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1. 穿孔域中线性弹性方程的自适应多尺度模型降阶
Adaptive multiscale model reduction for linear elasticity equation in perforated domains
作者: Wei Xie, Eric Chung, Yin Yang, Yunqing Huang
类别: math.NA
发布日期: 2026-06-05
arXiv ID: 2606.06839v1
摘要翻译: 本文提出了一种约束能量最小化广义多尺度有限元方法（CEM-GMsFEM），用于求解非均匀穿孔域中的线性弹性问题。大量穿孔的存在为计算域引入了多个尺度，使得直接细网格模拟在计算上非常昂贵。所提出的方法遵循 CEM-GMsFEM 的标准离线——在线分解。在离线阶段，在粗单元上求解局部谱问题以构造辅助空间，然后在过采样区域上计算局部能量最小化基函数，以捕捉由穿孔引起的细尺度几何信息。在线阶段，在扩大的粗邻域中构造残差驱动的基函数，以纳入源项信息并自适应地提高多尺度近似的精度。我们为离线和在线阶段都建立了收敛性结果。具体地，我们推导了局部多尺度近似的误差估计，并证明了自适应在线增强算法的收敛性。此外，我们证明了在线阶段使用的过采样区域可以在局部确定，从而在保持收敛性的同时降低计算成本。对不同几何构型的穿孔介质的数值实验展示了所提出方法的准确性和效率。
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链接: http://arxiv.org/abs/2606.06839v1
1. 研究对象和背景
本文研究穿孔域（perforated domain）上的线性弹性（linear elasticity）问题。穿孔域是指从有界区域 $\Omega\subset\mathbb{R}^d$（$d=2,3$）中移除大量随机分布的小穿孔后得到的区域 $\Omega^\varepsilon$，穿孔的并集记为 $B^\varepsilon$，其特征尺度 $\varepsilon$ 远小于区域直径。在实际中，此类构型广泛存在于多孔介质流动、复合材料、油气开采等领域，其中固相骨架的力学响应可通过线性弹性方程描述。由于穿孔引入复杂的多尺度几何特征，直接数值模拟（如有限元法）需要极细的网格解析所有穿孔边界，导致巨大的计算量甚至无法承受。因此，开发能够有效降阶并保持精度的多尺度计算方法具有重要的理论与实际意义。
控制方程为非线性弹性问题简化后的线性化形式，具体为：
$$ \begin{cases} -\nabla\cdot \sigma(u) = f, & \text{in } \Omega^\varepsilon,\\ \sigma(u) = 2\mu \varepsilon(u) + \lambda (\nabla\cdot u)\mathbf{I},\\ \varepsilon(u) = \frac{1}{2}(\nabla u + (\nabla u)^T),\\ \sigma(u)\cdot n = 0, & \text{on } \partial B^\varepsilon \cap \partial \Omega^\varepsilon,\\ u = 0, & \text{on } \partial \Omega \cap \partial \Omega^\varepsilon, \end{cases} $$其中 $u$ 为位移向量场，$\sigma$ 和 $\varepsilon$ 分别为应力和应变张量，$\mu, \lambda > 0$ 为拉梅（Lamé）常数，$\mathbf{I}$ 为单位张量，$n$ 为单位外法向量。在穿孔边界上施加零牵引条件（自由表面），外部边界施加齐次 Dirichlet 条件。定义函数空间 $V = \{v\in [H^1(\Omega^\varepsilon)]^d : v|_{\partial\Omega^\varepsilon\cap\partial\Omega}=0\}$，则弱形式为：求 $u\in V$ 满足
$$a(u, v) = \int_{\Omega^\varepsilon} \big(2\mu\varepsilon(u):\varepsilon(v) + \lambda \nabla\cdot u \,\nabla\cdot v\big) = \int_{\Omega^\varepsilon} f v, \quad \forall v\in V.$$设细网格尺寸 $h$ 充分小以解析穿孔几何，对应的有限元空间 $V_h$ 用分片双线性（$Q_1$）元构成。但求解细网格代数系统代价高昂。为此，本文采用约束能量最小化广义多尺度有限元法（CEM-GMsFEM）框架，在粗网格 $T^H$（$h\ll\varepsilon\ll H$）上构造多尺度逼近空间 $V_{ms}\subset V_h$，并求解 $u_{ms}\in V_{ms}$ 使得 $a(u_{ms},v)=(f,v), \forall v\in V_{ms}$。
已有研究主要沿两条路线展开：一类是推导等效宏观模型，如渐近均质化（homogenization）方法 [1, 25, 8]、异质多尺度方法（HMM）[35, 36, 24] 和多连续体均质化 [18, 9, 37, 40, 27]；另一类是构造特殊的多尺度基函数直接对粗网格离散，例如广义有限元法（GFEM）[4, 2]、多尺度有限元法（MsFEM）[26, 28, 29, 6]、广义多尺度有限元法（GMsFEM）[17, 13, 15, 32, 39, 38]、局部正交分解（LOD）[31, 7] 以及 CEM-GMsFEM [11, 21, 10, 33, 34, 41, 16]。其中，CEM-GMsFEM 通过在每个粗网格块上求解局部谱问题生成辅助空间，再在延拓区域上求解能量最小化问题得到多尺度基函数，能够较精确地刻画局部异质特征，且具备指数衰减的局部化性质。为提升精度，残差驱动的在线基函数（online basis functions）被引入以捕捉未解析的源项与误差信息 [12, 20]。
