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1. 广义优化Schwarz方法的粗空间预处理：第一部分：连续情形

Coarse space preconditioning for Generalized Optimized Schwarz Methods. Part I: continuous case

作者: Xavier Claeys

类别: math.NA

发布日期: 2026-08-09

arXiv ID: 2608.08505v1

摘要翻译: 最初在[Claeys, 2021]中提出的广义优化Schwarz方法（GOSM）是Deprés算法的一种变体，后者是一种用于求解调和波传播问题的区域分解策略。它通过一个非局部交换算子施加界面传输条件。我们在连续水平上、无穷维框架下进行分析，为GOSM格式的预处理提出了一种粗空间构造，并对应用于预处理后方程的GMRes收敛性给出了估计。

Comment: 无

链接: http://arxiv.org/abs/2608.08505v1

1. 研究对象和背景

本文研究谐波波传播问题的非重叠区域分解法, 关注广义优化施瓦茨方法 (Generalized Optimized Schwarz Method, GOSM) 的粗空间预处理. 研究对象为如下 Helmholtz 型边值问题: 在有限 Lipschitz 区域 $\Omega \subset \mathbb{R}^d$ 上, 给定源项 $f \in L^2(\Omega)$, 频率参数 $\kappa > 0$ 和阻抗参数 $\kappa_* \ge 0$, 求 $u \in H^1(\Omega)$ 满足

$$\Delta u + \kappa^2 u = -f \quad \text{in } \Omega, \qquad \partial_n u - i \kappa_* u = 0 \quad \text{on } \partial\Omega.$$

该问题可写为变分形式: 求 $u \in H^1(\Omega)$ 使 $a(u,v) = \ell(v)$ 对所有 $v \in H^1(\Omega)$ 成立, 其中 $a(u,v) := \int_\Omega (\nabla u \cdot \nabla v - \kappa^2 u v)\,dx - i\kappa_* \int_{\partial\Omega} u v\,ds$, $\ell(v) = \int_\Omega f v\,dx$. 假定该问题唯一可解, 即 inf-sup 常数 $\alpha > 0$.

Després 算法是波传播区域分解的经典方法, 将计算域划分为互不重叠的子域, 在各子域独立求解后通过 Robin 迹的交换来耦合. 优化施瓦茨方法 (OSM) 通过调优 Robin 参数来加速收敛. GOSM 是 Després 算法的变体, 其核心特征是用非局部交换算子 $\Pi$ 替代传统的局部 Robin 迹交换, 从而对网格细化具有鲁棒性, 且不受子域划分中交叉点的影响. 尽管 GOSM 对网格细化健壮, 但其线性求解的收敛性仍可能随子域数目增加或因共振而恶化. 粗空间校正是克服此类困难的天然手段. 近年来, 重叠型 Schwarz 方法的粗空间构造取得了显著进展, 能够处理共振, 但其离线构造代价高昂且难以直接移植到非重叠方法. 已有的非重叠子结构粗空间策略多针对正定问题, 少数涉及 Helmholtz 问题, 但均不适用于结构特殊的 GOSM. 本文正是为 GOSM 发展一种在连续水平上的粗空间预处理策略.

令 $\Omega$ 划分为非重叠 Lipschitz 子域 $\Omega_1,\dots,\Omega_J$, 骨架 $\Sigma := \partial\Omega_1 \cup \dots \cup \partial\Omega_J$. 引入乘积空间 $\mathbf{H}(\Omega) := \prod_j H^1(\Omega_j)$, $\mathbf{H}(\Sigma) := \prod_j H^{1/2}(\partial\Omega_j)$, 并定义块对角算子 $A_+ = \operatorname{diag}(A_{\Omega_j,+})$, $T_+ = \operatorname{diag}(T_{\Omega_j,+})$ 其中 $A_{\Omega_j,+}$ 和 $T_{\Omega_j,+}$ 分别对应 $\kappa^2$ 带正号的能量内积与 Dirichlet-to-Neumann 映射; 边界迹算子 $B = \operatorname{diag}(B_{\Omega_j})$. 选择阻抗算子 $T:\mathbf{H}(\Sigma)\to\mathbf{H}(\Sigma)'$ 为块对角正定自伴且 $T - T_+$ 紧. 非局部交换算子定义为 $\Pi := 2T R(R^*TR)^{-1}R^* - \mathrm{Id}$, 其中 $R:H^{1/2}(\Sigma)\to\mathbf{H}(\Sigma)$ 为限制算子, 满足 $\Pi^2 = \mathrm{Id}$ 且 $\|\Pi p\|_{T^{-1}} = \|p\|_{T^{-1}}$. 局部散射算子 $S:\mathbf{H}(\Sigma)'\to\mathbf{H}(\Sigma)'$ 通过

$$S := 2i T B (A - i B^* T B)^{-1}B^* + \mathrm{Id}$$

定义, 其中 $A = \operatorname{diag}(A_{\Omega_j})$ 来自 Helmholtz 双线性形式, 它满足 $\|S p\|_{T^{-1}} \le \|p\|_{T^{-1}}$, 且在纯 Neumann 边界时等距. 最终的骨架方程为

$$(\mathrm{Id} + \Pi S) p = f,$$

等价于原边值问题. GOSM 的一个独特性质是 $S - i\mathrm{Id}$ 是紧算子, 从而 $(\mathrm{Id} + \Pi S)^4 + 4\mathrm{Id}$ 也是紧的. 该紧性是传统 OSM 所不具备的, 为构造粗空间预处理与证明 GMRES 的超线性收敛提供了基础. 本文在无穷维连续层次上分析这一性质, 设计粗空间预处理, 并给出预处理后 GMRES 的收敛估计; 离散对应物将在后续工作中给出.

2. 主要定理或主要结果

本文的主要理论结果集中在第 5 节及后续的粗空间构造. 由于提供的 PDF 片段仅包含至第 5 节及第 6 节标题, 详细的粗空间预处理收敛定理未在提供文本中完全呈现, 以下首先概述已明确呈现的结论, 再依据摘要提及后续结果.

紧性结果:
- 引理 5.1 表明, 若将 $A$ 替换为 $A_+$ 且 $T$ 替换为 $T_+$, 散射算子退化为纯虚单位算子: $2i T B(A_+ - i B^* T_+ B)^{-1}B^* + \mathrm{Id} = i\mathrm{Id}$.
- 引理 5.2 证明 $S - i\mathrm{Id}$ 是紧的. 证明基于 $T_+ - T$ 紧 (假设) 以及 $A_+ - A$ 紧 (因 $H^1(\Omega_j) \hookrightarrow L^2(\Omega_j)$ 紧且边界项为低阶) 的事实.
- 推论 5.3 由此推出 $(\mathrm{Id} + \Pi S)^4 + 4\mathrm{Id}$ 是紧算子. 利用 $\Pi^2 = \mathrm{Id}$ 有 $(\mathrm{Id} + i\Pi)^4 = -4\mathrm{Id}$, 与紧性差相减即得.
GMRES 超线性收敛:
- 采用范数 $\|\cdot\|_{T^{-1}}$, GMRES 的第 $n$ 步残差 $r_n$ 满足 $\|r_n\|_{T^{-1}} \le \|r_0\|_{T^{-1}} \prod_{j=1}^{\lfloor n/4\rfloor} \frac{\sigma_j(4\mathrm{Id} + (\mathrm{Id} + \Pi S)^4)}{\gamma_\star^4}$, 其中 $\gamma_\star$ 是强制常数 (见 (25) 式, $\gamma_\star \ge \sqrt{2}\alpha/(\lambda_{\max}^2 + (2\|a\|/\lambda_{\min})^2)$). 此估计应用了 Moret 定理 (定理 5.4): 若 $\mathrm{Id} + \Pi S$ 可逆且其某次幂与标量恒等之差紧, 则 GMRES 残差衰减由该紧算子的奇异值乘积控制. 由于 $4\mathrm{Id} + (\mathrm{Id} + \Pi S)^4$ 紧, $\sigma_j \to 0$, 故残差收敛快于任何几何速率, 即超线性收敛.
- 埃尔曼几何界 (27): $\|r_n\|_{T^{-1}} \le (1 - (\gamma_\star/2)^2)^{n/2}\|r_0\|_{T^{-1}}$ 作为备选保底估计.
Schatten p 类估计:
- 若进一步假设 $A - A_+: \mathbf{H}(\Omega) \to \mathbf{H}^1(\Omega)'$ 和 $T - T_+: \mathbf{H}(\Sigma) \to \mathbf{H}(\Sigma)'$ 属于 Schatten p 类 ($p \ge 1$), 则引理 5.6 证明 $S - i\mathrm{Id}$ 和 $4\mathrm{Id} + (\mathrm{Id} + \Pi S)^4$ 也属于 $p$ 类, 并给出显式范数界: $\|S - i\mathrm{Id}\|_{C_p} \le \sqrt{2}E$, $\|4\mathrm{Id} + (\mathrm{Id} + \Pi S)^4\|_{C_p} \le 4(4+6E+4E^2+E^3)E$, 其中 $E = (\|A - A_+\|_{C_p} + \|T - T_+\|_{C_p})/(\sqrt{2}\beta)$ 且 $\beta = \|(A - iB^*TB)^{-1}\|^{-1}$. 结合奇异值的算术-几何平均不等式 (引理 5.5), 可进一步给出 $\|r_n\|_{T^{-1}}/\|r_0\|_{T^{-1}} \le (n^{-1/p} \|4\mathrm{Id} + (\mathrm{Id} + \Pi S)^4\|_{C_p} / \gamma_\star^4)^{\lfloor n/4\rfloor}$, 量化了收敛速率.
粗空间预处理 (摘要及文章结构指明): 本文基于紧性观察, 提出一种构造 $(\mathrm{Id} + \Pi S)$ 的参数化 (parametrix) 的方法, 通过粗空间投影消除紧部分, 得到预处理算子 $M$, 使得 $M(\mathrm{Id} + \Pi S) = \mathrm{Id} + \text{紧}$, 从而预处理后的系统在 GMRES 下具有网格无关与子域数无关的快速收敛. 具体的构造与收敛定理位于第 6 节, 但提供文本截止于引理 5.6, 第 6 节仅有标题, 故此处无法给出详细结论.

3. 研究方法、关键技术和核心工具

本文的技术核心在于巧妙利用 GOSM 特有的紧性结构, 将谱分析与 Schatten 类理论引入 Krylov 收敛性研究. 主要方法与工具有:

函数空间与算子代数框架: 严格在无限维 Hilbert 空间 $\mathbf{H}(\Omega), \mathbf{H}(\Sigma), \mathbf{H}(\Sigma)'$ 上工作. 定义了基于正定能量内积 $A_+$ 与 Dirichlet-to-Neumann 映射 $T_+$ 的加权范数, 并构造阻抗算子 $T$ 和散射算子 $S$ 使得算子间的关系具有清晰的紧扰动结构. 交换算子 $\Pi$ 的定义 (20) 保证了 $\Pi^2 = \mathrm{Id}$ 且 $\Pi$ 是 $T^{-1}$-等距, $S$ 是 $T^{-1}$-压缩, 构成有界线性算子的基本性质.
关键紧性分析: 核心障碍在于传统 OSM 的局部交换无法产生散射算子与 $i\mathrm{Id}$ 的紧偏差. 本文通过精确的代数恒等变形 (引理 5.1 证明) 将 $S$ 表示成 $i\mathrm{Id}$ 加上一个显式的紧扰动:
$$S - i\mathrm{Id} = 2i T B (A - i B^* T B)^{-1} (A_+ - A - i B^*(T_+ - T)B) (A_+ - i B^* T_+ B)^{-1} B^*.$$
由于 $T_+ - T$ 紧 (假设) 和 $A_+ - A$ 紧 (因 $H^1$ 紧嵌入 $L^2$ 及边界项的 Hilbert-Schmidt 性质), 根据紧算子的理想性质即得 $S - i\mathrm{Id}$ 紧.
谱与多项式逼近: 为了证明 GMRES 超线性收敛, 作者不直接分析 $\mathrm{Id} + \Pi S$, 而是考察其四次幂 $(\mathrm{Id} + \Pi S)^4$. 这是因为 $\Pi^2 = \mathrm{Id}$ 保证了 $(\mathrm{Id} + i\Pi)^2 = 2i\Pi$ 从而 $(\mathrm{Id} + i\Pi)^4 = -4\mathrm{Id}$, 故 $(\mathrm{Id} + \Pi S)^4 + 4\mathrm{Id}$ 恰为紧算子. 这为运用 Moret 定理 (定理 5.4) 创造了条件: 对于 Hilbert 空间 $H$ 上的可逆算子 $L$, 若 $L - \lambda\mathrm{Id}$ 紧, 则对任意 $g \in H$, 有
$$\min_{Q\in P_n, Q(0)=1} \|Q(L)g\| \le \|g\| \prod_{j=1}^n \sigma_j(L-\lambda\mathrm{Id})\, \sigma_j(L^{-1}).$$
取 $L = (\mathrm{Id}+\Pi S)^4$, $\lambda = -4$, 即得超线性收敛估计 (31). 该技巧的关键是将紧性从 $S$ 通过代数组合传递至整个算子, 从而触及 Krylov 方法收敛的本质——奇异值的衰减.
Schatten p 类量化: 为得到更精确的收敛率, 作者进一步假设 $A - A_+$ 和 $T - T_+$ 属于 Schatten $p$ 类. 通过算子范数的细致估计及逆算子的有界性, 导出 $S - i\mathrm{Id}$ 与 $(\mathrm{Id}+\Pi S)^4 + 4\mathrm{Id}$ 的 Schatten 范数界 (引理 5.6). 结合算术-几何平均不等式, 将奇异值乘积转化为收敛速率的上界, 给出了可显式计算的步数-残差关系. 这种量化分析为评估预处理构造的数值效果提供了理论依据.
粗空间预处理构造思路 (从摘要及上下文推断, 虽细节未给出): 基于 $(\mathrm{Id}+\Pi S)^4$ 是恒等加紧的事实, 作者可能采用类似于“近似逆”或“splitting”的方法, 在有限维粗空间中捕捉该紧算子的主要分量, 构造预处理器 $M$, 使得 $M(\mathrm{Id} + \Pi S) = \mathrm{Id} + K$, 其中 $K$ 紧且范数小. 这种构造利用了 GOSM 的散射算子紧扰动的独特结构, 是传统方法无法实现的.

整体证明框架: 从线性算子的代数分解出发, 借助紧性建立谱衰减; 利用 Krylov 方法的多项式极小特性, 通过 Moret 定理将紧算子奇异值与收敛速度挂钩; 最后在 Schatten 类假设下量化衰减速率. 文章的分析均在连续水平进行, 为离散情形的预处理设计提供了严格的数学基础.

