benchmarks recover second-order temporal convergence and the expected conservative or dissipative behavior, while incompressibility remains near round-off level during long-time vortex shedding. The matrix-free updates require (\mathcal{O}(N\log N)) work and (\mathcal{O}(N)) memory. Device-resident transform workloads reach (94.9\times) GPU acceleration, and the complete embedded-cylinder solver achieves a (42.1\times) CPU–GPU speedup under matched numerical settings. These results establish OptiXDE as a deterministic and extensible operator-centric framework for structured and embedded-domain differential equations. Asymptotic behavior of Eckhoff’s method for convergence acceleration of Dirac eigenfunction expansions 作者: R. H. Barkhudaryan, G. G. Gevorkyan, L. D. Poghosyan 类别: math.NA 发布日期: 2026-09-01 arXiv ID: 2609.00895v1 链接: http://arxiv.org/abs/2609.00895v1 摘要: The current paper considers the problem of recovering a vector-function on $[-1,1]$ from a limited number of coefficients of its expansion into a series of eigenfunctions of a one-dimensional Dirac system. The Krylov–Lanczos–Eckhoff–Gottlieb acceleration method is examined in the situation when the boundary values it requires have to be computed from the generalized Fourier coefficients themselves. This leads to a $2q\times 2q$ linear system whose matrix is a block Vandermonde matrix; its determinant and inverse are computed explicitly, and the asymptotic $L_2$-error constant of the method is found, paralleling the classical trigonometric case. DOFFO_TR: a Decentralized Objective Function-Free Optimization method with Trust-Region 作者: Stefania Bellavia, Greta Malaspina, Benedetta Morini 类别: math.OC, math.NA 发布日期: 2026-09-01 arXiv ID: 2609.00878v1 链接: http://arxiv.org/abs/2609.00878v1 摘要: In this paper, we propose a novel objective function-free trust-region method designed to solve optimization problems over decentralized networks. Unlike traditional approaches that often rely on stepsize tuning, our framework employs a function-free trust-region procedure that enables adaptive selection of the step length. Our approach accommodates first- and second-order models and eliminates the need to share local function values and gradients among agents, thereby enhancing privacy and computational efficiency. On the theoretical side, we establish provable iteration complexity guarantees that, for some variants, match those established for classical centralized trust-region methods. Numerical evaluations demonstrate that our approach achieves a favorable trade-off between performance and efficiency, requiring only moderate communication overhead compared to state-of-the-art methods in the literature. Sharp Mixed Spectral Barron Regularity of Coulombic Many-Electron Wave Functions 作者: Pingbing Ming, Hao Yu 类别: math.AP, cs.LG, math.NA 发布日期: 2026-09-01 arXiv ID: 2609.00872v1 链接: http://arxiv.org/abs/2609.00872v1 摘要: We establish sharp mixed spectral Barron regularity for eigenfunctions of molecular Coulomb Hamiltonians. The mixed norm is a Fourier $L^1$ norm with one isotropic weight and coordinate-product weights, and therefore detects regularity invisible to the isotropic Barron scale. For a nonempty set $I$ of electron indices on which the wave function is antisymmetric, we derive an explicit admissible region for the isotropic order $s$ and the coordinate orders $α,β$. This region is optimal as a uniform statement over the class of clamped-nuclei Coulomb Hamiltonians. For fixed-spin components with two occupied spin blocks, it reduces to $s+α+β<1$; in the fully spin-polarized class it reduces to $s+α<1$. In particular, if $\mathcal I_σ$ denotes the family of occupied same-spin blocks determined by $σ$, then every fixed-spin spatial component $ψ_σ$ satisfies, for every $0\leqα<1$, [ \left(\sum_{I\in\mathcal I_σ}\prod_{i\in I}\langleξ_i\rangle^α\right)\widehat{ψ_σ}\in L^1(\mathbb{R}^{3N}). ] For a fully spin-polarized state, $\mathcal I_σ=\{\{1,\ldots,N\}\}$. Assessment of Numerical Lift Coefficient Data for a Circular Cylinder with Application to Bladeless Turbines 作者: L. Ridgway Scott, Henry von Wahl 类别: physics.flu-dyn, math.NA, physics.comp-ph 发布日期: 2026-09-01 arXiv ID: 2609.00850v1 链接: http://arxiv.org/abs/2609.00850v1 摘要: We assess computed lift