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1. 混合边界交界面处梯度依赖半线性方程中的对数共振

Logarithmic Resonance at Mixed Boundary Junctions in Gradient-Dependent Semilinear Equationsv

作者: Marin Mišur

类别: math.NA

发布日期: 2026-08-03

arXiv ID: 2608.01790v1

摘要翻译: 椭圆型偏微分方程的解在混合 Dirichlet-Neumann 边界交界处的正则性严重退化, 经典上表现为一个 $\mathcal{O}(r^{1/2})$ 的首项奇异函数。虽然这种线性行为已有充分记录, 但引入梯度依赖的半线性扰动会改变局部渐近剖面。本文证明, 与 $|\nabla u|$ 成线性关系的梯度惩罚项会诱导出一个高度局域化的 $\mathcal{O}(r^{-1/2})$ 源项, 该源项与主齐次微分算子的半整数谱发生共振。这种非正交共振导致标准的可分离多项式假设在 $\mathcal{O}(r^{3/2})$ 阶失效。我们建立了一个广义共振定理, 它强制出现对数异常, 并给出了所产生的 $r^{3/2} \ln(r)$ 剖面的精确解析形式, 以及对余项的严格局部 Sobolev 正则性界。通过计算 $\ell_1$, $\ell_2$, $\ell_\infty$ 以及任意 $\ell_q$-范数惩罚的对数系数和角偏移, 我们证明了这种障碍的普适性。最后, 我们形式化了相应的丰富连续 Galerkin 空间 (XFEM), 建立了其拟最优性, 并给出了有限元实验, 这些实验确认了所预测的局部污染, 在无曲率的平坦交界上针对 $\ell_1$, $\ell_2$ 和 $\ell_\infty$ 惩罚恢复出了所预测的对数系数, 并表明解析交界可恢复标准有限元求解器的最优自由度效率; 最后我们简要指出了完全丰富实现所需的目标软件架构。

Comment: Work in progress

链接: http://arxiv.org/abs/2608.01790v1

1. 研究对象和背景

本文研究椭圆偏微分方程在混合 Dirichlet-Neumann 边界结点处的解的正则性。考虑有界区域 $\Omega \subset \mathbb{R}^2$，其边界 $\partial\Omega$ 分为 Dirichlet 部分 $\Gamma_D$ 和 Neumann 部分 $\Gamma_N$，在结点 $P \in \overline{\Gamma_D \cap \Gamma_N}$ 处边界条件发生跃变。假设在 $P$ 附近边界局部平坦，即内角为 $\pi$，并建立局部极坐标 $(r,\theta)$，使得 $\Gamma_D$ 对应于 $\theta=0$，$\Gamma_N$ 对应于 $\theta=\pi$。方程形式为

$$ -\Delta u + \mu u = f(x, u) + g(\nabla u) \quad \text{在 } \Omega \text{ 中}, $$

带有混合边界条件 $u=0$ 在 $\Gamma_D$，$\nabla u\cdot \boldsymbol{\nu}=0$ 在 $\Gamma_N$。这里 $\mu > 0$ 保证强单调性，$g:\mathbb{R}^2 \to \mathbb{R}$ 是梯度依赖的惩罚项，满足全局 Lipschitz 连续且正齐次度为一 (即 $g(\alpha \mathbf{p}) = \alpha g(\mathbf{p})$，$\forall \alpha>0$)，例如 $\ell_1$、$\ell_2$、$\ell_\infty$ 范数以及各向异性罚项等。$f(x,u)$ 为较温和的源项，具有足够的可积性。

经典的线性 Poisson 方程在混合结点附近的奇异性理论已确立：解的主导奇性函数为 $c_0 r^{1/2}\sin(\theta/2)$，这导致标准有限元方法的收敛率退化。为恢复最优收敛，需要在离散空间中显式引入此奇性函数（如 XFEM）。然而，引入梯度依赖的半线性项后，由于 $\nabla u_0 \sim r^{-1/2}$，$g(\nabla u)$ 会产生一个强奇异源项 $O(r^{-1/2})$，其径向指数 $-1/2+2 = 3/2$ 恰好与齐次 Sturm-Liouville 算子 $-\partial_\theta^2$ 在混合边界条件下的特征值 $\lambda=3/2$（对应的特征函数为 $\sin(3\theta/2)$）重合。根据 Kondrat’ev 的角点渐近理论，当强迫项的径向指数与算子铅笔的特征值匹配时，标准的分离变量展开会失效，并且必须引入对数项来修正。然而，以往的工作主要关注线性问题或右端强迫直接由几何或显式非齐次项引起的共振，本文首次证明，在梯度依赖半线性方程中，共振可以由非线性自身内在产生。这一机制揭示了此类方程在混合结点附近的一个全新的对数反常现象，并具体确定了其函数形式，填补了半线性椭圆方程角点渐近分析的空白。

2. 主要定理或主要结果

本文的主要结果是对数共振定理（定理1）及其对不同梯度罚范数的应用。定理1断言：在上述几何与假设下，若 $c_0 \neq 0$ 且非正交条件 $\kappa = \int_0^\pi G(\theta) \sin(3\theta/2) \mathrm{d}\theta \neq 0$ 成立，则解在结点 $P$ 附近不能展开为纯幂次函数 $\{r^{k+1/2}\Theta_k\}$，而必须包含对数项，其局部渐近展开为

$$ u(r,\theta) = c_0 r^{1/2} \sin\frac{\theta}{2} - \frac{2\kappa}{3\pi} r^{3/2} \ln r \, \sin\frac{3\theta}{2} + r^{3/2} \Bigl( \Psi(\theta) + c_1 \sin\frac{3\theta}{2} \Bigr) + R(r,\theta). $$

其中 $G(\theta) = g(\boldsymbol{v}(\theta))$，$\boldsymbol{v}(\theta)=\frac{c_0}{2}(-\sin(\theta/2), \cos(\theta/2))$ 是 $\nabla u_0$ 的角向量场，$\kappa$ 为上述非正交积分。$\Psi(\theta)$ 是边值问题 $\Psi'' + \frac{9}{4}\Psi = -G(\theta) + \frac{2\kappa}{\pi} \sin(3\theta/2)$ 在 $\Psi(0)=\Psi'(\pi)=0$ 且与 $\sin(3\theta/2)$ $L^2$-正交条件下的唯一解。$c_1$ 是第二个应力强度因子，对应于齐次模 $r^{3/2}\sin(3\theta/2)$ 的振幅，由全局解确定。剩余项 $R$ 满足正则性引理1：若 $f(x,u_0) \in L^p_{\mathrm{loc}}(\Omega)$，$p>2, p\neq 4$，则 $R \in W^{2,p}_{\mathrm{loc}}(\Omega)$，特别地 $R \in C^{1,\gamma}$（$\gamma=1-2/p$）；当 $p=4$ 时，$R \in W^{2,p'}_{\mathrm{loc}}$ 对一切 $p'<4$ 成立，从而 $R \in C^{1,\gamma}$ 对所有 $\gamma<1/2$ 成立；若 $f(x,u_0) \in L^\infty_{\mathrm{loc}}$，则逐点有 $R=O(r^2)$。此外，文中具体计算了 $\ell_1$ 范数（Mišur 格式）的对数系数：$G(\theta)=\frac{c_0}{2}(\sin\frac{\theta}{2}+\cos\frac{\theta}{2})$，得到 $\kappa = \frac{c_0}{3}$，$A=-\frac{2c_0}{9\pi}$，并给出 $\Psi$ 的解析表达式。对于 $\ell_2$ 范数，$\kappa = \frac{c_0}{2}$，$A=-\frac{c_0}{3\pi}$；对于 $\ell_\infty$ 范数，$\kappa = \frac{2c_0}{3}$，$A=-\frac{4c_0}{9\pi}$。作者还给出了一般 $\ell_q$ 范数的结果，以及包含对流项的罚函数的例子，证明了共振的普遍性，但同时也指出通过调节横向漂移参数可使共振抵消（$\kappa=0$），此时对数项消失。在数值方面，文章构造了包含 $\{r^{1/2}\sin(\theta/2), r^{3/2}\ln r \sin(3\theta/2), r^{3/2}\Psi(\theta)\}$ 等的增强连续 Galerkin 空间（XFEM），证明了逼近的拟最优误差估计。有限元实验在单位圆盘上进行，数值提取的对数系数与理论预测一致，并展示了仅增强 $r^{1/2}$ 项无法恢复最优收敛率，而包含对数项后，结点附近的污染被消除，自由度效率恢复最优。

3. 研究方法、关键技术和核心工具

本文的证明核心是处理梯度惩罚项带来的奇异源项与齐次算子谱的共振。整体分析框架基于 Kondrat’ev 加权 Sobolev 空间理论，并结合了算子铅笔、Fredholm 二择一以及角点渐近展开的标准技术。具体技术路线和创新点如下：

局部展开与方程分解：首先利用先验正则性（命题1）将解写为 $u = u_0 + u_{\mathrm{reg}}$，其中 $u_0 = c_0 r^{1/2}\sin(\theta/2)$，$u_{\mathrm{reg}} \in H^2_{\mathrm{loc}}$。然后将方程右端分离为 $r^{-1/2}G(\theta)$ 与一个更正则的剩余项 $\tilde{F}_{\mathrm{reg}}$。这里的关键是利用 $g$ 的正齐次性得到 $g(\nabla u_0)=r^{-1/2}G(\theta)$。
共振的识别与对数 ansatz：对奇异项应用分离变量，设 $u_1 \propto r^{3/2}\Theta_1(\theta)$，代入极坐标拉普拉斯算子导致常微分方程 $\Theta_1'' + \frac{9}{4}\Theta_1 = -G$。由于齐次问题的解 $\sin(3\theta/2)$ 是特征函数，Fredholm 二择一要求右端与特征函数正交。由非正交假设 $\kappa \neq 0$，此方程无解，标志着纯幂次展开的失效。为突破这一障碍，作者采用标准的对数-幂次合形变：$u_1 = r^{3/2}\ln r\,\Phi(\theta) + r^{3/2}\Psi(\theta)$。代入后，令 $\ln r$ 项的主部为零导出 $\Phi''+\frac{9}{4}\Phi=0$，结合边界条件得 $\Phi = A\sin(3\theta/2)$。再处理 $O(r^{3/2})$ 项得到 $\Psi$ 的非齐次方程，其可解性条件 $-\kappa - 3A(\pi/2)=0$ 唯一确定了 $A$。这一步骤巧妙地运用了 Fredholm 二择一两次，第一次说明无对数项则不可解，第二次确定对数系数。$\Psi$ 的确定需再次使用齐次方程的解作为自由度，并通过正交化条件固定特解，余下的齐次解振幅 $c_1$ 由全局匹配决定。
剩余项的正则性分析：扣除 $u_0$ 和 $u_1$ 后，剩余项 $R$ 满足 $-\Delta R + \mu R = F_R$，其中右端已不含 $r^{-1/2}$ 奇性，且 $F_R$ 的可积性取决于 $f$ 和 Lipschitz 估计。通过自举（bootstrap）论证和加权椭圆估计，将 $R$ 提升到 $W^{2,p}_{\mathrm{loc}}$。具体地，先由 $H^1$ 到 $H^2$ 利用 $L^2$ 数据，然后因 $R$ 不含低于 $5/2$ 的奇异模，且 $2-2/p$ 不在特征值集上（避开 $p=4$），加权估计保证 $R$ 的 $W^{2,p}$ 正则性。当 $p=4$ 时，由于 $2-2/p=3/2$ 恰为特征值，不能直接得到 $W^{2,4}$，但通过略小的指数获得 $W^{2,p'}$。若数据有界，则进一步得到 $O(r^2)$ 点估计。
数值方法设计：基于渐近展开，作者构造了增广有限元空间，将奇异函数和对数项作为节点富集函数纳入，并证明了 Galerkin 逼近的拟最优性（在加权范数下）。有限元实验验证了共振引起的局部污染、对数系数的提取以及最优收敛率的恢复。

