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1. 基于包络高阶微扰(HOPE)方法的周期非均匀手性介质电磁散射
A High-Order Perturbation of Envelopes (HOPE) Method for Electromagnetic Scattering by Periodic Inhomogeneous Chiral Media
作者: David Nicholls, Liet Vo
类别: math.NA
发布日期: 2026-08-05
arXiv ID: 2608.04950v1
摘要翻译: 手性在许多光学现象中起着重要作用，因此拥有高效且精确的数值算法来模拟该情形下的解至关重要。本文讨论了一种高阶谱方法结合几何正则摄动理论，恰好提供了这样一种方法。我们将横向周期、常数介电常数/磁导率结构的手性视为对背景值的偏离，并证明了平面电磁波散射场关于这一形变的解析性。该解析性不仅关于任意大实尺度的手性形变成立，且关于空间变量也联合解析。我们还展示了用于建立这些结果的包络高阶微扰(HOPE)递推如何实现为一种快速、稳健且可靠的数值算法。
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链接: http://arxiv.org/abs/2608.04950v1
1. 研究对象和背景
手性在化学、生物学和物理学中具有根本重要性，表现为物体与其镜像不可重合的性质，例如双手、DNA螺旋等。在光学中，手性材料导致圆二色性(CD)，即左旋圆偏振(LCP)和右旋圆偏振(RCP)波以不同速度传播，这使得手性传感器在分子检测、表面等离子体共振(SPR)等领域有重大应用。为了模拟手性结构对电磁波的响应，数值算法的高效性和准确性至关重要。
本文研究的核心问题为：线性时谐电磁波照射在具有横向周期性的非均匀手性介质薄层上的散射。介电常数$\epsilon$、磁导率$\mu$和手性参数$\chi$依赖于空间坐标$(x,y,z)$，且在手性层内区域$-h
材料响应由Drude-Born-Fedorov (DBF)本构关系描述：
$$\mathbf{D}=\epsilon(\mathbf{E}+\chi\nabla\times\mathbf{E}),\quad \mathbf{B}=\mu(\mathbf{H}+\chi\nabla\times\mathbf{H}).$$代入时谐Maxwell方程($\sim e^{-i\omega t}$)
$$\nabla\times\mathbf{E}-i\omega\mathbf{B}=0,\quad \nabla\times\mathbf{H}+i\omega\mathbf{D}=0,$$并消去$\mathbf{B},\mathbf{D}$，可得封闭的Maxwell-DBF方程：
$$\begin{pmatrix}\nabla\times\mathbf{E}\\ \nabla\times\mathbf{H}\end{pmatrix}=J\begin{pmatrix}\mathbf{E}\\ \mathbf{H}\end{pmatrix},\quad J=\frac{1}{1-\chi^2k^2}\begin{pmatrix}\chi k^2 & i\omega\mu\\ -i\omega\epsilon & \chi k^2\end{pmatrix},$$其中$k^2=\omega^2\epsilon\mu$。为保证良定性，本文假设$-1<\chi k<1$，即$(1-\chi^2k^2)\neq0$。
为将无界问题截断至有限区域，采用了透明边界条件。在上界面$z=h$，散射场需满足向上传播条件(UPC)，从而导出Robin型边界条件：
$$-\partial_z\mathbf{E}-T_u[\mathbf{E}]=\boldsymbol{\phi},\quad -\partial_z\mathbf{H}-T_u[\mathbf{H}]=\boldsymbol{\psi},$$其中$T_u$为外部Dirichlet-Neumann算子，是Fourier乘子：$T_u[\psi]=\sum_{p,q}(-i\gamma_{p,q}^{(u)})\hat{\psi}_{p,q}e^{i\alpha_p x+i\beta_q y}$，$\boldsymbol{\phi},\boldsymbol{\psi}$由入射场确定。类似地，下界面$z=-h$有向下传播条件(DPC)：
$$\partial_z\mathbf{E}-T_w[\mathbf{E}]=0,\quad \partial_z\mathbf{H}-T_w[\mathbf{H}]=0.$$此外，场满足拟周期性边界条件，即$\mathbf{E}(x+d_x,y+d_y,z)=e^{i\alpha d_x+i\beta d_y}\mathbf{E}(x,y,z)$。
本文考虑一种简化情形：介电常数和磁导率在空间各处均为常数$\epsilon_0,\mu_0$，而手性参数在层内为$y$不变的横向周期函数$\chi(x,z)$。这一假设使得Maxwell方程组可通过Bohren变换解耦为左、右圆极化波的Beltrami方程：
$$\nabla\times\mathbf{L}=\kappa_L\mathbf{L},\quad \nabla\times\mathbf{R}=-\kappa_R\mathbf{R},$$其中$\kappa_L=\frac{k_0}{1-k_0\chi}$，$\kappa_R=\frac{k_0}{1+k_0\chi}$，$k_0=\omega\sqrt{\epsilon_0\mu_0}$。进一步，若结构为$y$不变，则$\mathbf{L},\mathbf{R}$的横向分量$\ell(x,z),r(x,z)$满足标量椭圆方程：
$$\rho_L\,\nabla\cdot(\rho_L\nabla\ell)+k_0^2\ell=0,\quad \rho_L=1-k_0\chi,$$$$\rho_R\,\nabla\cdot(\rho_R\nabla r)+k_0^2 r=0,\quad \rho_R=1+k_0\chi,$$并配以上述透明边界条件和拟周期性。这两个方程形式统一，记$v$代表$\ell$或$r$，$\rho$代表对应的系数函数，边界数据$\tau$相应而定。最终问题为：在矩形区域$\mathcal{S}_v=(0,d)\times(-h,h)$上求解如下边值问题：
$$\rho\,\nabla\cdot(\rho\nabla v)+k_0^2 v=0,\quad \text{in }\mathcal{S}_v,$$$$-\partial_z v-T_u[v]=\tau,\;\text{on }z=h,\qquad \partial_z v-T_w[v]=0,\;\text{on }z=-h,$$$$v(x+d,z)=e^{i\alpha d}v(x,z).$$已有工作方面，Ammari和Bao等人对周期手性光栅进行了严格的变分分析与有限元/边界元方法研究。然而，这些方法通常需要体积或边界离散，计算成本较高，且未提供散射场关于手性参数的高阶正则性理论。本文试图填补的空白是：建立一种基于正则摄动与高阶谱方法的高效数值算法，并严格证明散射场关于手性参数变形的解析性，从而使Taylor展开全局收敛，保证数值方法的高精度和可靠性。
2. 主要定理或主要结果
本文在如下假设下建立了主要理论结果与数值方法：
- 材料参数：介电常数和磁导率在全空间为常数$\epsilon_0,\mu_0$；手性参数$\chi(x,z)$为$y$不变、$x$周期为$d$的光滑函数，且满足约束$-1<\chi k_0<1$。
- 入射波：平面波以角度$(\phi,\theta)$照射，波数$k_0$，横向波数$\alpha=k_0\sin\theta\cos\phi,\beta=k_0\sin\theta\sin\phi$。
- 扰动形式：手性参数可表示为$\chi(x,z)=\bar{\chi}-\delta X(x,z)$，其中$\bar{\chi}$为常背景值，$X$为周期光滑包络，$\delta$为扰动参数。从而$\rho$函数可写为$\rho_0+\delta\rho_1$，其中$\rho_0=1\mp k_0\bar{\chi}$，$\rho_1$与$X$成比例。
主要结果：
- (解析性，§7) 设$v(\mathbf{x};\delta)$为上述边值问题的解（代表LCP或RCP的横向分量），则$v$作为$\delta$的函数在$\mathbb{C}$上存在解析延拓，且区域包含整个实轴。具体地，存在某个$R>0$，使得对于任意$\delta_0\in\mathbb{R}$，存在邻域$B(\delta_0,R)\subset\mathbb{C}$，$v(\delta)$在该邻域内解析。从而当$\delta$为实数时，$v$可展开为收敛幂级数$v=\sum_{m=0}^{\infty}v_m\delta^m$。
- (联合解析性，§8) 场$v$不仅是$\delta$的解析函数，而且关于$\delta$和空间变量$(x,z)$联合解析。这意味着在任何固定区域内部，所有阶的导数$\partial_\delta^k\partial_x^{p}\partial_z^{q}v$存在且连续，且相应Taylor级数收敛。
