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1. 多边形网格上椭圆界面问题的混合扩展虚拟元方法

A Mixed Extended Virtual Element Method for Elliptic Interface Problems on Polygonal Meshes

作者: Xianyan Zheng, Jinru Chen, Feng Wang

类别: math.NA

发布日期: 2026-06-07

arXiv ID: 2606.08526v1

摘要翻译: 我们针对非适配多边形网格上的椭圆界面问题，提出了一种最低阶 (H(\operatorname{div}))-协调的混合扩展虚拟元方法。通量和压力分别由子区域扩展的 (H(\operatorname{div}))-VEM 空间和分片常数逼近。在切割单元上，可计算的多项式投影定义在整个背景单元上，然后限制到两个子区域。与 BDM 型多项式空间相比，混合 VEM 空间包含非多项式分量，这会在切割单元上产生额外的相容项。为了控制这些项，我们在切割单元上使用增强核稳定化，并在混合耦合中使用界面法向通量平均。添加了修正的界面通量惩罚和局部散度鬼点惩罚，以获得与切割位置无关的稳定性，而无需使用体积散度-散度增广。我们证明了连续性、离散 inf-sup 条件以及在网格依赖范数下的最优一阶误差估计。常数与网格尺寸以及界面相对于背景网格的位置无关，但可能依赖于系数反差。

Comment: 29 pages

链接: http://arxiv.org/abs/2606.08526v1

1. 研究对象和背景

本文研究一类椭圆界面问题：设 $\Omega\subset\mathbb{R}^2$ 为有界凸区域，被一条光滑界面 $\Gamma$ 划分为两个子区域 $\Omega_1$ 和 $\Omega_2$。扩散系数 $\alpha$ 在 $\Omega_i$ 上分别为正常数 $\alpha_i$，即 $\alpha|_{\Omega_i}=\alpha_i>0$。未知函数 $p$ 满足

$$ \begin{cases} -\operatorname{div}(\alpha\nabla p)=f & \text{在 }\Omega_1\cup\Omega_2,\\ [![\alpha\nabla p\cdot\mathbf{n}]!]=0 & \text{在 }\Gamma,\\ [![p]!]=0 & \text{在 }\Gamma,\\ p=0 & \text{在 }\partial\Omega. \end{cases} $$

其中 $[![\cdot]!]$ 表示跨越界面的跳跃。通过引入通量 $\mathbf{u}=\alpha\nabla p$，可将上述二阶问题改写为等价的一阶混合形式：

$$ \begin{cases} -\operatorname{div}\mathbf{u}=f & \text{在 }\Omega_1\cup\Omega_2,\\ \mathbf{u}=\alpha\nabla p & \text{在 }\Omega_1\cup\Omega_2,\\ [![\mathbf{u}\cdot\mathbf{n}]!]=0, \ [![p]!]=0 & \text{在 }\Gamma,\\ p=0 & \text{在 }\partial\Omega. \end{cases} $$

对应的弱形式为：求 $(\mathbf{u},p)\in\mathbf{V}\times Q=H(\operatorname{div};\Omega)\times L^2(\Omega)$，使得

$$ \begin{aligned} (\alpha^{-1}\mathbf{u},\mathbf{v})_\Omega+(p,\operatorname{div}\mathbf{v})_\Omega&=0 & \forall\mathbf{v}\in\mathbf{V},\\ -(q,\operatorname{div}\mathbf{u})_\Omega&=(f,q)_\Omega & \forall q\in Q. \end{aligned} $$

该问题在物理和工程中广泛出现，如复合材料热传导、非均质多孔介质渗流、电磁场传输问题等。由于材料参数在界面上间断，解的全局光滑性丢失（通常仅在各子区域内部光滑），且必须在界面上准确施加法向通量连续与压力连续的传输条件。

传统的界面匹配有限元方法（interface-fitted FEM）通过使网格贴合界面来保持逼近精度，但生成匹配网格的代价高昂，尤其当界面弯曲、复杂或移动时。非匹配（unfitted）方法避免了这一困难，直接在不受界面影响的背景网格上离散，通过弱方式施加界面条件。代表性的非匹配方法包括扩展有限元（XFEM）、Nitsche 方法、CutFEM 等。针对混合形式，已有基于 BDM 多项式空间的扩展混合有限元方法（如文献 [15]），利用适当的稳定项在非匹配三角网格上获得了 inf-sup 条件和与界面位置无关的最优收敛。

虚拟元方法（Virtual Element Method, VEM）是近年来兴起的一种可处理多边形/多面体网格的灵活数值框架。其核心思想是：局部基函数无需显式构造，方法完全通过自由度、多项式投影和稳定项来定义和计算。这为网格生成提供了极大便利。混合虚拟元方法（mixed VEM）作为经典混合有限元在多边形网格上的推广，已经建立了 $H(\operatorname{div})$-协调的 VEM 空间系统和相应的投影计算理论 [10,4]。然而，将混合 VEM 应用于非匹配界面问题尚未见研究。这一扩展面临本质性的新困难：在 VEM 空间中，局部速度函数 $\mathbf{v}_h$ 包含一个显式未知的非多项式核部分。记 $Q_K^1$ 为向 $[\mathbb{P}_1(K)]^2$ 的 $L^2$ 投影，在每个单元上成立正交分解 $\mathbf{v}_h=Q_K^1\mathbf{v}_h+(I-Q_K^1)\mathbf{v}_h$。在切割单元上，可计算的投影 $Q_K^1\mathbf{v}_h$ 是在整个背景单元 $K$ 上计算的，然后限制到切割产生的子区域 $K_i=K\cap\Omega_i$ 上，而真实的积分却在这些子区域上进行。这种不一致性产生了额外的相容误差项，这些项在基于多项式的 BDM 方法中并不存在（因为那儿 $Q_K^1\mathbf{v}_h=\mathbf{v}_h$）。此外，由于 VEM 函数在单元内部无显式表达式，其在界面段上的法向通量也无法直接求值，这对混合耦合项（涉及通量散度与界面项）的构造造成障碍。