整个证明的核心创新在于：将非线性产生的自适应源项与线性算子的谱特征联系起来，揭示了非线性效应可以直接激发对数共振，而不需要外加的边界奇性或几何参数聚合。这扩展了 Kondrat’ev 理论的适用范围，并为梯度依赖半线性方程的数值解提供了必要的先验信息。

4. 与之前工作的比较

本文的工作与已有研究在以下几个方面有明显的推进和区别：

与线性混合边值问题角点渐近理论的比较：Kondrat’ev 的经典理论（见文献 [8,9,13]）指出，当右端项的径向指数与算子铅笔的特征值重合时，解的展开中会出现对数项。以往这种共振通常出现在右端项本身具有匹配的径向奇性，或由于角点角度改变导致特征值重合（如重新进入角 $\alpha > \pi$ 时的双重强奇性）。本文的新颖之处在于，共振完全由非线性项 $g(\nabla u)$ 内在产生：齐次解 $u_0$ 的梯度提供了 $O(r^{-1/2})$ 因子，而 $g$ 的正齐次性将其转化为与特征值 $3/2$ 匹配的源项。这与预先指定的右端项无关，是方程自洽性的结果。因此，即使几何上是最简单的平坦结点 ($\alpha=\pi$)，共振依然发生，这在纯线性理论中并非典型现象。
梯度依赖半线性方程相关工作的比较：对带有 $g(\nabla u)$ 的方程的研究常聚焦于解的存在性、正则性和泛函极小化（例如文献 [2,11]）。Mišur (2021) 通过变分方法证明了解的存在唯一性，并提出了 XFEM 格式进行数值模拟，但其关注点在于离散收敛性，并未分析局部渐近行为的奇异性转变，也没有意识到 $\ell_1$ 罚项会导致对数污染。本文明确揭示了这一机制，并针对 Mišur 的模型给出了精确的对数系数和角依赖函数，从而为其数值方案中观察到的某些次优收敛行为提供了理论解释，并指明了必须增强哪些局部函数才能恢复最优阶。
关于角点角度的一般性讨论：本文明确指出，对数共振在角度为 $\pi$ 的平坦结点上并非特例，而在更一般的角度 $\alpha \neq k\pi$ 时不发生，因为匹配条件 $\pi/(2\alpha)+1 = (k+1/2)\pi/\alpha$ 要求 $\alpha = k\pi$。因此，这种共振是“在梯度罚泛函中普遍的，但在几何上是例外的”。这一定性结论与其他角点奇异性研究不同：以往对数项的出现往往与角点角度密切相关（如 $2\pi$ 角的裂缝），而此处平坦结点竟然出现对数项，更出人意料。
数值处理策略的拓展：经典的 XFEM 通常只增强已知的局部奇性函数（如 $r^{1/2}\sin(\theta/2)$）。本文指出对于这类方程，仅增强 $O(r^{1/2})$ 和 $O(r^{3/2})$ 的幂函数是不充分的，必须显式包含 $r^{3/2}\ln r$ 项才能在图集上消除污染。这与对具有对数奇性的界面问题或角点问题的增强方法精神一致，但针对的是由非线性产生的特殊对数项。文中实验证实了这点，并给出了包含对数项的增强空间设计及拟最优误差分析。
假设与代价：本文的共振结论建立在几个关键假设之上：(a) $g$ 严格正齐次度为 1（线性增长）；若 $g(\nabla u) \sim |\nabla u|^p$ 且 $p \neq 1$，则无共振，除非存在空间权函数调节尺度（Remark 3）；(b) 应力强度因子 $c_0 \neq 0$ 且非正交条件成立，否则展开可回归纯幂次形式。此外，分析局限于局部平坦的结点，且尚未考虑弯曲边界情形（弯曲边界会导致有限半径修正，使测量的对数系数偏离理论值，如数值实验所示）。文章结尾也提到了向三维碰撞边的推广只是概述，尚未完成。

总之，本文在理论上揭示了半线性方程中一种由非线性与边界条件相互作用产生的新型对数奇性，并为其数值处理提供了系统框架，填补了该领域的空白。

2. 面向温度相关非线性热辐射问题的复合材料结构高阶多尺度方法及其收敛性分析

A high-order multi-scale method and its convergence analysis for temperature-dependent nonlinear thermal radiation problems of composite structures

作者: Hao Dong, Yongfei Hu, Jiale Linghu, Yaochuang Han

类别: math.NA, math.AP

发布日期: 2026-08-02

arXiv ID: 2608.01214v1

摘要翻译: 精确预测具有温度相关特性的复合材料结构中的非线性辐射热传递在材料的高温应用中具有重要意义。本研究建立了一种高精度多尺度计算模型，该模型引入了新颖的高阶修正项，用于复合材料结构中非线性热辐射的高保真模拟，并能保持热量的局部平衡。此外，针对所得到的高阶多尺度解，还推导了显式的收敛率。进一步，发展了一种包含离线和在线计算阶段的高效多尺度算法，用于复合材料结构中非线性热辐射行为的高精度模拟，并获得了相应的收敛性分析。通过二维和三维数值算例验证了所提出多尺度方法的竞争性优势，不仅数值精度出色，而且在存储需求和计算时间方面均降低了计算成本。

Comment: 无

链接: http://arxiv.org/abs/2608.01214v1

1. 研究对象和背景

本文研究具有周期性微观结构的复合材料在高温环境下的非线性辐射热传导问题。这类问题在航空航天工程中具有重要应用，因为复合材料在高温服役时，其热物性参数（密度 $\rho$，比热容 $c$，导热系数 $k_{ij}$，吸收系数 $\beta$）强烈依赖于温度，并且辐射换热成为主导传热机制，从而对流场的精确预测提出了挑战。所考察的数学模型为如下含快速振荡系数的非线性热辐射方程：

$$ \begin{cases} \rho^\varepsilon(x, T^\varepsilon) c^\varepsilon(x, T^\varepsilon) \frac{\partial T^\varepsilon}{\partial t} - \frac{\partial}{\partial x_i} \left( k_{ij}^\varepsilon(x, T^\varepsilon) \frac{\partial T^\varepsilon}{\partial x_j} \right) + \beta^\varepsilon(x, T^\varepsilon) \sigma_B (T^\varepsilon)^4 = h(x,t) , & \text{在} \ \Omega \times (0,t^*) \text{ 中},\\[4pt] T^\varepsilon(x,t) = \hat{T}(x,t), & \text{在} \ \partial\Omega_T \times (0,t^*) \text{ 上},\\[4pt] k_{ij}^\varepsilon(x, T^\varepsilon) \frac{\partial T^\varepsilon}{\partial x_j} n_i = \hat{q}(x,t), & \text{在} \ \partial\Omega_q \times (0,t^*) \text{ 上},\\[4pt] T^\varepsilon(x,0) = \tilde{T}(x), & \text{在} \ \Omega \text{ 上}. \end{cases} $$

这里 $\Omega \subset \mathbb{R}^d \ (d=2,3)$ 是有界凸区域, 边界 $\partial\Omega = \partial\Omega_T \cup \partial\Omega_q$ Lipschitz 连续且 ${{\partial\Omega_T}}$ 正测度; $\varepsilon$ 为微观结构周期, 表征非均匀性尺度; $T^\varepsilon$ 为待求温度场; $\sigma_B$ 为 Stefan-Boltzmann 常数; $h$, $\hat{T}$, $\hat{q}$, $\tilde{T}$ 分别为内热源、边界给定温度、边界热流和初始温度. 所有材料系数均依赖于温度且具有周期为1的微观振荡:

$$\rho^\varepsilon(x,T) = \rho(x/\varepsilon, T),\quad c^\varepsilon(x,T) = c(x/\varepsilon, T),\quad k_{ij}^\varepsilon(x,T) = k_{ij}(x/\varepsilon, T),\quad \beta^\varepsilon(x,T) = \beta(x/\varepsilon, T),$$

其中 $y=x/\varepsilon\in Q:=(0,1)^d$ 为微观坐标. 基本假设包括: (A1) $k_{ij}$ 对称且一致椭圆: $k|\xi|^2 \le k_{ij}(y,T)\xi_i\xi_j \le \bar{k}|\xi|^2$, 常数 $k,\bar{k}>0$ 不依赖 $\varepsilon$ 和 $T$; (A2) 所有物性系数一致有界远离零; (A3) 物性系数关于温度 Lipschitz 连续; (A4) 数据函数具有足够的光滑性以保证后续分析.

直接的精细数值模拟因需要网格尺度远小于 $\varepsilon$ 而计算开销过大. 经典的均化方法 (Cioranescu, Babuška, Oleinik 等) 通过平均化处理得到宏观有效方程, 能够预测整体行为, 但无法恢复微观尺度的振荡细节. 为克服这一缺陷, 多种多尺度数值方法被提出, 如多尺度有限元法 (MsFEM), 异质多尺度方法 (HMM), 变分多尺度方法 (VMS), 局部正交分解方法 (LOD) 等. 然而, 这些方法对于高反差材料复合结构的模拟往往精度不足. 近年来, 崔俊芝团队系统性地建立了面向复合材料热、力、多物理场问题的高阶多尺度渐近方法, 通过引入高阶校正项显著提高了多尺度解的精度, 同时保持计算效率.