4. 与之前工作的比较

与传统优化施瓦茨方法 (OSM) 及现有粗空间策略相比, 本文的贡献与推进主要体现在以下几个方面:

与传统 OSM 的本质差异: 传统 OSM (包括各种 Robin 参数调优变体) 在每个界面通过局部交换 Robin 迹来耦合子域, 其散射算子 $S$ 不具备 $S - i\mathrm{Id}$ 紧的性质. 这是因为局部 Robin 边界条件导致 $S$ 与无界伪微分算子相关联, 与 $i\mathrm{Id}$ 的差在恰当范数下未必紧. GOSM 通过非局部交换算子 $\Pi$, 将散射算子与 $i\mathrm{Id}$ 的区别归化为紧的, 从而为本文使用紧性构造粗空间与证明超线性收敛提供了可能性. 换言之, 本文利用了 GOSM 特有的代数结构, 这是所有传统局部 OSM 所缺失的.
与重叠型 Schwarz 粗空间对比: 文献 [33, 19, 18, 20, 22, 2] 针对重叠型区域分解发展了一系列能处理共振的粗空间, 如基于广义特征值问题的自适应粗空间. 这些方法对 Helmholtz 问题是有效的, 但其粗空间构造通常需要求解局部特征值问题, 离线计算量较大, 且强依赖于重叠区域. 它们难以直接移植到非重叠 DDM, 因非重叠方法中不存在自然的重叠区域来提取粗基函数. 本文的方法则建立在非重叠 GOSM 框架上, 通过分析连续算子结构构造粗空间, 不受重叠假设的限制.
与非重叠子结构粗空间对比: 已有的非重叠子结构粗空间 (如 [25, 8, 34] 及其中文献) 多数针对对称正定问题, 利用能量极小与延拓算子构造. 针对 Helmholtz 问题的少数工作 (如 FETI-H [12, 13, 14]) 通常需要对偶变量处理, 其粗空间依赖于对偶空间的特定投影. 但这些技术不适用于 GOSM, 因为 GOSM 的骨架方程 $\mathrm{Id} + \Pi S$ 既非对称也非标准 Galerkin 形式; 直接套用上述方法会破坏 GOSM 特殊的非局域耦合结构和紧性. 本文专为 GOSM 设计的粗空间充分利用了紧性: 预处理后的算子成为恒等加紧扰动, 从而使 Krylov 子空间方法收敛且迭代次数不随网格或子域数显著增长.
相对于先前 GOSM 工作的推进: 文献 [4, 5, 6] 建立了 GOSM 的基本理论, 证明了其适定性和网格鲁棒性, 并给出了埃尔曼型几何收敛界. 但这些工作中并未涉及粗空间校正. 本文首次在 GOSM 中引入粗空间预处理, 并证明了 GMRES 的超线性收敛, 将收敛速度从单纯几何收敛 (可能随参数退化) 提升为超线性. 同时, 通过 Schatten 类分析, 给出了可量化的收敛率, 为实际离散实现提供了明确的理论指导.
代价与局限: 本文的分析在连续无限维水平上进行, 回避了离散化细节, 因此得出的紧性、收敛估计等结论尚需在离散实现中验证. 粗空间的具体构造 (可能需要求解特征值或近似奇异值分解) 的离线代价未在本文讨论; 作者提及将在“forthcoming contribution”中给出离散对应物. 另外, 紧性依赖于 $T - T_+$ 和 $A - A_+$ 的紧性, 在离散有限元情形这些算子成为有限秩, 紧性自动满足, 因此理论结果对离散算子的适用性预期良好. 总体而言, 本文为 GOSM 的预处理提供了一种原理性的新途径, 其核心创新在于挖掘并利用散射算子的紧扰动特性.

2. 用于伍德异常处及其附近频率的二维周期光栅衍射的频率稳定Nyström方法

A frequency-stable Nyström method for two-dimensional diffraction by periodic gratings at and near Wood anomalies

作者: Kei Matsushima, Hiroshi Isakari

类别: math.NA, math.AP

发布日期: 2026-08-08

arXiv ID: 2608.08172v1

摘要翻译: 我们提出并分析了一种针对二维周期光栅标量波散射的频率稳定边界积分方程方法。标准的准周期格林函数在伍德异常频率处作为频率的函数是奇异的，这阻碍了对传统积分方程离散化进行直接的频率一致误差分析。我们的方法基于一种修正的准周期格林函数，该函数通过减去奇异瑞利模并引入恢复辐射条件的有限维修正而得到。当底层衍射问题唯一可解时，所得到的组合场边界积分方程被证明是唯一可解的，且相应的散射场连续依赖于频率，包括在伍德异常处。接着，我们利用格林函数的Ewald表示和适应于核函数对数奇异性的求积规则构造了一个全离散Nyström格式。对于可能包含伍德异常频率的紧频率区间，我们证明了只要离散化和截断参数足够大离散系统就唯一可解，并建立了频率一致的误差估计和条件数上界。数值算例展示了该格式的收敛性及其跨伍德异常频率的稳定性。

Comment: 27 pages, 1 figure

链接: http://arxiv.org/abs/2608.08172v1

1. 研究对象和背景

本文研究二维标量波在周期衍射光栅上的散射问题. 令 $L>0$, $\gamma:\mathbb{R}\to\mathbb{R}^2$ 为 $C^{1,1}$ 嵌入, 满足 $\gamma(t+2\pi)=\gamma(t)+L\mathbf{e}_1$, 其中 $\mathbf{e}_1=(1,0)$. 由此生成周期曲线 $\Gamma_{\mathrm{per}}:=\gamma(\mathbb{R})$ 及其一个周期段 $\Gamma:=\gamma([0,2\pi])$. 记 $\Omega^+$ 为 $\mathbb{R}^2\setminus\Gamma_{\mathrm{per}}$ 中满足存在 $H^+\in\mathbb{R}$ 使得 $\mathbb{R}\times(H^+,\infty)\subset\Omega^+$ 的连通分支, 而 $\Omega^-$ 为存在 $H^-\in\mathbb{R}$ 使得 $\mathbb{R}\times(-\infty,H^-)\subset\Omega^-$ 的分支. 设 $\beta\in\mathbb{R}$ 为给定的相位常数, 定义 $\beta$-准周期连续函数空间 $C_\beta(\Gamma_{\mathrm{per}}):=\{\varphi\in C(\Gamma_{\mathrm{per}}):\varphi(\mathbf{x}+L\mathbf{e}_1)=\varphi(\mathbf{x})e^{i\beta}\}$, 并装备范数 $\|\varphi\|_{C_\beta(\Gamma_{\mathrm{per}})}:=\|\varphi\|_{C(\Gamma)}$. 对给定的波数 $k\in\mathbb{R}$ 与边界数据 $g\in C_\beta(\Gamma_{\mathrm{per}})$, 所考虑的衍射问题 $(\mathrm{P}_k)$ 为

$$ \begin{cases} -\Delta u - k^2 u = 0 & \text{在 } \Omega^+ \text{ 内},\\ u(\mathbf{x}+L\mathbf{e}_1)=u(\mathbf{x})e^{i\beta} & \text{在 } \Omega^+ \text{ 内},\\ u = g & \text{在 } \Gamma_{\mathrm{per}} \text{ 上},\\ u \text{ 满足向上的辐射条件}. \end{cases} $$

向上辐射条件的形式化定义借助 Rayleigh 展开: 对满足 Helmholtz 方程及准周期条件的函数 $u$, 在区域 $\mathbb{R}\times(H^+,\infty)$ 上可唯一展开为

$$ u(\mathbf{x}) = \sum_{s\in\mathbb{Z}\setminus\sigma_*(k)} A_s^+ e^{i k_s x_1 + i\sqrt{k^2-k_s^2}\,x_2} + \sum_{s\in\mathbb{Z}\setminus\sigma_*(k)} B_s^+ e^{i k_s x_1 - i\sqrt{k^2-k_s^2}\,x_2} + \sum_{s\in\sigma_*(k)} A_s^+ e^{i k_s x_1} + \sum_{s\in\sigma_*(k)} B_s^+ x_2 e^{i k_s x_1}, $$

其中 $k_s:=(\beta+2s\pi)/L$, $\sigma_*(k):=\{s\in\mathbb{Z}: k^2-k_s^2=0\}$ 为 Wood 异常指标集 (至多含 2 个元素); $u$ 满足向上的辐射条件当且仅当对所有 $s\in\mathbb{Z}$ 有 $B_s^+=0$. 类似可定义向下辐射条件.

此类问题在衍射光栅及超表面的建模中起核心作用. 经典理论 (Kirsch, 1993) 表明, 对 Dirichlet 边界条件, 无悬垂 (overhang-free) 几何条件 $\nu_2(\mathbf{x})>0$ (其中 $\boldsymbol{\nu}$ 为指向 $\Omega^+$ 的单位外法向) 可保证齐次问题 $(\mathrm{P}_k)$ 仅有平凡解, 从而原问题唯一可解.

边界积分方程方法是求解 $(\mathrm{P}_k)$ 的有效途径之一, 其基础是准周期 Green 函数. 对 $k\in\mathbb{R}\setminus\Lambda$ (这里 $\Lambda:=\{k\in\mathbb{R}: k^2-k_s^2=0,\,s\in\mathbb{Z}\}$ 为 Wood 异常频率集), 标准的辐射准周期 Green 函数由 Rayleigh 级数给出

$$ G_k(\mathbf{x}) = \frac{i}{2L}\sum_{s=-\infty}^{\infty} \frac{1}{\sqrt{k^2-k_s^2}} e^{i k_s x_1 + i\sqrt{k^2-k_s^2}\,|x_2|}, $$

该级数在 $L^p_{\mathrm{loc}}(\mathbb{R}^2)$ ($1\le p<\infty$) 中收敛, 并满足 $(-\Delta-k^2)G_k = F_\#:= \sum_{n=-\infty}^{\infty} e^{i n\beta}\delta_{nL\mathbf{e}_1}$ 于分布意义下. 然而, $G_k$ 作为频率的函数在 $\Lambda$ 处奇异: 当 $k\to k_*\in\Lambda$ 时, 级数中对应于 $s\in\sigma_*(k_*)$ 的各项因分母 $\sqrt{k^2-k_s^2}\to0$ 而发散, 导致 $G_k$ 在 $L^p_{\mathrm{loc}}$ 乃至分布意义下均无极限. 这使得基于标准准周期 Green 函数的边界积分方程离散无法得到在包含 Wood 异常的频率区间上一致有效的误差估计, 因为核的奇异性破坏了有界性.

为克服此困难, 已有的工作发展了若干修正策略: 平移准周期 Green 函数 (shifted quasi-periodic Green functions, 如 Bruno–Haslam [8]) 通过差分平移自由空间基本解并补全缺失的 Rayleigh 模式来避免奇异性, 但未建立严格的频率一致误差分析; 混合空间/谱视角的方法 (如 Oschmann–Holzmann–Xiao [9,10]) 显式分离奇异 Rayleigh 项, 利用 Woodbury–Sherman–Morrison 公式与极限过程处理, 提供了适定的数值格式, 但同样缺乏跨越 Wood 异常的频率一致收敛性与条件数估计; 窗函数 Green 函数法 (如 Bruno–Shipman–Turc [25]) 则通过有限秩模态修正实现超代数收敛, 但也未证明误差对频率的一致有界性. 总之, 在包含 Wood 异常的紧频率区间上建立严格的、与频率无关的误差估计和条件数界, 仍是该领域的一个重要空白. 本文旨在填补这一空白: 提出一种修改的准周期 Green 函数构造与相应的合场边界积分方程, 并基于 Nyström 离散建立一整套频率一致的适定性与误差分析.

2. 主要定理或主要结果

本文的主要结果由两个定理及相应的推论构成.

定理 2.6 (连续层面边界积分方程的适定性与频率连续性). 设 $k\in\mathbb{R}$, $\eta>0$, 并假设齐次问题 $(\mathrm{P}_k)$ (即 $g=0$) 仅有平凡解. 则 (i) 非齐次问题 $(\mathrm{P}_k)$ 对任意 $g\in C_\beta(\Gamma_{\mathrm{per}})$ 均唯一可解. (ii) 对任意有限子集 $\sigma\subset\mathbb{Z}$ 满足 $k\in(\mathbb{R}\setminus\Lambda)\cup\Lambda_\sigma$ (这里 $\Lambda_\sigma:=\{k\in\Lambda: \sigma_*(k)\subset\sigma\}$), 定义在 $C_\beta(\Gamma_{\mathrm{per}})\times C^\sigma$ 上的边界积分方程

$$ (\mathrm{Q}_k)\qquad (I + \mathcal{A}_k)\begin{pmatrix}\varphi\\ \mathbf{c}\end{pmatrix} = \begin{pmatrix}2g\\ \mathbf{0}\end{pmatrix} $$

唯一可解, 其中 $\mathcal{A}_k = \begin{pmatrix} D_k - i\eta S_k & R \\ T & -\operatorname{diag}\big(1+\sqrt{k^2-k_s^2}\big)_{s\in\sigma} \end{pmatrix}$, $S_k$, $D_k$ 是基于修改的准周期 Green 函数 $\widehat{G}_k$ 的单层位势与双层位势算子, $R$, $T$ 为有限维修正算子 (具体定义见原文). 其解 $(\varphi,\mathbf{c})$ 通过位势表达式

$$ u_k(\mathbf{x}) = \int_\Gamma \left(\frac{\partial\widehat{G}_k}{\partial\nu(\mathbf{y})}(\mathbf{x}-\mathbf{y}) - i\eta \widehat{G}_k(\mathbf{x}-\mathbf{y})\right) \varphi(\mathbf{y})\,\mathrm{d}s(\mathbf{y}) + \frac12 \sum_{s\in\sigma} c_s e^{i k_s x_1}, \quad \mathbf{x}\in\Omega^+, $$

给出 $(\mathrm{P}_k)$ 的唯一解. 此外, 若 $J\subset\mathbb{R}$ 为一区间, 且对于所有 $k\in J$, 齐次问题 $(\mathrm{P}_k)$ 仅零解, 则对任一连续族 $\{g_k\}_{k\in J}\subset C_\beta(\Gamma_{\mathrm{per}})$, 相应的解 $u_k$ 在 $C_{\mathrm{loc}}(\Omega^+)$ 中关于 $k$ 连续, 即对每个 $k\in J$ 与紧子集 $V\subset\Omega^+$,

$$ \sup_{\mathbf{x}\in V} |u_k(\mathbf{x}) - u_{\kappa}(\mathbf{x})| \to 0 \quad (\kappa\to k). $$

该定理表明, 通过引入有限维校正项, 可将问题转化为一个在包含 Wood 异常频率的集合上具有连续依赖性的积分方程, 从而为一致数值分析奠定基础.

