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1. 使用反射广义源的良态电场表面积分方程

Well-conditioned Electric Field Surface Integral Equations using Reflective Generalized Sources

作者: Yossi Dahan, Suryakumar Sivakumar, Yaniv Brick, Simon B. Adrian

类别: math.NA

发布日期: 2026-07-27

arXiv ID: 2607.24526v1

摘要翻译: 本文使用广义源方法发展了一类无需结合公式即无内共振的良态积分算子。广义源积分方程（GSIE）核是通过用辅助贡献增强传统电场积分方程（EFIE）核，以增强相应矩量矩阵块的秩亏损而得到的。本文首次研究了由内部散射凸屏蔽产生的辅助核的GSIE算子谱。利用同心圆散射体和屏蔽的闭式表达式，表明具有适合增强压缩性的屏蔽参数时，横磁（TM）和横电（TE）GSIE算子无内部真谐振和准谐振。辅助分量被证明是其EFIE对应物的紧扰动。因此这些GSIE继承了它们的稠密离散化崩溃，这仍可通过Calderón型预处理来治愈。控制高频崩溃的机理被证明受辅助分量影响，在某些情况下导致更强的稳定性。这些观察结果被证明对于使用非圆形模板屏蔽设计的矩量矩阵和GSIE仍然有效。对于GSIE对偶和Yukawa核预处理器，这些公式表现出良好的谱性质，同时又保持其可压缩性。这使得这些公式对于快速迭代求解器的设计具有吸引力。关于二维屏蔽算子的结果为将该方法推广至三维问题以及具有更广泛几何适用性的辅助核提供了基础。

Comment: 无

链接: http://arxiv.org/abs/2607.24526v1

1. 研究对象和背景

本文研究二维电磁散射问题的电场型表面积分方程(EFIE)在迭代求解时的条件数与谱性质。对于理想导电体(PEC)的时谐散射，表面感应电流$\boldsymbol{j}$满足EFIE

$$ \hat{\boldsymbol{n}}\times\left(-j k \int_S \boldsymbol{G}_f(\boldsymbol{r},\boldsymbol{r}')\cdot\boldsymbol{j}(\boldsymbol{r}')\,\mathrm{d}t'\right) = -\frac1{\eta}\,\hat{\boldsymbol{n}}\times\boldsymbol{e}^{\text{inc}},\quad \boldsymbol{r}\in S, $$

其中$\boldsymbol{G}_f$为自由空间并矢格林函数，$\eta$为波阻抗，$k=2\pi/\lambda$为波数。对二维$z$不变问题，TM极化退化为标量方程

$$ T^{\text{TM}}(j_z)(\boldsymbol{r}) = -j k\int_S g_f(\boldsymbol{r},\boldsymbol{r}')\,j_z(\boldsymbol{r}')\,\mathrm{d}t' = -\frac1{\eta}e_z^{\text{inc}}, $$

TE极化简化为

$$ T^{\text{TE}}(j_t)(\boldsymbol{r}) = -j k\int_S g_f(\boldsymbol{r},\boldsymbol{r}')\,j_t(\boldsymbol{r}')\,\mathrm{d}t' -\frac1{j k}\,\partial_t\int_S g_f(\boldsymbol{r},\boldsymbol{r}')\,\partial_{t'} j_t(\boldsymbol{r}')\,\mathrm{d}t' = -\frac1{\eta}e_t^{\text{inc}}. $$

矩量法(MoM)离散得到稠密线性系统$\mathbf{T}\mathbf{j}=\mathbf{e}$。传统EFIE存在三类病态：内部共振解不唯一；稠密离散化导致条件数以$\mathcal{O}(1/h)$恶化（$h$为网格尺寸）；高频时条件数以$\mathcal{O}(k^{1/3})$增长。组合场方程(CFIE)或组合源方程(CSIE)可消除共振，但往往丧失低秩压缩性。

另一方面，为加速求解，代数压缩方法（如低秩分解）被广泛应用，其效率取决于矩阵块秩的增长。传统EFIE核的宽边辐射使大块间相互作用秩与电尺寸$kR$线性增长，限制了压缩效果。广义源积分方程(GSIE)通过修改格林函数，引入位于散射体内部的辅助源贡献，衰减宽边辐射，增强掠射分量，从而有效降低秩的增长。具体地，将核修改为$\boldsymbol{G}_m(\boldsymbol{r},\boldsymbol{r}') = \boldsymbol{G}_f(\boldsymbol{r},\boldsymbol{r}') + \boldsymbol{G}_{\text{aux}}(\boldsymbol{r},\boldsymbol{r}')$，其中$\boldsymbol{G}_{\text{aux}}$由内部凸屏蔽表面的散射场给出。例如反射型GSIE中，对于TM问题，辅助标量格林函数$g_{\text{aux}}$使得$g_m = g_f+g_{\text{aux}}$，TE问题则引入辅助并矢。由此得到GSIE

$$ T_m^{\text{TM}}(\tilde{j}_z) = T^{\text{TM}}(\tilde{j}_z)+T_{\text{aux}}^{\text{TM}}(\tilde{j}_z) = -\frac1{\eta}e_z^{\text{inc}}, $$

以及TE类似形式。

已有的GSIE工作（[40]–[47]）主要专注于通过不同辅助源机制提升压缩性，但其作为电场方程，谱性质与条件数未被充分研究，尚缺乏可同时保证良好条件数与压缩性的GSIE格式，因而无法直接用于快速迭代求解器的设计。本文正是填补这一空白，系统分析基于反射凸屏蔽的GSIE算子的谱，包括内部共振、准共振、稠密离散化崩坏及高频崩坏，并提出有效的预条件策略，最终给出可同时满足无共振、可压缩且条件数受控的GSIE格式。

2. 主要定理或主要结果

本文聚焦于同心圆散射体和屏蔽构型，并在屏蔽上施加Leontovich阻抗边界条件（表面阻抗$\eta_{\text{sh}}\in\mathbb{C}$）。主要结果如下：

(i) 闭合形式特征值谱 对于TM极化和TE极化，在极坐标谐波基函数$\tilde{j}_z^{(n)}(\varphi)=\mathrm{e}^{j n\varphi}$下，GSIE算子对角化，得到特征值

$$ \lambda_{\text{TM}}^n = -\frac{\pi R k}{2}\left(\frac{H_n^{(2)}(kR)}{A_{\text{TM}}^n}+J_n(kR)\right)H_n^{(2)}(kR), $$

$$ \lambda_{\text{TE}}^n = -\frac{\pi R k}{2}\left(\frac{H_n^{(2)\prime}(kR)}{A_{\text{TE}}^n}+J_n'(kR)\right)H_n^{(2)\prime}(kR), $$

其中$R$为散射体半径，$a$为屏蔽半径，$n\in\mathbb{Z}$，

$$ A_{\text{TM}}^n = -\frac{\eta H_n^{(2)}(ka)+j\eta_{\text{sh}} H_n^{(2)\prime}(ka)}{\eta J_n(ka)+j\eta_{\text{sh}} J_n'(ka)},\quad A_{\text{TE}}^n = -\frac{\eta H_n^{(2)\prime}(ka)-j\eta_{\text{sh}} H_n^{(2)}(ka)}{\eta J_n'(ka)-j\eta_{\text{sh}} J_n(ka)}. $$

(ii) 无内部共振条件 特征值为零当且仅当关于$J_n$和$Y_n$的齐次线性系统非零解，其系数矩阵行列式正比于Wronskian $W\{J_n,Y_n\}(ka)=2/(\pi ka)$。只要$\eta_{\text{sh}}$具有非零实部$\eta_R\neq0$，该行列式不为零，从而杜绝了严格共振。对于径向准共振，将频谱按连续阶$\nu$分析，划分为三个区域：$\nukR$ (III)。当屏蔽距离$\upsilon = R-a<\lambda$时，区域II宽度不足以容纳准共振凹陷；若$\upsilon>\lambda$，区域II出现接近零但非零的凹陷（准共振），但仍可通过适当$\eta_{\text{sh}}$抑制。选择$\upsilon<\lambda$和损耗$\eta_{\text{sh}}$可使GSIE在离散频率上实际无共振。

(iii) 稠密离散化崩坏与Calderón预条件 辅助算子$T_{\text{aux}}$是相应$T^{\text{EFIE}}$的紧扰动，故在区域III（对应细网格极限）辅助贡献可忽略，GSIE继承EFIE的稠密离散化崩坏：TM条件数$\propto 1/h$，TE特征值无界。但满足Calderón恒等式

$$ D_{\text{TE}}T_m^{\text{TM}} = -\frac14 I + K,\qquad D_{\text{TM}}T_m^{\text{TE}} = -\frac14 I + K,\quad \text{$K$为紧算子} $$

其中$D$可取为对偶GSIE算子或Yukawa核算子。离散后$\mathbf{G}^{-T}\mathbf{D}\mathbf{G}^{-1}\mathbf{T}^m = -\frac14\mathbf{I}+\text{紧压缩}$，特征值簇集于$-1/4$附近，条件数与$h$无关。数值实验证实，椭圆屏蔽GSIE同样适用。

(iv) 高频行为 TM-GSIE条件数随$k$增长极缓，呈$\propto k^{0.05}$；Yukawa对偶预条件后几乎不随$k$变化。TE-GSIE则保持传统EFIE的$\propto k^{1/3}$增长，无论采用GSIE对偶还是Yukawa对偶预条件。GSIE对偶预条件TM-GSIE时重新引入$\propto k^{1/3}$增长，原因在于TE对偶算子的扫掠谱最小值下降更快。TM-GSIE结合Yukawa对偶预条件可同时消除稠密离散化崩坏且高频极好。

3. 研究方法、关键技术和核心工具

本文的技术核心在于利用同心圆构型获得解析的可对角化GSIE算子，从而严格分析谱机制。具体创新点与技术框架如下：

1. 修正格林函数的柱谐波展开与特征值闭合形式 对圆形屏蔽施加Leontovich边界条件，利用分离变量法得到TM-MGF(16)和TE-MGD(19)的傅里叶级数表示。当源点和场点均位于散射体圆$r=R$上时，算子自然对角化在谐波基$\mathrm{e}^{j n\varphi}$下，直接读出特征值(24)和(26)。这避免了复杂的数值特征值求解，为后续分析提供了显式表达式。

2. 内部共振与准共振的解析判别 为判明特征值零点，从$\lambda_n=0$导出差分方程系统(30)。系数矩阵行列式可化为Bessel函数的Wronskian，利用$W\{J_n,Y_n\}=2/(\pi x)$非零的性质，严格证明只要$\eta_R\neq0$即无严格共振。进一步，将整数阶推广为连续阶$\nu$考察频谱的整体结构。根据柱函数的渐近行为将频谱分成$\nukR$三个区域，解释准共振成因：当$\upsilon$较大时，区域II内辅助项与自由空间项不同步抵消，产生凹陷；当$\upsilon<\lambda$，区域II消失，准共振避免。

3. 紧扰动分析与Calderón预条件 证明$T_{\text{aux}}$是$T$的紧算子：通过MGF级数在大阶数$n$时辅助项迅速衰减（因$J_n$和$H_n^{(2)}$的渐近性质），而自由空间项主导高阶模，故$T_{\text{aux}}$的高阶特征值趋于零。因此，GSIE与EFIE共享稠密离散化崩坏的本质（TM特征值趋于零，TE趋于无穷）。直接套用EFIE的Calderón预条件框架：对TM，$D_{\text{TE}}$可以是$T_m^{\text{TE}}$或Yukawa-TE；对TE，$D_{\text{TM}}$同理。利用对偶算子乘积满足紧扰动恒等式(14)，离散后通过Gram矩阵$\mathbf{G}$的兼容性，广义特征值问题$\mathbf{G}^{-T}\mathbf{D}\mathbf{G}^{-1}\mathbf{T}^m$的谱簇集在$-1/4$，从而消除$h$依赖性。

4. 高频崩坏的局部波机制分析 通过跟踪扫掠谱（$\nu\approx ka$附近）的局部极值，发现TM-GSIE的极值衰减速率接近，而TE-GSIE的最小值衰减更快（$\propto k^{-4/3}$对比$\propto k^{-1}$），造成比值增长$\propto k^{1/3}$。屏蔽的反射改变了TM的局部场结构，使其对飞回波不敏感。Yukawa对偶算子由于指数衰减，不引入自身扫掠极值，故预条件后TM-GSIE条件数几乎水平。TE-GSIE的崩坏无法通过Yukawa对偶消除，因Yukawa-TE自身的高频行为同样恶化。

5. 数值验证与推广 所有解析预测均通过Galerkin MoM离散的数值实验验证。对于椭圆屏蔽和非圆散射体，通过将圆形情形导出的参数准则迁移，矩量矩阵的广义特征值谱与理论一致。预条件效果通过$h$扫描和$k$扫描确认。

4. 与之前工作的比较

与GSIE先前工作的比较 已有的反射型GSIE[40][41][44][45]、吸收型[42][43]及多极子型[46][47]均集中于提升矩量矩阵块的压缩性，即降低低轶秩。这些工作展示了GSIE在快速直接求解器中的潜力，但从未系统考察离散后方程的条件数性质及迭代求解适配性，更未提出良好条件的GSIE格式。本文首次聚焦于GSIE算子的谱分析，填补了这一空白，并明确给出了同时可实现压缩与良好条件数的参数准则（$\upsilon<\lambda$、$\eta_R\neq0$等）。

与传统EFIE及CFIE/CSIE的比较 EFIE内部共振需要结合MFIE形成CFIE或引入外部源构造CSIE来消除。本文提出的反射型GSIE通过屏蔽表面的欧姆损耗自然避免共振，无需磁流体方程，从而保留了纯电场方程结构，这有利于单极化的快速求解器设计。稠密离散化崩坏方面，GSIE与EFIE同源性使得经典Calderón预条件可直接移植，论文验证了GSIE双对偶及Yukawa双对偶的有效性。对于高频崩坏，传统CFIE/CSIE经Yukawa–Calderón预条件后仍保留$\propto k^{1/3}$增长，只有特殊局部修正波数方法[30]可部分治愈。本文发现TM-GSIE高频率异常温和，且经Yukawa预条件后近乎频率无关，这是传统电场方程首次通过辅助核设计获得显著改善，而TE-GSIE仍遗留崩坏，提示屏蔽设计可能对不同极化有选择性影响。

与同心圆特例（GEIE）的比较 早期GEIE[41]假设存在固定内部散射体，需修改右侧项。本文GSIE基于Love等效原理，仅将屏蔽作为辅助源场引入核中，不改变激励项，更符合通用电磁建模。论文证明封闭形式特征值同样适用，并揭示了频谱分区现象。进一步，将结果推广至非同心、椭圆屏蔽和任意非圆散射体，验证了谱性质的鲁棒性，而GEIE文献未涉此点。

预条件策略的比较 文中对比了两种对偶预条件子：GSIE对偶和Yukawa核对偶。对于TM-GSIE，Yukawa对偶给出最低条件数且无频率恶化；GSIE对偶虽降低条件数但重新引入$\propto k^{1/3}$增长。对于TE-GSIE，两者均无法消除高频崩坏，但GSIE对偶在低频段条件数更优，直到高频才被Yukawa对偶超越（交叉点见于图9c）。这表明预条件器选择需根据极化和频段权衡。

局限性与未来工作 本文限定于二维散射体及凸反射屏蔽，文中指出三维推广和基于更通用辅助核（如多极子等[46][47]）的设计将是下一步研究方向。此外，TE-GSIE的高频崩坏尚未解决，可能需要探索新的屏蔽配置或预条件器。

2. 基于柱谐函数的Helmholtz方程显式Runge逼近

Explicit Runge approximation for Helmholtz equation with cylindrical harmonics

作者: Yu Chen, Jin Cheng, Tingyue Li

类别: math.NA

发布日期: 2026-07-26

arXiv ID: 2607.23699v1

摘要翻译: 起源于全纯函数的复逼近，椭圆型方程的Runge逼近自Lax、Malgrange提出以来，已发展成为反问题乃至基于学习的数值方法的基本工具，其定量刻画由Rüland和Salo进一步建立。这里需要指出，Runge逼近是不适定的。在数值分析中，需要显式的定量估计来刻画逼近解的增长对原解向外延拓距离的依赖关系。本文利用柱谐函数研究Helmholtz方程的空间依赖定量Runge逼近，同时考虑内部和外部边值问题。我们针对三圆构型显式推导了相关指标，并针对一般几何情形获得了渐近指标。推导的结果给出了展开系数的范数估计，这对于正则化方法的实现至关重要。此外，所建立的界使得能够构造Helmholtz方程解的谱精度数值逼近。

Comment: 无

链接: http://arxiv.org/abs/2607.23699v1

1. 研究对象和背景

Runge逼近起源于全纯函数的复多项式逼近, 经 Lax 和 Malgrange 推广至椭圆型方程, 已成为数学物理反问题和基于学习的数值方法的基础工具. 其基本思想是: 定义在较小区域 $\Omega$ 内的椭圆方程解, 可以用定义在更大区域中的解来逼近, 代价是逼近解在扩大区域上的范数剧烈增长. Rüland 和 Salo 给出的定量刻画表明, 若原解不能向外延拓, 则逼近解的 $H^{1/2}$ 半范数以指数形式增长; 若能向外延拓, 则增长退化为多项式, 其指数 $\mu$ 与延拓距离 $\delta$ 有关, 即存在 $C>0$ 使得 $\|u_\varepsilon\|_{H^{1/2}(\partial\widetilde{\Omega})}\le C \varepsilon^{-\mu}\|u\|_{L^2(\partial\Omega)}$. 这一结论对反问题稳定性估计和基于学习的数值方法(如文献 [4] 中的亥姆霍兹方程学习算法)具有关键意义. 然而, 在数值分析层面, 需明确 $\mu$ 对 $\delta$ 的依赖关系, 以指导逼近样本的选取和正则化参数的设计.