对于考虑热辐射的传热问题, Amosov 研究了非稳态辐射-导热耦合系统的半离散格式与渐近近似, Bakhvalov 给出了周期性介质中辐射传热的平均化方法及温度场的渐近展开, Allaire 等对非线性辐射热传导问题进行了均化分析并指出二阶校正项对于重构多尺度解必不可少. Huang 等利用 Rothe 方法证明了辐射传热模型非负弱解的整体存在唯一性, 并导出了含 $\varepsilon^{1/2}$ 阶显式收敛率的多尺度渐近展开; 进一步针对非齐次初边值条件获得了 $\varepsilon$ 阶收敛率, 并发展了相应的数值算法. Han 等提出了三维半透明介质辐射传热的快速多极边界元法. 上述工作均未考虑材料物性参数对温度的依赖性, 而高温环境中物性变化剧烈, 温度依赖性不可忽略.

本文的核心创新在于: 首次将温度依赖性引入非线性热辐射方程的多尺度渐近分析框架, 建立了包含新颖高阶校正项的高精度多尺度计算模型, 能够保持局部热量的精确平衡; 推导了高阶多尺度解的显式收敛率; 发展了离线微观计算与在线宏观及多尺度计算相结合的两阶段高效算法. 本文的工作填补了温度依赖复合材料非线性辐射传热高阶多尺度模拟的理论空白, 为高温结构设计提供了可靠且高效的计算手段.

2. 主要定理或主要结果

本文的主要结果包括三个方面: 高阶多尺度渐近展开模型的构造, 多尺度解的误差分析以及高效多尺度算法设计. 具体地:

基于经典多尺度渐近展开框架, 假设真解 $T^\varepsilon$ 具有形式展开 $T^\varepsilon = T_0 + \varepsilon T_1 + \varepsilon^2 T_2 + O(\varepsilon^3)$, 其中 $T_0$ 为宏观温度, $T_1$, $T_2$ 分别为一阶和二阶微观校正项. 利用 Taylor 展开处理系数与温度的非线性依赖, 代入原始方程并收集 $\varepsilon$ 的同次幂, 得到一系列递推方程. 从 $O(\varepsilon^{-2})$ 方程得出 $T_0$ 与微观变量 $y$ 无关; 从 $O(\varepsilon^{-1})$ 方程定义了
$$ T_1(x,y,t) = M_{\alpha_1}(y,T_0) \frac{\partial T_0}{\partial x_{\alpha_1}}, $$
其中一阶辅助单胞函数 $M_{\alpha_1}$ 满足单元胞上的周期边值问题 (13). 通过体积平均 $O(\varepsilon^0)$ 方程, 导出宏观均化方程 (14), 均化系数 (15) 由一阶单胞函数和原始系数决定, 且均显式依赖于 $T_0$. 对 $O(\varepsilon^0)$ 方程进一步分解, 得到二阶校正项的具体表达式
$$ \begin{aligned} T_2(x,y,t) &= M_{\alpha_1\alpha_2}(y,T_0) \frac{\partial^2 T_0}{\partial x_{\alpha_1}\partial x_{\alpha_2}} + G(y,T_0) \frac{\partial T_0}{\partial t} + N(y,T_0) (T_0)^4 \\ &\quad + N_{\alpha_1}(y,T_0) \frac{\partial T_0}{\partial x_{\alpha_1}} + N_{\alpha_1\alpha_2}(y,T_0) \frac{\partial T_0}{\partial x_{\alpha_1}}\frac{\partial T_0}{\partial x_{\alpha_2}}, \end{aligned} $$
其中 $M_{\alpha_1\alpha_2}$, $G$, $N$, $N_{\alpha_1}$, $N_{\alpha_1\alpha_2}$ 为五类二阶辅助单胞函数, 分别满足与均化系数、热容、辐射源、非齐次项以及非线性耦合项对应的单胞问题 (18)–(22). Lax-Milgram 定理结合假设 (A1) 保证了所有单胞问题对每个固定 $T_0$ 存在唯一解. 最终得到了显式的高阶多尺度渐近解 (23).
对多尺度解进行了点态局部误差分析和全局积分意义下的误差估计. 引入一阶逼近 $T^{(1\varepsilon)} = T_0 + \varepsilon T_1$ 和二阶逼近 $T^{(2\varepsilon)} = T^{(1\varepsilon)} + \varepsilon^2 T_2$, 并定义相应残差. 点态分析表明, 所构造的高阶多尺度解能够在局部保持热流平衡, 这是对经典均化方法的重要改进. 全局误差分析给出了显式收敛率: (从摘要中可知) 推导了高阶多尺度解的显式收敛阶, 但提供的内容第3.1节仅开始定义残差, 后续第3.2节以及具体定理未在摘录中显示, 因此具体收敛阶 (如 $\varepsilon^2$ 或 $\varepsilon$) 无法从提供内容中确认.
设计了一种高效的两阶段数值算法: 离线阶段针对离散温度采样点预求解所有单胞问题, 计算并存储均化系数; 在线阶段求解宏观均化方程, 并利用预存单胞函数重构多尺度解. 该算法将微观计算与宏观计算解耦, 显著降低了计算存储量和时间成本. 数值算例验证了该方法在二维和三维问题中的高精度与高效率.

3. 研究方法、关键技术和核心工具

本文的核心技术路径是: 通过多尺度渐近展开结合系数在宏观温度附近的 Taylor 展开, 将温度依赖性非线性转化为一系列在周期胞元上的拟线性单胞问题. 主要技术障碍及创新点如下:

非线性系数温度依赖的处理. 由于物性系数 $\Psi^\varepsilon(x,T^\varepsilon)$ 直接依赖于未知温度 $T^\varepsilon$, 经典的多尺度展开 (系数视为仅依赖于 $y$) 不再适用. 作者巧妙地将系数的微观表达 $\Psi(y,T^\varepsilon)$ 在宏观温度 $T_0$ 处进行 Taylor 展开:
$$ \Psi(y,T^\varepsilon) = \Psi(y,T_0) + \varepsilon T_1 D^{(0,1)}\Psi(y,T_0) + \varepsilon^2 \bigl[ T_2 D^{(0,1)}\Psi(y,T_0) + \tfrac12 T_1^2 D^{(0,2)}\Psi(y,T_0) \bigr] + O(\varepsilon^3), $$
这样就将非线性依赖转化为关于 $T_0$ 幂级数的形式, 每一阶的系数仅依赖于 $y$ 和已知的 $T_0$, 从而使得整个多尺度展开可以逐阶进行. 这是本文处理温度依赖性的关键创新.
高阶校正项的系统构造. 与传统二阶校正仅引入二阶空间导数项不同, 本文的 $T_2$ 表达式中出现了时间导数项 $G\, \partial_t T_0$ 和辐射源项 $N\, T_0^4$, 以及由系数温度依赖性产生的额外耦合项 $N_{\alpha_1\alpha_2} \partial_{x_{\alpha_1}}T_0 \partial_{x_{\alpha_2}}T_0$. 这些项分别源自 $O(\varepsilon^0)$ 方程中与瞬态效应、辐射源以及 Taylor 展开中导数项相关的剩余项. 作者通过将 $T_2$ 的贡献分成五部分, 每部分对应一个特定的单胞问题, 实现了整个校正项谱系的完整解耦. 这种分解不仅具备清晰的物理意义, 而且保证了计算时各单胞问题相互独立, 可并行求解.
单胞问题的适定性分析. 所有一阶和二阶单胞问题在形式上均为带有周期边界条件的椭圆型方程, 右端项依赖于已求得的宏观量和一阶单胞函数. 利用假设 (A1) 的一致椭圆性以及 Lax-Milgram 定理, 可以证明对任意固定宏观温度 $T_0$, 每个单胞问题存在唯一弱解. 求解这些单胞问题时, 宏观坐标 $x$ 和温度 $T_0$ 以参数出现, 这意味着在实际计算中需要为整个温度变化区间采样并求解一系列单胞问题, 但成本仍在可接受范围内.
误差分析框架. 尽管提供的内容中未给出完整的误差估计证明, 但根据文献惯例及本文摘要, 可以推断作者采用了能量方法或比较原理, 将原始方程减去宏微观展开所满足的近似方程, 得到关于误差的方程, 然后通过先验估计获得收敛阶. 核心困难在于控制因系数温度依赖产生的非线性项, 这需要利用系数的 Lipschitz 连续性以及先验梯度估计. 在点态意义下的局部误差分析中, 作者定义了残差函数, 通过分析单元胞上的局部残差证明了二阶多尺度解能够满意地保持局部热平衡, 这为全局收敛奠定了基础.
高效算法的实现. 算法的离线性在于预计算单胞函数对应的微观矩阵, 并组装均化材料参数表. 在线阶段则仅需求解宏观尺度的非线性抛物方程 (均化方程 (14)), 其规模与 $\varepsilon$ 无关, 然后利用预先计算出的单胞函数进行后处理重构出精细尺度解. 这种策略充分利用了多尺度方法分离尺度的天然优势, 避免了反复在细网格上求解原问题.