本文发展了一套最低阶 $H(\operatorname{div})$-协调的混合扩展虚拟元方法，填补了这一理论空白。通量在子区域扩展的 VEM 空间中逼近，压力采用分片常数。为了克服上述困难，文中提出了一系列创新的稳定化技术：在切割单元上引入增强核稳定（enhanced kernel stabilization）来控制非多项式核部分；利用散度定理与 VEM 可计算量重构界面法向通量平均值（interface normal-flux average），使其能精确参与混合耦合；设计修正界面通量惩罚（corrected interface-flux penalty）以控制多项式投影法向通量的零均值部分；并加入局部散度鬼项惩罚（local divergence ghost penalty）来实现散度的扩展区域估计，从而避免使用体积形式的 $\operatorname{div}$-$\operatorname{div}$ 增广项。最终在合理的网格与界面几何假设下，证明了格式的连续性、离散 inf-sup 条件以及网格依赖范数下的最优一阶收敛。所有常数不依赖于网格尺寸和界面相对背景网格的位置，但可能与系数反差有关。

2. 主要定理或主要结果

论文建立的理论结果可归纳为以下核心定理（均基于假设 3.1 且参数 $\eta_{\ker},\eta_{\operatorname{div}},\gamma_F$ 充分大）：

离散双线性型的连续性：定义 $\mathcal{L}_h(\mathbf{u}_h,p_h;\mathbf{v}_h,q_h)=a_h(\mathbf{u}_h,\mathbf{v}_h)+b_h(\mathbf{v}_h,p_h)-b_h(\mathbf{u}_h,q_h)+\gamma_1 J_1(p_h,q_h)+\gamma_2 J_2(p_h,q_h)$，其中 $a_h$ 包含投影项、增强 VEM 稳定项 $S_{\alpha,i}^{K,+}$、发散鬼项 $G_{\operatorname{div}}$ 和修正界面通量惩罚 $P_F$；$b_h$ 利用界面法向通量平均 $M_\Gamma$ 定义。则在网格依赖范数 $\lVert(\mathbf{v}_h,q_h)\rVert_h$ 下，存在与 $h$ 及切割位置无关的常数 $C$，使得对任意 $(\mathbf{u}_h,p_h),(\mathbf{v}_h,q_h)\in\mathbf{V}_h\times Q_h$，有
$$ |\mathcal{L}_h(\mathbf{u}_h,p_h;\mathbf{v}_h,q_h)|\le C\,\lVert(\mathbf{u}_h,p_h)\rVert_h\,\lVert(\mathbf{v}_h,q_h)\rVert_h. $$
离散 inf-sup 条件（压力提升引理，Lemma 4.6）：对任意 $q_h\in Q_h$，存在提升速度 $\mathbf{v}_{p,h}\in\mathbf{V}_h$，满足
$$ \begin{aligned} b_h(\mathbf{v}_{p,h},q_h)&\ge c_p\lVert\alpha^{1/2}q_h\rVert_{0,\Omega_1\cup\Omega_2}^2-C_p\bigl(J_1(q_h,q_h)+J_2(q_h,q_h)\bigr),\\ \lVert\mathbf{v}_{p,h}\rVert_h&\le C_p\Bigl(\lVert\alpha^{1/2}q_h\rVert_{0,\Omega_1\cup\Omega_2}^2+J_1(q_h,q_h)+J_2(q_h,q_h)\Bigr)^{1/2}, \end{aligned} $$
其中 $c_p,C_p>0$ 为不依赖于 $h$ 和切割位置的常数。这表明由 $b_h$ 和稳定项 $J_1,J_2$ 组成的“正则化”混合形式满足一致 inf-sup 条件，结合连续性即得离散问题解的存在唯一性。
最优一阶误差估计（Section 5 的最终结果）：若真解具有最低正则性 $\mathbf{u}_i\in H^1(\operatorname{div};\Omega_i)$、$p_i\in H^1(\Omega_i)$ $(i=1,2)$，则离散解 $(\mathbf{u}_h,p_h)$ 满足
$$ \lVert\mathbf{u}-\mathbf{u}_h\rVert_h+\lVert\alpha^{1/2}(p-p_h)\rVert_{0,\Omega_1\cup\Omega_2}\le C\,h\sum_{i=1}^2\bigl(\lVert\mathbf{u}_i\rVert_{1,\Omega_i}+\lVert\operatorname{div}\mathbf{u}_i\rVert_{1,\Omega_i}+\lVert p_i\rVert_{1,\Omega_i}\bigr), $$
其中常数 $C$ 仅依赖于网格形状正则性、界面几何假设 (A1)-(A5) 以及系数反差 $\alpha_{\max}/\alpha_{\min}$，而与 $h$ 和界面相对背景网格的切割位置无关。范数 $\lVert\cdot\rVert_h$ 涵盖了对通量多项式投影的 $L^2$ 误差、增强核稳定项、发散鬼项和修正界面通量惩罚的共同度量，因而是一阶意义下的完全控制。

3. 研究方法、关键技术和核心工具

本文的证明体系围绕虚拟元非多项式核带来的核心障碍展开，主要创新点和关键技术可归纳如下。

1. 切割单元上的增强核稳定（Enhanced Kernel Stabilization）

标准的最低阶 $H(\operatorname{div})$-VEM 稳定项基于缩放自由度平方和：$S_{\alpha,i}^K(\mathbf{v}_h,\mathbf{v}_h)=\alpha_i^{-1}|K|\sum_{r=1}^{N_K}|\hat{\chi}_r(\mathbf{v}_h)|^2$，它在 $\ker Q_K^1$ 上导出一个与 $L^2$ 范数等价的量（Lemma 4.1）。然而，该等价性是在整个背景单元 $K$ 上成立的，当实际积分区域仅为切割子区域 $K_i$（其面积 $|K_i|$ 可能远小于 $|K|$）时，稳定项的 $|K|$ 因子无法有效控制 $(I-Q_K^1)\mathbf{v}_h$ 在 $K_i$ 上的 $L^2$ 模。为此，在切割单元上，增强稳定取为 $S_{\alpha,i}^{K,+}(\mathbf{v}_h,\mathbf{v}_h)=(|K|+\eta_{\ker}h_K)D_{\alpha,i}^K(\mathbf{v}_h,\mathbf{v}_h)$，其中 $D_{\alpha,i}^K$ 是去除了 $|K|$ 因子的自由度平方和。附加项 $\eta_{\ker}h_K D_{\alpha,i}^K$ 提供了额外的控制，其尺度正比于 $h_K$ 而非 $|K|$，因而在估计切割域上的相容误差时能给出恰当的放缩。这一处理是本文区别于标准 VEM 分析的关键，它使非多项式核部分在切割单元上的积分误差被稳定项吸收，而无需进一步了解其内部结构。

