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1. 时域中理想导体电磁散射的区域导数
A Domain Derivative for Electromagnetic Scattering by Perfect Conductors in the Time Domain
作者: Marvin Knöller
类别: math.NA
发布日期: 2026-09-21
arXiv ID: 2609.24822v1
摘要翻译: 建立了理想导体时变电磁散射的区域导数。通过 Laplace 域，推导了 Maxwell 方程解的频率相关界。这些界既用于建立区域导数的时间正则性性质，也用于证明所提出的时间上 Runge–Kutta 卷积求积半离散化的收敛性。当该时间离散与空间上的 Galerkin 方法相结合时，还对区域导数的逐点求值进行了完整的收敛性分析。最后，区域导数被应用于一个迭代形状重构算法，其中测量了远离理想导体的若干接收器位置处的电场近场。数值例子表明了该算法的可行性，并特别突出了当对数据施加额外噪声时该算法的鲁棒性。
Comment: 无
链接: http://arxiv.org/abs/2609.24822v1
1. 研究对象和背景
本文研究三维自由空间中完美导体障碍对时变电磁波的散射, 并建立相应的时域区域导数. 设 $D\subset\mathbb{R}^3$ 为有界 $C^2$ 区域, 其边界为 $\Gamma=\partial D$, 外域为 $\Omega=\mathbb{R}^3\setminus\overline{D}$, 外法向为 $\nu$. 时域 Maxwell 方程描述总电场 $E$ 与总磁场 $H$ 在完美导体外部的传播:
$$\varepsilon_0\partial_t E-\operatorname{curl}H=0,\quad \mu_0\partial_t H+\operatorname{curl}E=0\quad \text{in }\Omega\times(0,\infty),$$$$\nu\times E=0\quad \text{on }\Gamma\times(0,\infty).$$这里 $\varepsilon_0$ 与 $\mu_0$ 分别为真空介电常数与磁导率. 入射场 $(E^i,H^i)$ 在初始时刻 $t=0$ 的支撑远离 $D$, 随着时间推进传播到障碍物, 在 $\Gamma$ 上诱导电流并产生散射场 $(E^s,H^s)$. 总场满足 $(E,H)=(E^i,H^i)+(E^s,H^s)$. 散射场本身满足
$$\varepsilon_0\partial_t E^s-\operatorname{curl}H^s=0,\quad \mu_0\partial_t H^s+\operatorname{curl}E^s=0\quad \text{in }\Omega\times(0,\infty),$$$$\nu\times E^s=-\nu\times E^i\quad \text{on }\Gamma\times(0,\infty),$$也可以化为散射电场单独满足的二阶方程
$$c^{-2}\partial_t^2 E^s+\operatorname{curl}^2 E^s=0\quad \text{in }\Omega\times(0,\infty),$$$$\nu\times E^s=-\nu\times E^i\quad \text{on }\Gamma\times(0,\infty),$$其中 $c=(\varepsilon_0\mu_0)^{-1/2}$ 为光速, 通过时间尺度变换可设 $c=1$. 本文关注的核心对象是上述散射问题关于障碍物几何形状的区域导数. 令 $h\in C^1(\Gamma,\mathbb{R}^3)$ 为边界变形向量场, 将其延拓为 $h\in C_0^1(D_0,\mathbb{R}^3)$, 定义微分同胚 $\zeta(x)=x+h(x)$, 变形后的障碍物为 $D_h=\zeta(D)$, 其边界为 $\Gamma_h=\zeta(\Gamma)$. 给定远离 $D_h$ 的接收点 $z_1,\dots,z_M$, 定义算子 $\hat{F}_\Gamma(h)=\hat{X}(\Gamma_h)=(\hat{E}^s(z_j))_{j=1}^M\in\mathbb{C}^{3M}$, 其中 $\hat{E}^s$ 是 Laplace 域中的散射电场. 区域导数问题就是求 Fréchet 导数 $\hat{F}_\Gamma'(0):C^1(\Gamma,\mathbb{R}^3)\to\mathbb{C}^{3M}$, 使得
$$\frac{1}{\|h\|_{C^1(\Gamma)}}\bigl|\hat{F}_\Gamma(h)-\hat{F}_\Gamma(0)-\hat{F}_\Gamma'(0)h\bigr|\to 0\quad \text{as }\|h\|_{C^1(\Gamma)}\to 0.$$在 Laplace 域中, 散射电场 $\hat{E}^s=\mathcal{L}\{E^s\}$ 满足
$$\operatorname{curl}^2\hat{E}^s+s^2\hat{E}^s=0\quad \text{in }\Omega,$$$$\nu\times\hat{E}^s=-\nu\times\hat{E}^i\quad \text{on }\Gamma.$$该问题之所以重要, 是因为许多真实电磁源是宽频或多色光, 其时域数据拟合和形状优化不能简单归结为单一频率的时谐问题. 在时谐情形, 电磁散射的区域导数已有大量研究, 例如完美导体的频域形状导数见 [21,23,37], 可穿透介质见 [24,16], 高阶形状导数见 [7], 声软障碍物的早期工作见 [27]. 在时域情形, 完美导体散射问题通过 Laplace 域已有适定性分析, 见 [3,6,31], 但区域导数的严格推导、时间正则性以及数值离散误差分析仍不完整. 已有的时域 Maxwell 形状导数工作 [12,13,45] 处理的是有界区域上的内问题, 而不是本文的自由空间散射外问题. 时域形状优化在完美导体上的数值实现可见 [39,41], 但它们分别使用时间域间断 Galerkin 方法和时间域边界元方法, 并未建立本文所考虑的 Runge–Kutta 卷积求积半离散与全离散的逐点误差理论. 近年来卷积求积也被用于逆散射 [17,42,43,44], 其思路通常是把时域数据拟合转化为一组 Laplace 域问题, 再在 Laplace 域计算形状导数; 本文则直接建立时域区域导数, 并对其时间离散和空间离散进行收敛分析. 因此, 本文试图补上的空白是: 在完美导体时域电磁散射外问题中, 建立可计算的区域导数, 证明其时间正则性, 给出 Runge–Kutta 卷积求积时间半离散与 Galerkin 空间全离散的收敛结果, 并将该导数用于带噪近场数据的迭代形状重构.
2. 主要定理或主要结果
在提供的内容中, 可核实的定量结果主要集中在 Laplace 域频域界限、时域映射性质以及区域导数的一阶扰动估计. 完整的时间域区域导数定理、Runge–Kutta 卷积求积误差阶和全离散误差阶的定理编号与精确常数在提供的 PDF 文本中未出现, 因此下面只列出已有明确公式的结论, 并明确区分摘要中声称但细节未展开的部分.
- 时域散射解的 Laplace 域表示与时间映射. 对 Laplace 域算子 $S(s)V^{-1}(s)$ 有估计
由此得到时域散射电场
$$E^s=-(S V^{-1})(\partial_t)\gamma_t E^i,$$并且对任意 $r\in\mathbb{R}$ 有有界映射
$$(S V^{-1})(\partial_t):H^{r+2}_0(0,T;H^{-1/2}(\operatorname{Div},\Gamma))\to H^r_0(0,T;H(\operatorname{curl},\Omega)).$$这是把频域显式 $s$ 界限转化为时域正则性的关键结论.
- 点值估计. 对固定 $x\in\Omega$, 令 $d_x=\operatorname{dist}(x,\Gamma)$. 单层势与双层势的点值算子满足
常数 $C(\sigma,d_x)$ 依赖 $\sigma^{-1}$ 与 $d_x^{-1}$, 但不依赖 $s$. 该指数衰减因子 $e^{-d_x\operatorname{Re}s}$ 对远离导体的接收点处逐点评估区域导数至关重要.
- 频域正则性. 设 $D$ 为 $C^2$ 区域, $\Omega=\mathbb{R}^3\setminus\overline{D}$, $O\in\{D,\Omega\}$, $\hat{g}\in H^{1/2}_t(\Gamma)$. 则边值问题
的任意弱解 $\hat{v}\in H(\operatorname{curl},O)$ 满足 $\hat{v}\in H^1(O)^3$ 且
$$\|\hat{v}\|_{H^1(O)^3}\le C\frac{|s|^2}{\operatorname{Re}s}\|\hat{g}\|_{H^{1/2}(\Gamma)^3}.$$进一步, 若 $\hat{g}\in H^{3/2}_t(\Gamma)$, 则 $\operatorname{curl}\hat{v}\in H^1(O)^3$ 且
$$\|\operatorname{curl}\hat{v}\|_{H^1(O)^3}\le C\frac{|s|^3}{\operatorname{Re}s}\|\hat{g}\|_{H^{3/2}(\Gamma)^3}.$$相应的单层算子逆满足
$$V^{-1}(s):H^{3/2}_t(\Gamma)\to H^{1/2}_t(\Gamma),\quad \|V^{-1}(s)\|_{H^{1/2}(\Gamma)^3\leftarrow H^{3/2}(\Gamma)^3}\le C\frac{|s|^2}{\operatorname{Re}s}.$$更一般地, 若 $\Gamma$ 是 $C^{m+1}$ 区域边界, $m\in\mathbb{N}$, 则
$$V^{-1}(s):H^{m+1/2}_t(\Gamma)\to H^{m-1/2}_t(\Gamma),\quad \|V^{-1}(s)\|_{H^{m-1/2}(\Gamma)^3\leftarrow H^{m+1/2}(\Gamma)^3}\le C\frac{|s|^2}{\operatorname{Re}s}.$$- 区域导数的一阶扰动估计. 对边界变形 $h\in C^1(\Gamma,\mathbb{R}^3)$, 通过变量变换回到未扰动域后, 有
该估计给出了区域导数关于边界扰动 $h$ 的线性化误差控制, 是证明 $\hat{F}_\Gamma$ 在 $0$ 处 Fréchet 可微的基础.
- 摘要中声称的结果. 作者在摘要中声称建立了完美导体时域电磁散射的区域导数, 推导了 Maxwell 方程解的频域界限, 并用这些界限建立了区域导数的时间正则性以及 Runge–Kutta 卷积求积时间半离散的收敛性. 摘要还声称对区域导数的逐点求值进行了完整收敛分析, 其中时间离散为 Runge–Kutta 卷积求积, 空间离散为 Galerkin 方法. 最后, 区域导数被用于迭代形状重构算法, 测量远离完美导体的若干接收点处的电场近场, 数值例子显示了算法可行性以及在附加噪声下的鲁棒性. 不过, 具体的收敛阶、时间步长与空间网格的耦合关系、Runge–Kutta 方法的阶数假设、全离散误差中的常数依赖以及逆问题中的正则化参数选择, 从提供的 PDF 文本中无法确认.
3. 研究方法、关键技术和核心工具
本文的技术路线不是直接在时域对边界变形做形状微分, 而是先进入 Laplace 域, 在频域建立显式依赖于 $s$ 的界限, 再通过 Laplace 域卷积算子与时间域映射定理回到时域. 这一选择的核心障碍在于: 时域 Maxwell 解是时间卷积型算子作用在边界数据上的结果, 边界变形会同时改变算子核与积分曲面, 直接估计会遇到时间正则性与空间迹正则性的耦合. Laplace 域把时间导数变为乘以 $s$, 把时间卷积变为算子乘积, 从而使频域边值问题可以逐点、逐阶地控制.
第一项关键工具是 [32, Lem. 1] 给出的 Laplace 域到时间域的映射定理. 若解析算子族 $K(s)$ 满足多项式有界性
$$\|K(s)\|_{Y\leftarrow X}\le M_\sigma\frac{|s|^\kappa}{(\operatorname{Re}s)^\nu},\quad \operatorname{Re}s\ge\sigma>0,$$则卷积型算子 $K(\partial_t)g=\mathcal{L}^{-1}\{K\}*g$ 可延拓为
$$K(\partial_t):H^{r+\kappa}_0(0,T;X)\to H^r_0(0,T;Y),\quad \forall r\in\mathbb{R}.$$本文正是通过证明单层势、双层势、单层算子逆 $V^{-1}(s)$ 以及区域导数算子的频域界限具有上述形式, 来获得时域正则性和离散误差估计. 因此, 频域界限中的 $|s|^\kappa/(\operatorname{Re}s)^\nu$ 结构不是技术装饰, 而是后续时间映射阶数的直接来源.
第二项关键工具是频域 Maxwell 解的高阶正则性. 作者没有只使用标准的 $H(\operatorname{curl})$ 能量估计, 而是通过紧嵌入 $X_N(D)\hookrightarrow H^1(D)^3$ 和外域截断分解, 证明弱解 $\hat{v}$ 属于 $H^1(O)^3$, 并得到 $\|\hat{v}\|_{H^1(O)^3}\le C|s|^2/(\operatorname{Re}s)\|\hat{g}\|_{H^{1/2}(\Gamma)^3}$. 进一步, 利用 $\nu\cdot\operatorname{curl}\hat{v}=\operatorname{Div}(\nu\times\hat{v})=\operatorname{Div}\hat{g}$ 以及另一个紧嵌入 $X_T(D)\hookrightarrow H^1(D)^3$, 作者在 $\hat{g}\in H^{3/2}_t(\Gamma)$ 时得到 $\operatorname{curl}\hat{v}\in H^1(O)^3$ 与 $\|\operatorname{curl}\hat{v}\|_{H^1(O)^3}\le C|s|^3/(\operatorname{Re}s)\|\hat{g}\|_{H^{3/2}(\Gamma)^3}$. 这些估计既用于边界积分算子的可逆性与范数控制, 也用于区域导数中 Jacobian 变换后的误差项. 作者还指出, 若边界更光滑, 则 $V^{-1}(s):H^{m+1/2}_t(\Gamma)\to H^{m-1/2}_t(\Gamma)$ 且范数仍为 $C|s|^2/(\operatorname{Re}s)$, 这为更高阶空间离散或高阶形状导数预留了正则性框架.
