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1. 高对比度多尺度偏微分方程中的拓扑谱隙、多尺度Weyl律与均匀化
Topological spectral gap, multiscale Weyl’s law, and homogenization in high-contrast PDEs
作者: Eric T. Chung, Lijian Jiang, Yajun Wang
类别: math.NA, math.AP
发布日期: 2026-06-15
arXiv ID: 2606.16251v1
摘要翻译: 本文引入一个统一的抽象变分与拓扑框架, 用以刻画高对比度多尺度偏微分方程(PDEs)中的谱隙与特征值分布。我们严格证明, 谱隙的精确位置由与高对比度包含体关联的局部零空间的维数万有地决定。对于具有无穷维核的系统, 这一位置由拓扑Betti数严格确定。此外, 我们通过谱解耦定理建立了多尺度Weyl律, 证明当对比度趋于无穷时, 多尺度谱分岔为两个独立分量: 背景介质的Dirichlet谱与包含体的内部Neumann谱。利用谱均匀化理论, 我们还证明在周期趋于零的极限下, 这一扩张的拓扑特征空间渐近地张成宏观均匀化算子的整个谱空间。这些理论结果通过扩散、弹性、四阶板、Maxwell和grad-div算子的数值实验得到了全面验证。
Comment: 27 pages
链接: http://arxiv.org/abs/2606.16251v1
1. 研究对象和背景
本文研究具有高振荡和高对比度系数的多尺度偏微分方程的谱分布问题。这类方程广泛出现于复合材料、超材料、多孔介质等复杂物理系统的建模中，例如标量扩散、线性弹性、四阶板弯曲、时谐 Maxwell 电磁场以及 grad-div 算子。数值模拟的瓶颈在于必须精确分辨微观几何与极端材料对比度，导致计算量巨大。为降低成本，过去二十年发展了多种基于局部谱问题的多尺度模型降阶方法和区域分解预条件技术，如广义多尺度有限元方法 (GMsFEM)、局部正交分解 (LOD)、广义有限元方法 (MS-GFEM)、约束能量最小化广义多尺度有限元 (CEM-GMsFEM) 以及基于广义特征值问题构造粗空间的 GenEO 和自适应 BDDC/FETI-DP 方法。这些方法的鲁棒性高度依赖于能否捕获局部低能物理模态，而这又需要严格理解局部谱行为以及谱间隙的存在性。
对于标量扩散方程，Yalchin 和 Galvis (2010) 已严格证明：当介质中含有 M 个高对比度内含物时，前 M 个特征值以 $\mathcal{O}(\eta^{-1})$ 阶趋于零，而第 $M+1$ 个特征值下方有一致正下界，从而在 $n = M+1$ 处形成尖锐谱间隙。其本质原因是每个内含物内梯度算子的零空间仅由常数函数构成，维数为 1。然而，对于向量值方程或高阶算子，如线弹性（应变算子的零空间为刚体位移，2D 下维数为 3）、四阶板（Hessian 算子的零空间为线性多项式，2D 下维数为 3）、Maxwell 方程和 grad-div 算子，局部零空间可能具有更高的维数甚至无穷维。现有谱间隙理论尚未被系统地推广到此类一般情况，这构成了本文试图填补的空白。
具体而言，考虑定义在有界区域 $\Omega\subset\mathbb{R}^d$ 上的抽象广义特征值问题：求 $u\in V(\Omega)$ 和 $\lambda^\eta$ 使得
$$ a^\eta(u,v) = \lambda^\eta m^\eta(u,v), \quad \forall v\in V(\Omega), $$其中双线性形式按背景矩阵 ($\Omega_0$) 和 $M$ 个不连通高对比度内含物 $D_i$ 分解为
$$ a^\eta(u,v) = a_0(u,v) + \eta a_1(u,v), \quad m^\eta(u,v) = m_0(u,v) + \eta m_1(u,v). $$这里 $a_0, a_1, m_0, m_1$ 与对比度参数 $\eta\gg 1$ 无关。算子 $L$ 取决于具体物理模型：
- 标量扩散：$Lu = \nabla u$，原方程 $-\nabla\cdot(\kappa^\eta\nabla u)=\lambda^\eta\rho^\eta u$；
- 线弹性：$Lu = \varepsilon(u)=\frac12(\nabla u+\nabla u^\top)$，方程 $-\nabla\cdot(\kappa^\eta\varepsilon(u))=\lambda^\eta\rho^\eta u$；
- 四阶板：$Lu = \nabla^2 u$，方程 $\nabla\cdot\nabla\cdot(\kappa^\eta\nabla^2 u)=\lambda^\eta\rho^\eta u$；
- 时谐 Maxwell：$L = \nabla\times$，属于 $H(\mathrm{curl})$ 空间，方程 $\nabla\times(\kappa^\eta\nabla\times u)=\lambda^\eta\rho^\eta u$；
- grad-div 算子：$L = \nabla\cdot$，属于 $H(\mathrm{div})$ 空间，方程 $-\nabla(\kappa^\eta\nabla\cdot u)=\lambda^\eta\rho^\eta u$。
已有理论仅覆盖标量扩散情形（$L=\nabla$，$R=1$）。本文的核心洞察是：谱间隙的位置并非特定于某个方程，而是普适地由局部零空间 $\mathcal{N}(D_i)=\{u\in V(D_i)\mid L u=0\}$ 的代数维数 $R$ 决定。对于有限维零空间，谱间隙出现在 $n = R M + 1$。对于 Maxwell 或 grad-div 等具有无穷维零空间 (因 $\mathrm{Im}(\mathrm{curl})\subset\mathrm{Ker}(\mathrm{div})$) 的系统，直接应用上述框架会导致物理离散谱被无穷多个零本征值淹没。为解决这一根本性困难，本文引入 Hilbert 复形与代数拓扑工具。通过 De Rham 复形
$$ H^1(\Omega)\xrightarrow{\nabla} H(\mathrm{curl};\Omega)\xrightarrow{\nabla\times} H(\mathrm{div};\Omega)\xrightarrow{\nabla\cdot} L^2(\Omega), $$及 Hodge 分解，将冗余的无穷维正合形式商去，揭示出真正有效的局部无能量模式同构于有限维调和空间，其维数恰好由拓扑不变量——Betti 数 $\beta_k$ 给出。至此，谱间隙的存在性可通过纯拓扑信息预测。
此外，在极高对比度极限 $\eta\to\infty$ 下，高频谱的分布尚未被清晰刻画。经典 Weyl 律无法描述因材料对比度导致的能量局域化和内部共振效应。本文进一步建立谱解耦定理，证明多尺度谱在对比度无穷大时分岔为两个独立不变分量：背景介质的 Dirichlet 谱与内含物的内部 Neumann 谱，从而导出新的多尺度 Weyl 律。最后，本文还探讨周期高对比度 PDE 中拓扑低频谱与均匀化理论的关系，证明当周期性趋于零时，拓扑模式的新增本征空间渐近地张满宏观均匀化算子的整个无穷维谱空间。这些成果为多尺度粗空间构造、误差分析和预条件设计提供了严格的理论基础，并将谱分析方法从标量方程推广到复杂拓扑物理系统和高频区域。
2. 主要定理或主要结果
本文的核心成果包括以下严格证明的定理，在适当的 Hilbert 空间框架和对比度 $\eta\to\infty$ 的渐近假设下成立。
1. 通用谱间隙定理 (Theorem 2.2)：
设抽象变分问题 (2.5) 中算子 $L$ 对应的局部零空间 $\mathcal{N}(D_i)$ 具有有限维数 $R$。令 $N = R\times M$ 为由 $M$ 个内含物产生的全局零空间总维数，并假设抽象强制条件 (Assumption 2.1) 成立，即存在与 $\eta$ 无关的常数 $C_{\rm coer}>0$，使得对任意 $u\in V(D_i)$ 且 $u\perp\mathcal{N}(D_i)$ 相对于 $m_1$ 内积，有 $a_1(u,u)\ge C_{\rm coer} m_1(u,u)$。则当 $\eta\to\infty$ 时，前 $N$ 个特征值满足 $\lambda_n^\eta \le \mathcal{O}(\eta^{-1})\to 0$，并且存在另一常数 $C_{\rm gap}>0$ 与 $\eta$ 无关，使得第 $N+1$ 个特征值下界严格为正：$\lambda_{N+1}^\eta \ge C_{\rm gap}$。从而谱间隙严格出现在 $n = N+1 = RM+1$ 处。
该定理直接给出各物理系统的谱间隙位置：标量扩散 ($R=1$) 为 $M+1$；2D 线弹性 ($R=3$) 为 $3M+1$，3D 线弹性 ($R=6$) 为 $6M+1$；2D 四阶板 ($R=3$) 为 $3M+1$，3D 板 ($R=4$) 为 $4M+1$。所需强制常数分别由 Poincaré 不等式、第二 Korn 不等式和 Bramble-Hilbert 引理提供，从而保证 $C_{\rm coer}$ 存在且仅依赖于内含物的几何形状。
2. 拓扑谱间隙定理 (Theorem 4.1)：
对于 Maxwell 方程 ($k=1$, $L=\nabla\times$) 或 grad-div 算子 ($k=2$, $L=\nabla\cdot$)，其自然函数空间 $V_k$ 的局部零空间均为无穷维。为提取离散谱，定义 gauge-fixed 子空间 $W := \{ u\in V_k(\Omega) \mid (u, d_{k-1} p)_{L^2(\Omega)} = 0,\,\forall p\in V_{k-1}(\Omega) \}$，其中 $d_{-1}=0$。在每个内含物 $D_i$ 上，第 $k$ 阶上同调群 $H^k(D_i) := \mathrm{Ker}(d_k|_{D_i}) / \mathrm{Im}(d_{k-1}|_{D_i})$ 的维数等于 $k$ 阶 Betti 数 $\beta_k(D_i)$（即 $D_i$ 中 $k$ 维“孔洞”的数量）。定义总拓扑指标 $N_{\rm topo} = \sum_{i=1}^M \beta_k(D_i)$。则当 $\eta\to\infty$ 时，限制在 $W$ 内的特征值问题前 $N_{\rm topo}$ 个特征值满足 $\lambda_n^\eta \le \mathcal{O}(\eta^{-1})$，而 $\lambda_{N_{\rm topo}+1}^\eta \ge C_{\rm coer} > 0$，其中 $C_{\rm coer}$ 为 generalized coercivity constant (如 Maxwell-Poincaré 或 Friedrichs 常数)。于是，谱间隙的位置由 Betti 数严格决定。
实例：对于 $\Omega$ 内含有两个实心球体和一个环形内含物（每个环形 $\beta_1=1$, 球体 $\beta_1=0$），则 $N_{\rm topo}=1$，谱间隙出现在 $n=2$。
3. 多尺度 Weyl 律 (Section 5)：
引入新颖的 contrast-weighted $L^2$-normalization 技术，建立谱解耦定理 (Spectral Decoupling Theorem)。其核心结论为：当 $\eta\to\infty$ 时，原问题的多尺度谱渐近地分裂为两个互不耦合的分量，即
其中 $\sigma^{\rm Dir}(\Omega_0)$ 表示背景介质在零边值条件下的 Dirichlet 谱，而 $\sigma^{\rm Neu}(D)$ 为所有内含物内部 Neumann 谱的并集。基于此，建立多尺度 Weyl 律：特征值计数函数满足
$$ \mathcal{N}_{\rm ms}(\lambda) = \mathcal{N}_{\Omega_0}^{\rm Dir}(\lambda) + \sum_{i=1}^M \mathcal{N}_{D_i}^{\rm Neu}(\lambda) + o(\lambda^{d/2}), \quad \lambda\to\infty, $$其中每一项计数函数遵循经典 Weyl 渐近公式 $\mathcal{N}(\lambda) \sim C_{\rm Weyl}\,\lambda^{d/2}$，且常数由对应区域和边界条件下的导波速度决定。该定律提供了高频谱分布的量化叠加描述。
4. 周期谱均匀化与谱完备性 (Section 6)：
考虑周期高对比度介质，令微观周期长度 $\varepsilon\to 0$，利用双尺度收敛和谱均匀化理论证明：拓扑低频谱间隙的相对比例与 $\varepsilon$ 无关（鲁棒带隙分离）。进一步建立谱完备性定理：当 $\varepsilon\to 0$ 时，由 $N_{\rm topo}$ 个拓扑低阶模态张成的本征空间在适当的极限意义下渐近地张满宏观均匀化算子的整个无穷维谱空间。这意味着微观拓扑信息通过均匀化极限被提升为宏观算子的谱基底完备性。
3. 研究方法、关键技术和核心工具
本文的主要创新点和技术方法体现在以下几个方面：
1. 抽象变分流形与谱间隙的极大极小刻画