coefficient data for flow past a circular cylinder to evaluate their suitability for practical applications. Specifically, we consider lift coefficient data for a circular cylinder over Reynolds numbers from 120 to 8000. The results are obtained from two-dimensional finite element simulations of the incompressible Navier-Stokes equations using pressure robust discretizations. We compare the computed lift coefficients with published experimental and numerical results, finding good agreement in some cases but significant disagreement in others. Because lift fluctuations are central to vortex-induced vibration concepts, these data therefore provide input for the analysis and preliminary design of bladeless turbines. Fast computation and convergence analysis of the infinite-product representation of the Schottky–Klein prime function 作者: Shuntaro Yamamoto, Hiroyuki Miyoshi 类别: math.NA 发布日期: 2026-09-01 arXiv ID: 2609.00785v1 链接: http://arxiv.org/abs/2609.00785v1 摘要: The Schottky–Klein prime function is a standard tool for boundary-value problems on multiply connected circular domains. Because this function is represented as an infinite product over a Schottky group, numerical evaluation requires truncation to finitely many factors. The standard word-length truncation grows exponentially in cost and becomes inefficient when the boundary circles nearly touch one another or the unit circle. To address this difficulty, we assign to each group element a cross-ratio potential measuring the size of its contribution, and retain only terms below a prescribed threshold. We establish uniform closed-form bounds on the change in this potential when prepending Schottky-group generators, and from these bounds we derive an efficient enumeration algorithm. The resulting relative error decays exponentially with the threshold at a rate determined by the Hausdorff dimension of the limit set of the Schottky group. Numerical experiments demonstrate that the proposed formulation achieves substantial computational speedups over word-length truncation in challenging geometric configurations. FTU-Seek: Foundation Model-Guided Hard-Negative Learning for Sparse Functional Tissue Unit Segmentation 作者: Zonghao Liu, Lei Su, Jiguang Yu, Xuqing Geng, Louis Shuo Wang, Jianmin Wang, Jingfeng Liu 类别: cs.CV, math.NA 发布日期: 2026-09-01 arXiv ID: 2609.00704v1 链接: http://arxiv.org/abs/2609.00704v1 摘要: Functional tissue units (FTUs), including tertiary lymphoid structures (TLSs), blood vessels, and glands, encode localized immune, vascular, and epithelial organization in histopathology. Accurate quantification of these structures is important for studying tissue architecture and disease-associated tissue organization. However, FTUs are frequently sparse, heterogeneous, and surrounded by large amounts of morphologically similar background tissue, making automated segmentation in whole-slide images (WSIs) challenging. We therefore developed FTU-Seek, a pathology foundation model-guided framework that treats morphology-aware negative-patch selection as a key component of sparse FTU segmentation. FTU-Seek uses frozen multi-depth features from the UNI pathology foundation model to train a patch-level classifier that distinguishes FTU-containing from FTU-absent tissue. Target-absent patches are subsequently ranked according to their predicted target-containing probabilities, and the highest-scoring hard negatives are selected through a static Top$K$ strategy to construct compact segmentation training sets. The framework was evaluated using five-fold cross-validation and internal test cohorts across TLS, blood-vessel, and gland segmentation tasks, with an additional independent 30-WSI held-out cohort for TLS. Positive-only, all-tissue, random-negative, and matched random Top$K$ sampling strategies served as comparators. Segmentation-derived phenotypes were further explored in external TCGA cohorts. Comparison of Algebraic Block Multi-Coloring and Leiden Methods for Parallel Preconditioning in the ICCG Method 作者: Tomohiro Suzuki 类别: cs.DC, math.NA 发布日期: 2026-09-01 arXiv ID: 2609.00561v1 链接: http://arxiv.org/abs/2609.00561v1 摘要: In the application of incomplete Cholesky preconditioning to the incomplete Cholesky-conjugate gradient (ICCG) method, forward and