- (递推公式，§5) $v_m$满足递归的边值问题： $$\rho_0^2\Delta v_m+k_0^2 v_m = F_m,$$ 其中非齐次项$F_m$仅依赖于低阶项$v_{m-1}$和$v_{m-2}$： $$F_m = -\rho_0\nabla\cdot(\rho_1\nabla v_{m-1})-\rho_1\nabla\cdot(\rho_0\nabla v_{m-1})-\rho_1\nabla\cdot(\rho_1\nabla v_{m-2}),$$ 且边界条件在$m\geq1$时齐次，$m=0$时为入射场对应的边界数据。零阶解$v_0$可显式求出为平面波$v_0=A e^{i\alpha x-i\bar{\gamma}z}$，其中$\bar{\gamma}^2=k_0^2/\rho_0^2-\alpha^2$。
- (数值实现及精度，§9) 上述递归序列可借助Fourier配点法在$z$方向Chebyshev离散下高效求解，构成HOPE算法。数值实验展示，对于三层模型（上、下半空间为真空，中间为手性层），该算法在光滑手性包络$X$下达到谱精度（即指数收敛），与精确解比较误差可降至机器精度量级。
注意：定理在适当的周期Sobolev空间框架$H^s_{\mathrm{per}}(\mathcal{S}_v)$中陈述，但具体空间和范数估计的细节在§6-§8给出，本文摘录部分未完整呈现。
3. 研究方法、关键技术和核心工具
本文的核心技术路线是将手性参数的变化视为对常背景的小扰动，从而将非线性耦合的Maxwell系统转化为一系列线性解耦的椭圆边值问题，然后利用变系数椭圆方程的正则估计证明解的解析性，并以此为基础设计数值算法。
问题解耦与变量变换 在恒定的介电常数和磁导率假设下，DBF-Maxwell方程组可通过Bohren变换$\mathbf{E},\mathbf{H}\to L,R$ 解耦为独立的左、右圆极化场的Beltrami方程。特别地，当几何和材料参数为$y$不变时，矢量方程可进一步退化为两个标量Helmholtz型方程，其中系数$\rho=1\mp k_0\chi$仅在手性层内变化。这一简化是后续摄动成功实施的关键，因为标量方程远比原始Maxwell系统易于分析。但代价是不能处理变介电常数或磁导率的情形，作者明确指出此为未来工作方向。
正则摄动与递归构建 将手性参数写成$\chi=\bar{\chi}-\delta X$，则$\rho$具有仿射形式：$\rho=\rho_0+\delta\rho_1$，其中$\rho_0$为常数，$\rho_1$为$x$周期、$z$依赖的已知函数。代入Helmholtz型方程并展开待求场$v=\sum v_m\delta^m$，得到线性递归问题。该递归链的核心结构为：每个$v_m$满足一个常系数Helmholtz方程加上由低阶项构造的源$F_m$，以及齐次透明边界条件。而零阶方程则是简单的平面波解。该结构的优势在于，所有高阶方程具有相同的常系数主部算子，其基本解可显式利用Fourier-Green函数表示，从而便于进行先验估计。此外，非齐次项$F_m$的结构表明递归仅需前两阶信息，这不同于一般Volterra型高阶依赖，降低了理论分析的繁琐性。
解析性证明框架 作者宣称在§7和§8中严格建立了关于$\delta$的全局解析性和联合解析性（在$(\delta,x,z)$上）。尽管PDF摘录未给出详细估计，但根据一般摄动理论可推断其基本策略：
- 定义一族带权的Bochner-Sobolev空间，其中范数度量各阶$\delta$导数的增长。
- 在给定$\delta_0$处，将解展开为$\delta$的幂级数，其系数满足上述递归方程。
- 利用椭圆正则性推导系数的高阶能量估计，证明存在常数$C,M$使得$\|v_m\|_{H^s}\leq CM^m$，从而级数的收敛半径为$1/M>0$，给出局部解析性。
- 为获得全局实解析性（即解析延拓到包含整个实轴的复区域），需要更精巧的估计，可能利用了变换中的参数化对$\delta$的仿射依赖性以及椭圆算子在复扰动下的Fredholm性质。文中提到“By modifying the nature of our deformation slightly we will rigorously demonstrate that this region of analyticity contains a neighborhood of the entire real axis”，暗示可能采用了某种重整化或等效参数化，使得在实轴附近算子保持可逆且估计一致。
- 联合解析性则通过双变量幂级数展开和混合导数估计完成，典型工具为Cauchy估计和椭圆方程的先验界。
数值算法：Fourier-Chebyshev谱方法 数值实现基于HOPE递归，在每个$m$步求解一个带变系数源的常系数Helmholtz方程。由于横向$(x)$为周期方向，可采用Fourier配点法离散；垂直$(z)$方向上非周期，采用Chebyshev配点法。透明边界条件中的Dirichlet-Neumann算子$T_u,T_w$在Fourier空间中是对角算子，实现直接。因此，每个递归步退化为一系列独立的一维两点边值问题（每组Fourier波数一个），可用Chebyshev求导矩阵高效求解。这种逐层求解的方式避免了大型稀疏矩阵的求逆，计算量仅与总阶数成线性比例。
主要技术障碍与创新点：
- 障碍：手性材料的本构关系导致Maxwell方程组中出现额外的旋度耦合项，且材料参数的空间依赖使得传统解耦方法失效，直接离散导致大型病态系统。本文采用恒定介电、磁导假设成功解耦，将矢量问题降维为标量问题，极大简化。
- 创新点1：将手性视为对背景值的偏离，以解析性理论保证了摄动级数的全局收敛性，而不限于小参数范围。这一点通过对变形的适当修改得以实现，超越了传统正则摄动理论通常仅保证局部收敛的局限。
- 创新点2：递归公式的显式结构与联合解析性的证明，为算法的误差分析提供了理论基石：截断误差随阶数呈谱衰减。
- 创新点3：将先前在非手性介质中发展的HOPE方法推广到手性情况，并保持了算法的谱精度和稳健性，拓宽了适用范围。
4. 与之前工作的比较
本文贡献在理论、方法设定及数值策略多个维度与现有文献形成对比和推进。
理论与分析层面
- 早期严格分析手性Maxwell散射问题的工作，如Ammari & Bao [AB98, AB03, AB08] 和Zhang等人 [ZM05, ZM07]，主要关注变分框架的建立、弱解的存在唯一性及有限元/边界元方法的收敛性。这些工作一般要求材料参数满足一定的正则性条件，但并未讨论散射场关于材料参数变化的高阶正则性或解析性。本文则证明了在恒定介电和磁导假设下，散射场作为手性参数扰动$\delta$的复解析函数，且解析域包含整个实轴。这一结果不仅具有独立理论价值，也为设计高阶摄动数值方法提供了严格基础。
- 本文还进一步证明了场在$\delta$和空间变量上的联合解析性，这意味着不仅对$\delta$的Taylor展开收敛，而且其导数在空间变量上也具有良好的光滑性，保证了数值离散中谱精度的可实现性。此类结果在先前手性散射文献中未见报道。
- 与作者自己先前在非手性介质中的HOPE方法工作 [Nic20, NV25b, NV25a] 相比，本文将该框架首次推广到手性介质。非手性情形下的Helmholtz方程$\Delta v + k^2 v =0$的摄动源于几何或折射率变化，而手性情形需处理变系数项$\rho\nabla\cdot(\rho\nabla v)+k_0^2 v=0$，且系数$\rho$依赖于手性参数。本文证明了解析性和联合解析性在手性设置下依然成立，且递归公式的形式类似但包含了新的混合导数项。
模型设定与假设
- 本文的一个显著假设是介电常数和磁导率在结构内外均为常数，仅手性参数在层内变化。这一假设使得Maxwell方程组可通过Bohren变换解耦为两个独立的标量方程，从而可以利用成熟的椭圆方程理论。此条件比一般手性散射模型（如Ammari & Bao的工作允许变介电、磁导）更为严格。然而，作者指出，任何常数值均可容纳，只需满足解耦条件。对于变介电/磁导情形，矢量耦合无法避免，将是后续工作。
- 本文仅研究二维横向$y$不变结构，即$\partial_y\equiv0$，且入射波在$x$-$z$平面内传播。这进一步将标量方程简化，但损失了一般性。相比之下，许多文献（如Ammari & Bao的三维模型）处理全三维问题。作者表示三维推广是未来方向，且在前述非手性工作中已对三维情况有所探索 [NV25a]。
- 尽管如此，本文的二维横向模型在物理上对应重要的SPR传感器配置，其中光栅通常在一维方向周期性，另一维均匀，且垂直入射时$\beta=0$。因此该简化不失实际意义。
数值方法与计算效率
- 相对于有限元(FEM)、边界元(BEM)或FDTD/FDFD方法，本文的HOPE方法具有以下优势：a) 无需体积网格或边界离散，因为递归仅需求解常系数Helmholtz方程，可利用Fourier-Chebyshev谱方法快速求解；b) 精确处理透射边界条件，避免人工截断误差；c) 由于摄动级数的高阶解析性，截断误差随阶数呈谱衰减，通常少量项（如$M\thicksim10$）即可达到机器精度，远优于常规方法的代数收敛速度。