对于有限维核的通用框架，关键在于将谱间隙位置与局部零空间维数关联。证明采用 Courant-Fischer Min-Max 和 Max-Min 原理：
- 上界构造：直接使用全局零空间 $S_N = \bigoplus_{i=1}^M \tilde{\mathcal{N}}(D_i)$（其中 $\tilde\cdot$ 表示零延拓）作为试探空间。由于在 $D_i$ 内 $Lu=0$，高对比度惩罚项 $a_1(u,u)=0$，Rayleigh 商退化为 $\frac{a_0(u,u)}{m_0(u,u)+\eta m_1(u,u)}$，从而前 $N$ 个特征值 $\le \mathcal{O}(\eta^{-1})$。
- 下界构造：利用 Max-Min，选 $V_N = S_N$，则任何与 $S_N$ 正交的试探函数 $u^*$ 自然满足局部正交条件 $u^*|_{D_i}\perp \mathcal{N}(D_i)$。由 Assumption 2.1，$a_1(u^*,u^*)\ge C_{\rm coer} m_1(u^*,u^*)$，再适当处理分母中的背景质量项，得到 $\lambda^\eta_{N+1}\ge C_{\rm coer}$（当 $\eta\to\infty$ 时）。
该框架将谱间隙的证明彻底转化为对局部强制常数的依赖，而强制常数由标准泛函不等式保证。
2. 强制常数的物理化归约
文本将抽象强制条件实例化为三类经典分析结果，从而直接给出不同物理系统的间隙位置。这些并非简单的引用，而是揭示了各不等式的物理对应：
- 标量扩散：Poincaré-Wirtinger 不等式 $\int_{D_i}|\nabla u|^2\ge C_P \int_{D_i} |u|^2$ 要求均值零，这正是正交于零空间（常数函数）。
- 线弹性：第二 Korn 不等式 $\int_{D_i}|\varepsilon(u)|^2\ge C_K \int_{D_i}|u|^2$ 要求位移正交于刚体运动，其验证涉及 Killing 向量场的 $L^2$ 正交。
- 四阶板：Bramble-Hilbert 引理或高阶 Poincaré 不等式 $\int_{D_i}|\nabla^2 u|^2\ge C_{BH}\int_{D_i}|u|^2$ 要求与 $P_1$ 多项式正交，这是通过混合泛函分析技术可严格成立的。
这些结果为各种物理系统的粗空间设计提供了显式指南。
3. De Rham 复形与拓扑不变量的融合
对于 Maxwell 和 grad-div 算子，核心障碍是无界正合形式产生的无穷维谱。作者的真正创新在于将代数拓扑工具引入多尺度谱分析，具体路径如下：
- 利用 de Rham 上同调群 $H^k(D_i) = \mathrm{Ker}(d_k)/\mathrm{Im}(d_{k-1})$ 提取“有效”零空间，通过 Hodge 分解 $V_k(D_i)=\mathrm{Im}(d_{k-1})\oplus \mathfrak{H}^k(D_i)\oplus \mathrm{Im}(d_k^*)$ 得到标准调和代表，使 $\dim\mathfrak{H}^k(D_i) = \dim H^k(D_i)$。
- 引用 de Rham 定理，指出 $\dim H^k(D_i) = \beta_k(D_i)$，即 Betti 数。由此证明有限维调和空间正是真正的局部无能量模式，且其维数完全由内含物拓扑（连通分量个数、隧道数、空腔数）决定。
- 在 gauge-fixed 空间 $W$ 内重新定义特征值问题，商掉了正合分量，从而恢复了离散谱。定理 4.1 的证明沿袭了有限维情况的 Min-Max 策略，但强制常数变为 Maxwell-Poincaré 型不等式，其成立依赖于闭值域定理和复形性质。这体现了功能分析中标准工具（Hilbert 复形、Hodge 分解）在多尺度问题中的巧妙融合。
4. 对比度加权 $L^2$ 归一化与谱解耦
高频分析的传统难点在于 $\eta$ 同时影响能量和质量对称双线性形式，导致能量不均衡。本文引入一种新颖的归一化技巧：考虑重新标度的特征值问题
通过考虑 $\eta\to\infty$ 的极限，并将其与两个极限问题（背景 Dirichlet 问题 $a_0(u,v)=\lambda m_0(u,v)$ 在 $\Omega_0$ 上和内含 Neumann 问题 $a_1(u,v)=\lambda m_1(u,v)$ 在各 $D_i$ 上）匹配，作者利用质量集中现象（mass concentration dichotomy）证明任何特征函数的极限或者集中在背景中，或者集中在内含物内，进而得到谱解耦。详细证明使用了变分收敛理论和特征值分支扰动技术（如 Kato 扰动理论），最终导出多尺度 Weyl 律的特征值计数叠加公式。
5. 双尺度收敛与谱均匀化
在周期高对比介质中，作者采用双尺度收敛和谱均匀化标准工具，但引入了拓扑低频谱在 $\varepsilon\to 0$ 下的渐近行为分析。关键点是证明拓扑谱间隙的相对比值 $\lambda_{N_{\rm topo}+1}^\eta / \lambda_{N_{\rm topo}}^\eta \to \infty$ 均匀地关于 $\varepsilon$ 成立，从而确保了带隙分离的鲁棒性。谱完备性定理的证明则依赖于将宏观均匀化算子的任意模函数用微观拓扑调和的有限维近似来逼近，利用了 $H$-收敛或双尺度收敛意义下的模性质。
4. 与之前工作的比较
本文在多个层次上显著推进了既有文献，具体对比分析如下：
1. 谱间隙理论的扩展：从标量扩散到一般算子族
先期工作 [21] (Yalchin and Galvis, 2010) 仅针对标量扩散方程建立了谱间隙位于 $n=M+1$ 的结果，其证明依赖于梯度算子零空间仅含常数这一事实被显式使用。本文定理 2.2 将该结果扩展至任意具有有限维局部零空间的算子，包括向量值线弹性、高阶板问题等。核心推广在于识别出间隙位置 = $(\dim\mathcal{N}(D_i))\times M + 1$，并提供一套基于抽象强制条件的统一证明。此外，作者通过明确给出不同物理问题的强制常数来源（Poincaré、Korn、Bramble-Hilbert 不等式），提供了具体的谱间隙位置表格 (Table 1)，这是此前文献未系统整理过的。与 [21] 相比，本文的假设更加抽象，但给出了更全面的适用范围。
2. 突破有限维核限制：拓扑谱间隙与前人方法
对于 Maxwell 或 grad-div 系统，由于核的无穷维性，传统的维度计数法失败。此前在多尺度方法中（如 GMsFEM 用于 Maxwell 问题 [11, 17]），通常通过选取一组局部多项式或特定函数来构造粗空间，但缺乏关于“应选多少个模态”的先验理论，且未建立类似谱间隙的严格理论。本文定理 4.1 是首次将谱间隙的存在性建立在拓扑不变量（Betti 数）之上。这一工作与利用 De Rham 复形分析有限元空间稳定的 H(div) 和 H(curl) 方法的文献（如 [3, 5]）有本质不同：后者关注离散模板的稳定性，而本文将连续上同调信息用于多尺度谱的零空间刻画。通过 Hodge 分解商掉正合形式，作者成功提取出真正决定低频谱行为的调和空间，并论证其维数等于 Betti 数，这在多尺度方法领域是全新的。该结果也暗示了对于复杂拓扑，为获得鲁棒收敛，多尺度粗空间的维度不应由任意截断决定，而应由介质拓扑决定，这对方法设计与误差分析具有指导意义。
3. 高频谱分布和多尺度 Weyl 律
经典 Weyl 律 [21] 中曾将标量扩散的特征值计数与区域体积联系，但通常假设系数光滑或温和振荡。对于高对比度多尺度问题，谱刚性的局部能量集中使得全局 Weyl 律失效。本文的谱解耦定理和多尺度 Weyl 律将总计数显式拆分为背景 Dirichlet 谱和内含 Neumann 谱的叠加。这与既有的多尺度特征值估计方法（如 [2, 33] 中考虑的局部特征值问题）截然不同：后者旨在通过加边修正近似低频特征，而本文直接给出了极限谱的完整分布，并保留了前三阶 Weyl 渐近项。代价是假设对比度 $\eta\to\infty$ 的渐近极限，但该极限在许多实际超材料中是可接近的。在数值验证部分（第7节），作者对五种物理模型进行了实验，证实了谱间隙、谱分裂以及特征值计数与理论的吻合。
4. 谱均匀化中的拓扑模式
已有谱均匀化理论 (如 Allaire 等, [14] 相关文献) 主要关注于证明不同物理系统在 $\varepsilon\to 0$ 时谱的收敛性，但很少讨论介质拓扑对极限谱基底的影响。本文证明在内含物拓扑模式下，当 $\varepsilon\to 0$ 时这些低频本征空间将覆盖整个均匀化算子谱空间，这为均匀化理论提供了基底完备性视角。同时，带隙鲁棒性结果（相对比例恒定）表明，即使微观周期变化，粗空间的维数要求（$N_{\rm topo}$）不会因 $\varepsilon$ 减小而失去比例，这对多尺度算法中粗空间选取的稳定性具有实际意义。这一部分相比传统的双尺度均匀化工作，鲜明地突出了拓扑信息在谱分析中的超越性。
5. 对计算方法的启示与局限
本文明确指出其理论可直接服务于多尺度降阶方法的先验粗空间维数选择（例如 GMsFEM 的局部基函数数量），并为局部子空间迭代 (LSI) 等方法的收敛率提供分析工具。然而，作为纯理论工作，它未涉及实际粗空间构造的具体离散化选择（如基函数局部边界条件的优化）。此外，对于多个内含物靠近导致相互作用较强的情况，本文假设内含物彼此分离且对比度足够大，未讨论近场耦合的影响，这可能是在实际应用中需要进一步考虑的。与前人相比，本文的谱解耦限于线性谱，对非线性问题或非自伴算子的扩展尚未涉及。但总体而言，本文为一大类高对比度多尺度 PDE 的数值算法提供了坚实且统一的数学基础。
了解领域论文（摘要翻译）
1. 约束Helfrich流的保几何结构参数化有限元逼近
Geometric structure-preserving parametric finite element approximations for the constrained Helfrich flow
作者: Xiaoxiao Liu, Quan Zhao
类别: math.NA
发布日期: 2026-06-15
arXiv ID: 2606.16697v1
链接: http://arxiv.org/abs/2606.16697v1
摘要翻译
中文标题: 约束Helfrich流的保几何结构参数化有限元逼近
摘要翻译: 我们针对闭合曲线和曲面的约束Helfrich流提出了一种保结构参数化有限元方法。该方法基于两阶段速度分裂策略。在第一阶段，通过曲率演化方程计算法向速度，其中体积和表面积约束以法向速度形式软施加。这一步骤近似了Helfrich流的梯度流结构，其全离散参数化有限元逼近导致线性系统，并在全离散层面给出了无条件能量耗散估计。在第二阶段，通过将计算得到的法向速度与BGN型切向速度相结合来更新曲面网格。还使用了时间加权界面法向和面积修正乘子来精确保持封闭体积和表面积。该修正步骤导致非线性系统，可通过牛顿迭代高效求解。由此得到的方法同时实现了能量衰减、精确几何守恒和良好的网格质量。对二维曲线和三维曲面的数值实验，包括非光滑初始数据和非零自发曲率，展示了所提方法的精度、稳健性和保结构性质。
2. 完整Ericksen-Leslie模型的几何重述与结构保持旋转离散梯度格式
A geometric reformulation and structure-preserving rotational discrete gradient scheme for the full Ericksen-Leslie model
作者: Hanbin Wang, Jie Xu, Zhiguo Yang
类别: math.NA
发布日期: 2026-06-15
arXiv ID: 2606.16563v1
链接: http://arxiv.org/abs/2606.16563v1
摘要翻译
中文标题: 完整Ericksen-Leslie模型的几何重述与结构保持旋转离散梯度格式
摘要翻译: 本文针对具有一般各向异性Oseen-Frank弹性与完整Leslie应力的向列型液晶流动的完整Ericksen-Leslie模型，提出了一种结构保持的旋转离散梯度方法。主要困难在于在离散层面上同时保持指向矢场的逐点单位长度约束和能量耗散，这涉及长度约束、各向异性弹性能量与流体动力学耦合之间相互作用的高非线性。为解决这一问题，我们首先将Ericksen-Leslie系统重述为一个等价的旋转形式，其中指向矢的演化本质上切于单位球面，且Leslie应力和Ericksen应力中的耦合项被重新组织，使得能量交换与耗散结构显式化。基于这一重述，我们构造了保持指向矢长度且满足无条件离散能量律的旋转离散梯度格式，以及一个基于精确无散谱近似的全离散格式。数值实验验证了所提方法的精度、单位长度保持性和能量稳定性，并展示了完整Ericksen-Leslie模型中各向异性弹性系数与剪切流所诱导的动力学效应。
3. 多孔弹性介质中显式波传播的等几何分析
Isogeometric Analysis for Explicit Wave Propagation in Poroelastic Media
作者: Maarten M. Hodzelmans, René R. Hiemstra, Joris J. C. Remmers, Clemens V. Verhoosel
类别: math.NA
发布日期: 2026-06-15
arXiv ID: 2606.16293v1
链接: http://arxiv.org/abs/2606.16293v1