4. 与之前工作的比较

与前人工作的直接对比. 在辐射热传导多尺度分析领域，Allaire 等[32,33,34]建立了非线性辐射传递的均化理论与相应的二阶校正, 但其分析假设物性参数不随温度变化. Huang 等[35,36]采用 Rothe 方法得到了非稳态辐射传热方程的弱解存在唯一性与多尺度渐近展开, 并导出了 $\varepsilon^{1/2}$ 和 $\varepsilon$ 的显式收敛率, 同样未考虑温度依赖性. 本文首次将温度依赖性系统地纳入多尺度渐近框架, 这是最主要的推进. 相比 Huang 等的渐近展开只包含梯度校正, 本文的二阶展开项因温度依赖性而引入更多类型的校正 (时间导数项、辐射源项以及交叉导数项), 从而在同等周期尺度 $\varepsilon$ 下能够捕获更丰富的微观信息. 从收敛阶角度看, 摘要声称推导了显式收敛率, 若该收敛率优于 $\varepsilon$, 则本文在精度理论上比 Huang 等[36]的 $\varepsilon$ 阶结果更进一步, 但具体阶数需要从全文获取.
假设条件的差异. 本文对物性系数的基本假设 (A1)–(A4) 与已有工作类似 (一致椭圆性、有界性、光滑性), 但新增了 (A3) 中系数关于温度的 Lipschitz 连续性, 这是为了在 Taylor 展开中控制余项. 该假设在工程实际中通常是合理的, 且不会造成过强限制. 此外, 对初边值和外源数据的光滑性要求 (A4) 比单纯均化问题略高, 这是因为需要保证二阶校正项的有界性和误差估计的进行. 相较于 Allaire 等处理非线性辐射时的假设, 本文的条件更具针对性, 但本质上是相容的.
适用范围与能力边界. 本文的方法针对周期性复合结构, 因此适用于具有明显周期微结构的材料, 如编织复合材料、点阵结构等. 均化系数与单胞函数均显式依赖于宏观温度 $T_0$, 这意味着在模拟过程中, 若温度场变化剧烈, 必须为不同的温度区间求解不同的单胞问题, 这增加了离线阶段的存储需求. 尽管如此, 得益于温度依赖的 Lipschitz 连续性, 可通过插值技术减少采样点. 此外, 本工作局限于非耦合的辐射-导热问题, 未考虑对流或相变, 并且辐射项的处理采用简化的 $\beta\sigma_B T^4$ 形式, 未涉及复杂介质的散射或光谱效应. 与 Allaire 等的方法相比, 本文的框架可直接推广至更一般的辐射传输方程, 但需要解决更高维度的单胞问题.
计算代价与优势. 本文所提两阶段算法在计算效率上显然优于直接细网格求解, 也优于仅利用一阶均化的方法. 与 Allaire 等[34]的二阶校正相比, 本文需要额外求解 $G, N, N_{\alpha_1}, N_{\alpha_1\alpha_2}$ 等单胞问题, 离线成本增加. 然而, 这些单胞问题只需在每个温度采样点上求解一次, 且相互独立, 可并行计算, 一旦前处理完成, 在线模拟的额外开销仅在于后处理中对解的重构, 这是可以忽略的. 数值实验 (摘要提及) 表明, 在计算时间和存储消耗方面, 所提多尺度方法具有显著竞争优势. 总体而言, 本文的工作在精度-代价平衡上为温度依赖复合材料辐射传热问题提供了目前最优的选择之一, 尤其适合需要反复参数化分析的设计优化问题.

摘要翻译: 我们针对带输运噪声的随机Vlasov-Poisson方程提出保结构的动态低秩方法。我们首先推导了连续的低秩演化方程，并证明通过包含与守恒量相关的固定速度模态，低秩动力学继承了原始随机模型的质量、动量和能量平衡律。然后，基于两种随机离散化，我们开发了两种增广基更新Galerkin（BUG）积分器：应用于等价Itô公式的Euler-Maruyama格式和直接应用于Stratonovich公式的Heun格式。这两种选择使我们能够研究随机时间离散化如何与保守的低秩框架相互作用。在两种情况下，基增广和保守的秩截断保留了相关的矩空间，并在降秩下提供了鲁棒性。数值实验展示了所提方法的守恒性质，并比较了两种随机公式引起的增广需求、随机修正项以及动量和能量行为。

不相关论文（基本信息）

1. 关于在泊松-玻尔兹曼理论中计算非刚性蛋白质的极化溶剂化能

On calculating polar solvation energy of nonrigid proteins in the Poisson-Boltzmann theory

作者: Matthias Dogbatsey, Yuanzhen Shao, Emil Alexov, Shan Zhao

类别: math.NA, math-ph

发布日期: 2026-08-03

arXiv ID: 2608.02525v1

链接: http://arxiv.org/abs/2608.02525v1

摘要: The Poisson-Boltzmann (PB) theory is a cornerstone of implicit solvent models for electrostatic analysis, and has found a great success in various biomolecular applications. However, in calculating polar solvation energy, one should consider that the structure of the protein changes upon transition from vacuum to water phases. To address this, here we report for the first time a generalized PB framework capable of accommodating nonrigid conformational changes without suffering from self-energy artifacts. For regularized PB models, in which the charge singularities are captured by the Green’s functions, self-energies in the water and vacuum states will be analytically canceled. For non-regularized PB solvers, such as APBS and DelPhi, a simple thermodynamic cycle is proposed for nonrigid proteins by adding a Coulombic correction in vacuum. The generalized PB theory is validated using a perturbed two-atom system and a diverse set of proteins with different structures in vacuum and water, demonstrating its accuracy and robustness, regardless of the choice of sharp-interface and diffuse-interface PB models and different numerical solvers.

2. 未调整朗之万算法的 Wasserstein 混合时间

Wasserstein mixing time of the unadjusted Langevin algorithm

作者: Francesco Pedrotti, Peter A. Whalley

类别: stat.CO, cs.LG, math.NA, math.PR

发布日期: 2026-08-03

arXiv ID: 2608.02430v1

链接: http://arxiv.org/abs/2608.02430v1

摘要: We provide new estimates in Wasserstein distance for the asymptotic bias of the unadjusted Langevin algorithm, in the classical setting of log-smooth strongly log-concave measures. Our bound implies a Wasserstein mixing time of order $κ\sqrt{d}/\varepsilon$, where $κ$ is the condition number, $d$ is the dimension, and $\varepsilon$ is the target precision: this improves by a factor of $\sqrt{d}/\varepsilon$ over the previous state-of-the-art results.

3. 无方程多尺度建模中的广义投影积分格式

Generalised projective integration scheme in equation-free multiscale modelling

作者: Tanay Kumar Karmakar, Durga Charan Dalal

类别: math.NA

发布日期: 2026-08-03

arXiv ID: 2608.02419v1

链接: http://arxiv.org/abs/2608.02419v1

摘要: When the spectrum of a system varies significantly over time, fixed choices of macro-, meso-, and micro-time steps, as well as burst lengths, become inadequate, necessitating adaptive and locally informed strategies. To address these challenges, this article proposes a novel and flexible generalised projective integration (GPI) scheme, designed to accommodate time-dependent spectral variation and dynamically evolving scale separation. The proposed framework unifies and extends several existing multiscale methodologies, thereby offering a more general and adaptable computational paradigm. A comprehensive stability analysis of the GPI scheme is carried out, including a detailed investigation of the splitting of the stability region, which forms a central component of this work. Furthermore, problem-dependent strategies for selecting the micro-, meso-, and macro-time steps, as well as the burst length, are developed and their impacts are systematically validated through numerical experiments. To assess the effectiveness of the proposed scheme, three representative problems with distinct types of spectral evolution are considered. The performance of the proposed GPI scheme is evaluated and compared with several existing projective integration methods as well as some widely used stiff solvers, based on (i) number of micro time steps, (ii) accuracy, (iii) computational time, (iv) memory usage and (v) proportion of micro-scale simulations. The results demonstrate that the GPI scheme consistently outperforms the existing methods in terms of performance.

4. 基于熵产生率的生成扩散模型统一 Kullback–Leibler 散度分析

A Unified Kullback–Leibler Divergence Analysis of Generative Diffusion Models via Entropy Production Rate

作者: Han Wu, Zhiwen Zhang

类别: math.NA, stat.ML

发布日期: 2026-08-03

arXiv ID: 2608.02406v1

链接: http://arxiv.org/abs/2608.02406v1

摘要: We introduce a unified framework for the error analysis of generative models based on the entropy production rate of the forward-reverse diffusion process pair. For a pair of continuity equation flows, the rate admits a closed velocity form identity whose time integral decomposes the terminal Kullback–Leibler (KL) divergence into the sum of an initialization error, a score approximation error, and a time-discretization error. By analyzing the entropy production at the level of marginal distributions, rather than in path space, our framework yields a sharp convergence rate of $\mathcal{O}(h^2)$ for the Euler-Maruyama sampler, where $h$ is the step size. This improves upon the $\mathcal{O}(h)$ rates typically obtained from Girsanov’s path-space analyses. Furthermore, our framework unifies the analysis of score-based SDEs, probability-flow ODEs, and stochastic interpolants by varying diffusion coefficients within a single inequality, revealing the trade-off between deterministic and stochastic sampling. Numerical experiments confirm the predicted scaling with step size and terminal time.

5. 双调和Cauchy问题的最小二乘弱Galerkin方法

A Least-Squares Weak Galerkin Method for the Biharmonic Cauchy Problem

作者: Chunmei Wang, Shangyou Zhang

类别: math.NA

发布日期: 2026-08-03

arXiv ID: 2608.02317v1

链接: http://arxiv.org/abs/2608.02317v1

摘要: We develop a least-squares weak Galerkin (LS-WG) finite element method for the Cauchy problem of the biharmonic equation. The proposed approach reformulates the fourth-order equation as a coupled system of two second-order equations, which are discretized using discrete weak Laplacian operators on weak finite element spaces. The resulting least-squares formulation yields a symmetric positive definite linear system, thereby eliminating the discrete inf-sup condition required by mixed finite element methods while avoiding the construction of globally $C^1$-conforming finite element spaces. Furthermore, the weak Galerkin framework naturally accommodates general polygonal meshes, offering considerable flexibility in mesh generation and approximation. Under the assumption that the continuous biharmonic Cauchy problem admits a unique solution, we establish the uniqueness of the discrete LS-WG solution and derive optimal-order error estimates in a discrete energy norm. Numerical experiments confirm the theoretical convergence rates and demonstrate the accuracy, robustness, and effectiveness of the proposed method.

6. 用于动态形状的连续鲁棒比较的推前变换

The Push-Forward Transform for Continuous and Robust Comparison of Dynamic Shapes

作者: Roua Rouatbi, Juan-Esteban Suarez Cardona, Ivo F. Sbalzarini

类别: cs.CV, cs.LG, math.NA

发布日期: 2026-08-03

arXiv ID: 2608.02306v1

链接: http://arxiv.org/abs/2608.02306v1

摘要: We introduce a mathematical framework for shape comparison based on mapping functions from the shape domain to a common reference domain. This Push-Forward Transform enables invariant and robust comparison of shapes, preserving intrinsic geometric information. Quantitatively comparing shapes and their temporal evolution is a fundamental challenge in image analysis. Meaningful shape comparison requires representations that are invariant to transformations that do not alter shape itself, such as translation, rotation, reflection, re-parametrization, and uniform scaling, while remaining sensitive to intrinsic geometric variation. Existing approaches often rely on sensitive parameterizations, landmark correspondence, or learned representations that are difficult to interpret and reproduce. We show that the Push-Forward Transform (PF-T) applied to Signed Distance Functions (SDFs) yields a continuous representation that captures both boundary and interior geometry. We derive an interpretable morphometric that quantifies shape similarity and reveals features such as skeletal topology and rotational symmetries. The push-forward transform applies consistently to two- and three-dimensional shapes, extends to time-evolving geometries, and supports the joint analysis of shape and additional scalar fields defined over shapes, such as intensity or molecular signals. We present the mathematical formulation, describe an efficient algorithm, and benchmark the approach on 2D, 3D, and temporal data sets.