第三项关键工具是点值势估计. 对固定 $x\in\Omega$, 作者直接估计基本解及其一阶、二阶、三阶导数, 得到 $|\partial_{x_i}\partial_{x_j}\Phi_s(x,y)|\le C(\sigma,d_x)|s|^2e^{-d_x\operatorname{Re}s}$ 以及三阶导数对应的 $|s|^3e^{-d_x\operatorname{Re}s}$ 界. 由此推出单层势和双层势的点值算子在 $H^{-1/2}(\operatorname{Div},\Gamma)$ 上有界, 且具有 $|s|^2e^{-d_x\operatorname{Re}s}$ 的衰减. 这一指数衰减因子使得接收点远离导体时, 逐点评估区域导数的 Laplace 域表达式具有良好可积性, 并允许在时间域中通过逆 Laplace 变换获得点值误差估计. 若没有这个估计, 逐点评估区域导数会受到边界近场奇性的严重干扰.
第四项关键工具是边界积分表示与 Stratton–Chu 公式. 作者引入 Maxwell 单层势 $S(s)$, 双层势 $D(s)$ 和单层算子 $V(s)=\gamma_t S(s)$, 并利用 $V(s)$ 的可逆性得到散射电场的表示. 通过 Stratton–Chu 公式
$$a_O\bigl(-(D(s)\gamma_t\hat{v})(x)+\frac{1}{s}(S(s)\gamma_t\operatorname{curl}\hat{v})(x)\bigr)=\begin{cases}\hat{v}(x),&x\in O,\\0,&x\notin O,\end{cases}$$其中 $a_D=1$, $a_\Omega=-1$, 作者推出 $\hat{E}^s(x)=-\frac{1}{s}(S(s)\gamma_t\operatorname{curl}\hat{E})(x)$ 以及 $\frac{1}{s}\gamma_t\operatorname{curl}\hat{E}=V^{-1}(s)\gamma_t\hat{E}^i$. 这些表示把散射场写成边界积分算子作用于入射迹的形式, 为区域导数提供了可以对 $\Gamma$ 和 $h$ 做微分的显式结构.
第五项关键工具是区域导数的弱形式与变量变换. 为了比较未扰动域 $\Omega$ 与扰动域 $\Omega_h$ 上的解, 作者引入截断函数 $\chi$, 使得 $\hat{E}^s+\chi\hat{E}^i$ 与 $(\hat{E}^s_h)^*+\chi\hat{E}^i$ 在相应边界上具有消失的切向迹, 从而可以放在 $H_0(\operatorname{curl},\Omega)$ 中比较. 弱形式为
$$a(\hat{z},\hat{\varphi})=\int_\Omega \operatorname{curl}\hat{z}\cdot\operatorname{curl}\hat{\varphi}+s^2\hat{z}\cdot\hat{\varphi}\,dx,$$$$f(\hat{\varphi})=\int_\Omega \operatorname{curl}(\hat{E}^i\chi)\cdot\operatorname{curl}\hat{\varphi}+s^2(\hat{E}^i\chi)\cdot\hat{\varphi}\,dx =-\int_{D_1\setminus D_0}\nabla\chi\cdot(\operatorname{curl}\hat{\varphi}\times\hat{E}^i+\hat{\varphi}\times\operatorname{curl}\hat{E}^i)\,dx.$$随后通过 $\zeta(x)=x+h(x)$ 的 Jacobian $J_\zeta$ 把扰动域上的弱形式拉回参考域, 得到双线性形式
$$a_h(\hat{z}_h^*,\hat{\varphi})=\int_\Omega \frac{1}{\det(J_\zeta)}\operatorname{curl}(\hat{z}_h^*)^\top J_\zeta^\top J_\zeta\operatorname{curl}\hat{\varphi}+s^2(\hat{z}_h^*)^\top J_\zeta^{-1}J_\zeta^{-\top}\hat{\varphi}\det(J_\zeta)\,dx.$$在证明中, 作者用 Riesz 表示定理引入有界可逆算子 $T,T_h:H_0(\operatorname{curl},\Omega)\to H_0(\operatorname{curl},\Omega)$, 满足 $\langle T\hat{u},\hat{\varphi}\rangle_{|s|,\Omega}=a(\hat{u},\hat{\varphi})$ 与 $\langle T_h\hat{u},\hat{\varphi}\rangle_{|s|,\Omega}=a_h(\hat{u},\hat{\varphi})$. 通过比较 $a$ 与 $a_h$ 中 Jacobian 因子的扰动, 并控制 $\det(J_\zeta)$ 与 $J_\zeta$ 在 $C^1$ 范数下的偏差, 最终得到 Lemma 3.1 中的 $|s|^2/(\operatorname{Re}s)^2$ 估计. 这是全文区域导数存在性与误差分析的核心技术步骤.
第六项是时间离散与空间离散. 作者采用 Runge–Kutta 卷积求积作为时间半离散, 并计划用 Laplace 域界限和卷积求积的稳定性与收敛理论证明误差估计. 空间离散采用 Galerkin 方法, 对区域导数的逐点求值进行全离散误差分析. 由于提供的 PDF 文本只到第 10 页, Runge–Kutta 卷积求积的具体阶数假设、卷积权重的构造、半离散与全离散误差中的时间步长 $\Delta t$ 和网格尺寸 $h$ 的耦合关系, 以及空间 Galerkin 的离散空间选择, 从提供内容中无法确认. 最后, 作者把全离散区域导数用于迭代形状重构: 在若干接收点测量散射电场近场, 通过最小化数据残差更新障碍物边界, 数值例子表明算法在附加噪声下仍具有鲁棒性.
4. 与之前工作的比较
与已有工作相比, 本文的推进可以从以下几个维度理解.
第一, 与频域电磁形状导数相比. 时谐电磁散射的形状导数已有成熟理论, 引言中列举了 [15,28,35,36] 等时谐分析, [27] 的声软障碍物区域导数, [21,23] 的完美导体频域区域导数与高阶形状导数, [7] 的高阶形状导数系统研究, [37] 的通过边界积分方程的完美导体频域形状导数, [16] 的可穿透介质频域形状导数, 以及 [25] 的微分形式方法. 这些工作的共同点是固定频率, 方程是 Helmholtz 型或时谐 Maxwell 型, 时间变量被频率参数取代. 本文处理的是时域多频散射, 通过 Laplace 变换把频率 $s$ 作为复参数引入, 并在频域得到显式依赖于 $s$ 的界限. 因此, 本文不是简单重复频域形状导数, 而是把频域结果提升为时间域卷积算子的映射性质, 并进一步分析时间离散. 代价是必须处理 $s$ 的复参数依赖性、Laplace 逆变换以及时间正则性, 而不是只处理单一实频率.
第二, 与时域 Maxwell 形状导数相比. 引言中明确指出, 时域 Maxwell 系统的形状导数已在 [12,13,45] 中获得, 但它们处理的是有界区域上的内问题, 而不是自由空间中的外散射问题. 内问题通常有紧区域和体积积分, 外问题则需要处理辐射条件、边界积分算子和远场衰减. 本文针对完美导体外散射, 使用 Laplace 域中的单层势、双层势和边界积分算子, 并利用远离导体的接收点处的指数衰减估计. 因此, 本文把时域形状导数的适用范围从有界内问题推进到完美导体自由空间散射外问题, 同时保留了时间卷积求积的数值可计算性.
第三, 与卷积求积用于逆散射的工作相比. 引言提到 [17,42,43,44] 将卷积求积方法用作时间离散, 把时域数据拟合转化为一组 Laplace 域问题, 然后在 Laplace 域计算形状导数. 这些工作通常把时间离散和形状导数计算分开: 先离散时间数据, 再在频域做形状优化. 本文的不同在于直接建立时域区域导数, 并对其 Runge–Kutta 卷积求积半离散与 Galerkin 全离散进行收敛分析, 包括区域导数逐点求值的误差. 这意味着本文不仅提供算法, 还提供离散误差理论. 代价是分析更复杂, 需要 Laplace 域频域界限、点值势估计以及区域导数一阶扰动估计的共同配合. 从提供内容中无法确认本文是否在数值实验中与 [17,42,43,44] 做了直接对比.
第四, 与时间域形状优化数值工作相比. [39] 使用时间域间断 Galerkin 方法, [41] 使用时间域边界元方法进行完美导体形状优化. 这些工作侧重于数值实现和优化效果. 本文则采用 Runge–Kutta 卷积求积作为时间离散, 采用 Galerkin 方法作为空间离散, 并给出半离散与全离散误差分析. 与 [39,41] 相比, 本文的推进在于理论收敛保证和区域导数的显式构造, 但代价是可能需要更强的边界正则性假设 $D\in C^2$ 和变形场 $h\in C^1(\Gamma,\mathbb{R}^3)$, 以及接收点远离导体以保证点值估计中的指数衰减. 从提供内容中无法确认本文是否与这些数值方法在相同算例上进行了效率比较.
第五, 与波动方程区域导数相比. [29] 和 [20] 分别建立了声软障碍物和可穿透障碍物的波动方程区域导数, 其中 [20] 还数值求解了三维逆散射问题. 这些工作与本文共享时域区域导数的基本困难, 但对象是标量波动方程或声学散射, 边界条件为声软或穿透型. 本文处理的是矢量 Maxwell 系统, 边界条件为完美导体条件 $\nu\times E=0$, 涉及切向迹空间 $H^{-1/2}(\operatorname{Div},\Gamma)$、$H^{-1/2}(\operatorname{Curl},\Gamma)$ 以及表面散度与表面旋度. 因此, 本文在函数空间、边界积分算子和正则性分析上都需要新的处理. 从提供内容中无法确认作者是否把 [29,20] 的方法直接推广到 Maxwell 系统, 还是发展了新的估计.
第六, 与直接散射问题的数值分析相比. 引言提到直接散射问题的相关研究已在 [3] 中进行. 本文研究的是区域导数问题, 即散射场关于几何变形的灵敏度, 而不是直接散射场的正问题. 两者共享 Laplace 域频域界限和边界积分表示, 但区域导数还需要对边界和计算域做变量变换、引入截断函数并控制 Jacobian 扰动. 因此, 本文在 [3] 的基础上增加了形状微分的层次. 从提供内容中无法确认本文是否完全复用了 [3] 中的空间离散和误差分析框架.
第七, 假设、适用范围与代价. 本文的频域界限在 $\operatorname{Re}s\to 0^+$ 时以 $|s|^2/(\operatorname{Re}s)$ 或 $|s|^3/(\operatorname{Re}s)$ 的速率增长, 在空间点值估计中则带有指数衰减 $e^{-d_x\operatorname{Re}s}$. 这些尺度关系说明: 接收点离导体越远, 高频分量的衰减越快; 而当 Laplace 参数实部很小时, 低频或长时行为需要更仔细的估计. 本文的边界正则性假设为 $D\in C^2$, 在需要 $\operatorname{curl}\hat{v}\in H^1$ 时进一步要求 $\hat{g}\in H^{3/2}_t(\Gamma)$, 更一般地要求 $\Gamma$ 为 $C^{m+1}$ 以得到 $V^{-1}(s):H^{m+1/2}_t(\Gamma)\to H^{m-1/2}_t(\Gamma)$. 这些假设比某些仅要求 Lipschitz 边界或低正则性的正问题分析更强, 但为形状导数的 Fréchet 可微性和边界积分算子的可逆性提供了基础. 代价是本文的方法可能不直接适用于非光滑边界、角点或裂缝, 也不直接覆盖可穿透介质、非完美导体或非线性材料. 从提供内容中无法确认作者是否讨论了这些扩展或给出了数值例子来验证更弱正则性下的行为.
总体而言, 本文相对于已有文献的主要推进是: 在完美导体时域电磁散射外问题中, 通过 Laplace 域建立频域显式界限, 进而得到区域导数的时间正则性和 Runge–Kutta 卷积求积半离散与 Galerkin 全离散收敛分析, 并将区域导数用于带噪近场数据的迭代形状重构. 与频域形状导数相比, 它把问题推进到时域多频; 与内问题时域形状导数相比, 它处理外散射; 与卷积求积逆散射相比, 它直接分析时域区域导数的离散误差; 与声学波动方程区域导数相比, 它处理矢量 Maxwell 系统和完美导体边界条件. 这些推进的代价是需要更强的正则性假设、精细的频域界限和点值势估计, 以及更复杂的全离散误差分析.