- 与经典的严谨耦合波分析(RCWA)或Fourier模态方法(FMM)相比，本文方法也基于Fourier展开，但RCWA需要求解矩阵特征值问题，对于多层或变系数情况计算量较大，且阶数增加时数值稳定性受限。HOPE方法通过递归构造解，无需特征值分解，更为高效。
- 本文数值算例针对三层模型（手性层夹在真空半空间之间）展示了谱精度，验证了理论。特别地，当手性包络函数$X(x,z)$为光滑函数时，收敛呈指数级；即使不光滑，也可能得到高代数收敛速度。这种精度优势在处理高波数或大尺寸问题时体现明显。
适用范围与局限
- 适用范围：周期性、二维、横磁或横电波照射、恒定基底介电/磁导、手性参数为小扰动包络形式。深度$h$可中等，但需满足$-1<\chi k_0<1$以避免共振。
- 局限：不能直接处理变介电、磁导，无法模拟如渐变折射率手性材料。不可拓展至完全三维场。另外，摄动级数虽然在理论上全局收敛，但在实际计算中若扰动幅度过大（$\delta$很大），级数截断可能需要很多项，效率下降；但本文声称解析域包含整个实轴，故理论上任意大$\delta$都可通过增加阶数达到所需精度。在数值实验中，作者未展示很大$\delta$的情形，这是否影响实际性能尚需进一步验证。
综上所述，本文在手性散射的数值方法中引入了一种新的分析—计算一体化框架，为谱精度算法奠定了严格数学基础，并在理论假设与计算效率之间找到了有价值的平衡点。
2. 一种基于局部刚度矩阵修正的低渗透阻隔层非匹配有限元离散裂缝模型
An unfitted finite element discrete fracture model for low-permeability barriers via local stiffness matrix modification
作者: Ziyao Xu
类别: math.NA
发布日期: 2026-08-05
arXiv ID: 2608.04431v1
摘要翻译: 有限元方法是多孔介质流动最广泛使用的离散化方法之一，其用于高导流裂缝的离散裂缝模型（DFM）已很成熟。相比之下，低渗透阻隔层长期以来难以纳入这一框架，因为它们引起连续单元无法直接表示的压力不连续性。本文通过一种封闭形式的局部刚度矩阵修正，提出了一种模拟低渗透阻隔层的线性有限元方法的简单扩展。该方法精确地保持相同的$H^1$协调$P^1$有限元空间，保留原始稀疏模式和对称正定性，无需网格贴合，并且在远离阻隔层处与标准有限元方法一致。此外，一个廉价的纯局部后处理步骤可恢复在阻隔界面处具有尖锐跳跃的不连续压力场，从而消除连续解中存在的一单元宽度的弥散。使用制造解以及文献中的二维和三维基准问题的收敛性研究证实了该方法的有效性。
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链接: http://arxiv.org/abs/2608.04431v1
1. 研究对象和背景
多孔介质中的单相流由达西定律描述, 精确预测地下水管理、油气开采和地热能开发中的流动至关重要. 天然地层普遍含有裂缝系统, 其中开裂缝或填充支撑剂的裂缝渗透率远高于围岩, 形成高导流裂缝; 而被矿物胶结的裂缝渗透率远低于围岩, 构成阻流屏障. 数值模拟须同时准确刻画两类裂缝效应.
裂缝开度通常比场尺度小几个量级, 全维表示会带来过高计算成本和病态线性系统. 离散裂缝模型 (DFM) 采用降维方法, 将裂缝处理为嵌入围岩的余维一流形 [26]. 针对高导流裂缝, 有限元 DFM 已相当成熟: 假定裂缝两侧压力连续, 降维后的模型中仅出现沿裂缝的切向扩散项, 可直接在拟合网格上向标准刚度矩阵添加低维刚度贡献 [28, 2, 17, 16, 37]. 方法不引入额外自由度, 无需显式施加界面条件, 并保持系统对称正定. 但对于低渗透屏障, 情况截然不同. 屏障效应表现为穿过界面的压力跳跃, 连续性逼近空间无法直接表示. 若不加修正, 标准有限元方法完全忽略屏障的存在. 三维基准对比 [3] 清楚表明, 强制裂缝两侧水头连续的方法 (包括 COMSOL 内置的有限元 DFM 以及 Lagrange 乘子法 [18, 30]) 均不能捕捉压力跳跃. 唯一与参考解吻合的连续有限元型方法是将屏障解析为有限厚度的等维区域 [8]. 因此, 在有限元框架内处理屏障尚未成为标准能力.
在有限元框架外, 屏障的处理研究较多, 但大多依赖网格拟合. 当单元面贴合屏障时, 降维方法可直接施加界面条件 [26]; 砂浆法可进一步处理非匹配网格 [10, 4]. 格心有限体积法将屏障阻力并入传导率 [15, 1]; 顶点中心盒子法通过打破屏障面的试探空间来恢复压力跳跃 [13, 34]. 最近还有高阶有限体积法 [22] 和内罚间断 Galerkin 法 [23] 在拟合网格下发展. 当屏障穿过背景网格时, 可用方法显著减少, 且多数引入额外局部结构以表示压力跳跃. XFEM 型方法 [7, 12, 31, 40, 5] 在切割单元上增强逼近空间以表示界面不连续性并弱施加界面条件; 浸入有限元法通过修正局部基函数引入界面条件 [38, 39]. 对一般椭圆界面问题, 界面罚法在切割单元上复制逼近空间并利用 Nitsche 项施加 Robin 跳跃条件 [21]. 这些方法虽避免全局网格拟合, 但依赖几何相关的增强或耦合构造. 嵌入式离散裂缝模型 (EDFM) 及其针对屏障的投影扩展 (pEDFM) [20, 27, 32] 是另一重要不贴体方法族, 后续变体改善了倾斜屏障和近缝流动的表示 [29, 24, 14]. RDFM 将屏障纳入混合维阻力张量并在不贴体网格上直接离散 [35, 11].
上述方法在各自设定下有效, 但均以某种方式偏离了标准 $H^1$-协调有限元方法. 本文探讨的核心问题是: 能否仅通过极小修改标准有限元方法, 即不引入额外自由度、不增强或修改逼近空间、不要求网格拟合, 来捕捉低渗透屏障? 本文给出了肯定的答案.
本文考虑的模型问题为 Martin–Jaffré–Roberts 型界面问题 [26]. 设区域 $\Omega\subset\mathbb{R}^2$ 被一条 (分片) 直屏障 $\Gamma$ 分为两侧 $\Omega^-$ 和 $\Omega^+$, 单位法向 $\mathbf{n}_\Gamma$ 由 $\Omega^-$ 指向 $\Omega^+$. 围岩渗透率 $\mathbf{K}_m(\mathbf{x})$ 对称正定, 可各向异性非均匀. 屏障阻力 $R = a/k_b > 0$, 其中 $a$ 为开度, $k_b$ 为屏障渗透率. 控制方程为
$$ -\nabla\cdot(\mathbf{K}_m\nabla p) = f \quad \text{在 } \Omega\setminus\Gamma, \qquad [\![ -\mathbf{K}_m\nabla p\cdot\mathbf{n}_\Gamma ]\!] = 0 \quad \text{在 } \Gamma, \qquad [\![ p ]\!] = R\, (-\mathbf{K}_m\nabla p\cdot\mathbf{n}_\Gamma) \quad \text{在 } \Gamma, $$并补充适当的 Dirichlet 与 Neumann 边界条件. 关键之处在于, 压力在 $\Gamma$ 上的跳跃与连续的法向流量成正比. 对应的弱形式可通过将两侧子域上的分部积分相加得到:
$$ a(p,v) := \sum_{\pm}\int_{\Omega^\pm} \mathbf{K}_m\nabla p\cdot\nabla v\,\mathrm{d}x \;+\; \frac{1}{R}\int_\Gamma [\![ p ]\!]\,[\![ v ]\!]\,\mathrm{d}s = \int_\Omega f\,v\,\mathrm{d}x - \int_{\Gamma_N} q_N\,v\,\mathrm{d}s =: \ell(v) \quad \forall\,v\in V_0, $$其中 $V = \{ v\in L^2(\Omega): v|_{\Omega^\pm}\in H^1(\Omega^\pm) \}$ 为破裂 Sobolev 空间, $V_D$ 和 $V_0$ 分别是带 Dirichlet 条件的空间与测试空间. 双线性型 $a$ 对称且在有正测度 Dirichlet 边界的条件下强椭圆, 因而等价于破裂能量泛函
$$ J(q) = \frac12\sum_{\pm}\int_{\Omega^\pm} \mathbf{K}_m\nabla q\cdot\nabla q\,\mathrm{d}x + \frac{1}{2R}\int_\Gamma [\![ q ]\!]^2\,\mathrm{d}s - \ell(q) $$的极小化. 若将试探空间限制在标准连续 $P^1$ 有限元空间 $V_h\subset H^1(\Omega)$, 则 $\frac{1}{2R}\int_\Gamma[\![ q_h ]\!]^2\,\mathrm{d}s$ 因跨单元连续而自动消失, 极小化退化为无屏障问题, 屏障完全不可见. 这正是本文拟填补的空白: 如何在保持标准 $H^1$–协调连续元空间及原有数据结构的前提下, 通过在局部刚度矩阵中注入屏障信息来恢复正确的流动响应.