本文在该框架下发展了一种自适应多尺度模型降阶方法，并给出了完整的理论分析。具体地，离线（offline）阶段在每个粗块 $K_i$ 上求解谱问题提取 $l_i$ 个特征函数形成辅助空间 $V_{aux}$，然后在过采样区域 $K_{i,k_i}$ 上求解约束能量最小化问题生成离线多尺度基函数（包括松弛版本与约束版本）；在线（online）阶段则根据上一轮多尺度解的残差，在过采样粗邻域 $\omega_{i,m_i}$ 上构造在线基函数，并通过自适应算法（算法1）基于局部残差指标选择富集区域。本文的主要贡献在于：为上述离线与在线两阶段建立了收敛性理论，特别地，证明了在线阶段所需的过采样区域可仅用局部信息确定（即衰减性质），从而在保持收敛的同时降低在线计算开销；数值实验也验证了方法的有效性与高效性。
2. 主要定理或主要结果
本文严格证明了离线与在线两阶段的收敛性，主要结论为下列定理（编号沿用原文）：
定理1（离线收敛） 设 $u$ 为问题 (2) 的精确解，$u_{ms}$ 为由离线多尺度空间 $V_{ms}$（采用松弛基函数 (6) 或约束基函数 (8) 构造）所得的逼近解，则存在正常数 $C_1, C_2$ 以及由过采样层数决定的指数衰减因子 $E$，使得
$$ \|u - u_{ms}\|_a \le C_1 \Lambda_{\Omega^\varepsilon}^{-1/2} \|\tilde{\kappa}^{-1/2} f\| + C_2 E \|u_{glo}\|_s, $$其中 $\Lambda_{\Omega^\varepsilon} = \min\{ \lambda_{l_i+1}^i : K_i \subset \Omega^\varepsilon \}$ 为未被选入辅助空间的最小局部特征值，$\tilde{\kappa} = \sum_j |\nabla\chi_j|^2$ 涉及单位分解函数 $\chi_j$，$u_{glo}$ 为全局多尺度空间的投影。若进一步每个过采样参数 $k_i$ 充分大，且 $\{\chi_i\}$ 为双线性单位分解函数，则误差界可简化为
$$ \|u - u_{ms}\|_a \le C H \Lambda_{\Omega^\varepsilon}^{-1/2} \|f\|. $$定理2（在线自适应收敛） 设 $u$ 为细网格解，$\{u_{ms}^n\}$ 为自适应在线富集算法1输出的多尺度解序列，则对于第 $n+1$ 次迭代，满足递归误差估计
$$ \|u - u_{ms}^{n+1}\|_a^2 \le 3\bigl(1+\Lambda_{\Omega^\varepsilon}^{-1}\bigr)\bigl(1+D_{\Omega^\varepsilon}\bigr)\bigl( C_1 E + C_2 (1-\theta) \bigr) \|u - u_{ms}^n\|_a^2, $$其中 $E$ 为离线与在线基函数指数衰减因子的一个上界，$D_{\Omega^\varepsilon} = C_T (1+\Gamma_{\Omega^\varepsilon})$，$C_T$ 为粗单元顶点数的平方的最大值，$\Gamma_{\Omega^\varepsilon} = \max\{\lambda_{l_i}^i : K_i \subset \Omega^\varepsilon\}$，$\theta\in[0,1]$ 为算法1中的自适应参数（当 $\theta=1$ 时对应一致富集，$\theta=0$ 时不添加任何在线基函数）。此估计表明，若 $C_1 E + C_2 (1-\theta) < 1$，误差将严格递减。
此外，引理4 给出了在线基函数的局部化误差控制：对于由粗邻域 $\omega_i$ 向外扩展 $m_i$ 层得到的过采样域 $\omega_{i,m_i}$ 上的逼近基函数 $\beta_{ms}^i$ 与全局基函数 $\beta_{glo}^i$，成立
$$ \|\beta_{glo}^i - \beta_{ms}^i\|_a^2 + \|\pi(\beta_{glo}^i - \beta_{ms}^i)\|_s^2 \le E_i \bigl( \|\beta_{glo}^i\|_a^2 + \|\pi(\beta_{glo}^i)\|_s^2 \bigr), $$其中衰减因子为
$$ E_i = 3\bigl(1+\Lambda_{\Omega^\varepsilon\setminus\omega_{i,m_i-1}}^{-1}\bigr)\left(1+ \bigl(2(1+\Lambda_{\omega_{i,m_i-1}}^{-1/2})\bigr)^{-1}\right)^{1-m_i}. $$该衰减性质保证了在线阶段过采样区域可仅由局部信息确定。
3. 研究方法、关键技术和核心工具
本文的证明框架主要依赖 CEM-GMsFEM 的几个核心工具：局部谱问题构造的辅助空间、约束能量最小化生成的全局-局部基函数的指数衰减性质、以及将误差分解为谱截断与局部化两部分的技术。主要障碍在于穿孔域几何的复杂性以及在线自适应收敛性的严格论证。作者的创新点在于将上述框架推广到穿孔域线性弹性方程，并发展了自适应在线策略，同时证明了在线基函数过采样区域可仅用局部信息决定，从而降低了在线计算成本。下面分块说明关键技术。
1. 辅助空间与投影算子 $\pi$ 的构造 在每个粗块 $K_i$ 上定义局部内积 $a_i$ 和 $s_i$（$s_i$ 含单位分解梯度加权项 $\tilde{\kappa}$），求解谱问题 $a_i(\phi_j^i, v) = \lambda_j^i s_i(\phi_j^i, v)$。选取前 $l_i$ 个特征函数形成局部辅助空间 $V_{aux}^i$，并拼成整体 $V_{aux}$。定义投影 $\pi: V\to V_{aux}$ 为 $\pi(u) = \sum_{i,j} \frac{s_i(u, \phi_j^i)}{s_i(\phi_j^i, \phi_j^i)} \phi_j^i$。该投影满足稳定性 $\|\pi(v)\|_s^2 \le C_0 \|v\|_a^2$。辅助空间的正交性（单位正交基）使得后续的约束条件 $s$‑正交容易实现。