本文研究对象为二维亥姆霍兹方程:

$$\Delta u + k^2 u = 0,$$

其中 $k>0$ 为波数, $k^2$ 不是相关区域上 $-$Dirichlet 特征值. 考虑两类典型问题:

内问题: $u$ 定义在有界区域 $\Omega$ 内, 且能延拓至更大的区域 $\Omega_\delta = \{x: \mathrm{dist}(x,\Omega)<\delta\}$ 中满足方程.
外问题: $u$ 为辐射解, 定义在 $\mathbb{R}^2\setminus\overline{\Omega}$ 中, 能延拓至更小的区域 $\mathbb{R}^2\setminus\overline{\Omega}_\delta$ 内.

已有的定量 Runge 逼近结果(如 Rüland 与 Salo[18])多针对调和函数或一般椭圆系统, 且通常限于有界区域. 对于亥姆霍兹方程, 其 Green 函数涉及振荡行为, 且外问题需满足 Sommerfeld 辐射条件, 这为显式定量估计带来新的挑战. 本文旨在:

显式给出指数 $\mu$ 与延拓距离 $\delta$ 及几何形状的关系, 特别是对三圆构型导出精确的调和测度公式, 对一般几何得到渐近估计 $\mu\sim C/\delta$.
基于柱谐函数(Fourier–Bessel 级数 $\sum c_n J_n(k|x|)e^{in\theta}$ 内问题, 或 Hankel 级数 $\sum c_n H_n^{(1)}(k|x|)e^{in\theta}$ 外问题)构造逼近, 这种基函数在数值计算中具有天然优势, 且便于分离变量.
给出展开系数 $c_n$ 的范数估计, 其显式的指数界是实施正则化方法(如截断或 Tikhonov 正则化)并选取正则化参数的理论基础.

技术上的空白在于, 虽然调和函数情形的指数与调和测度的关系已在 [3,10] 中明确, 但亥姆霍兹方程因包含波数 $k$, 基函数的渐近行为更为复杂, 且外问题尚无相关定量 Runge 逼近的结果. 本文通过精细的渐近分析, 首次将指数显式地与环域上的调和测度相联系, 并证明 $\mu$ 与 $\delta$ 成反比, 从而为在实际数值计算中可靠地应用柱谐逼近提供了理论保证.

2. 主要定理或主要结果

本文的主要结论包含针对圆域基本情形的估计, 以及推广到一般几何的内/外问题定量 Runge 逼近定理.

引理 3.1 (外问题圆域情形) 设 $u$ 在 $\mathbb{R}^2\setminus O_R$ 中满足亥姆霍兹方程及辐射条件, 且 $f=u|_{\partial O_R}\in H^{1/2}(\partial O_R)$. 则存在系数 $\{c_n\}$ 使得 $u(x)=\sum_{n\in\mathbb{Z}} c_n H_n^{(1)}(kr)e^{in\theta}$ 对 $r>R$ 收敛. 对任意的 $\varepsilon\in(0,1)$, i) 若 $u$ 不能向外延拓, 则存在辐射解 $\tilde u$ 在 $\mathbb{R}^2\setminus O_\rho$ ($0<\rho $$\|u-\tilde u|_{\partial O_R}\|_{L^2(\partial O_R)}\le \varepsilon\|u\|_{H^{1/2}(\partial O_R)},\qquad \|\tilde u\|_{H^{1/2}(\partial O_\rho)}\le Ce^{\bigl(\log\frac{R}{\rho}\bigr)\varepsilon^{-2}}\|u\|_{L^2(\partial O_R)}.$$

ii) 若 $u$ 可向外延拓至 $\mathbb{R}^2\setminus O_\sigma$ ($0<\rho<\sigma $$\|u-\tilde u|_{\partial O_R}\|_{L^2(\partial O_R)}\le \varepsilon\|u\|_{H^{1/2}(\partial O_\sigma)},\qquad \|\tilde u\|_{H^{1/2}(\partial O_\rho)}\le C\varepsilon^{-\frac{\log(\rho/R)}{\log(\sigma/R)}}\|u\|_{L^2(\partial O_R)}.$$

这里指数 $\frac{\log(\rho/R)}{\log(\sigma/R)}$ 正是环域 $O_R\setminus O_\rho$ 上的调和测度(在 $\partial O_\rho$ 上为 1, 在 $\partial O_R$ 上为 0).

命题 3.2 (内问题圆域情形) 类似地, 若 $u$ 在 $O_R$ 内满足亥姆霍兹方程且 $k^2$ 非特征值, 则存在 Fourier–Bessel 展开 $u=\sum c_n J_n(kr)e^{in\theta}$. 可延拓至 $O_\sigma$ ($R<\sigma<\rho$)时, 有

$$\|u-\tilde u|_{\partial O_R}\|_{L^2(\partial O_R)}\le \varepsilon\|u\|_{H^{1/2}(\partial O_\sigma)},\quad \|\tilde u\|_{H^{1/2}(\partial O_\rho)}\le C\varepsilon^{-\frac{\log(\rho / R)}{\log(\sigma / R)}}\|u\|_{L^2(\partial O_R)}.$$

注意到当 $\delta=\sigma-R\ll1$ 时, $\frac{\log(\rho/R)}{\log(\sigma/R)}\sim \frac{R\log(\rho/R)}{\delta}$, 因此 $\mu\sim C/\delta$.

定理 2.1 (内问题一般域) 设 $\Omega$ 为星形光滑边界域, $\Omega_\delta\supset\Omega$ 满足 $\mathrm{dist}(\partial\Omega_\delta,\Omega)\ge\delta>0$, $u$ 在 $\Omega_\delta$ 中满足亥姆霍兹方程并具有 $L^2$ 单层位势表示: $u(x)=\int_{\partial\Omega_\delta}\Phi_k(x,y)\mu(y)\,\mathrm{d}y$, $\|\mu\|_{L^2(\partial\Omega_\delta)}0$, 存在由柱谐函数组成的逼近解 $u_\varepsilon(x)=\sum_{n\in\mathbb{Z}}c_n J_n(k|x|)e^{in\theta}$ (在 $O_R\supset\Omega_\delta$ 中收敛), 满足

$$\|u|_{\partial\Omega}-u_\varepsilon|_{\partial\Omega}\|_{L^2(\partial\Omega)}\le \varepsilon M,\qquad \|u_\varepsilon\|_{H^{1/2}(\partial O_R)}\le C\varepsilon^{-c/\delta}M,$$

其中 $c$ 和 $C$ 与 $u$ 及 $\delta$ 无关, $c$ 取决于 $\partial\Omega$ 的曲率等几何量.

定理 2.2 (外问题一般域) 设 $\Omega$ 光滑, $u$ 为 $\mathbb{R}^2\setminus\Omega$ 中的辐射解, 且在 $\partial\Omega_\delta$ 上具单层位势表示 (密度 $\mu\in L^2$, $\|\mu\|_{L^2} $$\|u-u_\varepsilon|_{\partial\Omega_\delta}\|_{L^2(\partial\Omega_\delta)}\le \varepsilon M,\qquad \|u_\varepsilon\|_{H^{1/2}(\partial O_\rho)}\le C\varepsilon^{-c/\delta}M.$$

两个定理中的指数 $c/\delta$ 显式体现了延拓距离 $\delta$ 对逼近解增长的控制: $\delta$ 越大, 可延拓性越好, 增长越慢. 此外, 借助于边界 $L^2$ 误差控制内部误差的经典估计, 上述结果可直接导出区域内部的逼近精度, 如 $\|u-u_\varepsilon\|_{L^2(\Omega)}\le C_\Omega d^{-1}\varepsilon M$, 其中 $d=\min_j|k^2-E_j|/E_j$ 与 Dirichlet 特征值有关.

3. 研究方法、关键技术和核心工具

本文的核心创新在于: (1) 利用柱谐函数的显式表达式和渐近性质, 将定量 Runge 逼近的指数 $\mu$ 与环域上的调和测度精确等同, 从而给出依赖延拓距离 $\delta$ 的反比关系; (2) 发展一套“圆链”传播技巧, 将局部圆域上的估计组合为一般域上的全局估计; (3) 首次将定量 Runge 逼近推广至亥姆霍兹方程的外问题.

关键技术一: 圆域上的截断误差与系数增长显式计算. 对于圆对称区域, 亥姆霍兹方程的解可展开为 Fourier 级数. 以外问题为例, $u(r,\theta)=\sum c_n H_n^{(1)}(kr)e^{in\theta}$, 其中 $c_n = \hat f_n / H_n^{(1)}(kR)$, $\hat f_n$ 是边界迹 $f=u|_{\partial O_R}$ 的 Fourier 系数. 取有限项逼近 $\tilde u_N = \sum_{|n|\le N} c_n H_n^{(1)}(kr)e^{in\theta}$, 则截断误差为

$$\|u-\tilde u_N\|^2_{L^2(\partial O_R)} = 2\pi \sum_{|n|>N} \frac{1}{R}\sqrt{1+\tfrac{n^2}{R^2}}\, R \sqrt{1+\tfrac{n^2}{R^2}}\,|\hat f_n|^2 \le \frac{2\pi R}{N}\sum_{n\in\mathbb{Z}}\sqrt{1+\tfrac{n^2}{R^2}}|\hat f_n|^2 = \frac{R}{N}\|f\|^2_{H^{1/2}},$$

因而取 $N\approx 1/\varepsilon^2$ 即可满足精度 $\varepsilon$. 在该截断下, $\tilde u_N$ 在小圆 $\partial O_\rho$ 上的范数可借助 Hankel 函数的渐近式 $H_n^{(1)}(t) = \frac{2^n(n-1)!}{\pi i t^n}(1+O(\frac1n))$ 估计:

$$\|\tilde u_N\|_{H^{1/2}(\partial O_\rho)}^2 \lesssim N\Bigl(\frac{R}{\rho}\Bigr)^{2N}\sum_{|n|\le N}|\hat f_n|^2 \lesssim N e^{2N\log(R/\rho)}\|f\|^2_{L^2}.$$

由此得到 $\|\tilde u_N\|_{H^{1/2}(\partial O_\rho)} \lesssim e^{c\varepsilon^{-2}\log(R/\rho)}\|f\|_{L^2}$, 对应不可延拓的情形. 当解可延拓至 $r=\sigma>0$ 时, 系数变为 $c_n = \hat f_n^{(")}/H_n^{(1)}(k\sigma)$, 此时 $H_n^{(1)}(kR)/H_n^{(1)}(k\sigma) \sim (\sigma/R)^n$, 故截断误差含因子 $(\sigma/R)^N$. 为达到 $\varepsilon$ 误差, 需 $N\approx \frac{\log\varepsilon}{\log(\sigma/R)}$, 代入范数增长估计得

$$\|\tilde u_N\|_{H^{1/2}(\partial O_\rho)} \lesssim N^{1/2} e^{N[\log(R/\rho)+\log(\sigma/R)]}\|f\|_{L^2} \lesssim \varepsilon^{-\frac{\log(R/\rho)}{\log(\sigma/R)}-o(1)}.$$

该指数恰好是环域上调和测度 $\phi$ 满足 $\Delta\phi=0$, $\phi|_{\partial O_\rho}=1$, $\phi|_{\partial O_R}=0$ 的解 $\phi=\log(r/R)/\log(\rho/R)$, 在 $r=\sigma$ 处的值. 内问题的 Bessel 函数情形完全类似, 差异在于 $J_n$ 的增长方向相反. 这一显式关联是本文的关键洞察.

关键技术二: 局部单层位势的“圆链”传播. 对于一般域, 将边界 $\partial\Omega_\delta$ 分割为 $N=O(1/\delta)$ 段小弧, 每段弧用外接圆 $O(y_i,\delta/2)$ 覆盖. 将解 $u$ 写为单层位势 $u=\int_{\partial\Omega_\delta}\Phi_k\mu$ 的和, 每一局部贡献 $u_i=\int_{l_i}\Phi_k\mu$ 在某个外接圆外区域满足方程. 对单个 $u_i$ 应用圆域上的定量 Runge 逼近 (引理 3.1), 得到一个以 $y_{i,0}$ 为中心的 Hankel 展开逼近 $u_{i,0}$, 其误差为 $O(\varepsilon\sqrt{\delta}\|\mu\|_{L^2(l_i)})$, 范数增长 $\sim\varepsilon^{-C/\delta}$. 接着, 利用星形域的几何性质, 通过一系列半径 $h/3$ 的圆将 $y_{i,0}$ 逐步移动到原点附近. 每一步应用圆域逼近并保持相对误差较上一步放大, 形成递增的指数序列 $\alpha_{i+1} = \alpha + (1+\alpha)\alpha_i$, 其中 $\alpha\sim C/h$. 最终, 经过 $K=O(1/h)$ 步传播后, 范数增长累积为 $\varepsilon^{-C'/\delta}$. 这一圆链传播技术使得局部估计可以整合为全局柱谐展开, 而无需对整个区域直接构造全局上的逼近解. 对非星形域, 可通过折线段连接, 结论仍适用.

关键技术三: 内/外问题的统一处理与正则化参数准则. 利用柱谐函数的完备性, 可将原点处的展开系数的范数与逼近解的 $H^{1/2}$ 范数建立等价关系, 从而量化的系数界直接对应正则化参数的选取. 如对于内问题, 逼近解 $u_\varepsilon=\sum c_n J_n(k|x|)e^{in\theta}$, 其系数满足 $\sum(1+|n|)^{1/2}|c_n J_n(kR)|^2 \approx \|u_\varepsilon\|_{H^{1/2}(\partial O_R)}$, 结合定理 2.1 的界可导出 $\sum(1+|n|)^{1/2}|c_n|^2 \le C\varepsilon^{-c/\delta}M$, 从而直接控制截断正则化或 Tikhonov 正则化解的误差.

综合来看, 本文的分析框架将复杂的定量 Runge 逼近问题归结为对柱谐函数系数的精细估计, 并通过调和测度连接了几何与指数, 得到了对数值计算具有指导意义的显式界.

4. 与之前工作的比较

本文在定量 Runge 逼近的框架下做出了多方面的实质性推进, 与已有工作形成鲜明对比.

1. 对亥姆霍兹方程及外问题的推广. 经典的定量 Runge 逼近大多处理调和方程或具有光滑系数的椭圆方程 (如 [18,19]), 且常见的误差增长指数多表现为定性形式, 未显式联系几何参数. 对于亥姆霍兹方程, 由于基本解涉及 $H_0^{(1)}(k|x-y|)$, 其振荡特性使得渐近分析更为复杂. 本文不仅成功地将定量估计推广到亥姆霍兹方程, 还首次建立了外问题的定量 Runge 逼近. 外问题要求逼近解满足辐射条件, 这在以往文献中鲜有涉及. 该结果对散射问题中通过内部数据重建远场模式具有潜在应用价值.

2. 显式调和测度关系的建立. 在调和函数情形, 文献 [3,10] 已经指出 Runge 逼近的指数与调和测度有关, 但其针对的是基础解或者调和函数的 Cauchy 问题稳定性. 本文在亥姆霍兹方程下, 通过圆域上的直接计算, 明确证明了可延拓解情形下的指数 $\mu$ 恰好为环域上的调和测度: $\mu = \log(\rho/R)/\log(\sigma/R)$. 这一公式是精确而非渐近的, 彻底澄清了指数与几何的定量关系. 进而, 对于小 $\delta$, 利用 Taylor 展开得到 $\mu\sim C/\delta$, 其中 $C$ 与域边界曲率相关. 这一定量关系使数值分析中的正则化参数选择有了直接的理论依据.

3. 对柱谐系数界的显式推导. 已有的定量 Runge 逼近结论通常只给出逼近解在边界上的范数增长, 没有深入到级数展开系数的界. 本文基于 Fourier-Bessel/Hankel 展开, 得到了系数 $c_n$ 的范数估计, 例如 $\|\{c_n (1+|n|)^{1/2}\}\|_{\ell^2}\le C\varepsilon^{-c/\delta}$. 这对数值求解至关重要: 在实际算法中, 往往直接处理系数, 通过截断或正则化抑制病态, 此时系数的先验界决定了正则化参数的合理选取区间. 这一应用层面的结论是本文对数值计算的主要贡献.

4. 数值应用与谱精度. 基于所得估计, 作者构造了可延拓解的柱谐逼近, 并声称可以达到谱精度. 这是因为柱谐函数是分离变量解, 其逼近精度仅受解的正则性限制. 若解可延拓, 则其 Fourier 系数指数衰减, 结合本文确定的误差-截断长度关系, 即可确保数值方法获得高阶精度. 这在声学仿真等实际问题中有应用潜力, 且相较于 [4] 中使用基本解作为样本, 系统性的柱谐基函数可能在收敛速度上更具优势.