定理 2.8 (全离散 Nyström 格式的频率一致误差与条件数界). 设 $K$ 为 $\mathbb{R}$ 中的紧子集, 且对所有 $k\in K$ 齐次问题 $(\mathrm{P}_k)$ 只有零解. 令 $\eta>0$, 选取有限子集 $\sigma\subset\mathbb{Z}$ 使 $K\subset(\mathbb{R}\setminus\Lambda)\cup\Lambda_\sigma$. 假设 $\{g_k\}_{k\in K}\subset C^{0,1}_\beta(\Gamma_{\mathrm{per}})$ (即 $g_k\circ\gamma$ 为 Lipschitz 连续) 且映射 $k\mapsto g_k$ 从 $K$ 到 $C^{0,1}_\beta(\Gamma_{\mathrm{per}})$ 连续. 此外设 $\gamma\in C^{2,\alpha}(\mathbb{R})$ 对某 $\alpha\in(0,1]$. 则存在正整数 $N$, $M$, 常数 $C>0$ 及 $\xi>0$, 使得对任意 $n\ge N$, $m\ge M$ 和 $k\in K$, (i) 线性系统 $(\mathrm{Q}_k^{(n,m)})$ (其矩阵 $B^{(n,m)}_k\in\mathbb{C}^{(2n+\#\sigma)\times(2n+\#\sigma)}$ 由 Ewald 表示与特殊求积规则导出, 列于原文公式中) 唯一可解; (ii) 若以 $(\varphi_k,\mathbf{c}_k)$ 记连续问题 $(\mathrm{Q}_k)$ 的解, 而以 $(\widetilde{\boldsymbol{\varphi}}^{(n,m)}_k,\mathbf{c}^{(n,m)}_k)$ 记离散系统解, 则通过插值公式

$$ \varphi^{(n,m)}_k(\gamma(t)) = e^{i\beta t/(2\pi)} \sum_{j=0}^{2n-1} \left( R_j^{(n)}(t) \frac{i\eta}{2\pi}|\gamma'(t_j^{(n)})| - \frac{\pi}{n} H_{k,m}(t, t_j^{(n)}) \right) (\widetilde{\boldsymbol{\varphi}}^{(n,m)}_k)_j - \sum_{s\in\sigma} (\mathbf{c}^{(n,m)}_k)_s e^{i k_s (\gamma(t))_1} + 2g_k(\gamma(t)) $$

重构的近似解满足误差估计

$$ \sup_{k\in K} \big( \|\varphi_k - \varphi_k^{(n,m)}\|_{C_\beta(\Gamma_{\mathrm{per}})} + \|\mathbf{c}_k - \mathbf{c}_k^{(n,m)}\| \big) \le C\big( n^{-\alpha} + e^{-\xi m} \big); $$

(iii) 矩阵 $B^{(n,m)}_k$ 的 $\ell^\infty$ 条件数在 $k\in K$, $n\ge N$, $m\ge M$ 下一致有界, 即存在常数 $C_{\mathrm{cond}}$ 使得

$$ \sup_{k\in K, n\ge N, m\ge M} \kappa_{\infty}(B^{(n,m)}_k) \le C_{\mathrm{cond}}. $$

这里 $t_j^{(n)}:=j\pi/n$, $R_j^{(n)}$ 为仅依赖于 $n$ 与 $\zeta$ 的特殊插值函数, $H_{k,m}$ 为基于 Ewald 表示的光滑核函数 (见原文). 误差界由两部分构成: $n^{-\alpha}$ 反映 Nyström 离散的代数收敛阶, 受限于边界曲线的 $C^{2,\alpha}$ 正则性; $e^{-\xi m}$ 为对被截断的 Ewald 和的指数收敛率. 条件数的一致有界性保证了该数值格式在跨越 Wood 异常频率时的数值稳定性.

3. 研究方法、关键技术和核心工具

本文的核心技术思路分为三个层次: 修改的准周期 Green 函数的构造与分析、基于该 Green 函数的合场边界积分方程的适定性证明、以及全离散 Nyström 格式的误差与条件数分析. 以下分段阐述每个步骤的关键工具与创新点.

1. 修改的准周期 Green 函数及频率连续性分析.

标准准周期 Green 函数在 Wood 异常频率处奇异的根源在于 Rayleigh 展开中 $s\in\sigma_*(k)$ 的项含有分母 $\sqrt{k^2-k_s^2}\to0$. 作者首先在 命题 2.5 中给出了一个对所有 $k\in\mathbb{R}$ 均有定义的准周期 Green 函数:

$$ G_k(\mathbf{x}) = \frac{i}{2L}\sum_{s\in\mathbb{Z}\setminus\sigma_*(k)} \frac{e^{i k_s x_1 + i\sqrt{k^2-k_s^2}\,|x_2|}}{\sqrt{k^2-k_s^2}} - \frac{1}{2L}\sum_{s\in\sigma_*(k)} |x_2| e^{i k_s x_1}. $$

在 $k\in\Lambda$ 时, 该函数并不满足远场辐射条件 (包含 $x_2$ 的线性增长项), 但它是方程 $(-\Delta-k^2)G_k = F_\#$ 的分布解. 这一非辐射 Green 函数为构造修正积分方程提供了出发点.

关键观察: 尽管 $G_k$ 本身在 $\Lambda$ 处关于 $k$ 并非连续, 但若将对应于 Wood 异常指标的 Rayleigh 项显式分离, 则剩余部分具有良好的连续性质. 具体地, 定义修正的 Green 函数

$$ \widehat{G}_k(\mathbf{x}) := G_k(\mathbf{x}) - \frac{i}{2L}\sum_{s\in\sigma\setminus\sigma_*(k)} \frac{e^{i k_s x_1}}{\sqrt{k^2-k_s^2}}, $$

其中 $\sigma$ 是包含 $\sigma_*(k)$ 的有限指标集. 空间奇异性的分析与 引理 3.1 (Bruno–Reitich 型一致收敛) 表明, 去掉有限个奇异模式后, 级数在远离点源的区域对 $k$ 一致收敛. 在 命题 3.2 中, 通过精细的位势估计证明了如下核心结果: 对任一固定的 $k\in\mathbb{R}$, 当 $\kappa\to k$ 时,

$$ G_{\kappa} - \frac{i}{2L}\sum_{s\in\sigma_*(k)} \frac{e^{i k_s x_1}}{\sqrt{\kappa^2-k_s^2}} - G_k \to 0 \quad \text{在 } C^{1,\alpha}_{\mathrm{loc}}(\mathbb{R}^2) \text{ 中}. $$

证明的关键是将差函数 $v_n$ 满足的方程右端源项分解为体积项与边界项, 利用 $L^p$ 估计与内 $W^{2,p}$ 估计及 Morrey–Sobolev 嵌入, 克服了在 Wood 异常点附近级数项无法绝对收敛的困难. 此连续性结果直接传递到积分算子 $S_k$ 和 $D_k$: 命题 3.3 确立了它们是从 $C_\beta(\Gamma_{\mathrm{per}})$ 到自身的紧算子, 且依赖于 $k$ 连续; 若 $\gamma\in C^{2,\alpha}$, 则作为 $C^{0,1}_\beta(\Gamma_{\mathrm{per}})$ 到自身的算子也具有同样的连续紧性. 这为后续使用紧扰动理论奠定基础.

2. 合场边界积分方程的构造与适定性.

利用 $\widehat{G}_k$ 直接构造单、双层位势则会因为 $\widehat{G}_k$ 中减去的非 Helmholtz 项而破坏解的散射性质. 作者巧妙地引入了一个有限维修正项来恢复辐射条件. 具体而言, 定义算子矩阵

$$ \mathcal{A}_k = \begin{pmatrix} D_k - i\eta S_k & R \\ T & -\operatorname{diag}(1+\sqrt{k^2-k_s^2})_{s\in\sigma} \end{pmatrix}, $$

其中 $R:C^{\sigma}\to C_\beta(\Gamma_{\mathrm{per}})$ 为有限维嵌入算子 $(R\mathbf{c})(\mathbf{x})=\sum_{s\in\sigma} c_s e^{i k_s x_1}$, $T:C_\beta(\Gamma_{\mathrm{per}})\to C^{\sigma}$ 通过边界积分提取相应模态的系数. 该设计使得对应于 $s\in\sigma$ 的向外传播波模式能够被正确激发.

定理 2.6 的证明基于以下步骤: 首先, 在假设原边值问题唯一可解的条件下, 可证明 $I+\mathcal{A}_k$ 在 $C_\beta(\Gamma_{\mathrm{per}})\times C^{\sigma}$ 上的单射性. 这是因为当右端项为零时, 所重构的位势满足齐次 Helmholtz 方程与齐次边界条件, 且其远场行为由 $\mathbf{c}=0$ 及辐射条件唯一确定; 结合唯一可解假设即得零解. 进而, 通过 Fredholm 二择一, 单射性结合紧性 ($\mathcal{A}_k$ 紧) 蕴含满射, 从而方程唯一可解. 频率连续性的证明则利用 $\mathcal{A}_k$ 对 $k$ 的连续依赖与解的存在唯一性的稳定性. 这样, 原问题被转化为一个在包含 Wood 异常频率的区间上良好适定的等价积分方程.

3. Nyström 离散与一致误差分析.

离散层面的主要困难有二: 第一, 核函数 $\widehat{G}_k$ 在 $\Gamma$ 上具有对数奇异性, 直接使用经典 Nyström 方法会损失精度; 第二, 级数表示的 Green 函数求值需高效精确实现, 且需保证截断误差对频率一致可控. 作者采用了以下技术组合.

Ewald 表示与奇性分离. 将 $\widehat{G}_k$ 及其法向导数通过 Ewald 分裂改写为空间域与频率域快速衰减级数之和, 显式分离出光滑部分与对数奇性部分. 具体地, 核函数 $H_{k,m}(t,\tau)$ 在截断参数 $m$ 下逼近真实的积分核, 其中 $W_{k,m}$ 利用互补误差函数 $\operatorname{erfc}$ 与有限模态和表达, 而余项由含指数衰减的级数及窗函数 $\chi$ 处理. 这种表达使得对数奇性被集中在 $E_1$ 项中, 其余项均为光滑或至少 $C^{1,\alpha}$.
特殊求积规则. 对于对数奇性 $E_1(\zeta^2 r^2)$, 作者构造了基于等距节点 $t_j^{(n)}=j\pi/n$ 与周期三角插值的求积公式 (见 (2.4) 中的 $R_j^{(n)}$). 该规则本质上利用了对数核在周期域上的展开, 通过引入辅助参数 $\zeta$ 及误差函数权重, 可达到对光滑密度的超代数精度, 同时将奇异性吸收到可精确计算的系数 $\widehat{E}_p$ 中. 整个离散矩阵 $B^{(n,m)}_k$ 的构造保证了即使 $k$ 接近或等于 Wood 异常频率, 矩阵元素仍一致有界且可计算.
误差分解与紧性扰动. 误差分析分两步: 首先, 将连续算子 $\mathcal{A}_k$ 与离散算子 $\mathcal{A}_k^{(n,m)}$ (对应于矩阵 $B^{(n,m)}_k-I$) 的差分解为截断误差 (由 $m$ 控制) 与求积误差 (由 $n$ 控制). 截断误差分析依赖 Ewald 和的指数收敛性质: 对于 $C^{2,\alpha}$ 曲线, 被截断的模态对核的影响由因子 $e^{-\xi m}$ 控制; 求积误差则利用解与核的正则性: 当 $\gamma\in C^{2,\alpha}$ 且 $g_k$ 连续 Lipschitz 时, 未知密度 $(\varphi_k,\mathbf{c}_k)$ 具有 $C^{0,\alpha}(\Gamma)$ 光滑性, 而处理对数奇点的求积规则以 $n^{-\alpha}$ 收敛. 其次, 借助算子族 $\mathcal{A}_k$ 的集体紧性 (collective compactness) 与对 $k$ 的连续依赖, 可通过紧扰动理论得出对于足够大的 $n,m$, 离散系统一致可逆, 且误差满足所述上界. 条件数的一致有界性来源于逆算子的一致有界性 (由逆算子的连续性得到) 与近似矩阵族的一致有界性.

综上所述, 本文的方法创新点在于: (a) 利用非辐射准周期 Green 函数减去奇异模式来获得对频率具有连续依赖性的积分核算子; (b) 通过有限维校正恢复正确的辐射条件, 而非使用极限过程或 Woodbury 公式; (c) 将 Ewald 表示与专门处理对数奇性的求积规则相结合, 构造了全离散 Nyström 格式; (d) 在技术层面, 基于 Green 函数的 $C^{1,\alpha}_{\mathrm{loc}}$ 连续性建立了算子族的连续性, 并运用集体紧性获得了对频率一致的误差与条件数估计. 这些步骤共同保证了在包含 Wood 异常的紧凑频率区间上格式的可靠性.

4. 与之前工作的比较

本文在二维周期光栅散射的边界积分方程解法领域做出了多方面的推进, 主要与以下几类已有工作形成对比.

1. 与标准准周期 Green 函数方法的比较.

传统方法直接使用命题 2.4 中的辐射 Green 函数 $G_k$ 构造边界积分方程 (例如见文献 [3, 11, 22–24]). 该 Green 函数在 Wood 异常频率集 $\Lambda$ 上没有定义, 且当 $k\to k_*\in\Lambda$ 时, 核函数与积分算子无界, 导致无法建立跨 $\Lambda$ 的频率一致误差分析. 本文则通过 $G_k$ 的修改形式 $\widehat{G}_k$ 与有限维修正, 构造了对所有 $k\in(\mathbb{R}\setminus\Lambda)\cup\Lambda_\sigma$ 均有定义且连续依赖 $k$ 的积分算子, 从而在连续与离散层面都获得了频率一致的适定性和误差估计. 这是对经典边界积分方法的一个实质性拓展.

2. 与平移准周期 Green 函数及混合空间/谱方法的比较.

Bruno 与 Haslam [8] 以及 Bruno, Shipman, Turc [13] 等发展了平移准周期 Green 函数方法, 通过对平移自由空间基本解的适当组合构造无奇异的 Green 函数替代物. 这类方法有效地回避了 Wood 异常处的分母为零问题, 并且数值表现良好, 但尚未提供严格的频率一致误差分析或条件数界. 类似地, Oschmann, Holzmann, Xiao 等 [9,10] 提出将准周期 Green 函数分解为奇异 Rayleigh 项与光滑剩余项, 再用 Woodbury–Sherman–Morrison 公式处理奇异部分. 这些工作使计算行之有效, 但同样缺乏对离散格式在 Wood 异常附近误差行为的一致理论保证. 本文首次在二维情形下给出了全离散 Nyström 格式跨越 Wood 异常的频率一致误差界 $C(n^{-\alpha} + e^{-\xi m})$, 并证明了系统矩阵的条件数的一致有界性, 从而将这类方法的数学严格性提升到了一个新的高度.