摘要翻译
中文标题: 多孔弹性介质中显式波传播的等几何分析
摘要翻译: 对于显式动力学问题的高阶离散，与经典有限元分析（FEA）相比，等几何分析（IGA）具有若干有利性质。FEA 在超过线性的阶数下会产生虚假模态，而 IGA 则不会出现这种情况。因此，达到同等精度所需的自由度更少，可采用更大的时间步长，且该方法对非线性问题更为稳健。若移除异常模态，时间步长甚至几乎与阶数无关。本文研究这些优势如何适用于多孔弹性连续体模型。我们同时考虑原始形式，其中变量为基体材料位移、流体位移和压力，以及消去了压力的简化形式。在离散化中，我们采用散度协调样条空间。流体位移的协调样条空间保证了原始形式的 inf-sup 稳定性，并在简化形式中给出了正确的零空间。此外，我们证明并展示当两种位移均采用协调样条空间离散时，两种形式等价。通过谱分析，我们发现 IGA 的上述优势确实可直接推广到多孔弹性领域。在一维情形，我们将离散谱分解为快波和慢波。当用解析解进行归一化时，每个子谱都极为类似于弹性力学中已知的结果。因此，当采用无异值 IGA 离散多孔弹性问题时，时间步长本质上与阶数无关。我们同样在二维情形展示了这种时间步长缩放规律。
4. 用于二维非线性弹性的一阶兼容应变混合四边形有限元
First-Order Compatible-Strain Mixed Quadrilateral Finite Elements for 2D Nonlinear Elasticity
作者: Mohsen Jahanshahi, Damiano Pasini, Arash Yavari
类别: math.NA
发布日期: 2026-06-14
arXiv ID: 2606.15925v1
链接: http://arxiv.org/abs/2606.15925v1
摘要翻译
中文标题: 用于二维非线性弹性的一阶兼容应变混合四边形有限元
摘要翻译: 兼容应变混合有限元方法（CSMFEs）利用非线性弹性的微分复形来构造保持底层拓扑结构的离散格式。现有的CSMFEs专注于可压缩与不可压缩非线性弹性的单纯形网格。本文针对可压缩与不可压缩固体，发展了适用于四边形单元的兼容应变混合格式。对于一般四边形单元，Piola变换仅在矩形、平行四边形和梯形等特殊几何形状下才能保持切向和法向于单元边的向量场，因此无法用于构造兼容形函数。为克服这一限制，形函数直接通过数值积分在物理空间中计算得出。采用第一类Nédélec形函数插值位移梯度，并引入一类新的、与位移和位移梯度离散相协调的应力形函数。可压缩格式沿用了之前为单纯形CSMFEs建立的框架；而对于不可压缩情形，提出了一种采用压力场单元层面凝聚的新格式，从而避免了引入额外的整体自由度。这些进展将兼容应变混合有限元框架从单纯形网格推广至一般四边形网格，同时适用于可压缩与不可压缩非线性弹性。数值算例表明，所提出的四边形单元能够以更少的自由度求解原本需要二阶单纯形CSMFEs处理的问题。
5. PML-BIE方法在阻抗半空间中声散射问题的收敛性
Convergence of the PML-BIE Method for Acoustic Scattering in an Impedance Half-Space
作者: Wangtao Lu
类别: math.NA
发布日期: 2026-06-14
arXiv ID: 2606.15878v1
链接: http://arxiv.org/abs/2606.15878v1
摘要翻译
中文标题: PML-BIE方法在阻抗半空间中声散射问题的收敛性
摘要翻译: 基于完美匹配层的边界积分方程(PML-BIE)方法(Lu等人, \emph{SIAM J. Appl. Math.} 78 (2018))已成为无界区域中波散射问题的有效工具。尽管该方法在实际应用中取得了成功，但在许多物理相关的设置下，严格的收敛理论仍不完整。本文提出了一个建立PML-BIE方法收敛性的一般框架，并以阻抗半空间中的声散射作为第一个示例。该框架将分析分解为三个部分：PML截断偏微分方程的收敛性、PML问题与精确PML-BIE之间的等价性、以及通过用拉伸自由空间核替换精确核而得到的可计算PML-BIE的收敛性。针对阻抗半空间问题，我们证明了精确和可计算PML-BIE格式的收敛性。最终的误差界随PML吸收强度的增加呈指数衰减。所得到的理论为PML-BIE方法提供了严格基础，并为层状和非均匀介质中更一般的散射问题提供了参考模型。
6. 具有保正性的强稳定性保持积分因子Runge-Kutta方法求解微分Lyapunov方程
Strong Stability Preserving Integrating Factor Runge-Kutta Methods for Differential Lyapunov Equations with Positivity Preservation
作者: Wenhui Zhang, Rihui Lan, Xiaobo Jing
类别: math.NA
发布日期: 2026-06-14
arXiv ID: 2606.15703v1
链接: http://arxiv.org/abs/2606.15703v1
摘要翻译
中文标题: 具有保正性的强稳定性保持积分因子Runge-Kutta方法求解微分Lyapunov方程
摘要翻译: 本文提出了二阶和三阶积分因子强稳定性保持Runge-Kutta方法用于求解微分Lyapunov方程。提出的格式突破了传统的阶数障碍，同时严格保持数值解的对称性和半正定性（SPSD）——这是稳定性分析和控制理论应用中的本质性质。此外，我们为二阶格式提供了严格的误差估计。数值实验验证了方法的精度及其保持SPSD性质的能力。
7. 一种面向多面体网格上线弹性问题的交错间断伽辽金方法
A Staggered Discontinuous Galerkin Method for linear elasticity problem on Polytopal Meshes
作者: Long Chen, Xuehai Huang, Ruishu Wang, Ran Zhang
类别: math.NA
发布日期: 2026-06-13
arXiv ID: 2606.15223v1
链接: http://arxiv.org/abs/2606.15223v1
摘要翻译
中文标题: 一种面向多面体网格上线弹性问题的交错间断伽辽金方法
摘要翻译: 本文基于Hellinger-Reissner变分原理，提出了一种用于线弹性的新型交错间断伽辽金（SDG）方法。我们在任意多面体网格上构造了具有单元边界法向连续性的对称应力空间，同时使用在相同网格上定义的分片多项式函数来逼近位移场。该方法无锁且满足线动量和角动量的局部平衡。我们给出了全面的理论分析，包括稳定性证明和误差估计。该格式具有可杂交化结构，显著简化了数值实现。数值实验验证了理论结果，并展示了所提方法的有效性。
8. 物理一致的潜变量孪生
Physics-conforming Latent Twins
作者: Matthias Chung, Yutong Bu, Deepanshu Verma
类别: cs.LG, math.NA
发布日期: 2026-06-13
arXiv ID: 2606.15053v1
链接: http://arxiv.org/abs/2606.15053v1
摘要翻译
中文标题: 物理一致的潜变量孪生
摘要翻译: 代理模型是科学机器学习的核心，它为复杂物理系统提供快速预测、仿真、推断和控制。然而，对于含时问题，仅对训练轨迹进行精确插值是不够的：可靠的代理模型还应尊重赋予这些轨迹物理意义的守恒律、不变量、容许性条件和耗散结构。我们引入物理一致的潜变量孪生（Physics-conforming Latent Twins），一种用于学习潜变量代理解算子的框架，其动力学通过设计满足选定的物理原理。该方法建立在潜变量孪生（Latent Twin）表述之上，联合学习编码器、解码器和任意时间索引状态之间的潜变量流映射，同时约束潜变量动力学以保持或耗散指定的结构量。我们发展了一种约束转移观点，将原始状态空间中的物理结构与潜空间中兼容的约束联系起来，并证明了结构保持界，表明潜变量强制执行如何改善解码后对物理缺陷的控制。我们还推导了潜变量流映射保持线性与二次不变量或施加耗散不等式的代数条件。在代表性常微分方程和偏微分方程基准上的数值实验表明，在保持准确代理预测的同时，约束满足度、结构保真度和定性的长时间行为均得到改善。
9. 一种能量守恒的非交错电磁势粒子云网格方法，第一部分：非相对论广义动量表述
An Energy-Conserving Unstaggered Electromagnetic-Potential Particle-in-Cell Method, Part I: Non-relativistic Generalized-Momentum Formulation
作者: Andrew J. Christlieb, Luis Chacon, Sining Gong
类别: math.NA
发布日期: 2026-06-13
arXiv ID: 2606.15035v1
链接: http://arxiv.org/abs/2606.15035v1
摘要翻译
中文标题: 一种能量守恒的非交错电磁势粒子云网格方法，第一部分：非相对论广义动量表述
摘要翻译: 我们针对Lorenz规范下的非相对论Vlasov-Maxwell系统，发展了一种基于势的非交错粒子云网格方法。场更新写成关于标量势、矢量势及其时间导数的一阶波动方程组的Crank-Nicolson离散。电荷密度不直接沉积；而是利用粒子沉积的电流，通过离散连续性方程推进。这为算法带来了灵活性和一系列创新，包括在离散层面保持Lorenz规范和高斯定律的非交错网格布局。在此势表述中，这种源序还允许在离散层面保持Lorenz规范和高斯定律。为将该范式扩展至能量守恒表述，我们引入了一致的轨道平均散射、收集与粒子推进。为了保证能量一致性，通过用与轨道平均映射相容的网格插值矢量势的轨道平均离散梯度替换矢量势的逐点中点导数，修改了正则动量的更新。该构造沿着每条粒子轨道满足精确的有限差分链式法则。因此，粒子功等于出现在Crank-Nicolson场能平衡中的网格功，从而在非线性求解器容差和舍入误差范围内实现了精确的总能量守恒。我们在冷双流不稳定性问题上，展示了该方法在三维情况下的精确能量守恒。
10. 一种针对常系数时谐波传播问题的带传输变量的 HDG 方法
A HDG method with transmission variables for time-harmonic wave propagation problems with constant coefficients
作者: Simone Pescuma, Gwénaël Gabard, Théophile Chaumont-Frelet, Axel Modave
类别: math.NA
发布日期: 2026-06-12
arXiv ID: 2606.14950v1
链接: http://arxiv.org/abs/2606.14950v1
摘要翻译
中文标题: 一种针对常系数时谐波传播问题的带传输变量的 HDG 方法
摘要翻译: 时谐波问题的迭代有限元求解器因其问题的基本性质而收敛速度极慢。我们提出了一种更适合于快速迭代求解的可杂交间断伽辽金 (HDG) 方法的变体。与通过引入单元面上辅助数值通量来消去物理未知量的标准杂交策略不同，我们的方法是在这些面上引入一个传输变量。对于 Helmholtz 问题，这种称为 CHDG 的重新表述已被证明相对于标准 HDG 能够显著加速迭代格式的收敛。本工作将 CHDG 扩展到一个涵盖常系数波传播问题的通用框架中，能够以统一的方式处理多种波型。我们证明了所得的杂交系统是适定的，且适用于不动点迭代。作为一个实际应用，我们将该方法应用于具有均匀亚声速平均流的时谐线性化欧拉方程。通过涉及声波和涡旋波的二维数值基准测试说明了该方法，并对多种迭代格式在一系列配置下的收敛行为进行了系统比较。
不相关论文（基本信息）
1. 基于 Nitsche 方法的拉普拉斯特征值问题有限元法：谱逼近与后验误差分析
Nitsche-based FEM for the Laplace eigenvalue problem: spectral approximation and a posteriori error analysis
作者: Arbaz Khan, David Mora, Jesus Vellojin
类别: math.NA
发布日期: 2026-06-15
arXiv ID: 2606.17052v1
链接: http://arxiv.org/abs/2606.17052v1