2. 离线基函数的构造与衰减性质 对每个 $\phi_j^i$，定义全局基函数为能量最小化问题 (10) 的解 $\psi_{glo}^{i,j}$，它满足正交性 $a(\psi_{glo}^{i,j}, v)=0$ 对任意 $v\in\tilde{V} = \ker(\pi)$，从而保证直和分解 $V = V_{glo}\oplus\tilde{V}$。这一正交性是 CEM 框架的基石，使得误差分析中可将误差分为 $\tilde{V}$ 部分（受谱间隙控制）和 $V_{glo}$ 部分（受局部化控制）。关键的引理2证明了全局基函数具有指数衰减性质，即在过采样区域 $K_{i,k_i}$ 上局部求解的基函数 $\psi_{ms}^{i,j}$ 能近似全局基函数，误差呈 $E_i = O(e^{-c k_i})$ 衰减（具体形式依赖于松弛或约束版本）。该衰减是利用能量最小化与谱间隙 $\Lambda$ 导出的，本质源于椭圆型方程的局部性。通过引理3的叠加估计，衰减误差可控。
3. 离线误差分析（定理1） 将误差 $u - u_{ms}$ 用投影分解，得到 $\|u - u_{ms}\|_a \le \|u - \pi(u)\|_a + \|\pi(u) - u_{ms}\|_a$。第一项由谱截断估计为 $\Lambda_{\Omega^\varepsilon}^{-1/2} \|\tilde{\kappa}^{-1/2} f\|$（利用特征值性质），第二项则利用全局-局部基函数的衰减性以及 $V_{glo}$ 的逼近性质，结合单位分解得到 $CH\Lambda_{\Omega^\varepsilon}^{-1/2}\|f\|$ 的最终形式。
4. 在线基函数的自适应构造与收敛性（定理2） 在线阶段的创新在于：以局部残差 $r_i(v) = \langle a(u_{ms}^n, \chi_i v) - (f, \chi_i v) \rangle$ 的算子范数 $\delta_i = \sup_{v\in V_0(\omega_i)} |r_i(v)|/\|v\|_a$ 作为自适应指标，仅对满足 $\sum_{i>q} \delta_i^2 \le \theta \sum_i \delta_i^2$ 的 $q$ 个具有最大 $\delta_i$ 的邻域构造在线基函数 $\beta_{ms}^i$。其局部化形式 (15) 来自全局方程 (16) 的局部化求解。引理4证明了在线基函数同样具有指数衰减，因而过采样层数 $m_i$ 可按需选取，且该衰减仅依赖局部谱信息，故过采样域可局部确定。
证明定理2的关键步骤是构造恒等式：由残差定义有 $a(u - u_{ms}^n, v) = \sum_i r_i^n(v)$。而在线基函数满足 $a(\beta_{glo}^i, v) + s(\pi(\beta_{glo}^i), \pi(v)) = r_i(v)$，求和可得
$$a(u - u_{ms}^n + \sum_i \beta_{glo}^i, v) = s(-\pi(\sum_i \beta_{glo}^i), \pi(v)), \quad \forall v\in V_h.$$特别地，对 $v\in\tilde{V}$ 有 $\pi(v)=0$，于是 $a(u - u_{ms}^n + \sum_i \beta_{glo}^i, v)=0$，即 $u - u_{ms}^n + \sum_i \beta_{glo}^i \in V_{glo}$。接着利用 $V_{glo}$ 与 $\tilde{V}$ 的正交性以及提升引理1（可将辅助空间元素提升为能量范数受控的全空间函数），可将误差 $u - u_{ms}^{n+1}$ 约束到新增在线基函数所扩张的子空间上，最终通过代数操作和自适应选择的标准推出带 $(1-\theta)$ 因子的递归估计。该估计清晰地显示了过采样衰减因子 $E$ 和自适应参数 $\theta$ 的共同作用。
4. 与之前工作的比较
本文针对穿孔域线性弹性问题，在 CEM-GMsFEM 框架下建立了自适应多尺度方法并给出了完整的收敛分析。与已有工作的比较可从以下几点说明：
1. 针对的问题类型 大部分多尺度方法（如 GMsFEM [17, 13, 15]、LOD [31, 7]、CEM-GMsFEM [11, 21]）最初针对的是含高对比度系数的扩散或弹性方程，其多尺度来源于材料系数而非几何穿孔。本文将 CEM-GMsFEM 应用于穿孔域，需处理内部自由边界 $\sigma(u)\cdot n =0$ 带来的非标准边界条件，且局部区域的几何连通性与常规块域不同。文中的理论推导未显式依赖穿孔的具体分布，仅要求穿孔尺寸与粗网格之间满足 $h<\epsilon \ll H$，这使得方法对一般穿孔构型具有普适性。
2. 与均质化及宏观模型方法的区别 均质化 [1, 25, 8] 通过渐近分析导出孔径趋于零时的等效方程，适用于周期或统计均匀介质，但要求明确的尺度分离；HMM [35, 36] 则耦合宏观与微观模型。本文的方法不需要渐近极限，可在有限穿孔尺寸下直接求解粗网格上的离散问题，且能通过自适应在线富集逐步提高精度，避免了宏观模型在非均匀情况下的模型误差。
3. 与离线 CEM-GMsFEM 的对比 原始 CEM-GMsFEM [11] 完全依赖离线基函数，精度由所选特征函数个数和过采样层数决定。为了提高精度，须增加 $l_i$ 或 $k_i$，但会迅速增大离线计算量和在线求解规模。本文引入自适应在线基函数 [12, 20] 后，可在离线阶段采用较少基函数，再通过少量在线迭代达到所需精度，收敛分析明确给出了误差缩减因子。文中的算法 1 与 [12] 的残差驱动在线基函数相比，主要改进在于自适应选择富集区域：仅对局部残差较大的部分邻域构造在线基函数，从而平衡精度与计算量。
4. 过采样区域灵活性的推广 文献 [41] 证明了 CEM-GMsFEM 的离线基函数在过采样区域选择上具有灵活性，即区域向外扩展 $k_i$ 层即可保证指数衰减。本文进一步证明了在线基函数也具备同样的局部衰减性质（引理4），即在线阶段所需的过采样区域 $\omega_{i,m_i}$ 可仅由局部谱信息决定其扩展层数 $m_i$，而无需全局信息。这一结论降低了在线计算的通信与存储开销，因为每个在线基函数可独立求解，且区域大小可由局部谱间隙自适应调整。