5. 假设与局限性比较. 本文假设解具有单层位势表示, 且边界为光滑星形域. Rüland 与 Salo[18] 对一般椭圆算子仅要求 $u$ 属于 $H^1$ 且方程系数光滑, 未必修要求位势表示; 本文的假设相对更强, 但其目的在于获得显式指数关联. 此外, 本文要求 $k^2$ 不是 Dirichlet 特征值, 这是椭圆方程内问题的标准要求. 值得指出的是, 文中未讨论通解 (非单层位势) 的处理, 对于更为一般的延拓方式, 估计是否仍成立需进一步研究. 在几何上, 星形假设在证明中用于构造连接原点与边界点的线段链; 对于更一般的单连通域, 作者称可通过折线适应, 但常数可能会改变. 这些局限为未来的推广留下了空间.

综上, 本文在定量 Runge 逼近的显式估计方面迈出了重要一步, 特别是将经典调和分析思路与亥姆霍兹方程的特殊函数论紧密结合, 为数值方法的严格误差分析提供了理论基础.

3. 面向裂纹尖端与角点奇异性的 B 样条分析的力学训练神经坐标映射

Mechanics-trained neural coordinate mapping for B-spline analysis of crack-tip and corner singularities

作者: Hyunju Kim

类别: math.NA

发布日期: 2026-07-25

arXiv ID: 2607.23229v1

摘要翻译: 在裂纹尖端或凹角附近，分数阶径向幂函数可能具有无界导数并降低高阶样条的收敛速度。奇异映射对计算半径进行渐变，使得拉回场更光滑而不改变物理区域。经典映射需要事先给定奇异指数和渐变。本文从力学问题中训练径向坐标。它是一个正神经密度的归一化积分，既固定了径向边界，又确保在退化尖端之外 r’(s)>0。径向渐变指数和密度修正权重是从离散平衡处计算的 Galerkin 能量推断出来的，无需指数标签或精确内部场。我们在标量和平面应变 B 样条公式中测试了该映射，并将其与恒等映射、径向分级节点向量、预设幂映射以及自适应增强 B 样条方法进行了比较。在 156 个向量自由度下，与恒等映射相比，力学训练的映射将相对能量范数误差降低了 33.42 倍。在 3 条积分路径上恢复的混合模式应力强度因子的最大误差为 1.923 x 10^(-5)。对于直裂纹，学习到的密度修正消失，预设的 r=s^2 映射给出了相同的改进。对于一个非线性 Robin 族，其测试参数位于训练区间之外，密度修正保持非零，并至少给出了 4.839 的固定 q 增量能量增益。因此，对于本文所考虑的问题，当所需坐标无法由预设幂映射表示时，神经修正是有用的。

Comment: 30 pages, 19 figures

链接: http://arxiv.org/abs/2607.23229v1

1. 研究对象和背景

裂纹尖端与凹角处的弹性力学及标量场问题中, 解常包含分数次幂径向项 $r^\lambda\Psi(\theta)$, 其中 $0<\lambda<1$ 为奇异指数, $\Psi$ 为角向函数, 例如裂纹尖端位移场具有 $\lambda=1/2$ 的平方根奇异性. 这类分数次幂在 $r=0$ 处导数无界, 严重降低高阶 B 样条或 NURBS 逼近的收敛阶. Williams 展开 [1] 揭示了该分数次幂是导致正则性丧失的根本原因. 为恢复高精度, 一类方法通过奇异映射将物理径向坐标 $r$ 替换为计算半径 $s$, 使得拉回后的场 $u\circ\chi(s,\theta)$ 在参数空间内足够光滑, 而物理域不变. 经典映射需预先给定奇异指数 $\lambda$ 以及合适的幂律关系, 如 $r=s^q$, 其中 $q$ 为径向分级指数, 使得 $u\sim (s^q)^\lambda = s^{q\lambda}$, 当 $q\lambda$ 为整数时分数次幂被多项式替换. 在等几何分析 (IGA) 中, Jeong 等 [4] 通过重复 NURBS 控制网格的一列控制点使参数边塌陷到奇异点, 此时几何映射满足 $\boldsymbol{F}(\xi,\eta)-\boldsymbol{x}_0 = \eta^q\tilde{\boldsymbol{F}}(\xi)+o(\eta^q)$, 于是 $r\sim \eta^q$, 拉回场为 $\eta^{q\lambda}\tilde{\Psi}(\xi)$. 匹配 $q$ 与 $\lambda$ 后, 奇异分量在参数域中变为多项式, 从而恢复样条逼近的预期阶数. 此类方法中, 奇异性结构（包括指数 $\lambda$ 和映射形式）需在求解前给定. 然而, 当问题参数化变化时, 真实的奇异指数可能未知, 且纯粹的幂律映射可能不足以完全正则化非齐次或非线性材料中的径向变化. 本文试图填补这一空白: 直接从待求解的力学问题出发, 通过 Galerkin 离散平衡能量, 在不使用指数标签或精确内部场的前提下, 学习一个由神经网络参数化的径向坐标映射. 该映射自动满足边界条件和单调性, 并能够同时捕捉径向分级指数 $q$ 与超越幂律的密度修正. 研究对象为标量圆域上的 Laplace 方程 (单位裂缝盘) 及平面应变线弹性断裂问题, 以及一个具有径向变化模量和 Robin 边界的非线性特征值问题. 论文旨在回答: 能否通过改变坐标 (而非扩展试空间或逼近物理场) 来提升 Galerkin 空间的逼近质量, 并使之适用于未知 $\lambda$ 或非幂律情形?

2. 主要定理或主要结果

在标量单位裂缝盘 ($-\Delta u=0$, $u(1,\theta)=\sin(\theta/2)$, $\partial_\theta u(r,\pm\pi)=0$, $u(0)=0$) 上, 力学训练映射使用 $156$ 个向量自由度时, 相对能量范数误差相比恒等映射 ($r=s$) 降低 $33.42$ 倍. 对于平面应变混合模式裂纹问题 (Young 模量 $E=1$, Poisson 比 $\nu=0.3$, 应力强度因子 $K_I=1.25$, $K_{II}=-0.45$, T 应力 $T=0.08$), 在三个围线上恢复的应力强度因子最大误差为 $1.923\times10^{-5}$. 当裂纹为直线型且尖端场严格为平方根奇异性时, 学习得到的密度修正权重 $w_j$ 趋于零, 此时训练的映射退化到预设幂律 $r=s^2$, 与经典方法提升一致. 在径向变模量 $\mu(r)=\exp(c_\mu r(1-r))$, 楔形角 $2\alpha=0.75\pi$ 的非线性 Robin 特征值问题 (控制方程 $-\nabla\cdot(\mu(r)\nabla u)=0$, 边界条件 $\partial_n u + \kappa u/r = 0$, 奇异性指数由 $\lambda\tan(\lambda\alpha)=\kappa$ 确定) 中, 对训练区间外的参数 $\kappa$, 密度修正保持非零, 并在固定 $q$ 的前提下带来至少 $4.839$ 的增量能量增益. 上述结果均在相同的 B 样条试空间和力学算符下获得, 训练过程仅使用组装离散能量 $J_h(\phi) = \frac12 \boldsymbol{d}^T \boldsymbol{K}(\phi) \boldsymbol{d}$ 作为目标, 无需参考解或奇异指数标签. 与径向分级节点向量、自适应富集 B 样条及预设幂律映射的比较进一步证实: 当所需坐标无法由单一幂律表达时, 神经密度修正是必要的; 而当幂律足够时, 训练自动退化为经典结果.

3. 研究方法、关键技术和核心工具

核心技术创新在于将径向坐标 $r_\phi(s)$ 的导数 $r'_\phi$ 参数化为一个正密度函数, 并通过归一化积分恢复坐标本身, 从而天然满足边界锚定 $r_\phi(0)=0$, $r_\phi(1)=1$ 和严格单调性 $r'_\phi(s)>0\;(s>0)$. 密度函数定义为

$$\rho_\phi(s) = s^{q-1} \exp\bigg(\sum_j w_j \tanh(\beta_j(s-c_j))\bigg),$$

其中 $q>0$ 控制径向分级指数, $w_j$ 为可训练的密度修正权重, $c_j, \beta_j$ 为固定的特征中心和斜率. 该形式在 $s=0$ 附近给出 $\rho_\phi(s)\sim C s^{q-1}$, 从而 $r_\phi(s) \sim \tilde{C} s^q$, 因此物理奇异幂 $r^\lambda$ 被拉回为 $s^{q\lambda}$, 通过调节 $q$ 可使指数整数化. 指数外因子 $\exp(\sum w_j\tanh(\cdots))$ 对远离尖端的坐标进行局部调整, 而不破坏正性.

训练循环仅更新 $q$ 和 $w$, 而位移系数通过求解组装后的 Galerkin 系统 $\boldsymbol{K}_{ff}(\phi)\boldsymbol{d}_f = \boldsymbol{f}_f - \boldsymbol{K}_{fc}(\phi)\boldsymbol{d}_c$ 得到, 目标泛函为平衡时的 Dirichlet 能量 $J_h(\phi)=\frac12 \boldsymbol{d}^T\boldsymbol{K}\boldsymbol{d}$. 梯度回传通过隐式自动微分或解析导数实现. 该设计确保网络仅作用于“坐标”, 而不逼近物理场, 从而保持刚度算子和 SIF 提取过程不变.

对参数化问题族, 引入条件规则: 通过一个浅层神经网络 (如 $q_\vartheta(\alpha)$ 预测器或 $q(\kappa), w_j(\kappa)$ 预测器) 在给定物理参数 (楔形角 $\alpha$ 或 Robin 系数 $\kappa$) 时输出 $\phi$. 对于非线性 Robin 族, 损失函数扩展为

$$\mathcal{L}(\vartheta) = \mathrm{mean}_\kappa\bigg[ \frac{E_c}{E_0} + \gamma_E\frac{\max(E_c-E_e,0)}{E_0} + \gamma_R\operatorname{Var}_m(\ell_m) + \gamma_A\bigg(\frac{A_c-A_e}{\max(|A_e|,\epsilon_A)}\bigg)^2 \bigg] + \gamma_\vartheta\|\vartheta\|^2,$$

其中 $E_c, E_e$ 为粗、细离散能量, $\ell_m$ 为径向一致性残差, $A_c, A_e$ 为近尖端幅值预估. 层次能量项防止映射通过降低粗能量而牺牲细精度, 径向惩罚项迫使径向模态与离散特征指数一致.

证明框架的核心障碍在于: 如何从离散能量中推断出连续场的奇异结构, 同时确保映射的容许性. 作者的真正创新在于: (1) 利用密度参数化将可容许坐标集闭式表征, 避免在使用优化器时因边界违反导致求解失败; (2) 将训练目标完全建立在离散 Galerkin 能量上, 而不是点值误差, 从而无需参考解; (3) 将 $q$ 与 $w$ 分离, 使二者分工明确: $q$ 捕获主导奇异行为, $w$ 修饰非幂律偏差. 引理或分析框架方面, 作者并未给出理论收敛阶证明, 而是通过数值实验 (包括不同网格、围线积分、Jacobian 审计) 验证了方法的稳定性、一致性和泛化能力.

4. 与之前工作的比较

与既有基于等几何分析的奇异映射方法相比, 本文的推进主要体现在以下几方面:

映射的先验知识需求: 经典的塌陷边映射 [4] 需要预先指定奇异指数 $\lambda$ 和径向分级指数 $q$, 且通常仅采用幂律形式 $r=s^q$. 虽然匹配后能恢复最优收敛阶, 但当问题参数变化导致 $\lambda$ 未知或径向分布非齐次时, 固定的幂律映射可能效果不佳. 本文方法则完全从力学平衡能量学习 $q$ 和密度修正, 不要求 $\lambda$ 的标签, 也无需精确场. 例如, 在非线性 Robin 族中, 即使使用正确的 $q$ (由 $\kappa$ 预测), 预设的 $r=s^q$ 映射也无法完全正则化因模量 $\mu(r)$ 引入的非幂律径向变化, 而密度修正 $w_j$ 能够以非零形式提供额外增益.
训练信号与监督: 已有工作中的神经断裂方法 (如 PINNs 裂纹模型 [9,10]) 通常将网络嵌入到物理场逼近中, 需要参考解或强形式残差作为学习目标. 本文则仅利用离散系统的应变能, 这是一种无监督的力学驱动训练. 由此, 训练得到的坐标可转用于同一几何的不同物理问题 (标量传热转至弹性断裂), 而无需重新训练.
与径向分级节点向量的比较: 通过重新分布节点位置 (如向裂纹尖端聚集), 可在参数域实现类似幂律的效果, 但其要求事先知道合适的分级规律, 且缺乏处理非幂律偏差的灵活性. 本文的映射可视为一种连续分级, 且 $w_j$ 能调整中间半径处的分辨力.
与试空间富集方法的比较: 另一类路线是在 B 样条空间中添加奇异基函数或使用自适应细化 [5,6,7]. 此类方法修改了试空间, 而本文方法保持试空间不变, 仅改变坐标, 因此保留了标准刚度矩阵结构和 SIF 提取流程. 另一方面, 在相同自由度下, 本文的力学训练映射在能量范数误差上优于恒等映射和典型富集方法, 但当奇异结构已精确已知时, 富集方法可能仍具竞争力.
适用范围与代价: 本文方法适用于二维裂纹尖端、凹角奇异性以及径向非齐次特征值问题. 代价是一次性的训练计算 (粗离散下若干次 Galerkin 求解及梯度回传) 和训练后对映射的 Jacobian 审计, 但重复求解时仅需执行标准等几何分析的装配与求解, 与预设幂律映射的计算量相当. 容错性方面, 正密度构造自动保证了 $r'_\phi>0$ 和边界锚定, 无需在训练中施加惩罚或投影. 相比之下, 一些基于网络的几何变换方法 [14,15] 可能需要额外的惩罚项来确保双射性, 增加了训练难度.

总体上, 本文的核心贡献在于证明了: 当幂律映射不足以描述奇异坐标时, 从离散能量学习一个保持容许性的神经坐标修正能够显著提升逼近精度, 并且这种学习不需要指数标签或精确场, 这为处理更复杂的、奇异性结构不确定的问题提供了新的范式.

4. AlgMortar：一种全代数的多尺度 mortar 预条件子

AlgMortar: a fully algebraic multiscale mortar preconditioner

作者: Luan F. Santos, Fabricio S. Sousa, Roberto F. Ausas, Rafael T. Guiraldello, Felipe Pereira

类别: math.NA

发布日期: 2026-07-24

arXiv ID: 2607.22905v1

摘要翻译: 求解二阶椭圆型方程离散产生的大规模对称正定线性系统具有挑战性，特别是在系数高度非均质的应用中（例如多孔介质中的流动），这可能导致系统严重病态。在此背景下，近来多尺度方法凭借其良好的并行可扩展性被用于加速 Krylov 子空间方法。本文提出了 AlgMortar，它是多尺度 mortar 混合有限元方法（MMMFEM）的一种全代数实现。该方法仅使用从细网格系统矩阵中提取的信息，从而便于在现有求解器中实施。AlgMortar 利用图划分直接从矩阵图定义区域分解，在每个子区域上构造模拟 Dirichlet 问题的局部线性系统，并通过一个代数界面条件耦合所得局部解，该条件复原了 MMMFEM 的弱通量连续性机理。我们证明：当细网格矩阵对称正定且非对角元非正时（这一结构常见于椭圆问题的离散），该方法是适定的。对达西流问题有限体积离散所得细网格线性系统进行的数值实验表明，当作为共轭梯度法的预条件子时，所提方法在具有挑战性的非均质、高反差测试用例（包括高度不规则角点网格）中表现出良好的可扩展性，并可与当前最先进的代数多重网格方法相媲美。

Comment: 无

链接: http://arxiv.org/abs/2607.22905v1

1. 研究对象和背景

本文考虑求解由二阶椭圆型方程

$$-\nabla\cdot(K\nabla p) = f \quad \text{in}\ \Omega\subset\mathbb{R}^d,\ d=2,3,$$

经细尺度离散化得到的大规模对称正定线性系统 $A\mathbf{p}=\mathbf{f}$. 这里 $K$ 是对称正定张量, $p$ 代表压力, $f$ 为源项, 此类问题广泛存在于多孔介质流、地下储层模拟、地下水流动以及非均质材料热传导等领域. 当介质高度非均质时, 离散系统往往严重病态, 求解十分困难. 为利用大规模并行架构, 预条件 Krylov 子空间方法(如 PCG)需要具有良好并行可扩展性的预条件子. 区域分解预条件子天然适合并行, 而多尺度方法作为一类重要的区域分解手段, 通过构造能捕捉局部非均质性的基函数, 借助粗问题耦合得到全局近似. 多尺度 Mortar 混合有限元法(MMMFEM)通过求解局部 Dirichlet 问题构建多尺度基, 并用弱形式强加界面法向通量连续性. 多尺度混合-混合方法(MHM)使用局部 Neumann 问题并弱强加压力连续性. 多尺度 Robin 耦合方法(MRCM)则引入 Robin 边界条件, 统一了前两者. 然而, 上述方法在构造粗空间和基函数时通常需要几何信息(子域网格、界面几何)、PDE 系数 $K$ 以及具体离散格式, 而这些信息在工业模拟中可能高度复杂、专有、不可获取, 或在代码间传输代价高昂. 代数预条件子仅依赖全局离散矩阵和右端项, 能够适应任意网格、非结构化离散、混合维模型及剧烈变化的系数, 免去粗网格或几何邻域的手动重构, 简化集成并减少网格转换误差. 已有多项代数多尺度工作, 如两阶段代数多尺度方法(TAMS)、代数多尺度求解器(AMS)、多尺度两点通量逼近(MsTPFA)等, 但它们多与特定离散格式关联. 鉴于 MRCM 预条件在 MMMFEM 极限下展现出优异的计算性能, 以及对代数求解器的实际需求, 本文提出 AlgMortar——一种完全代数化的 MMMFEM 实现. 该方法仅利用 SPD 矩阵 $A$, 通过图划分定义域分解, 构造局部代数 Dirichlet 算子, 并依据矩阵连通性建立界面弱通量连续性条件, 从而在仅使用矩阵信息的条件下复现 MMMFEM 的机制. 当 $A$ 源于一致笛卡尔网格上 $K=I$ 的 TPFA 离散时, AlgMortar 精确还原 MMMFEM; 对于非均质问题, 该方法仍可作为 PCG 预条件子高效使用.