摘要: In this paper, we present the numerical analysis of an elliptic eigenvalue problem in which the essential boundary condition is imposed weakly by means of the Nitsche method. The resulting discrete eigenvalue problem is studied within the framework of compact operator theory. We prove norm convergence of the discrete solution operator and derive error estimates for the eigenvalues and eigenfunctions, with rates depending on the chosen Nitsche variant. In addition, we develop an a posteriori error analysis and propose a residual-based estimator suitable for adaptive refinement. Several numerical experiments are presented to assess the convergence, stability and robustness of the method, including the influence of the Nitsche stabilization parameter and the performance of the adaptive strategy.
2. 线性算子的最优多尺度学习
Optimal Multiscale Learning of Linear Operators
作者: Jiaheng Chen, Daniel Sanz-Alonso
类别: math.ST, math.NA, stat.ML
发布日期: 2026-06-15
arXiv ID: 2606.16913v1
链接: http://arxiv.org/abs/2606.16913v1
摘要: We study the statistical and computational limits of learning bounded linear operators between Sobolev spaces from noisy input-output data. In wavelet coordinates, the problem is recast as an infinite-dimensional matrix regression problem with a heterogeneous two-sided multiscale structure. We establish minimax rates under Sobolev operator-norm loss and construct a finite-resolution blockwise least-squares estimator attaining these rates. The analysis reveals a nonuniform local estimation difficulty across scales, which can be exploited algorithmically: by assigning scale-adaptive sample sizes, the estimator achieves the optimal computational cost among dense least-squares implementations.
3. PF-DIC: 相场数字图像相关方法用于全场位移、应变与损伤一体化测量
PF-DIC: Phase field digital image correlation for integrated full-field displacement, strain, and damage measurements
作者: Dingxiang Zhu, Ye Lu
类别: math.NA, cond-mat.mtrl-sci, physics.optics
发布日期: 2026-06-15
arXiv ID: 2606.16850v1
链接: http://arxiv.org/abs/2606.16850v1
摘要: This work presents a novel digital image correlation (DIC) framework for full-field measurements of displacement, strain, and damage, based on a phase field (PF) approach. The idea is to take advantage of the ability of the PF method to track complex crack morphologies and to provide a natural way in DIC to perform damage and crack measurements from experimental speckle images, in addition to displacement and strain fields. Moreover, incorporating the damage variable into DIC can improve the displacement accuracy near the crack tip, and can avoid the need of user-defined masks when dealing with cracked samples, which is advantageous when cracks become complex and the manual application of masks becomes challenging. The theoretical formulation of the proposed framework, namely PF-DIC, was presented in detail in the paper, along with a finite element implementation. Numerical examples have demonstrated the capability of the proposed PF-DIC in terms of capturing different types of cracks while improving the measurement accuracy of displacement fields. Additionally, it is shown that the PF-DIC can be easily adapted to selectively identify critical cracks under specific loading conditions or mechanisms for damage assessment and diagnostic purposes. The proposed DIC framework offers a powerful and automatic damage measurement technique and paves the way for the unification of PF simulations and experimental measurements of fracture and opens numerous opportunities in science and engineering, e.g., materials defect characterization and structural health monitoring.
4. 基于零空间方法的任意截面梁变偏移连接公式
A variable-offset joint formulation for beams with arbitrary cross-sections using a null space method
作者: Myung-Jin Choi, Roger A. Sauer, Simon Klarmann, Sven Klinkel
类别: math.NA
发布日期: 2026-06-15
arXiv ID: 2606.16755v1
链接: http://arxiv.org/abs/2606.16755v1
摘要: In this paper, we present a variational formulation of local configurational constraints that couple multiple beams with arbitrarily shaped cross-sections. Since this formulation requires no explicit interface to rotational degrees-of-freedom, it applies to any beam kinematics and finite element discretization. Here, we define the offset coordinates in a moving frame to constrain or release the relative position between connected beams. The present method is based on a first-order approximation of the Lagrange multiplier field in the cross-section, which limits the transferability of the joint to the resultant force and moment only. The multipliers are eliminated using a discrete null space method, which provides size reduction and improved conditioning of the system matrix. Further, we apply the developed formulation to a beam element based on extensible directors and to a brick element in nonlinear elastostatics. Several numerical examples are presented.
5. 混合边界条件下椭圆问题有限元法中数值积分的影响
The effect of numerical integration in the FEM for elliptic problems with mixed boundary conditions
作者: Juan Han, Haijun Wu
类别: math.NA
发布日期: 2026-06-15
arXiv ID: 2606.16675v1
链接: http://arxiv.org/abs/2606.16675v1
摘要: This paper investigates the impact of quadrature accuracy for volume and face integrals in the finite element method using $p$-th order polynomial shape functions for elliptic problems with mixed Dirichlet and Robin boundary conditions. The optimal $p$-th order $H^1$-convergence is maintained when numerical integration with algebraic precision at least $2p-2$ for volume terms and at least $2p-1$ for face terms is adopted. For $L^2$-error, we achieve optimal $O(h^{p+1})$ convergence when using quadrature rules of precision no less than $\max\{p,2p-2\}$ for volume terms and no less than $2p-1$ for face terms. Of particular significance, we present two examples to show that the above result on $L^2$-error is sharp for the linear FEM ($p=1$). When reduced to the case of Dirichlet boundary condition, our results yield improved dependence on the given data compared to the classical results established by Ciarlet, \textit{et al}. Numerical experiments are provided to illustrate the theoretical findings and confirm the necessity of specified quadrature accuracy and data regularity.