5. 理论分析的贡献 在收敛性方面，离线误差估计（定理1）的形式与 [11, 41] 类似，但针对弹性问题给出了明确的常数依赖。在线部分的递归误差估计（定理2）是本文的核心理论新意，它将收敛因子表达为离线/在线衰减因子 $E$ 与自适应参数 $\theta$ 的线性组合，清晰地揭示了收敛速度与过采样层数、自适应比例之间的权衡关系。该结果在形式上比单纯的一致富集分析更为精细，也为实际计算中参数选择提供了理论指引。
6. 局限性与代价 从定理 2 可知，收敛速度依赖于 $E$ 和 $1-\theta$。当 $\theta$ 较小时（即仅富集少量邻域），收敛因子可能接近 1，导致收敛缓慢；增大 $\theta$ 会改善收敛但增加在线计算量。此外，速率还依赖于 $\Lambda_{\Omega^\varepsilon}^{-1}$，若局部谱间隙很小，则需要选取更大的 $l_i$ 来降低 $\Lambda_{\Omega^\varepsilon}^{-1}$。在论文提供的数值实验中，作者验证了不同穿孔构型下的精度，但从提供内容中无法确认是否对计算代价随 $\theta$ 的变化进行了详细比较。另外，两种离线基函数（松弛版本 (6) 与约束版本 (8)）的差异仅被理论分析涵盖，但数值上的优劣未在文中展现对比。
了解领域论文（摘要翻译）
1. 面向随机浅水方程的保结构间断伽辽金方法
Structure-Preserving Discontinuous Galerkin Methods for Stochastic Shallow Water Equations
作者: Yekaterina Epshteyn, Akil Narayan, Yinqian Yu
类别: math.NA
发布日期: 2026-06-05
arXiv ID: 2606.07155v1
链接: http://arxiv.org/abs/2606.07155v1
摘要翻译
中文标题: 面向随机浅水方程的保结构间断伽辽金方法
摘要翻译: 浅水方程（SWE）是流体动力学中的基本模型，对于研究广泛的地球物理和工程现象至关重要。在许多实际应用中，必须考虑初始条件和底部地形所引起的不确定性，这推动了为随机SWE发展稳定且精确的数值方法。基于SWE的保持双曲性随机伽辽金公式 [Dai, Epshteyn, Narayan, 2021 SISC] 以及针对反对称SWE的熵稳定间断伽辽金方法的随机扩展 [Fu, 2022 JSC]，我们为随机浅水系统开发了一种保结构、熵守恒且熵稳定的间断伽辽金–随机伽辽金方法，并具有良好平衡性质。我们通过几个数值实验展示了所提出的保结构算法的精度、适用性和鲁棒性。
2. Cahn-Hilliard 方程的高阶协调有限元：相对能量稳定性与能量缺陷
High-order conforming finite elements for the Cahn-Hilliard equation: Relative-energy stability and energy defects
作者: Aaron Brunk, Marvin Fritz
类别: math.NA
发布日期: 2026-06-04
arXiv ID: 2606.06719v1
链接: http://arxiv.org/abs/2606.06719v1
摘要翻译
中文标题: Cahn-Hilliard 方程的高阶协调有限元：相对能量稳定性与能量缺陷
摘要翻译: 我们研究使用高阶协调有限元空间的 Cahn-Hilliard 方程的半离散单场 Galerkin 逼近。更具体地，具有 $H^2$ 协调试探空间的全局 $C^1$ 有限元，包括 Argyris、Bell 和 Bogner-Fox-Schmit 元，允许对四阶格式直接离散化并精确保持质量。主要的结构性结果是物理 Cahn-Hilliard 能量的精确能量平衡。除了预期的离散耗散外，该平衡还包含一个显式可计算的能量缺陷。该缺陷对于 Laplace 不变的周期空间（如 Fourier 空间）消失，但对于经典的 $C^1$ 有限元通常非零。因此，它量化了离散梯度流结构的精确损失。我们通过相对能量论证证明了半离散先验误差估计。该估计通过使用增广相对能量和逆离散 Laplace 算子的离散椭圆重构界来封闭。所得的收敛速度与预期的逼近阶数吻合。使用 Bell 和 Argyris 元的数值实验证实了这些收敛速度，并通过与 Fourier 参考离散化进行比较，展示了缺陷机制。
不相关论文（基本信息）
1. p-Laplace方程的混合虚拟元方法
A Mixed Virtual Element Method for the p-Laplace equation
作者: Kirubell B. Haile, Giuseppe Vacca
类别: math.NA
发布日期: 2026-06-05
arXiv ID: 2606.07477v1
链接: http://arxiv.org/abs/2606.07477v1
摘要: We introduce and analyze a mixed Virtual Element Method for the $p$-Laplace equation in a non-Hilbertian setting, covering the full range $p \in (1, \infty)$. The discrete framework combines standard mixed Virtual Element spaces with a novel non-linear stabilization term designed to mimic the power-law structure of the continuous operator. We establish discrete inf-sup stability under non-Hilbertian norms and rigorously prove the continuity and coercivity of the discrete form. This guarantees the well-posedness of the problem and allows us to derive a priori error estimates for the primal variable and the flux. A set of numerical tests supports the theoretical derivations.
