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1. 基于CEM-GMsFEM的高对比度非均匀系数问题多尺度建模

Multiscale modeling for problems with high contrast heterogeneous coefficients by the CEM-GMsFEM

作者: Eric T. Chung, Yalchin Efendiev, Xingguang Jin, Wing Tat Leung, Changqing Ye

类别: math.NA

发布日期: 2026-06-09

arXiv ID: 2606.10524v1

摘要翻译: 本文综述了约束能量最小化广义多尺度有限元方法（CEM-GMsFEM），用于求解具有高度非均匀、高对比度系数的椭圆型偏微分方程。我们通过谱辅助空间详细阐述了多尺度基函数的构造，并结合过采样策略实现局部计算并保证指数误差衰减。我们概述了严格的误差估计，以验证该方法的最优收敛性和稳健性。数值模拟验证了多尺度基函数的指数衰减性质。此外，我们讨论并评述了CEM-GMsFEM的若干最新应用。

Comment: 无

链接: http://arxiv.org/abs/2606.10524v1

1. 研究对象和背景

许多实际应用涉及具有非均匀系数的偏微分方程，例如非均匀或断裂介质中的达西定律，以及复合材料中的弹性系统。这些应用通常导致扩散系数 $\kappa$ 呈现高对比度和通道状结构，这使得传统数值方法面临巨大挑战：一方面，精细网格需要极大自由度；另一方面，高对比度导致线性代数系统的求解收敛恶化。针对此类问题，多尺度计算方法应运而生，最早由 Hou 和 Wu 提出了多尺度有限元法 (MsFEM) 的概念，将模型信息嵌入有限元空间构造，利用粗细两层嵌套网格。然而，当系数不满足尺度分离假设时，MsFEM 的精度可能严重下降。为了克服这一局限性，Efendiev, Galvis 和 Hou 提出了广义多尺度有限元法 (GMsFEM)，通过在粗子域上求解局部谱问题来降维，对高对比度和通道状系数表现良好。另一方面，Målqvist 和 Peterseim 提出了局部正交分解 (LOD) 方法，利用拟插值算子分离宏观与微观部分，首次实现了对一般 $L^\infty$ 系数的最佳理论逼近。Chung, Efendiev 和 Leung 结合 GMsFEM 与 LOD 的思想，发展了约束能量最小化广义多尺度有限元法 (CEM-GMsFEM)，其核心创新在于用子域上的特征空间投影替代拟插值算子，并通过求解松弛的能量最小化问题构造多尺度基函数，从而避免了解鞍点线性系统。该方法在面对高对比度系数时具有对比度稳健性 (contrast robustness)，且通过局部化计算保证了指数衰减的逼近精度。

本文是一篇综述，旨在全面介绍 CEM-GMsFEM 的构造原理、理论分析和应用进展。研究的模型问题为具有 Dirichlet 边界条件的标准非均匀椭圆方程：

$$-\operatorname{div}(\kappa \nabla u) = f \quad \text{in } \Omega, \qquad u = 0 \quad \text{on } \partial\Omega,$$

其中 $\Omega \subset \mathbb{R}^d$ ($d=2,3$) 为有界 Lipschitz 域，$\kappa$ 满足椭圆性条件，$f$ 为源项。本文详细阐述了 CEM-GMsFEM 的核心模块：两层嵌套网格、基于谱分解的辅助空间构造、过采样 (oversampling) 技术以及基函数的能量最小化构造；给出了严格的理论误差估计，包括全局多尺度空间的最优逼近性以及局部化导致的指数衰减性质；并通过数值实验验证了多尺度基函数的指数衰减特性；最后评论了 CEM-GMsFEM 在当前多个前沿方向的扩展和应用。本文旨在为读者提供一份完整的参考，以理解和实施这一方法。

2. 主要定理或主要结果

本文综述了 CEM-GMsFEM 已有的主要理论结果，这些结果在适当假设下保证了方法的收敛性和稳健性。设 $\Lambda = \min_j \lambda_{j,k+1}$ 为所有粗子域上第 $k+1$ 个特征值的最小值，其中 $k$ 为每个子域所选取的局部特征函数个数。全局多尺度空间 $V_{H,\infty}$ 由算子 $G_\infty$ 的像构成，它精确满足正交分解 $V = V_{H,\infty} \oplus W$，其中 $W = V \cap \ker \pi$。利用这一分解，可得如下最优误差估计：若 $u$ 为原问题 (1) 的解，$u_{H,\infty}^{ms}$ 为相应的全局多尺度 Galerkin 逼近，则

$$\|u - u_{H,\infty}^{ms}\|_a \le \frac{1}{\sqrt{\Lambda}} \|f\|_{s^*},$$

其中 $\|\cdot\|_a$ 为能量范数，$\|\cdot\|_{s^*}$ 为对偶范数，且 $\|f\|_{s^*} \le H \frac{1}{\min \kappa} \|f\|_{L^2(\Omega)}$，表明误差关于粗网格尺寸 $H$ 至少是一阶收敛，且与对比度无关（当 $\Lambda$ 与对比度无关时）。这一结果与 Kolmogorov $n$-宽度理论相符，显示出空间 $V_{H,\infty}$ 的逼近最优性。