6. 黑塞变化下的切比雪夫精确加速，第一部分：正弦-雅可比方法
Chebyshev-Exact Acceleration under Hessian Variation, I: Sine-Jacobi Method
作者: Dmitry Pasechnyuk-Vilensky, Martin Takáč
类别: math.OC, math.NA
发布日期: 2026-06-15
arXiv ID: 2606.16671v1
链接: http://arxiv.org/abs/2606.16671v1
摘要: We study finite-horizon one-gradient realizations with the Chebyshev minimax terminal residual on $[μ,L]$. Under time-dependent Hessian perturbations, the terminal first variation is governed by a time-ordered spectral kernel $K_s(λ,ν)$; its sharp $\ell_2$ gain is $A_N$. For the prefix-exact Chebyshev recurrence,
$$ A_N^{\rm pref} =\frac{ε_N^\star}{L-μ} \left(4N^2+16\sum_{m=1}^{N-1}m^2\right)^{1/2} =\frac4{\sqrt3}\frac{N^{3/2}}{L-μ}ε_N^\star(1+o(1)), $$and this is sharp in the causal two-term class with Chebyshev exactness at every prefix. For terminal-only exactness, Jacobi coordinates give $P_N=2^{1-N}T_N$: the spectrum is fixed at the midpoint Chebyshev nodes, while the spectral weights parametrize the realizations. The sine weights give a final-exact Jacobi method with the same terminal residual and
$$ A_N(J_N^{\sin}) =2\sqrt{c_{\sin}}\frac{N^{3/2}}{L-μ}ε_N^\star(1+o(1)), \; 2\sqrt{c_{\sin}}\approx2.137936<4/\sqrt3. $$Thus the Chebyshev terminal polynomial does not determine the first-order Hessian-drift gain. The experiments show the finite-horizon effect: lower stochastic curvature overhead, larger admissible-block frontiers, accurate time-varying quadratic predictions, and lower restart cost on an endpoint-coupled smooth strongly convex GLM.
7. CREATOR 项目：迈向计算电机实验室
The CREATOR Project: Towards a Computational Electric Machine Laboratory
作者: Sebastian Schöps, Annette Muetze, Herbert De Gersem, Herbert Egger, Manfred Kaltenbacher
类别: cs.CE, math.NA
发布日期: 2026-06-15
arXiv ID: 2606.16653v1
链接: http://arxiv.org/abs/2606.16653v1
摘要: The Collaborative Research Centre TRR 361/F90 CREATOR (2022-2030) aims at establishing a new paradigm for the simulation-driven design of electric machines. Increasing demands on efficiency, power density and sustainability require the integration of multiphysical effects, advanced materials and complex geometries into the design process. Traditional sequential workflows are no longer sufficient to address these challenges. CREATOR therefore combines expertise from electrical engineering, applied mathematics, fluid dynamics and materials science to establish integrated modelling, simulation and optimisation methodologies in a single large-scale project funded by the German and Austrian national funding agencies. This article provides an overview of the research vision, key achievements from the first funding period (2022-2026) and current developments towards a computational electric machine laboratory.
8. 核基多层插值的节点表示
Nodal Representations for Kernel-Based Multilevel Interpolation
作者: Lorenz Gollwitzer, Rüdiger Kempf, Holger Wendland
类别: math.NA
发布日期: 2026-06-15
arXiv ID: 2606.16643v1
链接: http://arxiv.org/abs/2606.16643v1
摘要: We study the kernel-based multilevel method for approximating or learning a multivariate function from scattered data, motivated in part by recent applications in sparse grid methods. For nested families of point sets, we derive a nodal representation of the multilevel interpolant that depends explicitly on the values of the target function. This representation allows us to characterize the range of the associated interpolation operator and to construct a cardinal basis for this space. We prove that the resulting basis functions exhibit exponential decay, analogous to the localization properties known for certain kernel-based Lagrange functions. We further analyze the computational cost of the resulting formulation. For non-nested families of point sets, we derive a generalized nodal representation.
9. PhysGuard：面向仿真到真实神经偏微分方程代理模型的Fisher引导梯度投影
PhysGuard: Fisher-Guided Gradient Projection for Sim-to-Real Neural PDE Surrogates
作者: Changjian Zhou, Junfeng Fang, Negin Yousefpour, Peng Wu, Bin Yan, Guillermo A Narsilio
类别: cs.LG, math.NA, physics.comp-ph
发布日期: 2026-06-15
arXiv ID: 2606.16602v1
链接: http://arxiv.org/abs/2606.16602v1
摘要: Neural operator models trained on simulation data often lose accuracy when applied to experimental measurements due to the sim-to-real gap. Standard fine-tuning with limited real data can reduce this gap, but it may also damage the core physics-relevant representations learned during pretraining. Although knowledge-preserving adaptation has been widely investigated in vision or language tasks, it remains unclear whether these methods are suitable for neural operators whose architectures and protected knowledge are fundamentally different. Neural operators need to preserve core-scale physical structures rather than semantic or visual features. We propose PhysGuard, a physics-preserving framework for accurate sim-to-real adaptation of neural operators. Specifically, PhysGuard uses the empirical Fisher Information Matrix computed on simulation data to identify physics-critical parameter directions, then restricts fine-tuning updates to directions that do not interfere with them. A layer-wise Gram-matrix formulation makes this efficient for models with millions of parameters, while an adaptive threshold automatically determines the protected subspace size. A spectral probe experiment shows that the dominant Fisher directions are strongly associated with low-frequency output structures. Experiments on benchmark across four neural operator architectures and different physical systems show that PhysGuard performs strongly on most evaluation metrics compared to baselines. The benefits are most evident under severe domain shift, where it reduces low-frequency error by up to 32% compared to standard fine-tuning while maintaining adaptability. Our code is available at https://github.com/ZhouChaunge/PhysGuard.
10. 具有诺伊曼边界条件的抛物型热方程的热星游走
Walking on Heat Stars for Parabolic Heat Equations with Neumann Boundary Conditions
作者: Anchang Bao, Enya Shen, Jianmin Wang
类别: cs.GR, math.NA
发布日期: 2026-06-15
arXiv ID: 2606.16578v1
链接: http://arxiv.org/abs/2606.16578v1
摘要: Monte Carlo methods have proven highly effective for elliptic partial differential equations through algorithms such as Walk on Spheres and Walk on Stars, which evaluate solutions at individual points without volumetric meshing or global linear solves. Extending these methods to the transient regime has remained an open challenge: parabolic equations couple space and time through an anisotropic scaling, requiring joint sampling of spatial displacements and backward time steps whose distribution was not previously available in a unified, exact form. We present Walk on Heat Stars, a grid-free Monte Carlo solver that closes this gap by extending the boundary integral framework of Walk on Stars to the parabolic setting. Our method introduces a non-cylindrical boundary integral formulation that accommodates the time-varying domains induced by heat-ball sampling. The heat ball geometry is parameterized by a logarithmic time coordinate and a spatial direction, revealing that the double-layer kernel factorizes into independent Gamma and uniform components. This factorization enables exact directional importance sampling of the recursive next walk position, the Neumann flux contribution, and the volumetric source term. We further derive a decoupled gradient estimator that expresses spatial derivatives as weighted boundary integrals of the solution, requiring no recursion on the gradient, and adapt a heteroscedastic regression-based denoiser to the space-time domain for variance reduction. We validate our method on analytical solutions across a range of geometries and spatial frequencies, confirm convergence at the expected Monte Carlo rate, and demonstrate practical applicability on heat sink and cooling scenes with mixed or pure Neumann boundary conditions.
11. 用于二维奇异摄动问题的 Petrov-Galerkin 变分物理信息神经网络框架
Petrov-Galerkin Variational Physics-Informed Neural Network Framework for Two-Dimensional Singularly Perturbed Problems
作者: Vijay Kumar, Gautam Singh
类别: math.NA, cs.LG
发布日期: 2026-06-15
arXiv ID: 2606.16510v1
链接: http://arxiv.org/abs/2606.16510v1
摘要: This study proposes a Petrov-Galerkin based Variational Physics-Informed Neural Network (VPINN) for efficiently solving two-dimensional singularly perturbed problems (SPPs) with one and two small perturbation parameters. The approach employs neural networks to construct the trial solution space, while tensor-product hat functions are adopted as test functions to enforce the variational form. To accurately resolve of sharp boundary layers, the variational form is implemented using a Petrov-Galerkin formulation. Dirichlet boundary conditions are imposed directly, while the source terms are computed using automatic differentiation. Computational experiments on standard two-dimensional problems demonstrate that the proposed method achieves high accuracy in both the maximum and L_2 norms. These results confirm the efficiency and robustness of the Petrov-Galerkin VPINN approach in accurately capturing the multiscale features of two-dimensional SPPs.
12. Grassmann流形上的测地线插值：GLERP与递归GIDER插值法
Geodesic Interpolation on the Grassmann Manifold: GLERP and Recursive GIDER Interpolants
作者: Shingyu Leung
类别: math.NA
发布日期: 2026-06-15
arXiv ID: 2606.16463v1
链接: http://arxiv.org/abs/2606.16463v1
摘要: This article develops a geodesic interpolation framework for data on the Grassmann manifold. The motivation is that many matrix-valued data sets represent subspaces rather than fixed bases: if the columns of two matrices differ only by a right orthogonal transformation, then they describe the same point on $Gr(r,m)$. Interpolation should therefore be invariant under this basis ambiguity. We first introduce $GLERP$, a Grassmann analogue of spherical linear interpolation defined by the Grassmann exponential and logarithm maps. The method follows the constant-speed geodesic joining two nearby subspaces and is second-order accurate for smooth Grassmann-valued curves under the usual normal-neighborhood condition. We then define $GIDER_n$, a recursive higher-order construction obtained by replacing each affine interpolation step in Neville’s algorithm by $GLERP$. The resulting interpolant matches $n+1$ subspace data exactly, is basis-invariant, and is locally of order $n+1$ for sufficiently smooth curves. We compare the construction with tangent-space interpolation and projection-matrix interpolation, discuss intrinsic and extrinsic error measures, and present numerical tests. The results confirm the expected convergence orders and show that $GIDER_n$and tangent-space interpolation are nearly indistinguishable in a smooth local regime, while projection-matrix interpolation provides a useful extrinsic baseline. We also outline how the recursive construction can be used as the basis of a Grassmann ENO procedure.
13. 非均质井筒系统中断裂的相场分析：套管偏心率与水泥环-地层界面强度的影响
Phase-field analysis of fracture in heterogeneous wellbore systems: effects of casing eccentricity and cement-formation interface strength
作者: Tharunsarathy, Wasim Niyaz Munshi, Chandrasekhar Annavarapu, Birendra Jha
类别: cs.CE, cond-mat.mtrl-sci, math.NA, physics.geo-ph
发布日期: 2026-06-15
arXiv ID: 2606.16346v1
链接: http://arxiv.org/abs/2606.16346v1
摘要: Predicting the initiation and propagation of cracks in heterogeneous wellbore systems under complex in-situ conditions remains challenging. We present a hybrid phase-field fracture framework to model crack growth in heterogeneous wellbore systems with weak interfaces. The framework is first validated against benchmark problems with available analytical and numerical solutions. Subsequently, numerical experiments are conducted to isolate the effects of interface strength and casing eccentricity on crack growth. The results show that casing eccentricity strongly influences both the pressure at crack initiation and the resulting crack paths, reducing the crack initiation pressure by up to 30% relative to the concentric configuration. Beyond a critical eccentricity threshold of 50%, localized shear stresses drive the nucleation of inclined cracks in the formation in addition to radial cracking – a failure mode absent in concentric configurations. For sufficiently weak interfaces (i.e., interfaces with 30% of the strength of the surrounding bulk material), radially propagating cracks in the cement sheath are deflected along the interface rather than penetrating into the formation. This deflection delays stress relaxation within the sheath, promotes the nucleation of additional radial cracks, and increases the risk of sustained casing pressure and wellbore failure. Finally, a three-dimensional simulation reveals depth-dependent crack nucleation, stress-shadow effects that suppress full-depth crack growth and crack coalescence along the cement-formation interface – phenomena that are fundamentally inaccessible under plane-strain assumptions - demonstrating the applicability of the framework to realistic heterogeneous wellbore systems.