2. Helmholtz 问题的自适应局部正交分解方法
An adaptive localized orthogonal decomposition method for Helmholtz problems
作者: Yuwen Li, Chuxin Que
类别: math.NA
发布日期: 2026-09-21
arXiv ID: 2609.24776v1
摘要翻译: 我们针对具有局部振荡和奇异性的 Helmholtz 问题，发展了一种自适应局部正交分解（ALOD）方法。主要结果是首个在标准分辨率、过采样与离散稳定性范围内，针对 Helmholtz LOD 离散的波数鲁棒残差型后验误差估计。该估计相对于细网格解是双侧的，并包含校正子局部化误差的可计算界，从而能够分别自适应在线粗空间、细校正子网格和过采样水平。对于给定载荷族，我们复用指示子已经计算出的局部残差 Riesz 代表元，作为区域加性 Schwarz 修正量。它们张成的空间构成共享富集，随后通过能量加权本征正交分解（POD）压缩以供在线使用。因此，约化空间直接由自适应误差控制机制生成。对于在原始 LOD 检验空间上满足 Galerkin 正交性的富集解，该误差估计仍然有效。核提升检验函数在统一过采样条件下提供了稳定、拟最优的耦合，且无需额外的全局细网格求解，并通过逐成员检查控制压缩引起的扰动。二维实验展示了针对给定载荷族的在线压缩与低秩重用。
Comment: 24 pages, 5 figures
链接: http://arxiv.org/abs/2609.24776v1
1. 研究对象和背景
本文研究二维有界多边形 Lipschitz 区域 Ω⊂R^2 上带混合 Dirichlet、Neumann 与 Robin 阻抗边界条件的 Helmholtz 方程。设波数 k≥k_min>0，边界 Γ_D、Γ_N、Γ_R 互不相交至端点，且 |Γ_R|>0。强形式为
$$ -Δu - k^2 u = f \quad 在 Ω 内, \qquad u=0 \quad 在 Γ_D 上, \qquad ∂_ν u=0 \quad 在 Γ_N 上, \qquad ∂_ν u - i k u=0 \quad 在 Γ_R 上, $$其中 ν 为外法向。相应弱形式在 V={v∈H^1(Ω;C): v|{Γ_D}=0} 中寻找 u，使 $$ a(u,v)=F(v) \quad 对所有 v∈V, $$ 其中 $$ a(v,w)=(∇v,∇w)_Ω - k^2(v,w)_Ω - i k (v,w)_{Γ_R}, \qquad F(v)=(f,v)_Ω. $$ 能量内积与能量范数定义为 $$ b_k(v,w)=(∇v,∇w)_Ω + k^2(v,w)_Ω, \qquad \|v\|_k^2=b_k(v,v). $$ 这类问题出现在声学、电磁学、地震成像、逆问题、不确定量化与设计中，核心困难是波传播是全局现象，必须全局解析波长；但角点、局部源或介质奇异又只在局部要求细分辨率。对标准有限元，解析波长本身不能避免大波数下的 pollution 误差，即累积相位误差；即使名义上每个波长有若干单元，误差仍可能随 k 增长，因此需要越来越严格的全局网格。Wavenumber-explicit 先验分析已对线性与高阶方法、非捕获与变系数散射问题给出定量估计，但这些估计通常要求全局过分辨，多查询场景下代价很高。\n\n局部正交分解（LOD）把上述两种分辨率需求分离：局部细尺度 corrector 编码未解析振荡，算子自适应的粗空间保持在线维度。对 Helmholtz 方程，已有先验理论在粗网格解析波长、过采样深度随波数对数增长、局部 corrector 问题被准确求解的条件下，证明了稳定性和一阶能量误差估计，且常数不依赖波数；multiresolution LOD 进一步解耦离散尺度；super-localized LOD 改善了 localization 误差的衰减并放宽过采样要求；并行 patch 求解器和 reduced corrector 模型用于降低 setup 成本。然而，这些结果都是先验地规定分辨率和 localization 参数，没有提供可计算指标来区分粗尺度分辨率不足、细尺度分辨率不足和过采样深度不足。\n\n标准 Helmholtz 自适应有限元后验指标通过残差或平衡通量控制 conforming、stabilized 与 HDG 有限元离散误差，但它们不控制 LOD 截断 corrector 导致的 localization 误差，也不能把粗尺度误差、细尺度误差与 localization 误差分开。本文要补的正是这一空白：在标准 LOD 分辨率、过采样和离散稳定性 regime 内，为 Helmholtz LOD 离散建立首个 wavenumber-robust 残差型后验误差估计。估计相对于当前细网格解是双边的，并给出 corrector-localization 误差的可计算上界，从而使在线粗空间、用于 corrector 与残差的细网格以及过采样层数可以分别自适应。\n\n在离散层面，论文采用嵌套形状正则三角网格 T{H,n} ⪯ T_{h,n}，对应连续分片仿射空间 V_{H,n}⊂V_{h,n}⊂V。粗空间决定在线维度，细空间用于求解 corrector 和残差。细网格 Galerkin 解 u_h 满足 a(u_h,v_h)=F(v_h)，并用 inf-sup 常数
$$ γ_h(k)= inf_{0≠v_h∈V_h} sup_{0≠w_h∈V_h} |a(v_h,w_h)|/(\|v_h\|_k\|w_h\|_k)>0 $$度量稳定性。LOD 通过准插值 I_H:V→V_H 定义细尺度核 W=ker I_H，W_h=W∩V_h，分解 V_h=V_H⊕W_h。在分辨率条件
$$ χ_res=C_app C_patch(1)^{1/2} \|kH\|_{L∞(Ω)} ≤ χ_0<1 $$下，核上有强迫性 Re a(w,w)≥c_W|w|_k^2，其中 c_W=(1−χ_res^2)/(1+χ_res^2)。这些是本文后验分析的基础假设。\n\n论文还处理 prescribed load family：同一 Helmholtz 形式 a 下反复求解不同右端 F_μ，μ∈P。这类问题出现在多入射场逆散射、多候选源定位、多发射器地震或超声成像、以及不确定量化与设计中。为每个右端重复自适应构造代价高，而只针对一个名义载荷自适应的空间可能遗漏其他载荷激活的特征。作者因此把训练族上已经由后验指标计算出的局部残差 Riesz 代表回收为区域加性 Schwarz 修正，其张成形成共享 enrichment；再用能量加权 POD 压缩供在线使用。这样 reduced space 直接由自适应误差控制机制产生，而不是用独立全局代理模型替代 LOD。
2. 主要定理或主要结果
论文在标准 LOD regime 内建立了局部残差等价、可计算的 corrector-localization 控制和覆盖 enrichment 解的细网格误差界。核心假设是：核强迫性成立，即 χ_res≤χ_0<1；当前细网格 Galerkin 问题满足 γ_h(k)>0；并且近似解 U 在原始局部 LOD 测试空间 Y_{H,ℓ}=T*ℓ V_H 上满足 Galerkin 正交 R_U(y)=0。\n\n1. 残差局部化与效率，定理 3.2。对每个粗顶点 z，令 W{h,z}={w∈W_h: supp w⊂N^2(z)}。对 G∈V_h^，定义 ξ_z(G)∈W_{h,z} 为 $$ b_k(ξ_z(G),w)=G(w), \quad w∈W_{h,z} $$ 的解，并令 η(G)^2=Σ_z |ξ_z(G)|k^2。则有 $$ C_syn^{-1} η(G) ≤ \|G\|_{W_h^*} ≤ C_sd η(G). $$ 因此对任意 U∈V_h，有 $$ η_H(U) ≤ C_a C_syn \|u_h-U\|_k, $$ 且局部形式 η{H,z}(U)≤C_{a,z}|u_h-U|{k,N^2(z)}。这里 η_H(U)=η(R_U)，R_U=F-a(U,·)。该结果把细尺度核上的残差范数等价地表示为局部可计算 Riesz 代表之和，并给出效率。\n\n2. corrector-localization 误差，定理 3.4。设 D_ℓ=Q∞−Q_ℓ，δ_ℓ=|D_ℓ|_{L(V_H,V_h)}=|Dℓ|{L(V_H,V_h)}。定义 G_ℓ(v_H)(w)=a(w,T*ℓ v_H)，以及 $$ Θ_ℓ= sup_{0≠v_H∈V_H} η(G_ℓ(v_H))/\|v_H\|_k. $$ 则 $$ Θ_ℓ/(C_a C_syn) ≤ δ_ℓ ≤ (C_sd/c_W) Θ_ℓ. $$ 与通常先验衰减估计 δ_ℓ≤C_loc C_patch(ℓ+1)^{1/2}β^ℓ，0<β<1 相比，Θ_ℓ 是可计算的，且不需要对 C_patch(ℓ) 的增长规律作假设。\n\n3. 双边可靠性，定理 3.5。若 U∈V_h 满足 R_U(y)=0 对所有 y∈Y{H,ℓ}，则
$$ \|u_h-U\|_k ≤ C_sd (c_W^{-1}+ C_I C_F/γ_h(k) δ_ℓ) η_H(U). $$结合 Θ_ℓ 的可计算界，得到
$$ η_H(U)/(C_a C_syn) ≤ \|u_h-U\|_k ≤ C_sd (c_W^{-1}+ C_I C_F C_sd/(γ_h(k)c_W) Θ_ℓ) η_H(U). $$该估计对不位于基 LOD 试探空间中的 enrichment 与压缩解同样成立，只要其测试空间包含 Y_{H,ℓ}，并在接受解处重算残差。若基正交仅近似成立，即
$$ ϵ_0= sup_{0≠v_H∈V_H} |R_U(T*_ℓ v_H)|/\|v_H\|_k, $$则右端增加 (C_I C_F/γ_h(k))ϵ_0。\n\n4. 波数鲁棒性，注 3.6。该估计中唯一显式波数依赖通过 δ_ℓ/γ_h(k) 或 Θ_ℓ/γ_h(k)。若 δ_ℓ≲γ_h(k) 或充分条件 Θ_ℓ≲γ_h(k) 成立，则可靠性常数关于 k 一致。若 γ_h(k)^{-1} 关于 k 至多多项式增长，且先验局部化衰减中的 patch 因子关于 ℓ 至多多项式增长，则指数局部化衰减给出 ℓ=O(log k) 足够。\n\n5. 局部化稳定性与过采样，命题 3.7 及 (3.12)。令 A_ℓ(p_H,q_H)=a(T_ℓ p_H,T*ℓ q_H)，A∞ 在 V_H 上的能量 inf-sup 常数为
$$ bα_{H,∞} ≥ γ_h(k)/(C_F C_I^2). $$展开差得
$$ A_ℓ(p_H,q_H)-A_∞(p_H,q_H)=a(D_ℓ p_H,D*_ℓ q_H). $$若 C_a δ_ℓ^2<bα_{H,∞}，则局部问题唯一可解且
$$ \|U_{H,ℓ}-U_{H,∞}\|_k ≤ C_os(δ_ℓ) δ_ℓ \|u_h\|_k, $$$$ C_os(δ)= (C_I+C_a(C_F+δ)(1+C_Iδ))/(bα_{H,∞}-C_aδ^2). $$作者进一步给出可计算的稳定性与鲁棒性条件
$$ C_a^2/c_W (C_sd/c_W Θ_ℓ)^2 ≤ ρ_stab γ_h(k)/(C_F C_I^2), \qquad (C_sd/c_W) Θ_ℓ ≤ C_rob γ_h(k), $$其中 0<ρ_stab<1。失败时增加过采样层 ℓ。为在自适应细化中区分 localization 截断误差与当前 LOD 误差，还引入平衡指标
$$ eq_ℓ = d_{μ_nom} Θ_ℓ/η_{H,μ_nom}, $$其中 d_μ=max{|U_μ|k,d_min}，μ_nom 为指定名义参数；超过阈值则增加 ℓ。\n\n6. 载荷族共享 enrichment 与压缩。对给定训练族，作者把局部残差 Riesz 代表 ξ_z(R{μ,U}) 作为区域加性 Schwarz 修正，其张成形成共享 enrichment，然后用能量加权 POD 进行在线压缩。误差估计仍适用于满足原始 LOD 测试空间 Galerkin 正交的 enriched 解。Kernel-lifted tests 由现有 LOD 基构造，在统一过采样条件下提供稳定且准最优的耦合，无需额外全局细网格解；memberwise checks 控制压缩带来的扰动。二维实验，包括光滑波包与重入角算例，展示了在线压缩和有限载荷族上的低秩复用。
3. 研究方法、关键技术和核心工具
技术障碍主要有三层。第一，Helmholtz 形式 a 是不定的，没有全局强制性，标准残差后验估计依赖的全局能量 coercivity 不成立；已有 LOD 先验分析虽然能在核 W_h 上得到强迫性，但要把核上的残差控制转化为全局细网格误差，必须利用 LOD 正交性和细网格 inf-sup 稳定性。第二，LOD 误差由粗尺度离散误差、细尺度表示误差和 corrector-localization 截断误差耦合而成；后验指标必须能区分这三者，并能分别驱动在线粗空间、细 corrector 网格和过采样层。第三，多查询载荷族要求重用已经计算的局部信息，但复用压缩不能破坏误差控制的正交性。\n\n作者的第一项核心工具是粗顶点斑块上的细尺度核局部残差 Riesz 代表。为了在 W_h 上做局部化，论文证明稳定局部分解：任意 w∈W_h 可写成 w=Σ_z w_z，w_z∈W_{h,z}={w∈W_h: supp w⊂N^2(z)}，且
$$ Σ_z \|w_z\|_k^2 ≤ C_sd^2 \|w\|_k^2, \qquad \|Σ_z w_z\|_k^2 ≤ C_syn^2 Σ_z \|w_z\|_k^2. $$构造取 w_z=Π_f I_h(λ_z w)，其中 λ_z 为粗节点帽函数，I_h 为细网格节点插值，Π_f=I−I_H。关键观察是 I_H 的局部性使 w_z 落在 N^2(z) 上，而 λ_z w 在每个细三角形上是二次多项式，故多项式 scaling 与 product rule 控制能量；再用 I_H w=0 和准插值估计消去 H^{-1} 项。这个分解避免了依赖粗斑块中细单元数目的常数。\n\n第二项工具是把局部残差指标解释为加性 Schwarz 预条件诱导的残差范数。定义 ξ_z(G) 为