局部多尺度空间 $V_{H,l}$ 通过引入 $l$ 层过采样构造而成，对应的算子记为 $G_l$。误差分析的关键在于证明全局基函数的指数衰减性质以及局部与全局算子之差的指数衰减。具体地，对于任意 $v \in L^2(K_j)$，有引理 3.3：

$$\|G_{j,\infty}v\|_{a(\Omega\setminus K_{j,l})}^2 + \|\pi G_{j,\infty}v\|_{s(\Omega\setminus K_{j,l})}^2 \le \theta^l \big( \|G_{j,\infty}v\|_a^2 + \|\pi G_{j,\infty}v\|_s^2 \big),$$

其中 $\theta = c^*/(c^*+1) < 1$，而 $c^*(\Lambda)$ 由下式给出：

$$c^*(\Lambda) = \max_{x\in[0,\pi/2]} (\cos x + \sin x) \left( \frac{\cos x}{\sqrt{\Lambda}} + \sin x \right).$$

进一步，局部算子与全局算子的差满足引理 3.4：

$$\|(G_{j,\infty} - G_{j,l})v\|_a^2 + \|\pi (G_{j,\infty} - G_{j,l})v\|_s^2 \le c^\star \theta^{l-1} \big( \|G_{j,\infty}v\|_a^2 + \|\pi G_{j,\infty}v\|_s^2 \big),$$

其中 $c^\star(\Lambda) = \max_{x\in[0,\pi/2]} \big[ (\frac{1}{\sqrt{\Lambda}}+1)\cos x + \sin x \big]^2 + \big( \frac{\cos x}{\sqrt{\Lambda}} + \sin x \big)^2$。由此可组合得出局部多尺度逼近 $u_{H,l}^{ms}$ 的误差估计：

$$\|u - u_{H,l}^{ms}\|_a \le C \left(\frac{1}{\sqrt{\Lambda}} + \sqrt{c^\star}\, \theta^{\frac{l-1}{2}} \right) \|f\|_{s^*}.$$

上述估计仅依赖于粗网格正则性、过采样层数 $l$ 和特征值比 $\Lambda$，而不涉及 $\kappa$ 的对比度，从而保证了方法对高对比度系数的稳健性。数值模拟进一步验证了多尺度基函数的指数衰减特性，与理论结果一致。

3. 研究方法、关键技术和核心工具

证明中的核心技术障碍在于：如何对一般高对比度系数构造具有全局连续性的局部基函数，并满足最优逼近性，同时避免系数的高低对比度影响估计常数。CEM-GMsFEM 的解决方案可概括为以下关键创新：

  1. 谱辅助空间与投影的逼近性质：在每个粗子域 $K_j$ 上，求解带有 Neumann 边界条件的特征值问题

    $$a_j(\Phi_{j,k}, v) = \lambda_{j,k} s_j(\Phi_{j,k}, v), \quad \forall v\in H^1(K_j),$$

    其中 $a_j(\Phi,v)=\int_{K_j} \kappa \nabla \Phi \cdot \nabla v$, $s_j(\Phi,v)=\int_{K_j} \tilde{\kappa} \Phi v$, 且权函数 $\tilde{\kappa}$ 选为 $\tilde{\kappa} = \kappa C_{\rm msh} H^{-2}$。这样的尺度使得特征值 $\lambda_{j,k}$ 无量纲，并且与对比度无关（若 $\kappa$ 与 $H$ 无关）。取前 $L$ 个特征函数张成局部辅助空间 $V_j^{\rm aux}$，定义投影 $\pi_j:L^2(K_j)\to V_j^{\rm aux}$，它满足两个关键估计：

    $$\|\pi_j u\|_{s_j} \le \|u\|_{s_j}, \qquad \|u - \pi_j u\|_{s_j} \le \lambda_{j,L+1}^{-1/2} \|u\|_{a_j}, \quad \forall u\in H^1(K_j).$$