14. symveig: 对称分解的埃尔米特矩阵的验证特征值包络
symveig: Verified eigenvalue enclosures for symmetry-decomposed Hermitian matrices
作者: Sarang Vehale, Ritu Goel
类别: cond-mat.str-el, math.NA, physics.comp-ph
发布日期: 2026-06-15
arXiv ID: 2606.16217v1
链接: http://arxiv.org/abs/2606.16217v1
摘要: Exact diagonalization of quantum lattice Hamiltonians returns floating-point eigenvalues whose accuracy is not certified: rounding error, eigensolver behaviour, and ill-conditioning can corrupt a result without warning. We present \texttt{symveig}, a pure NumPy/SciPy package that computes rigorous, machine-checkable enclosures of all eigenvalues of a Hermitian matrix, with an optional symmetry-sector decomposition. For a matrix that commutes with an abelian conserved quantity diagonal in the working basis (for example the total magnetization of a spin model), the package verifies each symmetry sector independently. Because each sector block is much smaller than the full matrix, this yields enclosures that are both tighter (by a factor of $3$-$9$ across system sizes $L = 4$-$12$) and dramatically faster (a wall-clock speedup of up to $130\times$ at $L = 12$) than verifying the full matrix, while never forming or diagonalizing it. Every enclosure half-width is a guaranteed upper bound on the distance from a computed eigenvalue to the nearest true eigenvalue under IEEE~754 round-to-nearest arithmetic, obtained by explicit floating-point error analysis with no heuristic slack. The implementation requires neither INTLAB nor MATLAB, bringing rigorously verified eigenvalue enclosure into the standard scientific-Python stack used in computational physics. We validate the package on $1$D Heisenberg (open and periodic), $J_1$-$J_2$ Heisenberg, and $2$D Heisenberg lattices, confirming that every computed eigenvalue is contained in its enclosure across all tested configurations.
15. 基于晶格格林函数的无穷晶格离散 Calderón 投影在主动噪声屏蔽与约束中的应用
Infinite-lattice discrete Calderón projection via the lattice Green’s function for active noise shielding and confinement
作者: Qing Xia
类别: math.NA, physics.comp-ph
发布日期: 2026-06-15
arXiv ID: 2606.16192v1
链接: http://arxiv.org/abs/2606.16192v1
摘要: We construct an infinite-lattice discrete Calderón projection for the Helmholtz equation by convolution with the lattice Green’s function (LGF), and apply it to active noise shielding and confinement on Cartesian grids with arbitrary geometry. The LGF fixes the outgoing radiation condition and removes the geometry-dependent auxiliary Helmholtz problem and artificial outer boundary from the projection and control synthesis; its finite numerical tabulation depends only on $(h,k)$ and is reusable across geometries. We prove idempotence, characterize the range as the trace space of interior lattice-Helmholtz solutions, and establish range equivalence with a well-posed Tsynkov-type projection. The two projectors coincide as operators when the auxiliary problem reproduces the exact lattice radiation condition. A capacity-matrix realization yields closed-form shielding and confinement densities supported on the exterior and interior sublayers of a single lattice boundary strip, respectively. For pure-noise shielding, exterior-sublayer measurements suffice under explicit invertibility assumptions; preservation of an unknown wanted interior field requires the full strip trace. Experiments on circular, L-shaped, and star-shaped regions verify machine-precision cancellation for LGF-consistent sources and near-second-order convergence for analytic plane waves and point sources. Conditioning and measurement noise tests quantify the configuration dependence of the reconstruction.
16. 针对具有 Dzyaloshinskii–Moriya 相互作用的 Landau–Lifshitz–Gilbert 方程的质量集中中点有限元方法与保结构求解器的先验误差分析
A priori error analysis of a mass-lumped midpoint finite element method with a structure-preserving solver for the Landau–Lifshitz–Gilbert equation with Dzyaloshinskii–Moriya interaction
作者: Agus L. Soenjaya
类别: math.NA
发布日期: 2026-06-15
arXiv ID: 2606.16142v1
链接: http://arxiv.org/abs/2606.16142v1
摘要: The Landau–Lifshitz–Gilbert (LLG) equation is a fundamental model in micromagnetics for the magnetisation dynamics of ferromagnets at temperatures well below the Curie temperature. In chiral magnetic materials, the bulk Dzyaloshinskii–Moriya interaction (DMI) contributes a first-order term to the effective field, induces a natural chiral boundary condition, and plays an important role in the dynamics of skyrmions. In this paper, we analyse a mass-lumped midpoint finite element scheme for the LLG equation with exchange interaction and bulk DMI. The fully discrete scheme preserves the nodal unit-length constraint exactly and satisfies a discrete energy law. Under suitable regularity assumptions on the exact solution, we prove an optimal-order convergence estimate in the energy norm, first order in space and second order in time. To our knowledge, this is the first such a priori error estimate for the mass-lumped midpoint finite element method, even in the exchange-only case. Since the midpoint scheme is nonlinear, its practical implementation requires an algebraic solver that is compatible with the geometric structure of the method. To this end, we propose a structure-preserving fixed-point iteration which preserves the nodal unit-length constraint at every iterate. We further prove that, under an appropriate residual stopping criterion, the resulting inexact scheme retains the same convergence rate as the ideal midpoint scheme, up to the contribution of the solver tolerance. Numerical experiments support the theoretical results and illustrate the robustness and structure-preserving properties of the method.
17. 用于线性偏微分方程数据同化的随机特征卡尔曼滤波
Random-Feature Kalman Filtering for Linear PDE Data Assimilation
作者: Xi’an Li, Jiale Linghu, Yangshuai Wang
类别: math.NA
发布日期: 2026-06-15
arXiv ID: 2606.16086v1
链接: http://arxiv.org/abs/2606.16086v1
摘要: Data assimilation for time-dependent partial differential equations (PDEs) requires Bayesian updates of an evolving field from streaming, sparse, and noisy observations, while keeping the filtering state finite dimensional. We introduce a random-feature Kalman filtering framework for linear PDE data assimilation. Once the random features are frozen and the linear PDE is Galerkin discretized, the coefficient vector satisfies a finite-dimensional linear-Gaussian state-space model, so the Kalman recursion gives the exact posterior for the chosen coefficient model. For non-orthogonal random-feature draws, we construct a mass-whitened effective-rank coordinate system that removes near-null mass directions and identifies the posterior dimension $r$. For the heat equation with implicit-Euler time stepping, we prove a high-probability posterior-contraction and PDE-consistency theorem in these mass-whitened coordinates. The mean-square $L^2$ reconstruction error separates into an effective-rank feature approximation term, a deterministic time-consistency term, and a Bayesian estimation term. In the high-information regime, the leading posterior contribution scales as $rσ^2/N_o$, where $σ^2$ is the observation-noise variance and $N_o$ is the number of observations per analysis time. Thus the analysis distinguishes the exact coefficient-space posterior from deterministic PDE approximation errors, and gives a checkable uncertainty-quantification guarantee for random-feature filtering of a representative parabolic PDE.
18. PyTorch 中的有限维 I 型 von Neumann 代数：面向随机块对角算子的 GPU 加速框架
Finite-Dimensional Type I von Neumann Algebras in PyTorch: A GPU-Accelerated Framework for Random Block-Diagonal Operators
作者: Irina Nikolaeva, Andrej Novikov
类别: cs.MS, math.NA, math.OA, math.QA, math.SP, quant-ph
发布日期: 2026-06-14
arXiv ID: 2606.15882v1
链接: http://arxiv.org/abs/2606.15882v1
摘要: We present \texttt{torch_vn_algebra}, an open-source Python library built on PyTorch for numerical experiments with finite-dimensional Type I von Neumann algebras (direct sums of matrix algebras). The library provides: $\bullet$ a compact batched tensor representation $(B,C,k_{\max},k_{\max})$ that handles both Monte Carlo samples and multiple direct summands; $\bullet$ lazy evaluation of operators to avoid unnecessary memory allocation; $\bullet$ generation of random operators with arbitrary eigenvalue distributions (user-provided samplers) and various unitary ensembles (Haar, $\mathrm{SU}(n)$, COE, CSE, diagonal phases); $\bullet$ functional calculus via SVD (absolute value, square root, inverse, entropy) and a hybrid method for extreme eigenvalues (exact diagonalisation for $k_{\max}\le256$, otherwise power iteration); $\bullet$ three trace functionals (blunt, normalised subspace trace, and the von Neumann tracial state); $\bullet$ GPU-accelerated batched linear algebra for moderate-scale Monte Carlo studies (e.g., $2\times10^4$ samples of $100\times100$ operators). The library is validated against analytical expectations (Haar moments, trace properties). Performance benchmarks on a Tesla P100 GPU are presented and discussed. Limitations and future work are outlined. The code is open-source.
19. DLRA对SDEs的指数收敛性
Exponential Convengence of DLRA for SDEs
作者: Jianhai Bao, Haitao Wang, Yue Wu
类别: math.PR, math.NA
发布日期: 2026-06-14
arXiv ID: 2606.15843v1
链接: http://arxiv.org/abs/2606.15843v1
摘要: We study dynamical orthogonal (DO) approximations of stochastic differential equations and investigate their long-time behaviour. The DO formulation represents the solution by a low-rank decomposition and leads to a coupled system consisting of an evolution equation on the Stiefel manifold and a reduced stochastic process. We establish the well-posedness of the strong DO system and derive quantitative error estimates between the original stochastic differential equation and its low-rank approximation in the Wasserstein distance. Our main contribution is the analysis of invariant probability measures for the DO dynamics. Under suitable dissipativity, Lipschitz continuity, and non-degeneracy assumptions on the coefficients, we prove the existence of an invariant probability measure for the strong DO system. The proof combines uniform moment estimates, a Krylov–Bogoliubov argument for an associated frozen system, and a Kakutani-Fan-Glicksberg fixed-point theorem to recover the self-consistent dynamics. We further show that the induced low-rank process admits an invariant probability measure and discuss the structure of invariant measures through several illustrative examples. These results provide a rigorous foundation for the use of dynamical low-rank approximations in the approximation of long-time statistical properties of stochastic dynamical systems.