2. 主要定理或主要结果

在矩阵 $A$ 对称正定, 且其界面耦合权重(非对角元)非正(即 M-矩阵结构)的假设下, AlgMortar 方法是适定的.

唯一可解性: 系统(3.34)存在唯一解. 将系统重写为对称形式 $\mathcal{Q}\begin{pmatrix} \mathbf{p} \\ \boldsymbol{\eta} \end{pmatrix} = \begin{pmatrix} \frac14 \mathbf{f} \\ 0 \end{pmatrix}$, 其中 $\mathcal{Q} = \begin{pmatrix} \frac14 \widehat{A} & M^T W Z \\ Z^T W M & -Z^T W Z \end{pmatrix}$, $\widehat{A} = \operatorname{diag}(A_1,\dots,A_m)$, $A_i = A_{ii} - \sum_{j\sim i} D_{ij}$, $D_{ij} = \operatorname{diag}(A_{ij} \mathbf{1})$, $W$ 为负定对角权矩阵(其元素为矩阵 $A$ 的对应非对角元), $Z$ 满足 $\operatorname{range}(Z) = \Lambda$ (mortar 空间). 在所述假设下 $\mathcal{Q}$ 正定, 从而解唯一.
条件数估计: 对 PCG 预条件, 证明了谱聚类性质依赖于界面空间 $\Lambda$ 逼近图分区界面上平均压力的精度. 具体估计形式在现有摘录中未能完整呈现, 但论文指出条件数与界面空间的逼近性质相关.
代数等价性: 若 $A$ 源自均匀笛卡尔网格上 $K=I$ 的全局 TPFA 离散, 则 AlgMortar 精确恢复 MMMFEM (见注 3.4).
数值实验: 在笛卡尔网格和非规则角点网格上, 将 AlgMortar 作为 PCG 预条件子应用于达西流有限体积离散系统. 结果表明, 与先进的代数多重网格 (AMG) 方法相比, 在高对比度、高度非均质渗透率场的大规模问题上, AlgMortar 具有竞争力甚至更优; 方法展现出良好的并行可扩展性 (具体迭代次数与加速比数据未在摘录中提供).

3. 研究方法、关键技术和核心工具

AlgMortar 的核心创新在于完全代数的构造, 避免了对网格、系数或离散格式的依赖. 主要技术步骤与分析工具如下:

1. 图分区与代数界面定义. 从矩阵图 $G(A)$ 出发, 利用图划分工具 METIS 将未知量集合 $\mathcal{V}$ 划分为 $m$ 个非重叠子域 $\mathcal{V}_i$, $i=1,\dots,m$. 界面边集 $E_{ij} = \{(k,l): k\in\mathcal{V}_i, l\in\mathcal{V}_j, a_{kl}\neq 0\}$ 仅由矩阵非零模式确定. 代数骨架 $E = \bigcup_{i

2. 代数算子的系统构造. 定义了一系列仿照 MMMFEM 几何对应物的代数算子:

关联矩阵 $C_{ij}\in\mathbb{R}^{\mathcal{V}_i\times E_{ij}}$: 若边 $e=(k,l)\in E_{ij}$ 与结点 $k\in\mathcal{V}_i$ 关联, 则 $[C_{ij}]_{k,e}=1$, 否则为 $0$.
限制矩阵 $R_{E_{ij}}$: 从 $E$ 提取 $E_{ij}$ 对应的分量.
权重矩阵 $W_{ij}\in\mathbb{R}^{E_{ij}\times E_{ij}}$: 对角元 $[W_{ij}]_{e,e} = a_{kl}$, 对应透射率; 整体 $W = \sum_{i
迹算子: $L = \sum_{i
平均算子 $M = \frac12 (L+U)$, 满足 $(M\mathbf{p})_e = (p_k + p_l)/2$.
界面耦合算子 $B_{ij} = 2 C_{ij} W_{ij} R_{E_{ij}}$, 用于将界面变量映射到子域上的贡献; 总界面算子 $B_i = \sum_{j\sim i} B_{ij}$, $B = (B_1^T,\dots,B_m^T)^T$. 特别地, 命题 3.1 证明 $B^T = 4 W M$, 这一关系在后续推导弱耦合条件时起关键作用.

3. 修改 Dirichlet 算子的构造. 将原系统 $A\mathbf{p} = \mathbf{f}$ 按子域拆分: $A_{ii}\mathbf{p}_{\mathcal{V}_i} + \sum_{j\sim i} A_{ij} \mathbf{p}_{\mathcal{V}_j} = \mathbf{f}_{\mathcal{V}_i}$. 利用 $A_{ij}$ 的行和定义 $D_{ij} = \operatorname{diag}(A_{ij}\mathbf{1}_{\mathcal{V}_j})$, 构造局部算子 $A_i = A_{ii} - \sum_{j\sim i} D_{ij}$. 在 TPFA 离散下, $A_i$ 精确对应子域上齐次 Dirichlet 边界条件的离散算子. 由于 $D_{ij}$ 仅依赖矩阵元素, 这一修改完全是代数的. 当 $A$ SPD 且 $W$ 对角元为负时, 每个 $A_i$ 也 SPD.

4. 弱通量连续性的代数实现. 将原系统等价改写为

$$\widehat{A}\mathbf{p} + B\boldsymbol{\lambda} = \mathbf{f},\quad \boldsymbol{\lambda} = M\mathbf{p},$$

其中 $\widehat{A} = \operatorname{diag}(A_1,\dots,A_m)$. 引入有限维 mortar 空间 $\Lambda\subset\mathbb{R}^E$ 作为界面压力的逼近空间, 并在权 $W$ 下弱强加 $\boldsymbol{\lambda} \approx M\mathbf{p}$:

$$\boldsymbol{\mu}^T W (\boldsymbol{\lambda} - M\mathbf{p}) = 0,\quad \forall\boldsymbol{\mu}\in\Lambda.$$

注 3.3 指出, 该条件正是 MMMFEM 通量连续性条件(2.3)的代数翻版, 因为 $a_{kl}$ 扮演透射系数角色.

5. 多尺度基函数与界面问题. 在每个子域上求解特解 $A_i\bar{\mathbf{p}}_{\mathcal{V}_i} = \mathbf{f}_{\mathcal{V}_i}$ 及对每个基函数 $\psi_m\in\Lambda$ 的局部 Dirichlet 问题 $A_i\hat{\mathbf{p}}^{(m)}_{\mathcal{V}_i} = -B_i\psi_m$. 将解表达为 $\mathbf{p} = \bar{\mathbf{p}} + \hat{\mathbf{p}}(\boldsymbol{\lambda})$, $\hat{\mathbf{p}}(\boldsymbol{\lambda}) = -\widehat{A}^{-1} B\boldsymbol{\lambda}$, 代入弱连续性条件得到关于 $\boldsymbol{\lambda}$ 的界面系统:

$$\boldsymbol{\mu}^T \bigl( W M \widehat{A}^{-1} B + W \bigr) \boldsymbol{\lambda} = \boldsymbol{\mu}^T W M \bar{\mathbf{p}},\quad \forall\boldsymbol{\mu}\in\Lambda.$$

应用基函数测试后, 得到一个低维对称正定系统(推论 4.2), 该系统的形成与求解均基于预计算的局部基函数.

6. 适定性证明框架. 引入矩阵 $Z$ 使 $\Lambda=\operatorname{range}(Z)$, 并将系统(3.34)等价转化为对称形式 $\mathcal{Q} \begin{pmatrix} \mathbf{p} \\ \boldsymbol{\eta} \end{pmatrix} = \begin{pmatrix} \frac14 \mathbf{f} \\ 0 \end{pmatrix}$ (式 4.1). 通过在假设下验证 $\mathcal{Q}$ 的正定性, 证明了存在唯一性. 正定性证明利用了 $A$ 的 SPD 性质、$W$ 对角元的严格负定性以及 $Z$ 的满列秩, 核心引理是 $\widehat{A}$ 的正定性以及 $\mathcal{Q}$ 的舒尔补正定. 条件数估计则通过分析 $\mathcal{Q}$ 的谱分布与界面逼近性质的关系得到.

这些技术共同构成了 AlgMortar 的理论基础, 使得方法在仅使用矩阵信息的条件下保持 MMMFEM 的核心机制, 并能严格证明其适定性和预处理效果.

4. 与之前工作的比较

AlgMortar 与已有工作的区别与推进主要体现在以下几个方面:

1. 与几何多尺度方法的比较. MMMFEM, MHM 和 MRCM 等方法在构建粗空间和基函数时严重依赖几何信息(子域网格、界面几何)、PDE 系数 $K$ 以及特定离散格式(如 Raviart-Thomas 元). 这在实际工业代码中往往需要额外传输或重构几何与系数数据, 增加了预处理集成的复杂度. AlgMortar 完全避免了对这些信息的依赖: 它只从矩阵 $A$ 出发, 图划分自动生成子域, 界面由矩阵非零模式定义, 局部 Dirichlet 算子通过矩阵元素修正产生. 这种“黑箱” 特性使得 AlgMortar 可以方便地嵌入现有模拟器, 无需网格转化或系数重构.

2. 与其他代数多尺度方法(TAMS, AMS, MsTPFA)的比较. 这些方法同样致力于代数构造粗空间, 但它们往往与 TPFA 或有限体积格式绑定, 且构造过程通常需要特定假设(如通量连续性). AlgMortar 则建立在更一般的 SPD 系统上, 仅要求矩阵具有 M-矩阵结构(非对角元非正). 其适定性证明和条件数分析为代数多尺度预条件提供了严谨的理论基础, 而许多同类代数方法缺少这种分析. 此外, AlgMortar 明确地以 MMMFEM 为蓝本, 在代数框架下复现了弱通量连续性, 这使其在理论上与已有几何方法直接衔接.

3. 与代数多重网格(AMG)的性能对比. 论文数值实验将 AlgMortar 作为 PCG 预条件子与 BoomerAMG 等先进 AMG 方法比较. 在二维/三维高对比度渗透率场、不规则角点网格等挑战性算例上, AlgMortar 的迭代次数和计算时间优于或至少与 AMG 相当. 尤其在涉及大跳越高对比度时, AMG 可能因粗化策略失效而性能下降, 而 AlgMortar 受益于局部多尺度基函数的精确求解, 保持了健壮性. 同时报告的强/弱可扩展性实验表明, AlgMortar 在数千处理器核心上具有良好的并行效率.

4. 作为预条件子的理论贡献. 文献中已有将 MMMFEM 或 MRCM 用作预条件子的尝试(如 [9]), 但多基于几何构造. 本文首次给出完全代数实现的 MMMFEM 预条件子, 并证明其界面 Schur 补的正定性, 从而保证 PCG 可应用. 同时, 推导的条件数估计显示, 收敛率依赖于 mortar 空间逼近界面平均压力的能力, 为 mortar 空间的选取提供了理论指导.

5. 适用范围与局限. AlgMortar 假设 $A$ 是 SPD 且非对角元非正, 这覆盖了一大类椭圆型方程离散(如 TPFA, 某些混合有限元离散). 但若离散产生非 M-矩阵(例如部分有限元离散或各向异性强), 方法可能失效或需要修正. 此外, 局部求解 $A_i$ 并进行基函数计算需在每个子域上分解矩阵, 当子域较大时内存和计算开销增加; 不过子域独立并行可缓解这一问题. 与 AMG 相比, AlgMortar 需要用户提供 mortar 空间的维度或基函数分布, 不具备 AMG 的完全“黑箱” 自适应能力.

了解领域论文（摘要翻译）

1. DGM与PINN算法求解非线性偏微分方程的全局收敛性

Global Convergence of DGM and PINN Algorithms for Solving Nonlinear PDEs

作者: Justin Sirignano, Konstantinos Spiliopoulos, Samuel Cohen

类别: cs.LG, math.NA

发布日期: 2026-07-27

arXiv ID: 2607.24726v1

链接: http://arxiv.org/abs/2607.24726v1

摘要翻译

中文标题: DGM与PINN算法求解非线性偏微分方程的全局收敛性

摘要翻译: 深度 Galerkin 方法（DGM）与物理信息神经网络（PINNs）已成为快速发展的科学机器学习领域中求解偏微分方程（PDEs）的广泛使用方法。在这些方法中，通过（随机）梯度下降最小化神经网络的 PDE 残差来训练神经网络，以逼近 PDE 解。由于 PDE 残差目标函数的非凸性，训练后的神经网络原则上可能仅收敛到目标函数的局部极小值点（该点并不是 PDE 的解）。因此，关于这些算法的数学基础存在一个长期存在的问题，确立训练后的神经网络将收敛到 PDE 解具有重要价值。针对一类半线性 PDE（关于解及其一阶导数非线性），我们证明了用梯度下降训练以最小化 PDE 残差目标函数的神经网络将收敛到 PDE 解。

2. 微扰-NeuSA: 面向时间依赖偏微分方程的结构化谱框架

Perturbative-NeuSA: A Structured Spectral Framework for Time-Dependent PDEs

作者: Xianli Zhu, Jia Yin

类别: cs.LG, math.NA

发布日期: 2026-07-27

arXiv ID: 2607.24345v1

链接: http://arxiv.org/abs/2607.24345v1

摘要翻译

中文标题: 微扰-NeuSA: 面向时间依赖偏微分方程的结构化谱框架

摘要翻译: 神经网络谱偏微分方程求解器通常学习整个未解的向量场，即使一个廉价的近似模型已经可以捕捉到大部分轨迹。在这里，我们引入微扰-NeuSA，一种将目标解分解为低保真背景和高分辨率扰动的残差公式，从而只学习未解析的动力学。从精确的扰动方程出发，该方法结合了固定的谱算子、依赖背景的修正项、目标偏微分方程中的背景缺陷和一个可选的神经封闭项。这种构造使得物理结构和神经封闭项的作用可以分别度量。在二维Burgers方程、Klein-Gordon方程和非均匀二维波动方程上，确定性结构化求解器在无需任何神经网络训练的情况下优于训练后的NeuSA基线模型。最大的增益出现在Burgers方程上，其中确定性修正将训练误差和外推误差分别降低了24倍和44倍。此外，在Klein-Gordon方程上对七种背景分辨率进行的扫描表明，封闭项的作用是有条件的：它可以使较差的背景改善3.6倍，在中等分辨率时变为中性，并降低良好分辨背景的性能。然而，对于波动方程，当残留残差局限于界面时，封闭项提供了额外的18%的误差降低。多初始条件诊断进一步表明，有用的封闭项机制取决于初始条件谱，并且当结构化修正已经捕捉到主导的Burgers动力学时，封闭项在外推中可能会消失。因此，微扰-NeuSA将神经封闭重新定义为一种有条件的、可诊断的修正，受背景保真度、残差组织形式以及与封闭模型的兼容性所控制。

3. 强磁场下相对论带电粒子动力学的多物理结构保持积分器的一致误差界

A multi-physics structure-preserving integrator with uniform error bounds for relativistic charged-particle dynamics under strong magnetic fields

作者: Mengting Hu, Yifa Tang, Bin Wang

类别: math.NA

发布日期: 2026-07-27

arXiv ID: 2607.24058v1

链接: http://arxiv.org/abs/2607.24058v1

摘要翻译

中文标题: 强磁场下相对论带电粒子动力学的多物理结构保持积分器的一致误差界

摘要翻译: 本文针对最大阶标度下强磁场中的四维相对论带电粒子动力系统，发展了一种显式的多物理结构保持Strang分裂格式。该格式不仅保体积、能量和Lorentz不变性，而且对位置和平行速度分量具有二阶一致误差界。我们为这些几何性质及误差收敛提供了严格的理论证明，随后通过若干数值实验进行了验证。

4. 双调和算子多重与聚簇特征值的 $C^0$ IPG 逼近的后验误差估计子

An a Posteriori Error Estimator for $C^0$ IPG Approximation of Multiple and Clustered Eigenvalues of the Biharmonic Operator