    这为将解分解为“好”的部分（可被辅助空间捕捉）和“坏”的部分（能量控制 $L^2$ 加权范数）奠定了基础。

  2. 能量最小化构造与松弛公式：在每个 oversampled 域 $K_{j,l}$ 上，为每个特征函数 $\Phi_{j,k}$ 构造基函数 $\Psi_{j,k}$ 时，采用能量最小化：

    • 约束形式：求解 $\min \int_{K_{j,l}} \kappa |\nabla v|^2$，满足正交约束 $s(\pi v, \Phi_{j',k'}) = \delta_{jj'}\delta_{kk'}$，这导致鞍点系统。
    • 松弛形式：求解 $\min \int_{K_{j,l}} \kappa |\nabla v|^2 + s(\pi v - \Phi_{j,k}, \pi v - \Phi_{j,k})$，通过罚方法将约束融入目标函数，转化为标准正定系统，显著简化求解。松弛形式是理论分析的主要对象，但其误差同样可控制。

  3. 正交分解与最优性：全局多尺度空间 $V_{H,\infty}$ 由所有全局算子 $G_{j,\infty}$ 的像生成，通过分析证明其与 $W = V\cap \ker \pi$ 构成 $V$ 的直和分解。核心在于：对任一 $v\in W$，利用特征值不等式有 $\|v\|_s^2 = \sum_j \|v\|_{s_j}^2 \le \sum_j \lambda_{j,L+1}^{-1} \|v\|_{a_j}^2 \le \Lambda^{-1} \|v\|_a^2$，因此 $W$ 中函数的能量范数控制 $s$ 范数。进一步，由 Galerkin 正交性，误差 $e$ 必然落入 $W$，从而：

    $$\|e\|_a^2 = a(e,e) = (f,e) \le \|f\|_{s^*} \|e\|_s \le \frac{1}{\sqrt{\Lambda}} \|f\|_{s^*} \|e\|_a,$$

    立即得到定理 3.2。这一估计清晰展示了 $\Lambda$ 的作用：$\Lambda$ 越大（即取更多的局部特征函数），逼近误差越小。

  4. 指数衰减与局部化：引入 cut-off 函数 $\chi_j^{l,l'}$，满足 $\kappa|\nabla \chi|^2 \le \tilde{\kappa}$（得益于 $\tilde{\kappa}$ 的选取）。证明链如下：

    • 第一步（引理 3.3）：在 (4) 中取测试函数 $(1-\chi_{j}^{l-1,l}) G_{j,\infty}v$，将 $\Omega\setminus K_{j,l}$ 上的能量用 $K_{j,l}\setminus K_{j,l-1}$ 上的量表示，利用 $\chi$ 的性质和 Cauchy-Schwarz 不等式，结合 $W$ 空间中的范数控制，导出递推不等式： $$A_{l-1} \ge (1+1/c^*) A_{l},$$ 其中 $A_l = \|G_{j,\infty}v\|_{a(\Omega\setminus K_{j,l})}^2 + \|\pi G_{j,\infty}v\|_{s(\Omega\setminus K_{j,l})}^2$。由此迭代得 $A_l \le \theta^l A_0$，即指数衰减。
    • 第二步（引理 3.4）：将差 $z_j = G_{j,\infty}v - G_{j,l}v$ 分解为 $z_j = z_j' + z_j''$，其中 $z_j' = (1-\chi_j^{l-1,l}) G_{j,\infty}v$，$z_j''$ 属于 $V_{j,l}$。利用变分等式 $a(z_j, z_j'') + s(\pi z_j, \pi z_j'') = 0$ 和 $z_j'$ 的估计，结合第一步的结果，得到 $z_j$ 的误差界，同样呈现指数衰减。

  5. 常数优化：特别注意到在不等式的推导中，通过引入参数 $x$（代表范数比例）并取最大值，得到了与 $\Lambda$ 相关但形式上统一的最优常数 $c^*$ 和 $c^\star$，使得 $\theta$ 尽可能小。这一技巧保证了估计精确。

这些技术综合起来，使得 CEM-GMsFEM 能够在最小假设下（只需 $\kappa$ 满足椭圆性，网格正则性）获得与对比度无关的误差估计和指数衰减，为方法的实际应用提供了坚实的理论基础。