$$ b_k(ξ_z(G),w)=G(w), \quad w∈W_{h,z} $$的局部 Riesz 代表，η(G)^2=Σ_z |ξ_z(G)|k^2。则 M_h G=Σ_z ξ_z(G) 给出 η(G)^2=G(M_hG)。在 W_h 上 b_k 强制，因此 η(G) 等价于 |G|{W_h^}。这与预条件型后验估计和 smoother 型指标有联系，但本文的实现是在细尺度核子空间上做 subspace correction。由于全局 Helmholtz 形式不定，作者先用该 Schwarz 构造控制 W_h 上的残差，再用 LOD Galerkin 正交性、localization 控制和细网格 inf-sup 把受限估计转成全局误差界。\n\n第三项工具是可计算的 corrector-localization 控制。传统 LOD 先验证明直接估计 Q_∞−Q_ℓ 的算子范数，得到 δ_ℓ≤C_loc C_patch(ℓ+1)^{1/2}β^ℓ；但该界含不可计算常数且需要 patch 因子增长规律。作者改为考察 G_ℓ(v_H)(w)=a(w,Tℓ v_H)，并利用理想正交性得到对 w∈W_h 有 G_ℓ(v_H)(w)=a(w,D*ℓ v_H)。核强迫性与连续性给出 $$ c_W \|D*_ℓ v_H\|_k ≤ \|G_ℓ(v_H)\|_{W_h^*} ≤ C_a \|D*_ℓ v_H\|_k. $$ 再结合残差等价 η，得到 Θ_ℓ/(C_a C_syn)≤δ_ℓ≤(C_sd/c_W)Θ_ℓ。因此 Θ_ℓ 完全由局部 Riesz 问题和已构造的测试空间决定，可作为 localization 的在线控制量。对于先验衰减，作者也给出 kernel-preserving cut-off 论证：用 Π_f I_h(ϑ w) 截断 corrector 尾，其中 ϑ 为粗分片仿射截断函数；稳定局部分解的 product estimate 控制截断函数，核强迫性导出固定宽度环上的 corrector 尾收缩，求和得到 δ_ℓ≤C_loc C_patch(ℓ+1)^{1/2}β^ℓ。\n\n可靠性证明的框架是误差分解 e=u_h−U=e_W+e_H，e_W=Q∞e∈W_h，e_H=T∞e∈X_{H,∞}。细尺度部分由 corrector 方程和核强迫性直接得 |e_W|k≤c_W^{-1}|R_U|{W_h^}。粗尺度部分是关键：假设 U 在 Y_{H,ℓ} 上正交，则对 y_∞=T∞v_H 有 $$ a(e_H,y_∞)=R_U(y_∞)=R_U(y_∞−T*_ℓ v_H)=−R_U(D*_ℓ v_H). $$ 这里只要求 U 在局部测试空间上正交，不要求 U 属于基 LOD 试探空间。于是 v_H=I_H y∞，由准插值稳定性和理想 LOD inf-sup 常数得 |e_H|k≤C_I C_F/γ_h(k) δ_ℓ|R_U|{W_h^}。相加并用定理 3.2 得可靠性。这个结构解释了为什么 enriched 与 POD 压缩解仍可被同一估计覆盖：只要保留原始 LOD 测试空间上的正交性，压缩只通过重算残差和 ϵ_0 项进入。\n\n局部化稳定性分析则利用 A_ℓ−A_∞ 的一次项因理想正交而消失，只剩二次扰动 a(D_ℓ p_H,Dℓ q_H)。这给出 δ_ℓ 小条件下的唯一可解性和系数误差界。作者把先验的 δ_ℓ 衰减与可计算的 Θ_ℓ 阈值分开：稳定性与波数鲁棒性条件用 Θ_ℓ 表述，避免依赖 C_patch(ℓ) 的增长规律。为适应自适应细化，作者另设平衡指标 eq_ℓ=d{μ_nom}Θ_ℓ/η_{H,μ_nom}，因为局部网格细化会缩小斑块物理尺寸，使 localization 可能成为限制误差源。\n\n在载荷族部分，作者把已经算出的局部残差 Riesz 代表直接作为区域加性 Schwarz 修正，而不是重新构造全局 reduced basis。共享 enrichment 的张成携带训练族激活的局部特征；能量加权 POD 仅用于压缩该残差生成空间。Kernel-lifted tests 从已有 LOD 基构造，在统一过采样条件下保持稳定和准最优，并避免额外全局细网格求解；memberwise checks 逐成员验证压缩扰动。整体上，作者的技术路线可概括为：局部核残差等价，然后可计算 localization 控制，再用 LOD 正交性转全局可靠性，最后以同一局部代表生成载荷族 enrichment 并压缩。
4. 与之前工作的比较
与已有 Helmholtz LOD 先验理论相比，本文的主要推进是把后验误差控制引入 LOD，并且在标准 LOD regime 内做到 wavenumber-robust。已有工作，如文中引用的 [19,36]，在粗网格解析波长、过采样深度随 k 对数增长、局部 corrector 问题准确求解的条件下，证明稳定性和一阶能量误差估计，常数不依赖波数；但这些结论是先验的，参数必须预先给定。本文不否定这些假设，而是在同一 regime 内给出双边残差估计，并增加可计算的 corrector-localization 界，使粗空间、细 corrector 网格和过采样层可分别自适应。代价是估计依赖核强迫性 χ_res≤χ_0<1、细网格 inf-sup γ_h(k)>0 以及近似解在原始 LOD 测试空间上的 Galerkin 正交；对于 enriched 解，需要额外监控正交缺陷 ϵ_0。\n\n与标准 Helmholtz AFEM 后验估计，如文中提到的 [13,10,9]，相比，那些指标针对 conforming、stabilized 或 HDG 有限元离散，通过残差或平衡通量控制离散误差。它们没有 LOD 的 corrector-localization 环节，因此不能直接区分粗尺度误差、细尺度误差和截断 corrector 误差。本文的指标 η_H 定义在细尺度核 W_h 上，并额外给出 Θ_ℓ 控制 localization；因此可把三种误差源纳入同一自适应循环。与预条件型后验估计 [28,29] 和 smoother 型指标 [27] 相比，本文借用了加性 Schwarz 预条件诱导残差范数的思想，但针对 Helmholtz 不定形式，先在核 W_h 上控制残差，再通过 LOD 正交性转全局；这与直接在全局空间上做指标不同。\n\n与 multiresolution LOD [20] 和 super-localized LOD [17] 相比，后者分别解耦离散尺度和改善 localization 衰减、放松过采样要求，但仍然是先验设计。本文的自适应策略可以随误差分布调整在线粗空间、细网格和过采样层数；同时可计算的 Θ_ℓ 不要求 C_patch(ℓ) 的增长规律，而先验衰减界仍含 patch 因子。与并行 patch 求解器、reduced corrector 模型 [14,1,23] 相比，本文不是主要降低 setup 成本，而是把局部残差代表同时用于误差指示和载荷族 enrichment。POD 压缩只压缩残差生成空间，不替代 LOD 或引入独立全局代理。\n\n与问题自适应 enrichment [33,2]、reduced spaces [25,39,40,5] 和局部自适应降阶 [35,8] 相比，本文的特点是 reduced space 直接由自适应误差控制机制产生：局部残差 Riesz 代表先作为加性 Schwarz 修正，其张成形成共享 enrichment，再经能量加权 POD 压缩。误差估计对 enriched 或压缩解仍有效，只要它们在原始 LOD 测试空间上满足 Galerkin 正交；这使压缩误差通过 memberwise checks 和 ϵ_0 项被显式控制。与 hybrid FEM-LOD [46] 相比，后者可用有限元处理局部奇异但可能牺牲 LOD 的自由度减少；本文则识别局部奇异区域并用区域残差修正 enrichment，目标是在保留 LOD 在线降维的同时改善局部奇异性分辨率。\n\n适用范围与局限方面，论文正文分析和数值实验限于二维；备注 3.8 声称构造可推广到三维形状正则四面体网格，但常数可能额外依赖空间维数和三维形状正则界。波数鲁棒性不是无条件的：可靠性常数中显式出现 δ_ℓ/γ_h(k) 或 Θ_ℓ/γ_h(k)，要得到关于 k 一致的常数需要 δ_ℓ≲γ_h(k) 或充分条件 Θ_ℓ≲γ_h(k)；若 γ_h(k)^{-1} 至多多项式增长且 patch 因子至多多项式增长，则指数局部化衰减给出 ℓ=O(log k) 足够。从提供内容无法确认作者是否给出完整的三维数值验证、POD 压缩误差的显式先验界、以及载荷族规模与低秩压缩的定量理论保证。作者也未显式展开与所有已有 Helmholtz LOD 变体的逐项数值比较，因此关于实际计算代价和参数选择的优势仍需以文中二维实验为准。
了解领域论文（摘要翻译）
1. GradAgent：一种用于保结构梯度流计算的知识引导多智能体系统及其在多组分囊泡动力学中的应用
GradAgent: A Knowledge-Guided Multi-Agent System for Structure-Preserving Gradient-Flow Computation with an Application to Multicomponent Vesicle Dynamics
作者: Zhenlin Guo, Jiale Meng, Shuqi Tang, Haiyan Su, Maosheng Jiang, Kaiwen Shi, Meng Zhao
类别: math.NA
发布日期: 2026-09-21
arXiv ID: 2609.24871v1
链接: http://arxiv.org/abs/2609.24871v1
摘要翻译
中文标题: GradAgent：一种用于保结构梯度流计算的知识引导多智能体系统及其在多组分囊泡动力学中的应用
摘要翻译: 高阶微分算子和非线性耦合使得为耦合梯度流系统构造守恒且能量稳定的格式具有挑战性。我们提出 GradAgent，一种知识引导的多智能体系统，它协调三个智能体，分别涵盖模型分析、算法设计与证明，以及数值实现与验证。独立审计通过发现数学错误和证明缺口、指导修订并保持各阶段之间的一致性，从而增强了可靠性。在重构模式中，GradAgent 重构了 20 项已发表研究，并将经审计的知识组织在一个可扩展的知识图谱（KG）GradAgent-KG 中，连接模型结构、离散化策略与证明、实现以及数值证据。在设计模式中，智能体评估检索到的知识的适用性，并在相关离散化策略的启发下开发新格式。应用于完全耦合的多组分囊泡相场-流体模型，GradAgent 在三个算法族中产生了三个一阶和三个二阶格式，其中包括四个线性、解耦格式。在所述假设下，所有六个格式都守恒膜组分质量和囊泡体积，并无条件耗散其各自的时间离散能量。使用与不使用 GradAgent-KG 的对比表明，它促进了针对该目标模型的保结构格式设计的多样性。数值测试证实了二阶空间精度、预期的时间阶、守恒性和时间离散能量耗散，而三维剪切流模拟与实验定性一致。这些结果证明了 GradAgent 能够结合可重用知识、协调推理和独立审计，为复杂耦合系统开发和验证保结构算法。
2. 耗散哈密顿系统的四阶能量一致平均向量场离散梯度方法
Fourth-Order Energy-Consistent Average Vector Field Discrete Gradient Methods for Dissipative Hamiltonian Systems
作者: Lucas Lautwein, Nicole Marheineke, Håkon Noren Myhr, Kevin Schäfers
类别: math.NA, math-ph
发布日期: 2026-09-21
arXiv ID: 2609.24692v1
链接: http://arxiv.org/abs/2609.24692v1
摘要翻译
中文标题: 耗散哈密顿系统的四阶能量一致平均向量场离散梯度方法
摘要翻译: 离散梯度方法为耗散哈密顿系统的能量一致时间积分提供了一种保结构途径。然而，在其传统表述中，它们通常至多达到二阶收敛。将此前为 Poisson 系统发展的扰动原理推广到耗散情形，我们推导了四阶平均向量场离散梯度格式。基于合同变换的补全保持了耗散矩阵的半正定性，从而保证对所有正步长的无条件能量耗散性。这一补全概念为高阶能量一致格式提供了系统性途径。对于纯耗散系统，推导了另一种指数补全，以及保结构的有理逼近。特别地，我们刻画了保持半正定性且可纳入能量关联分裂格式的 Padé 逼近。对阻尼物理摆和耗散 Fermi–Pasta–Ulam 系统的数值实验表明，半正定补全对于避免非物理数值不稳定性和捕捉正确的定性动力学至关重要。它们进一步表明，所提方法在与现有 Galerkin 型方法相当的计算成本下达到了所需的精度和能量一致性。此外，尽管四阶格式的每步成本更高，但在所考虑的精度范围内，其在计算效率上优于传统的二阶离散梯度方法。
3. 鞍点问题中的转置方法
The transposition method in saddle-point problems
作者: Dietmar Gallistl, Zhen Liu
类别: math.NA
发布日期: 2026-09-21
arXiv ID: 2609.24478v1
链接: http://arxiv.org/abs/2609.24478v1
摘要翻译
中文标题: 鞍点问题中的转置方法
摘要翻译: 来自二阶椭圆偏微分方程理论的转置方法，在由 Brezzi 分裂已知的某些结构假设以及反映带边界条件微分算子具体情形中椭圆正则性的映射性质下，被表述用于鞍点问题。构建了一个框架，该框架允许通过标准的、未经修改的 Galerkin 型方法进行逼近，而这些方法相对于转置算子所允许的更一般数据可能是非协调的。数据无需正则化是鞍点方法所特有的优势。该框架涵盖的例子包括 Laplacian、Stokes 算子、线弹性、双 Laplacian 以及具有 $L^2$ 边界数据的稳态 Maxwell 问题。
4. 线性应变梯度弹性问题的 Morley 单元
Morley element for the linear strain gradient elasticity problem
作者: Xiao Li, Shudan Tian, Hua Wang
类别: math.NA
发布日期: 2026-09-21
arXiv ID: 2609.24340v1
链接: http://arxiv.org/abs/2609.24340v1
摘要翻译
中文标题: 线性应变梯度弹性问题的 Morley 单元
摘要翻译: 我们建立了一个离散 $H^1$-Korn 不等式，适用于满足二阶弱连续性条件的有限元空间。这意味着经典的 $H^2$-非协调元满足离散 Korn 不等式。基于此结果，我们仅使用分片二次多项式，为线性应变梯度弹性问题发展了一种稳定化 Morley 有限元方法。得益于离散 $H^1$-Korn 不等式，该方法不需要任何罚参数。我们证明该方法对 $λ$ 和 $ι$ 均具有鲁棒性。数值实验验证了理论结果。
5. 一种带有基于卷积的算子压缩的基于物理的非线性降阶模型
A Nonlinear Physics-based Reduced Order Model with Convolutional-based Operator Compression
作者: Anna Ivagnes, Giovanni Stabile, Gianluigi Rozza
类别: math.NA
发布日期: 2026-09-21
arXiv ID: 2609.24300v1
链接: http://arxiv.org/abs/2609.24300v1
摘要翻译
中文标题: 一种带有基于卷积的算子压缩的基于物理的非线性降阶模型
摘要翻译: 降阶模型（ROMs）被广泛用于加速参数化偏微分方程的求解，其中基于投影的方法依赖于在低维空间中对控制方程的线性表示。然而，其效率从根本上受限于解流形的 Kolmogorov N-宽度：当 N-宽度 衰减缓慢时，精确近似需要大量线性模态。这一局限促使了非线性 ROMs，这类方法通常用通过自编码器架构学习到的非线性潜表示来替代解场的线性表示。在本工作中，我们通过将非线性直接引入控制算子的表示，提出了一种不同的视角。我们不是仅压缩解场，而是学习全阶算子的非线性低维表示，同时保留显式的降阶方程组。采用卷积架构来利用微分算子的局部性和空间结构，所得压缩算子通过一种同时考虑重构精度和降阶解误差的训练策略，与解的非线性表示耦合。为处理由细密和非结构化离散化产生的大型稀疏算子，使用连续卷积直接作用于其非零元。最后，径向基函数插值能够预测未见过参数配置下的压缩算子，从而提供一个无需组装相应全阶系统的高效在线阶段。在三个参数化问题上的数值实验展示了非线性算子压缩作为基于方程的 ROMs 的一种替代策略的潜力。
6. 基于杂交分裂形式微分算子与广义熵投影的离散熵守恒/稳定通量重构格式