backward substitutions exhibit sequential dependencies that constitute a major bottleneck for parallelization in multicore environments. To alleviate this bottleneck, the algebraic block multi-coloring (ABMC) method achieves both parallelism and data locality through block-wise coloring. However, ABMC requires the number of blocks to be specified as an input parameter in advance. This study evaluates the Leiden method as an alternative blocking approach for parallel preconditioning in the ICCG method. As a community detection technique that maximizes a quality function for graph partitioning, the Leiden method automatically generates blocks that reflect the matrix structure without requiring the number of blocks a priori. We partition the adjacency graphs of sparse matrices using the Leiden method and utilize the resulting blocks for parallel preconditioning. We implement the Leiden method using modularity and the constant Potts model as quality functions and compare its performance with that of the ABMC method in terms of the number of iterations, execution time, and L2 cache efficiency across eight symmetric positive definite matrices. The experimental results demonstrate that the Leiden method with the constant Potts model achieves performance comparable to that of the ABMC method configured with an optimized number of blocks. Spectral Bounds for Kernel Quadrature 作者: A. Cloninger, Q. T. Le Gia, H. N. Mhaskar 类别: math.NA 发布日期: 2026-09-01 arXiv ID: 2609.00553v1 链接: http://arxiv.org/abs/2609.00553v1 摘要: A bottleneck in the theory of kernel methods in machine learning is the storage requirement. To ameliorate this, a standard trick is to replace the kernel with an explicit feature map. Perhaps, the most well known example is the Gaussian kernel which can be expressed in terms of the Fourier features. Analytically, the kernel $K$ can be expressed in terms of an integral expression that involves a possibly asymmetric kernel $G$ representing the feature map. Numerically, one needs to approximate this integral by a suitable numerical integration scheme, typically Monte Carlo. In this paper, we demonstrate that the eigenvalues of $K$ are approximated much better by the eigenvalues of the kernel obtained by discretizing the integral using suitable quadrature formulas instead. We illustrate this fact in the case of the Gaussian kernel and neural tangent kernels on the unit sphere of a four-dimensional Euclidean space corresponding to the sigmoid and ReLU activation functions. Single- and Multilevel Quadrature with Error Control for Fourier Pricing under the Rough Heston Model 作者: Chiheb Ben Hammouda, Abderrahmene Ben Romdhane, Michael Samet, Raul F. Tempone 类别: q-fin.CP, math.NA 发布日期: 2026-08-31 arXiv ID: 2609.00438v1 链接: http://arxiv.org/abs/2609.00438v1 摘要: Unlike the classical Heston model, Fourier pricing under the rough Heston model requires solving a fractional Riccati equation at every quadrature point. Since the required resolution varies with model parameters and quadrature point, a single uniform time discretization can be inefficient. We develop single- and multilevel Gauss-Laguerre quadrature methods that balance the time discretization and Fourier quadrature errors. Both methods scale the laguerre weight to the estimated Fourier integrand decay. The single-level method allocates a prescribed tolerance between the two errors. The multilevel method splits the integrand into a level-zero term and level differences, selecting quadrature points separately at each level. Suppose that the Fourier integrand discretization error is $O(Δt^p)$, that evaluating the characteristic function once costs $O(Δt^{-β})$, and that the algebraic Gauss-Laguerre quadrature error is $O(N^{-s_{SL}/2})$, where $s_{SL}$ is the smoothness index. Under this estimate and assumptions on the regularity and decay of level differences, we prove that the proposed single-level method requires $O(ε^{-(β/p+2/s_{SL})})$ computational work to achieve accuracy $ε$, whereas the proposed multilevel method requires $O(ε^{-β/p})$ computational work. We also study root-exponential Gauss-Laguerre error models for practical multilevel quadrature allocation. Numerical experiments support the observed fractional Riccati and Fourier integrand convergence