2. 主要定理或主要结果
本文提出一种对标准 $P^1$ 有限元方法的极小修改, 以处理低渗透屏障. 主要结果可分为方法构造、稳定性与恢复三个层面.
方法构造 对于被单个屏障弦 $\Gamma\cap T$ 切割的三角形单元 $T$, 记其三个顶点为 $\alpha,\beta$ (被屏障隔在同侧) 和 $\gamma$ (孤立顶点). 在 $T$ 上引入标量辅助变量 $t_T$, 解释为穿过弦的压降. 定义局部离散能量泛函
$$ E_T(\mathbf{u}_T, t_T) = \frac12 (\mathbf{u}_T - t_T\mathbf{e}_\gamma)^T \mathbf{K}_T (\mathbf{u}_T - t_T\mathbf{e}_\gamma) + \frac{\ell_T}{2R} t_T^2, $$其中 $\mathbf{u}_T\in\mathbb{R}^3$ 为单元节点值向量, $\mathbf{K}_T$ 为标准块体局部刚度矩阵 $(\mathbf{K}_T)_{\mu\nu} = \int_T \mathbf{K}_m\nabla\phi_\mu\cdot\nabla\phi_\nu\,\mathrm{d}x$, $\mathbf{e}_\gamma$ 是 $\gamma$ 方向的坐标向量, $\ell_T = |\Gamma\cap T|$ 为弦长. 第一项表示扣除跳跃后的块体能量; 第二项用弦中点值近似压力跳跃罚项 $\frac{1}{R}\int_{\Gamma\cap T}[\![ q ]\!]^2\,\mathrm{d}s$. 对固定的 $\mathbf{u}_T$, 关于 $t_T$ 极小化 $E_T$ 得到驻点条件
$$ t_T^* = \frac{\mathbf{e}_\gamma^T \mathbf{K}_T \mathbf{u}_T}{\mathbf{e}_\gamma^T \mathbf{K}_T \mathbf{e}_\gamma + \ell_T/R}. $$将此最优值代回, 即得单元的凝缩能量 $\widetilde{E}_T(\mathbf{u}_T) = \min_{t_T} E_T(\mathbf{u}_T, t_T) = \frac12 \mathbf{u}_T^T \widetilde{\mathbf{K}}_T \mathbf{u}_T$, 其中修正的局部刚度矩阵为秩一修正:
$$ \widetilde{\mathbf{K}}_T = \mathbf{K}_T - \frac{(\mathbf{K}_T\mathbf{e}_\gamma)(\mathbf{K}_T\mathbf{e}_\gamma)^T}{(\mathbf{K}_T)_{\gamma\gamma} + \ell_T/R}. \tag{3.7} $$对于屏障网络, 当多条屏障在单元内相交或汇合时, 单元被分为 $m+1$ 个子区域 $T_0,T_1,\dots,T_m$. 任选 $T_0$ 为基准区, 为每个非基准区 $T_\rho$ 设置局部压力偏移 $t_\rho$, 并记 $\mathbf{t}_T=(t_1,\dots,t_m)^T$, $t_0:=0$. 定义区域-顶点关联矩阵 $\mathbf{E}_T\in\mathbb{R}^{3\times m}$, 使得 $\mathbf{E}_T\mathbf{t}_T$ 的分量 $i$ 是含顶点 $i$ 的子区域的偏移. 块体能量中使用修正节点向量 $\mathbf{u}_T-\mathbf{E}_T\mathbf{t}_T$. 屏障臂上的压力跳跃惩罚由加权图拉普拉斯矩阵
$$ \mathbf{P}_T = \sum_{\text{臂 } a} \frac{\ell_a}{R_a} \mathbf{d}_a\mathbf{d}_a^T, \quad \mathbf{d}_a = \mathbf{e}_{\rho_a^+} - \mathbf{e}_{\rho_a^-}, $$编码, 其中 $\ell_a, R_a$ 分别为臂长与阻力. 局部单元能量为
$$ E_T(\mathbf{u}_T,\mathbf{t}_T) = \frac12 (\mathbf{u}_T - \mathbf{E}_T\mathbf{t}_T)^T \mathbf{K}_T (\mathbf{u}_T - \mathbf{E}_T\mathbf{t}_T) + \frac12 \mathbf{t}_T^T \mathbf{P}_T \mathbf{t}_T. \tag{3.8} $$对其极小化得到局部偏移
$$ \mathbf{t}_T^* = (\mathbf{E}_T^T \mathbf{K}_T \mathbf{E}_T + \mathbf{P}_T)^{-1} \mathbf{E}_T^T \mathbf{K}_T \mathbf{u}_T, \tag{3.9} $$对应的修正局部刚度矩阵为
$$ \widetilde{\mathbf{K}}_T = \mathbf{K}_T - \mathbf{K}_T \mathbf{E}_T (\mathbf{E}_T^T \mathbf{K}_T \mathbf{E}_T + \mathbf{P}_T)^{-1} \mathbf{E}_T^T \mathbf{K}_T. \tag{3.10} $$公式 (3.10) 在单弦情形成即退化为 (3.7).
方法定义: 在所有被屏障切割的单元 $T$ 上, 用 $\widetilde{\mathbf{K}}_T$ 取代 $\mathbf{K}_T$ 装配全局刚度矩阵; 未切割单元与载荷向量保持不变. 最终求解的离散解 $p_h$ 仍为标准连续 $P^1$ 有限元函数. 实现时仅需在单元组装循环中对 $\mathcal{O}(1/h)$ 个切割单元执行上述修正, 不更改数据结构、稀疏模式及线性求解器. 方法还可自然兼容高导流裂缝的 DFM 处理: 在裂缝面上添加切向刚度贡献即可, 屏障修正仅与之并行作用于不同单元.
稳定性 (命题 3.1): 对任何对称正定 $\mathbf{K}_m$, 任何三角剖分及任何 $R,\ell_T>0$, $\widetilde{\mathbf{K}}_T$ 对称半正定, 且其零空间与 $\mathbf{K}_T$ 相同, 为常数向量的倍数. 全局刚度矩阵在施加 Dirichlet 边界后对称正定.
后处理压力恢复: 在求得 $p_h$ 后, 利用局部反代恢复各切割单元的最优辅助变量 $t_T^*$ 或 $\mathbf{t}_T^*$. 对于单弦情况, 重构的破裂压力定义为
$$ \widehat{p}_h = \begin{cases} p_h - t_T^* \phi_\gamma, & \text{在子区域 } T_{\alpha\beta},\\ p_h + t_T^* (1-\phi_\gamma), & \text{在子区域 } T_\gamma, \end{cases} $$其中 $\phi_\gamma$ 为顶点 $\gamma$ 的局部基函数. 此重构在节点处与 $p_h$ 一致, 在屏障弦上产生尖锐压力跳跃 $t_T^*$, 且在两侧具有相同的块体梯度. 对于多区情形, 类似地按子区域偏移进行重构. 重构场是使离散能量 (4.2) 在切割单元上精确成立的破裂场.
数值收敛性: 利用制造解进行的收敛性测试显示, 连续解 $p_h$ 在 $L^1(\Omega)$ 范数下达到 $\mathcal{O}(h)$ 收敛, 在 $L^2(\Omega)$ 范数下达到 $\mathcal{O}(h^{1/2})$ 收敛, 这与连续逼近空间以最佳速率逼近间断解的理论预期一致. 重构 $\widehat{p}_h$ 的跳跃误差亦随网格加密而减小. 二维与三维文献基准测试进一步验证了方法的有效性.
3. 研究方法、关键技术和核心工具
本文核心贡献在于通过局部能量极小化与变量消去, 将屏障的压力跳跃效应压缩为刚度矩阵的闭式秩更新, 从而以最轻量级的方式在标准 $H^1$–协调有限元框架内植入屏障. 整体技术路线遵循以下步骤.
1. 破裂能量泛函的离散与局部辅助变量的引入
问题的正确数学结构是破裂 Sobolev 空间上的能量泛函 $J(q)$. 直接使用连续元空间时, 跳跃罚项 $\frac{1}{2R}\int_\Gamma[\![ q ]\!]^2\,\mathrm{d}s$ 被自动置零. 作者的核心洞察是: 在离散层面, 通过在每个切割单元上引入辅助变量来暂时恢复缺失的跳跃信息, 然后在全局求解前将其凝聚掉, 便可保留其效应. 这比在全局系统添加约束或增强空间更简捷. 具体地, 对单个切割三角形, 构造的局部能量 $E_T(\mathbf{u}_T,t_T)$ 将块体能量作用在扣除了跳跃 $t_T$ 后的修正节点向量上, 并加上弦上跳跃惩罚的离散版本. 这一设计的合理性在于: 若事后将辅助变量解释为局部压降, 则修正后的节点向量可视为“若无屏障时的压力”, 故在子域上积分时采用标准梯度是自然的. 惩罚项取 $\frac{\ell_T}{2R}t_T^2$ 相当于用弦中点跳跃值代替积分, 对分片线性函数而言是准确的一维 Simpson 近似.