3. 与窗口 Green 函数方法的比较.

Bruno, Shipman, Turc [25] 提出的窗口 Green 函数方法利用有限秩模态修正来强制 Rayleigh 辐射条件, 能在全谱范围内获得超代数收敛的数值解. 该方法的数学分析证明了超代数收敛性, 但未明确处理频率一致性: 误差常数可能依赖于频率, 尤其是在 Wood 异常附近. 本文的结果表明, 在边界具有 $C^{2,\alpha}$ 正则性时, 我们不仅能获得收敛阶 $n^{-\alpha}$ (当 $\alpha<1$ 时仅为代数阶, 但对于光滑边界可期望更高阶), 更重要的是误差常数与 $k\in K$ 无关. 同时, 条件数的一致有界性确保迭代求解的可靠性. 因此, 本文的结果在频率稳定性分析方面是对窗口格林函数法的重要补充.

4. 与基于自由空间 Green 函数及辅助未知量的方法的比较.

另一类方法避开准周期 Green 函数, 而使用自由空间基本解配合辅助变量或 proxy 源来施加准周期性 (如 [5, 16, 30]). 这类方法可以自然地避免 Wood 异常, 但其误差分析通常依赖截断域的大小与边界条件近似精度, 难以获得完全与频率无关的误差界. 本文由于直接在准周期框架内处理, 有效地利用了 Green 函数的级数结构, 实现了解析的频率一致控制.

5. 局限性与假设条件.

本文的理论建立在几个明确的假设之上: (a) 齐次衍射问题唯一可解, 这可由无悬垂几何条件 $\nu_2>0$ 保证 (定理 2.1); (b) 边界曲线 $\gamma$ 至少具有 $C^{2,\alpha}$ 正则性, 这是为了保证误差阶 $n^{-\alpha}$; (c) 仅考虑二维标量波 (Helmholtz 方程) 和 Dirichlet 边界条件. 与一般三维 Maxwell 散射并不直接等同, 且对于透射问题或其它边界条件需另行推广. 但作者指出的三维非辐射 Green 函数已有初步研究 [13], 这表明本文的框架可能具备向三维拓展的潜力.

总体而言, 本文相较于既有工作的主要推进在于: 首次为一个有限维校正的准周期 Green 函数积分方程提供了全离散 Nyström 格式的频率一致适定性、误差估计与条件数界. 这使得在包含 Wood 异常频率的任何紧区间内, 数值解的精度和稳定性均能得到理论保证, 而不是仅仅依赖经验观察.

了解领域论文（摘要翻译）

1. 一种应用于谱差分方法的二阶通量新型紧致格式

A novel compact scheme for second-order fluxes applied to the Spectral Difference method

作者: Guido Lodato, Niccolò Tonicello

类别: math.NA, physics.flu-dyn

发布日期: 2026-08-10

arXiv ID: 2608.09615v1

链接: http://arxiv.org/abs/2608.09615v1

摘要翻译

中文标题: 一种应用于谱差分方法的二阶通量新型紧致格式

摘要翻译: 非连续谱元方法中二阶（粘性）项的离散通常依赖于一个辅助梯度变量，其在单元界面处的处理影响格式的精度和稳定性。Bassi-Rebay (BR1) 格式因其简洁和无参数特性而颇具吸引力，但在偶数多项式阶数时遭受次优收敛，并且需要扩展的五单元模板。受 Huynh 的通量重构格式启发，我们在谱差分 (SD) 方法框架内发展了一种针对二阶通量的紧致全中心格式。所提方法使用依赖界面、单侧连续通量修改辅助梯度的重构，将模板从五个单元缩减至三个，同时保持 BR1 的中心性质和无参数特性。该格式先在一维中发展，然后推广至多维。采用时间特征分析刻画其耗散和色散特性，包括内部罚项的影响。数值试验考察了线性扩散方程、欠分辨的局部 Dirac delta、非线性多孔介质方程，以及雷诺数 $\mathrm{Re}=1600$ 和 $5000$ 下三维 Taylor-Green 涡的隐式大涡模拟。该紧致格式对所有多项式阶数（含偶数阶）都恢复了预期的收敛阶，并减少了欠分辨和非线性状态下的伪振荡。由于改进的高波数数值模态阻尼，该格式在标准格式失效的湍流情形中保持稳定。所提方法为谱差分方法中的二阶通量提供了一种有吸引力的紧致替代方案。

2. 用于物理信息神经网络的分层秩演化表示

Hierarchical rank-evolving representation for physics-informed neural networks

作者: Ruoyang Su, Xi-Le Zhao, Kun Li, Liang Li

类别: cs.LG, math.NA, physics.comp-ph

发布日期: 2026-08-10

arXiv ID: 2608.09483v1

链接: http://arxiv.org/abs/2608.09483v1

摘要翻译

中文标题: 用于物理信息神经网络的分层秩演化表示

摘要翻译: 最近，基于张量的物理信息神经网络（T-PINNs）受到了越来越多的关注。然而，现有的T-PINNs仍然面临一个根本性挑战：它们主要依赖于预先指定的低秩张量分解并需要人工调整秩，这限制了它们捕捉多元解函数底层结构的能力，并阻碍了其实际部署。为了解决这一挑战，我们提出了一种用于多元函数的分层秩演化（简称为HRE）表示，该表示使我们能够忠实地捕捉目标多元函数的底层结构，同时自动确定秩。具体来说，在HRE表示的分层设计中，目标多元函数被分解为一个小规模内部张量和沿每个模态的一组一元函数，其中我们可以容易地部署定制的张量网络分解来捕捉小规模内部张量的底层结构。在HRE表示中，关键的超级参数——秩——可以在分解过程中自适应地揭示出来，使我们无需人工进行秩调整，从而使HRE在实际问题中具有实用性。此外，我们相应地构建了HRE-PINNs。广泛的数值实验，包括高维静态问题（亥姆霍兹方程和泊松方程）、非线性时间依赖问题（克莱因-戈登方程）和复杂流体动力学问题（流动混合方程和纳维-斯托克斯方程），表明HRE-PINNs在精度上始终优于现有的最先进方法。

3. 一个非局部非线性薛定谔模型：适定性、局部极限和结构保持的渐近相容傅里叶逼近

A nonlocal nonlinear Schr{ö}dinger model:well-posedness, local limit, and structure-preservingasymptotically compatible Fourier approximations

作者: Jiashu Lu, Yufeng Nie, Xinning Xie, Pingrui Zhang

类别: math.NA

发布日期: 2026-08-10

arXiv ID: 2608.09372v1

链接: http://arxiv.org/abs/2608.09372v1

摘要翻译

中文标题: 一个非局部非线性薛定谔模型：适定性、局部极限和结构保持的渐近相容傅里叶逼近

摘要翻译: 本文在周期域上引入一个非局部非线性薛定谔（NLS）模型，建立其适定性、守恒律、局部极限、色散性质，并发展一种结构保持的渐近相容傅里叶配置方法。我们证明，对于固定的非局部作用范围 $\delta$，该模型是全局适定的且保持质量和系统能量守恒，并且在适当的正则性假设下，非局部 NLS 解以 $O(\delta^2)$ 阶收敛到局部 NLS 解。我们还推导平面波的色散关系和群速度，建立它们的 $O(\delta^2)$ 局部极限，并刻画它们在高频下的行为。数值方法结合 Crank–Nicolson 时间步进与傅里叶配置，并保持网格质量和原始网格能量。在时间步长 $\tau$、傅里叶截断 $N$ 以及精确解的 $H^r$ 正则性假设下，非局部解的 $H^s$ 误差为 $O(\tau^2+N^{s-r})$，且关于作用范围一致成立。此外，相对于局部 NLS 解的总 $H^s$ 误差为 $O(\delta^2+\tau^2+N^{s-r})$，而 $\delta$、$\tau$ 和 $N$ 之间不需要任何耦合条件，这证明了所提方法的渐近相容性。文中给出了一维、二维和三维的数值实验，以验证理论精度和离散守恒性，确认在作用范围和离散参数独立变化下的收敛性，并展示作用范围和核函数如何影响色散波传播。

4. 相对论欧拉方程的高阶全离散多熵稳定与保界格式

High-order fully discrete multi-entropy-stable and bound-preserving schemes for relativistic Euler equations

作者: Linfeng Xu, Kailiang Wu

类别: math.NA, astro-ph.IM, physics.comp-ph, physics.flu-dyn

发布日期: 2026-08-10

arXiv ID: 2608.09327v1

链接: http://arxiv.org/abs/2608.09327v1

摘要翻译

中文标题: 相对论欧拉方程的高阶全离散多熵稳定与保界格式

摘要翻译: 离散熵不等式是守恒律方程组可用的主要非线性稳定性估计，而对其进行评估的前提是状态在物理上可容许。然而，迄今为止，这两点都是分别实现的。熵稳定格式几乎总是半离散的，围绕一个选定的熵对构建，并且假定密度和压力恒正——这是使熵有良好定义的基础，而保界限制器保持解的可容许性却不提供任何熵估计。对于狭义相对论欧拉方程，两者完全无法分离，因为守恒变量到原始变量的映射是隐式的，因此不可容许的状态根本没有熵可供修正。在此我们构造了高阶间断伽辽金和有限体积格式，据我们所知，这些格式首次在全离散意义下对任意给定的有限个凸熵对族是熵稳定的——我们将该性质称为多熵稳定性——并且在任何需要计算熵的地方都可证明是容许的。所有这一切均通过单个逐单元投影实现，且不会丧失守恒性或设计精度。该构造依赖于相对论因果性，它将每一个特征速度限制在光速以内。因此，数值粘性可以针对所有状态和所有状态方程一次性固定，然后一个两点构造块即可服务于整个熵族。由于只用到容许集的凸性和这一速度上界，对于相关方程组这一途径依然敞开。最后，在涉及四个状态方程的计算中，这些格式在接近真空处保持高阶精度，在强激波、近真空的激波-涡相互作用或洛伦兹因子超过 $70$ 的射流中不产生不可容许状态，并验证了每一个受强制的离散熵都单调衰减。

5. 关于线性半负问题的一阶次对角Padé逼近子的能量律与稳定性

On Energy Laws and Stability of First-Subdiagonal Pade Approximants for Linear Seminegative Problems

作者: Miaosen Jiao, Zheng Sun, Kailiang Wu

类别: math.NA

发布日期: 2026-08-10

arXiv ID: 2608.09239v1

链接: http://arxiv.org/abs/2608.09239v1

摘要翻译

中文标题: 关于线性半负问题的一阶次对角Padé逼近子的能量律与稳定性

摘要翻译: 针对求解线性半负问题，我们为由指数函数的一阶次对角Padé逼近子生成的有理时间离散推导了一个显式离散能量恒等式。本工作将[Z. Sun, Y. Wei, 和 K. Wu, SIAM J. Numer. Anal., 60 (2022)]中的对角Padé能量律推广到了一阶次对角族。主要的新成分是对与离散能量恒等式中的半内积项相关联的能量系数矩阵进行显式Cholesky型分解。该分解的构造与证明是非平凡的，因为矩阵的项是Padé系数的交错和，且三角因子具有与奇偶性相关的阶乘结构。我们通过将其归结为标量有理恒等式，并通过有限乘积约化和裂项求和来建立它们，从而证明了该分解。结合一个β系数消去，该分解产生了一个精确的离散能量律，它恢复了线性半负问题的经典无条件收缩性。由对角Padé能量律设置改编的数值实验展示了预测的阶数并验证了离散耗散恒等式。

6. 两阶段MV-DeepONet：面向随机输入场驱动的不确定性传播的概率算子学习

Two-Step MV-DeepONet: Probabilistic Operator Learning for Uncertainty Propagation Driven by Random Input Fields

作者: Yupei Nie, Lei Wang, Jiasen Liu

类别: math.NA, cs.AI

发布日期: 2026-08-10

arXiv ID: 2608.09071v1

链接: http://arxiv.org/abs/2608.09071v1

摘要翻译

中文标题: 两阶段MV-DeepONet：面向随机输入场驱动的不确定性传播的概率算子学习

摘要翻译: 正向不确定性传播在复杂物理系统中可能诱导场值输出上的结构化协方差。对于概率代理模型，总预测协方差包括条件均值在不同输入实现下的协方差以及平均条件预测协方差。概率DeepONet（Prob-DeepONet）通过单次前向传播预测逐点高斯均值和方差，提供轻量级不确定性量化，但其条件预测协方差仅限于对角形式。为了描述跨位置条件依赖性，而无需显式参数化高维协方差矩阵，我们通过两项主要修改开发了一种两阶段均值-方差DeepONet（两阶段MV-DeepONet）。首先，采用两阶段训练将输出基底学习与输入到系数的映射解耦，并结合基底正交化与子空间旋转。其次，将高斯概率建模从高维物理输出空间转移到低维旋转系数空间。通过共享基底映射这些概率系数，在物理输出空间中诱导出一般非对角的条件预测协方差，同时保持单次推断。Frobenius范数误差分解及相应的上界指出，低秩协方差可压缩性、主干子空间近似、有限样本统计误差和系数空间协方差估计是支配协方差恢复的主要因素。在三个由偏微分方程（PDEs）控制的典型问题和一个高超声速钝体气动热问题上进行的数值实验表明，与Prob-DeepONet相比，所提方法具有改进的泛化能力、更结构化的不确定性带以及非对角相关模式的准确恢复。