7. 非协调有限元方法的局部逐点后验误差估计

Localized pointwise a posteriori error estimates for nonconforming finite element methods

作者: Yongxing Guo, Yuwen Li

类别: math.NA

发布日期: 2026-08-03

arXiv ID: 2608.02132v1

链接: http://arxiv.org/abs/2608.02132v1

摘要: This paper establishes localized pointwise a posteriori error estimates for nonconforming finite element discretizations of the Poisson and biharmonic equations. For the Poisson problem, we derive localized estimates for the function-value and broken gradient errors of the Crouzeix–Raviart method. For the Morley discretization of the biharmonic equation, we derive a localized a posteriori estimate that controls the local Hessian error. The key ingredient is the design of two novel weight functions that facilitate sharp estimates of the (L^1) norms of derivatives of a regularized Green’s function for the biharmonic equation.

8. Landau-Lifshitz-Gilbert方程的二阶BDF方法的后验误差分析

A posteriori error analysis for the second-order BDF method for the Landau-Lifshitz-Gilbert equation

作者: Stefan Karch

类别: math.NA

发布日期: 2026-08-03

arXiv ID: 2608.02021v1

链接: http://arxiv.org/abs/2608.02021v1

摘要: The tangent plane scheme (TPS) is a well-established discretization of the Landau-Lifshitz-Gilbert (LLG) equation. However, rigorous a posteriori error estimates have not been established. In this work, we derive a rigorous a posteriori error estimate for the TPS based on the second-order backward differentiation formula (BDF(2)) in time and finite elements of arbitrary polynomial degree in space. The proposed estimators provide computable upper bounds for the temporal and spatial discretization errors on adaptive meshes with variable time-step sizes. This result establishes the mathematical foundation for fully adaptive algorithms for the LLG equation.

9. 水平集函数定义区域上的 $H^{-1}$ 最小二乘 UnCut 有限元法

An $H^{-1}$ least-squares UnCut FEM on domains defined by a level set function

作者: Jiashun Hu, Buyang Li, Han Yang

类别: math.NA

发布日期: 2026-08-03

arXiv ID: 2608.02015v1

链接: http://arxiv.org/abs/2608.02015v1

摘要: We propose a novel UnCut finite element method (FEM) for the Poisson and Stokes equations on domains with curved boundaries represented by a level set function. Like the $φ$-FEM, the method avoids numerical integration over cut subregions of boundary elements, but introduces a novel least-squares formulation that minimizes an $H^{-1}$ residual of the governing equations. This formulation ensures stability without requiring large stabilization parameters, thereby eliminating the need for user-tuned penalty parameters and improving the robustness of the computation. Optimal-order convergence of the UnCut FEM solutions is rigorously established in the $H^1$ norm for both the Poisson and Stokes equations, and numerical experiments are presented to support the theoretical analysis.

10. 度量图上分数扩散方程的数值逼近

Numerical approximation of fractional diffusion equations on metric graphs

作者: David Bolin, Lenin Riera-Segura, Alexandre B. Simas

类别: math.NA

发布日期: 2026-08-03

arXiv ID: 2608.01932v1

链接: http://arxiv.org/abs/2608.01932v1

摘要: We study fractional diffusion equations on compact metric graphs, where the nonlocal dynamics is governed by fractional powers of the shifted Kirchhoff-Laplacian. Building on recent advances in the analysis of fractional operators on metric graphs, we establish a rigorous mathematical framework and propose a fully discrete scheme based on backward Euler time-stepping and finite element discretization. To approximate the action of the fractional operator, we employ rational approximations, reducing the problem to a sequence of sparse elliptic solves for efficient implementation. We derive error estimates for the temporal, spatial, and rational discretizations, and confirm convergence through numerical experiments.

11. 关于与辐射传热相关的边值问题

On a boundary value problem related to radiative heat transfer

作者: Mihai Halic

类别: math.NA

发布日期: 2026-08-02

arXiv ID: 2608.01472v1

链接: http://arxiv.org/abs/2608.01472v1

摘要: We consider the boundary value problem $y''=κ^2y^n, y(0)=1, y(1)=0,$ relevant to radiative heat transfer, chemical reactions, etc. We use upper/lower envelopes to study the qualitative and quantitative dependence of its solution on parameters, and transform it into an easily solvable initial value problem.

12. Emden-Fowler方程的解析-数值方法

An analytical-numerical approach to the Emden-Fowler equation

作者: Mihai Halic

类别: math.NA, gr-qc

发布日期: 2026-08-02

arXiv ID: 2608.01465v1

链接: http://arxiv.org/abs/2608.01465v1

摘要: We investigate the Emden-Fowler equation $u''=-x^ru^p, u(0)=1, u'(0)=0$, with roots in astrophysics, and study the qualitative and quantitative dependence of its solution on the parameters; the analytical work is paralleled by numerical simulations. A special attention is given to estimating the first zero $x_0$ of $u$ in terms of $r, p$. The results are used to address two apparently new issues: first, we solve EF backwards starting from $x_0$; second, we transform it into an overdetermined boundary value problem and decide when is this solvable.

13. 随机反问题中的结构和概率不确定性

Random Inverse Problems with Structural and Probabilistic Ambiguities

作者: Wolfgang Hoegele

类别: stat.ME, math.NA, stat.CO

发布日期: 2026-08-02

arXiv ID: 2608.01439v1

链接: http://arxiv.org/abs/2608.01439v1

摘要: In this concise paper, we investigate a computational class of random inverse problems that incorporates model uncertainties through random variable parameters nonlinearly in the forward model as well as additive observational uncertainty. Random inverse problems with nonlinear parameter dependencies may arise in engineering, geophysics, image processing or uncertainty quantification. We introduce a new perspective on structural ambiguities due to the non-injectivity of the forward model together with probabilistic ambiguities by assigning mixture model densities with separate components to the parameters, which leads to a possibly complex forward model, observation model and posterior. As a result, the mixture-model parameters in the forward model can be interpreted as simultaneously describing aspects of both the nonlinear ambiguity and the uncertainty of the inverse problem. The underlying solution algorithm is presented based on Bayesian inversion for three observation scenarios leading to posterior densities for the input for given output samples or an observed output density. By applying the derived algorithms to 1D and 2D quadratic models, we numerically demonstrate in which scenarios the proposed algorithm can resolve probabilistic ambiguities in the solution of the random inverse problem. It is demonstrated that making the residual structural ambiguities visible in the posterior and showing the interplay with probabilistic ambiguities is a relevant perspective, including cases with a finite and an infinite number of solutions.

14. 径向基函数逼近与计算的结果、挑战与新进展

Results, challenges and new steps on RBF approximation and computation

作者: Stefano De Marchi, Nadaniela Egidi, Josephin Giacomini, Pierluigi Maponi

类别: math.NA

发布日期: 2026-08-02

arXiv ID: 2608.01362v1

链接: http://arxiv.org/abs/2608.01362v1

摘要: We present an up-to-date overview of approximation methods based on Radial Basis Function (RBF) techniques, which have recently attracted attention across various computational tools and application fields. This review study presents relevant results on RBF techniques, highlighting the associated computational challenges and stability issues that must be addressed when high-performance or parallel computation is required.

15. 基于神经代理的控制变量法用于动理学方程的不确定性量化

Control variates with neural surrogates for uncertainty quantification in kinetic equations

作者: Wei Chen, Giacomo Dimarco, Lorenzo Pareschi

类别: math.NA

发布日期: 2026-08-02

arXiv ID: 2608.01360v1

链接: http://arxiv.org/abs/2608.01360v1

摘要: Efficient uncertainty quantification for kinetic equations with random inputs is challenging because it requires repeated simulations of high-dimensional models, such as the Boltzmann, Landau, and related collisional equations, whose computational cost can quickly become prohibitive. Multifidelity control variates address this difficulty by coupling a small number of high-fidelity simulations with many evaluations of lower-complexity reduced models. In this work, we analyze the case in which the reduced model is replaced by a neural surrogate rather than evaluated through a classical numerical scheme. We show that the resulting estimator remains unbiased and that the change in the optimal variance induced by the neural approximation is controlled by the error between the exact low-fidelity observable and its neural approximation. This estimate is then combined with residual stability estimates for inhomogeneous Fokker–Planck and Bhatnagar–Gross–Krook surrogates. We also extend the analysis to several control variates and to an asymptotic-preserving (AP) hierarchy containing the limiting Euler observable. In the fluid limit, the optimal hierarchical variance converges to the variance associated with the limiting Euler control, while the contribution of the intermediate kinetic correction vanishes. Numerical tests based on micro–macro neural surrogates illustrate the predicted variance stability and the behavior of the AP hierarchy.

16. 基于高阶WENO的半隐式牛顿型快速扫描方法用于静态Hamilton-Jacobi方程

High-order WENO-based semi-implicit Newton-type fast sweeping methods for static Hamilton-Jacobi equations

作者: Yuan Liu, Jianliang Qian

类别: math.NA

发布日期: 2026-08-02

arXiv ID: 2608.01304v1

链接: http://arxiv.org/abs/2608.01304v1

摘要: In this paper, we propose high-order weighted essentially non-oscillatory (WENO)-based semi-implicit Newton-type Gauss-Seidel Lax-Friedrichs fast sweeping methods for solving the generalized Eikonal equation arising in wave propagation through a moving fluid. Building upon the Newton-type framework of Li and Qian (2020), which updates the solution line-wise using Newton’s method with a tridiagonal and strictly diagonally dominant Jacobian, we extend the local solver to fifth-, seventh-, and ninth-order accuracy by incorporating high-order WENO approximations of the spatial derivatives into the numerical Hamiltonian. Three alternating sweeping strategies, namely column-wise, row-wise, and column-row-wise, are considered within the Gauss-Seidel iteration framework. Numerical examples in both two and three spatial dimensions demonstrate the efficiency and accuracy of the proposed schemes.

17. 关于一类用于不定最小二乘问题的块正则化预处理迭代方法

On a Class of Block-Regularized Preconditioned Iterative Methods for Indefinite Least-Squares Problems

作者: Pinki Khatun

类别: math.NA

发布日期: 2026-08-02

arXiv ID: 2608.01273v1

链接: http://arxiv.org/abs/2608.01273v1

摘要: This paper proposes a novel block-regularized splitting (BRS) framework for the efficient solution of indefinite least-squares (ILS) problems. Based on a block-wise regularization strategy incorporated into a matrix splitting scheme, we develop a BRS iterative method together with an effective BRS preconditioner. The convergence of the proposed iterative method is rigorously analyzed, and spectral bounds for the BRS-preconditioned matrix are established. To further enhance computational performance, we introduce a relaxed BRS (RBRS) preconditioner, which provides improved spectral properties and significantly accelerates the convergence of Krylov subspace methods. Extensive numerical experiments on both dense and sparse test problems demonstrate that the proposed BRS and RBRS preconditioners consistently outperform existing approaches in terms of iteration count, computational time, and overall efficiency. These results highlight the effectiveness and robustness of the proposed block-regularized splitting framework for solving large-scale ILS problems.