A Discretely Entropy Conserving/Stable Flux Reconstruction Scheme via Hybridized Split Form Differential Operator and Generalized Entropy Projection
作者: Geng Liang, Junjie Wang, Hui Xu
类别: math.NA, physics.comp-ph
发布日期: 2026-09-21
arXiv ID: 2609.24257v1
链接: http://arxiv.org/abs/2609.24257v1
摘要翻译
中文标题: 基于杂交分裂形式微分算子与广义熵投影的离散熵守恒/稳定通量重构格式
摘要翻译: 本文针对非线性守恒律提出了一种新颖的离散熵守恒/稳定通量重构（FR）格式。该格式将杂交分裂形式微分算子与广义熵投影相结合，能够在无需修改原始求积权重的情况下，对任意节点分布和任意校正函数实现离散熵守恒。该构造直接建立在经典 FR 框架之上，并被证明在保持守恒性质的同时，在半离散意义下实现熵守恒。使用等熵涡问题的数值实验证实了理论预测的精度，并展示了在熵耗散界面通量下稳健的熵行为。
7. 用于 Stokes 方程的一种基于笛卡尔网格的实现友好、具有压力鲁棒性和超收敛性的 SDG 格式
An Implementation-Friendly SDG Scheme based on Cartesian Grids for Stokes Equations with Pressure Robustness and Superconvergence
作者: Bohan Yang, Eric Chung
类别: math.NA, math-ph
发布日期: 2026-09-21
arXiv ID: 2609.24232v1
链接: http://arxiv.org/abs/2609.24232v1
摘要翻译
中文标题: 用于 Stokes 方程的一种基于笛卡尔网格的实现友好、具有压力鲁棒性和超收敛性的 SDG 格式
摘要翻译: 本文针对 Stokes 方程发展了一种基于笛卡尔网格的交错间断 Galerkin (SDG) 格式，该格式易于实现、内蕴压力鲁棒，并且对所有变量均超收敛。我们不是使用标准 SDG 中的复合网格，而是由笛卡尔网格构造交错四边形网格。该格式以速度、压力和速度梯度为未知量，并采用具有精心设计的交错连续性的分片常数空间。额外的梯度未知量通过静力凝聚被局部消元，并且可以通过质量集中进一步消去而不损失精度。推导了该格式的显式逐点形式，这便于实现，并使得详细的逐点分析成为可能。我们严格证明了压力鲁棒性和二阶超收敛性，其中二阶超收敛性在一般的非均匀笛卡尔网格上成立。该格式通过引入一个具有二阶相容性的新颖离散对流项，进一步推广到 Navier-Stokes 方程。结合标量辅助变量 (SAV) 方法和 Crank-Nicolson (CN) 格式，这给出了一个无条件能量稳定且二阶精确的格式。数值实验验证了理论，并展示了精度和鲁棒性。
8. 成本-精度权衡：神经算子与经典数值求解器
Cost-Accuracy Trade-offs: Neural Operator vs Classical Numerical Solver
作者: Daniel Zhengyu Huang, Andrew M. Stuart
类别: math.NA, cs.LG
发布日期: 2026-09-21
arXiv ID: 2609.24021v1
链接: http://arxiv.org/abs/2609.24021v1
摘要翻译
中文标题: 成本-精度权衡：神经算子与经典数值求解器
摘要翻译: 神经算子是数据驱动的模型，学习从参数化偏微分方程的输入——例如空间变化系数、初始条件、强迫项、边界条件或几何——到解场或感兴趣量的映射。一旦训练完成，在需要对变化输入进行重复评估的多查询场景中，它们可以作为经典数值求解器的代理。我们研究神经算子代理在给定精度下于成本方面何时、以及为何优于经典数值求解器。我们关注训练后的多查询极限，其中数据获取和训练成本被视为固定且完全摊销。即使在这一刻意有利于神经算子的情形中，仍然存在经典求解器优于代理模型的情形。我们通过一项可复现的基准研究，在计算科学与工程中的代表性问题上将神经算子与问题匹配的经典求解器进行比较，关注预测误差、每次查询的浮点成本以及墙钟运行时间，从而比较神经算子代理与经典数值求解器的成本-精度性能。神经算子在低到中等精度要求下最具竞争力。其浮点成本优势强烈依赖于问题结构：当它们避免时间迭代或非线性迭代，或者预测约化的感兴趣量而非完整解场时，这种优势就会出现。额外的墙钟加速来自非常适合现代硬件的稠密张量运算。随着目标精度收紧，用神经算子达到所需精度变得越来越具有挑战性，而在此情形下经典求解器优于代理模型；因此，经典求解器对于验证和高精度计算仍将重要。
9. 平均曲率流的动态 Ritz 投影与参数有限元的最优 ${\bf L^2}$ 收敛性
Dynamic Ritz projection of mean curvature flow and optimal ${\bf L^2}$ convergence of parametric FEM
作者: Buyang Li, Rong Tang
类别: math.NA
发布日期: 2026-09-20
arXiv ID: 2609.23361v1
链接: http://arxiv.org/abs/2609.23361v1
摘要翻译
中文标题: 平均曲率流的动态 Ritz 投影与参数有限元的最优 ${\bf L^2}$ 收敛性
摘要翻译: 本文提出了一种新方法，用于研究三维空间中闭曲面平均曲率流的参数有限元逼近的收敛性。在该方法中，误差分析通过将数值解与本文引入的平均曲率流的动态 Ritz 投影进行比较来进行，而不是与文献中通常使用的平均曲率流的插值进行比较。在 $L^2$ 和 $W^{1,p}$ 范数下建立了动态 Ritz 投影逼近平均曲率流的相关误差。利用这些结果，证明了闭曲面平均曲率流的参数有限元方法在 $L^\infty(0,T;L^2)$ 范数下的最优阶收敛性，包括采用分片线性有限元的参数有限元方法的收敛性。
10. 非线性 Schrödinger 方程的高阶质量、能量与动量守恒方法
High-order mass-, energy- and momentum-conserving methods for the nonlinear Schrödinger equation
作者: Georgios Akrivis, Buyang Li, Rong Tang, Hui Zhang
类别: math.NA
发布日期: 2026-09-20
arXiv ID: 2609.23344v1
链接: http://arxiv.org/abs/2609.23344v1
摘要翻译
中文标题: 非线性 Schrödinger 方程的高阶质量、能量与动量守恒方法
摘要翻译: 本文提出了一种新颖的 formulation 以及相应的时空有限元方法，用于模拟非线性 Schrödinger 方程的解。所提算法的一个主要优点是其内在能力，即在离散层面保持质量、能量和动量的守恒。这一点已对由全离散隐式格式确定的数值解得到证明。基于一种等价 formulation，本文提出了一种有效的迭代格式来求解该非线性系统；该等价 formulation 表明，对解使用 Newton 迭代，而对非线性系统中的 Lagrange 乘子不进行迭代。本文提供了大量数值算例，以验证所提算法在模拟一维 Ma-孤子和双孤子以及由非线性 Schrödinger 方程支配的二维孤子时的高阶收敛性，以及在保持质量、能量和动量守恒方面的有效性。数值结果表明，与不守恒这些量的方法相比，本文设计的质量、能量和动量守恒方法在长时间模拟中也显著减小了数值解的误差。
11. 采用分片线性参数有限元的稳定化 BGN 方法用于曲线缩短流的超逼近与最优 $L^2$ 收敛性
Super-approximation and optimal $L^2$ convergence of the stabilized BGN method with piecewise linear parametric finite elements for curve-shortening flow
作者: Genming Bai
类别: math.NA
发布日期: 2026-09-20
arXiv ID: 2609.23326v1
链接: http://arxiv.org/abs/2609.23326v1
摘要翻译
中文标题: 采用分片线性参数有限元的稳定化 BGN 方法用于曲线缩短流的超逼近与最优 $L^2$ 收敛性
摘要翻译: 我们发展了一个双同伦论证，用于估计相容性泛函的离散物质时间导数。结合分部求和，当用法向误差速度作测试时，该论证给出相容性贡献的一个方向性超收敛估计。结合反向加权离散法向的超逼近，这些技巧建立了采用分片线性参数有限元的曲线缩短流稳定化 BGN 方法的投影误差 $L_t^\infty H_x^1$ 超收敛性以及轨迹误差的最优 $L_t^\infty L_x^2$ 收敛性。
12. 面向扩散主导偏微分方程的 TASE 稳定化时间演化自然梯度方法
TASE-Stabilized Time-Evolving Natural Gradient Methods for Diffusion-Dominated PDEs
作者: Zihao Shi, Dongling Wang
类别: math.NA
发布日期: 2026-09-19
arXiv ID: 2609.23054v1
链接: http://arxiv.org/abs/2609.23054v1
摘要翻译
中文标题: 面向扩散主导偏微分方程的 TASE 稳定化时间演化自然梯度方法
摘要翻译: 时间演化自然梯度 (TENG) 方法通过局部自然梯度迭代将时间离散目标状态投影到神经网络流形上，来演化时间相关偏微分方程的神经网络逼近。然而，对于刚性扩散主导问题，显式 Runge–Kutta 方法面临严重的稳定性限制。高频扩散分量可能驱使级目标超出神经网络流形的局部可达区域，从而恶化局部最小二乘投影的条件数，并使参数更新失稳。我们提出 TASE–TENG，它在投影之前对每个 Runge–Kutta 级增量应用高稳定的显式算子 $T_p(hW_0)$，其中 $W_0$ 近似主导的刚性扩散算子。为了在无网格设置中构造该替代算子，我们发展了一个可学习算子，其局部相互作用系数由共享的依赖几何的神经规则生成。结构化分解强制满足离散自伴性、耗散性和常数保持性。该算子在离线阶段训练，并在在线积分期间保持固定，其中它仅用于 TASE 稳定化算子内部。在刚性扩散和反应–扩散问题上的数值实验表明，在相同时间步长下，稳定性和精度得到提高，并且来自高频扩散模态的误差降低。学到的规则还允许在不同空间点分辨率下重构稳定化算子，而无需重新训练。
13. NeSTR：面向高斯 SPDEs 的神经 $S$-变换重构
NeSTR: Neural S-Transform Reconstruction for Gaussian SPDEs
作者: Nacira Agram, Fred Espen Benth, Jan Rems
类别: math.NA, math.PR
发布日期: 2026-09-19
arXiv ID: 2609.23001v1
链接: http://arxiv.org/abs/2609.23001v1
摘要翻译
中文标题: NeSTR：面向高斯 SPDEs 的神经 $S$-变换重构
摘要翻译: 我们提出了用于高斯随机偏微分方程（SPDEs）的神经 $S$-变换重构（NeSTR）。所学习的对象不是随机解本身，而是其在白噪声检验空间的一个有限维子空间上的 $S$-变换。对于 Wick 型 SPDEs，NeSTR 对受限 $S$-变换给出一个确定性参数化 PDE，并且随机解由该变换在原点处的 Taylor 系数恢复。因此，神经近似直接与解的 Wiener 混沌展开相联系。NeSTR 的主要创新点是将逆 $S$-变换重构原理与有限模态学习相结合：我们证明恢复的混沌系数的一致性，分离噪声模态误差、混沌截断误差、确定性求解器误差和神经近似误差，并给出受限变换的确定性学习表述。对加性随机热方程和乘性 Wick 热方程的数值实验表明，变换变量中的多项式谱表示为高阶混沌系数提供了稳定的获取途径。所得重构将依赖于方程的确定性系数场与可重用的 Gaussian–Hermite 基分离。因此，NeSTR 学习一个解析生成函数，其导数恢复 Wiener 混沌系数，而不是直接学习随机轨迹。
不相关论文（基本信息）
1. 多参数特征值问题的一种围道方法
A Contour Method for Multiparameter Eigenvalue Problems
作者: Emil Graf, Alex Townsend
类别: math.NA
发布日期: 2026-09-21
arXiv ID: 2609.24943v1
链接: http://arxiv.org/abs/2609.24943v1
摘要: Multiparameter eigenvalue problems arise in boundary value problems, stability analysis, and delay-differential equations. Despite their importance, existing methods either require solving extremely large global problems or rely on local iterative techniques that only recover a few eigenvalues at a time. In this work we develop the first contour method for analytic multiparameter eigenvalue problems. The key theoretical ingredient is a new residue formula for multivariate matrix-valued analytic functions, extending Beyn’s Keldysh-based residue theorem for meromorphic operator functions to several complex variables. Using this, we derive a multidimensional analogue of Beyn’s contour method that computes all the eigenvalues of a multiparameter eigenvalue problem contained in a prescribed region of $\mathbb{C}^d$. The resulting algorithm targets eigenvalues in a region without constructing preposterously large matrices and is embarrassingly parallelizable. Numerical experiments demonstrate that we can now successfully solve large multiparameter problems arising in applications where existing approaches struggle.