2. 随机平衡中的统计收敛与数值收敛
Statistical and Numerical Convergence in Stochastic Equilibrium
作者: David Staines
类别: econ.EM, econ.TH, math.NA, math.PR
发布日期: 2026-06-05
arXiv ID: 2606.07469v1
链接: http://arxiv.org/abs/2606.07469v1
摘要: This paper sets out the most general computational and econometric implications of the rigorous stochastic equilibrium theory from SELCKE (Staines (2024a)) arXiv:2312.16214. The analytical backbone is the discovery that the system converges geometrically to long-run equilibrium, at a rate given by the greater of the eigenvalue or inverse eigenvalue (from outside) closest to the unit circle and the maximum shock persistence. High-order shocks converge faster. I develop a simulation procedure to test, with asymptotic power, whether stochastic equilibrium exists for a particular model. The fundamental approximation result asserts that, whatever the order of expansion or loss function, the stochastic steady state delivers the most accurate perturbation solution. I also show that super-consistent parameter estimators $O(1/T)$ arise whenever second-order terms vanish. Besides Calvo, I study stochastic equilibrium in two alternative pricing models. Dynamics simplify considerably. I bound the time the impulse response peaks, by the maximum lag in the errors. This lends empirical support to Taylor contracts, although there are issues surrounding unit roots and the strong cost-channel. For menu costs, I demonstrate that the initial price distribution decays away super-exponentially, producing a system equivalent to Calvo with an endogenous reset probability. The impact of idiosyncratic disturbances appears as an additional wedge between actual and efficient output. Blow-up of the objective function at the boundary is proven, with the help of new distributional arguments, so the model meets existing eigenvalue existence conditions for the recursive equilibrium. Along the way, new light is shone on existing theoretical models and statistical procedures.
3. 非插值无网格空间中本质边界条件的自然分解方法
A Natural Decomposition Method for Essential Boundary Conditions in Noninterpolatory Meshfree Spaces
作者: Jingkai Zhang, Tiexiang Li, Shuo Zhang
类别: math.NA
发布日期: 2026-06-05
arXiv ID: 2606.07281v1
链接: http://arxiv.org/abs/2606.07281v1
摘要: This paper develops a natural decomposition method (NDM)for imposing essential boundary conditions in noninterpolatory meshfree Galerkin spaces without boundary parameter tuning or auxiliary constraint construction. In such spaces, algebraic coefficients generally do not coincide with boundary values; hence coefficient assignment or nodal boundary prescription is not equivalent to imposing the continuous trace required by the variational problem. NDM introduces boundary data before discretization through a natural transfer mechanism: a source subproblem accounts for the forcing term, a weighted curl correction transfers the remaining trace mismatch, and a scalar recovery step reconstructs the solution from the corrected weighted gradient. For topologically trivial single domains with connected boundary, the reconstructed solution is equivalent, at the continuous level, to the solution satisfying the prescribed essential boundary data. The discrete analysis separates the approximation defect of the recovery space from the upstream transfer error visible to that space. Numerical experiments on benchmark problems evaluate the proposed transfer mechanism and report the associated conditioning, computational cost, and boundary perturbation behavior.
4. 一种求解平面多片几何上双调和方程的自适应对偶-原始等几何撕裂与互连（IETI-DP）方法
An adaptive Dual-Primal Isogeometric Tearing and Interconnecting (IETI-DP) method for solving the biharmonic equation over planar multi-patch geometries
作者: Mario Kapl, Aljaž Kosmač, Vito Vitrih
类别: math.NA
发布日期: 2026-06-05
arXiv ID: 2606.07220v1
链接: http://arxiv.org/abs/2606.07220v1
摘要: We present a novel adaptive isogeometric method for solving the biharmonic equation over planar multi-patch domains with possibly extraordinary vertices, parametrized by analysis-suitable G^1 multi-patch geometries. The proposed technique relies on the concept of Dual-Primal Isogeometric Tearing and Interconnecting (IETI-DP), which enforces the required C^1-smoothness of the solution across a common edge of two neighboring patches by imposing appropriate continuity conditions by means of Lagrange multipliers. The resulting saddle point problem is solved using a dual-primal formulation, first by a small linear problem for the Lagrange multipliers and then by local, parallelizable linear problems on the individual patches for the coefficients of the numerical solution. While for the local problems on the single patches standard diagonally scaling is used as preconditioner, a suitable preconditioner for the problem of finding the Lagrange multipliers is introduced. To perform adaptive refinement, the solution of the biharmonic equation on the single patches of the multi-patch domain is discretized by employing (truncated) hierarchical B-splines, and an appropriate refinement strategy of the underlying mesh is presented. Finally, the potential of the developed adaptive IETI-DP method for solving the biharmonic equation over planar multi-patch geometries is numerically tested on the basis of several numerical examples. Thereby, the numerical results show on the one hand optimal convergence behavior with respect to adaptive refinement, and on the other hand a good performance of the proposed preconditioner for the linear problem of determining the Lagrange multipliers.