作者: Jianing Guo, Qigang Liang

类别: math.NA

发布日期: 2026-07-25

arXiv ID: 2607.23215v1

链接: http://arxiv.org/abs/2607.23215v1

摘要翻译

中文标题: 双调和算子多重与聚簇特征值的 $C^0$ IPG 逼近的后验误差估计子

摘要翻译: 本文基于 $C^0$ 内部惩罚伽辽金（$C^0$IPG）离散，针对双调和算子的多重与聚簇特征值所对应的特征函数，提出并分析了一个后验误差估计子。所提出的估计子能够高效地捕捉目标特征函数的局部奇性，并在自适应过程中发挥重要作用。我们发展了一套严格的基于聚簇投影的分析，并引入一个辅助的理论误差估计子，将不变子空间误差与一个可计算的残差估计子联系起来。最重要的是，所得到的可靠性与有效性界关于网格尺寸、网格层次以及目标聚簇内部的谱间隔是鲁棒的。数值实验支持理论结果，并展示了所提估计子的鲁棒性。

5. 基于投影环境联络拉普拉斯算子的微分形式数据驱动扩散过程

Data-Driven Diffusion Processes on Differential Forms via the Projected Ambient Connection Laplacian

作者: Alvaro Almeida Gomez, Jorge Duque Franco

类别: math.NA, cs.LG

发布日期: 2026-07-25

arXiv ID: 2607.23192v1

链接: http://arxiv.org/abs/2607.23192v1

摘要翻译

中文标题: 基于投影环境联络拉普拉斯算子的微分形式数据驱动扩散过程

摘要翻译: 我们针对由点云采样的光滑黎曼流形上的微分形式，开发了一种投影环境联络拉普拉斯算子的数据驱动逼近。所提出的构造将经典的扩散映射和向量扩散映射的框架从标量函数和切向量场扩展到了任意阶的微分形式。我们的方法基于微分形式的一种新表示，即通过扩展经典的音乐同构得到的交替微分阵列。该表示使得能够从点云数据直接构造一个矩阵值扩散算子来逼近投影环境联络拉普拉斯算子，而无需网格或单纯复形。所提出的离散化继承了扩散映射的渐近最优核带宽缩放，从而比先前数据驱动的霍奇拉普拉斯算子逼近具有更强的收敛保证。基于该算子，我们导出了微分形式上热方程的全数据驱动显式欧拉格式，并通过单位球面上的数值实验验证了所提出的方法。实验证实了解析解预期的衰减，并证明了所提出离散化的有效性。所提出的框架将向量扩散映射自然地推广到任意阶微分形式，并为直接从点云数据数值逼近几何偏微分方程奠定了实用基础。

6. 用于非线性平衡律的物理信息令牌变换器方法论. I. 施瓦西-伯格斯流体流动

Physics-informed token transformer methodology for nonlinear balance laws. I. Schwarzschild–Burgers fluid flows

作者: Philippe G. LeFloch, Shuyang Xiang

类别: math.NA, physics.flu-dyn

发布日期: 2026-07-25

arXiv ID: 2607.23143v1

链接: http://arxiv.org/abs/2607.23143v1

摘要翻译

中文标题: 用于非线性平衡律的物理信息令牌变换器方法论. I. 施瓦西-伯格斯流体流动

摘要翻译: 我们针对一维空间中的非线性双曲平衡律, 引入了一种物理信息令牌变换器 (PITT) 方法论, 该方法使用分段稳态剖面来表示近似弱解。该方法结合了符号方程令牌化、傅里叶神经算子编码器、用于激波运动的显式 Rankine–Hugoniot 定律以及一个学习的修正项。为了清晰起见, 我们在此针对相对论施瓦西-伯格斯方程进行阐述, 这是施瓦西背景下球对称流体流动的一个标量模型。对于该模型, 稳态不变量和广义黎曼解是显式的, 因此它们可以被嵌入到神经演化中。特别地, 主导间断由解析的跳跃条件推进, 而学习部分则重建光滑区域、稀疏波扇、几何依赖性和有限分辨率效应。该方法旨在准确追踪波前并保持相关的稳态。我们在运动激波、驻定激波和稀疏波上测试了 PITT 方法, 并将其与标准的高阶有限体积近似进行了比较。我们还分析了标准伯格斯极限 (当施瓦西质量趋于零时)。Rankine–Hugoniot 先验在这些测试中起到了主导作用, 而方程令牌化则提供了系统性的额外增益。该方法适用于涉及几何效应和/或复杂激波动力学的问题, 并且在此用于研究稳态解的扰动的长时间动力学。特别地, 我们给出了受扰稳态流的激波位置的渐近传播律。

7. 用于任意域上偏微分方程学习的几何感知 LegONet

Geometry-aware LegONet for PDE Learning on Arbitrary Domains

作者: Jiahao Zhang, Yueqi Wang, Guang Lin

类别: math.NA

发布日期: 2026-07-25

arXiv ID: 2607.23069v1

链接: http://arxiv.org/abs/2607.23069v1

摘要翻译

中文标题: 用于任意域上偏微分方程学习的几何感知 LegONet

摘要翻译: 习得的偏微分方程求解器常将控制算子与训练所用的几何、边界条件和离散格式纠缠在一起。这限制了相同物理规律在新域上的复用，也使得物理定律的发现依赖于具体几何。我们提出几何感知 LegONet（gLegONet），它是 Lego 式算子学习在边界流形上的推广。物理机制先在一个环境谱域上以模块化变分块的形式一次性预训练。对于目标几何，采样得到的边界约束定义了一个仿射容许流形，其质量正交归一的切向坐标被直接用于演化动力学和计算候选定律发现特征。因此，改变求解域仅改变代数坐标接口，而不改变已学得的算子块。这将任意域上的偏微分方程学习从针对特定几何的重训练或软惩罚强制，转变为可复用机制的边界保证式组装。在未见域上的正演模拟和稀疏辨识测试中，该方法能将边界残差保持在代数容差附近，并从短时间观测中给出预测性的控制方程。

不相关论文（基本信息）

1. 近端点法的安德森加速：精确自适应极小极大、谱相变与最优保护

Anderson acceleration of the proximal point method: the exact adaptive minimax, a spectral phase transition, and optimal safeguarding

作者: Zheng Jia, Yekini Shehu, Yonghong Yao

类别: math.NA, math.OC

发布日期: 2026-07-27

arXiv ID: 2607.24643v1

链接: http://arxiv.org/abs/2607.24643v1

摘要: \noindent We study residual-polynomial acceleration of the proximal point method (PPM) for maximal monotone inclusions, with Anderson acceleration (AA) as the prototypical adaptive scheme. We answer three questions exactly. (i)~The minimax complexity over all adaptive methods is precisely $d_0/(K+1)$ per $K$ resolvent evaluations. The upper bound is attained by the averaged-reflection estimator; the matching lower bound uses an explicit skew-adjoint instance with resolvent eigenvalues at the roots of $u^{K+1}=-1$ and $\csc^2$-distributed masses, on which every degree-$K$ polynomial method satisfies $\|r(y_K)\|\ge 1/(K+1)$. The optimal polynomial is uniquely the Fejér kernel, and the same instance certifies a per-step floor. (ii)~A sharp phase transition separates regimes: Jackson-kernel polynomials achieve $O(d_0/(K^2 s))$ when the spectral floor $s$ satisfies $sK\to\infty$, while at the critical scale $s\asymp 1/K$ the barrier is exactly $1/(K+1)$. The picture extends to normal operators and the nonlinear family $M=S+N_C$. (iii)~On linear problems AA-PPM needs no safeguarding; on nonlinear problems certification of the $O(1/k)$ envelope requires exactly two oracle evaluations per iteration, and this factor is optimal. We also correct and complete the theory for structured problems—affine, strongly monotone, piecewise-affine, and Hölderian growth—and confirm all predictions numerically.

2. 双曲系统的自适应多物理场耦合

Adaptive Multiphysics Coupling for Hyperbolic Systems

作者: Simon Candelaresi, Erik Faulhaber, Michael Schlottke-Lakemper

类别: math.NA

发布日期: 2026-07-27

arXiv ID: 2607.24600v1

链接: http://arxiv.org/abs/2607.24600v1

摘要: For the discontinuous Galerkin code Trixi.jl we implement capabilities for adaptively coupling arbitrarily many domains with different physics. The coupling of the systems is realized through the exchange of boundary information and user-definable coupling functions. This gives us the ability to couple systems that do not share a single variable or have a very different number of variables. This is particularly useful when we have a hierarchy of systems. Our implementation is such that we can adaptively select the model of the coupled systems in the course of the simulation. This allows for highly dynamical scenarios as can be found e.g. in astrophysics. The criteria for adaptivity can be user defined and tailored to the physical problem. Compared to computing the most complex model in the entire simulation domain, our method of coupled systems of high and low complexity leads to a significant reduction in computational time with very little overhead.

3. 任意阶 Padé 封闭锚定双导数时间离散: $s$ 个有效级, $2s$ 阶, 与 $L$-稳定性

Arbitrary-Order Padé-Closed Anchored Two-Derivative Time Discretizations: $s$ Active Stages, Order $2s$, and $L$-Stability

作者: Zhixin Huo

类别: math.NA

发布日期: 2026-07-27

arXiv ID: 2607.24592v1

链接: http://arxiv.org/abs/2607.24592v1

摘要: An arbitrary-order family of implicit two-derivative one-step methods is constructed in an anchored active-stage formulation. At each information node the method uses both the vector field and its first total time derivative, enriching the local Hermite data without increasing the number of unknown stage states. With the known initial value retained as an anchor and $s$ unknown active stages, $2s$ Hermite moment conditions yield global order $2s$. The two remaining coefficients in each stage row are fixed by the second-subdiagonal Padé approximant $[s-1/s+1]_{e^z}$. For every ordered real node set, a Padé–Hermite basis theorem proves that the closure is unique, preserves all moment conditions, and gives $\det(I-zA-z^2\widehat A)=Q_s(z)$ and $R_s(z)=P_s(z)/Q_s(z)$. Hence the coupled stage system has no hidden poles and the accepted one-step map is $L$-stable (and therefore $A$-stable) for every positive integer $s$. Exact symbolic verification is reported through $s=6$, and high-precision computations confirm orders $2,4,6,$ and $8$ for the first four members. At equal active-stage count, comparisons with Gauss–Legendre and Radau IIA methods demonstrate the combined high-order accuracy and strong stiff damping of the construction over broad step-size ranges.

4. 强非恒定磁场下三维带电粒子动力学的曲线参考指数积分器

A Curve-Reference Exponential Integrator for Three-Dimensional Charged-Particle Dynamics under a Strong Nonconstant Magnetic Field

作者: Zhirui Shen, Bin Wang

类别: math.NA

发布日期: 2026-07-27

arXiv ID: 2607.24523v1

链接: http://arxiv.org/abs/2607.24523v1

摘要: We study a three-dimensional charged-particle dynamics model in the nonrelativistic momentum formulation and construct a curve-reference exponential integrator (CREI) for this system. Since the zero eigenvalue of linear section is not semisimple, the system after usual change of variables does not meet the spectral condition used in the classical locally linearized extended exponential integrator (LLEEI). A kernel-range decomposition of the dominant magnetic matrix and a further linear transformation reduce the equation to a system with a semisimple zero block and one constant skew-symmetric oscillatory block. The CREI expands the nonlinear term along the exact oscillatory curve rather than at a fixed point. Retaining monomials through degree k defines CREI(k + 1). We prove uniform local and global error bounds and present reproducible numerical experiments for convergence in h and uniformity in ε.

5. 基于加速对称非负矩阵分解的低秩依赖分解

Low-Rank Dependence Decomposition via Accelerated Symmetric Non-negative Matrix Factorization

作者: Lavinia Ghita, Dhruv Desai, Jake Goldberg, Roman Yokunda Enzmann

类别: cs.LG, math.NA

发布日期: 2026-07-27

arXiv ID: 2607.24518v1

链接: http://arxiv.org/abs/2607.24518v1

摘要: Symmetric non-negative matrix factorization (SymNMF) recovers latent group structure from a dependence matrix, but its dense, quadratic-memory objective has confined prior work to moderate sizes. We present a large-scale GPU study of seven algorithm families (over 30 configurations) on absolute Pearson correlation and tail pairwise dependence matrices from Extreme Value Theory, two proxies for empirical risk-factor estimation on large portfolios. A trace-identity reformulation eliminates all $n \times n$ intermediates, so a single GPU reaches $n \approx 10^5$ and multi-node distribution scales to $n = 10^6$ and beyond. Under a two-phase protocol, eleven methods converge at moderate scale; six remain efficient enough at $n = 10^5$ (five AdaGrad-family plus ADMM), and five AdaGrad-family methods still converge at $n = 10^6$: AdaGrad, RMSprop, and three we introduce (Piecewise AdaGrad, Row-Stochastic SVRG, Block-SVRG AdaptGrow). At $n = 10^6$ the fastest solver tracks the matrix spectrum: Block-SVRG AdaptGrow wins on the flat, ill-conditioned tail-dependence spectrum, where its lower per-iteration cost decides a long factorization, and full-batch AdaGrad wins on the dominant-low-rank correlation spectrum, where the run is short. We also benchmark spherical K-means as a hard-label baseline: cheaper when angular cluster structure is present, yet provably degenerate once the matrix collapses toward a single common factor, where the soft factorization remains necessary.

6. Gompertz 生长下肿瘤总体的优化

Optimization of the total tumor population under Gompertz growth

作者: Iulia Martina Bulai, Francesca Gladiali, Benedetta Pellacci

类别: math.AP, math.NA

发布日期: 2026-07-27

arXiv ID: 2607.24468v1

链接: http://arxiv.org/abs/2607.24468v1

摘要: We study optimal control problems for a stationary reaction–diffusion model describing the spatial distribution of a tumor cell population with Gompertz growth. The control $m(x)$ represents a treatment term acting as a density-dependent removal rate and it is subject to $L^{1}-L^{\infty}$ constraints. When the intrinsic growth rate is constant, the uniform distribution of the treatment is shown to be the unique minimizer. For the maximization problem, we prove that every optimal control is of bang-bang type. In addition, we show that in the one dimensional case and for sufficiently large diffusion rates, the positivity set of optimal controls is an interval sticking to one of the extrema of the domain. Finally, numerical simulations complement the theoretical analysis and explore regimes that are not fully covered by the results proved in the paper. The computations confirm the bang-bang structure of maximizers, and illustrate how the shape of optimal controls and the associated states are affected by spatial heterogeneity in the growth rate, localized admissible treatment regions, and the diffusion coefficient. Moreover, they reveal a monotone dependence of the optimized total population on the diffusion coefficient: this is a new phenomenon with respect to the logistic setting.

7. 用于双曲守恒律的具有统一模板的五阶有限体积基于导数的Hermite WENO格式

Fifth-order finite volume derivative-based Hermite WENO scheme with unified stencils for hyperbolic conservation laws

作者: Peiwen Chen, Zhuang Zhao

类别: math.NA

发布日期: 2026-07-27

arXiv ID: 2607.24325v1

链接: http://arxiv.org/abs/2607.24325v1

摘要: In this paper, we propose a derivative-based finite volume Hermite WENO (HWENO) scheme for hyperbolic conservation laws, where both the solution and its first-order derivatives are evolved in time and utilized in spatial reconstructions. The key challenge for solving hyperbolic conservation laws is the possible emergence of discontinuities in the numerical solutions. When facing discontinuities, the derivatives can become excessively large, which may compromise the robustness of HWENO schemes. In the first HWENO scheme, different sets of stencils were adopted for reconstructing the governing equation and the derivative equation, respectively, aiming to reduce the influence of the derivatives while preserving high-order accuracy. However, this approach not only substantially increases computational cost but also introduces considerable algorithmic complexity. To overcome these limitations, we exclude the information of the target cell’s derivatives from spatial reconstructions, while employing the same reconstructed polynomial during temporal evolution to limit the derivatives. This strategy enhances the robustness of traditional HWENO schemes and allows unified stencils within the derivative-based HWENO framework. Furthermore, the proposed scheme supports arbitrary positive linear weights that sum to one and maintains a compact stencil. Numerical results demonstrate the high-order accuracy, efficiency, high resolution, and robustness of the proposed HWENO scheme.

8. 带压缩的坐标下降法用于全组态相互作用的收敛性与误差分析

The Convergence and Error Analysis of Coordinate Descent Methods with Compression for Full Configuration Interaction

作者: Yingzhou Li, Qiang Wu

类别: math.NA, math.OC

发布日期: 2026-07-27

arXiv ID: 2607.24319v1

链接: http://arxiv.org/abs/2607.24319v1

摘要: We study the effect of compression in Coordinate Descent Full Configuration Interaction (CDFCI) within an unconstrained optimization formulation of the full configuration interaction ground-state problem. Under suitable local assumptions, we prove that the compressed iteration converges linearly to the solution of an associated restricted problem. We also characterize the convergence point of the compressed algorithm. Under an additional exponential decay assumption on the target eigenvector, we show that the resulting eigenvalue error is of order $τ^2$, where $τ$ denotes the compression threshold. Numerical results support the analysis.

9. 具有非负权重的$L_1$-偏差遭受维度灾难

The $L_1$-Discrepancy with Nonnegative Weights Suffers from the Curse of Dimensionality

作者: Josef Dick

类别: math.NA, math.PR

发布日期: 2026-07-27

arXiv ID: 2607.24290v1

链接: http://arxiv.org/abs/2607.24290v1

摘要: We prove that the $L_1$-discrepancy with arbitrary nonnegative weights suffers from the curse of dimensionality. More precisely, for every $\varepsilon \in (0,1)$ and $d \in \mathbb{N}$, the inverse of the $L_1$-discrepancy satisfies [ N_{1,+}(\varepsilon, d) \ge \frac{(1-\varepsilon)^2}{1 + \varepsilon} \left( \frac{3+2 \sqrt{3}}{6}\right)^d, ] where $(3+2\sqrt{3})/6 = 1.07735\ldots$. The proof combines a change to a volume-biased probability measure with a fractional-moment estimate for the normalized discrepancy function. The lower bound applies, in particular, to equally weighted point sets. The argument uses the nonnegativity of the weights in an essential way and does not cover arbitrary signed weights.