2. 界面法向通量平均（Interface Normal-Flux Average）与混合耦合

混合形式要求在界面 $\Gamma$ 上计算通量测试函数 $\mathbf{v}_h$ 的法向分量，但 VEM 函数在单元内部无显式表达式，无法直接插值。本文巧妙地利用散度定理和 VEM 的可计算量（边上的法向矩和常数散度）定义了两组常数 $M_{\Gamma,K}^1(\mathbf{v}_h)$ 和 $M_{\Gamma,K}^2(\mathbf{v}_h)$（式(3.16)-(3.17)），使得

$$ |\Gamma_K|M_{\Gamma,K}^i(\mathbf{v}_h)=\pm\int_{\Gamma_K}\mathbf{v}_{h,i}\cdot\mathbf{n}\,\mathrm{d}s,\quad i=1,2, $$

且在 $\Gamma_K$ 上为常数（Lemma 3.1）。于是，混合耦合项中原本需要的界面项 $\int_{\Gamma_K}[![\mathbf{v}_h\cdot\mathbf{n}]!]q_h\,\mathrm{d}s$ 可等价替换为 $[![M_\Gamma\mathbf{v}_h]!]\{q_h\}|\Gamma_K|$，因为压力 $q_h$ 在 $\Gamma_K$ 两侧均为常数。这一定义完全基于自由度，不依赖 $\mathbf{v}_h$ 的显式形式，且保持了积分的精确性，从而保证了格式的相容性。

3. 修正界面通量惩罚（Corrected Interface-Flux Penalty）与局部散度鬼项惩罚

标准的 Nitsche 型惩罚通常直接惩罚法向通量的跳跃，但本文除需控制常数部分外，还需控制多项式投影 $Q_K^1\mathbf{v}_h\cdot\mathbf{n}$ 的零均值部分（即 $(I-\Pi_\Gamma^0)[![Q_K^1\mathbf{v}_h\cdot\mathbf{n}]!]$）。因此，定义了修正界面通量 $F_{\Gamma,K}(\mathbf{v}_h)=[![M_\Gamma\mathbf{v}_h]!]+(I-\Pi_\Gamma^0)[![Q_K^1\mathbf{v}_h\cdot\mathbf{n}]!]$，其中 $\Pi_\Gamma^0$ 是向 $\Gamma_K$ 上常数的 $L^2$ 投影。惩罚项 $P_F$ 即为 $h_K^{-1}\alpha_\Gamma^{-1}\lVert F_{\Gamma,K}(\mathbf{v}_h)\rVert_{0,\Gamma_K}^2$ 的加权和。由于两项在 $L^2(\Gamma_K)$ 中正交，该惩罚提供了对完整法向通量跳跃的控制，且只用到可计算的多项式投影和平均常数。与此同时，为了在不引入体积 $\operatorname{div}$-$\operatorname{div}$ 增广项的情况下获得对离散散度的全局控制，文中引入局部散度鬼项惩罚 $G_{\operatorname{div}}$（式(3.15)），在切割单元附近的传输边 $e\in\mathcal{E}_{\Gamma,i}$ 上惩罚散度跳跃。结合典型鬼项论证（Lemma 4.2 的扩展），可得到扩展域上的 $L^2$ 散度估计，从而使稳定性与界面位置无关。

4. 压力提升构造与 inf-sup 条件证明

为了建立离散 inf-sup 条件，论文使用了“压力提升”（pressure lifting）技术。对给定的分片常数压力 $q_h\in Q_h$，先用连续散度提升获得 $\mathbf{z}_p\in H^1(\Omega)^2$ 满足 $\operatorname{div}\mathbf{z}_p=\alpha_{\min}q_h$ 及 $\lVert\mathbf{z}_p\rVert_{1,\Omega}\le C\lVert\alpha^{1/2}q_h\rVert_{0,\Omega_1\cup\Omega_2}$；然后取 VEM 插值 $\mathbf{v}_{p,h}=I_h^{\operatorname{div}}\mathbf{z}_p$。利用插值的交换性质（Lemma 4.4）得 $\operatorname{div}\mathbf{v}_{p,h}=P_h^0(\operatorname{div}\mathbf{z}_p)=\alpha_{\min}P_K^0 q_h$。对于切割单元，$P_K^0 q_h$ 是面积加权平均 $\bar{q}_K$，于是 $b_h(\mathbf{v}_{p,h},q_h)=\alpha_{\min}M_h(q_h,q_h)$。此时，$\mathbf{v}_{p,h}$ 的构造使得 $F_{\Gamma,K}(\mathbf{v}_{p,h})=0$，故界面惩罚项不贡献，提升了稳定性。结合标量扩展估计（Lemma 4.5）通过 $J_1+J_2$ 将 $M_h(q_h,q_h)$ 与实际 $L^2$ 范数联系起来，即得所需的下界和范数控制。

5. 误差分析中的相容项处理

最终的误差界通过 Strang 引理推导，将误差分为插值误差和离散相容项误差。插值误差通过子区域解的先验延拓（$E_i$ 与 $E_i^{\operatorname{div}}$）和 VEM 逼近性质得到 $O(h)$ 估计。关键难点在于相容项：对于任意充分光滑的 $\mathbf{v}$，有

$$ \mathcal{R}_K(\mathbf{v}) = \int_{K_i} (\alpha_i^{-1}\mathbf{v}\cdot\mathbf{v}_h - \alpha_i^{-1}Q_K^1\mathbf{v}\cdot Q_K^1\mathbf{v}_h) - S_{\alpha,i}^{K,+}((I-Q_K^1)\mathbf{v},(I-Q_K^1)\mathbf{v}_h) + \text{界面及散度项}, $$