4. 与之前工作的比较

CEM-GMsFEM 的提出和发展是建立在多尺度有限元领域的持续进展之上，与多种已有方法既有联系又有显著改进。下面从几个维度进行比较：

  1. 与 MsFEM 的比较：经典的 MsFEM 在粗网格单元上直接求解局部椭圆问题（通常施加 Dirichlet 或 Neumann 边界）来构造基函数，其精度依赖于系数与基函数边界的尺度分离。当存在强通道或高对比度时，边界条件可能引入严重误差，且没有系统性的误差控制 [36, 37, 66]。CEM-GMsFEM 通过过采样和约束能量最小化克服了边界影响：基函数在更大的子域上构造，并以全局协调的方式满足近似正交性，从而在理论上保证了指数衰减的局部化，即使没有尺度分离也成立。

  2. 与 GMsFEM 的比较：GMsFEM [28] 同样利用局部谱分解获得辅助空间，但通常直接使用这些特征函数（或其简单组合）作为多尺度基函数，或者通过特定的局部插值来形成全局基函数。这样的基函数可能不是最佳的，且缺乏类似 LOD 的严格误差分析框架。CEM-GMsFEM 对这些特征函数施加能量最小化，得到了满足全局连续性的最优逼近基函数，并建立了如同 LOD 的完整理论保证，同时保留了 GMsFEM 处理高对比度的优势。在实现上，GMsFEM 的需求可以通过 CEM-GMsFEM 的子模块直接实现。

  3. 与 LOD 的比较：LOD [46, 47] 依赖一个细致的拟插值算子 $\mathcal{I}$（如通过对节点值进行分片线性插值再修正）将解分解为 $u = \mathcal{I}u + (u-\mathcal{I}u)$，然后通过求解细尺度子问题获得校正项。拟插值算子的构造需要利用系数 $\kappa$ 的信息，且对于高对比度情形，可能不是对比度稳健的——实际上，标准的拟插值算子产生的逼近界可能依赖于对比度 [47]。相比之下，CEM-GMsFEM 用谱投影 $\pi$ 代替拟插值，而 $\pi$ 由局部特征值问题定义，其所满足的逼近估计（$\|u-\pi u\|_s \le \lambda_{L+1}^{-1/2}\|u\|_a$）自动与对比度无关。此外，LOD 需要求解鞍点问题或罚公式，而 CEM-GMsFEM 的松弛形式直接导向正定系统，实现更简单且避免 inf-sup 稳定性问题。

  4. 对比度稳健性：本文多处强调 CEM-GMsFEM 的对比度稳健性并非无代价——它依赖于特征值问题中权函数 $\tilde{\kappa}$ 与 $\kappa$ 成比例，从而特征函数能很好地分离高低传导区域。LOD 在最初的分析中并未显式探讨对比度，但通过选用特定的权函数，LOD 也可达到对比度稳健。从这个角度看，CEM-GMsFEM 提供了更系统的框架。文中还指出，若仅关心多尺度逼近而不考虑对比度，则可以使用齐次 Poisson 算子构造预条件，但此时迭代次数会随对比度增长而发散；CEM-GMsFEM 本质上构造了一个对比度稳健的粗空间，可以用作预条件或直接 Galerkin。

  5. 计算代价与灵活性：由于每个粗子域需要求解特征值问题和局部能量最小化问题，CEM-GMsFEM 的离线计算量显著高于传统方法，但这些计算是完全并行的。松弛公式为减轻计算提供了途径，尤其是当精细自由度可以无重叠分区时，可通过 Sherman-Morrison 公式将求解规模降至与特征函数总数成比例的小稠密系统。与 LOD 相比，CEM-GMsFEM 不要求构造显式的拟插值，对网格和系数分布的适应性更强。在非协调网格、非多项式有限元或有限体积法框架下的推广也较为自然，但相关研究尚处早期阶段。

  6. 与预条件技术的联系：近年来发展的一些对比度稳健的多水平预条件（如谱 AMG 方法）同样利用了局部谱空间构造粗空间 [68, 69]，其精神与 CEM-GMsFEM 异曲同工。然而，CEM-GMsFEM 提供了完整的 Galerkin 多尺度近似框架，而不仅是预条件中的粗空间校正，故可直接用于降阶建模或不确定性量化等任务。

总体而言，CEM-GMsFEM 成功融合了 GMsFEM 的谱降维、LOD 的局部化与最优逼近理论，以及能量最小化带来的实现灵活性，成为一类面向高对比度多尺度问题的有力工具。本综述通过系统梳理这些联系与创新，帮助读者理解为何 CEM-GMsFEM 能够达到目前的理论与实践效果。

不相关论文（基本信息）

1. 一阶轨迹匹配：混沌、湍流、随机系统的快速集合预测

First-Order Trajectory Matching: Fast Ensemble Predictions of Chaotic, Turbulent, Stochastic Systems