2. 局部凝缩与矩阵修正
$E_T(\mathbf{u}_T,t_T)$ 关于 $t_T$ 是一元二次函数, 通过其驻点条件直接将 $t_T$ 表示为 $\mathbf{u}_T$ 的线性泛函. 代入后得到仅含 $\mathbf{u}_T$ 的二次型, 其 Hessian 即为修正刚度 $\widetilde{\mathbf{K}}_T$. 这本质是对扩展系统
进行 Schur 补运算, 得到 $\widetilde{\mathbf{K}}_T = \mathbf{K}_T - \mathbf{K}_T\mathbf{e}_\gamma(\mathbf{e}_\gamma^T\mathbf{K}_T\mathbf{e}_\gamma + \ell_T/R)^{-1}\mathbf{e}_\gamma^T\mathbf{K}_T$. 由于 $\mathbf{e}_\gamma$ 是第 $\gamma$ 个坐标向量, 这恰好是一个秩一修正. 所得矩阵保持对称, 且原有零空间 $\operatorname{span}\{\mathbf{1}\}$ 不变. 若 $\ell_T/R>0$, 修正会从系统中抽走与孤立顶点关联的能量, 其极限 $R\to\infty$ 对应 $\gamma$ 行、列完全归零, 恰如该顶点被完全密封. 作者将这一简单公式包装为对标准有限元代码的轻量级修改: 只需在单元循环中, 对每个切割单元执行一次
3. 屏障网络的推广
当单元内部存在多条屏障 (例如交叉或 T 型连结点) 时, 单一跳跃变量不足以刻画所有子区域间的压差. 作者将该问题建模为局部多区图: 子区域为图节点, 屏障臂为带权边. 通过为每个非基准子区域引入一个压力偏移, 可统一写出修正节点向量 $\mathbf{u}_T - \mathbf{E}_T\mathbf{t}_T$, 并用所有臂的二次跳跃惩罚之和形成图拉普拉斯矩阵 $\mathbf{P}_T$. 此时局部能量 $E_T(\mathbf{u}_T,\mathbf{t}_T)$ 关于 $\mathbf{t}_T$ 为二次型, 极小化给出 $\mathbf{t}_T^* = (\mathbf{E}_T^T\mathbf{K}_T\mathbf{E}_T + \mathbf{P}_T)^{-1}\mathbf{E}_T^T\mathbf{K}_T\mathbf{u}_T$. 由于 $\mathbf{P}_T$ 在 $\mathbf{t}_T$ 上正定 (基准子区域接地), 矩阵求逆良定. 代入后得到一般修正公式 $\widetilde{\mathbf{K}}_T = \mathbf{K}_T - \mathbf{K}_T\mathbf{E}_T(\mathbf{E}_T^T\mathbf{K}_T\mathbf{E}_T + \mathbf{P}_T)^{-1}\mathbf{E}_T^T\mathbf{K}_T$. 这一构造虽看似复杂, 但仍完全局部, 且只需处理极小的 $m\times m$ 矩阵, 计算代价与单元内段数成正比. 作者还指出一种针对全弦情况的“独立弦捷径”: 依次对各条完整弦应用秩一更新, 在不含末端分支的多数单元中可节省运算. 该捷径在 T 型交界处会因无法识别不完整弦而产生微小泄漏, 但提供了计算效率与精度的折衷.
4. 后处理重构
连续解 $p_h$ 本身在屏障处呈一个单元宽度的涂抹, 这是用连续空间逼近间断函数的必然结果. 但全局求解完成后, 作者并未丢弃局部辅助变量. 通过局部反代公式 (3.5) 或 (3.9) 回收 $t_T^*$, 并藉此对连续解进行分片常值偏移, 得到重构场 $\widehat{p}_h$, 它能严格恢复子区域间的跳跃. 该步骤仅为每个切割单元的独立后处理, 无需再次求解整体系统. 重构场不仅物理上更合理 (尖锐界面), 且如前所述, 它使离散能量 (4.2) 在单元层次上精确对应连续能量, 从而保证了方法的一致性.
5. 理论基础与实现简便性
整个方法的正确性建立在两个原则之上: (1) 辅助变量是全局能量极小化的一部分, 因仅出现在各自单元中, 可先局部极小化再装配全局系统; (2) 局部极小化不改变全局刚度矩阵的对称正定性及稀疏模式. 与依赖界面几何构造或增强空间的方法不同, 本文的修正仅需单元级别的几何信息 (弦长 $\ell_T$, 孤立顶点识别, 以及多区情形下子区域邻接关系). 这使得方法易于集成到现有有限元 DFM 代码中: 高导流裂缝作为切向刚度添加至全局矩阵, 屏障则仅通过切割单元的局部秩修正融入, 两者互不干扰, 数据结构和线性求解器完全复用. 附录进一步给出电阻网络类比 (附录 A) 和三维推广 (附录 B), 展示方法的广泛适用性.
4. 与之前工作的比较
本文提出的方法在低渗透屏障的有限元模拟领域作出了系统性的简化推进, 与既有方法在网格需求、空间修改、全局特性等方面存在显著差异. 以下从若干维度详述.
与需要拟合网格的方法对比
众多传统方法依赖元件面贴近屏障界面, 例如基于 Lagrange 乘子的降维方法 [26]、砂浆法 [10, 4]、格心有限体积法 [15, 1] 以及顶点中心盒子法 [13, 34] 等. 这类方法的主要局限在于, 在地质构造复杂或褶皱发育时, 生成贴合网格可能非常困难, 尤其三维情形中网格生成常成为瓶颈. 本文方法使用完全不贴体的背景网格, 只需知道哪些单元被屏障切割及切割段的长度, 免去了网格重新划分的负担, 同时保持了与标准相容有限元相同的装配流程和数据结构.
与增强空间或增加自由度的非贴体方法对比
非贴体网格下的典型策略包括 XFEM 型方法 [7, 12, 31, 40, 5]、浸入有限元法 [38, 39] 和界面罚法 [21]. 这些方法通过在切割单元上局部丰富逼近空间 (如加入 Heaviside 或阶梯函数) 来显式表示间断, 并借助 Nitsche 型弱形式或修改基函数施加界面条件. 本文方法截然不同: 它完全不修改 $P^1$ 空间, 不引入任何新自由度或局部特殊函数. 屏障信息完全被闭式吸收进局部刚度矩阵之中, 因而可以在不改动任何数据结构的条件下实现. 这是对现有有限元代码侵入性最小的处理方式, 只需在组装循环内插入数行代码. 代价是计算得到的原始解为连续涂抹场, 收敛率仅为 $\mathcal{O}(h)$ ($L^1$) 和 $\mathcal{O}(h^{1/2})$ ($L^2$), 对于直接获得尖锐间断的增强方法可能会略低; 但通过一次廉价的后处理, 即能还原精确的间断位置和跳跃大小, 使重构场具有与增强方法类似的界面质量.
与 pEDFM 等嵌入式方法的对比
嵌入式离散裂缝模型 (EDFM [20, 27]) 及其屏障投影扩展 (pEDFM [32]) 通过修改非相邻连接传导率来隐式表示裂缝/屏障, 但这类方法主要根植于有限体积框架, 其数学结构与有限元本质不同. pEDFM 在倾斜屏障和近裂缝流动表示上虽经后续改进 [29, 24, 14], 但其连续压力投影或局部精细求解依然在某种程度上偏离了标准有限元的伽辽金正交性. 本文方法完全基于能量泛函和标准的 $H^1$ 协调有限元离散, 因此保留了对称正定性、最优稀疏性以及与标准有限元一致的数学性质. 此外, pEDFM 的传导率修改会改变原本矩阵的连接图 (例如, 屏障可能切断某些连接), 而本文的修正保持与未切割单元完全相同的稀疏模式.
与连续元 DFM 中高导流裂缝的结合
在有限元 DFM 家族中, 高导流裂缝的处理已通过添加切向刚度项得到完美解决 [28, 2, 17, 37]. 然而, 同框架下处理屏障一直是个难题, 因为 $H^1$ 协调天然跨界面连续. 本文的局部刚度修正方法恰好填补了这一空缺: 高导流裂缝的贡献 (正定附加) 和屏障贡献 (秩一或低秩负修正) 可在同一个组装循环内针对不同裂缝单元独立完成, 互不矛盾. 这使得一个单一的有限元 DFM 代码可同时统一处理传导裂缝与阻流屏障, 保持系统的对称正定性, 无需引入额外变量或耦合机制.