18. 广义模滞编码

Generalized Modulo Hysteresis Encoding

作者: Matthias Beckmann, Jürgen Jeschke

类别: math.NA, eess.SP

发布日期: 2026-08-02

arXiv ID: 2608.01270v1

链接: http://arxiv.org/abs/2608.01270v1

摘要: Unlimited sensing and its extension via modulo hysteresis provide an efficient encoding scheme for high dynamic range signals by folding the signal’s amplitude into the range of the analog-to-digital converter prior to sampling. Motivated by shortcomings of the modulo hysteresis model, in this work we introduce generalized modulo encoders that flexibly design the folding by incorporating a general transition function and allowing the folding times to depend on the nonideality of the folding transition. We rigorously study mathematical properties of the respective operators and derive conditions under which a bandlimited signal is uniquely determined by its generalized modulo samples. Finally, we propose a recovery algorithm backed by mathematical guarantees and illustrate our theoretical results through numerical experiments.

19. 重访多面体上的分解不变条件梯度法

Revisiting Decomposition-Invariant Conditional Gradient Methods for Polytopes

作者: Dan Garber

类别: math.OC, math.NA

发布日期: 2026-08-02

arXiv ID: 2608.01243v1

链接: http://arxiv.org/abs/2608.01243v1

摘要: We revisit Decomposition-Invariant Conditional Gradient methods, originally introduced by Garber and Meshi in 2016, for minimizing a convex and $β$-smooth function over a polytope in $\reals^n$, under an $α$-quadratic growth condition. For 2-level polytopes we design a simple and parameter-free dyadic step-size rule that yields a linear convergence rate which scales with the dimension of the optimal face and not with the ambient dimension as in standard away-step-based conditional gradient methods for polytopes. For general polytopes, under a slightly stronger condition of $α_{\mathrm{F}}$-\textit{facial quadratic growth}, we introduce a method whose number of iterations to reach an $ε$-approximate solution is of the order $n+\frac{βD^2}{αr^{*2}} + \frac{(d^*+1)βD^2}{α_{\mathrm{F}}}\log(1/ε)$, where $d^*$ is the dimension of the optimal face, $r^*$ is a separation parameter between the optimal set and faces that do not contain an optimal solution, and $D$ is the diameter of the polytope. This method is also parameter-free and only relies on standard line-search computations. The second result improves upon previous conditional gradient methods, whose number of iterations to $ε$-approximation scales with $\frac{βD^2n}α\log(1/ε)$, in a meaningful regime $\max\{\fracα{α_{\rm F}}(d^*+1), \frac{1}{r^{*2}}\} \ll n$

20. 一种新的$n$维流形上的双射保角与保体积平衡参数化方法

A Novel Bijective Angle and Volume-preservation Balanced Parameterization for $n$-dimensional Manifolds

作者: Tiexiang Li, Wen-Wei Lin, Zhong-Heng Tan, Xiao Wan, Junxin Zhang

类别: math.NA

发布日期: 2026-08-02

arXiv ID: 2608.01073v1

链接: http://arxiv.org/abs/2608.01073v1

摘要: We propose a unified framework for balanced and bijective parameterizations of $n$-dimensional manifolds. The proposed energy combines conformal and volume-preserving terms to control both local anisotropy and volumetric distortion. At the continuous level, both energies are nonnegative and their zero-energy mappings are characterized. At the discrete level, the conformal, volume-preserving, and logarithmic barrier energies are formulated on oriented simplicial manifolds. A key result is that all their gradients admit a unified cotangent Laplacian-type representation, enabling sparse and dimension-independent computation. Bijectivity is enforced through signed simplex Jacobians, feasibility restoration, and a strictly orientation-preserving logarithmic barrier. The framework applies uniformly to spherical boundary parameterizations and parameterizations of discrete $n$-manifolds onto ball-like canonical domains.

21. 基于积分的$f(A)b$预处理框架

An Integral-Based Framework for Preconditioning $f(A)b$

作者: Gustavo Ramirez-Hidalgo

类别: math.NA, hep-lat

发布日期: 2026-08-02

arXiv ID: 2608.01003v1

链接: http://arxiv.org/abs/2608.01003v1

摘要: The computation of the action of a matrix function on a vector, $f(A)b$, is a major computational bottleneck for large, sparse matrices, particularly when unfavorable spectral distributions cause standard Krylov subspace methods to stagnate. In this work, we propose a unified framework for preconditioning $f(A)b$ based on the Cauchy integral representation of the matrix function. By exploiting shift-invariance properties, we decouple the preconditioner evaluation from the Krylov subspace generation. We develop this framework in two distinct directions. First, for rational shift-and-invert preconditioning, we resolve a fundamental trade-off between optimal spectral compression and finite-precision instability. We achieve this by formulating a closed-form extraction stabilized via Double Modified Gram-Schmidt reorthogonalization, which eliminates the formation of spurious phantom poles. Second, we present a matrix-free polynomial approach. To ensure numerical stability, we isolate the continuous numerical quadrature step using a Schur decomposition of the projected Hessenberg matrix. To further stabilize the integration near contour singularities and accelerate overall convergence, we incorporate an exact LR-deflation scheme targeting the critical low modes of the preconditioned operator. We analyze the asymptotic stability and proximity to singularity of these methods, and present numerical experiments demonstrating their efficiency on the 2D Laplacian with $f=\textrm{exp}$, and a highly ill-conditioned Wilson-Dirac operator from lattice quantum chromodynamics with $f=\textrm{sign}$, although the framework can be in principle used with any $f$ and it is particularly beneficial when applying $f(A)b_{i}$ with many different vectors $b_{i}$.

22. 非等温变密度糊状斯特凡问题的自相似结构与一种改进的低马赫数焓法

Self-similar structure of non-isothermal variable-density mushy Stefan problems and an improved low-Mach enthalpy method

作者: Yavkreet Swami, Amneet Pal Singh Bhalla

类别: math.NA, physics.flu-dyn

发布日期: 2026-08-02

arXiv ID: 2608.00960v1

链接: http://arxiv.org/abs/2608.00960v1

摘要: The enthalpy method was introduced in the late 1970s to simulate phase-change problems on fixed grids without explicitly tracking the moving phase-change front. It remains one of the most widely used approaches in academic and commercial software for the simulation of industrial melting and solidification processes. For pure phase-change materials (PCMs) that melt or solidify at a single temperature, the enthalpy method introduces an artificial mushy region bounded by the solidus temperature, $T^{\rm sol}$, and the liquidus temperature, $T^{\rm liq}$. As the numerical parameter $ΔT=T^{\rm liq}- T^{\rm sol}$ approaches zero, the solution obtained with the enthalpy method is generally assumed to converge to that of the classical Stefan problem, in which the phase-change front is infinitesimally thin. This assumption is largely based on benchmark studies performed under the simplifying assumption of equal solid and liquid densities. In this work, we systematically investigate the accuracy and spatio-temporal convergence properties of the enthalpy method for both low- and high-density-ratio phase-change problems. Because the limiting behavior $ΔT\rightarrow0$ is difficult to realize numerically, owing to the diminishing thickness of the mushy region, we formulate and analyze the finite-$ΔT$ mushy Stefan problem solved by the enthalpy method. We show that this problem possesses a self-similar structure that reduces the governing equations to a boundary-value problem involving two unknown parameters. The theoretical analysis also enables improvements to our previously developed low-Mach enthalpy method, enhancing its stability and accuracy as the density ratio between the two phases increases from $\mathcal{O}(1)$ to $\mathcal{O}(3)$.

23. 使用Polyak步长的随机迭代方法求解广义绝对值方程

Randomized iterative methods with Polyak step-size for solving generalized absolute value equations

作者: Jiayun Chen, Qiye Zhang, Deren Han, Jiaxin Xie

类别: math.NA

发布日期: 2026-08-02

arXiv ID: 2608.00952v1

链接: http://arxiv.org/abs/2608.00952v1

摘要: In this paper, we systematically incorporate the Polyak step-size into the randomized iterative method to improve its efficiency for solving generalized absolute value equations. In particular, we adopt the Polyak step-size within a stochastic iterative setting where the objective function updates dynamically at every step, unlike the classical Polyak step-size designed for deterministic optimization with fixed objective functions. Consequently, this novel implementation differs from the conventional Polyak scheme and demands a dedicated convergence analysis. We rigorously analyze the convergence properties of the proposed method and establish its linear convergence in expectation. Numerical experiments demonstrate that the incorporation of the Polyak step-size substantially improves the computational performance of randomized iterative methods with constant step-sizes.

24. 局部Chan–Vese分割的MBO格式

MBO Scheme for Local Chan–Vese Segmentation

作者: Kevin Bui, Adina Ciomaga

类别: cs.CV, eess.IV, math.NA

发布日期: 2026-08-01

arXiv ID: 2608.00893v1

链接: http://arxiv.org/abs/2608.00893v1

摘要: Robust to intensity inhomogeneity, the local Chan–Vese (LCV) model extends the classical Chan–Vese (CV) image segmentation method by incorporating local statistical information around each pixel. Originally, the LCV model was solved using a finite difference scheme, following the approach used for the CV model. As an alternative to the finite difference scheme, a more efficient algorithm based on the Merriman-Bence-Osher (MBO) scheme was later developed for the CV model. In this paper, we derive a similar MBO-based algorithm to solve the LCV model and propose an efficient implementation. The algorithm is developed for both two-phase and multiphase segmentation, and an extension to color images is also discussed. To demonstrate the effectiveness of the proposed approach, we apply it to a variety of grayscale and color images, including medical and microscopy images.