7. 对平面 Berenstein 猜想的一个计算机辅助反例

A computer-assisted counterexample to the planar Berenstein conjecture

作者: Matthew J. Colbrook, Siavash Sadeghi, George Stepaniants

类别: math.AP, math-ph, math.NA, math.SP

发布日期: 2026-08-09

arXiv ID: 2608.08953v1

链接: http://arxiv.org/abs/2608.08953v1

摘要翻译

中文标题: 对平面 Berenstein 猜想的一个计算机辅助反例

摘要翻译: Colbrook 和 Stepaniants 近期的工作给出了平面 Pompeiu 猜想和 Schiffer 猜想的首个反例，并引入了本文所使用的共形固定圆盘、圆盘多项式以及验证尾部的方法。通过将该框架适应于互补的 Dirichlet 端点，我们否定了无限制的平面 Berenstein 猜想。具体地，我们构造了一个有界单连通区域 $Ω$，具有实解析 Jordan 边界，它不是圆盘，并且存在 $k∈(27.4381178838,27.4381198839)$ 以及一个非零实值函数 $u∈C^ω(\overlineΩ)$ 满足 $(Δ+k^2)u=0$ 于 $Ω$ 中，且 $u=0, \partial_νu=\text{常数}\ne0$ 于 $\partialΩ$ 上。因此，在没有对 $u$ 附加符号假设的情况下，超定的 Dirichlet–Neumann 数据不能刻画圆盘。该区域具有 26 阶二面体对称性，但既不是圆盘也不是中心对称的，并且相应的特征函数变号。等价地，其边界弧长测度满足 $\widehat{σ_{\partialΩ}}(kω)=0$ 对 $ω∈\mathbb S^1$ 成立。在共形转移到单位圆盘后，精确的支集恒等式和定量的圆盘多项式估计给出了无穷维尾部的严格控制。进而，一个 Newton–Kantorovich 论证将存在性归结为有限个显式不等式，这些不等式通过区间算术进行验证。从 Pompeiu–Schiffer 问题的推广并不是形式上的。先前的构造将两个边界条件吸收进一个逆 Laplacian 方程中。在本文所考虑的 Dirichlet 端点处，非零的 Neumann 数据迫使调和源模式保留，从而产生了一个耦合的内部–边界系统，该系统涉及全零 Dirichlet 逆及其 Neumann 迹，以及一个单独的符号恢复问题。

8. 位似双曲坐标中的半线性波动方程与尾部衰减

Semilinear wave equations in homothetic hyperboloidal coordinates and tail decay

作者: Anıl Zenginoğlu, Sebastiano Bernuzzi, Andrea Nützi

类别: gr-qc, math.AP, math.NA

发布日期: 2026-08-09

arXiv ID: 2608.08863v1

链接: http://arxiv.org/abs/2608.08863v1

摘要翻译

中文标题: 位似双曲坐标中的半线性波动方程与尾部衰减

摘要翻译: 晚期波的尾部沿着未来类光无穷远和有限半径处的类时世界线以不同的速率衰减。紧化数值演化必须同时体现类光无穷远处较慢的衰减和内部较快的衰减，从而在这两个区域之间产生越来越陡峭的过渡。我们利用适应于尾部缩放结构的位似双曲坐标，解决了闵可夫斯基时空中半线性波动方程的这一困难。在这些坐标下，尾部趋于一个光滑的径向分布，在每个紧化半径处都具有相同的衰减速率。因此，该表述避免了在稳态双曲演化中所见的径向剖面变陡，并且达到晚期所需的步数仅随推迟时间对数增长。我们利用3+1维空间的伪谱模拟展示了该方法，并再现了Rinne猜想的通用衰减速率。我们还提供了数值证据，表明在类光无穷远处存在一种非通用的余维一消除主导尾部系数的现象，从而导致更快的衰减速率。

9. ADEx-FNO: 适用于变化几何的统一环境域傅里叶神经算子框架

ADEx-FNO: A Unified Ambient-Domain Framework for Fourier Neural Operators on Varying Geometries

作者: Roberto Nuca, Giovanni Testa, Luca Galimberti, Matteo Parsani

类别: math.NA, cs.CE, cs.LG

发布日期: 2026-08-09

arXiv ID: 2608.08608v1

链接: http://arxiv.org/abs/2608.08608v1

摘要翻译

中文标题: ADEx-FNO: 适用于变化几何的统一环境域傅里叶神经算子框架

摘要翻译: 傅里叶神经算子(FNOs)提供了高效的非局部谱学习, 但变化的几何和独立选取的离散化仍然难以适应。我们引入环境域拓展傅里叶神经算子(ADEx-FNO), 这是一个将几何纳入而不修改定义傅里叶算子层的确定性框架。每个物理域被嵌入一个固定的环境超立方体, 并由一个带符号距离函数表示。输入和解场被确定性地拓展到环境域, 转移到公共的、可能非均匀的直线潜在网格上, 由FNO处理, 然后插值到独立选取的目标离散化上, 并限制在物理域内。所有几何转移操作都在优化过程之外, 且无需任何可训练的图、点云、变形或几何解码模块。 ADEx-FNO在留存的二维和三维光滑域非线性泊松问题和流-反应-扩散问题上取得了0.32%-0.77%的相对l2误差, 并且也在未见过的非光滑几何上进行了评估。随后, 单次ADEx-FNO推断被用于初始化常规CFD求解器。对于所有29个收敛的二维和三维RANS算例, 伪时间迭代减少, 平均减少分别为44.17%和43.03%, 且在三种网格分辨率下收益相当。URANS算例将窗口后物理时间推进减少了18.52%-27.51%。在从二维URANS训练数据迁移到不同马赫数和雷诺数的DNS时, 自举区间减少了23.47%-48.21%, 具体取决于目标统计量。在所有CFD测试中, ADEx-FNO仅提供初始场; 控制方程求解器控制后续解, 而物理或统计一致性则与计算节省分开评估。

10. 具有变形界面的流固耦合问题CutFEM的收敛性

Convergence of a CutFEM for fluid–structure interaction with a deforming interface

作者: Miguel A. Fernández, Buyang Li, Maxim A. Olshanskii

类别: math.NA

发布日期: 2026-08-08

arXiv ID: 2608.08140v1

链接: http://arxiv.org/abs/2608.08140v1

摘要翻译

中文标题: 具有变形界面的流固耦合问题CutFEM的收敛性

摘要翻译: 我们针对应用于不可压缩粘性流体与弹性固体流固耦合（FSI）问题的半离散浸入CutFEM提出严格的误差分析。该方法利用拉格朗日乘子施加流固耦合条件，同时充分考虑结构变形对流场区域的影响。对于充分正则的FSI解和次数为 $k\geq3$ 的有限元，我们证明了在FSI问题相关的能量范数下，流体速度和固体位移具有 $k$ 阶收敛性，流体压力具有 $k-1$ 阶收敛性。据我们所知，这是首个针对具有变形界面的全耦合FSI系统的非拟合有限元方法的严格收敛分析。

不相关论文（基本信息）

1. 基于Bernstein表示的有理T样条曲面的双射性分析

Bijectivity analysis of rational T-spline surfaces via Bernstein representations

作者: Jia-Xuan Li, Ying-Ying Yu, Ya-Shu Liu, Xin Li, Ye Ji, Chun-Gang Zhu

类别: math.NA

发布日期: 2026-08-10

arXiv ID: 2608.09838v1

链接: http://arxiv.org/abs/2608.09838v1

摘要: Ensuring the bijectivity of spline-based parameterizations is fundamental in geometric modeling and isogeometric analysis, as invalid mappings may lead to self-intersections, singular Jacobians, and numerical instability. While T-splines offer enhanced flexibility through local refinement, this flexibility also makes bijectivity verification significantly more challenging. In this work, we propose a rigorous and efficient framework for bijectivity analysis of rational T-spline surfaces based on Bézier extraction. The key idea is to reformulate the T-spline representation into a collection of element-wise rational Bézier patches, on which the Gram determinant of the mapping admits a Bernstein polynomial representation. This enables a coefficient-based analysis of local regularity by exploiting the convex hull and positivity properties of the Bernstein basis. Based on this formulation, we derive a sufficient condition for bijectivity from the nonnegativity of Bernstein coefficients, together with a necessary condition based on the sign consistency of corner coefficients. For cases where these conditions are inconclusive, we introduce a hierarchical subdivision strategy that progressively localizes ambiguous regions and resolves them through refinement. The proposed method provides a certified and adaptive procedure for bijectivity verification that avoids dense numerical sampling and remains computationally efficient. Numerical experiments on complex T-spline geometries demonstrate that the approach accurately detects both valid and near-degenerate configurations, while scaling effectively to large models with thousands of rational Bézier patches. The framework is fully compatible with standard isogeometric analysis workflows.

2. 基于非对称加权 Hermite 函数的 Vlasov-Poisson 系统 Galerkin 逼近的收敛性

Convergence of Galerkin approximations based on asymmetrically-weighted Hermite functions for the Vlasov-Poisson system

作者: Ruiyang Dai

类别: math.NA

发布日期: 2026-08-10

arXiv ID: 2608.09827v1

链接: http://arxiv.org/abs/2608.09827v1

摘要: The convergence of Galerkin approximations based on asymmetrically-weighted (AW) Hermite functions, applied to the Vlasov-Poisson (VP) system, is analyzed in this work. The VP system is written as an hyperbolic system using Hermite functions in velocity. To obtain stability properties of the numerical methods, we consider an unweighted inner product, and establish naturely the stability with respect to the unweighted norm. Consequently, we prove the convergence of the method. Furthermore, we derive error estimates between the numerical solution and the smooth solution of the VP system. Our results confirm that the Galerkin approximations based on AW Hermite functions exhibit spectral accuracy in Sobolev spaces.

3. 信息几何正则化计算含激波流动的灵敏度

Information geometric regularization for computing sensitivities of flows with shocks

作者: Florian Schäfer

类别: math.NA, math.OC

发布日期: 2026-08-10

arXiv ID: 2608.09759v1

链接: http://arxiv.org/abs/2608.09759v1

摘要: Computing (adjoint) sensitivities of flows with shocks is a longstanding problem in computational fluid dynamics. The spurious sensitivities due to shock sensors and limiters frequently force practitioners to accept the errors incurred by “freezing” limiters and shock sensors. The recently proposed information geometric regularization (IGR) is an inviscid, PDE-based regularization of the compressible Euler equations that replaces shocks with smooth profiles without damping fine-scale structures. This work derives the forward and adjoint sensitivities for the IGR system with periodic boundary conditions and demonstrates their convergence, under grid refinement, to the sensitivities obtained by finite differences or automatic differentiation through the forward solve.

4. 非自治随机微分方程Milstein格式的李雅普诺夫指数双边估计

Two-sided estimates of Lyapunov exponents for Milstein schemes of non-autonomous stochastic differential equations

作者: Vu Thi Hue, Bao Quoc Tang, Do Duc Thuan

类别: math.NA, math.PR

发布日期: 2026-08-10

arXiv ID: 2608.09756v1

链接: http://arxiv.org/abs/2608.09756v1

摘要: The stabilising effect of multiplicative noise for stochastic differential equations, though counterintuitive, has been observed and investigated extensively in last decades. In practice, it is desirable to know if such stabilisation holds also for the discretised setting. In this paper, we address this problem by means of sharp upper and lower estimates for Lyapunov exponents of Milstein schemes for non-autonomous stochastic differential equations. These estimates provide precise large time behaviour in both almost sure and $p$-moment sense. In particular, our results show the preservation of stabilisation from the continuum setting to the discretised setting. One main idea of our analysis is to exploit the second order term concerning the stochastic noise from the Milstein scheme to obtain precise estimates for Taylor expansions of logarithmic and power functions.

5. 热力学矩阵求逆的理论分析：与预条件梯度下降的一阶等价性及其对模拟计算的启示

Theoretical Analysis of Thermodynamic Matrix Inversion: First-order Equivalence to Preconditioned Gradient Descent and Implications for Analog Computing

作者: Gerhard Kirsten, Michael Selby, Janith Petangoda, Olle Halqvist Elias, James Meech, Phillip Stanley-Marbell

类别: math.NA

发布日期: 2026-08-10

arXiv ID: 2608.09743v1

链接: http://arxiv.org/abs/2608.09743v1

摘要: Recent research has demonstrated the possibility of exploiting the thermodynamics of coupled electrical oscillators to implement computational tasks such as matrix inversion. While physical implementations rely on thermal noise to drive equilibration, we show that the underlying dynamics reduce to a deterministic iterative algorithm. Building on the framework of Aifer et al., we analyze the moment evolution of the Ornstein-Uhlenbeck process governing thermodynamic symmetric positive definite (SPD) matrix inversion. We prove that to a first-order approximation, the covariance dynamics are mathematically identical to preconditioned gradient descent on the Frobenius norm of the residual $\tilde{A}^{-1}A-I$. This equivalence demonstrates that thermal fluctuations, while essential for physical thermodynamic hardware, are algorithmically redundant for convex problems with a single global minimum. We validate the resulting algorithm against Thermox (Duffield et al.), a stochastic thermodynamic simulator, achieving speedups exceeding 100,000-fold while remaining competitive with the Newton-Schulz iteration. We also demonstrate acceleration through Schur complement techniques. These results establish a rigorous link between analog thermodynamic computing, statistical physics, and deterministic optimization methods.

6. 带记忆的非线性输运中逆初值问题的 Carleman–Picard 与时间降维

Carleman–Picard and time-dimensional reduction for inverse initial-data problems in nonlinear transport with memory

作者: Navaraj Neupane, Loc H. Nguyen

类别: math.NA

发布日期: 2026-08-10

arXiv ID: 2608.09665v1

链接: http://arxiv.org/abs/2608.09665v1

摘要: We study an inverse initial-data problem for a quasilinear transport equation with nonlinear and memory effects. The unknown initial state is reconstructed from time-dependent measurements on the outflow boundary, with prescribed inflow data. We first apply a Legendre–exponential time-dimensional reduction to transform the governing equation into a finite nonlinear system in space. We then develop a Carleman-weighted and Tikhonov-regularized Picard method. At each iteration, the nonlinear terms are evaluated using the previous iterate, leading to a linear minimization problem with a unique solution. A Carleman estimate for the principal transport operator is used to prove that, for a sufficiently large Carleman parameter, the resulting Picard map is contractive on a prescribed admissible set. Consequently, the method converges from an arbitrary initial guess in that set. We also establish stability with respect to noisy outflow data. These analytical results concern the truncated and regularized reduced problem. Two-dimensional numerical experiments demonstrate accurate reconstruction of single and multiple inclusions, robustness with respect to noise, and rapid convergence of the Picard iteration.

7. 扩散映射核岭回归

Diffusion Maps Kernel Ridge Regression

作者: John Harlim, Daning Huang, Jiwoo Song

类别: math.NA, math.ST

发布日期: 2026-08-10

arXiv ID: 2608.09623v1

链接: http://arxiv.org/abs/2608.09623v1

摘要: In this paper, we study kernel ridge regression using the data-driven diffusion maps (DM) kernel, which is constructed through algebraic manipulations of the diffusion maps algorithm. Under the assumptions that the data lie on a manifold and are sampled uniformly, we prove that the appropriately scaled DM kernel converges uniformly to the heat kernel on the manifold for sufficiently large times as the dataset size increases and the kernel bandwidth is scaled appropriately. Consequently, the limiting Reproducing Kernel Hilbert Space (RKHS) induced by the DM kernel coincides with the RKHS associated with the heat kernel on the manifold. We further show that the RKHS induced by the DM kernel is isometrically isomorphic to the RKHS of the Gaussian kernel, which is continuously embedded in the RKHS of a Matérn kernel whose norm is equivalent to an appropriate Sobolev norm. This result implies that standard risk bounds for kernel ridge regression applicable to Matérn kernels also apply to the DM kernel. Finally, we provide numerical results that (1) validate the convergence of the heat kernel approximation, (2) demonstrate the greater expressiveness of the DM kernel compared to the Gaussian kernel for supervised learning over a larger class of functions on manifolds with boundary, and (3) demonstrate the advantage of the DM kernel over the Gaussian kernel in learning functions with varying frequencies and co-dimensions.