20. 优化高阶IMEX-RK格式求解退化扩散反应问题及其在行波现象中的应用
Optimized high-order IMEX-RK schemes for degenerate diffusion-reaction problems with application to travelling waves phenomena
作者: Paola F. Antonietti, Mattia Corti, Giuseppe Orlando
类别: math.NA
发布日期: 2026-06-14
arXiv ID: 2606.15726v1
链接: http://arxiv.org/abs/2606.15726v1
摘要: We study a class of IMplicit-EXplicit Runge–Kutta (IMEX-RK) schemes for the numerical approximation of reaction and diffusion-reaction problems arising in a variety of biological and physical applications. Such models may admit travelling wave solutions, with the Fisher–Kolmogorov equation representing a prototypical example. Motivated by this feature, the proposed time integration schemes are designed to accurately capture sharp propagating fronts. We also investigate a less standard use of IMEX-RK methods that circumvents a splitting of reaction terms into linear and nonlinear components, while still requiring the solution of linear systems at each stage. This semi-implicit formulation, referred to as SI-IMEX-RK, enables a targeted treatment of stiffness by isolating its relevant contributions. The time discretization is coupled with a high-order polygonal discontinuous Galerkin method for space discretization, resulting in a flexible and robust framework for the treatment of multiscale dynamics in complex geometries. A comprehensive validation strategy is presented to assess the accuracy and stability properties of the proposed schemes across a hierarchy of increasingly challenging test problems.
21. 基于张量列车随机向量的随机迹估计
Stochastic trace estimation with tensor train random vectors
作者: Zvonimir Bujanović, Daniel Kressner, Hrvoje Olić
类别: stat.ML, cs.LG, math.NA
发布日期: 2026-06-14
arXiv ID: 2606.15679v1
链接: http://arxiv.org/abs/2606.15679v1
摘要: Stochastic trace estimation is a standard tool for approximating the trace of a large-scale matrix available only through matrix-vector products. However, in tensor-structured settings, unstructured Gaussian or Rademacher test vectors may be prohibitively expensive to store and compute with, while cheaper rank-one tensor-product vectors can require sample complexities that grow exponentially with the tensor order. This work studies Gaussian random tensor train vectors as a structured alternative for stochastic trace estimation. We show that, with a suitable choice of the tensor train rank, random tensor train vectors recover dimension-independent guarantees for the Girard–Hutchinson estimator. In particular, a median-of-means variant with tensor train rank $r \geq d-1$ achieves the same dependence on the accuracy $\varepsilon$ and failure probability $δ$ as the classical estimator based on unstructured Gaussian vectors. We further prove an oblivious subspace injection result for sketches formed from independent Gaussian random tensor train vectors: tensor train rank $r\geq d-1$ and $\mathcal{O}(\varepsilon^{-2}(k+\log(1/δ)))$ samples suffice for a $k$-dimensional target subspace. Finally, we investigate the use of such sketches within the Nyström++ framework. We show that the resulting estimator can achieve the desired $\mathcal{O}(\varepsilon^{-1})$ sample complexity under an additional spectral-tail condition. These results provide clarififcation on both the potential and the limitations of random tensor train vectors in stochastic trace estimation.
22. 阻抗圆柱体斜入射散射的耦合边界积分方程的高阶Nyström方法
A High-Order Nyström Method for Coupled Boundary Integral Equations in Oblique-Incidence Scattering by Impedance Cylinders
作者: Haochen Liu, Qinghao Yu
类别: math-ph, math.NA
发布日期: 2026-06-14
arXiv ID: 2606.15616v1
链接: http://arxiv.org/abs/2606.15616v1
摘要: We study the numerical solution of electromagnetic scattering by an infinitely long impedance cylinder under oblique incidence. After separation of the axial phase factor, the axial electric and magnetic components satisfy a pair of coupled two-dimensional Helmholtz equations. The Leontovich impedance condition couples these components through tangential derivatives, and the associated boundary integral system contains both logarithmic kernels and principal-value tangential derivative terms. Building on existing coupled integral-equation formulations for oblique-incidence cylinder scattering, we construct a high-order Nystrom implementation based on Kress-type logarithmic kernel decomposition, periodic product quadrature, Fourier differentiation for the tangential derivative contribution, and a block diagonal preconditioner associated with the scalar impedance subproblem. Under uniqueness of the continuous scattering problem and a uniform discrete stability assumption, we formulate a high-order convergence framework for the boundary densities and far-field patterns. Numerical experiments include a manufactured Fourier-Bessel benchmark, a plane-wave circular-cylinder validation, a smooth non-circular boundary test, condition-number and GMRES comparisons, and a variable-impedance scattering-width reduction example in a prescribed backward angular sector. The results indicate that the method provides a stable high-accuracy forward solver for the coupled impedance system, rather than a new physical model.
23. 用于时间相关偏微分方程的液体随机特征方法
Liquid Random Feature Methods for Time-Dependent Partial Differential Equations
作者: Jiale Linghu, Yangshuai Wang
类别: physics.comp-ph, math.NA
发布日期: 2026-06-14
arXiv ID: 2606.15571v1
链接: http://arxiv.org/abs/2606.15571v1
摘要: A central challenge in mesh-free space–time approximation for time-dependent partial differential equations is to represent evolving temporal scales while keeping residual minimization computationally tractable. Random feature methods simplify this algebraic problem by freezing nonlinear trial functions and fitting only a linear readout, but standard static space–time activations provide no explicit relaxation-scale mechanism, making temporal-scale resolution a finite-dimensional bottleneck in stiff, dispersive, or multi-scale regimes. We introduce liquid random feature methods (L-RFM), which replace static temporal activations by closed-form liquid time-constant responses with sampled relaxation scales. The resulting frozen features form temporally structured local or global trial spaces with analytic space–time derivatives for residual least-squares assembly. A density theorem proves density of the deterministic trial spaces in the continuous space–time function class, and a temporal-rank calculation clarifies the role of sampled relaxation scales. Ablation and finite-feature tests identify the liquid temporal response as the primary source of the observed accuracy improvement. Across stiff reaction–diffusion, nonlinear transport, dispersive, complex-valued, and multidimensional benchmarks, L-RFM improves finite-feature accuracy in regimes where temporal-scale representation controls the approximation. By embedding relaxation scales directly into frozen trial functions, L-RFM provides a route to high-accuracy continuous space–time surrogates for evolutionary PDEs while preserving the simplicity of linear least-squares solvers.
24. 概率论、数论与动力系统交叉中的数值积分
Numerical Integration at the Intersection of Probability Theory, Number Theory and Dynamical Systems
作者: Zicheng Wang, Haisen Zhang, Xu Zhang
类别: math.DS, math.NA
发布日期: 2026-06-13
arXiv ID: 2606.15375v1
链接: http://arxiv.org/abs/2606.15375v1
摘要: This paper is devoted to the study of uniformly distributed (u.d.) sequences on general non-compact topological spaces. Convergence-determining classes and the equivalent characterizations of u.d. sequences on $T_4$ spaces are discussed. For Polish spaces endowed with regular probability measures, the existence of a countable convergence-determining class for u.d. sequences is proved by virtue of the tightness of Borel probability measures. It is shown that, under the condition that a countable convergence-determining class exists, both independent identically distributed (i.i.d.) randomly sampled sequences and orbit sequences of ergodic transformations are u.d.\ with full measure.
25. 基于张量网络降维的指数尺度线性代数
Linear algebra at exponential scale via tensor network dimension reduction
作者: Chris Camaño, Ethan N. Epperly, Raphael A. Meyer, Joel A. Tropp
类别: math.NA, cond-mat.str-el, cs.DS, quant-ph
发布日期: 2026-06-13
arXiv ID: 2606.15350v1
链接: http://arxiv.org/abs/2606.15350v1
摘要: Many problems in modern scientific computing are challenging because of a \emph{curse of dimension}, where their mathematical formulation involves objects whose dimension is \emph{exponential} in the nominal “size” of the problem. Tensor networks can provide a compact representation for exponentially large vectors and matrices that arise in applications, but these representations do not always lead to reliable algorithms. This paper develops and analyzes techniques for randomized dimension reduction of tensor network data. These techniques support a suite of efficient algorithms for provably solving exponential-scale linear algebra problems, including trace estimation and eigenvalue approximation. The paper includes several stylized illustrations from quantum many-body physics with ambient dimension up to $2^{200}$.
26. 最小正则性下双调和Steklov特征值问题的协调和非协调虚拟元法
Conforming and non-conforming virtual element methods for the biharmonic Steklov eigenvalue problem with minimum regularity
作者: Dibyendu Adak, Daniele Boffi, Francesca Gardini, Gianmarco Manzini, Jesus Vellojin
类别: math.NA, math.FA
发布日期: 2026-06-13
arXiv ID: 2606.15340v1
链接: http://arxiv.org/abs/2606.15340v1
摘要: In this work, we analyze the conforming and $C^0$-non-conforming Virtual Element Method for a fourth-order Steklov eigenvalue problem on a generally shaped, possibly nonconvex, polygonal domain. By employing an {\it enriching } operator, we derive the convergence analysis using the discrete $H^2$ seminorm, and the $H^1$ and $L^2$ norms. We use the Babuška–Osborn spectral theory \cite{BO} to prove that the numerical scheme approximates the spectrum without introducing any spurious eigenvalue. Moreover, we derive the optimal order of convergence for eigenfunctions and double order for eigenvalues. We assess the performance of the method on several numerical tests using different families of polygonal meshes.
27. 面向有限体积环境网格自适应的各向同性恢复型误差估计器算法及其在大气流动中的应用
An isotropic recovery-based error estimator algorithm for mesh adaptation in a finite volume environment with application to atmospheric flows
作者: Lander Besabe, Michele Girfoglio, Simona Perotto, Annalisa Quaini, Gianluigi Rozza
类别: math.NA
发布日期: 2026-06-13
arXiv ID: 2606.15296v1
链接: http://arxiv.org/abs/2606.15296v1
摘要: We develop an Isotropic Recovery-based Error Estimator (IREE) to drive mesh adaptation within a finite volume framework. Recovery-based error estimators are widely used in practice thanks to their simplicity, relying solely on the available discrete solution and a suitable post-processing step. While recovery-based estimators are well established in the finite element framework, their application to finite volume methods, despite the widespread use of the latter in industrial and commercial codes, remains largely unexplored and motivates the present study. We assess the performance of the proposed IREE-driven mesh adaptation procedure through well-known benchmarks for dry atmospheric flows modeling and compare it against a widely-used plain mesh adaptation algorithm. Both qualitative and quantitative results show that the IREE algorithm is more effective at suppressing numerical instabilities compared with the plain approach and, for most of the simulation time, provides a superior accuracy, though at a moderately higher computational cost. In addition, both the adaptive approaches offer substantial computational savings (between 35% and 94%) relative to simulations based on fixed uniformly fine meshes. These findings demonstrate that IREE-based mesh adaptation is a promising and effective strategy for atmospheric flow simulations and has the potential to substantially reduce the computational effort of generating reanalysis-quality data.