25. 通过列采样的次线性时间特征向量近似

Sublinear Time Eigenvector Approximation via Column Sampling

作者: Rajarshi Bhattacharjee, Cameron Musco, Dominic Rutkowski

类别: cs.DS, math.NA

发布日期: 2026-08-01

arXiv ID: 2608.00840v1

链接: http://arxiv.org/abs/2608.00840v1

摘要: We study sublinear time sampling methods for approximating the outlying eigenvectors of large matrices. Our main result is an algorithm that uniformly samples just $\tilde{O}(\log n/ε^4)$ columns of a symmetric matrix $A \in \mathbb{R}^{n \times n}$ with entries bounded in magnitude by $1$, and, for any eigenvalue $λ$ of $A$ with $|λ| \ge εn$, outputs an approximate eigenvector $v$ satisfying $\|Av - λv\|_2 \le εn$. For approximating just the eigenvector of the largest magnitude eigenvalue, our algorithm samples only $\tilde{O}(\log n/ε^2)$ columns. Given the ability to sample rows and columns of $A$ proportional to their squared norms, we give a similar result with an improved error bound of $ε\|A\|_F$. For top eigenvector approximation, we show our bound is tight up to logarithmic terms. A key feature of our algorithms is that the output eigenvectors are spanned by a small number of $A$’s columns, and individual entries can be computed rapidly, in poly(log n, 1/epsilon) time per entry. This makes them applicable in the quantum-inspired algorithms framework of [Tang, STOC 2019], where we give the first sublinear time classical algorithms for eigenvector approximation with additive error $ε\|A\|_F$. Finally, we present an alternative approach, based on a truncated Nystrom method, that, while not allowing poly(log n, 1/epsilon) time entrywise computation of the approximate eigenvectors, achieves near optimal sample complexity for general symmetric matrices, and improved bounds for positive semidefinite matrices. Technically, our bounds build on recent work on approximating the outlying eigenvalues of symmetric matrices via random sampling in [Bhattacharjee et al. ‘22] and [Swartworth and Woodruff ‘25]. We demonstrate for the first time that these approaches extend to the problem of eigenvector estimation.

26. 论保正性在高阶逼近 Dean–Kawasaki 方程中的作用

On the role of positivity preservation for high order approximations of the Dean–Kawasaki equation

作者: Ana Damnjanović, Ana Djurdjevac, Nicolas Perkowski

类别: math.PR, math.NA

发布日期: 2026-08-01

arXiv ID: 2608.00838v1

链接: http://arxiv.org/abs/2608.00838v1

摘要: We study a spectral regularization of the Dean–Kawasaki equation and quantify how the failure of positivity preservation affects its weak approximation of the empirical measure of independent Brownian particles. For initial densities bounded away from zero, we prove a uniform-in-time weak error measured through the Laplace transform and prove a superpolynomial convergence rate for smooth test functions. When the initial density is allowed to vanish, the negative part of the regularized solution leads to weaker upper bounds, and a one-dimensional example gives a lower error bound ruling out superpolynomial convergence. We also present numerical experiments that confirm the theoretical results and illustrate main observations.

27. 使用可微多重网格线性求解器的变分推断

Variational Inference Using a Differentiable Multigrid Linear Solver

作者: Andrés Ramírez, Philipp Haim, David Föger, Torsten A. Enßlin, Philipp Frank, Philipp Gschwandtner, Dominik Jüstel, Philipp Mertsch, Vo Hong Minh Phan, Laurin Söding, Hanieh Zandinejad, Ralf Kissmann

类别: math.NA, physics.med-ph, stat.CO

发布日期: 2026-08-01

arXiv ID: 2608.00760v1

链接: http://arxiv.org/abs/2608.00760v1

摘要: Gradient-based Bayesian inference methods require efficient access to Jacobian and adjoint-Jacobian operators of high-dimensional forward models. While multigrid solvers provide near-optimal complexity for elliptic partial differential equations, they are rarely available in forms compatible with automatic differentiation (AD). We develop a differentiable multigrid solver for steady-state diffusion-absorption problems and derive its adjoint operations analytically through the full multigrid hierarchy. The resulting solver, DMGS, is implemented in C++ and interfaced with JAX to provide efficient Jacobian-vector and vector-Jacobian products for variational inference in the NIFTy framework. We validate the approach on a 3D inverse problem involving diffuse radiative transfer in tissue, reconstructing an effective radiative source from Monte Carlo-simulated data. The reconstruction reproduces the data at a reduced chi-squared of 1.1 and generalizes to 32 independent validation datasets. Benchmarks against a JAX-native multigrid implementation show comparable runtimes and consistently lower peak memory for the hand-derived adjoint, with modest reverse-mode overhead. These results establish differentiable multigrid solvers as practical building blocks for variational inference in PDE-constrained problems.

28. 针对 $\mathbb{R}^d$ 中一般共振流形的三波动力学与 $C_{12}$ 量子玻尔兹曼方程的谱算法

Spectral Algorithms for 3-Wave Kinetic and $C_{12}$ Quantum Boltzmann Equations with General Resonance Manifolds in $\mathbb{R}^d$

作者: Thanh Trung Le, Minh-Binh Tran

类别: math.NA

发布日期: 2026-08-01

arXiv ID: 2608.00603v1

链接: http://arxiv.org/abs/2608.00603v1

摘要: Following recent developments in numerical schemes for 3-wave kinetic equations [2, 7, 42, 44, 43], we develop spectral algorithms for multidimensional 3-wave kinetic equations and $C_{12}$ quantum Boltzmann equations with general polynomial dispersion relations. The principal numerical difficulty arises from the resonance constraint, supported on a nonlinear manifold in wave-vector space. We approximate the Dirac distribution by a truncated Fourier representation and derive two spectral discretizations of the collision operator. The first is a direct spectral method with complexity $\mathcal{O}\big(L(2N)^{3d}\big)$, while the second exploits multidimensional FFTs to reduce the complexity to $\mathcal{O}\big(L(2N)^{2d}\log(2N)\big)$. Numerical tests show excellent agreement between the two methods, with the fast algorithm providing substantial computational savings. To suppress unresolved high-frequency modes, we combine the classical $2/3$-rule with exponential spectral filtering. Simulations in two and three dimensions capture the gain–loss dynamics of the $C_{12}$ quantum Boltzmann equation for both rapidly and algebraically decaying initial data. For the 3-wave kinetic equation, the computations exhibit strong oscillations and rapid spectral broadening, providing numerical evidence of an apparent energy cascade toward high frequencies. The results also show that the dispersion relation and spatial dimension strongly influence the transient resonant dynamics.

29. 基于稳定化半隐式物质点法的滑坡研究

Study of Landslides through a Stabilised Semi-Implicit Material Point Method

作者: Mian Xie, Pedro Navas, Susana López-Querol

类别: math.NA

发布日期: 2026-08-01

arXiv ID: 2608.00592v1

链接: http://arxiv.org/abs/2608.00592v1

摘要: In this research, a new semi-implicit two-phase double-point material point method is proposed, in which the soil and water phases are modelled using two distinct sets of material points, both being stabilised with a novel approach. The Nor-Sand constitutive model is implemented to simulate more realistic soil behaviour. Some landslide numerical examples are presented to investigate the performance of the proposed method and highlight the importance of using the double-point approach. The formulation with two sets of material points shows significantly different but more reliable results in the cases of landslides, compared with the conventional single-point approach. Furthermore, this research shows that the additional computational cost given by the additional water material points is acceptable. Therefore, it is recommended to use two sets of material points for some large deformation geotechnical problems.

30. 可变形多孔介质中自由表面流与渗流的半隐式双点物质点法

A semi-implicit double-point Material Point Method for both free-surface flow and seepage in deformable porous media

作者: Mian Xie, Pedro Navas, Susana López-Querol

类别: math.NA, physics.flu-dyn

发布日期: 2026-08-01

arXiv ID: 2608.00578v1

链接: http://arxiv.org/abs/2608.00578v1

摘要: A new semi-implicit, two-phase, double-point formulation of the Material Point Method (MPM) for soil-water interaction with seepage and free-surface flows under large deformation is presented in this paper. The approach advances the water phase implicitly while keeping the soil phase explicit, enabling stable, efficient time integration in problems that involve rapid seepage and strong free-surface motion. The proposed framework models high-Reynolds-number interphase drag through a non-linear Darcy’s law implemented for the first time within an incremental fractional step MPM formulation without enlarging the implicit solve. This methodology also enhances the numerical stability for fast flows and wave breaking via a hyperelastic constitutive treatment of slightly compressible viscous water, and mitigates spurious oscillations through a new stabilisation approach for the velocity. Robustness of soil-water interface is achieved by combining nodal-based, free-surface detection, suited for higher-order spline functions with smooth porosity-permeability transitions that avoid constitutive divergence at sharp material boundaries. Validation against laboratory benchmark cases reported in the literature, including pure-water dam break, dam-break seepage through a porous barrier, two granular-collapse tsunami experiments, and a dam-break wave over a movable granular bed, shows accurate and stable free-surface evolution, pressure time histories, seepage fronts, and wave-gauge records. Using an advanced critical-state soil model (NorSand) further improves the reproduction of granular flow kinematics. The results demonstrate that the proposed formulation is a reliable and computationally efficient tool for geotechnical hazards involving intense soil-water coupling, seepage, sediment transport and free water.

31. 从波利亚条件到超插值 $L^2$ 收敛性的完全刻画

From Pólya’s Conditions to a Complete Characterization of the $L^2$ Convergence of Hyperinterpolation

作者: Congpei An, Xiannan Hu, Xiaoming Yuan

类别: math.NA

发布日期: 2026-08-01

arXiv ID: 2608.00525v1

链接: http://arxiv.org/abs/2608.00525v1

摘要: It has remained open to identify the necessary and sufficient conditions for the $L^2$ convergence of hyperinterpolation since it was introduced by Sloan in 1995. We show that the $L^1$-$L^2$ Marcinkiewicz-Zygmund (MZ) condition, together with the asymptotic functional approximation property for polynomials, is the answer. We further prove that the optimal $L^1$-$L^2$ MZ constant coincides with the operator norm of the hyperinterpolation operator, and it admits a natural Banach space duality interpretation. With an explicit construction, we also show that Pólya’s classical conditions for quadrature convergence are not sufficient for the $L^2$ convergence of hyperinterpolation. This reveals a fundamental distinction between the convergence of linear functionals (quadrature formulas) and that of linear operators (hyperinterpolation operators). We establish a strict logical hierarchy for the stability and accuracy conditions governing the convergence of quadrature and hyperinterpolation.

32. 通过流映射学习建模未知的非局部偏微分方程系统

Modeling Unknown Nonlocal PDE Systems via Flow Map Learning

作者: Zhongshu Xu, Ying Li, Yanzhi Zhang, Dongbin Xiu

类别: cs.LG, math.DS, math.NA

发布日期: 2026-08-01

arXiv ID: 2608.00400v1

链接: http://arxiv.org/abs/2608.00400v1

摘要: Nonlocal partial differential equations arise in many applications but are often difficult to model and learn because of the presence of nonlocal operators. We present a flow-map learning (FML) framework for modeling unknown nonlocal PDEs directly from solution data. Rather than learning or approximating the underlying nonlocal operators, the proposed approach learns the finite-time evolution operator in either modal or nodal space. Two complementary formulations are developed for spectral and grid-based solution representations. Numerical experiments on one- and two-dimensional fractional diffusion and wave equations demonstrate accurate and stable long-time prediction using only short observation windows. The proposed approach provides an effective data-driven framework for learning unknown nonlocal dynamics without explicit evaluation of nonlocal operators.