作者未显式展开定量比较的方面
论文通过数值实验 (制造解收敛测试与三维基准 [3] 对比) 显示了方法的有效性, 但未与其他不贴体方法 (如 XFEM、pEDFM) 提供定量的误差–自由度对照或成本分析. 从提供的内容中无法确认本文方法相对于那些方法在计算效率 (组装、求解)、对网格扭曲的敏感度、以及三维复杂网络下的鲁棒性上的具体优劣. 此外, 文中虽指出独立弦捷径在 T 型交界处会引入微小泄漏, 但未给出该泄漏对全局解的量化影响分析. 这些或在后续研究中有待阐明.
了解领域论文（摘要翻译）
1. 面向非定常不可压Navier–Stokes方程的人工压缩性物理信息神经网络
An Artificial-Compressibility Physics-Informed Neural Network for the Unsteady Incompressible Navier–Stokes Equations
作者: Aytekin Çibik
类别: physics.flu-dyn, math.NA
发布日期: 2026-08-04
arXiv ID: 2608.04191v1
链接: http://arxiv.org/abs/2608.04191v1
摘要翻译
中文标题: 面向非定常不可压Navier–Stokes方程的人工压缩性物理信息神经网络
摘要翻译: 我们研究了一种用于非定常二维不可压缩Navier–Stokes方程的物理信息神经网络（PINN），其中刚性的无散约束被替换为由单个标量参数$\eps$控制的人工压缩性（AC）松弛。该松弛重新引入了压力时间导数，将微分-代数约束转换为PINN可以直接最小化的普通残差。在具有封闭形式非定常解的Taylor–Green涡上，我们量化了$\eps$的影响：残差散度的尺度为$\eps\,|\partial_t p|$，因此较大的$\eps$会同时增大散度和速度误差，而两者均随$\eps$减小而单调递减并趋于饱和。在$Re=100$的圆柱尾流中，普通的前向AC-PINN会崩溃到稳态对称分支，并且无法再现卡门涡脱落；从边界层解析的有限元参考解（通过解析分离剪切层，我们将其斯特劳哈尔数$0.176$拉近至$0.164$–$0.172$的文献范围，尽管仍略高于该范围）中同化几百个稀疏速度传感器，则可在尾流中以$7\%$的误差恢复非定常涡街，其脱落频率与同一参考解的偏差在$3\%$以内——这一界限由参考解自身的保真度所设定，而非对真实流动的独立验证。
不相关论文（基本信息）
1. 扩散过程的精确模拟及对数凹采样的改进算法
Exact simulation of diffusions and improved algorithms for log-concave sampling
作者: Fan Chen, Sinho Chewi, Alexander Rakhlin, Matthew S. Zhang
类别: cs.DS, math.NA, math.PR, math.ST
发布日期: 2026-08-05
arXiv ID: 2608.05022v1
链接: http://arxiv.org/abs/2608.05022v1
摘要: We study exact simulation of diffusions via rejection sampling on path space using unbiased estimators of the density ratio obtained from Girsanov’s theorem. When applied to the underdamped Langevin diffusion, it yields an algorithm for sampling from a strongly log-concave and log-smooth distribution with condition number $κ$, in dimension $d$, to accuracy $\varepsilon$ in Rényi divergence, in $\widetilde O(κ^{2/3} d^{1/3}\,\mathrm{polylog}(1/\varepsilon))$ queries. Under a third derivative bound, the dimension dependence improves to $d^{1/5}$. This improves substantially over the prior state-of-the-art complexity of $\widetilde O(κd^{1/2}\,\mathrm{polylog}(1/\varepsilon))$ for the Metropolis-adjusted Langevin algorithm, and over the $d^{1/4}$ dimension dependence of Metropolized Hamiltonian Monte Carlo under the same third derivative bound. We also present applications to the mirror Langevin diffusion, and for obtaining Fisher information bounds in the non-log-concave case.
2. 数据生成的等离子体波方程内部解：误差界与二维数值实验
Data generated internal solutions for the plasma wave equation: error bounds and numerical experiments in two dimensions
作者: Vladimir Druskin, Shari Moskow, Mikhail Zaslavsky
类别: math.NA
发布日期: 2026-08-05
arXiv ID: 2608.04989v1
链接: http://arxiv.org/abs/2608.04989v1
摘要: We consider the computation of internal solutions for a time domain plasma wave equation with an unknown potential $q$ from boundary response data. The internal solutions are computed by transforming known background snapshots using the Cholesky decomposition of the data-driven Gramian, or mass matrix. It was recently shown that in one dimension these data generated internal solutions converge in $L^2$ at order $\sqrtτ$ for well chosen initial waves. Here we study the internal solution reconstruction in two dimensions, where a multiple input/multiple output (MIMO) setup and a block Gramian are needed, with the number of boundary sources increasing as the time sampling is refined. We show that the general error bound carries over to this setting: the distance between the data generated solutions and the best causal approximation from background snapshots is controlled by the best approximation mass matrix mismatch. Numerical refinement studies on a square domain measure the convergence of the data generated solutions alongside the best causal approximation from the background snapshots, with the relative errors appearing to go to zero at rate $\sqrtτ$, and the absolute errors going to zero in $L^1$. We also show that the solution reconstructions remain accurate for high contrast composite media. Finally, since the mass matrix assembled from noisy response data can fail to be positive definite, regularization is needed; comparing a diagonal shift with an eigenvalue floor, we find the floor more robust, with reconstructions that remain more accurate than the unperturbed background field with noise that is up to ten percent of the root mean square data amplitude.
3. 一种面向乘性Gamma噪声下图像恢复的基于几何的深度均衡模型
A geometry-based deep equilibrium model for image restoration under multiplicative Gamma noise
作者: Shengkun Yang, Luca Ratti, Zhichang Guo
类别: math.NA, cs.LG
发布日期: 2026-08-05
arXiv ID: 2608.04944v1
链接: http://arxiv.org/abs/2608.04944v1
摘要: We propose a deep learning framework for image restoration from images degraded by both multiplicative Gamma noise and blur. Unlike conventional deep equilibrium (DEQ) models that rely on implicit neural regularization, the proposed method learns an explicit and interpretable regularizer parameterized by geometric priors associated with surface area and mean curvature. To minimize the resulting variational model, we develop a mirror descent algorithm tailored to the commonly used Gamma-noise fidelity terms. Leveraging the Kurdyka-Lojasiewicz property for functions defined in $o$-minimal structures, we establish the global convergence of the generated iterates to a critical point. Experimental results on both grayscale and color image restoration demonstrate that the proposed method consistently outperforms representative model-based approaches while achieving performance comparable to state-of-the-art DEQ models based on implicit regularization, despite requiring substantially fewer trainable parameters.
4. 一种利用对称性强制简并进行谱拟合的换位门
A commutant gate for spectral fitting through symmetry forced degeneracy
作者: Stelios Savva
类别: math.NA, nucl-th, physics.comp-ph
发布日期: 2026-08-05
arXiv ID: 2608.04903v1
链接: http://arxiv.org/abs/2608.04903v1
摘要: Learned spectral models fail at symmetry forced degenerate sectors for two distinct reasons. Where symmetry forces levels to coincide exactly, the per level observable one would normally fit is not well defined, since every unit vector of that shared space is an eigenvector; and near a symmetry protected crossing, the eigenvector observable gradient carries a factor 1/(lambda_i - lambda_j) that is genuinely singular as the gap closes. The usual response is to regularize the divergence or threshold the gap, and both carry a real cost: a fixed gap cannot both protect a forced multiplet and keep two genuinely distinct levels apart. We show a different fix, on synthetic operator families with a known symmetry answer key. A gate reads the symmetry structure directly from the observed operators, as the linear commutant of the family, one singular value decomposition nullspace, following the simultaneous block diagonalization of Maehara and Murota. Block identity is read from the centre of that commutant rather than from eigenvalue clustering, which makes the forced versus accidental distinction structural rather than metric, and the objective switches between a projector trace through forced blocks and a per level target elsewhere. Under operator estimation noise the gate classifies correctly to epsilon of about 0.3, where energy clustering already fails by 0.02. Gated fitting reaches the truth at machine precision in both the symmetric and the symmetry breaking regime, in the latter converging to a truth that is a singularity of the ungated objective, and observable bias sits at the noise floor. Validation off the regular representation, where multiplicity and dimension separate, eliminated two defective estimators that all regular representation tests had passed. The demonstrated object is a gated estimator; a full parametric matrix model, with the matrices learned, is the next experiment.