2. Rayleigh-残差流 I：正规矩阵
Rayleigh-Residual Flow I: Normal Matrices
作者: Eric Shen
类别: math.FA, math.DS, math.NA
发布日期: 2026-09-21
arXiv ID: 2609.24901v1
链接: http://arxiv.org/abs/2609.24901v1
摘要: We study the squared Rayleigh-residual functional on complex projective space and its negative gradient flow. For an arbitrary complex matrix, every positive-residual critical point has a strictly descending tangent direction; consequently, the local minima are precisely the eigenlines. For normal matrices, we describe the critical set in terms of circles through spectral points and identify the amplitude dynamics with a replicator equation whose payoff matrix has rank at most four. This gives explicit logarithmic first integrals and a reduced representation of interior trajectories. We compute transverse escape rates near critical components and analyze how polynomial reweighting changes unstable coordinates. Finally, we distinguish local saddle trapping from concentration of Haar-distributed initial states and derive the density of the filtered ensemble. Numerical examples with random spectra and a Jordan block illustrate convergence, nonmonotone Rayleigh motion, and the improvement of finite-time spectral exploration by low-degree random polynomial filters.
3. 一种用于时域阻抗壁的、耦合辅助微分方程的高阶虚拟点浸入边界方法
A high-order ghost-point immersed boundary method coupled with auxiliary differential equations for time-domain impedance walls
作者: Abdoulaye Ouattara
类别: physics.flu-dyn, math.NA, physics.comp-ph
发布日期: 2026-09-21
arXiv ID: 2609.24785v1
链接: http://arxiv.org/abs/2609.24785v1
摘要: A ghost-point immersed boundary method is coupled with auxiliary differential equations to impose a locally-reacting impedance condition on a wall that does not conform to a Cartesian grid, in the framework of the linearized Euler equations. The ghost-point reconstruction follows the normal-stencil approach: an elliptical least squares cloud of fluid nodes is used to build a high-order polynomial along the local wall normal, which is then extrapolated onto the ghost points by one-dimensional La grange interpolation. The impedance, modeled as a massspringdamper oscillator, is advanced in time through two auxiliary state variables per ghost point, coupled to the reconstruction at every RungeKutta stage. The method is validated against exact and semi-analytical references on a flat immersed wall, a cylindrical wall reflecting an acoustic pulse, and a cylinder scattering a harmonic monopole. Convergence orders ranging from about three on the curved wall to about four and a half on the flat wall are measured, and a cross-check against an exact harmonic reference shows that a persistent offset observed against a semi-analytical reference is a property of that ref erence rather than of the coupled solver. A spectral stability analysis of the linearized semi-discrete operator further shows that the normal-stencil reconstruction reduces, without eliminating, a known instability that occurs when the immersed wall becomes tangent to the grid.
4. 关于 GMRES 应用于三对角 Toeplitz 系统的容许行为的约束
Constraints on admissible behavior of GMRES applied to tridiagonal Toeplitz systems
作者: Fei Chen, Kirk M. Soodhalter
类别: math.NA
发布日期: 2026-09-21
arXiv ID: 2609.24751v1
链接: http://arxiv.org/abs/2609.24751v1
摘要: The result of Greenbaum and Strakoš that for a given set of eigenvalues, any convergence curve is possible [SIMAX 1996] and the subsequent parameterization of such matrix-right-hand side pairs $(A,\mathbf{b})$ of Arioli, Pták, and Strakoš [BIT 1998] demonstrated that the behavior of the \gmres could not be completely characterized by the eigenvalues of $A$ alone. In this paper, we consider how to use this theory to understand the admissible and attainable \gmres behavior for matrices with constrained structure, focussing on non-Hermitian (non-symmetric) tridiagonal Toeplitz matrices. We show that Toeptliz structure necessarily constrains the how the theory from these papers can manifest but that a continuum of admissible behaviors is still attainable.
5. 面向时空多重网络的基于边的Katz中心性
Edge-based Katz centralities for spatio-temporal multiplex networks
作者: Kai Bergermann, Francesco Gravili, Valeria Simoncini, Martin Stoll
类别: math.NA, cs.SI
发布日期: 2026-09-21
arXiv ID: 2609.24638v1
链接: http://arxiv.org/abs/2609.24638v1
摘要: Katz centrality is a well-established measure to identify and rank the most important nodes in complex networks by means of a linear system solve. Recent works have developed notions of Katz centrality for temporal, i.e., time-evolving networks. Their drawback is that small changes in the network structure may drastically change centralities across large parts of the network. Moreover, an unproportional effort in the network science community has been devoted to the study of node-based quantities while edge-based measures are far less explored. In this manuscript, we introduce a novel failure-aware temporal multiplex network model for computing edge-based Katz centralities for spatial networks. This class of networks admits the use of the line graph as network representation. We use a block-triangular and -banded supra-adjacency matrix representation, modeling inter-layer connections as low-rank matrices, which assigns a spatially constrained emphasis on up- or downdated edges. These could represent, e.g., breakages of water/gas pipes or obstructions in street networks in real-time infrastructure monitoring applications. We analyze the structure of the of the block matrix inverse with respect to entry decay in the blocks above the block-diagonal. This gives rise to two truncation approaches that allow drastic computational runtime reductions at the cost of introducing a controlled truncation error. Numerical experiments on a range of real-world spatio-temporal networks of size up to $2\cdot 10^8$ illustrate accuracy and efficiency with runtime gains of up to a factor of $679$.
6. 逆热流的先验正则性与高阶扩散采样器的维度依赖复杂度
A priori regularity of the reverse heat flow and dimension-dependent complexity of higher-order diffusion samplers
作者: Xixian Wang, Zhongjian Wang
类别: math.NA, math.AP
发布日期: 2026-09-21
arXiv ID: 2609.24622v1
链接: http://arxiv.org/abs/2609.24622v1
摘要: We establish arbitrary-order a priori regularity estimates for the reverse heat flow in the Ornstein–Uhlenbeck setting, that is, for the probability-flow ODE of score-based diffusion models. For compactly supported, possibly singular targets satisfying a dimension-uniform doubling condition on supporting-cap masses, we bound the iterated material derivatives of the flow and their first spatial derivatives pointwise, with constants independent of the dimension $d$ and explicit in time and support radius. The proof rests on two ingredients: a spatially uniform bound on normal-direction posterior fluctuations, which is the only point where the geometry of the target enters, and a centered posterior-moment normal form closed under material differentiation. This algebraic structure, together with a weighted regularity calculus, yields quantitative bounds at every order, including the mixed space–time directional derivatives required by Runge–Kutta stages. As an application, for exact-score sampling stopped at forward time $δ>0$, order-$p$ Taylor and explicit Runge–Kutta schemes reach accuracy $\varepsilon$ in $\widetilde O(d^{1/p}\varepsilon^{-1/p})$ steps in total variation, and the Taylor scheme in $\widetilde O(d^{1/(2p)}\varepsilon^{-1/p})$ steps in $W_2$.
7. AWE：用于在 FP4 张量核心上以更少 GEMM 实现精确整数矩阵乘积的自适应权重编码
AWE: Adaptive Weight Encoding for Exact Integer Matrix Products with Fewer GEMMs on FP4 Tensor Cores
作者: Shun-ichiro Hayashi, Daichi Mukunoki, Tetsuya Hoshino, Takahiro Katagiri
类别: cs.MS, cs.AR, math.NA
发布日期: 2026-09-21
arXiv ID: 2609.24519v1
链接: http://arxiv.org/abs/2609.24519v1
摘要: Emulation of high-accuracy floating-point matrix multiplication, as in the Ozaki scheme, splits the inputs into low-precision components and multiplies them pairwise. These products must be error-free, and each is an integer matrix product times a scale factor. FP4 Tensor Cores are the fastest on the NVIDIA B200 and B300 but cannot hold INT8 operands. The FP4 values scaled by 2 form the set $S = \{0, \pm1, \pm2, \pm3, \pm4, \pm6, \pm8, \pm12\}$, which contains every residue modulo 13, so with carries any integer splits into base-13 digits that FP4 can store. Prior work splits each INT8 operand into 3 such digits (limbs) with weights $(1, 13, 169)$ and multiplies them pairwise, 9 FP4 matrix multiplications (GEMMs) for INT8$\times$INT8. The classical ways to reduce products, such as the Karatsuba and Toom–Cook methods, do not apply as they stand: sums of limbs reach $\pm 24$ and leave $S$. This paper asks how many FP4 GEMMs are needed for one integer matrix product. We propose Adaptive Weight Encoding (AWE): the limbs take freely chosen integer weights, the stored planes are linear combinations of limbs, and the exact product is the sum of the FP4 GEMMs scaled by reconstruction coefficients. For each input range, we searched these choices for encodings with fewer products and found INT8$\times$INT8 in 6 products and INT4$\times$INT8 in 4. The formulation also holds modulo $m$, which covers the residue number systems of Ozaki scheme II: for the FP64 significand, the 75 products of prior work are reduced to 59. The boundary in product count between encodings with and without residues lies near input width 15. We release the encodings found.
8. 混合精度DEIM-CUR分解的误差分析与精度选择
Error Analysis and Precision Selection for Mixed-Precision DEIM-CUR Decompositions
作者: Ioannis Thanasis, Erin Carson
类别: math.NA
发布日期: 2026-09-21
arXiv ID: 2609.24509v1
链接: http://arxiv.org/abs/2609.24509v1
摘要: CUR approximation is a popular approach for constructing an interpretable low-rank approximation of a matrix. There are many approaches for constructing the CUR approximation, many of which are based on the SVD. Results in the literature bounding the quality of the approximation usually rely on the assumption that all or part of the SVD is computed exactly. In this work, we relax this assumption and show how the quality of the approximation is affected by an inexact SVD. Moreover, we analyze the effect of finite precision arithmetic on the DEIM-CUR method. Our analysis suggests guidelines for selecting the working precisions for the SVD and the DEIM index selection algorithm. In particular, it suggests that in many cases one can use very low precisions for these computations without significantly impacting the approximation quality. A set of numerical experiments is presented to demonstrate the theoretical results.
9. 基于熵的选择原理在开尔文-亥姆霍兹不稳定性示例中的数值实现
Numerical Realization of an Entropy-Based Selection Principle in the Example of Kelvin-Helmholtz Instability
作者: Shaoshuai Chu, Michael Herty
类别: math.NA
发布日期: 2026-09-21
arXiv ID: 2609.24479v1
链接: http://arxiv.org/abs/2609.24479v1
摘要: In this paper, we extend numerical schemes based on parameterized Young measures and linear programming to two space dimensions and apply them to the Kelvin-Helmholtz instability governed by the two-dimensional Euler equations of gas dynamics. We construct first-, second-, third-, fifth-, seventh-, and ninth-order Young-measure schemes and compare them with corresponding standard local Lax-Friedrichs (LLF) flux-splitting schemes and LLF schemes equipped with an entropy-based local characteristic decomposition (LLF-ELCD). We examine instantaneous and time-averaged density profiles, cumulative averages across spatial reconstruction orders, regional empirical density distributions, averaged density marginals, and several selection criteria for dissipative weak solutions. The numerical results reveal scheme-dependent flow patterns, particularly in the small-scale structures generated during the roll-up of the shear layers. At every reconstruction order considered, the Young-measure schemes yield the largest time-averaged physical entropy. By contrast, the LLF-ELCD schemes do not systematically yield larger entropy than the standard LLF schemes and therefore do not constitute a consistent maximum-entropy selection mechanism. The averaged density marginals of the Young measures are concentrated on several neighboring density states, and the later cumulative averages are closer for all Young-measure schemes. These results demonstrate the feasibility of the two-dimensional Young-measure formulation and show that the objective function in the linear-programming problem can act as an effective selection mechanism. The observed entropy preference is consistent with the local optimization of the expected physical entropy. It cannot be reproduced by merely incorporating entropy into a conventional numerical construction.
10. 由线性热孔隙弹性支配的分布最优控制问题的数值逼近
Numerical approximation of distributed optimal control problems governed by linear thermo-poroelasticity
作者: Jeonghun J. Lee
类别: math.OC, math.NA
发布日期: 2026-09-21
arXiv ID: 2609.24448v1
链接: http://arxiv.org/abs/2609.24448v1
摘要: In this paper, we propose and analyze a novel three-field symmetric formulation for optimal control problems governed by linear thermo-poroelasticity. The state variables are the solid displacement $\boldsymbol{u}$, the fluid pressure $p$, and the temperature $θ$, and both the distributed fluid source $m_p$ and the distributed heat source $m_θ$ serve as control variables. We prove well-posedness of the symmetric differentiated formulation, establish the existence and uniqueness of an optimal control pair $(\bar{m}_p,\bar{m}_θ)$, and derive the first-order necessary optimality conditions via a coupled three-field adjoint system. For the fully discrete scheme based on dG(0) time discretization, we employ a variational discretization approach for both controls and derive a priori error estimates of order $\mathcal{O}(h^s + Δt)$ for the state, adjoint, and control errors. We also present manufactured-solution experiments that validate the fully discrete optimality system and illustrate the storage-degenerate regime covered by our analysis.