其中第一行包含切割域上的投影误差和核部分。利用正交分解 $\mathbf{v}=Q_K^1\mathbf{v}+(I-Q_K^1)\mathbf{v}$ 并反复使用 Cauchy–Schwarz 不等式，可将每一项与 $\lVert\mathbf{v}_h\rVert_h$ 或 $h$ 的正则性范数相乘而得以控制。增强核稳定 $S_{\alpha,i}^{K,+}$ 在此扮演核心角色：它提供了 $(I-Q_K^1)\mathbf{v}_h$ 在 $K$ 上的 $L^2$ 控制，再通过迹不等式和投影误差估计转化为对切割域积分的控制。修正界面通量惩罚 $P_F$ 则用于吸纳因 $Q_K^1\mathbf{v}_h\cdot\mathbf{n}$ 与真实法向通量的差异造成的界面相容项。发散鬼项 $G_{\operatorname{div}}$ 处理散度边界的估计。整套分析精细区分了各稳定项的角色，且无需引入体积散度惩罚。

4. 与之前工作的比较

本文的工作在多个层面上与已有文献形成对比与推进，主要体现在以下几方面。

1. 与 BDM 扩展混合有限元（文献 [15]）的关系

[15] 提出了一种基于 $\mathbb{BDM}_1$–$\mathbb{P}_0$ 对的使用非匹配三角网格的扩展混合方法，是混合形式下较早的非匹配做法。二者的思想基调相似：通量逼近采用 $H(\operatorname{div})$ 协调空间，压力为分片常数，并通过鬼项惩罚（ghost penalty）和 Nitsche 型处理获得与切割位置无关的稳定性与最优收敛。然而，最本质的区别在于离散通量空间的构成。BDM 空间是纯多项式的，因此 $Q_K^1\mathbf{v}_h=\mathbf{v}_h$，$(I-Q_K^1)\mathbf{v}_h\equiv0$；VEM 空间则包含非多项式核部分，这导致了切割单元上额外的相容项，必须通过增强核稳定和修正界面通量惩罚来应对。可以说，本文将 [15] 的策略适配到了 VEM 框架下，使得方法适用于更一般的多边形网格。代价是公式和分析更加复杂。目前两者均只给出最低阶 $O(h)$ 精度；BDM 方法可自然地提升阶次，而本文的低阶 VEM 若要高阶推广，需要设计更高次的 $H(\operatorname{div})$-VEM 空间并解决投影限制带来的新困难，这仍有待研究。

2. 与其他扩展/immersed 虚拟元方法（例如 [29, 30, 17]）的比较

已有的扩展 VEM 文献几乎都面向 primal 形式：$H^1$-协调 VEM [29]、非协调 VEM [30]、immersed VEM [17]。它们均以原始变量 $p$ 为直接未知量，依赖加强的稳定性（如 Nitsche 项、切向或法向跳跃惩罚）来施加界面条件。本文则首次采用混合 $H(\operatorname{div})$-VEM 格式，直接计算通量 $\mathbf{u}$。这样做的好处是数值通量散度在局部严格守恒，且无需后处理即可获得高精度的通量场，对某些应用（如多孔介质流）尤为重要。在弱施加界面条件的方式上，primal VEM 常惩罚压力的跳跃和通量的法向跳跃，而本文则结合了通量法向分量的平均重构、修正的界面通量惩罚以及对压力跳跃的 $J_2$ 惩罚。特别地，非协调 VEM [30] 中为了处理非多项式部分也引入了特殊边项，但其结构依赖于非协调空间的边自由度，与本文的 $H(\operatorname{div})$ 空间的边矩处理策略完全不同。

3. 与一般 CutFEM 及鬼项惩罚技术（如 [12, 13, 22]）的联系与发展

本文大量使用了鬼项惩罚（ghost penalty）。压力鬼项 $J_1$ 是 CutFEM 中常见的用于恢复扩展区域控制的工具，散度鬼项 $G_{\operatorname{div}}$ 则是专门为混合形式定制的。许多切割单元混合方法，因担心界面附近散度控制不足，常常添加体积形式的 $(\varepsilon\operatorname{div}\mathbf{u}_h,\operatorname{div}\mathbf{v}_h)_\Omega$ 项，其中 $\varepsilon$ 为一小参数。本文却完全摒弃了这一做法，仅通过 $G_{\operatorname{div}}$ 在界面单元附近的传输边上施加跳惩罚，便得到了扩展域散度的一致估计，并结合压力提升获得了 inf-sup 条件。这样避免了参数 $\varepsilon$ 对切割位置的敏感依赖，在理论上更干净。然而，这也要求网格满足连通假设 (A5)，并且需要在实施中找出并存储额外的传输边。

4. 系数反差与解的延拓

文中所有稳定性与误差估计的常数都明确声明可能依赖于系数反差 $\alpha_{\max}/\alpha_{\min}$，并未追求关于反差的一致鲁棒性。这与一些专门采用加权 Sobolev 空间或特殊界面权重以做到反差鲁棒的方法（例如某些 CutFEM 变体）存在差距。当然，本文使用的调和平均权重 $\alpha_\Gamma$ 和相应的 Nitsche 权重对中等反差通常表现良好；但面对极大反差时，常数可能急剧增大，这是本方法的一个公认局限。在解的延拓方面，使用了标准的 $H^m$ 和 $H(\operatorname{div})$ 延拓算子（$E_i$ 和 $E_i^{\operatorname{div}}$），与 [15, 16] 等做法一致，且延拓常数仍与切割位置无关，这保证了插值误差估计的品质。

5. 网格与界面几何假设

本文沿用了非匹配文献中常见的界面几何假设：局部最多两个交点、局部可表为单值函数等 (A3-A4)，以及鬼项分析所需的子区域连通性条件 (A5)。这些假设在界面充分光滑的前提下易于满足，例如 [15] 中也有类似要求。相比一些可以处理更复杂界面拓扑的 immersed 方法（如允许单元被多次切割），本文的当前设定仍比较经典，但为分析提供了坚实基础。

综上，本文的贡献在于：在虚拟元框架下，首次为椭圆界面问题构造并分析了混合 $H(\operatorname{div})$ 格式，解决了由 VEM 非多项式核引发的特有相容性障碍，在不使用体积散度惩罚的条件下获得了切割位置独立的 inf-sup 稳定性和最优一阶收敛。其技术路线丰富了混合扩展非匹配方法的理论，并为多边形网格上的通量-压力耦合计算提供了新的可行方案。局限在于目前仅覆盖最低阶、对反差不完全鲁棒，且分析要求界面具有一定光滑性。