5. 基于神经形变场的工业 NURBS 几何约束驱动优化与参数化
Constraint-driven Optimization and Parametrization of Industrial NURBS Geometries via Neural Deformation Field
作者: Federico Tamburlin, Giovanni Canali, Giuseppe Alessio D’Inverno, Nicola Demo, Andrea Mola, Gianluigi Rozza
类别: math.NA
发布日期: 2026-06-05
arXiv ID: 2606.07198v1
链接: http://arxiv.org/abs/2606.07198v1
摘要: This work presents a differentiable framework for the parametrization and shape optimization of industrial CAD geometries represented by multi-patch NURBS surfaces. The method enables the deformation of complex CAD models through a physics-informed geometric parametrization, allowing direct morphing driven by physical constraints without the need to prescribe a predefined deformation strategy. A neural displacement field, implemented as a multi-layer perceptron acting on the NURBS control points, provides a compact parametrization of the admissible design space while preserving patch connectivity. Global geometric quantities relevant to hydrostatic design, including displaced volume, wetted surface area and buoyancy centroid, are formulated as differentiable integral operators evaluated on the parametric domain. These quantities are computed through Gauss-Legendre quadrature combined with analytical B-spline derivatives for surface metric evaluation, allowing gradient propagation to the deformation parameters while limiting the computational overhead of automatic differentiation. The proposed framework operates directly on CAD representations without intermediate mesh generation. Numerical experiments on a modified KVLCC2 hull demonstrate the capability of the method to satisfy competing hydrostatic constraints while producing smooth CAD-compatible geometries and showing stable convergence across multiple random initializations.
6. 基于伴随方法的动态对比增强超声灌注估计：对流-扩散模型与双室模型
Adjoint-based Perfusion Estimation from Dynamic Contrast-Enhanced Ultrasound: Advection-Diffusion and Two-Compartment Models
作者: Sophie Externbrink, Ahmed El Kaffas, Dimitre Hristov, Sebastian Götschel
类别: math.NA
发布日期: 2026-06-05
arXiv ID: 2606.07195v1
链接: http://arxiv.org/abs/2606.07195v1
摘要: Tumor perfusion and vascular properties are important determinants of a cancer’s response to therapy. In this paper, we discuss the estimation of spatially varying blood flow velocities and perfusion parameters from time-resolved contrast agent concentration data. We compare a standard parabolic advection-diffusion model against a two-compartment model governed by a coupled system of hyperbolic advection-reaction equations, which is physiologically more sound. To address the inherent ill-posedness of this parameter identification problem, we employ Tikhonov regularization and derive continuous adjoint equations necessary for efficient, gradient-based minimization. We discuss the numerical discretization of the state and adjoint systems using state-of-the-art schemes, and demonstrate the efficacy of the proposed reconstruction algorithms through numerical experiments on synthetic data and in vivo dynamic contrast-enhanced ultrasound measurements.
7. 任意次数的$(\star,ε_1,ε_2)$-结构矩阵多项式的谱分解及其应用
Spectral decomposition of $(\star,ε_1,ε_2)$-structured matrix polynomials with arbitrary degree and its applications
作者: Kang Zhao, Shifang Yuan
类别: math.NA
发布日期: 2026-06-05
arXiv ID: 2606.07176v1
链接: http://arxiv.org/abs/2606.07176v1
摘要: This paper provides the spectral decompositions of $(\star,ε_1,ε_2)$-structured matrix polynomials $P(λ)$ in the unified form by a standard pair and parameter matrix. Using the recursive relationship between the coefficient matrices of $P(λ)$, equivalent expressions of these coefficient matrices are provided. And then the spectral decomposition is applied to solve the inverse eigenvalue problem and the eigenvalue embedding problem with no spill-over.
8. 具有奇异势的随机Cahn–Hilliard方程的数值逼近
Numerical Approximation of the stochastic Cahn–Hilliard equation with singular potential
作者: Ľubomír Baňas, Stefan Metzger
类别: math.NA
发布日期: 2026-06-05
arXiv ID: 2606.07164v1
链接: http://arxiv.org/abs/2606.07164v1
摘要: We discuss the numerical approximation of the stochastic Cahn–Hilliard equation with a singular double-obstacle potential and multiplicative conservative noise. We propose a regularised fully discrete finite element approximation scheme for the problem and show that it satisfies stability estimates which are uniform with respect to the discretization parameters. We show convergence of the approximation for vanishing discretization parameters towards a regularised version of the singular stochastic Cahn–Hilliard equation by monotonicity arguments. Hence, thanks to a uniform $H^1$-estimate for the regularised problem we show that the regularised solution converges towards the pathwise unique probabilistically strong solution of the original singular stochastic Cahn–Hilliard equation. We conclude by presenting numerical simulations where we compare the regularised numerical approximation to its unregularised counterpart and illustrate the effect of the conservative noise.
9. 基于残差校准不确定性的无损害物理信息逆学习
No-Harm Physics-Informed Inverse Learning with Residual-Calibrated Uncertainty
作者: Ronald Katende
类别: math.NA, cs.LG, math.OC
发布日期: 2026-06-05
arXiv ID: 2606.07153v1
链接: http://arxiv.org/abs/2606.07153v1
摘要: Physics-informed learning is increasingly used for partial differential equation (PDE)-governed inverse problems, but its reliability remains difficult to certify. This paper develops a no-harm certification-and-selection framework for physics-informed inverse learning. A learned reconstruction is accepted only when its residual-calibrated radius is no worse than the baseline radius, namely when
$$R_{\mathrm{learn}}\le R_{\mathrm{base}}+\varepsilon_{\mathrm{safe}};$$otherwise, the method returns the baseline. The certificate combines data, physics, boundary or initial-condition, and optimization residuals. Under a conditional stability estimate, these residuals yield an a posteriori reconstruction-error bound and a deterministic uncertainty radius. A high-probability certificate is also derived for physics residuals estimated from independent random collocation points. Numerical tests on Poisson source recovery, inverse heat reconstruction, limited-angle tomography, elliptic coefficient identification, and stochastic residual validation show that the selector accepts certified improvements, rejects shifted, hallucinated, or unfinished candidates, and becomes conservative in strongly ill-posed regimes. The framework is therefore a certification-and-selection layer, not another reconstruction architecture.