33. 一种用于具有三重结拖拽的网络曲率流的相场方法

A Phase-Field Method for Curvature Flow of Networks with Triple Junction Drag

作者: Yuchuan Yang, Selim Esedoglu

类别: math.AP, math.NA

发布日期: 2026-07-31

arXiv ID: 2608.00293v1

链接: http://arxiv.org/abs/2608.00293v1

摘要: We develop a new phase-field (diffuse interface) approximation of multiphase curvature motion with triple junction drag - an important sharp interface model for the evolution of microstructure in polycrystalline materials during heat treatment. This sharp interface model arises as gradient flow for the total length of the interfacial network with respect to a certain metric. Accordingly, we derive our diffuse interface approximation - a coupled system of partial differential equations that is a variant of the Allen-Cahn system - from a variational perspective, in the style of minimizing movements: starting from a discrete in time approximation that entails a convex optimization problem to advance from one time step to the next. In the process, we propose a simple integral expression that counts the number of junctions using the order parameter that appears to be new even for the standard multiphase Allen-Cahn system. The convergence of the resulting flow to the desired sharp interface limit is then verified via the method of matched asymptotic expansions. Numerical convergence studies against both known exact solutions as well as highly accurate benchmark solutions obtained via front tracking provide clear further evidence for this convergence. Moreover, the method retains the most desirable feature of diffuse interface methods: Automatic handling of topological changes in the network of interfaces.

34. 从非等距离散样本逼近傅里叶变换

Approximating the Fourier Transform from Non-equispaced Discrete Samples

作者: Daniel Potts, Laura Weidensager

类别: math.NA

发布日期: 2026-07-31

arXiv ID: 2608.00260v1

链接: http://arxiv.org/abs/2608.00260v1

摘要: We study the approximation of the Fourier transform of a function from finitely many samples. Departing from the equispaced setting, we sample the function at deterministic non-equispaced nodes obtained by transforming equispaced points on $[0,1]$ through the inverse cumulative distribution function of a probability density. This change of variables compactifies the real line, so that no truncation of the space domain is necessary. Combining an exact aliasing identity for the midpoint rule with stationary-phase estimates for the transformed oscillatory integrals, we derive deterministic error bounds for every frequency and in $L_p$. For functions with polynomial decay in space and frequency, an explicitly optimized density recovers the equispaced convergence rate, while an additional variance parameter substantially reduces the pre-asymptotic error constants. For (sub-)exponentially decaying functions a polynomially decaying density yields (sub-)exponential rates. Numerical experiments confirm the theory and demonstrate the reduced pre-asymptotic error compared with optimally scaled equispaced sampling.

35. 逆势平均场博弈的截断微分法

Truncated Differentiation for Inverse Potential MFGs

作者: Siting Liu, Yat Tin Chow, Samy Wu Fung

类别: math.OC, math.NA

发布日期: 2026-07-31

arXiv ID: 2608.00217v1

链接: http://arxiv.org/abs/2608.00217v1

摘要: We study inverse potential mean-field games (MFGs), in which an unknown spatial inverse-cost (mobility) map is inferred from observed population densities. We solve the forward MFG with a preconditioned primal-dual hybrid gradient (PDHG) method and develop Jacobian-free backpropagation (JFB-$r$), which records only the final $r$ iterations from a detached warm start while retaining the full forward solve. To analyze this truncated differentiation method, we show that the exact-proximal dual-extrapolated PDHG map is a metric resolvent of the maximal monotone KKT operator. This resolvent view shows that JFB-$r$ exactly differentiates a finite-trajectory surrogate and, under a locally fixed active set at an exact equilibrium detach point, converges to the implicit gradient as the tracked depth increases. Across several inverse-MFG settings, numerical experiments show that JFB-r at moderate tracked depths can achieve recovery accuracy comparable to full unrolling while reducing memory and runtime.

重点关注论文（完整分析）#

1. 混合边界交界面处梯度依赖半线性方程中的对数共振#

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. 面向温度相关非线性热辐射问题的复合材料结构高阶多尺度方法及其收敛性分析#

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. 信息几何正则化可压缩欧拉方程的间断 Galerkin 半离散#

摘要翻译#

2. Yang-Mills联络的有限元离散#

摘要翻译#

3. tFUSOperator: 用于经颅聚焦超声数字孪生的算子学习#

摘要翻译#

4. 一种带变分多尺度稳定的 Helmholtz-Leray 投影方法求解 Navier-Stokes 方程#

摘要翻译#

5. 保结构的随机Vlasov-Poisson方程动态低秩近似#

摘要翻译#

不相关论文（基本信息）#

1. 关于在泊松-玻尔兹曼理论中计算非刚性蛋白质的极化溶剂化能#

2. 未调整朗之万算法的 Wasserstein 混合时间#

3. 无方程多尺度建模中的广义投影积分格式#

4. 基于熵产生率的生成扩散模型统一 Kullback–Leibler 散度分析#

5. 双调和Cauchy问题的最小二乘弱Galerkin方法#

6. 用于动态形状的连续鲁棒比较的推前变换#

7. 非协调有限元方法的局部逐点后验误差估计#

8. Landau-Lifshitz-Gilbert方程的二阶BDF方法的后验误差分析#

9. 水平集函数定义区域上的 $H^{-1}$ 最小二乘 UnCut 有限元法#

10. 度量图上分数扩散方程的数值逼近#

11. 关于与辐射传热相关的边值问题#

12. Emden-Fowler方程的解析-数值方法#

13. 随机反问题中的结构和概率不确定性#

14. 径向基函数逼近与计算的结果、挑战与新进展#

15. 基于神经代理的控制变量法用于动理学方程的不确定性量化#

16. 基于高阶WENO的半隐式牛顿型快速扫描方法用于静态Hamilton-Jacobi方程#

17. 关于一类用于不定最小二乘问题的块正则化预处理迭代方法#

18. 广义模滞编码#

19. 重访多面体上的分解不变条件梯度法#

20. 一种新的$n$维流形上的双射保角与保体积平衡参数化方法#

21. 基于积分的$f(A)b$预处理框架#

22. 非等温变密度糊状斯特凡问题的自相似结构与一种改进的低马赫数焓法#

23. 使用Polyak步长的随机迭代方法求解广义绝对值方程#

24. 局部Chan–Vese分割的MBO格式#

25. 通过列采样的次线性时间特征向量近似#

26. 论保正性在高阶逼近 Dean–Kawasaki 方程中的作用#

27. 使用可微多重网格线性求解器的变分推断#

28. 针对 $\mathbb{R}^d$ 中一般共振流形的三波动力学与 $C_{12}$ 量子玻尔兹曼方程的谱算法#

29. 基于稳定化半隐式物质点法的滑坡研究#

30. 可变形多孔介质中自由表面流与渗流的半隐式双点物质点法#

31. 从波利亚条件到超插值 $L^2$ 收敛性的完全刻画#

32. 通过流映射学习建模未知的非局部偏微分方程系统#

33. 一种用于具有三重结拖拽的网络曲率流的相场方法#

34. 从非等距离散样本逼近傅里叶变换#

35. 逆势平均场博弈的截断微分法#

重点关注论文（完整分析）

1. 混合边界交界面处梯度依赖半线性方程中的对数共振

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. 面向温度相关非线性热辐射问题的复合材料结构高阶多尺度方法及其收敛性分析

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. 信息几何正则化可压缩欧拉方程的间断 Galerkin 半离散

摘要翻译

2. Yang-Mills联络的有限元离散

摘要翻译

3. tFUSOperator: 用于经颅聚焦超声数字孪生的算子学习

摘要翻译

4. 一种带变分多尺度稳定的 Helmholtz-Leray 投影方法求解 Navier-Stokes 方程

摘要翻译

5. 保结构的随机Vlasov-Poisson方程动态低秩近似

摘要翻译

不相关论文（基本信息）

1. 关于在泊松-玻尔兹曼理论中计算非刚性蛋白质的极化溶剂化能

2. 未调整朗之万算法的 Wasserstein 混合时间

3. 无方程多尺度建模中的广义投影积分格式

4. 基于熵产生率的生成扩散模型统一 Kullback–Leibler 散度分析

5. 双调和Cauchy问题的最小二乘弱Galerkin方法

6. 用于动态形状的连续鲁棒比较的推前变换

7. 非协调有限元方法的局部逐点后验误差估计

8. Landau-Lifshitz-Gilbert方程的二阶BDF方法的后验误差分析

9. 水平集函数定义区域上的 $H^{-1}$ 最小二乘 UnCut 有限元法

10. 度量图上分数扩散方程的数值逼近

11. 关于与辐射传热相关的边值问题

12. Emden-Fowler方程的解析-数值方法

13. 随机反问题中的结构和概率不确定性

14. 径向基函数逼近与计算的结果、挑战与新进展

15. 基于神经代理的控制变量法用于动理学方程的不确定性量化

16. 基于高阶WENO的半隐式牛顿型快速扫描方法用于静态Hamilton-Jacobi方程

17. 关于一类用于不定最小二乘问题的块正则化预处理迭代方法

18. 广义模滞编码

19. 重访多面体上的分解不变条件梯度法

20. 一种新的$n$维流形上的双射保角与保体积平衡参数化方法

21. 基于积分的$f(A)b$预处理框架

22. 非等温变密度糊状斯特凡问题的自相似结构与一种改进的低马赫数焓法

23. 使用Polyak步长的随机迭代方法求解广义绝对值方程

24. 局部Chan–Vese分割的MBO格式

25. 通过列采样的次线性时间特征向量近似

26. 论保正性在高阶逼近 Dean–Kawasaki 方程中的作用

27. 使用可微多重网格线性求解器的变分推断

28. 针对 $\mathbb{R}^d$ 中一般共振流形的三波动力学与 $C_{12}$ 量子玻尔兹曼方程的谱算法

29. 基于稳定化半隐式物质点法的滑坡研究

30. 可变形多孔介质中自由表面流与渗流的半隐式双点物质点法

31. 从波利亚条件到超插值 $L^2$ 收敛性的完全刻画

32. 通过流映射学习建模未知的非局部偏微分方程系统

33. 一种用于具有三重结拖拽的网络曲率流的相场方法

34. 从非等距离散样本逼近傅里叶变换

35. 逆势平均场博弈的截断微分法