2. 针对周期障碍物阵列散射的一致高精度 PML-BIE 方法：二维情形

A Uniformly High-Accuracy PML-BIE Method for Scattering by Periodic Arrays of Obstacles: The 2D Case

作者: Yan Tan, Carlos Pérez-Arancibia, Tao Yin

类别: math.NA, physics.comp-ph

发布日期: 2026-06-06

arXiv ID: 2606.07971v1

摘要翻译: 本文提出了一种新颖的频率鲁棒完美匹配层（PML）边界积分方程（BIE）方法，用于求解涉及周期障碍物阵列的二维电磁散射问题。在周期散射问题中，基于拟周期格林函数的标准BIE公式需要计算格点和或困难的索末菲型积分，这些积分在瑞利-伍德（RW）异常处发散。另一种方法是使用基于亥姆霍兹自由空间格林函数的BIE公式，但这些公式定义在无界单胞边界上，因此需要适当的截断策略，例如加窗格林函数（WGF）方法。尽管这类方法避免了使用昂贵的拟周期格林函数，但除非加入适当的模式修正，它们同样会在RW异常处失效。类似地，直接将PML-BIE技术应用于周期结构时，由于在RW异常附近对于固定的PML参数指数收敛性的丧失，也会遇到类似的困难。为了克服这一挑战，我们提出了一种修正的PML-BIE方法，该方法将PML技术与有限模式修正相结合，确保在RW异常处及其附近均具有高精度和鲁棒性。我们证明了经PML截断的边界积分算子的收敛性，并给出了若干数值算例，以验证所提方法的效率和性能。

Comment: 25 pages, 8 figures, 3 tables

链接: http://arxiv.org/abs/2606.07971v1

1. 研究对象和背景

本文研究二维空间中周期阵列障碍物的电磁散射问题，控制方程为Helmholtz方程：在介质区域 $D_j$（$j=1$ 为外部区域，$j=2$ 为障碍物内部）内，总场 $u$ 满足

$$ \Delta u + k_j^2 u = 0 \quad \text{in } D_j, $$

其中波数 $k_j = \omega \sqrt{\epsilon_j \mu_j}$，$\omega$ 为角频率，$\epsilon_j$ 和 $\mu_j$ 分别为介质的介电常数和磁导率。在物体边界 $\partial D_2$ 上，总场满足传输条件

$$ \gamma_{D,\partial D_2}^+ u = \gamma_{D,\partial D_2}^- u,\quad \gamma_{N,\partial D_2}^+ u = \eta \, \gamma_{N,\partial D_2}^- u, $$

其中 $\eta = \mu_1/\mu_2$（TE极化）或 $\eta = \epsilon_1/\epsilon_2$（TM极化），$\gamma_{D}^\pm$ 和 $\gamma_{N}^\pm$ 分别表示外/内Dirichlet和Neumann迹。入射波为平面波 $u^\mathrm{inc}(x) = e^{i(\alpha x_1 - \beta x_2)}$，且 $\alpha = k_1 \sin\theta^\mathrm{inc}$，$\beta = k_1 \cos\theta^\mathrm{inc}$。由障碍物的周期性和平面波入射，总场具有准周期性

$$ u(x_1 + \Lambda, x_2) = \zeta u(x_1, x_2), \quad \zeta = e^{i\alpha \Lambda}, $$

其中 $\Lambda$ 为周期长度。因此，问题可限制到单个单元域 $U = \{ (x_1, x_2) \in \mathbb{R}^2 : \hat f(t) < x_1 < \hat f(t) + \Lambda, \, x_2 = t \}$，其两侧边界为两条无限光滑曲线 $\Gamma_2$ 和 $\Gamma_3$。散射场 $u^\mathrm{sct}$ 还需满足向上/下的Rayleigh展开以描述辐射条件：

$$ u^\mathrm{sct}(x) = \sum_{n \in \mathbb{Z}} B_n^\pm e^{i(\alpha_n x_1 \pm \beta_n x_2)}, \quad \pm x_2 > \pm h^\pm, $$

其中 $\alpha_n = \alpha + 2\pi n/\Lambda$，$\beta_n = \sqrt{k_1^2 - \alpha_n^2}$（当 $|\alpha_n| \le k_1$）或 $\beta_n = i \sqrt{\alpha_n^2 - k_1^2}$（当 $|\alpha_n| > k_1$）。所谓Rayleigh–Wood（RW）异常发生当某 $\beta_n = 0$，即 $|\alpha_n| = k_1$；此时Rayleigh展开中的相应项为传播波沿阵列方向传播，标准的准周期Green函数（QGF）或相关的Sommerfeld积分发散，导致基于QGF的边界积分方程（BIE）方法失效。

为克服QGF的困难，研究者转而采用基于自由空间Green函数 $\Phi(k_j, x, y) = \frac{i}{4} H_0^{(1)}(k_j |x - y|)$ 的BIE，但由于这些BIE定义在无界的单元边界 $\Gamma_2,\Gamma_3$ 上，必须进行截断。常见的截断策略包括窗函数方法（WGF）和完全匹配层（PML）方法。WGF利用光滑窗函数将积分限制在有限窗口内，远离RW异常时可实现超代数收敛，但在RW附近需要引入模式校正才能保持精度。PML方法则通过复坐标拉伸

$$ \tilde x_2(x_2) = x_2 + i \int_0^{x_2} \sigma(t) \, dt, $$

使得散射场在PML区域指数衰减，从而可以简单地将积分截断。然而，直接应用PML-BIE到周期结构时，截断误差依赖于频率与RW异常的最小距离 $d_{\min} = \min_n |\beta_n|$。对于固定的PML参数，当 $d_{\min} \to 0$ 时，指数收敛速率退化：误差项的形式为 $O\big( d_{\min} / (e^{c d_{\min}} - 1) \big)$，此时退化为 $O(1)$，导致精度严重下降。这是因为PML逼近无法正确强化辐射条件：当存在传播或衰减模式时，非物理模式的系数会被错误地激活，且系统条件数恶化。