作者: Shreya Jha, Timo Schorlepp, Nicholas Geissler, Jules Berman, Benjamin Peherstorfer

类别: cs.LG, math.NA

发布日期: 2026-06-09

arXiv ID: 2606.11138v1

链接: http://arxiv.org/abs/2606.11138v1

摘要: We introduce First-Order Trajectory Matching (FTM), a surrogate-modeling method that learns the first-order local transport of probability mass from trajectories of stochastic systems. By matching the symmetric first-order motion of trajectories, FTM learns the probability current velocity, whose flow preserves time marginals to match ensemble averages, while also capturing current-like trajectory quantities such as fluxes, circulations, and barrier-crossing currents. FTM learns the current velocity directly from trajectories, avoiding drift, diffusion, and score estimation. Our stability analysis separates discretization error from sampling variance and shows that the one-step simulation-free FTM loss is stable when temporal resolution and sample size are properly balanced. Across stochastic dynamical systems and PDE examples, we empirically demonstrate that FTM provides trajectory-aware ensemble predictions at low, deterministic-rollout cost.

2. 论随机图延迟传播的表示

On the representation for stochastic graph delay propagation

作者: Shibo Zeng, Weiguo Gao, Yingzhou Li

类别: math.NA

发布日期: 2026-06-09

arXiv ID: 2606.11086v1

链接: http://arxiv.org/abs/2606.11086v1

摘要: In this work, we utilize the Hermite expansion to approximate the distributions of the sum and maximum of independent random variables. We model distributions with a three-segment representation, where the left and right tails are respectively modeled as combinations of Hermite functions, and the intermediate segment is approximated by piecewise polynomials. This approximation admits rigorous $L^2$- and pointwise convergence properties supported by classical results. We develop an algorithmic framework for applying our model to the graph delay propagation problem, where sum and max operations are performed on the proposed model structure. Numerical experiments demonstrate that our model can capture the quantile values with high accuracy compared to Monte Carlo simulation results, significantly outperforming classical Gaussian-based models.

3. Dmsh：一种用于全四边形网格生成的多智能体强化学习框架

Dmsh: A Multi-Agent Reinforcement Learning Framework for All-Quad Mesh Generation

作者: Anirudh Kalyan, Cosmin Anitescu, Xiaoying Zhuang, Timon Rabczuk, Somdatta Goswami, Sundararajan Natarajan

类别: math.NA, cs.AI, cs.LG

发布日期: 2026-06-09

arXiv ID: 2606.10601v1

链接: http://arxiv.org/abs/2606.10601v1

摘要: Generating high-quality meshes for arbitrary geometries remains a fundamental bottleneck in computational engineering, often demanding heuristic tuning and semi-manual workflows. In this paper, we introduce Dmsh, a first fully automated reinforcement learning pipeline that unifies geometric decomposition and quadrilateral mesh generation within a single learning-based framework. Dmsh decomposes the problem through three coordinated agents handling topology simplification, geometric regularization, and mesh generation. The meshing process is formulated as a Markov Decision Process and solved using a parametric Soft Actor-Critic architecture with decoupled critics, enabling efficient exploration of a hybrid discrete-continuous action space. A curriculum learning strategy ensures scalability from simple domains to highly complex geometries, suppressing seed variance. By design, the recursive decomposition enables parallel meshing of subregions, yielding globally conforming all-quadrilateral meshes without post hoc correction. Across a wide range of benchmarks, Dmsh consistently outperforms existing methods in automation, robustness, and mesh quality, establishing a new paradigm for learning-based mesh generation.

4. 观点对齐与环境反馈的非局部建模：空间聚集与非一致模式

Nonlocal modeling of opinion alignment and environmental feedback: Spatial aggregation and non-consensus patterns