rates and root-exponential quadrature behavior, and show substantial reductions in quadrature cost from the proposed scaling. The multilevel method provides clear computational savings over the single-level method. We further benchmark the multilevel fractional Riccati method against the BL2 Markovian approximation and report lower total CPU time in the tested configurations. Latent-Space No-Arbitrage Geometry of Generative Models for Implied Volatility Surfaces 作者: Jing Wang, Shuaiqiang Liu, Cornelis Vuik 类别: q-fin.CP, cs.AI, cs.LG, math.NA 发布日期: 2026-08-31 arXiv ID: 2609.00332v1 链接: http://arxiv.org/abs/2609.00332v1 摘要: Generative models for implied volatility surfaces must produce outputs that satisfy static no-arbitrage constraints. We study these constraints in latent space. For a fixed generator, we assign each latent code a scalar margin determined by the no-arbitrage conditions of the generated surface. The codes with nonnegative margin form the admissible latent set. We establish conditions under which strictly admissible codes remain admissible under small perturbations and the boundary of the admissible set is characterized by zero margin. For regular boundary components, we formulate a level-set equation whose local dynamics are directed toward the zero-margin set. The analysis treats the generator as a map from latent variables to surfaces and is therefore not restricted to a particular architecture. It applies to variational autoencoders, generative adversarial networks, and other generative models with a deterministic realization map. Numerical tests recover known boundaries in analytic examples. Experiments with a variational autoencoder trained on Heston surfaces show that similar reconstruction errors can correspond to different admissible regions and that the latent prior may be concentrated inside such a region. The computed boundary can also be used to modify latent codes that generate violating surfaces. Modeling of Mobility and Energy Policies in an Agent-Based Framework: Case Studies for Chicago Region in 2050 作者: Md Rakibul Alam, Omer Verbas, Taner Cokyasar, Hadi Bhidya, Jesse Altman, Nora Beck, Joshua Auld, Pedro Veiga de Camargo, Jamie Cook, Felipe de Souza, Gopindra Nair, Hyunseop Uhm, Jan Zill 类别: physics.soc-ph, math.NA 发布日期: 2026-08-31 arXiv ID: 2609.00327v1 链接: http://arxiv.org/abs/2609.00327v1 摘要: Metropolitan regions are simultaneously pursuing several interventions to improve mobility, accessibility, and energy efficiency, necessitating integrated tools to understand how these policies interact to affect travel behavior, energy use, and infrastructure needs. This paper evaluates the combined impacts of electrification, freight demand management, road pricing, parking reform, and transit expansion on the Chicago metropolitan transportation system in 2050, using a business-as-usual (BAU) scenario as the baseline. We employ POLARIS, a large-scale agent-based modeling framework calibrated to 2019 conditions, to simulate nine policy scenarios for the seven-county northeastern Illinois region. The framework co-simulates activity-based passenger demand, endogenous freight generation, multimodal traffic assignment, and transit operations, with charging infrastructure and freight operations optimized for each case. Our findings reveal that under the high electrification scenario, total fuel mass declines by 68% while total charging energy increases by approximately 4-8x from BAU, resulting in a peak power demand near 4 GW concentrated in the urban core. Furthermore, freight management policies reduce freight VMT by increasing trip frequency but shortening distances, smart road pricing most effectively reduces auto VMT, and transit expansion boosts ridership by 18% relative to BAU. By presenting the first integrated, agent-based scenario framework for Chicago that jointly evaluates these interventions, this study provides actionable insights for regional transportation planning, grid infrastructure investment, and emissions reduction, highlighting the value of targeted charger upgrades and coordinated policy bundles.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
了解领域论文（摘要翻译）
1. 点云离散的定向曲面上不可压缩Navier-Stokes方程的求解
摘要翻译
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