本文试图填补的空白是：设计一种频率鲁棒的PML-BIE方法，通过引入有限个传播/衰减模式的校正，使得方法在整个频率范围内——包括RW异常附近——均保持指数收敛，且PML参数的选择可以独立于RW异常。与已有的WGF方法相比，PML途径在相同厚度下可提供更高的指数收敛率，从而大幅减少计算自由度。

2. 主要定理或主要结果

本文的主要理论结果和数值贡献概括如下。

PML截断BIE系统的建立与误差分析：将无界单元上的PML-BIE系统（2.30）限制到有限区域 $\Gamma_j^b = \Gamma_j \cap U_{H,T}$（$j=1,2,3$）得到截断系统
$$ (E + \tilde T^b) [\hat\phi] = \phi^\mathrm{inc}, $$
其中 $\tilde T^b$ 由截断后的PML双层势、单层势及超奇异算子组成。原系统右端包含被忽略的尾项 $\psi = (\tilde T^+ + \tilde T^-)[\tilde\phi]$，其中 $\tilde T^\pm$ 是涉及 $\Gamma_j^\pm$（$|x_2| > H+T$ 的部分）的积分算子。通过分析PML复拉伸下Green函数的指数衰减性质，证明了尾项 $\psi$ 在一定的Sobolev范数下具有上界
$$ \|\psi\| \le C \frac{d_{\min}}{e^{c d_{\min}} - 1} e^{-\gamma T}, $$
其中 $C, c, \gamma > 0$ 为与PML参数相关的常数（$\gamma$ 依赖于 $S$ 和 $T$）。此估计显式刻画了当 $d_{\min} \to 0$ 时退化至 $O(e^{-\gamma T})$ 的行为，且与PML厚度 $T$ 呈指数衰减。
有限模式校正的提出与改进的PML-BIE：为了消除 $d_{\min}$ 退化，本文借鉴文[31]的思想，在截断BIE中显式引入对辐射条件的强化。具体而言，定义集合 $\mathcal{C}_\delta = \{ n \in \mathbb{Z} : 0 \le |\beta_n| \le \delta \}$，其中 $\delta$ 为某个与频率和PML参数无关的小量。通过将散射场的投影条件
$$ \frac{1}{\Lambda} \int_{-\Lambda/2}^{\Lambda/2} \big( \partial_{x_2} u^\mathrm{sct}(x_1, \pm h) \mp i\beta_n u^\mathrm{sct}(x_1, \pm h) \big) e^{-i\alpha_n x_1} dx_1 = 0, \quad n \in \mathbb{Z}, $$
与BIE系统耦合，构造了一个包含有限个额外未知数（非物理模式系数）的扩展系统。理论分析表明，修正后的系统在离散后条件数一致有界，且其解对真解的误差满足
$$ \|u - \hat u\| \le C e^{-\gamma T} $$
其中常数 $C$ 和 $\gamma$ 与 $d_{\min}$ 无关，从而在整个频率范围内（包括RW异常处）恢复了完全的指数收敛。数值实验进一步验证了该方法当 $T$ 增大时，误差呈 $O(e^{-c T})$ 下降，且PML参数 $S, P$ 可按与 $d_{\min}$ 无关的方式选取，证明方法具有频率鲁棒性。
数值算例：文中通过多个算例（包括不同入射角、包围圆形和椭圆障碍物的周期阵列）展示了所提方法的精度与效率。在计算散射场远场系数时，误差相对于PML厚度呈指数收敛，且在与WGF方法相同厚度的比较下，PML-BIE的误差显著更小。

3. 研究方法、关键技术和核心工具

本文的核心技术障碍在于：标准PML截断虽能带来指数衰减，但当频率靠近RW异常时，辐射条件的数值实现失效，导致截断BIE的系数矩阵病态，且误差退化。创新点集中于两点：一是对PML截断误差的精细分析，明确揭示其与RW异常距离 $d_{\min}$ 的依赖关系；二是在PML框架下引入有限模式校正，恢复指数收敛且使参数选择与RW异常无关。

1. PML截断误差的精确估计 为了定量描述PML截断引入的误差，作者将无界BIE系统中的积分算子对分解为两部分：在有限区域 $\Gamma_j^b$ 上的积分（对应 $\tilde T^b$）和在两侧无限远端 $\Gamma_j^\pm$ 上的积分（对应 $\tilde T^\pm$）。通过将积分沿 $x_2$ 方向参数化，利用PML复拉伸下自由空间Green函数 $\Phi(k_1, \tilde x, \tilde y)$ 的解析性质，证明其核函数在 $|x_2 - y_2|$ 很大时具有如下指数衰减界：

$$ |\Phi(k_1, \tilde x, \tilde y)| \le C \frac{e^{-\kappa |x_2 - y_2|}}{\sqrt{|x_2 - y_2|}}, \quad \kappa = \frac{S}{2} \min\{1, \frac{S}{k_1}\}, $$

其中 $S$ 为PML缩放参数。结合准周期因子的振荡，通过精细的Fourier分析将尾项 $\psi$ 表达为一系列模式贡献的和。对于每个模式 $n$，对应贡献含有因子 $e^{-|\beta_n| (T + t)}$（衰减模式）或 $e^{i\beta_n (T + t)}$（传播模式）。当 $\beta_n \to 0$ 时，传播模式的衰减速率趋于零，造成 $\psi$ 的范数失去对 $T$ 的指数依赖，而只以 $1/|\beta_n|$ 的形式发散。关键估计引理（如Lemma 3.2）建立了

$$ \| \tilde T^\pm \|_{H^s \to H^{s'}} \le C \frac{d_{\min}}{e^{c d_{\min}T} - 1} e^{-\gamma T} $$