10. 一维非局部非线性程函方程的一个格式的收敛性

Convergence of a scheme for a one dimensional nonlocal and nonlinear eikonal equation

作者: Diana Al Zareef, Ahmad El Hajj, Antoine Zurek

类别: math.NA

发布日期: 2026-07-27

arXiv ID: 2607.24216v1

链接: http://arxiv.org/abs/2607.24216v1

摘要: In this work, we study a one-dimensional nonlocal and nonlinear eikonal equation without sign restriction on its spatial gradient. The equation is characterized by weak regularity assumptions on both the nonlocal velocity field and the initial data. We derive a periodic version of this model and propose a semi-explicit (IMEX) scheme for its numerical approximation. We prove that the scheme preserves a discrete gradient entropy estimate and establish its convergence in the viscosity sense. Finally, we present numerical results illustrating the behavior of the model and the performance of the proposed scheme.

11. 后验停止准则下大规模不适定问题的随机化Krylov投影迭代Tikhonov正则化

Randomized Krylov-Projected Iterated Tikhonov Regularization for Large-Scale Ill-posed Problems Under A Posteriori Stopping Rule

作者: Ravi Verma, Harshit Bajpai, Ankik Kumar Giri

类别: math.NA

发布日期: 2026-07-27

arXiv ID: 2607.24138v1

链接: http://arxiv.org/abs/2607.24138v1

摘要: We introduce two novel randomized iterative regularization frameworks, termed \texttt{RIGKT} and \texttt{RIAT}, for solving large-scale linear ill-posed inverse problems governed by systems of equations. The proposed methods combine randomized iterated Tikhonov regularization with Krylov subspace projection techniques, utilizing Golub–Kahan bidiagonalization for general rectangular systems (\texttt{RIGKT}) and Arnoldi decomposition for square systems (\texttt{RIAT}). Unlike existing deterministic schemes that rely on fixed iteration counts, our framework incorporates randomized equation selection, an adaptive step-size strategy, and a global, discrepancy-based a posteriori early-stopping rule tailored specifically to the stochastic setting. We present a comprehensive regularization analysis establishing Bregman-distance monotonicity, finite termination, exact-data convergence, and pathwise stability under noise. Furthermore, we prove that the stopped iterates converge almost surely and in the mean-square sense to the true solution, establishing a rigorous regularization property. To the best of our knowledge, this is the first theoretical framework to simultaneously account for randomization, Krylov-subspace dimension reduction, and implementable early stopping. Numerical experiments involving two-dimensional X-ray computed tomography (CT) and image deblurring demonstrate that \texttt{RIGKT} and \texttt{RIAT} reliably reconstruct structural features across various noise regimes.

12. 一种用于忆阻器三组分漂移扩散模型的收敛 Scharfetter–Gummel 格式

A convergent Scharfetter-Gummel scheme for a three-species drift-diffusion model for memristors

作者: Ansgar Jüngel, Zhiwei Sun, Sara Xhahysa

类别: math.NA

发布日期: 2026-07-27

arXiv ID: 2607.24086v1

链接: http://arxiv.org/abs/2607.24086v1

摘要: A structure-preserving fully implicit Scharfetter-Gummel finite-volume scheme for a three-species drift-diffusion model for semiconductors is proposed and analyzed. The equations describe the evolution of the electron, hole, and oxygen vacancy densities in a (bounded) memristor device, coupled to the Poisson equation for the electric potential, with mixed-type boundary conditions. Recasting the Scharfetter-Gummel fluxes in an upwind form, a hidden Fisher information component is revealed. Owing to the degeneracy of the Bernoulli function appearing in the fluxes, additional edgewise coercivity estimates are required, leading to refined local and global dissipation estimates. Using these ideas, the existence of a discrete finite-volume solution, a discrete free energy inequality, and the convergence of the numerical scheme are established. Numerical simulations in two space dimensions confirm the structure-preserving properties of the scheme and illustrate the filament formation in a memristor device.

13. 关于非齐次波动方程离散无映射帐篷推进方法的一个注记

A note on the discrete Unmapped Tent Pitching for the heterogeneous wave equation

作者: Marcella Bonazzoli, Gabriele Ciaramella, Ilario Mazzieri

类别: math.NA

发布日期: 2026-07-27

arXiv ID: 2607.24066v1

链接: http://arxiv.org/abs/2607.24066v1

摘要: The Unmapped Tent Pitching (UTP) algorithm is a space-time domain decomposition method for the parallel solution of wave-type problems. We have recently extended UTP to heterogeneous settings and compared, at the continuous level, the computational cost of different space-time decompositions. In this note, we report discrete-level observations showing that the optimal decomposition strategy may differ from the one predicted by the continuous analysis.

14. 不可压缩Navier-Stokes方程的向量格子Boltzmann格式

A vectorial lattice Boltzmann scheme for the incompressible Navier-Stokes equations

作者: Denise Aregba-Driollet, Thomas Bellotti, Roberto Natalini, Tommaso Tenna

类别: math.NA, physics.flu-dyn

发布日期: 2026-07-27

arXiv ID: 2607.24059v1

链接: http://arxiv.org/abs/2607.24059v1

摘要: We introduce a second-order accurate vectorial lattice Boltzmann scheme for the incompressible Navier-Stokes system, inspired by a discrete-velocity kinetic approximation proposed by Carfora and Natalini [ESAIM: M2AN, 42(1), 93-112, 2008]. Advantages and drawbacks compared to relaxation schemes are investigated by providing spectral analyses in the linearized case, and numerical validations on the genuinely non-linear problem.

15. 用于高阶信赖域和球面多项式优化的齐次张量框架

A Homogeneous Tensor Framework for High-Order Trust-Region and Spherical Polynomial Optimization

作者: Wenqi Zhu, Haibin Chen, Guanglu Zhou

类别: math.OC, math.NA

发布日期: 2026-07-27

arXiv ID: 2607.24046v1

链接: http://arxiv.org/abs/2607.24046v1

摘要: High-order methods can improve worst-case evaluation complexity, but for orders $p\geq3$ their Taylor subproblems are nonconvex polynomial optimization problems and are generally difficult to solve. We develop a radius-controlled boundary approach based on homogeneous tensor representations. By augmenting the step with a constant coordinate, any $p$th-order Taylor polynomial can be represented exactly as an order-$p$ homogeneous tensor form; at a prescribed radius, the boundary model is a spherical polynomial optimization problem. The representation applies to arbitrary $p$, while the algorithmic development focuses on the cubic case $p=3$. For an inhomogeneous cubic on the sphere, we introduce a quadratic shift and prove, under an explicit shift bound, equivalence with a three-block multilinear formulation at global optimality. This motivates a proximal alternating minimization (PAM) method with closed-form block updates; its objective values decrease and every accumulation point is stationary. We embed the boundary-step mechanism in an Adaptive Homogeneous Tensor Method (Ada–HTM). Under explicit smoothness, safeguarded-decrease, weak-curvature nondegeneracy, and local-refinement conditions, Ada–HTM attains the adaptive-regularization-type (AR$p$-type) evaluation complexity $\mathcal{O}(ε^{-(p+1)/p})$ for first-order stationarity. Numerically, PAM matches order-$2$ moment–sum-of-squares (SOS) certificates on the structured cubic instances for which certification is tractable, scales particularly well for low-rank tensors, and makes Ada–HTM competitive with trust-region and cubic-regularization methods, with its largest gains on ill-conditioned and badly-scaled problems.

16. 估计协方差矩阵的特征向量和特征空间：一致性的最优界与条件

Estimating eigenvectors and eigenspaces of covariance matrices: Optimal Bounds and Conditions for Consistency

作者: Phuc Tran, Van Vu

类别: math.ST, math.NA, math.PR

发布日期: 2026-07-27

arXiv ID: 2607.23964v1

链接: http://arxiv.org/abs/2607.23964v1

摘要: Let $X = [ ξ_1, \,\, ξ_2,...\,\, ,ξ_d]^\top$ be a zero-mean random vector of large dimension $d$ ($d \rightarrow \infty$) with (hidden) covariance matrix $M = (m_{ij})_{1 \leq i, j \leq d},$ where $m_{ij} = m_{ji} = \textbf{Cov}(ξ_i, ξ_j).$ Let $X_1, X_2, \dots, X_n$ be $n$ iid samples of $X$. Consider the sample covariance matrix

$$\textstyle \tilde{M} := \frac{1}{n} \sum_{i=1}^{n} X_i X_i^\top.$$

In practice, one frequently uses the eigenvectors and eigenspaces of $\tilde M$ as estimators for those of $M$. A central task is to provide an error analysis for these estimators. In this paper, we provide an optimal error analysis, obtaining upper and lower bounds of matching order of magnitude, for a wide range of parameters $d$ and $n$, under mild assumptions on $M$. As corollaries, we obtain new necessary and sufficient conditions for the consistency of the estimators. In these conditions, we only require the number of samples $n$ to depend linearly on the effective rank of $M$, which can be much smaller than the dimension $d$.

17. 递归治理：一个用于自主智能体系统中风险传播与漂移检测的图论框架

Recursive Governance: A Graph-Theoretic Framework for Risk Propagation and Drift Detection in Agentic AI Systems

作者: Sriram Nagaraj, Advaith Nila Narayanan

类别: math.NA

发布日期: 2026-07-27

arXiv ID: 2607.23916v1

链接: http://arxiv.org/abs/2607.23916v1

摘要: As financial institutions transition from traditional predictive models to autonomous agentic systems, the static model inventory requirements of traditional model risk management (MRM) face structural obsolescence. This paper proposes a dynamic Inventory-as-Code (IaC) governance loop that treats the model inventory as a living architectural component rather than a periodic documentation artifact. We make four principal contributions. First, we introduce a calibrated Degree of Autonomy (DoA) materiality score with an explicit, taxonomized tool-complexity weighting scheme that addresses the dominance problem of naive additive risk formulations. Second, we construct the agent inventory as a Directed Acyclic Graph (DAG) and define a formal Composite Risk Propagation algorithm under which upstream validation failures induce risk penalties on all reachable descendants. Third, we develop a Trajectory Monitoring protocol based on distributional cosine drift across ensembled Chain-of-Thought (CoT) embeddings, with an explicit procedure for constructing and certifying the Golden Path baseline, a matched-bootstrap calibration that we show is necessary to avoid a severe false-positive artifact in the naive alternative, and a two-stage response that separates legitimate reasoning variation from detrimental drift without demanding that a single exceedance event trigger an irreversible action. Fourth, we address practical complications largely absent from the prior literature: LLM base-model version changes, latent feedback loops in nominally acyclic agent graphs, and a two-pass execution structure that stages validation ahead of risk propagation.

18. 面向目标的最小二乘有限元方法误差估计——基于物理上有意义的伴随偏微分方程

Goal-Oriented Error Estimation for Least-Squares Finite Element Methods via Physically Meaningful Adjoint PDEs

作者: Yueyao Wu, Shun Zhang

类别: math.NA

发布日期: 2026-07-26

arXiv ID: 2607.23850v1

链接: http://arxiv.org/abs/2607.23850v1

摘要: We develop a goal-oriented error-estimation framework for first-order system least-squares (FOSLS) finite element methods based on the physical PDE adjoint rather than the adjoint induced by the least-squares formulation. Identified explicitly from the differential operator, its mixed boundary conditions, and the output, the physical adjoint admits its own first-order flux system and hence a native, built-in least-squares estimator, which the least-squares-induced adjoint does not. Our error identities rest on a primal-dual corrected functional: they follow from the continuous primal and adjoint equations alone, hold for arbitrary conforming approximations, and require no Galerkin orthogonality. For outputs containing a weighted Dirichlet-boundary flux, whose weight becomes the essential datum of the adjoint, we construct two corrected approximations, one from the potentials and one also using the fluxes, and prove product-type error estimates in the primal and adjoint errors. The least-squares functionals then yield computable a posteriori bounds and a balanced marking indicator whose element sum equals the product of the two estimators exactly. Numerical experiments confirm the predicted convergence and the effectiveness of the marking.

19. 通过纤维依赖性消除的高维非线性偏微分方程的隐式张量列交叉积分

Implicit Tensor-Train Cross Integration of High-Dimensional Nonlinear PDEs via Fiber-Dependency Elimination

作者: Behzad Ghahremani, Hessam Babaee

类别: math.NA

发布日期: 2026-07-26

arXiv ID: 2607.23814v1

链接: http://arxiv.org/abs/2607.23814v1

摘要: Tensor-train (TT) representations have emerged as an effective framework for mitigating the curse of dimensionality in the numerical solution of high-dimensional tensor differential equations. Among existing approaches, TT-cross methods are particularly attractive because they require only pointwise evaluations of the governing equations, naturally accommodate arbitrary nonlinearities, and avoid tangent-space projections and the numerical difficulties associated with nearly singular low-rank factors. However, existing TT-cross rank-truncation methods have been restricted to explicit time integration. Extending TT-cross methods to implicit schemes presents an obstacle: the collocation equations associated with the cross-selected fibers depend on neighboring fibers that are not part of the unknown set. Consequently, the resulting nonlinear system is not closed, preventing the direct application of standard implicit solvers. In this work, we introduce a principled fiber-dependency elimination framework that resolves this obstacle by expressing neighboring fibers as linear combinations of the cross-selected fibers through cross interpolation identities. The resulting formulation produces a closed collocation system while preserving the principal advantages of TT-cross methods. The proposed framework applies to both linear and nonlinear high-dimensional partial differential equations and is naturally combined with Newton iterations and rank adaptivity. Numerical experiments demonstrate rapid convergence of the dependency-elimination iterations, preservation of the temporal accuracy of implicit multistep schemes, and efficient implicit integration of high-dimensional nonlinear problems with full-order discretizations containing up to $10^{55}$ degrees of freedom.

20. 基于图神经网络的稀疏车辆自组织网络中交通激波的多智能体控制

GNN-based Multi-Agent Control of Traffic Shockwaves in Sparse Vehicular Ad-hoc Networks

作者: Prachi Nandi, Madhuri Malakar, Sonakshi Satpathy, Pabitra Mohan Khilar

类别: cs.NI, cs.LG, math.NA

发布日期: 2026-07-26

arXiv ID: 2607.23792v1

链接: http://arxiv.org/abs/2607.23792v1

摘要: Traffic shockwaves are stop-and-go waves that propagate upstream through the streams of vehicles and are one of the major causes of traffic congestion, fuel inefficiency, and increased accident rates in modern transportation systems. Although Connected and Autonomous Vehicles (CAVs) offer a promising opportunity to mitigate such shockwaves, most existing control strategies rely on global traffic state information, making them impractical for early-stage deployment of Vehicular Ad-hoc Networks (VANETs). In this paper, we propose a decentralized Multi-Agent Reinforcement Learning (MARL) framework that integrates a Graph Neural Network (GNN) to enhance the control architecture of connected and autonomous vehicles. The proposed approach enables vehicles to learn cooperative control policies using locally available information and interaction with neighboring vehicles. The effectiveness of the proposed scheme is evaluated using a scalable simulation environment under realistic highway traffic conditions. Simulation results show that the proposed GNN-based MARL framework can reduce the propagation of traffic shockwaves by up to 80%, even when only 10% of the vehicles are connected.

21. Navier-Stokes方程高阶一致分裂格式的谱消失粘性稳定化

A spectral-vanishing-viscosity stabilization of a higher-order consistent splitting scheme for the Navier-Stokes equations

作者: M Nader Alhomsi, Akram Moustafa, Mohammad Al-Saqqa, Jiahong Wu, Xiaoming Zheng

类别: math.NA

发布日期: 2026-07-26

arXiv ID: 2607.23720v1

链接: http://arxiv.org/abs/2607.23720v1

摘要: Huang and Shen developed a novel class of high-order BDF-IMEX consistent-splitting schemes for the incompressible Navier-Stokes equations, giving the first rigorous stability and error analysis for a fully decoupled splitting scheme of temporal order higher than two. Extending their analysis from unit viscosity to arbitrary viscosity, this work reveals that the error upper bound coefficient contains inverse powers of the viscosity. Our numerical experiments show that the scheme can break down at high Reynolds number. To save the scheme from this failure, we stabilize it by adding to the velocity update a symmetric positive-semidefinite spectral vanishing viscosity operator, built from the directionally applied Maday-Kaber-Tadmor kernel, which selectively damps the high, under-resolved modes at no additional asymptotic cost and leaves the structure of the error analysis intact. We establish stability and error estimates for the stabilized scheme in which the spectral vanishing viscosity provides viscosity-independent coercive control of the high modes. Three two-dimensional tests demonstrate the robustness and accuracy of the stabilized scheme. For a manufactured solution, the stabilized scheme retains its design order for k=2,3,4, whereas the unstabilized scheme diverges. For a perturbed Kovasznay flow, it accurately resolves the boundary layer at Re=10^4 and drives the perturbation back to the steady state, while the unstabilized scheme blows up. For the Kelvin-Helmholtz instability problem, it reproduces the reference integral diagnostics throughout the reliable regime, whereas the unstabilized scheme produces spurious solutions or blows up.