11. 面向 Navier–Stokes 流的自编码器与数值分析指导的流形学习对比
Autoencoders vs. Numerical Analysis–Informed Manifold Learning for Navier–Stokes Flows
作者: Alessandro Della Pia, Lucia Russo, Ioannis Kevrekidis, Constantinos Siettos
类别: math.NA, physics.flu-dyn
发布日期: 2026-09-21
arXiv ID: 2609.24332v1
链接: http://arxiv.org/abs/2609.24332v1
摘要: Autoencoders (AEs) have become a dominant approach to nonlinear latent-space construction in data-driven reduced-order modelling (ROM), with their decoders lifting latent representations back to the ambient state space. Their prominence, however, has overshadowed an established alternative: manifold-learning methods grounded in classical numerical analysis. We revisit this alternative using Parsimonious Diffusion Maps (PDMs), benchmarking them against Proper Orthogonal Decomposition (POD)-based ROMs and several convolutional AE architectures for the two-dimensional incompressible flow past a rotating cylinder —a bifurcating Navier-Stokes (NS) system organized by a codimension-2 Bogdanov-Takens point and its associated Hopf, saddle-node, and homoclinic bifurcations. Our approach uses PDMs to identify a parsimonious and interpretable set of intrinsic latent coordinates and to estimate their dimension directly from data. Gaussian process regression then learns the latent dynamics, while convex K-nearest-neighbor (K-NN) interpolation in PDMs space constructs the pre-image map, for which we establish pointwise consistency. The resulting nonlinear ROM substantially outperforms POD-based ROMs and achieves reconstruction and prediction accuracy comparable to —and, in some bifurcating regimes, better than— that of AE-based ROMs. At the same time, latent-variable learning with PDMs requires orders of magnitude less computational time than AE training.
12. 体积约束非局部扩散问题有限元离散的网格与作用半径鲁棒预条件子
A mesh- and horizon-robust preconditioner for finite element discretizations of volume-constrained nonlocal diffusion problems
作者: Jiashu Lu, Yufeng Nie, Haolun Zhang, Pingrui Zhang
类别: math.NA
发布日期: 2026-09-21
arXiv ID: 2609.24143v1
链接: http://arxiv.org/abs/2609.24143v1
摘要: In this paper, we develop a preconditioner that is robust with respect to both the mesh size and the interaction horizon for finite element discretizations of volume-constrained nonlocal diffusion problems. Motivated by the Fourier multiplier of the nonlocal operator, we construct the preconditioner $B_h=G_h+βδ^2D_h^{-1}$, where $G_h$ is a symmetric local Poisson solver, $D_h$ is the lumped mass matrix, and $β$ is a kernel-dependent constant. We rigorously prove that the spectrum of the $B_h$-preconditioned system is bounded above and away from zero uniformly with respect to the mesh size and horizon. Moreover, we develop a pair-cell assembly for the finite element matrix that preserves the symmetry and positive definiteness required by the preconditioner and PCG, and establish an $L^2$-error estimate for the finite element discretization. Finally, we use a single symmetric geometric multigrid V-cycle for $G_h$ in the numerical experiments, and the corresponding numerical results confirm the predicted convergence rate and show bounded condition numbers and PCG iteration counts over the tested meshes, horizons, and kernels.
13. 浸入式流动分析中的精确壁面剪应力及其在基于点云的CFD中的应用
Accurate wall shear stress in immersed flow analysis with application to point cloud-based CFD
作者: Monu Jaiswal, Ming-Chen Hsu
类别: math.NA, cs.CE, physics.flu-dyn
发布日期: 2026-09-21
arXiv ID: 2609.24069v1
链接: http://arxiv.org/abs/2609.24069v1
摘要: Point cloud-based CFD enables flow analysis directly on discrete points obtained from 3D scanning and medical imaging, bypassing surface reconstruction, geometry cleanup, and boundary-fitted mesh generation. Derived from immersogeometric analysis, the method immerses the point cloud in a background mesh and enforces no-slip conditions on discrete points through a Nitsche-based weak boundary condition (BC). The framework delivers accurate velocity fields, pressure distribution, and integrated loads; however, accurate prediction of the local wall shear stress (WSS) has remained a critical challenge. The geometry intersects the background mesh arbitrarily, producing cut elements that lack the regularity required for consistent gradient evaluation. The issue is compounded by the stabilization term of the weak BC, whose parameter estimation in the symmetric Nitsche formulation is dependent on the cut configuration and affects the variationally consistent definition of traction from which the WSS is computed. In this work, we propose a new method to obtain accurate wall shear stress in immersed flow analysis with application to point cloud-based CFD, using a non-symmetric Nitsche’s formulation with near-wall modeling and a patch-based stress recovery approach with traction compatibility. The method is validated on canonical benchmarks and applied to turbulent flow past a sphere and to a patient-specific aorta, showcasing excellent agreement with reference results.
14. MGRD：用于方差感知跨域神经突起预测的紧凑形态门控残差扩散
MGRD: Compact morphology-gated residual diffusion for variance-aware cross-domain neurite forecasting
作者: Tsung Yeh Hsieh, Cosmin Anitescu, Chunghwan Kim, Victoria A. Webster-Wood, Yongjie Jessica Zhang
类别: cs.LG, math.NA, physics.bio-ph
发布日期: 2026-09-21
arXiv ID: 2609.23990v1
链接: http://arxiv.org/abs/2609.23990v1
摘要: Tracking neurite morphology over time helps characterize structural changes during neuronal development and deterioration, but long-term time-lapse imaging is resource-intensive and difficult to scale. Forecasting future morphology could reduce this burden. Existing neurite digital-twin models such as gated spatiotemporal attention (gSTA) produce a single deterministic forecast without representing variability among plausible futures. We introduce Morphology-Gated Residual Diffusion (MGRD), a compact stochastic surrogate that jointly forecasts twenty future neurite-morphology frames from ten observed frames while conditioning on morphology features derived from the latest observation. On controlled phase-field trajectories, MGRD reduces trajectory-wise mean MAE by 9.7% relative to a matched control while updating 4.46 times fewer parameters. On human iPSC-derived neuron microscopy, MGRD improves all four reported metrics over gSTA, including a 39.6% reduction in trajectory-wise mean MAE and a 45.3% increase in skeleton F1. Without mouse-domain retraining or fine-tuning, MGRD also improves MAE and skeleton F1 on mouse cortical-neurosphere microscopy across 10-40-min sampling intervals and forecast horizons beyond 13 hours. Repeated sampling provides a case-level variance score for ranking forecast difficulty. Retaining approximately 60% of the lowest-variance cases reduces mean MAE by 17.6% on iPSC microscopy and 16.8% on simulation data. MGRD uses 1.01% of gSTA’s parameters, requires less than one tenth of its training-update time, and generates a 50-step DDIM trajectory 7.9% faster when morphology features are cached. These results establish MGRD as a compact stochastic surrogate for neurite-morphology forecasting and case prioritization across simulation and microscopy datasets.
15. 带局地化的集合卡尔曼滤波的时间一致精度
Time-uniform accuracy of ensemble Kalman filters with localization
作者: Xiaoou Cheng, Daniel Sanz-Alonso, Nathan Waniorek
类别: math.NA, math.DS, math.OC, math.ST
发布日期: 2026-09-20
arXiv ID: 2609.23927v1
链接: http://arxiv.org/abs/2609.23927v1
摘要: This paper establishes time-uniform accuracy guarantees for deterministic square-root ensemble Kalman filters in linear–Gaussian state-space models under perfect-model dynamics. For hyperbolic, detectable systems, we prove that the ensemble means and covariances approximate their Kalman filter counterparts uniformly in time, with high probability and without inflation or resampling. The required ensemble size depends on the effective rank of the initial covariance and the dimension of the unstable subspace, rather than on the ambient state dimension. We also analyze a localized square-root ensemble Kalman filter for weakly coupled spatial systems. We show that the stabilizing Kalman covariance inherits spatial decay from the dynamics and derive accuracy bounds that separate sampling error from localization bias. For the localized filter, the required ensemble size depends on the local intrinsic dimension and only logarithmically on the number of spatial blocks, while the localization bias scales with the interaction strength. Our analysis combines stability and forgetting for possibly singular Riccati recursions, nonasymptotic covariance concentration, and perturbation estimates for localized covariance dynamics.
16. 半直线上函数在加权一致度量下的数值微分
Numerical differentiation of functions on the half-line in a weighted uniform metric
作者: S. G. Solodky, Y. A. Volynets
类别: math.NA
发布日期: 2026-09-20
arXiv ID: 2609.23833v1
链接: http://arxiv.org/abs/2609.23833v1
摘要: We consider the problem of numerical differentiation of functions defined on the half-line: the derivative $f^{(r)}$, $ r=1,2,\ldots$, is recovered from a finite set of perturbed Fourier–Laguerre coefficients of a function $f$, whose error is measured in $\ell_p$, while the accuracy of approximation is measured in the uniform metric with the weight $t^αe^{-t}$. It is established that the Wiener class $W^μ_{s}$ admits such a setting for $μ>r+1-1/s$. A class of methods $\mathcal{S}^θ$ is constructed that realize the optimal order of accuracy of numerical differentiation on $W^μ_{s}$ and use the smallest amount of input information in order.
17. CUR 逼近的盲误差估计
Blind error estimation for CUR approximation
作者: Lorenzo Lazzarino, Katherine J. Pearce, Nathaniel Pritchard
类别: math.NA
发布日期: 2026-09-20
arXiv ID: 2609.23810v1
链接: http://arxiv.org/abs/2609.23810v1
摘要: Low-rank approximation is a fundamental tool for scalable matrix computations. While such approximations have classically been formed via the truncated SVD, recent advances in randomized numerical linear algebra have produced methods of comparable accuracy at a fraction of the cost. A key advantage of these approaches is their ability to operate with limited access to the full matrix. A prominent example is the CUR decomposition, which builds an approximation from only a subset of columns and rows, making it particularly well-suited to settings where global matrix access is unavailable, such as in the design of spectro-microscopy experiments where even matrix-vector products cannot be computed. However, the randomness inherent to these methods introduces a key challenge: efficiently assessing approximation accuracy under the stringent constraint that only a subset of matrix entries can be observed. We derive a “blind” estimator of the Frobenius error of a CUR decomposition. For this estimator, we derive a worst-case lower bound on the number of required column queries, and provide techniques for quantifying its uncertainty. Finally, we demonstrate the performance of the estimator and the uncertainty sets on a mix of synthetic problems and real spectro-microscopy examples.
18. ELIPPS：基于部分自监督的反问题精确学习
ELIPPS: Exact Learning for Inverse Problems from Partial Self-supervision
作者: Benjamin Walder Markus Haltmeier, Lukas Neumann, Nadja Gruber, Gyeongha Hwang
类别: math.NA
发布日期: 2026-09-20
arXiv ID: 2609.23683v1
链接: http://arxiv.org/abs/2609.23683v1
摘要: In undersampled inverse problems (such as sparse-view computed tomography), only a small number of measurements are collected, which reduces radiation exposure and acquisition time and cost, and can also address the inaccessibility of certain acquisition arrangements. Most learning-based methods for such problems require supervision in the form of fully sampled measurements and ground-truth images, which are costly or even infeasible to acquire. To overcome this issue, we propose \emph{Exact Learning for Inverse Problems from Partial Self-supervision} (ELIPPS), an incomplete self-supervised training paradigm for undersampled inverse problems that requires neither ground-truth images nor fully sampled measurements. ELIPPS learns solely from incomplete forward measurements on a fixed incomplete supervision set. Our theory shows that when the data distribution is invariant under certain transformations, minimizing a masked empirical risk is equivalent to minimizing the full self-supervised risk, i.e. the risk against the complete, noise-free measurement. The equivalence is exact for arbitrary equivariant hypothesis classes and holds for signal-dependent noise such as the pre-log Poisson statistics of low-dose tomography, and for the log-transformed count model up to a quantifiable bias. When the underlying coverage condition is only approximately satisfied on a discrete grid, we give a stability estimate in terms of the associated frame constants and the irreducible error of the inverse problem. We realize ELIPPS for computed tomography by exploiting rotation and reflection invariance. In our experiments, ELIPPS substantially outperforms naive masked supervision and reaches the same order of accuracy as a reference model trained with full clean measurements.
19. 具有play型Prandtl–Ishlinskii迟滞的抛物型方程的数值分析
Numerical analysis of parabolic equations with Prandtl–Ishlinskii hysteresis of play type
作者: Shu Xu, Liqun Cao
类别: math.NA
发布日期: 2026-09-20
arXiv ID: 2609.23676v1
链接: http://arxiv.org/abs/2609.23676v1
摘要: Rigorous error analysis for numerical approximations of parabolic equations with hysteresis remains limited, even for the widely used Prandtl–Ishlinskii hysteresis of play type. In this work, we establish an $O(h+τ)$ error bound for an implicit Euler $P_1$ finite element discretization. The analysis requires neither higher-order temporal regularity of the hysteresis variables, which cannot in general be expected in hysteretic evolutions, nor additional spatial regularity of these variables. For the temporal discretization, we exploit a convex subgradient-flow structure in a weighted Hilbert space together with the associated dissipation and coercive subgradient remainder to obtain first-order convergence. For the spatial discretization, only the diffusive field is restricted to the finite element space, and a constraint-preserving comparison yields an $O(h)$ semidiscrete estimate. The analysis is developed for play-type Prandtl–Ishlinskii operators formulated directly on a spatial Hilbert space, encompassing the canonical pointwise model as well as more general spatially structured constraints.