这类技术对后续的校正设计至关重要，因为它表明退化主要源于传播模式（或近传播模式）的贡献。

2. 有限模式校正的构造与实现 受WGF方法中模式校正思想的启发，作者注意到辐射条件（2.15）等价于要求散射场在 $|x_2| > h^\pm$ 处不含“非物理”模式，即展开式（2.11）中所有 $C_n^\pm = 0$。在PML-BIE框架中，直接离散并求解截断系统（3.2）等价于隐式地允许边界数据 $\hat\phi$ 生成非零的 $C_n^\pm$，这正是误差退化的根源。因此，作者显式地将 $C_n^\pm$ 作为未知量添加到系统中。具体流程为： - 选取一个阈值 $\delta > 0$，定义集合 $\mathcal{C}_\delta = \{ n \in \mathbb{Z} : |\beta_n| \le \delta \}$，包含所有当前频率下须特殊处理的模式。 - 利用Green表示公式，将误差场在 $\Gamma^\pm$（即 $x_2 = \pm (H+T)$）上的迹或其导数用 $\hat\phi$ 和这些模式的系数表示出来。 - 将（2.15）中投影方程离散化，得到一组线性约束，可写为 $M \mathbf{c} = \mathbf{r}$，其中 $\mathbf{c}$ 为非物理模式系数，$M$ 为满秩或良条件矩阵。 - 最终组装一个扩展的线性系统，其中边界未知量 $\hat\phi$ 与 $\mathbf{c}$ 联合求解。由于 $M$ 的可逆性，该系统在离散后保持条件数一致有界，且理论上误差的指数界与 $d_{\min}$ 无关。

这种方法相比于直接在截断BIE上叠加有限模式校正的显式后处理，更加稳定；它通过增加极少的额外自由度（$|\mathcal{C}_\delta|$ 一般很小），消除了PML截断对RW异常的敏感性。

3. 算法实现细节 离散方面，对 $\Gamma_1$ 采用Nyström方法或边界元，对 $\Gamma_2^b$ 和 $\Gamma_3^b$ 采用参数化后利用分段多项式离散。PML拉伸函数选取分段光滑的幂函数形式（2.17），以兼顾理论分析与计算效率。快速多极子等加速技术虽未被强调，但框架可与之兼容。

4. 与之前工作的比较

本文的方法论和结果在以下维度与既有工作形成对比。

1. 与基于准周期Green函数的BIE方法比较 经典方法依赖对周期Green函数 $G^q(x,y)$（即QGF）的精确计算，通常通过Ewald求和或加速的格子求和实现[1,12,22,23]。此途径在远离RW异常时高效，但当频率穿越RW异常时 $G^q$ 本身发散，必须设计特殊的正则化策略，例如移动Green函数法[5,6,10,11,20]或混合谱/空间表示[7,18]。这些方法或者计算昂贵，或者需要针对不同的RW异常阶数进行自适应参数调整。本文提出的PML-BIE完全避免了QGF的直接评估，而是使用自由空间Green函数，并用PML截断处理无界边界，从而天然规避了QGF的奇点问题。同时，借助有限模式校正，方法在RW异常处完全不需参数调优，与频率彻底解耦。

2. 与窗函数方法（WGF）的比较 WGF方法[31]也是基于自由空间Green函数，但采用窗函数对无界积分截断，收敛性为超代数：对紧支撑密度的积分，当窗口尺寸增大时，误差以 $O(e^{-b N^2})$ 的形式衰减（$N$ 与窗口尺寸成比例），比指数收敛慢。文[31]还针对RW异常引入了同样的模式校正，使收敛性恢复。本文用PML代替窗函数后，在同等厚度下可获得指数收敛 $O(e^{-\gamma T})$，且理论上 $\gamma$ 可由PML参数控制，因而更高效。数值例子清楚地展示，当PML厚度与WGF半窗宽度相当时，前者的误差小两个数量级以上。代价是拉伸坐标可能引入对Green函数计算的轻微额外开销（复变元Bessel函数），但此开销可通过解析延拓的快速算法弥补，且在中等厚度下整体胜出。

3. 与直接PML-BIE方法的比较 直接PML截断方法已在层介质[3,24,26]和局部缺陷周期结构[17]等问题中获得应用，并在远离RW异常时表现出色。然而，正如本文分析所指出，其收敛性在靠近RW异常时严重退化：因为PML衰减来自 $e^{-\operatorname{Im}(\tilde x_2) \min |\beta_n|}$ 因子，当 $\min |\beta_n| \to 0$ 时，需要无限增加PML厚度 $T$ 才能维持准确度。本文首次通过定理形式证明了退化的精确速率，并且率先在PML框架下引入模式校正，完全解除收敛速率与 $d_{\min}$ 的关联。这相比文[13]中仅使用齐次Dirichlet边界条件（在PML端部设 $\tilde u=0$）的PML方法更精细，因为[13]的方法甚至不能准确刻画辐射条件，误差更早退化。

4. 理论分析的新颖性 本文对PML截断积分算子的衰减估计建立了与RW异常距离的显式定量关系，这是前人未给出的。文中引理与定理的证明综合利用了复拉伸下Hankel函数的渐近展开、傅里叶投影以及Sobolev空间中的连续性方法，形成一套分析PML-BIE在周期问题中有效性的严密框架。这一框架为将来向三维和非光滑障碍物的推广奠定了理论基础。

5. 与其他RW处理策略的对比 与直接基于Rayleigh展开的模式匹配法相比，本文的PML-BIE不依赖人工边界或DtN映射，对障碍物形状不敏感。与基于传播模式展开的混合方法相比，本文不需要提前知道入射波产生的传播模式数目，通过PML自动处理无穷模式。

中文标题: 曲面Stokes问题的无散度Scott-Vogelius有限元方法

摘要翻译: 我们构造并分析了一种用于曲面Stokes问题的精确无散度Scott-Vogelius有限元方法。所提格式同时精确地施加了切向性条件与不可压缩性，且具有与二维欧几里得离散化相同的未知量数目。我们的构造将[10,11]中的曲面有限元框架推广到了定义在弯曲Clough-Tocher三角剖分上的Scott-Vogelius离散化。与以往基于宏元构造的等参Scott-Vogelius方法不同，本文方法直接在细化后的曲面三角剖分上定义有限元空间，从而得到了一个显著更简单且更实用的实现。我们证明了该方法的inf-sup稳定性，并在等参范围内推导出了最优阶收敛。