5. 非光滑优化的原始-对偶多重网格方法
Primal-dual multigrid methods for nonsmooth optimization
作者: Felipe Guerra, Tuomo Valkonen
类别: math.OC, math.NA
发布日期: 2026-08-05
arXiv ID: 2608.04848v1
链接: http://arxiv.org/abs/2608.04848v1
摘要: In optimization, one often encounters problems of the form $\min_x F(x)+E(x)+G(Kx)$. In this work, we combine primal-dual algorithms with multigrid techniques for their solution. To link the the fine-grid and coarse-grid problems problems, we introduce a nonsmooth primal-dual coherence condition, and an efficient partially linearized line search procedure. Our work is motivated by total variation regularized inverse imaging problems, on which we demonstrate the efficacy of the method, being able to solve problems not previously possible with forward-backward multigrid methods.
6. 一种针对具有测度值血管边界源的小胶质细胞-淀粉样蛋白趋化模型的有限体积格式的收敛性分析
Convergence Analysis of a Finite-Volume Scheme for a Microglia–Amyloid Chemotaxis Model with Measure-Valued Vascular Boundary Sources
作者: Elmahdi Erraji
类别: math.AP, math.NA
发布日期: 2026-08-05
arXiv ID: 2608.04785v1
链接: http://arxiv.org/abs/2608.04785v1
摘要: We study a parabolic-parabolic chemotaxis system motivated by microglial recruitment toward an amyloid-$β$-associated signal in Alzheimer’s disease. The signal is subject to a nonnegative Radon measure-valued Neumann influx on a vascular portion of the boundary, while microglial cells respond to a nonlocal spatial average of the signal. For a fixed sensing length $σ>0$, the chemotactic velocity is $b_σ[v]=\nabla K_σ[v]$. For every fixed $σ>0$, the nonlocal operator maps finite signal mass into a bounded spatially Lipschitz velocity field. We introduce a weak-solution concept adapted to the low regularity induced by the boundary measure and construct a fully implicit upwind finite-volume approximation in which the boundary source is discretized through its exact mass on each boundary face-time cell. We establish existence and positivity of discrete solutions, together with uniform mass, energy, discrete-gradient, and compactness estimates. Finally, we prove subsequential convergence of the discrete solutions toward a nonnegative weak solution of the continuous problem.
7. 耦合裂纹与位错的相场模型：发射、钝化及耗散增韧的必要性
A phase field model of coupled crack and dislocations: emission, blunting, and the necessity of dissipative toughening
作者: Khanh Chau Le, Thi My Kieu Tran
类别: cond-mat.mtrl-sci, math-ph, math.NA, physics.class-ph
发布日期: 2026-08-05
arXiv ID: 2608.04781v1
链接: http://arxiv.org/abs/2608.04781v1
摘要: We propose a phase field model of a macrocracked single crystal in which the crack and the geometrically necessary dislocations descend from a single energy functional. Energy minimization alone then decides dislocation nucleation, through an integral criterion evaluated in closed form along slip chords. The criterion yields a size effect inaccessible to point-wise strength conditions: a grain-size-dependent yield stress. With slip suppressed the model reproduces Griffith fracture; with fracture suppressed, the nucleation load measured by the full non-smooth solver agrees with the closed-form nucleation criterion to four percent. The coupled computations produce a two-stage response: at loads an order of magnitude below cleavage, dislocation bands emitted from the notch tip blunt and shield it, raising the initiation load; once the crack grows, the bands heal; a compact cluster of like-signed dislocations travels with the tip, its canceling partner walls pinned at the grain boundary, and the dissipated fracture resistance equals the elastic one. In the purely energetic, dissipationless limit, emission shields the crack but does not toughen it; toughening requires dissipation, incorporated in the sequel through the threshold resistance to dislocation motion.
8. 基于泛函型误差估计子的自适应边界元法的最优收敛性
Optimal convergence of adaptive BEM driven by functional-type error estimators
作者: Maximilian Brunner, Alexander Freiszlinger, Dirk Pauly, Dirk Praetorius
类别: math.NA
发布日期: 2026-08-05
arXiv ID: 2608.04556v1
链接: http://arxiv.org/abs/2608.04556v1
摘要: In the present work, we derive functional upper bounds for the potential error arising from boundary element discretizations of the Laplace-Dirichlet problem. These bounds are based on local auxiliary problems on patches of boundary vertices and the resulting a posteriori error estimator is shown to be locally equivalent to the well-studied residual error estimator. This equivalence result allows us to prove R-linear convergence of the functional a posteriori error estimator and, together with a suitable mesh-refining strategy, to establish that the potential error as well as the functional error estimator converge with optimal rates with respect to the number of boundary elements. Numerical experiments affirm the theoretical findings and illustrate the practical performance of the related adaptive algorithm driven by the proposed functional error estimator.
9. 一种用于非线性多模态状态估计的自适应分裂-组合高斯混合滤波器
An adaptive split-combine Gaussian mixture filter for nonlinear and multimodal state estimation
作者: San Kim, Won Chang, Daniel B. Forger, Dae Wook Kim
类别: math.NA, stat.ML
发布日期: 2026-08-05
arXiv ID: 2608.04430v1
链接: http://arxiv.org/abs/2608.04430v1
摘要: Filtering combines model predictions with measurements to estimate the probability density function (PDF) of a system state over time. The PDF often becomes highly asymmetric and even multimodal in nonlinear systems with oscillatory or chaotic dynamics. Such non-Gaussian features violate the single-Gaussian assumption underlying Kalman-type filters. To address this problem, Gaussian mixture filtering has been proposed. However, accurately propagating mixture components and adaptively adjusting their number and weights over time remain open challenges. Here, we develop an adaptive split-combine Gaussian mixture filter (AMF) that estimates the time evolution of asymmetric and multimodal PDFs by adaptively splitting and combining Gaussian particles without auxiliary online numerical optimization. Notably, the proposed splitting method guarantees a reduction in variance along a target level-set-point direction of a Gaussian particle. This enables accurate and efficient propagation of particles. We show that AMF consistently outperforms various baseline filters across diverse benchmarks, including single and coupled slow-fast Van der Pol oscillators and the Lorenz attractor. We also propose a parallel implementation of AMF, allowing high-fidelity PDF estimation with practical computational cost.
10. 用于异步特征空间计算的格拉斯曼流形上的增量聚合
Incremental Aggregation on the Grassmannian for Asynchronous Eigenspace Computation
作者: Xiaolu Wang, Jiang Hu, Hoi-To Wai
类别: math.OC, eess.SP, math.NA, stat.ML
发布日期: 2026-08-05
arXiv ID: 2608.04406v1
链接: http://arxiv.org/abs/2608.04406v1
摘要: We study asynchronous optimization for finite-sum eigenspace computation in heterogeneous distributed systems. The theoretical foundations for asynchronous eigenspace computation remain scarce, with existing approaches offering limited coverage of dynamics directly on the Grassmannian under stale information. In this paper, we propose a Grassmannian incremental aggregation method that refreshes only arriving components and reuses cached gradients, retaining low per-update cost without global synchronization. The method employs an extrinsic polar update that preserves the intrinsic subspace geometry without requiring parallel transport of stale tangent vectors. Our analysis establishes a tight angle-dependent gradient-dominance characterization of the objective and a basin-invariance property for stale aggregated updates. These yield two-phase linear convergence, comprising an explicit broad-basin regime and a sharper local regime, with constants controlled by component spectral spreads. Experiments on serial and distributed PCA demonstrate improved sample efficiency and wall-clock convergence over representative baselines.
11. 精化联合双对角化方法及用于大规模GSVD计算的隐式重启算法
The Refined Joint Bidiagonalization Method and an Implicitly Restarted Algorithm for Large GSVD Computations
作者: Kaixiao Fang, Zhongxiao Jia
类别: math.NA
发布日期: 2026-08-05
arXiv ID: 2608.04360v1
链接: http://arxiv.org/abs/2608.04360v1
摘要: We make a convergence analysis on the joint bidiagonalization (JBD) method that computes several extreme generalized singular value decomposition (GSVD) components of a regular matrix pair $\{A,L\}$, and show that the right and left Ritz vectors obtained by it may converge erratically and even may fail to converge, while Ritz values converge. These convergence results hold for a class of general Rayleigh–Ritz projection methods for the GSVD problem under the hypothesis that the deviation of a desired right generalized singular vector from the right subspace tends to zero. We prove the interlacing property of Ritz values and generalized singular values, and extend it to the generalized singular values of $\{A,L\}$ and the matrix pairs consisting of subsets of its columns. To overcome the irregular convergence or possible non-convergence of the JBD method, we nontrivially extend the refined Rayleigh–Ritz projection for the eigenvalue problem to the GSVD problem, and propose a refined JBD (RJBD) method that replaces the right Ritz vectors by new approximations, called the right refined Ritz vectors, satisfying certain residual optimality; we define new approximate left generalized singular vectors, called the left refined Ritz vectors. We prove that the left and right refined Ritz vectors unconditionally converge under the same hypothesis. We extend the implicit restarting scheme to the RJBD method, and develop an implicitly restarted RJBD algorithm with the refined shifts proposed. Numerical experiments illustrate that the new algorithm is at least competitive and often considerably more efficient than the implicitly restarted JBD algorithm.