10. 通过平均与细化从非结构化样条构造$C^1$极限曲面
Constructing $C^1$ limit surfaces from unstructured splines via averaging and refinement
作者: Syeda Hijab Zahra, Thomas Takacs
类别: math.NA
发布日期: 2026-06-05
arXiv ID: 2606.07149v1
链接: http://arxiv.org/abs/2606.07149v1
摘要: In this paper we present a construction for unstructured splines over quadrilateral meshes by iterative averaging and refinement. We represent the spline as a multi-patch B-spline, where the degrees of freedom are those B-spline coefficients on the quadrilateral patches that are not associated with interior edges and vertices of the mesh, i.e., their corresponding Greville points lie inside the patches. In every averaging step, we replace the remaining B-spline coefficients associated with interior edges and vertices by suitable averages of neighboring degrees of freedom. In the refinement step we apply regular splits to all patches by knot insertion. This process results in a subdivision scheme that, for degree $p=2$, is similar to the almost-$C^1$ spline construction from (Takacs, Toshniwal. CMAME, 2023) and behaves similar to Doo-Sabin subdivision, cf. (Doo, Sabin. CAD, 1978), and that can be defined for arbitrary degrees and regularities inside the patches. We derive two families of spline constructions, based on simple and coplanar averaging, respectively, and analyze their spectral properties when interpreted as subdivision schemes. Using this interpretation, we show that they are $C^1$ in the limit. Moreover, the coplanar averaging scheme produces splines that are $C^1$ at all vertices for every level of refinement, whereas the simple averaging is $C^1$ only in the limit. For both constructions, we have control over the subdominant eigenvalue, which has multiplicity two and can range between $\frac{1}{4}$ and $1$, with $\frac{1}{2}$ often being the desired option. The resulting basis functions form a partition of unity. Moreover, they form a non-negative partition of unity for suitably selected averaging parameters.
11. 面向对数稳定无限维反问题的统一DeepONet框架
A Unified DeepONet Framework for Logarithmically Stable Infinite-Dimensional Inverse Problems
作者: Wen-Jie Wu, Tiexiang Li, Wen-Wei Lin
类别: math.NA
发布日期: 2026-06-05
arXiv ID: 2606.07122v1
链接: http://arxiv.org/abs/2606.07122v1
摘要: We develop a unified DeepONet framework for logarithmically stable infinite-dimensional inverse problems, with inverse acoustic scattering as a model application. The framework is formulated at the operator level by separating the learned inverse map into measurement encoding, finite-dimensional neural approximation, and functional reconstruction components. For inverse maps satisfying a logarithmic stability estimate, we establish quantitative a priori error bounds giving separate estimates for the encoder error, the neural approximation error, and the reconstruction error, thereby characterizing the dependence on the encoder dimension, the network size, and the reconstruction dimension. For comparison, we also record the corresponding Lipschitz-stable estimate arising from the same error decomposition. The abstract theory is then specialized to the recovery of a medium contrast from fixed-frequency far-field measurements. Numerical experiments in two and three dimensions illustrate stable reconstructions under measurement noise.
12. 大规模对数行列式估计的最优随机Krylov技术
Optimal Stochastic Krylov based Techniques for Large- Scale Log-Determinant Estimation
作者: Verlon Roel Mbingui, Antoine Tambue, Issa Karambal
类别: math.NA
发布日期: 2026-06-05
arXiv ID: 2606.07004v1
链接: http://arxiv.org/abs/2606.07004v1
摘要: Estimating the logarithm of the determinant of large sparse positive definite symmetric matrices is an important task in numerical linear algebra, machine learning, Gaussian processes, and uncertainty quantification. In this work, we introduce two scalable and efficient methods for large-scale log-determinant termed the Optimal Stochastic Arnoldi with Incomplete Orthogonalization Procedure (OSA-IOP) and the Optimal Stochastic Lanczos Quadrature (OSLQ). The OSA-IOP approach extends the Incomplete Orthogonalization Procedure (IOP), originally developed for matrix exponential functions for exponential time stepping integrators, to compute the action of the matrix algorithm on a vector. We observe that combining IOP with a randomized Hutch++ algorithm, the OSA-IOP significantly reduces computational cost while maintaining high accuracy. The OSLQ method estimates log-determinants by coupling Lanczos quadrature with Hutch++ and controlled orthogonalization, leveraging Krylov subspaces as efficient quadrature mechanisms to approximate quadratic forms involving the matrix logarithm. We derive error bounds for both methods. Extensive numerical experiments on large-scale sparse matrices from real-world applications demonstrate the accuracy, robustness, and scalability of the proposed approaches.
13. 多尺度松弛逼近：从宏观观测到微观动力学
Multiscale Nudging: From Macroscopic Observations to Microscopic Dynamics
作者: Liyao Lyu, Xinyue Yu, Hayden Schaeffer
类别: math.NA, physics.comp-ph
发布日期: 2026-06-05
arXiv ID: 2606.06809v1
链接: http://arxiv.org/abs/2606.06809v1
摘要: We introduce a measure-based nudging framework for assimilating macroscopic observations into microscopic mean-field particle dynamics. The central difficulty is a representation mismatch: the forecast is a labeled particle system, while the observations specify only a smoothed, permutation-invariant density. To address this mismatch, we define the forecast-observation discrepancy as a quadratic functional on probability measures after applying the same smoothing operator used by the observation process. The Wasserstein gradient of this functional induces a transport velocity on state space, which yields a particle-level correction without constructing particle-to-particle matching, linearizing the dynamics, or estimating ensemble covariances. For a fixed observation scale, we prove well-posedness of the assimilated McKean-Vlasov dynamics and propagation of chaos for the interacting particle approximation. Under exact smoothed observations and an observability condition at the kernel scale, we establish an $L^2$-stability estimate showing exponential decay up to a bias floor controlled by model misspecification. Numerical experiments on linear, bimodal, chaotic, kinetic, and collective-motion systems demonstrate that the method can recover macroscopic structure from incomplete density-level observations.