作者: Rui Wang, Yunfeng Xiong, Zhengru Zhang, Xiaofei Zhao

类别: math.NA

发布日期: 2026-06-09

arXiv ID: 2606.10599v1

链接: http://arxiv.org/abs/2606.10599v1

摘要: The formation of public opinion in modern information environments is shaped by the interplay between social conformity and information exposure. While social interactions promote opinion alignment, heterogeneous visibility and selective exposure may reinforce local agreement, a mechanism commonly associated with the echo chamber effects. To describe how such reinforcement influences spatially heterogeneous opinion activity and non-consensus patterns, we propose a spatial opinion dynamics model with attention-mediated feedback. The model couples nonlocal alignment with an evolving attention field and captures a self-reinforcing mechanism in which regions of high opinion activity attract greater visibility. Starting from agent-based jump mechanism inspired by bounded confidence interactions and biased random walks induced by environments, we formally derive a nonlocal advection-cross-diffusion system, where opinion transport is driven by nonlocal conformity and modulated by attention-dependent redistribution. We characterize the transition from spatially homogeneous consensus states to non-consensus clustered regimes through linear stability analysis of the homogeneous equilibrium. The results show that attention-mediated feedback has an explicit correction on the instability threshold and enlarges the parameter regime in which clustering occurs, thereby promoting persistent spatial heterogeneity and non-consensus patterns. Numerical simulations based on a structure-preserving IMEX spectral method support the theoretical predictions and quantify the resulting aggregation phenomena. These findings provide a macroscopic description of how nonlocal alignment and environmental feedback jointly shape spatial signatures of non-consensus patterns.

5. 通过随机低秩Hessian近似加速基于SAV的优化

Accelerating SAV-based optimization via randomized low-rank Hessian approximation

作者: Ryo Sagawa, Daisuke Furihata, Yuto Miyatake

类别: math.OC, cs.LG, math.NA

发布日期: 2026-06-09

arXiv ID: 2606.10562v1

链接: http://arxiv.org/abs/2606.10562v1

摘要: We propose a new optimization method, the Nyström-enhanced relaxed scalar auxiliary variable method (N-RSAV), which incorporates curvature information into the RSAV framework to accelerate convergence while preserving an unconditional modified energy dissipation law. Existing RSAV-based methods rely solely on first-order information and often suffer from slow convergence, particularly for ill-conditioned problems such as those arising in physics-informed neural networks (PINNs). To address this limitation, we design the linear operator in the RSAV scheme using approximate Hessian information obtained from a randomized low-rank Nyström approximation. To preserve the dissipation structure, we enforce positive semidefiniteness through eigenvalue truncation. Furthermore, we introduce an adaptive strategy that reuses the approximate Hessian based on the deviation between the original and modified energies, significantly reducing computational cost. We also provide a convergence analysis of the RSAV scheme with a general positive semidefinite operator under the Polyak-Lojasiewicz (PL) condition and establish corresponding convergence guarantees for N-RSAV under the PL condition and an additional convexity assumption. Numerical experiments on ill-conditioned problems with effectively low-rank structure, including convex quadratic problems and training of PINNs, demonstrate that the proposed methods achieve substantially faster convergence than conventional RSAV-based approaches.

6. 图上非局部扩散的正则化分数阶拉普拉斯算子的谱与计算方面

Spectral and computational aspects of a regularized fractional Laplacian for non-local diffusion on graphs

作者: Alessandro Filippo, Mariarosa Mazza

类别: math.NA

发布日期: 2026-06-09

arXiv ID: 2606.10480v1

链接: http://arxiv.org/abs/2606.10480v1

摘要: The fractional Laplacian has been widely used to model non-local diffusion on graphs, allowing interactions that extend beyond immediate neighbors. However, it suffers from a structural inconsistency as it breaks compatibility with the topology of the original network. To address this issue, a combination of the standard and fractional Laplacians aimed at restoring compatibility while retaining the spectral richness of the fractional operator was recently proposed. In this work, we provide a thorough analysis of the diffusion properties of the resulting regularized operator. We prove that it yields superdiffusive behavior independently of whether the underlying graph is weighted or unweighted -a property not generally satisfied by other nonlocal Laplacian variants. An efficient construction of the regularized operator which preserves the same asymptotic computational cost of the fractional Laplacian is also given. Our numerical experiments demonstrate the effectiveness and computational practicality of the regularized operator for modeling non-local diffusion on real-life networks.

7. 一种具有收敛性保证的不可分离优化随机梯度算法

A stochastic gradient algorithm for non-separable optimization with convergence guarantee

作者: Yingzhou Li, Ruofan Wu

类别: math.OC, math.NA

发布日期: 2026-06-09

arXiv ID: 2606.10383v1

链接: http://arxiv.org/abs/2606.10383v1

摘要: We study non-separable objectives in which the loss depend on dataset-level quantities. We introduce an SGD-style framework that employs two batch-gradient constructs: the ideal per-batch gradient $G$' and a cached surrogate $H$’ for cases where full-data terms are expensive. Notably, in the sample-wise separable case, our method reduces to standard mini-batch SGD. Our main contribution is a unified local convergence theory: under mild smoothness and Jacobian-boundedness assumptions, we prove local linear convergence under local strong convexity and local $O(1/k)$ sublinear convergence under local convexity for both $G$'-driven and $H$’-driven updates. Crucially, these guarantees hold for fixed step sizes within explicitly characterized ranges; we provide explicit bounds showing how cache staleness, surrogate approximation error, batch size, and step size influence the convergence constants and allowable step-size ranges.