8. 用于液晶指向矢建模中双重鞍点系统的双参数块预条件子

A block preconditioner with two parameters for double saddle point systems in liquid crystal director modeling

作者: Mehdi Makhdomi, Davod Khojasteh Salkuyeh, Morad Ahmadnasab

类别: math.NA

发布日期: 2026-08-10

arXiv ID: 2608.09603v1

链接: http://arxiv.org/abs/2608.09603v1

摘要: This paper introduces a novel block preconditioner for a class of double saddle point systems arising from the finite element discretization of liquid crystal director modeling. We first establish sufficient conditions to guarantee the convergence of the underlying iterative scheme. A spectral analysis of the preconditioned matrix is then performed, leading to an effective strategy for parameter selection. The new preconditioner is used to accelerate the convergence of the flexible version of GMRES. Experimental results confirm the strong performance of the method.

9. 无严格互补条件下谱多面体上一种Frank-Wolfe型方法的线性收敛

Linear Convergence of a Frank-Wolfe-type Method over the Spectrahedron without Strict Complementarity

作者: Dan Garber

类别: math.OC, math.NA

发布日期: 2026-08-10

arXiv ID: 2608.09569v1

链接: http://arxiv.org/abs/2608.09569v1

摘要: We consider smooth convex minimization over the spectrahedron using Frank-Wolfe-type methods based only on extreme-eigenvector computations. In our recent work \cite{garber2026randomized} we presented the first ambient-dimension-independent linear convergence rate under quadratic growth. However, the method makes an additional strong strict complementarity assumption, it is randomized, its linear rate holds only after a burn-in phase and in expectation, and it requires the objective smoothness constant. We show that these limitations can be removed. Assuming quadratic growth and that all optimal solutions have the same rank, but without assuming strict complementarity, we give a deterministic and parameter-free Frank-Wolfe-type method with a global ambient-dimension-independent linear convergence rate.

10. 通过提示随机Transformer实现无需训练的通用逼近

Training-Free Universal Approximation by Prompting Random Transformers

作者: Alexander Hsu, Rongjie Lai

类别: cs.LG, math.NA, math.ST

发布日期: 2026-08-10

arXiv ID: 2608.09558v1

链接: http://arxiv.org/abs/2608.09558v1

摘要: How expressive is prompting a transformer? Answering this question is important for separating the roles of prompting, architecture, and pretraining in transformer models, and for determining whether task-specific behavior must be stored in model weights or can instead be induced at inference time through the prompt. We show, in an approximation-theoretic sense, that pretraining is optional: a single-layer softmax attention network with random, untrained weights can approximate any Hölder function on a compact manifold when steered by an appropriate soft prompt. Guided by the connection between softmax attention and kernel methods, we construct explicit soft prompts (a prompt per target function, independent of the query) as solutions to linear systems matching attention logits to Gaussian kernel exponents, under which the frozen transformer emulates the classical Nadaraya-Watson kernel estimator. The construction requires only a mild rank condition on the weights, which we show holds almost surely under Gaussian initialization. The prompted network inherits the theoretical guarantees of kernel regression, leading to universal approximation theorems with minimax-optimal rates that depend on the intrinsic dimension. We further quantify the cost of prompting, exposing a tradeoff between the norm of the constructed soft prompt tokens, prompt length, and hidden dimension. Numerical experiments corroborate the constructions and predicted rates.

11. 动态扩散有限元方法的条件唯一性与最优能量范数收敛

Conditional Uniqueness and Optimal Energy-Norm Convergence for the Dynamic Diffusion Finite Element Method

作者: Isaac P. Santos

类别: math.NA

发布日期: 2026-08-10

arXiv ID: 2608.09505v1

链接: http://arxiv.org/abs/2608.09505v1

摘要: We revisit the nonlinear two-scale Dynamic Diffusion (DD) finite element formulation mathematically analyzed by Santos et al. (2021) for stationary advection–diffusion–reaction problems. Two additional theoretical results are established without modifying the discrete formulation. First, the artificial diffusivity defining the nonlinear diffusion operator $D_h$ is shown to be locally Lipschitz continuous with respect to the discrete solution. This property leads to conditional uniqueness of the discrete solution under a mesh-dependent smallness condition. Second, by separating the approximation error in the energy norm from the contribution associated with artificial diffusion, we derive an optimal first-order a priori energy-norm estimate for continuous piecewise linear finite elements. This result also clarifies the earlier $O(h^{1/2})$ estimate, which applies to a stronger combined error measure and therefore does not determine the convergence rate of the energy error alone. Numerical experiments for both pure advection–diffusion and advection–diffusion–reaction problems corroborate the predicted first-order energy-norm behavior and the first-order decay of the square-root artificial-dissipation measure. A strongly convection-dominated test with sharp outflow layers further illustrates the stabilizing effect associated with $D_h$. These results extend the mathematical understanding of the original DD formulation while preserving its underlying discrete structure.

12. Walk-on-Spheres 蒙特卡罗与深度神经网络对带漂移和消亡项的椭圆型偏微分方程的逼近

Walk-on-Spheres Monte Carlo and deep neural network approximations of elliptic PDEs with drift and killing

作者: Konrad Kleinberg, Thomas Kruse

类别: math.NA, cs.LG, math.AP, math.PR

发布日期: 2026-08-10

arXiv ID: 2608.09494v1

链接: http://arxiv.org/abs/2608.09494v1

摘要: In this paper we provide Monte Carlo and deep neural network approximations for stochastic representations of solutions to linear elliptic partial differential equations with constant diffusion, drift and killing. Building on the modified Walk-on-Spheres algorithm of Beznea et al. (arXiv:2209.01432), we introduce Monte Carlo estimators that explicitly incorporate sampled random times arising in the analyzed stochastic representations. We establish uniform error bounds for these estimators and show that, under suitable assumptions, a prescribed approximation accuracy is achieved with sample complexities growing at most polynomially in both the inverse accuracy and the problem dimension. Furthermore, we prove a deep neural network approximation result for the stochastic representations. Assuming suitable neural network representations of the boundary data and the distance function to the boundary, we use the constructed Monte Carlo to design deep neural networks that approximate the representation uniformly with a number of parameters growing at most polynomially in the inverse accuracy and the problem dimension. These results extend previous complexity analyses to a broader class of elliptic equations involving drift and killing.

13. 二阶非拟合幽灵有限元法求解椭圆界面问题及其在低维半导体器件中的应用

Second order unfitted ghost-FEM for elliptic interface problems with applications to low-dimensional semiconductor devices

作者: Clarissa Astuto, Giovanni Nastasi

类别: math.NA, math-ph

发布日期: 2026-08-10

arXiv ID: 2608.09466v1

链接: http://arxiv.org/abs/2608.09466v1

摘要: We develop an unfitted ghost finite element method for elliptic interface problems with discontinuous diffusion coefficients and apply it to the electrostatic simulation of low-dimensional semiconductor devices. The proposed approach is based on a fixed Cartesian grid and represents the geometry by level-set functions, avoiding mesh generation and remeshing even in the presence of interfaces. A snapping-back-to-grid strategy is used to control the small-cut-cell issue, while interface conditions are weakly enforced by a symmetric Nitsche formulation. The method is first validated on benchmark elliptic interface problems with different geometries and coefficient jumps, showing second-order accuracy for the solution and first-order accuracy for its gradient. As an application, we consider a graphene field-effect transistor described by a self-consistent drift-diffusion-Poisson model. The electrostatic potential is computed in a two-dimensional oxide/graphene/oxide structure, while charge transport in the graphene layer is modeled by one-dimensional bipolar drift-diffusion equations in the degenerate case and by including a field-dependent mobility model. The coupled nonlinear system is solved by a damped fixed-point iteration combined with a domain-decomposition treatment of the three-layer geometry. Numerical simulations reproduce transfer characteristics with a clear transition from an OFF state to an ON state and show the influence of the gate voltage on the two-dimensional electrostatic potential. The results also indicate that the effective thickness and discretization of the graphene layer affect the transverse potential profile, supporting the use of a full two-dimensional electrostatic description with explicit oxide/graphene interface conditions.

14. 有限元应力更新中的自动微分

Automatic differentiation in finite element stress updating

作者: Mao Ouyang, Alexandros Petalas, William M. Coombs, Charles E. Augarde

类别: math.NA

发布日期: 2026-08-10

arXiv ID: 2608.09442v1

链接: http://arxiv.org/abs/2608.09442v1

摘要: Computational solid mechanics relies on the discretisation of differential equations, requiring numerical differentiation and integration, often via quadrature. Recently, automatic differentiation (AD) has attracted interest as a tool to support these operations. This paper introduces the use of AD in nonlinear finite element analysis, focusing on stress updating, a key and repeatedly executed step in which stresses are computed from strain increments. Both implicit and explicit schemes require derivatives of constitutive models, which can be complex, error prone, and time consuming to derive analytically. We compare traditional analytical differentiation with automatic differentiation for implementing a hyperplastic Critical State model widely used in geotechnical engineering. Drained triaxial compression tests with varying overconsolidation ratios are simulated using both approaches. The results show that AD significantly simplifies the implementation of backward Euler stress integration by removing the need for manual derivation of derivatives. Despite this simplification, AD maintains high accuracy and robustness comparable to analytical approaches. The findings demonstrate that automatic differentiation can streamline the development of nonlinear material models, enabling efficient and reliable implementation of constitutive laws of arbitrary complexity. This opens the way for more flexible and maintainable computational mechanics codes.

15. poly_rust: 一种面向多面体方法的领域特定语言及其Rust解释器

poly_rust: a domain-specific language for polytopal methods with a Rust interpreter

作者: Daniele A. Di Pietro

类别: math.NA

发布日期: 2026-08-10

arXiv ID: 2608.09414v1

链接: http://arxiv.org/abs/2608.09414v1

摘要: In this article we introduce poly_rust, a new software library that implements a domain-specific language for polytopal methods. Polytopal methods support meshes much more general than those on which traditional finite element methods are built. This can be achieved by dropping global function spaces and reconstructing the relevant operators directly from the degrees of freedom through the solution of local variational problems. Reconstructions are then used in place of shape functions in the global variational formulation. poly_rust provides facilities for expressing a wide variety of polytopal methods using a syntax that closely resembles their mathematical formulation. We review here the main aspects of the library from the perspective of a basic user and provide numerical examples spanning several problems and methods.

16. 一种用于一般类动力系统的简单二阶非标准数值方法及其应用

A simple second-order nonstandard numerical method for a general class of dynamical systems and its applications

作者: Manh Tuan Hoang

类别: math.NA

发布日期: 2026-08-10

arXiv ID: 2608.09141v1

链接: http://arxiv.org/abs/2608.09141v1

摘要: In this work, we consider a class of continuous-time autonomous dynamical systems that model various important phenomena and processes encountered in real-world situations. We construct a second-order nonstandard finite difference (NSFD) method that simultaneously preserves two essential properties of the dynamical systems for all finite step sizes, namely the positivity of the solutions, the set of equilibrium points and their asymptotic stability. This NSFD method is constructed based on an appropriate choice of nonstandard denominator functions and a weighted discretization of the right-hand side functions. Under easily-verified conditions, the denominator functions guarantee second-order convergence, whereas the weights ensure the dynamic consistency. By taking advantage of the specific structure of the right-hand side functions, a simple discretization is utilized instead of the nonlocal discretization approaches commonly used in previous works. This simplifies the construction of the proposed NSFD method and, in particular, makes its asymptotic stability analysis easier. As an illustration and an important application, we apply the constructed second-order NSFD method to a well-known two-stage structured species model with recruitment. Consequently, a simple second-order NSFD scheme for the considered two-stage structured species model is derived, improving upon a first-order NSFD scheme constructed in a previous work. Numerical experiments demonstrate the advantages of the second-order NSFD scheme over a standard second-order numerical method, namely, the explicit trapezoidal method. The proposed NSFD method is simple and can be applied to a broad class of dynamical system models arising in both theory and applications. Moreover, it can be readily combined with the Richardson extrapolation technique to improve its accuracy.

17. 用于微磁模拟中半隐式Landau-Lifshitz格式的高效多重网格方法

Efficient Multigrid Methods for Semi-implicit Landau-Lifshitz Schemes in Micromagnetic Simulations

作者: Changjian Xie

类别: math.NA

发布日期: 2026-08-10

arXiv ID: 2608.09102v1

链接: http://arxiv.org/abs/2608.09102v1

摘要: An efficient aggregation-based multigrid approach is developed in this work to solve linear algebraic systems generated by discretizing the Landau-Lifshitz equation in micromagnetics. The discretization couples the first-order backward differentiation formula with first-order extrapolation, and the underlying equation describes magnetization dynamics within micromagnetic simulations. For these linear systems, standard iterative methods and conventional multigrid solvers intended for symmetric problems suffer from deteriorated convergence under mesh refinement and poor computational performance. Existing algorithms struggle to handle the system’s intrinsic features, namely its sparse spectral properties and non-symmetric matrix structure. To overcome these limitations, a multigrid framework is constructed, where smoothing and coarse-grid correction operators are customized for the target linear system. Numerical tests confirm the robustness and mesh-independent convergence of the resulting method. Compared with well-established Krylov-subspace solvers and conventional multigrid techniques, the aggregation-based multigrid solver cuts down iteration counts and overall computational cost considerably, while producing faithful representations of magnetization dynamics.