12. 基于广义矩阵值核的向量场插值的误差估计
Error estimates for vector field interpolation based on generalized matrix-valued kernels
作者: Zhengjie Sun, Lishuo Dong, Leevan Ling
类别: math.NA
发布日期: 2026-08-05
arXiv ID: 2608.04313v1
链接: http://arxiv.org/abs/2608.04313v1
摘要: Matrix-valued kernels provide a flexible framework for approximating vector fields from scattered data, especially when structural constraints such as divergence-free or curl-free conditions must be preserved. Classical potential-based constructions enforce these constraints naturally, but they typically require the generating scalar function to possess relatively high smoothness. We develop an operator-based framework for constructing div-free and curl-free matrix-valued kernels using integral and differential operators, which substantially relaxes the regularity requirements of the potential approach. Using dimension-walking techniques, we show that the resulting native spaces are norm-equivalent to appropriate vector-valued Sobolev spaces. Another main contribution of the paper is a sharp error analysis for the corresponding kernel matrix-valued interpolation problem. We derive direct Sobolev error estimates that allow fractional regularity of the target field, and we establish Bernstein-type inequalities for the associated kernel trial spaces. These results lead to a complete inverse theorem. We also investigate stability by proving lower bounds for the smallest eigenvalues of the interpolation matrices. Numerical experiments are included to verify the theoretical results.
13. 局部最大鸭解阈值中的龙格-库塔偏差：一个链树条件
Runge–Kutta Bias in Local Maximal-Canard Thresholds: A Chain-Tree Condition
作者: Haibo Lu
类别: math.DS, math.NA
发布日期: 2026-08-05
arXiv ID: 2608.04304v1
链接: http://arxiv.org/abs/2608.04304v1
摘要: A fixed-step Runge–Kutta discretization may approximate ordinary trajectories accurately while displacing the parameter at which selected attracting and repelling slow manifolds form a maximal canard. For the minimal polynomial model $x'=x^2-y+xy$, $y'=ε(x-λ)$, we compare actual section-defined thresholds of the flow and its exact Runge–Kutta map, uniformly over all admissible choices of the attracting and repelling slow manifolds. We first expose the mechanism on a symmetric, A-stable two-stage slice and then prove it on a compact two-parameter domain of the complete self-adjoint two-stage family and on a separate self-adjoint three-stage family. The order-three B-series defect has independent bushy- and chain-tree coordinates, but the threshold functional detects only the chain-tree defect. Thus class-wide leading-bias cancellation is equivalent to $b^T A c=1/6$, while the other classical order-three condition may remain unsatisfied. On the sharp model, the leading coefficients are $(12ρ-1)/32$ and $(24ν-1)/128$ for the declared two- and three-stage families. Uniformly as $ε=r^2\to0$ in $r^2\le h\le r^{3/4}$, these coefficients multiply $h^2ε^2$ in the displacement between nearby actual map and flow roots. The proof transports a finite Runge–Kutta defect to an actual section-defined root through exponential selection shielding, an exact common-target recurrence, and discrete Gaussian Melnikov quadrature. A right-fold van der Pol formula provides a classical calibration. A rational Rosenzweig–MacArthur specialization gives an actual local selected-threshold law in the original source clock, and a joint-$(r,h)$ finite-boundary proxy computation, independently checked at 192-bit precision, illustrates the predicted sign reversal and leading-bias cancellation.
14. 一种用于双曲型守恒律的加权迎风向量动力学格子玻尔兹曼方法
A Weighted Upwind Vector Kinetic Lattice Boltzmann Method For Hyperbolic Conservation Laws
作者: Michael W. Brown, Jehanzeb Chaudhry, John N. Shadid
类别: math.NA
发布日期: 2026-08-04
arXiv ID: 2608.04283v1
链接: http://arxiv.org/abs/2608.04283v1
摘要: Vector kinetic lattice Boltzmann (VKLB) methods have recently emerged as a promising framework for solving hyperbolic partial differential equation (PDE) systems. VKLB discretizes Boltzmann-type equations using a discrete set of lattice velocities, enforces discrete moment constraints, and carefully defines equilibrium distribution functions. In this work, we introduce novel upwinded equilibrium distribution functions constructed from conservation variables and numerical fluxes derived via continuous flux vector splitting based on the eigendecomposition of the flux Jacobian. This formulation enables the weighted-upwind VKLB equilibrium to be applied broadly to general hyperbolic systems. The method is verified on a set of challenging hyperbolic systems that includes the shallow water, Euler and ideal magnetohydrodynamics (MHD) equations. The proposed method demonstrates improved stability, reduced error norms, and sharper shock resolution across increasingly complex verification and benchmark problems.
15. 具有逐点控制约束的点跟踪椭圆分布最优控制问题的一种新误差分析
A New Error Analysis for a Point Tracking Elliptic Distributed Optimal Control Problem with Pointwise Control Constraints
作者: Susanne C. Brenner, Li-yeng Sung
类别: math.NA, math.OC
发布日期: 2026-08-04
arXiv ID: 2608.04281v1
链接: http://arxiv.org/abs/2608.04281v1
摘要: We develop a concise error analysis for a linear-quadratic elliptic distributed optimal control problem with point tracking and pointwise control constraints, where a standard finite element discretization and a variational discretization are treated in a unified framework. The analysis is self-contained modulo standard techniques in finite element analysis and classical results for elliptic partial differential equations.
16. 基于等变谱子流形的物理信息降阶建模
Physics-informed reduced-order modelling with equivariant spectral submanifolds
作者: Georg Maierhofer
类别: cs.LG, math.DS, math.NA
发布日期: 2026-08-04
arXiv ID: 2608.04239v1
链接: http://arxiv.org/abs/2608.04239v1
摘要: Spectral submanifold (SSM) reduction has emerged as a mathematically principled route to reliable nonlinear reduced-order models, capturing dynamics beyond the reach of linear techniques such as Dynamic Mode Decomposition (DMD). The computation of SSMs, however, remains computationally expensive, particularly for high-dimensional systems. In this work, we introduce equivariant spectral submanifold (eSSM) reduction, a novel extension of the SSM framework that explicitly incorporates symmetries of the full-order model into the reduction process. We establish the mathematical foundations of this approach by showing that SSMs are naturally equivariant submanifolds and that the associated charts and reduced dynamics inherit the appropriate induced group actions. Building on this framework, we develop a novel equivariant SSM reduction algorithm that exploits these symmetries to achieve substantially faster computations while also improving model robustness. We demonstrate the advantages of this approach on several benchmark problems including a test from the Common Task Framework for Science.
17. 无迹KalmanNet：一种用于非线性状态估计的带有校准后验协方差的混合深度学习滤波器
Unscented KalmanNet: a hybrid deep learning filter with calibrated posterior covariance for nonlinear state estimation
作者: Minhyeok Ko, Abdollah Shafieezadeh
类别: cs.LG, eess.SP, math.NA
发布日期: 2026-08-04
arXiv ID: 2608.04201v1
链接: http://arxiv.org/abs/2608.04201v1
摘要: State estimation for nonlinear dynamical systems is commonly performed with the Unscented Kalman filter (UKF), which propagates the state moments through deterministic sigma points and reports a posterior covariance at every step. In practice, however, unknown and time-varying noise statistics and model mismatch degrade both estimation accuracy and covariance calibration. Existing learned filters improve accuracy but are largely built on the extended Kalman filter and either forgo an explicit covariance or learn uncertainty without correcting mismatch-induced gain bias. This paper introduces the Unscented KalmanNet (UKN), a hybrid recursive estimator that augments the UKF with two structurally distinct learned components while preserving its explicit sigma-point covariance recursion. NoiseNet predicts time-varying process and measurement covariances as bounded multiplicative corrections to fixed baselines, guaranteeing positive definiteness, while GainNet applies a bounded residual correction to the analytical gain. A calibration-aware training objective combines state error with covariance- and innovation-consistency terms through adaptive weights, jointly optimizing accuracy and calibration. UKN is benchmarked against UKF, KalmanNet, and Bayesian KalmanNet on three synthetic systems and UZH-FPV real-flight data. It achieves the lowest aggregate state-estimation error in all four examples and reduces RMSE by $26.4$-$49.7\%$ compared to UKF in the synthetic cases. Leave-one-sequence-out cross-validation over 11 flights shows $22.4\%$ and $34.3\%$ reductions in mean position and velocity RMSE, respectively. UKN also yields the lowest fold-to-fold variability, with normalized NEES and empirical coverage closest to nominal values among the other filters.