8. 基于切比雪夫插值与高斯-赛德尔迭代的高阶扩散采样

Higher-order Diffusion Sampling via Chebyshev Interpolation and Gauss–Seidel Iterations

作者: Bingyuan Wei, Meng Huang

类别: math.NA, math.ST

发布日期: 2026-06-09

arXiv ID: 2606.10353v1

链接: http://arxiv.org/abs/2606.10353v1

摘要: Higher-order ODE solvers have shown strong empirical promise for accelerating diffusion models through the probability flow ODE, but rigorous non-asymptotic guarantees for such acceleration remain limited. In this paper, we develop a Chebyshev–Gauss–Seidel higher-order sampler and establish a non-asymptotic convergence guarantee that allows the approximation order to grow logarithmically with the number of outer iterations. In the exact-score setting, up to logarithmic factors, the proposed sampler requires at most [ d^{1+o_T(1)}\varepsilon^{-1/K_1} ] score functions to approximate the target distribution on (\mathbb{R}^d) within total variation distance (\varepsilon), where (o_T(1)\to 0) as (T\to\infty) and (K_1>0) is a sufficiently large constant. The analysis assumes only a polynomial second-moment bound on the target distribution, thereby relaxing the bounded-support condition imposed in existing higher-order theory. Moreover, the guarantee is robust to score and Jacobian estimation errors and does not require higher-order smoothness assumptions on the score estimates. Numerical experiments on anisotropic Gaussian mixture benchmarks support the predicted improvement in the accuracy–cost tradeoff under finite score-evaluation budgets.

9. 面向流动数据连续逼近的物理信息B样条框架

A Physics-Informed B-Spline Framework for Continuous Approximation of Flow Data

作者: Junoh Jung, David Lenz, Emil Constantinescu, Tom Peterka

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

发布日期: 2026-06-09

arXiv ID: 2606.10335v1

链接: http://arxiv.org/abs/2606.10335v1

摘要: Continuous approximations of flow data are useful for downstream analysis, differentiation, and visualization, but purely data-driven reconstructions do not, in general, preserve the governing physics. This limitation becomes particularly important when input data are physically inconsistent, whether due to low-fidelity discretizations or unmodeled discrepancies. In such cases, reconstructed fields may exhibit inaccurate PDE residuals, violated balance laws, or unreliable derived quantities. To address this, we propose a physics-informed B-spline framework that embeds physical constraints directly into the reconstruction process. The method constructs compact, continuously differentiable representations of discrete fields using tensor-product B-splines and determines spline control points by solving an optimization problem balancing data fidelity with residuals of the governing PDEs, alongside initial and boundary conditions. Leveraging exact analytical derivatives of the B-spline basis enables efficient and accurate evaluation of physical residuals without storing full-resolution fields. We refer to this approach as physics-informed multivariate functional approximation (PI-MFA). Numerical studies on the 1D convection-diffusion, 2D coupled Burgers, and 2D incompressible Navier-Stokes equations show PI-MFA reduces PDE residuals and improves global balance-law consistency. Compared with standard and regularized MFA, PI-MFA produces more physically faithful reconstructions and, for physically inconsistent data, lower approximation errors, while offering computational advantages over tested physics-informed neural networks. Overall, PI-MFA preserves the compactness, local support, and exact differentiability of classical spline spaces while producing reliable continuous flow fields for scientific analysis and visualization.

10. 通过张量记号改进矩阵李群运算的表示

Improved Representation of Matrix Lie Group Operations through Tensor Notation

作者: Clark Taylor

类别: cs.RO, math.NA

发布日期: 2026-06-09

arXiv ID: 2606.10289v1

链接: http://arxiv.org/abs/2606.10289v1

摘要: Several recent papers have demonstrated the utility of using Lie groups within estimation problems, yielding improved accuracy and consistency. This paper introduces a new tool for describing operations with matrix Lie groups: tensors and the Einstein summation notation. While tensors and Einstein notation are well-known in other research fields, applying this mathematical notation to represent and compute matrix Lie derivatives is novel. More importantly, this new notation greatly clarifies the derivatives and operations necessary to work with matrix Lie Groups in (gradient-based) estimation frameworks. Therefore, the main contribution of this paper is not a new capability, but a more perspicuous mathematical notation for working with matrix Lie groups.