20. 对流占优扩散问题的一种新颖单调有限体积元格式
A novel monotonous finite volume element scheme for convection dominant diffusion problem
作者: Cunyun Nie, Xiaoling Chen, Zhujun wang, Zhikun Tian, Chengjie Xia
类别: math.NA
发布日期: 2026-09-20
arXiv ID: 2609.23546v1
链接: http://arxiv.org/abs/2609.23546v1
摘要: One novel monotonous finite volume element (MFVE) scheme is put forward for convection dominant diffusion problem. The main contributions of this paper include four aspects. Firstly, one upwind sub-control volume, called upwind volume, is introduced for the discretization of the convection term similar to the upwind element, which leads to the upwind property. Secondly, one first-order linear discrete operator for the convection term in the balance equation is directly obtained by numerical integration in upwind volume, which is different from the classical finite volume methods.Thirdly, one asymptotic expansion of gradient functions in the dual element of each node is derived by one extension interpolation function. Then, one second-order nonlinear discrete operator for the convection term is constructed by the asymptotic expansion and error estimation of its FVE solution, where some perturbed coefficients are skillfully put into the expansion, which serves as a high-order correction. Fourthly, one novel two-order MFVE scheme, also accompanied by another one-order MFVE, is designed for convection-dominated diffusion problems together with one positivity-preserving finite volume element (PFVE) discrete diffusive operator. The L2 norm of the error of the approximate solution is derived. Finally, numerical results validate theoretical conclusions.
21. 反演基于通路的 Keller–Segel 模型中时空依赖的扩散与对流系数
Recovering the spatiotemporally dependent diffusion and advection coefficients in a pathway-based Keller–Segel model
作者: Yu’an Li, Lingyun Qiu, Min Tang, Hui Yu, Shenwen Yu, Siqin Zheng
类别: math.AP, math.NA
发布日期: 2026-09-20
arXiv ID: 2609.23364v1
链接: http://arxiv.org/abs/2609.23364v1
摘要: We investigate an inverse coefficient problem for an augmented Keller–Segel model arising in the description of phototactic and chemotactic population dynamics. The spatiotemporal evolution of the population density is governed by an advection-diffusion equation in which both the diffusion coefficient and the advection coefficient depend on space and time. Due to the underlying intracellular adaptation mechanism, even in a time-independent external environment, the diffusion coefficient (D(x,t)) and the scalar advection coefficient (K(x,t)) remain time-dependent and relax exponentially toward their respective steady-state profiles. Importantly, their relaxation is governed by the same rate, determined by the intracellular adaptation dynamics. From internal measurements of the density, we establish conditional Lipschitz stability for recovering either coefficient when the other is known, and derive cross-sensitivity estimates quantifying the compensation between diffusion and advection perturbations that produce the same density data. We further identify a finite-time separation between the transient and steady regimes: the full dynamics admit an exponentially accurate frozen-coefficient approximation at late times. When the steady-state coefficients are known, we also prove unique identification of the relaxation rate from the long-time behavior of the solution. Finally, we derive an explicit linearization of the forward map and, motivated by the temporal separation above, develop a two-stage gradient-based reconstruction strategy. Numerical experiments demonstrate the feasibility of the proposed method.
22. Lagrange 有限元函数的稀疏连接神经网络表示
Sparsely connected neural network representation of Lagrange finite element function
作者: Jiaxiong Hao, Yunqing Huang, Nianyu Yi
类别: math.NA
发布日期: 2026-09-20
arXiv ID: 2609.23299v1
链接: http://arxiv.org/abs/2609.23299v1
摘要: We construct a mesh-induced sparsely connected neural network framework that exactly reproduces arbitrary-order Lagrange finite element spaces over simplicial meshes. Unlike conventional black-box neural surrogates, the proposed network architecture is fully dictated by finite element discretization rules: local computations stem from simplex geometry and barycentric coordinate transformations, while global consistency is enforced through shared degrees of freedom. For linear Lagrange elements, local basis functions are directly implemented via affine barycentric layers, and high-order polynomial bases are explicitly decomposed into barycentric product compositions realized by specially designed $\mathrm{ReLU}^p$ modules. Equipped with element indicator branches and multiplication units, these modular local components are globally assembled into a sparsely connected neural network whose function space coincides exactly with the target finite element space, thereby inheriting the complete classical finite element approximation theory. By prescribing customized backward differentiation rules for piecewise activations, function values and their spatial gradients can be simultaneously extracted via automatic differentiation within a unified computational graph, eliminating the separate gradient calculation subroutines required in standard finite element implementations. Numerical experiments verify the accuracy of the neural network representation of Lagrange finite elements. Furthermore, by virtue of the intrinsic mesh-free nature of this neural network representation, finite element functions can be interpolated between non-matching meshes, and the proposed scheme can be applied to adaptive finite element methods for solving parabolic partial differential equations. An open-source code implementation of the proposed architecture is made publicly available.
23. K-格上三维镜像模型中的路由与记忆
Routing and memory in the three-dimensional mirror model on K-lattice
作者: Jian Gu, Paul Schaeffer, Qin Hao
类别: math-ph, math.NA
发布日期: 2026-09-19
arXiv ID: 2609.23244v1
链接: http://arxiv.org/abs/2609.23244v1
摘要: We study numerical observables relevant to delocalization in the three-dimensional Manhattan mirror model, also known as the K-lattice. Our principal experiment separates the locations of transmitting boundary channels from the routing between them. We derive an exact conditional comparison that preserves all transmission and reflection aperture sets, and find a statistically resolved deficit of physical transmission relative to this comparison. Nearby transmitting channels are themselves suppressed relative to a uniform subset with the same cardinality. A second experiment resolves the contribution of repeated vertices by their return age. The squared norm of the signed old-memory remainder is consistent with an inverse-square-root decay over the measured range. Annular experiments quantify decreasing continuation losses, while an initially apparent history dependence fails independent replication. Finally, a periodic configuration has a connected auxiliary contact graph but only finite routing cycles. These results identify estimates on reflection routing and adaptively selected revisit charges that may assist a rigorous proof. They do not establish an infinite-volume delocalized phase.
24. Landau 方程的一种快速谱粒子方法
A fast spectral particle method for the Landau equation
作者: Giacomo Borghi, Lorenzo Pareschi
类别: math.NA
发布日期: 2026-09-19
arXiv ID: 2609.23086v1
链接: http://arxiv.org/abs/2609.23086v1
摘要: We propose a fast deterministic particle method for the spatially homogeneous Landau equation. The method combines a particle representation of the solution with a Fourier approximation of the nonlinear collision flux. Nonuniform fast Fourier transforms are used to reconstruct the density from the particles and to evaluate the flux and density at the particle locations, while the convolutional structure of the flux enables its efficient computation by FFTs. For a fixed transform tolerance, the cost per time step is $\mathcal{O}(N+M^d\log M)$, where $N$ is the number of particles and $M$ the number of Fourier modes per velocity dimension. We establish a consistency estimate for the reconstructed velocity field on regions where the reference density is bounded away from zero. The estimate separates the spectral truncation error from the particle-density reconstruction error, showing how the latter can dominate for sufficiently smooth densities. Two-dimensional numerical experiments with Maxwellian interactions illustrate the accuracy and efficiency of the method, its sensitivity to particle and spectral resolution, and the effects of filtering. Comparisons with a direct blob implementation demonstrate improved accuracy at a strongly reduced computational cost.
25. 具有多个临界点的哈密顿快速平流下指数 Euler 方法的渐近保持性
Asymptotic preservation of an exponential Euler method under Hamiltonian fast advection with multiple critical points
作者: Jianbo Cui, Guozhen Li, Derui Sheng
类别: math.NA
发布日期: 2026-09-19
arXiv ID: 2609.23045v1
链接: http://arxiv.org/abs/2609.23045v1
摘要: We establish the asymptotic-preserving property of the exponential Euler method for stochastic reaction-diffusion-advection equations in $\mathbb R^2$ under fast Hamiltonian advection with multiple critical points. In the fast-advection limit, the dynamics reduces to a stochastic partial differential equation on a noncompact metric graph. A key ingredient is the strong convergence analysis of the limiting exponential Euler scheme, which is complicated by the graph’s noncompactness, its multiple edges and vertices, and the nonuniform ellipticity and vertex degeneracy of the graph operator. Using a weighted $L^2$ framework and analytic-semigroup smoothing estimates, we prove a temporal convergence order arbitrarily close to $1/2$ for weighted $L^2$ initial data and exactly $1/2$ under a half-order fractional-domain condition. Combining these estimates with the fast-advection limits yields the asymptotic-preserving property in the multiple-critical-point setting. Numerical experiments illustrate the asymptotic behavior and confirm the predicted temporal convergence rate.
26. 多面体网格上的拉链有限元方法
The Zipped Finite Element Method on polyhedral meshes
作者: Stefano Berrone, Gioana Teora
类别: math.NA
发布日期: 2026-09-19
arXiv ID: 2609.22955v1
链接: http://arxiv.org/abs/2609.22955v1
摘要: We extend the high-order Zipped Finite Element Method to three-dimensional polyhedral elements. The construction relies on a local sub-tetrahedralization obtained by triangulating the polygonal faces and connecting the resulting vertices to a suitable interior point. High-order shape functions are constructed as linear combinations of standard finite element basis functions, with coefficients chosen to ensure exact polynomial reproduction on the faces and in the element interior while preserving global C0-conformity. We further introduce a QR-based strategy as a systematic alternative to the heuristic procedure that was adopted for the two-dimensional case. The QR-strategy is more systematic and robust than the heuristic strategy, while being independent of the spatial dimension and avoiding geometric alignment tests and problem-dependent tolerances. The resulting method is formulated for a diffusion-reaction problem and numerically assessed through three-dimensional convergence tests, confirming the expected optimal polynomial order of accuracy.
27. 渐近保持指数积分器在双曲型 Korteweg-de Vries 系统中的应用
Asymptotic-Preserving Exponential Integrators applied to the Hyperbolic Korteweg-de Vries System
作者: Abhijit Biswas, Sebastiano Boscarino, David I. Ketcheson, Giovanni Russo
类别: math.NA
发布日期: 2026-09-19
arXiv ID: 2609.22864v1
链接: http://arxiv.org/abs/2609.22864v1
摘要: We study the application of exponential time integration methods, coupled with Fourier pseudospectral space discretization, to the hyperbolic Korteweg-de Vries (KdVH) system. We investigate the asymptotic preserving (AP) properties of such discretizations, showing that, in general, Lawson methods are not asymptotic preserving, because the auxiliary (derivative-approximating) variables do not satisfy the limit equilibrium manifold, whereas exponential time differencing (ETD) methods are AP for all components, including both the solution variable and the auxiliary variables. We also present an efficient numerical implementation based on an exact formula for the matrix exponential of this system, and compare it with previously-proposed ImEx Runge-Kutta time integration, showing that exponential methods can be competitive in this context.
28. 基于切比雪夫级数展开的高效严格延拓 I
Efficient Rigorous Continuation via Chebyshev Series Expansion I
作者: Maxime Breden, Olivier Hénot
类别: math.NA, math.DS
发布日期: 2026-09-19
arXiv ID: 2609.22689v1
链接: http://arxiv.org/abs/2609.22689v1
摘要: We study the global continuation of solution manifolds arising in dynamical systems. We present a rigorous continuation method based on a Chebyshev series expansion of the solution manifold. The branch is first approximated by a high-order Chebyshev interpolation polynomial, and an explicit error bound is then obtained by verifying the contraction of a quasi-Newton operator near this approximation. The contraction is formulated on a weighted $\ell^1$ space, giving a finer control than the typical $C^0$-error bound obtained from the uniform contraction theorem. In fact, the latter follows directly from our contraction operator. Furthermore, we discuss how our strategy applies naturally to pseudo-arclength continuation, where the continuation parameter fails to provide a valid local coordinate, and extends to multi-parameter continuation. Lastly, we detail two applications in which we compute a two-parameter family of steady-states for the Cahn–Hilliard equation, and a one-parameter family of steady-states undergoing saddle-node bifurcations for the Shigesada–Kawasaki–Teramoto system.
29. 悬臂剪切作用下加筋板的单元相关屈曲载荷
Element-dependent buckling loads of stiffened panels under cantilevered shear
作者: Lifeng Wang, Zhongli Qiu, Xuanhao Cheng, Run Du, Wenming Cheng, Min Xie, Xiong Rao
类别: cs.CE, math.NA
发布日期: 2026-09-19
arXiv ID: 2609.22656v1
链接: http://arxiv.org/abs/2609.22656v1
摘要: The linearized buckling load of a stiffened panel depends on the stress stiffness its shell element assembles. We read it from exported operators against three truncations of one second variation. The classic pass of ANSYS SHELL181 carries a rotation-rotation block pairing the drilling freedom with the bending rotations and its perturbation pass does not; removing the block recovers the perturbation load factor to 0.02%. SHELL281 carries block and couplings in both passes. Abaqus S4 matches the critical mode of the complete second variation to 1.0000 on the translations and its load factor to 1.7%, against 17% and 34% for the other two forms. On an optimized panel under cantilevered shear a 20-node continuum lies 3% to 6% above that form, S4 and SHELL281, 11% and 23% below both SHELL181 passes and 25% above Abaqus S8R, at the finest meshes. On a conventionally stiffened panel the SHELL181 passes stand 1.0% and 3.5% above the complete form, 9% and 21% at half the rib pitch; under a shear flow, on cylinders, open beams and under axial compression the three forms coincide and no pass parts by more than 0.3%.
30. 更新：谱集：数值域及其他
Update to: Spectral sets: Numerical range and beyond
作者: Michel Crouzeix, Anne Greenbaum
类别: math.NA
发布日期: 2026-09-18
arXiv ID: 2609.22460v1
链接: http://arxiv.org/abs/2609.22460v1
摘要: We use the results of Lorist and Schwenninger, {\em A solution to Crouzeix’s conjecture}, arXiv:2608.03841v2, https://arxiv.org/abs/2608.03841, to update the 2019 paper of Crouzeix and Greenbaum, {\em Spectral sets: Numerical range and beyond}, SIAM J.~Matrix Anal.~Appl., 40(3):1087-1101. For all regions considered in the 2019 paper, we are able to show that the region is a $K$-spectral set with a bound on $K$ that is smaller than that established in the 2019 paper. In particular, we show that various annular regions are $2$-spectral sets and that a more general convex region with a circular hole or cutout is a $4$-spectral set.