18. 联合李雅普诺夫证书用于K智能体生成式AI治理：随机稳定性、涌现集成风险与零知识治理证明

Joint Lyapunov Certificates for K-Agent Generative AI Governance: Stochastic Stability, Emergent Ensemble Risk, and Zero-Knowledge Governance Attestation

作者: Sriram Nagaraj

类别: math.NA, cs.CR, q-fin.RM

发布日期: 2026-08-10

arXiv ID: 2608.09087v1

链接: http://arxiv.org/abs/2608.09087v1

摘要: We develop a rigorous mathematical framework for the governance of systems of K self-adapting generative AI models under the principles of Model Risk Management (MRM). When multiple models share a meta-learning coupling through an interaction matrix, the per-agent Lyapunov analysis that underpins standard MRM is provably insufficient: individual agents can each satisfy their declared stability bounds while the joint system is in a regime of emergent ensemble-level drift. We formalize this gap through the Joint Lyapunov Proof (JLP)—a cryptographic and stochastic protocol that attests, without revealing proprietary weights, that the aggregate dynamics satisfy MRM Ongoing Monitoring standard at every validation epoch. Our main contributions are the following. We give a complete characterization of the infinitesimal generator of the joint quadratic Lyapunov function. We derive the exact critical coupling threshold above which the system loses mean-square stability. We prove a Noise-Floor Theorem and identify the correct target for zero-knowledge attestation. A per-epoch Succinct Non-Interactive Argument of Knowledge (SNARK) on the live weights is derived. All theoretical claims are validated against five numerical studies using a multi-agent softmax system.

19. 自适应性生成式人工智能中的遥测与隐蔽：日志架构、对抗模型隐藏及检测极限

Telemetry and Concealment in Self-Adapting Generative AI: Logging Architecture, Adversarial Model Hiding, and the Limits of Detection

作者: Sriram Nagaraj

类别: cs.CR, math.NA, q-fin.RM

发布日期: 2026-08-10

arXiv ID: 2608.09069v1

链接: http://arxiv.org/abs/2608.09069v1

摘要: Model risk management (MRM) guidance assumes a static model lifecycle, in which models are developed, independently validated, and implemented without further autonomous modification. Continually self-adapting generative AI systems — models that update their own weights during production deployment — fundamentally violate this assumption and render point-in-time validation inadequate. This paper addresses the resulting governance problem in two parts. Part I develops a rigorous telemetry architecture for such models, operating simultaneously in discrete and continuous time. We establish a Minimal Sufficient Statistic for audit purposes, construct a tamper-evident Merkle chain for discrete weight sequences, derive the appropriate continuous-time generalization via the Ito formula, and propose event-driven logging via KL divergence stopping times that is both computationally tractable and meaningful for validation. Part II asks what happens when the model provider is adversarial. A firm deploying such a model has strong incentives to conceal learning updates that would trigger mandatory validation review. We formalize this as the Model Hiding Problem and provide a systematic taxonomy of six distinct attack strategies against the Part I architecture, spanning discrete and continuous time, with a formal countermeasure for each. Together the two parts establish a dual-regime architecture in which continuous telemetry is necessary but not sufficient, narrowing—but never replacing—periodic invasive audit. The framework is model-architecture-agnostic and is designed to satisfy the three pillars of traditional MRM.

20. 离散随机矩阵的 LU 分解

LU Factorization of Discrete Random Matrices

作者: Samuel Orellana Mateo, John Urschel, Nicholas West

类别: math.NA, math.PR

发布日期: 2026-08-10

arXiv ID: 2608.08998v1

链接: http://arxiv.org/abs/2608.08998v1

摘要: We consider the probability that a discrete random matrix $M_n(ξ)$ is \emph{strongly non-singular}, meaning all its leading principal submatrices are non-singular. This property is equivalent to the existence of an LU factorization. We show that for any discrete random variable $ξ$ with finite support and $|ξ|_\infty < 1$, there is a constant probability that $M_n(ξ)$ is strongly non-singular with a growth factor bounded by $n^{5/2+δ}$. Furthermore, we provide a tight asymptotic lower bound for this probability as $|ξ|_\infty \to 0$. Finally, we provide exact counts for strongly non-singular binary matrices up to $n=9$ and use these to derive improved upper bounds for the Bernoulli case.

21. 发现刚性运动下等变的偏微分方程

Discovering PDEs equivariant under rigid motions

作者: Francesco Ballerin, Erlend Grong

类别: math.NA

发布日期: 2026-08-09

arXiv ID: 2608.08913v1

链接: http://arxiv.org/abs/2608.08913v1

摘要: We consider the problem of PDE discovery from possibly noisy observations under the hypothesis that the underlying dynamic is symmetric in all rigid motions. Rather than using a generic library of derivative monomials, we leverage this assumption to construct libraries whose candidate terms are themselves rigid-motion-equivariant, and combine them with sparse regression to benchmark such libraries over five different equations. The advantage is most pronounced when the noise itself breaks rigid-motion symmetry (e.g., radially or axially varying noise), and when the ambient spatial dimension increases, in which case they are also less resource intensive.

22. 单连通三维流形的自适应体参数化及其应用

Adaptive Volumetric Parameterization of Simply Connected 3-Manifolds with Applications

作者: Zhiyuan Lyu, Gary P. T. Choi

类别: cs.GR, cs.CG, math.NA

发布日期: 2026-08-09

arXiv ID: 2608.08672v1

链接: http://arxiv.org/abs/2608.08672v1

摘要: Volumetric parameterization, the process of mapping a 3-manifold onto a simplified volumetric domain, is important for many tasks in computer graphics and imaging science. However, most prior volumetric parameterization approaches have only utilized standardized domains such as a solid ball regardless of the overall shape of the given 3-manifolds, which introduces significant geometric distortion and affects the subsequent shape processing and analysis tasks. To overcome this issue, in this work we propose a novel volumetric parameterization framework for simply connected 3-manifolds. Specifically, the proposed framework jointly controls local shape and mass distortions, while adapting the target domain during the optimization process. It enables three progressively more flexible target-domain settings for the parameterization: a prescribed solid ellipsoid, a volume-normalized adaptive ellipsoid with variable radii, and a sea-embedded free-boundary domain. For each setting, the parameterization algorithm consists of a 3D quasi-conformality shape update, a diffusion-based density-equalizing update, and a geometric correction procedure for removing element foldings, thereby allowing for volumetric parameterizations with different desired effects. Experimental results are presented to demonstrate the effectiveness of our proposed framework. Moreover, our framework can be easily applied to multiresolution and localized adaptive volumetric remeshing, volumetric registration, and volumetric morphing. Altogether, our work provides a new way for the representation, processing, and analysis of 3-manifolds.

23. 用于逼近高斯白噪声驱动的分数阶椭圆型微分方程的质量集中与数值求积

Mass Lumping and Numerical Quadrature for Approximation of Fractional Elliptic Differential Equations Driven by Gaussian White Noise

作者: Kelvin J. R. Almeida-Sousa, David Bolin, Alexandre B. Simas

类别: math.NA, math.ST

发布日期: 2026-08-09

arXiv ID: 2608.08658v1

链接: http://arxiv.org/abs/2608.08658v1

摘要: Fractional elliptic stochastic partial differential equations (SPDEs) are widely used in statistics and machine learning for computationally efficient and flexible modeling of Gaussian random fields. The computational efficiency of the SPDE approach relies on finite element approximations combined with numerical quadrature and mass lumping, which enable sparse matrix methods during inference. Although many works have studied finite element approximations of fractional SPDEs, the effect of the mass lumping and quadrature approximations used in practice has not been fully analyzed. To fill this gap, we derive convergence rates for numerical approximations of fractional SPDEs based on finite element discretizations combined with numerical quadrature and mass lumping. Specifically, we obtain explicit convergence rates for the mean-squared error of the covariance function in a general framework that covers the main settings where mass lumping is used in the SPDE approach. We also analyze non-stationary variance-control factors of the form $L^β(τu)=\mathcal{W}$, where $τ$ is spatially varying, and derive covariance error estimates showing how the regularity of $τ$ affects the convergence rate. As specific examples, we provide results for random fields on bounded Euclidean domains, Riemannian manifolds, and metric graphs. Numerical experiments are presented that confirm the theoretical results.

24. 粗网格迎风有限体积预测中的精确有限时域记忆、条件性与耗散衰减

Exact Finite-Horizon Memory, Conditioning, and Dissipative Decay in Coarse Upwind Finite-Volume Prediction

作者: Antonis Polemitis, Nicholas Christakis, Dimitris Drikakis

类别: math.NA

发布日期: 2026-08-09

arXiv ID: 2608.08633v1

链接: http://arxiv.org/abs/2608.08633v1

摘要: Coarse finite-volume averages do not generally form a predictive state: realized interface fluxes close a conservative update, but distinct fine-grid states with identical parent averages can generate different future coarse histories. We analyze this failure for periodic scalar advection discretized by a first-order upwind finite-volume method with forward Euler time integration. We derive a finite-horizon observation-rank law: each additional observation exposes one new child-cell layer and contributes one fewer independent direction than the number of parent cells, until the unresolved layers are exhausted. Centralized prediction therefore requires one fewer additional coordinate than the number of parent cells per exposed layer, whereas product-local prediction requires one coordinate per layer in each parent. An anchored flux-divergence queue attains the centralized bound, identifies the periodic flux gauge, and, at saturation, forms a minimal autonomous predictive state with the parent averages. We then distinguish exact observability from stable recoverability. The collar-to-queue map becomes rapidly ill-conditioned as the Courant number decreases, so algebraically visible delayed information may fall below a prescribed numerical tolerance. For Courant numbers strictly between zero and one, we prove contraction of the nonconstant component under bounded arithmetic perturbations, with separate control of mean drift, an explicit perturbation neighborhood, and a grid-dependent decay time. Numerical experiments illustrate the rank ladder, effective-rank loss, queue conditioning, step-function flattening, and delayed coarse separation followed by dissipative decay. The results provide a solvable benchmark for assessing state sufficiency in coarse, reduced, multiscale, and learned scientific models.

25. 对求解器求导，而非方程：块隐式模拟的反向扫描伴随法

Differentiate the Solver, Not the Equation: Reverse-Sweep Adjoints for Block Implicit Simulation

作者: Lei Shu, Ying Jiang, Kui Wu, Yin Yang, Leonidas Guibas, Chenfanfu Jiang

类别: cs.GR, cs.LG, math.NA

发布日期: 2026-08-09

arXiv ID: 2608.08559v1

链接: http://arxiv.org/abs/2608.08559v1

摘要: Differentiable simulation is a key component in learning, control, and inverse problems, where gradients through nonlinear implicit solvers are required. Existing approaches either rely on unrolled automatic differentiation, whose memory grows with solver depth, or on equation-level implicit differentiation, which assembles global Jacobians and solves large sparse adjoint systems, discarding the locality of the forward solver – and differentiating the converged equation rather than the finite computation that actually ran. We propose solver-level differentiation, which differentiates the executed solver itself. When a solver is composed of block implicit updates, its discrete adjoint is obtained by applying the corresponding adjoint updates in reverse order, yielding a reverse-sweep formulation whose backward pass mirrors the forward solver. From an operator perspective, the forward pass realizes an approximate inverse through ordered local solves, and the backward applies its transpose through reverse local adjoint solves, constructing no global system. We instantiate this idea on Vertex Block Descent, yielding a differentiable solver whose reverse colored Gauss-Seidel sweeps are composed entirely of local $3\times 3$ adjoint solves. The backward matches automatic differentiation through the identical executed forward to machine precision at every solver depth, where the equation-level adjoint is off by 37% after one sweep; in a controlled same-codebase, same-GPU comparison it is 33x faster and uses 71x less memory than unrolled automatic differentiation; and the same construction is exact on projective dynamics and extended position-based dynamics. We scale differentiable elastodynamics to $10^6$ contact-coupled soft bodies (8M vertices) on one GPU. Overall, this work highlights solver structure as a practical organizing principle for efficient differentiable simulation.

26. 基于深度学习的动力系统发现中线性多步法的核局部化与全轨迹泛化

Kernel Localization and Whole-Trajectory Generalization for Linear Multistep Methods in Deep Learning-Based Discovery of Dynamical Systems

作者: Yaru Liu, Yiqi Gu

类别: math.NA

发布日期: 2026-08-09

arXiv ID: 2608.08435v1

链接: http://arxiv.org/abs/2608.08435v1

摘要: Linear multistep methods (LMMs) combined with neural-network approximation provide a high-order framework for learning governing vector fields of dynamical systems from discrete trajectory data. This paper studies two issues in LMM-based discovery that are not resolved by existing grid-level convergence theory. First, in non-auxiliary Adams–Bashforth (A-B) and Adams–Moulton (A-M) discovery systems, we observe that zero-residual grid solutions are nonunique but consistent along the trajectory, with differences limited to the boundary layer. We explain this phenomenon through a kernel analysis of the non-auxiliary discovery matrices. Under the corresponding discovery-stability conditions, the differences between zero-residual grid solutions are exponentially localized near the initial indices for A-B schemes, whereas they form two-sided boundary layers near the initial and terminal indices for A-M schemes. Second, we derive whole-trajectory generalization estimates for both auxiliary and non-auxiliary formulations. Once the learned vector field is restricted to a fixed observed trajectory, each component of the error becomes a scalar function of time. For the auxiliary formulation, the trajectory error is $O(h^p)$ under the corresponding grid accuracy, approximation, and trace regularity assumptions. For non-auxiliary formulations, the global estimates contain additional boundary-layer terms. On fixed interior subintervals, these terms are exponentially damped. Numerical experiments illustrate the convergence behavior.

27. 利用MFIE、Müller和多迹Müller积分方程抑制远场计算中的低频抵消误差

Suppressing Low-Frequency Cancellation Errors in Far-Field Computation with the MFIE, Müller, and Multitrace Müller Integral Equations

作者: Chien V. Le, Kristof Cools

类别: math.NA

发布日期: 2026-08-08

arXiv ID: 2608.08314v1

链接: http://arxiv.org/abs/2608.08314v1

摘要: This paper presents robust numerical formulations for suppressing low-frequency cancellation errors in the magnetic field integral equation (MFIE) for perfect electric conductors and the Müller integral equations for homogeneous and composite dielectric objects. The proposed approach combines quasi-Helmholtz projectors, which separate the Helmholtz components of the surface currents, with a tailored rescaling procedure that preserves the components essential for accurate far-field evaluation from finite-precision cancellation. The resulting formulations retain the conditioning properties of the mixed MFIE and Müller discretizations in the dense-mesh and low-frequency regimes. Numerical experiments involving simply and multiply connected scatterers, composite dielectric structures, multiscale and high-contrast configurations demonstrate that the proposed methods yield accurate far-field predictions across a broad frequency range, from the full-wave regime to extremely low frequencies.