14. 固体不可压缩线性孔隙弹性：相容格式与可扩展数值求解
Linear poroelasticity with solid incompressibility: consistent formulation and scalable numerical solution
作者: Nicolás A. Barnafi, Andrés E. Rubiano, Ricardo Ruiz-Baier
类别: math.NA
发布日期: 2026-06-04
arXiv ID: 2606.06750v1
链接: http://arxiv.org/abs/2606.06750v1
摘要: In this work we propose, by linearizing the equations of fully nonlinear poroelasticity, a consistent model in which only the solid phase is incompressible. This reformulation circumvents some inconsistency issues encountered in standard primal formulations of nonlinear poroelasticity while still retaining its key physical coupling mechanisms. We show a well-posed and consistent discretization strategy and also formulate scalable solvers based on a Schur complement formalism. A distinctive feature of the model is that it allows for a lowest order, inf-sup stable family of Finite Elements (FE) spaces. Numerical tests in two and three dimensions are provided to validate the proposed method and solver framework.
15. 一种基于谱的网络形成偏微分方程方法
A Spectral-Based Method for Network-Formation PDEs
作者: Pedro Aceves-Sanchez, Pierre Degond, Sara Merino-Aceituno, Claudia Wytrzens
类别: math.NA
发布日期: 2026-06-04
arXiv ID: 2606.06720v1
链接: http://arxiv.org/abs/2606.06720v1
摘要: We propose and study a simple and scalable Fourier-based spectral method for a continuum model of network formation under periodic boundary conditions. The model provides the evolution of the pressure $p$ and the conductivity $m$ over time. The evolution of $p$ is given by an anisotropic Poisson equation, while the equation for $m$ contains three terms corresponding to a diffusion and an activation term of the network – that depends on the gradient of the pressure – as well as a relaxation term that acts as a decaying term. This system arises as a formal $L^2$-gradient flow of a non-convex energy functional. Our algorithm combines two ingredients: (i) a splitting method for the equation for $m$, where the activation and relaxation parts are solved analytically, and the diffusion part is solved via Fast Fourier Transform (FFT), and (ii) an FFT combined with the Conjugate Gradient (CG) method applied to the equation for the pressure. This makes the scheme easy to implement compared to implicit schemes and naturally extensible to three dimensions on uniform periodic grids. To showcase the method, we recover the previously documented influence of the activation strength $c$, the diffusion coefficient $D$, and the metabolic exponent $γ$ on the morphology of emergent networks, and report grid convergence results.
16. 基于Stein无偏风险估计的奇异值平滑硬阈值方法
Smooth Hard-Thresholding for Singular Values with Stein’s Unbiased Risk Estimate
作者: Guanzhong Yang
类别: math.ST, eess.IV, eess.SP, math.NA
发布日期: 2026-06-04
arXiv ID: 2606.06632v1
链接: http://arxiv.org/abs/2606.06632v1
摘要: Low-rank matrix denoising is a central primitive in patch-based image restoration and many other inverse problems. Classical SVD-based image denoising methods often choose a truncation rank by matching residual singular-value energy with an estimated noise energy, but this rule is not a finite-sample risk principle because a fitted low-rank approximation inevitably absorbs part of the noise. This paper develops a mathematically rigorous alternative based on Stein’s unbiased risk estimate (SURE). Since singular value hard thresholding is discontinuous and does not satisfy the hypotheses of Stein’s lemma, we introduce a logistic smooth hard-threshold spectral estimator. We prove that the smooth shrinker satisfies the regularity conditions required by a spectral-estimator version of Stein’s lemma, and therefore admits an exactly unbiased fixed-threshold risk estimate under Gaussian noise. For a fixed observed matrix and a finite set of candidate thresholds separated from the observed singular values, the ordering of the fixed-threshold smooth SURE objective eventually agrees with a simple limiting score. The limiting score has the same algebraic form as the biased hard-threshold SURE formula, but here it is used only as a computational device for ranking finite candidates. Selecting the minimizing threshold is a data-adaptive tuning step; the selected SURE value should not be interpreted as an unbiased risk estimate of the finally selected estimator.
17. 扁长椭球波函数实现快速且指数感知的长程机器学习原子间势
Prolate spheroidal wave functions enable fast and exponent-aware long-range machine learning interatomic potentials
作者: Jiuyang Liang, Libin Lu, Yajie Ji, Shidong Jiang
类别: physics.chem-ph, math.NA
发布日期: 2026-06-04
arXiv ID: 2606.06617v1
链接: http://arxiv.org/abs/2606.06617v1
摘要: Long-range interactions such as electrostatics and dispersion remain a central bottleneck for machine learning interatomic potentials (MLIPs), especially in ionic, polar and interfacial systems. Ewald-based reciprocal-space mechanisms provide a physically grounded route for capturing these nonlocal effects, but often require dense Fourier grids and can become memory-limited at scale. This problem is particularly pronounced in molecular dynamics, where high efficiency requirements make accurate long-range modelling particularly costly. Here we introduce PSWF-LR, an exponent-aware long-range framework based on prolate spheroidal wave functions (PSWFs) that can be easily incorporated into existing model architectures. Its core components are PSWF-based mollification and atom-grid spreading, which enable compact and efficient representation of arbitrary inverse-power channels $1/r^p$ while treating the decay exponent as a physical prior. Across diverse long-range benchmarks, PSWF-LR reduces Fourier-mode requirements, improves energy and force accuracy, accelerates production-level simulations by about threefold, and extends long-range MLIP simulations beyond the memory limits of conventional MLIPs.