22. 时变边界条件下流动替代模型的无免费午餐: 双区域研究

No Free Lunch in Flow Surrogates under Time-Varying Boundary Conditions: A Two-Regime Study

作者: Georg Winkler, Martin Stoll

类别: math.NA, cs.LG, physics.flu-dyn

发布日期: 2026-07-26

arXiv ID: 2607.23667v1

链接: http://arxiv.org/abs/2607.23667v1

摘要: A flow surrogate validated on a simple regime is often taken as evidence that the approach will carry to a richer one. We test this assumption on two transient flows under time-varying boundary conditions emulating the process startup: the three-dimensional slurry film in chemical-mechanical planarisation (CMP), a core semiconductor-manufacturing process, and the two-dimensional Karman vortex street (KVS) behind a cylinder. Eight surrogate models are compared on one shared evaluation pipeline, differing in whether they learn the full field or a latent representation, and whether they predict trajectories in one shot or step by step. No single architecture wins both regimes. On the film, a one-shot full-field model reconstructs the process-relevant cumulative wall shear stress to 3.2% relative error. On the wake, a latent autoregressive DeepONet retains 96% of the shedding power that direct and one-shot models damp to almost zero. The deciding axis is the treatment of time. The self-sustained wake requires the phase memory that autoregressive feedback provides, while the boundary-driven film rewards a direct map. Pointwise RMSE picks the wrong model in both regimes, so the evaluation scores five physical questions instead, the field, its structure, invented motion, amplitude, and timing. The trained surrogates answer queries $10^3$ to $10^4$ times faster than the finite-element solver, but the offline cost of the training simulations means they pay off from the first query beyond the training set for CMP and the third for the KVS. The choice of surrogate should follow the dynamical character of the target flow, and its validation should use failure-mode-resolved metrics, since neither the winning architecture nor its validation transfers.

23. 不可压缩Navier-Stokes系统的IMEX-BDFk有限元格式的稳定性与误差分析

Stability and error analysis of IMEX-BDFk finite element schemes for the incompressible Navier-Stokes system

作者: Qianqian Ding, Yifan Luo, Shipeng Mao

类别: math.NA

发布日期: 2026-07-26

arXiv ID: 2607.23635v1

链接: http://arxiv.org/abs/2607.23635v1

摘要: In this paper, we propose and analyze a class of high-order numerical schemes within a fully discrete finite element framework for the incompressible Navier-Stokes equations with no-slip boundary conditions. The temporal discretization employs a kth-order (k=1,…,6) implicit-explicit backward difference formula (IMEX-BDFk), in which the nonlinear convection term is treated explicitly and the linear Stokes part implicitly, whereas the spatial discretization utilizes Taylor-Hood finite elements. We establish the stability and uniform boundedness of the numerical solution. We further establish optimal order error estimates in both space and time without any CFL-type condition, in the sense that the time step is independent of the spatial mesh size. In three dimensions, these include L2- and H1-norm error estimates for the velocity and L2-norm error estimates for the pressure, with temporal convergence rates up to sixth order for all variables. Numerical experiments are presented to demonstrate the effectiveness of the scheme and to confirm the theoretical convergence rates.

24. 中心Hermite感测与碰撞：D3Q125上的框架鲁棒阶解析松弛

Central-Hermite Sensing and Collision for Frame-Robust Order-Resolved Relaxation on D3Q125

作者: Bjørn Wu

类别: math.NA, physics.flu-dyn

发布日期: 2026-07-26

arXiv ID: 2607.23629v1

链接: http://arxiv.org/abs/2607.23629v1

摘要: Raw-Hermite sensing and collision on a fixed discrete-velocity set can convert a uniform translation into artificial coupling between nominally distinct nonequilibrium orders. We develop a central-Hermite formulation for a D3Q125 kinetic model with order-resolved log-Gaussian relaxation and compare three variants: raw sensing/raw collision (A), central sensing/raw collision (B), and central sensing/central collision (C). In homogeneous translated second-order perturbations, model C preserves third- and fourth-order modal purity to machine precision, whereas A and B develop boost-dependent cross-order content. Across a grid-CFL-boost matrix, model C reduces the post-transport collision frame discrepancy relative to A by 65.342-98.102% (median 81.131%) in the total relative L-infinity measure. Long-time calculations remain positive and conservative to numerical precision, although the accumulated benefit is configuration dependent because transport continually re-injects frame error. A transport study further reveals a clear trade-off: central-Hermite interface reconstruction strongly suppresses the third-order discrepancy but amplifies the fourth-order discrepancy. The fully central-Hermite collision therefore substantially reduces collision-induced cross-order frame discrepancy, while residual dependence remains due to discrete transport and finite velocity-space representation. This moment-space improvement does not by itself establish a comparable reduction in macroscopic Galilean transport error.

25. 用于图像变化检测的随机索引：一种距离阈值词汇方法

Random Indexing for Image Change Detection: A Distance-Threshold Vocabulary Approach

作者: Cristiano Tamborrino

类别: math.NA

发布日期: 2026-07-26

arXiv ID: 2607.23609v1

链接: http://arxiv.org/abs/2607.23609v1

摘要: Random Indexing (RI), used almost exclusively in text analysis (notably Temporal Random Indexing, TRI, for tracking word meaning over time), represents a discrete vocabulary with fixed random vectors and accumulates context by vector summation. We explore transplanting this mechanism to change detection in multitemporal images. A naive transplant using k-means to build the visual vocabulary is unstable: pixels are frequently reassigned between acquisition dates due to small radiometric shifts, destroying the pixel-to-vector correspondence RI depends on. We propose a distance-threshold (leader) clustering vocabulary instead, and give a short formal argument – a stability radius from the vocabulary covering/packing properties – for why this is provably more robust to acquisition noise than k-means. Combined with spatial context accumulation, this yields a training-free change detection pipeline, evaluated on four bi-temporal remote-sensing datasets (agriculture, river, urban PolSAR, wildfire) against a Change Vector Analysis baseline, which it consistently approaches but does not surpass. Multi-seed validation exposed and let us fix a degenerate-vector failure mode in a probabilistic-sparsity RI variant, and revealed a second, unresolved source of variability – sensitivity to the leader-clustering visitation order – which we characterize but, despite three attempted corrections, could not eliminate, reporting it as the main open problem. Results are cross-checked against an independent re-implementation. We report an ablation, a hyperparameter sensitivity analysis, an Otsu-vs-Gaussian-mixture thresholding comparison, and discuss RI incremental accumulation for future long time-series monitoring.

26. 逆星偏差的最优多项式可处理性指数

Optimal Polynomial Tractability Exponents for the Inverse Star Discrepancy

作者: Josef Dick

类别: math.NA

发布日期: 2026-07-26

arXiv ID: 2607.23571v1

链接: http://arxiv.org/abs/2607.23571v1

摘要: The inverse of the star discrepancy $n^\ast(d, \varepsilon)$ satisfies [ d \varepsilon^{-1} \lesssim n^{\ast}(d,\varepsilon)\lesssim d\varepsilon^{-2} ] for all $d \in \mathbb{N}$ and $0 < \varepsilon < \varepsilon_0$. The upper bound was shown by Heinrich, Novak, Wasilkowski and Woźniakowski (2001) and the lower bound was shown by Hinrichs (2004). A proof of this lower bound using elementary arguments was shown by Steinerberger (2023). These bounds show that the inverse of the star discrepancy depends linearly on the dimension, but the exact exponent of $\varepsilon^{-1}$ has remained open. In this paper we prove a lower bound which shows that the exponent $2$ of $\varepsilon^{-1}$ in the upper bound cannot be improved. More precisely, we show that for every $0<α<1$ and fixed $00$ and $\varepsilon_{α,A}>0$ such that for every integer $d$ satisfying [ A\varepsilon^{-α}\le d\le B\varepsilon^{-α} ] we have [ n^{\ast}(d,\varepsilon) \ge c_{α,B},d,\varepsilon^{-(2-α)}, ] for all $0 < \varepsilon < \varepsilon_{α, A}$. Along these polynomial strips the right-hand side is of order $\varepsilon^{-2}$. It follows that every uniform polynomial upper estimate $n^{\ast}(d,\varepsilon)\le C d^q\varepsilon^{-p}$ must satisfy $p \ge 2$. Together with the lower bound of Hinrichs (2004), which forces $q \ge 1$, this shows that the exponents $p=2$ and $q=1$ in the Heinrich–Novak–Wasilkowski–Woźniakowski upper bound are individually optimal.

27. 拟态有限差分的自适应一致性

Adaptive Consistency for Mimetic Finite Differences

作者: Yoo Jin Cha, Omar Duran, Nicola Castelletto, Victor A. P. Magri, Hamdi A. Tchelepi

类别: math.NA

发布日期: 2026-07-26

arXiv ID: 2607.23568v1

链接: http://arxiv.org/abs/2607.23568v1

摘要: The mimetic finite difference (MFD) method provides a robust discretization for flow simulation on general polyhedral meshes, but its computational cost can become significant due to dense local operators and reduced global sparsity. While the two-point flux approximation (TPFA) offers a substantially cheaper alternative, its accuracy is generally restricted to $K$-orthogonal grids. To balance these competing considerations, we present an adaptive MFD framework based on a residual-based consistency indicator derived from the discrete constitutive equations. The indicator measures local inconsistency and enables adaptive TPFA/MFD stencil selection through a user-prescribed tolerance $τ$. Because the adaptation is performed within a mimetic framework, arbitrary TPFA/MFD partitions remain stable and structure preserving. Theoretical analysis establishes uniform coercivity and proves explicit tolerance-controlled convergence of the relative flux error. Numerical experiments on challenging polyhedral reservoir benchmarks demonstrate accuracy comparable to full MFD discretizations while substantially reducing matrix density and computational cost.

28. 求解非凸优化问题的新的全局化牛顿型方法

New Globalized Newton-Type Methods for Nonconvex Optimization Problems

作者: Vo Thanh Phat, Tuyen Tran

类别: math.OC, math.NA

发布日期: 2026-07-26

arXiv ID: 2607.23433v1

链接: http://arxiv.org/abs/2607.23433v1

摘要: Newton’s method is one of the most effective second-order algorithms for smooth optimization because of its fast local convergence. However, existing globally convergent Newton-type methods typically require convexity or strong convexity of the objective function, while approaches for nonconvex optimization often rely on Hessian regularization at every iteration. In this paper, we propose a general line-search Newton framework for unconstrained optimization that avoids repeated Hessian regularization by exploiting the Newton direction only when it is well-defined and suitable. The proposed framework encompasses several existing hybrid gradient–Newton methods as special cases and naturally yields a new extragradient Newton method. We establish global convergence under mild assumptions, including the Polyak–Lojasiewicz–Kurdyka (PLK) condition, allowing both isolated and nonisolated accumulation points. We further prove local superlinear and quadratic convergence under appropriate regularity assumptions. Finally, we apply the proposed framework to strongly quasiconvex optimization and provide, to the best of our knowledge, the first Newton-type algorithm together with a comprehensive convergence analysis for this important class of nonconvex optimization problems. Numerical experiments demonstrate the effectiveness of the proposed methods.

29. 对称 V 循环在单元中心离散的多重网格公理下可能发散

The symmetric V-cycle can diverge under the multigrid axioms for cell-centred discretisations

作者: Ming Hei Wong

类别: math.NA

发布日期: 2026-07-25

arXiv ID: 2607.23391v1

链接: http://arxiv.org/abs/2607.23391v1

摘要: The axiomatic convergence theory for multigrid methods applied to cell-centred finite-difference and finite-volume discretisations rests on two hypotheses: an imbalanced Galerkin condition (G3), which states that $R_{\ell-1}A_\ell P_{\ell-1}=2A_{\ell-1}$ with $R_{\ell-1}=\frac{1}{2}P_{\ell-1}^T$, and a weak approximation property $(A2)_α$ of Bramble type. Under these hypotheses, together with Richardson smoothing, the symmetric W-cycle and the variable V-cycle are known to be uniformly convergent, while the uniform convergence of the standard symmetric V-cycle has remained open. We answer this in the negative by two constructions. First, for every smoothing count $m$ we exhibit hierarchies of every depth satisfying (G3), Richardson admissibility with $C_R=1$, and $(A2)_α$ for every $α\in(0,1]$ with the sharp level-independent constant $C_{A2}^2=4m$, whose symmetric $V(m,m)$-cycle error operator has spectral radius $θ_m(1+2θ_m)>1$ already on three levels, where $θ_m=(1-\frac{1}{4m})^{2m}$, and growing geometrically with the depth; the family shows that any smoothing-count threshold $m_0$ that could restore uniform V-cycle convergence must grow at least quadratically in $C_{A2}$. Second, we prove that the same failure occurs in a completely standard discretisation: the cell-centred finite-volume hierarchy for a one-dimensional diffusion equation with a mesh-aligned coefficient jump $1:κ$ and harmonic (Samarskii) interface averaging satisfies (G3) exactly and $(A2)_{1/2}$ with a level-independent constant $C_{A2}=O(κ)$, yet for every $κ\ge3$ its symmetric $V(1,1)$-cycle with any admissible Richardson parameter, including the optimal one, diverges geometrically in the number of levels. In both constructions the W-cycle remains uniformly contractive, so the hypotheses separate the two cycles. All claims are verified numerically.

30. 什么是 Jackson 常数？

What is Jackson’s constant?

作者: Rikhav Shah, John Urschel, Nicholas West

类别: math.NA

发布日期: 2026-07-25

arXiv ID: 2607.23375v1

链接: http://arxiv.org/abs/2607.23375v1

摘要: We prove a refinement of Jackson’s theorem on the approximation of Lipschitz functions by trigonometric polynomials. Our result precisely characterizes the leading error term associated with Jackson’s construction. We do the same for a related construction commonly used in the kernel polynomial method for spectral density estimation, which is slightly better than Jackson’s construction in this respect.

31. 可伸缩计算度量量级

Scalably computing metric magnitude

作者: Steve Huntsman, Jewell Thomas, Cynthia Ukawu

类别: math.NA, math.MG

发布日期: 2026-07-25

arXiv ID: 2607.23354v1

链接: http://arxiv.org/abs/2607.23354v1

摘要: Applications of metric magnitude often rely on numerically exact results in order to exploit a connection with information theory. We examine various approaches for scaling the dense linear algebra involved and identify hierarchical low-rank solvers as a preferred approach, with a clear path to scales of $10^5$ points on a single powerful workstation, and larger scales using our containerized CUDA-enabled C++/MPI pipeline.

32. 利用线性最小二乘学习具有收敛速率保证的渐近性质

Learning Asymptotics with Convergence-Rate Guarantees using Linear Least Squares

作者: Christos N. Efrem

类别: stat.ML, cs.LG, math.CO, math.NA

发布日期: 2026-07-25

arXiv ID: 2607.23287v1

链接: http://arxiv.org/abs/2607.23287v1

摘要: We introduce a new research area that is called Asymptotics Learning Theory (ALT) and combines optimization with asymptotic analysis. In particular, ALT provides a unified approach for computing unknown constants/parameters in proven asymptotic expansions using optimization theory. In this paper, we focus on a general asymptotic form which includes a broad class of asymptotics. Furthermore, we study two powerful numerical methods, namely, sliding Linear Least Squares (sLLSQ) and sliding Tikhonov Linear Least Squares (sT-LLSQ). For these techniques we rigorously prove asymptotic estimates that lead to sufficient conditions for convergence (to the correct values of unknown parameters) and convergence-rate guarantees. Despite their strengths, both methods have also limitations, e.g., slow convergence—or even, counterintuitively, divergence—in some cases. Moreover, we present fundamental applications in analytic combinatorics, a beautiful field of mathematics that deals with asymptotic enumeration of discrete structures using complex analysis. The proposed techniques complement existing approaches, such as the ratio method and its variants. Numerical examples also verify the theoretical results. Finally, we discuss interesting research directions in ALT.

33. 线性弹性问题在多胞形网格上的最小二乘弱 Galerkin 框架

A Least Squares Weak Galerkin Framework for Linear Elasticity on Polytopal Meshes

作者: Chunmei Wang, Shangyou Zhang

类别: math.NA

发布日期: 2026-07-25

arXiv ID: 2607.23279v1

链接: http://arxiv.org/abs/2607.23279v1

摘要: This paper develops and analyzes a least-squares weak Galerkin (LS-WG) finite element method for linear elasticity. By employing weak differential operators, specifically the weak gradient, weak strain tensor, and weak divergence, defined on weak finite element spaces, the proposed framework facilitates the treatment of complex boundary conditions and internal interfaces while avoiding the restrictive discrete inf-sup condition. The resulting formulation is symmetric and positive definite and exhibits robust numerical performance in the nearly incompressible regime. In addition, the proposed method offers exceptional geometric flexibility, allowing implementation on general polytopal (polygonal and polyhedral) meshes. We establish the uniqueness of the numerical solution and derive optimal-order error estimates with respect to a tailored discrete energy norm. Extensive numerical experiments confirm the theoretical convergence rates and demonstrate the method’s stability, efficiency, and locking-free performance for nearly incompressible materials.