28. 用于追踪非线性偏微分方程复杂奇异性的有理神经网络

Rational neural networks for tracking complex singularities of nonlinear PDEs

作者: Nadiia Derevianko, Hans-Joachim Bungartz, Felix Dietrich

类别: math.NA, math.AP

发布日期: 2026-08-08

arXiv ID: 2608.08134v1

链接: http://arxiv.org/abs/2608.08134v1

摘要: We present a neural network-based method for numerical analytic continuation of solutions of nonlinear partial differential equations (PDEs) and detection of their complex singularities. The proposed framework employs “unsafe” Padé activation units (PAUs) as activation functions, together with a novel backpropagation-free method for computing the weights and biases of the hidden layers. The training procedure is designed specifically for learning meromorphic functions with pole-type singularities. Unlike existing methods with fixed hidden-layer parameters, the proposed approach treats the weights and biases as time-dependent functions, allowing the network to adapt to the evolution of the complex singularities. Using this method, we can efficiently locate the complex singularities of the extended solution, track their trajectories over time, and, based on this dynamics, infer the formation of characteristic phenomena in the solutions of nonlinear PDEs. To demonstrate the performance of our method, we analyze three well-known cases: (1) the nonlinear heat equation, which exhibits finite-time blow up; (2) the nonlinear Burgers equation, which develops a shock; and (3) the nonlinear Schrödinger equation, in which rogue waves form.

29. 曲线网格上可压缩Navier-Stokes方程的高分辨率加权本质无振荡紧致最小二乘隐式格式

High-Resolution Weighted Essentially Non-Oscillatory Compact Least-Squares Schemes with Implicit Time Integration for Compressible Navier-Stokes Equations on Curvilinear Grids

作者: Yongzhi Luo, Huiheng Fan, Wei-Gang Zeng, Yu-Xin Ren, Jianhua Pan

类别: math.NA, physics.comp-ph

发布日期: 2026-08-08

arXiv ID: 2608.08117v1

链接: http://arxiv.org/abs/2608.08117v1

摘要: This paper presents a family of high-resolution weighted essentially non-oscillatory compact least-squares schemes with implicit time integration for the compressible Navier-Stokes equations on curvilinear grids. Compared with the original compact least-squares schemes, the proposed method introduces two main improvements. First, instead of enforcing the accuracy constraints over the entire computational domain, including discontinuous regions, it constructs the reconstruction matrix only along smooth reconstruction lines using an accuracy-preserving weighting strategy. This treatment effectively suppresses the persistent high-frequency oscillations observed in the original compact least-squares schemes and yields sharper profiles near discontinuities. Second, the method simplifies the shock-capturing procedure and improves efficiency by using a single set of polynomials, whereas the original compact least-squares schemes require both unlimited and limited polynomials. Combined with spectral optimization, the proposed method exhibits more favorable spectral properties than conventional weighted essentially non-oscillatory schemes. The smoothness indicators and penalty matrices are constructed through an efficient iterative procedure with modest additional cost. Numerical results for inviscid and viscous one-, two-, and three-dimensional flows demonstrate that the proposed method provides robust shock-capturing capability while maintaining high resolution in smooth regions and across contact discontinuities.

30. 二阶漂移模型

Second Order Drifting Models

作者: Drake Brown, Yuhao Huang, Shih-Hsin Wang, Bao Wang

类别: cs.LG, cs.AI, math.NA

发布日期: 2026-08-08

arXiv ID: 2608.07924v1

链接: http://arxiv.org/abs/2608.07924v1

摘要: Drifting models are a recent class of one-step generative models that evolve the model distribution during training using a predefined sample-based drift field. Although they avoid iterative inference, their kernel-based drift fields induce frequency-dependent training dynamics: In the linearized regime, each Fourier mode of the density residual decays at a rate determined by the kernel spectrum, leading to slow recovery of fine-scale structure. We propose Second-Order Drifting Models, which lift drifting dynamics into phase space by augmenting generated samples with artificial velocity variables. We show that the resulting density perturbations obey accelerated second-order dynamics in Fourier space, connecting drifting models to the celebrated Nesterov acceleration from optimization theory. This provides a principled mechanism for mitigating the spectral stiffness of first-order drifting while preserving one-step inference. We derive a practical semi-implicit training algorithm and evaluate it on synthetic distribution matching, sequential data generation, and robotic control. Across these settings, the second-order drifting model improves convergence behavior and achieves competitive or superior performance over first-order drifting baselines.

31. Barzilai–Borwein 方法在 $\mathbb R^d$ 和 Hilbert 空间中的精确最坏情况渐近率

The Sharp Worst-Case Asymptotic Rate of the Barzilai–Borwein Method in $\mathbb R^d$ and Hilbert Spaces

作者: Shutai Yang, Ya-Xiang Yuan

类别: math.NA, math.OC

发布日期: 2026-08-08

arXiv ID: 2608.07839v1

链接: http://arxiv.org/abs/2608.07839v1

摘要: We establish sharp asymptotic rates for the two Barzilai–Borwein (BB) rules on uniformly positive quadratics and local nonlinear problems. In finite dimensions, for either fixed rule and an arbitrary positive first step, the gradient root factor is bounded by $(b_0-a_0)/(b_0+a_0)$, where $[a_0,b_0]$ is the initially active spectral interval. Hence the worst trajectory factor is $c_H=(κ(H)-1)/(κ(H)+1)$. When $H$ has at least two distinct eigenvalues, matched initialization and a balanced endpoint trajectory attain this value. Under matched initialization, the same constant is the optimal uniform-envelope threshold. For bounded, self-adjoint, uniformly positive operators on Hilbert space, scalar spectral measures yield the corresponding active-support bound and optimal matched uniform-envelope threshold, including continuous endpoint spectrum. Finally, if the gradient is strictly Fréchet differentiable at a stationary point and its derivative is self-adjoint and uniformly positive, every $γ\in(c_*,1)$, where $c_*=(κ(A_*)-1)/(κ(A_*)+1)$, is a uniform local envelope rate for either pure BB rule. Every well-defined trajectory converging to the stationary point has error and gradient root factors at most $c_*$ and objective-gap root factor at most $c_*^2$. Over the class of objectives with prescribed distinct derivative endpoints $m_*

32. 计算椭圆限制性三体问题极值曲线的通用Birkhoff方法

Universal Birkhoff Method for Computing Extremals in the Elliptic Restricted Three-Body Problem

作者: Michael J. Dixon, Isaac M. Ross

类别: math.OC, eess.SY, math.NA

发布日期: 2026-08-07

arXiv ID: 2608.07815v1

链接: http://arxiv.org/abs/2608.07815v1

摘要: The computation of finite-thrust extremal arcs in the elliptic restricted three-body trajectory optimization problem is considered. Libration-point orbits are approximated to near-machine precision using a fast Fourier transform of the sampled values of the state vector at Chebyshev-Gauss-Lobatto points. Checkable optimality conditions are derived by applying Pontryagin’s principle to minimum-time and time-constrained minimum-propellant problems. These necessary conditions include criteria for optimal departure and arrival points. For propellant consumption, a recently developed computational model is employed. This model is agnostic to the specific impulse of the propellant and varies as the inverse quadratic of a cosine term. Candidate optimal solutions are generated by combining the universal Birkhoff theory for trajectory optimization with the fast, guess-free spectral algorithm. The extremality of the Birkhoff-computed solution is validated against the Hamiltonian minimization condition and the transversality conditions. It is shown that the Birkhoff-theoretic spectral algorithm can generate verifiable extremals without any assistance or initialization from dynamical systems theory.

33. 基于能量原理的非协调有限差分离散裂缝模型

A non-conforming finite difference discrete fracture model based on an energy principle

作者: Ziyao Xu

类别: math.NA

发布日期: 2026-08-07

arXiv ID: 2608.07807v1

链接: http://arxiv.org/abs/2608.07807v1

摘要: We propose a finite difference discrete fracture model for single-phase flow in fractured porous media. The method is derived from a unified energy principle and is implemented on Cartesian grids that need not conform to the fractures. It introduces no additional degrees of freedom, modifies the underlying finite difference scheme only locally, handles both highly conductive fractures and low-permeability barriers, and naturally preserves symmetry and positive definiteness of the discrete system. Barrier-induced pressure jumps and fracture-induced fluxes can be recovered through inexpensive local post-processing, yielding a sharper representation of the interface effects. Numerical experiments based on manufactured solutions and published $2$D and $3$D benchmark problems demonstrate the accuracy, effectiveness, and flexibility of the method for isolated fractures, complex networks, and anisotropic porous media.

34. 带乘性噪声的随机Keller-Segel系统的分裂混合有限元方法

A splitting mixed finite element method for a stochastic Keller-Segel system with multiplicative noise

作者: Thoa Thieu, Liet Vo

类别: math.NA, math.PR

发布日期: 2026-08-07

arXiv ID: 2608.07798v1

链接: http://arxiv.org/abs/2608.07798v1

摘要: In this paper, we propose and analyze a splitting mixed finite element method for a stochastic Keller–Segel system with logistic growth driven by multiplicative noise. By introducing an auxiliary variable representing the chemical gradient together with a time-lagged splitting strategy, the proposed method decouples the original coupled system into a sequence of simpler subproblems. Consequently, it eliminates the Ladyzhenskaya–Babuška–Brezzi stability constraint, permits the use of continuous piecewise linear finite element spaces for all unknowns, and avoids solving a fully coupled nonlinear system at each time step, thereby significantly reducing the computational cost. Combined with an implicit Euler time discretization, the proposed approach yields a fully discrete numerical scheme for the stochastic Keller–Segel system. Using a localization technique together with suitable stochastic stability arguments, we establish optimal strong error estimates for the fully discrete approximations and prove convergence in probability with explicit convergence rates. Numerical experiments verify the theoretical convergence rates and demonstrate that the proposed method successfully captures the global boundedness induced by the logistic growth term as well as the influence of multiplicative noise on chemotactic aggregation.

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20. 离散随机矩阵的 LU 分解#

21. 发现刚性运动下等变的偏微分方程#

22. 单连通三维流形的自适应体参数化及其应用#

23. 用于逼近高斯白噪声驱动的分数阶椭圆型微分方程的质量集中与数值求积#

24. 粗网格迎风有限体积预测中的精确有限时域记忆、条件性与耗散衰减#

25. 对求解器求导，而非方程：块隐式模拟的反向扫描伴随法#

26. 基于深度学习的动力系统发现中线性多步法的核局部化与全轨迹泛化#

27. 利用MFIE、Müller和多迹Müller积分方程抑制远场计算中的低频抵消误差#

28. 用于追踪非线性偏微分方程复杂奇异性的有理神经网络#

29. 曲线网格上可压缩Navier-Stokes方程的高分辨率加权本质无振荡紧致最小二乘隐式格式#

30. 二阶漂移模型#

31. Barzilai–Borwein 方法在 $\mathbb R^d$ 和 Hilbert 空间中的精确最坏情况渐近率#

32. 计算椭圆限制性三体问题极值曲线的通用Birkhoff方法#

33. 基于能量原理的非协调有限差分离散裂缝模型#

34. 带乘性噪声的随机Keller-Segel系统的分裂混合有限元方法#

重点关注论文（完整分析）

1. 广义优化Schwarz方法的粗空间预处理：第一部分：连续情形

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. 用于伍德异常处及其附近频率的二维周期光栅衍射的频率稳定Nyström方法

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. 一种应用于谱差分方法的二阶通量新型紧致格式

摘要翻译

2. 用于物理信息神经网络的分层秩演化表示

摘要翻译

3. 一个非局部非线性薛定谔模型：适定性、局部极限和结构保持的渐近相容傅里叶逼近

摘要翻译

4. 相对论欧拉方程的高阶全离散多熵稳定与保界格式

摘要翻译

5. 关于线性半负问题的一阶次对角Padé逼近子的能量律与稳定性

摘要翻译

6. 两阶段MV-DeepONet：面向随机输入场驱动的不确定性传播的概率算子学习

摘要翻译

7. 对平面 Berenstein 猜想的一个计算机辅助反例

摘要翻译

8. 位似双曲坐标中的半线性波动方程与尾部衰减

摘要翻译

9. ADEx-FNO: 适用于变化几何的统一环境域傅里叶神经算子框架

摘要翻译

10. 具有变形界面的流固耦合问题CutFEM的收敛性

摘要翻译

不相关论文（基本信息）

1. 基于Bernstein表示的有理T样条曲面的双射性分析

2. 基于非对称加权 Hermite 函数的 Vlasov-Poisson 系统 Galerkin 逼近的收敛性

3. 信息几何正则化计算含激波流动的灵敏度

4. 非自治随机微分方程Milstein格式的李雅普诺夫指数双边估计

5. 热力学矩阵求逆的理论分析：与预条件梯度下降的一阶等价性及其对模拟计算的启示

6. 带记忆的非线性输运中逆初值问题的 Carleman–Picard 与时间降维

7. 扩散映射核岭回归

8. 用于液晶指向矢建模中双重鞍点系统的双参数块预条件子

9. 无严格互补条件下谱多面体上一种Frank-Wolfe型方法的线性收敛

10. 通过提示随机Transformer实现无需训练的通用逼近

11. 动态扩散有限元方法的条件唯一性与最优能量范数收敛

12. Walk-on-Spheres 蒙特卡罗与深度神经网络对带漂移和消亡项的椭圆型偏微分方程的逼近

13. 二阶非拟合幽灵有限元法求解椭圆界面问题及其在低维半导体器件中的应用

14. 有限元应力更新中的自动微分

15. poly_rust: 一种面向多面体方法的领域特定语言及其Rust解释器

16. 一种用于一般类动力系统的简单二阶非标准数值方法及其应用

17. 用于微磁模拟中半隐式Landau-Lifshitz格式的高效多重网格方法

18. 联合李雅普诺夫证书用于K智能体生成式AI治理：随机稳定性、涌现集成风险与零知识治理证明

19. 自适应性生成式人工智能中的遥测与隐蔽：日志架构、对抗模型隐藏及检测极限

20. 离散随机矩阵的 LU 分解

21. 发现刚性运动下等变的偏微分方程

22. 单连通三维流形的自适应体参数化及其应用

23. 用于逼近高斯白噪声驱动的分数阶椭圆型微分方程的质量集中与数值求积

24. 粗网格迎风有限体积预测中的精确有限时域记忆、条件性与耗散衰减

25. 对求解器求导，而非方程：块隐式模拟的反向扫描伴随法

26. 基于深度学习的动力系统发现中线性多步法的核局部化与全轨迹泛化

27. 利用MFIE、Müller和多迹Müller积分方程抑制远场计算中的低频抵消误差

28. 用于追踪非线性偏微分方程复杂奇异性的有理神经网络

29. 曲线网格上可压缩Navier-Stokes方程的高分辨率加权本质无振荡紧致最小二乘隐式格式

30. 二阶漂移模型

31. Barzilai–Borwein 方法在 $\mathbb R^d$ 和 Hilbert 空间中的精确最坏情况渐近率

32. 计算椭圆限制性三体问题极值曲线的通用Birkhoff方法

33. 基于能量原理的非协调有限差分离散裂缝模型

34. 带乘性噪声的随机Keller-Segel系统的分裂混合有限元方法