28. 具有多项式势的欠阻尼Langevin方程的时间一致弱收敛显式数值方法
A uniform-in-time weakly convergent explicit numerical method for the underdamped Langevin equation with polynomial potentials
作者: Diancong Jin
类别: math.NA, math.PR
发布日期: 2026-06-13
arXiv ID: 2606.15175v1
链接: http://arxiv.org/abs/2606.15175v1
摘要: The underdamped Langevin equation is a fundamental model in statistical mechanics for sampling Gibbs measures and simulating molecular dynamics, for which numerical methods with uniform-in-time weak convergence are essential for accurately reproducing long-time statistical observables and invariant measures of the underlying dynamics. Currently, such uniform-in-time weak convergence is established for implicit schemes, but remains unknown for explicit ones under polynomially growing potentials. To improve efficiency in long-time simulations, we propose the first explicit numerical method for the underdamped Langevin equation with polynomially growing potentials that is proven to achieve uniform-in-time weak convergence. The explicit numerical method is constructed by introducing a dissipativity on the scalar auxiliary variable (SAV), which we call the DSAV method. The proposed DSAV method enables the approximation of the invariant measure for the underdamped Langevin equation with a precision of $\varepsilon$ at a significantly reduced computational cost of $\mathcal{O}(\varepsilon^{-1} \log(\varepsilon^{-1}))$. In addition, we establish the existence and positivity of the density function of the numerical solution without using the Malliavin calculus. Numerical experiments are performed to verify the theoretical findings and demonstrate the long-time stability of the proposed numerical method.
29. 双调和波动方程的空间混合高阶方法与时间隐式格式
Hybrid-high order method in space and implicit schemes in time for the biharmonic wave equation
作者: Raman Kumar, Neela Nataraj, Aamir Yousuf
类别: math.NA
发布日期: 2026-06-13
arXiv ID: 2606.15163v1
链接: http://arxiv.org/abs/2606.15163v1
摘要: This article presents the numerical analysis for the biharmonic wave equation with clamped boundary conditions employing two variants of the {hybrid high-order} method for the space discretization and two implicit time-stepping schemes for the time discretization. The Newmark scheme directly discretizes the second-order time derivative, while the Crank-Nicolson scheme discretizes a reformulated system where we introduce velocity as an independent variable to create coupled first-order equations. Optimal orders of convergence in space and time are achieved for both schemes. The numerical experiments validate the theoretical convergence rates and show the effectiveness of the proposed methods. To the best of our knowledge, this is the first work in literature that addresses hybrid-high order method and implicit time schemes for the biharmonic wave equation.
30. 一种高维Wigner-Poisson系统的保结构自适应秩方法
A Structure-preserving Adaptive-Rank Approach to the High-Dimensional Wigner-Poisson System
作者: Andrew J. Christlieb, Sining Gong, Jing-Mei Qiu, Nanyi Zheng
类别: math.NA
发布日期: 2026-06-13
arXiv ID: 2606.15067v1
链接: http://arxiv.org/abs/2606.15067v1
摘要: The Wigner-Poisson system is a deterministic phase-space model for quantum kinetic electron dynamics, but high-dimensional simulations are limited by the full 3D3V phase space and the nonlocal Wigner potential. We develop a structure-preserving, sampling-based adaptive-rank solver in hierarchical Tucker format for finite-$H$ regimes in which Wigner-Poisson solutions exhibit exploitable low-rank structure. The central difficulty is that adaptive compression can destroy the Fourier-Hermitian tensor symmetry required for a real inverse velocity transform and can break discrete global conservation laws. We address these issues with a Fourier-Hermitian-symmetry-aware sampling and mapping procedure and a global moment correction enforcing mass, momentum, and self-consistent total energy. Numerical tests for two-stream instability and strong Landau damping in 2D2V and 3D3V show roundoff-level conservation, preservation of the real-valued inverse transform, and approximately linear scaling with respect to the number of grid points per coordinate over the tested rank range. The results demonstrate that long-time 3D3V Wigner-Poisson simulations can be performed without assembling the full phase-space tensor.
31. 用于逆热点源问题中源计数的汉克尔行列式零阶原理
A Hankel determinant zero-order principle for source counting in an inverse heat point-source problem
作者: Zhiliang Deng, Ailin Qian, Xiaomei Yang
类别: math.NA
发布日期: 2026-06-13
arXiv ID: 2606.15065v1
链接: http://arxiv.org/abs/2606.15065v1
摘要: This paper studies the identification of an unknown number of stationary point sources in a two-dimensional heat equation from boundary flux measurements. Unlike many reconstruction approaches that assume the number of sources to be known in advance, we develop a determinant-based counting method that extracts this number directly from the measured data. In the unit disk, the Laplace-transformed and normalized boundary flux admits a Fourier moment representation whose low-frequency limit has a finite exponential-sum structure. This structure leads to a family of Hankel matrices and associated determinant characteristics. We prove that, under a generic determinant lifting condition, the vanishing order of the Hankel determinant at the zero Laplace frequency changes exactly when the Hankel order exceeds the true number of sources. Consequently, the source number is characterized by the first nonzero contour count of the determinant characteristic through the argument principle. We further establish a Rouché-type stability result showing that the determinant zero count is preserved under sufficiently small perturbations induced by measurement noise, boundary discretization, and time truncation. After the source number is identified, the source locations and strengths are recovered from the low-frequency moment sequence by an annihilating-polynomial and Vandermonde reconstruction procedure. Numerical experiments confirm the predicted count pattern, demonstrate robustness with respect to contour selection, illustrate the role of the Rouché margin under noise and near-degenerate configurations, and validate the subsequent recovery of source locations and strengths.
32. VQE 作为海森堡自旋玻璃哈密顿量上 QPE 的初始态制备
VQE as Initial State Preparation for QPE on Heisenberg Spin-Glass Hamiltonians
作者: Elijah Pelofske, Stephan Eidenbenz
类别: quant-ph, cond-mat.dis-nn, math.NA
发布日期: 2026-06-13
arXiv ID: 2606.15061v1
链接: http://arxiv.org/abs/2606.15061v1
摘要: Quantum Phase Estimation (QPE) is the quantum algorithmic workhorse for computing ground state energies of quantum Hamiltonians with quantum computers. Ground state energy calculation of physical systems is perhaps the most promising use case for quantum computing in terms of scientific and commercial value with a plausible path to outperformance of classical alternatives. This path, however, hinges on the availability of initial states for QPE with significant overlap with the true ground state. Using extensive (classical) numerical computations, we study whether the NISQ-era algorithm VQE (Variational Quantum Eigensolver) could be used to efficiently prepare high-overlap states of disordered fully-connected anisotropic Heisenberg spin glass quantum Hamiltonians with up to $15$ qubits. We find that (i) – consistent with widely held, but rarely numerically illustrated beliefs – VQE is generally unable to efficiently converge to the ground state for our Hamiltonians, which is a well-known issue with VQE due to a variety of factors including vanishing gradients and local minima; (ii) low energy states do not necessarily have large ground-state overlap, but there is typically a correlation between the two measures; (iii) adding more than three layers to the VQE ansatz neither improves overlap nor the energies found; and (iv) the best-found overlap scaling as a function of the Hamiltonian system size is not strongly exponentially decreasing, suggesting potential for VQE to be a heuristic state preparation algorithm for QPE.
33. 带有对流换热的最优隔热问题的有限元逼近
A Finite Element Approximation of an Optimal Insulation Problem with Convective Heat Transfer
作者: Harbir Antil, Alex Kaltenbach, Keegan L. A. Kirk
类别: math.NA
发布日期: 2026-06-12
arXiv ID: 2606.14991v1
链接: http://arxiv.org/abs/2606.14991v1
摘要: A finite element discretization of an optimal insulation problem with convective heat transfer is considered. The model is formulated as a non-smooth, two-variable convex minimization problem. It accounts for the temperature distribution in a thermally conducting body $Ω\subseteq\mathbb{R}^d$, with $d\in \{2,3\}$, and the distribution of a given amount of insulation material on an insulated boundary part $Γ_I\subseteq \partialΩ$. The surface integral over the insulated boundary $Γ_I$ is approximated by a mass-lumping quadrature that preserves the structure of the continuous setting and, in particular, yields discrete optimality conditions mirroring their continuous counterparts. Well-posedness, stability, and weak convergence of discrete solutions to the continuous ones are established. Furthermore, a block coordinate descent algorithm for the computation of the discrete solutions is formulated and its linear convergence is derived. Under suitable regularity assumptions, uniform $L^\infty(Γ_I)$-bounds and $\textit{a priori}$ error estimates for both the temperature distribution and the distribution of a given amount of insulation material are obtained. Numerical experiments are carried out that confirm the predicted error decay rates and demonstrate the method in a qualitative three-dimensional test on a realistic spacecraft crew module capsule geometry with idealized reentry-heating Robin data.
34. 高效多精度计算实阶与复变量贝塞尔函数并附Fortran实现——第三部分：第一类和第二类正则贝塞尔函数$J_ν(z)$和$Y_ν(z)$
Efficient Multi-Precision Computation of Bessel Functions for Real Orders and Complex Arguments with a Fortran Implementation – Part III: Regular Bessel Functions of the First and Second Kinds $J_ν(z)$ and $Y_ν(z)$
作者: Mofreh R Zaghloul, Steven G. Johnson
类别: math.NA, math-ph
发布日期: 2026-06-12
arXiv ID: 2606.14937v1
链接: http://arxiv.org/abs/2606.14937v1
摘要: This paper is the final part in a series devoted to the development of numerically stable and efficient algorithms, together with multi-precision Fortran implementations, for computing Bessel functions with real orders and complex arguments. PartsI andII presented stable and efficient algorithms for the modified Bessel functions I_nu(z) and K_nu(z); here we treat Bessel functions J_nu(z) and Y_nu(z). The proposed algorithms support complex arguments for both positive and negative real orders and are implemented in native double and quadruple precision. Quadruple precision substantially increases dynamic range and accuracy (by roughly an order of magnitude in reliably computable |nu| and |z|), thereby extending applicability to problems requiring 20-30 digits. Comprehensive accuracy and performance comparisons are carried out against both the widely used Algorithm644, restricted to double-precision arithmetic, and the more recent Algorithm912, which supports both double- and quad-precision arithmetic as well as complex orders. In double precision, the present implementation consistently outperforms Algorithm644, achieving execution times of approximately 35-67% for J_nu(z) and 44-72% for Y_nu(z), while also producing reliable results where Algorithm644 fails. In comparison with Algorithm912, the present algorithm achieves comparable accuracy in double-precision computations and significantly higher accuracy in quad-precision calculations. At the same time, it requires only a small fraction of the computational cost (from a few thousandths to a few hundredths) of the time taken by Algorithm912, depending on the precision and parameter regime. Furthermore, unlike Algorithm~912, whose applicability is restricted to a limited region of the (Re(nu),z) plane, the present algorithm remains stable and accurate over the full tested domain for which reliable reference values are available.