34. POD降阶模型对半线性发展方程的参数敏感性及其在G-方程中的应用

Parametric Sensitivity of POD Reduced-Order Models for Semilinear Evolution Equations with Applications to G-Equations

作者: Shengbo Ma, Luhao Xue, Zhiwen Zhang

类别: math.NA

发布日期: 2026-07-25

arXiv ID: 2607.23212v1

链接: http://arxiv.org/abs/2607.23212v1

摘要: Proper Orthogonal Decomposition (POD) provides low-dimensional surrogate models of evolution equations from solution snapshots. In parameterized problems, however, a basis computed at one parameter value need not remain accurate at another, while recomputing the basis for every parameter in a many-query study is costly. We develop a sensitivity analysis for POD reduced-order models of a class of parameterized semilinear evolution equations that includes the viscous G-equation and a viscous strain G-equation in their mean-free formulations. The analysis is based on a cross-space Lipschitz condition for the nonlinear operator, which accommodates the nonsmooth, gradient-dependent nonlinearities of these flame-propagation models, and on continuity moduli quantifying the parameter dependence of the bilinear form and of the nonlinearity. We prove that when a POD basis constructed at a reference parameter is applied to nearby query parameters, the resulting error variation is controlled by the corresponding parameter modulus, with constants independent of the POD dimension, the number of time steps, and the perturbation magnitude. Numerical experiments are consistent with the predicted modulus-dependent sensitivity behavior, and illustrate the practical robustness of basis reuse for nearby parameters.

35. 初等超越函数的不动点构造

A Fixed-Point Construction of the Elementary Transcendental Functions

作者: François Alouges, Giovanni Di Fratta, Alberto Fiorenza, Renato Fiorenza

类别: math.CA, cs.DC, cs.MS, math.FA, math.NA

发布日期: 2026-07-25

arXiv ID: 2607.23196v1

链接: http://arxiv.org/abs/2607.23196v1

摘要: We present a unified fixed-point construction of the elementary transcendental functions, encompassing the real exponential, the complex exponential (sine and cosine), and the natural logarithm. Each function is characterized as the unique solution of a duplication identity established through the Banach contraction principle. These foundational identities are $e(2x)=e^2(x)$ for the exponentials, and $\log(x^{2})=2\log x$ for the logarithm. Because a direct iteration of these identities is numerically unstable, owing to local expansiveness at the target, the central idea transfers the analysis to a residual function, on which the operator becomes a strict contraction with an explicit convergence rate. Beyond its theoretical economy, which dispenses with differential equations and power series, this characterization translates into efficient algorithms for the machine evaluation of elementary functions: the underlying framework yields floating-point kernels whose accuracy and iteration depth are governed by the theoretical contraction rate. We also present a computational study showing that, in a throughput-bound vectorized regime, these kernels are competitive with standard production libraries, and in favorable configurations exceed them, with the sine–cosine kernel faster at every tested iteration depth. These implementations operate without lookup tables or memory traffic, an architectural advantage for modern high-performance and energy-efficient computing.

36. 通过扰动启发的选主元加速预处理雅可比方法

Accelerating preconditioned Jacobi methods via perturbation-inspired pivoting

作者: Nian Shao, Yuji Nakatsukasa

类别: math.NA

发布日期: 2026-07-25

arXiv ID: 2607.23187v1

链接: http://arxiv.org/abs/2607.23187v1

摘要: Perturbation theory for symmetric matrices shows that eigenvalues with small spectral gaps are more sensitive to off-diagonal perturbation, implying that different entries affect the eigenvalues unevenly. Building on this insight, we incorporate spectral gap information into the Jacobi method for symmetric eigenvalue problems and propose a new pivoting strategy, which is completely different from classical ones governed solely by the magnitude of the off-diagonal entries. When combined with a mixed-precision preconditioner that diagonalizes the matrix to low precision, numerical experiments demonstrate that the resulting strategy can significantly outperform the classical greedy approach when the original matrix has clustered eigenvalues.

37. Vlasov-Poisson 系统的主动通量方法比较

A Comparison of Active Flux Methods for the Vlasov-Poisson System

作者: Lukas Hensel, Yanick Kiechle, Rainer Grauer, Gudrun Grünwald, Christiane Helzel, Katharina Kormann

类别: math.NA

发布日期: 2026-07-25

arXiv ID: 2607.23133v1

链接: http://arxiv.org/abs/2607.23133v1

摘要: Active Flux is a third-order accurate, fairly novel finite volume method for hyperbolic conservation laws that is becoming increasingly popular. It evolves additional nodal degrees of freedom (DOF) located on cell interfaces and shared by neighboring cells. The numerical fluxes are then computed from these DOFs. A crucial component of Active Flux methods is the evolution operator of the point values, which enables the natural use of semi-Lagrangian ideas and makes Active Flux an attractive candidate for a grid-based approach to the Vlasov equation. Here, we compare two recently proposed Active Flux methods for the 1D1V Vlasov-Poisson system: a split-step method and an unsplit method.

38. 谱Barron空间中双调和方程的混合残量法

A mixed residual method for biharmonic equations in spectral Barron spaces

作者: Mengjia Bai, Kuo Gai, Shuai Lu

类别: math.NA

发布日期: 2026-07-25

arXiv ID: 2607.23080v1

链接: http://arxiv.org/abs/2607.23080v1

摘要: We propose a mixed residual method (MIM) for numerically solving the biharmonic equation with nonhomogeneous clamped boundary conditions. By establishing the well-posedness of the biharmonic equation in spectral Barron spaces, we derive an error bound for MIM that relates shallow neural network approximations to the exact solution and overcomes the curse of dimensionality. This error bound consists of two components: the first corresponds to the approximation error of the neural network, while the second represents the generalization error arising from randomly sampled training data. Several numerical experiments are presented to demonstrate the effectiveness of the proposed method.

39. 范围控制匹配与散乱数据拟插值的等周组合界

Isoperimetric-Combinatorial Bounds for Range-Controlled Matchings and Quasi-Interpolation from Scattered Data

作者: Alexander Panchenko, Ben Hellwig, Oleh Rudenko

类别: math.NA, math.CA, math.CO, math.MG

发布日期: 2026-07-25

arXiv ID: 2607.22968v1

链接: http://arxiv.org/abs/2607.22968v1

摘要: We develop a mesoscopic framework for analyzing perturbations of finite point sets. Given a reference node set $Y$ with known cubature and approximation properties, we consider a disordered node set $Q$ that is observed only through its populations in cubes at scale $r>0$. By imposing Hall-type (HT) combinatorial constraints on these populations, we prove the existence of a perfect matching between $Y$ and $Q$ with $O(r)$ range. This allows integral approximation estimates on coarser cubes at scale $h>r$ to be transferred from $Y$ to $Q$ with explicit error control and anchors $Q$ to a periodic grid. We then use translation-invariant quasi-interpolation methods to obtain high-order estimates of order $h^s$ as in the quasi-uniform setting, but for a different class of geometries. The key restrictions are the HT conditions and the bound $r\le Ch$, where $C<1$ is scale independent.

40. Stokes 流的质量守恒混合应力格式的最优块预条件子

Optimal block preconditioners for a mass-conserving mixed stress formulation of Stokes flow

作者: Kaibo Hu, Jongho Park, Jindong Wang

类别: math.NA

发布日期: 2026-07-24

arXiv ID: 2607.22932v1

链接: http://arxiv.org/abs/2607.22932v1

摘要: We present optimal block diagonal and triangular preconditioners for a mass-conserving mixed stress formulation of Stokes flow. The algebraic formulation leads to a double saddle point system with unknowns corresponding to discrete stress, velocity, vorticity, and pressure. MINRES equipped with a block diagonal preconditioner for an augmented Lagrangian formulation of this system is analyzed and shown to be optimal, in the sense that the convergence rate is independent of key parameters such as mesh size and kinematic viscosity. GMRES equipped with a block triangular preconditioner is also analyzed using a field-of-values approach. Finally, we present numerical results for both two- and three-dimensional model problems to validate the parameter robustness of the proposed preconditioners.

41. 博弈动力学的参数化方法

Parameterized Methods for Game Dynamics

作者: Yijie Jin, Haomin Zhou

类别: math.NA

发布日期: 2026-07-24

arXiv ID: 2607.22899v1

链接: http://arxiv.org/abs/2607.22899v1

摘要: We introduce a parameterized computational framework for the evolution of strategic behavior in continuous games. We consider the collective dynamics of players through a time-dependent probability density over the strategy space, representing the likelihood of each strategy being chosen at any given time. Instead of directly solving the high-dimensional Fokker-Planck equation that governs this evolution, we represent the probability density as the pushforward of a reference distribution with parameterized pushforward maps and consider the evolution of the parameterized equations. The motivation for this work comes from a limitation of the parameterized Wasserstein gradient flow (PWGF) framework when it is applied to game dynamics. PWGF provides a parameterized approach for evolution equations of probability density that can be formulated as Wasserstein gradient flows. However, not all game dynamics admit such a gradient flow formulation. We generalize the parameterized pushforward map framework to the non-gradient flows and apply it to the game dynamics with provable error bound in Wasserstein metric. Numerical experiments with several non-gradient systems in economics are provided to demonstrate the effectiveness of this new framework.

42. 完美的瑞士阿尔卑斯长号能否设计？结合缩减基方法与机器学习的形状优化

Can the perfect Swiss alphorn be designed? A combination of reduced basis method and machine learning for shape optimization

作者: Fabio Marcinnò, Clément Froidevaux, Leonardo Bocchieri, Riccardo Tenderini, Gerald Pot, Simone Deparis

类别: math.NA

发布日期: 2026-07-24

arXiv ID: 2607.22823v1

链接: http://arxiv.org/abs/2607.22823v1

摘要: In this work, we investigate the shape optimization of a Swiss alphorn to achieve resonance frequencies as close as possible to prescribed target notes. We construct an accurate geometric model of the alphorn based on both experimental measurements and imaging techniques. The latter are used to reconstruct geometric information where direct measures are not available. The resulting 3D model is meshed using a fully structured approach. Then, we introduce a geometry-parametrized finite element (FE) formulation of the Helmholtz equation enabling the application of the Reduced Basis Method (RBM) to efficiently generate a large dataset of simulations, in which each set of geometric parameters corresponds to a specific set of resonance frequencies (notes). This dataset enables the training of machine learning (ML) models for forward prediction of resonance frequencies from geometric parameters, as well as inverse design, where geometries are estimated to achieve specified target notes.

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7. 用于双曲守恒律的具有统一模板的五阶有限体积基于导数的Hermite WENO格式#

8. 带压缩的坐标下降法用于全组态相互作用的收敛性与误差分析#

9. 具有非负权重的$L_1$-偏差遭受维度灾难#

10. 一维非局部非线性程函方程的一个格式的收敛性#

11. 后验停止准则下大规模不适定问题的随机化Krylov投影迭代Tikhonov正则化#

12. 一种用于忆阻器三组分漂移扩散模型的收敛 Scharfetter–Gummel 格式#

13. 关于非齐次波动方程离散无映射帐篷推进方法的一个注记#

14. 不可压缩Navier-Stokes方程的向量格子Boltzmann格式#

15. 用于高阶信赖域和球面多项式优化的齐次张量框架#

16. 估计协方差矩阵的特征向量和特征空间：一致性的最优界与条件#

17. 递归治理：一个用于自主智能体系统中风险传播与漂移检测的图论框架#

18. 面向目标的最小二乘有限元方法误差估计——基于物理上有意义的伴随偏微分方程#

19. 通过纤维依赖性消除的高维非线性偏微分方程的隐式张量列交叉积分#

20. 基于图神经网络的稀疏车辆自组织网络中交通激波的多智能体控制#

21. Navier-Stokes方程高阶一致分裂格式的谱消失粘性稳定化#

22. 时变边界条件下流动替代模型的无免费午餐: 双区域研究#

23. 不可压缩Navier-Stokes系统的IMEX-BDFk有限元格式的稳定性与误差分析#

24. 中心Hermite感测与碰撞：D3Q125上的框架鲁棒阶解析松弛#

25. 用于图像变化检测的随机索引：一种距离阈值词汇方法#

26. 逆星偏差的最优多项式可处理性指数#

27. 拟态有限差分的自适应一致性#

28. 求解非凸优化问题的新的全局化牛顿型方法#

29. 对称 V 循环在单元中心离散的多重网格公理下可能发散#

30. 什么是 Jackson 常数？#

31. 可伸缩计算度量量级#

32. 利用线性最小二乘学习具有收敛速率保证的渐近性质#

33. 线性弹性问题在多胞形网格上的最小二乘弱 Galerkin 框架#

34. POD降阶模型对半线性发展方程的参数敏感性及其在G-方程中的应用#

35. 初等超越函数的不动点构造#

36. 通过扰动启发的选主元加速预处理雅可比方法#

37. Vlasov-Poisson 系统的主动通量方法比较#

38. 谱Barron空间中双调和方程的混合残量法#

39. 范围控制匹配与散乱数据拟插值的等周组合界#

40. Stokes 流的质量守恒混合应力格式的最优块预条件子#

41. 博弈动力学的参数化方法#

42. 完美的瑞士阿尔卑斯长号能否设计？结合缩减基方法与机器学习的形状优化#

重点关注论文（完整分析）

1. 使用反射广义源的良态电场表面积分方程

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

2. 基于柱谐函数的Helmholtz方程显式Runge逼近

1. 研究对象和背景

2. 主要定理或主要结果

3. 研究方法、关键技术和核心工具

4. 与之前工作的比较

3. 面向裂纹尖端与角点奇异性的 B 样条分析的力学训练神经坐标映射

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

4. AlgMortar：一种全代数的多尺度 mortar 预条件子

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. DGM与PINN算法求解非线性偏微分方程的全局收敛性

摘要翻译

2. 微扰-NeuSA: 面向时间依赖偏微分方程的结构化谱框架

摘要翻译

3. 强磁场下相对论带电粒子动力学的多物理结构保持积分器的一致误差界

摘要翻译

4. 双调和算子多重与聚簇特征值的 $C^0$ IPG 逼近的后验误差估计子

摘要翻译

5. 基于投影环境联络拉普拉斯算子的微分形式数据驱动扩散过程

摘要翻译

6. 用于非线性平衡律的物理信息令牌变换器方法论. I. 施瓦西-伯格斯流体流动

摘要翻译

7. 用于任意域上偏微分方程学习的几何感知 LegONet

摘要翻译

不相关论文（基本信息）

1. 近端点法的安德森加速：精确自适应极小极大、谱相变与最优保护

2. 双曲系统的自适应多物理场耦合

3. 任意阶 Padé 封闭锚定双导数时间离散: $s$ 个有效级, $2s$ 阶, 与 $L$-稳定性

4. 强非恒定磁场下三维带电粒子动力学的曲线参考指数积分器

5. 基于加速对称非负矩阵分解的低秩依赖分解

6. Gompertz 生长下肿瘤总体的优化

7. 用于双曲守恒律的具有统一模板的五阶有限体积基于导数的Hermite WENO格式

8. 带压缩的坐标下降法用于全组态相互作用的收敛性与误差分析

9. 具有非负权重的$L_1$-偏差遭受维度灾难

10. 一维非局部非线性程函方程的一个格式的收敛性

11. 后验停止准则下大规模不适定问题的随机化Krylov投影迭代Tikhonov正则化

12. 一种用于忆阻器三组分漂移扩散模型的收敛 Scharfetter–Gummel 格式

13. 关于非齐次波动方程离散无映射帐篷推进方法的一个注记

14. 不可压缩Navier-Stokes方程的向量格子Boltzmann格式

15. 用于高阶信赖域和球面多项式优化的齐次张量框架

16. 估计协方差矩阵的特征向量和特征空间：一致性的最优界与条件

17. 递归治理：一个用于自主智能体系统中风险传播与漂移检测的图论框架

18. 面向目标的最小二乘有限元方法误差估计——基于物理上有意义的伴随偏微分方程

19. 通过纤维依赖性消除的高维非线性偏微分方程的隐式张量列交叉积分

20. 基于图神经网络的稀疏车辆自组织网络中交通激波的多智能体控制

21. Navier-Stokes方程高阶一致分裂格式的谱消失粘性稳定化

22. 时变边界条件下流动替代模型的无免费午餐: 双区域研究

23. 不可压缩Navier-Stokes系统的IMEX-BDFk有限元格式的稳定性与误差分析

24. 中心Hermite感测与碰撞：D3Q125上的框架鲁棒阶解析松弛

25. 用于图像变化检测的随机索引：一种距离阈值词汇方法

26. 逆星偏差的最优多项式可处理性指数

27. 拟态有限差分的自适应一致性

28. 求解非凸优化问题的新的全局化牛顿型方法

29. 对称 V 循环在单元中心离散的多重网格公理下可能发散

30. 什么是 Jackson 常数？

31. 可伸缩计算度量量级

32. 利用线性最小二乘学习具有收敛速率保证的渐近性质

33. 线性弹性问题在多胞形网格上的最小二乘弱 Galerkin 框架

34. POD降阶模型对半线性发展方程的参数敏感性及其在G-方程中的应用

35. 初等超越函数的不动点构造

36. 通过扰动启发的选主元加速预处理雅可比方法

37. Vlasov-Poisson 系统的主动通量方法比较

38. 谱Barron空间中双调和方程的混合残量法

39. 范围控制匹配与散乱数据拟插值的等周组合界

40. Stokes 流的质量守恒混合应力格式的最优块预条件子

41. 博弈动力学的参数化方法

42. 完美的瑞士阿尔卑斯长号能否设计？结合缩减基方法与机器学习的形状优化