11. 修剪背景网格上椭圆问题的虚拟单元法

A Virtual Element Method for elliptic problems on trimmed background meshes

作者: L. Beirão da Veiga, C. Canuto, R. H. Nochetto, G. Vacca, M. Verani

类别: math.NA

发布日期: 2026-06-08

arXiv ID: 2606.10218v1

链接: http://arxiv.org/abs/2606.10218v1

摘要: We consider a two-dimensional piecewise $C^2$ domain that cuts through a quasi-uniform fixed polygonal background mesh, for instance made of quadrilaterals. A simple procedure based on convex hulls gives rise to a rather small number of polygonal boundary elements of various shapes, including elements with small edges and large aspect ratios; this is the computational mesh for a virtual element method (VEM), a trimmed background mesh. We classify all possible geometric configurations and study their stability and approximability properties. This entails deriving robust stabilization mechanisms and interpolation estimates for anisotropic elements and elements with small cuts, as well as a weak maximum principle for enhanced virtual elements; these contributions have intrinsic interest for VEM theory on geometric flexibility. We prove that the resulting VEM is uniformly stable in $H^1$, and also show optimal order-regularity error estimates in $H^1$ and $L^2$. Insightful numerical experiments corroborate and complement our theory. The proposed method is suitable for treating ALE formulations of problems in moving domains.

12. 跨尺度桥接：渐近分析与人工智能辅助形式化

Bridging Scales: Asymptotic Analysis and AI-Assisted Formalization

作者: Xu Yang

类别: math.NA

发布日期: 2026-06-08

arXiv ID: 2606.10138v1

链接: http://arxiv.org/abs/2606.10138v1

摘要: Asymptotic analysis is one of the classical tools for bridging models across scales. Behind many such derivations lies a common symbolic structure: an ansatz, a substitution, an order-by-order matching procedure, and the extraction of effective equations or interface conditions. This article revisits that structure through two representative bridges: the kinetic-to-fluid limit, illustrated by radiative transfer with interface layers and by neural-network approximations of Boltzmann equations, and the quantum-to-classical limit, illustrated by the Frozen Gaussian Approximation and its Dirac extension. We then explain why such derivations are natural candidates for AI-assisted formalization: their recurring symbolic structures can be organized, verified, and reused. In this sense, a carefully structured expository paper may serve not only as a review, but also as a mathematical seed for future AI-assisted environments.

13. WING: 一种用于重复矩阵求解的简单加窗非正交化初始猜测程序

WING: A Simple Windowed Nonorthogonalized Initial Guess Procedure for Repeated Matrix Solves

作者: David Wells, Matthew G. Knepley, Boyce E. Griffith

类别: math.NA

发布日期: 2026-06-08

arXiv ID: 2606.10132v1

链接: http://arxiv.org/abs/2606.10132v1

摘要: Many numerical methods require solution of a sequence of linear systems with the same matrix and similar right-hand sides. Krylov subspace methods are a common tool for solving such linear systems, and a carefully chosen initial guess for the solution can reduce the total number of iterations, and thereby the total computational cost, required for convergence to a specified numerical tolerance. This paper introduces the WING algorithm, a modification of Fischer’s second algorithm, which lowers the cost of forming an acceptably close initial guess by skipping orthogonalization and solving the possibly singular normal equations with a pseudoinverse. We demonstrate the efficacy of the new algorithm, particularly for solving linear systems with coarse relative tolerances, with numerical benchmarks based on fluid-structure interaction, mantle convection, and earthquake models.

14. 二维多弧的积分形式

Integral Formulations for two-dimensional Multi-Arcs

作者: Jose Pinto, Ruben Aylwin

类别: math.NA

发布日期: 2026-06-08

arXiv ID: 2606.10016v1

链接: http://arxiv.org/abs/2606.10016v1

摘要: We study the Laplace equation with Dirichlet and Neumann boundary conditions posed on multi-arcs, i.e., collections of open arcs meeting at junction points. We begin by introducing a scale of Sobolev spaces constructed using the Sobolev spaces on open arcs as main building block and extend the definition of trace operators. We reformulate the boundary value problems using boundary integral formulations. We then establish a well-posed integral formulation for the Dirichlet problem, which can be discretized using standard numerical methods. We further investigate the singular behavior of the solution densities at branch points through numerical experiments and observe that these singularities are comparable to the corner singularities arising in polygonal domains. For the Neumann problem, we show that the associated hypersingular operator is not necessarily invertible on classical Sobolev spaces and provide numerical evidence that solutions may develop jump discontinuities at branch points.