不相关论文（基本信息）

1. 基于分层比率蒙特卡洛方法的戴德金数有限n估计

Finite-n Estimate of Dedekind Numbers by Layer-Ratio Monte Carlo

作者: Tian-Shun Chen, Hao Feng, Haozhe Wang, Kilar Zhang

类别: math.CO, cs.IT, hep-th, math-ph, math.NA

发布日期: 2026-06-08

arXiv ID: 2606.09795v1

链接: http://arxiv.org/abs/2606.09795v1

摘要: Dedekind’s problem counts monotone Boolean functions, equivalently downsets of a Boolean lattice. We recast this enumeration as a finite layer-ratio reconstruction problem for the Whitney numbers of the ranked ideal lattice. An exact adjacent-layer double count expresses each layer ratio through local averages of the number of addable elements and the number of removable elements. Reversible fixed-layer Markov chains estimate these averages and hence estimate the Dedekind number M(n). Backtests at M(8) and M(9) calibrate seed-level variability under the fixed protocol and measure the observed Monte Carlo budget scaling. The resulting estimate probes the Whitney-number sequence of the ideal lattice. Although these rows have previously been described empirically as unimodal, the high-precision n=9 estimate has a shallow two-shoulder feature around the central rank, contrary to that empirical description; n=11 and n=13 center-window estimates show a larger-contrast analogous pattern. The protocol estimate for M(10) is [ \widehat M(10)=(8.9360\pm0.0010)\times 10^{78}, ] where the displayed uncertainty is the budget-based forecast scale from the cross-n scaling law under the production budget.

2. 算子傅里叶变换的低秩加速

Low-Rank Acceleration of the Operator Fourier Transform

作者: Jack Kelley

类别: math.NA

发布日期: 2026-06-08

arXiv ID: 2606.09689v1

链接: http://arxiv.org/abs/2606.09689v1

摘要: We develop a numerical algorithm for the efficient solution or approximation of solutions to the Helmholtz equation on a structured grid in two dimensions. We make use of the Operator Fourier Transform (OFT) and a low-rank cross approximation scheme (Cross-DEIM) to decompose the problem into an integral over a pseudo-time of solutions to the Schrödinger equation. The OFT is a framework for solving operator equations like fractional Laplacian equations or the Helmholtz equation, when the latter is written as a product of two paraxial operators. The main computational cost in the OFT is the solution to the Schrödinger equation, especially when the dimension or mesh resolution is high. In this work, we alleviate this cost by utilizing a low-rank method. Such methods aim to beat the curse of dimensionality when low-rank structures are present in the solution. We show that the combination of these two approaches can have large cost reductions for certain classes of problems.

3. 波动方程的时空稀疏网格方法

A space-time sparse-grid method for the wave equation

作者: Matteo Ferrari, Andrea Moiola, Chiara Perinati, Ilaria Perugia

类别: math.NA

发布日期: 2026-06-08

arXiv ID: 2606.09688v1

链接: http://arxiv.org/abs/2606.09688v1

摘要: We develop a fast space-time numerical scheme for approximating solutions to the linear wave equation. The approach is based on the sparse-grid combination technique applied to a coercive space-time discretization. Designed for tensor-product space-time discretizations, the method enables efficient parallelization of the resulting solver. We provide a rigorous theoretical analysis establishing convergence rates and computational complexity estimates. Numerical experiments validate the theoretical estimates and demonstrate the efficiency of the proposed method.

4. 包含84种格式的数值目录与比特精确一致性向量：面向FP8、BF16、MXFP4及微缩放格式的供应商中立参考

An 84-Format Numeric Catalog with Bit-Exact Conformance Vectors: A Vendor-Neutral Reference for FP8, BF16, MXFP4, and Microscaling Formats

作者: Dmitrii Vasilev

类别: cs.AR, cs.AI, cs.MS, cs.PF, math.NA

发布日期: 2026-06-08

arXiv ID: 2606.09686v1

链接: http://arxiv.org/abs/2606.09686v1

摘要: Numeric format proliferation in machine learning hardware – FP8 (E4M3 and E5M2), BF16, MXFP4, microscaling block formats, and dozens of research variants – has outpaced the availability of vendor-neutral, bit-exact reference material. Engineers porting models across accelerators encounter silent divergences that are difficult to diagnose without a shared ruler. This paper describes a catalog of 84 numeric formats spanning 13 families, a suite of six bit-exact conformance packs covering GF16, MXFP4 element, BF16, FP8 E4M3, FP8 E5M2, and E8M0 block scale, and an IEEE P3109 v3.2.0 cross-walk that maps each pack to its corresponding standards-track configured format. Each pack is a self-contained JSON document with a SHA-256 fingerprint, a shared row schema, and an anchor vector that encodes 3.0 – the identity phi^2 + 1/phi^2 = 3 – as a cross-pack sanity check. Packs are cross-validated against ml_dtypes 0.5.4 (Google/JAX); any divergence is documented explicitly and interpreted as a spec-permitted interpretation gap rather than hidden. The work is framed as registry filling: it does not propose new formats, make model-accuracy claims, or assert superiority over any vendor’s implementation. All artifacts are publicly available at https://github.com/gHashTag/t27 under an open license.

5. 边界层引发的标准物理信息神经网络失效：奇异摄动输运问题的Legendre小波配点基准

Boundary-Layer-Induced Failure of Standard Physics-Informed Neural Networks: A Legendre Wavelet Collocation Benchmark for Singularly Perturbed Transport Problems

作者: Suvendu Nayak, Arun Kumar Gupta

类别: math.NA

发布日期: 2026-06-08

arXiv ID: 2606.09676v1

链接: http://arxiv.org/abs/2606.09676v1

摘要: Boundary layers provide a demanding test for numerical solvers because the solution may remain almost constant over most of the domain while changing rapidly in a narrow region near the boundary. This paper studies a singularly perturbed one-dimensional transport boundary-value problem with increasing Peclet number $(\mathrm{Pe})$. A local Legendre wavelet collocation method (LWM) is compared with a standard soft-boundary physics-informed neural network (PINN) for this benchmark. The wavelet approximation uses locally supported Legendre polynomial basis functions and converts the problem into a square algebraic collocation system with residual, boundary, and interface-continuity equations. Numerical experiments are performed for $\mathrm{Pe}=1,10,100,$ and $1000$. The LWM captures all four cases, with the largest error remaining below $5\times 10^{-3}$. The standard soft-boundary PINN performs well for the mild cases but fails to resolve the sharp boundary layer for the larger Peclet numbers. The results show that local wavelet collocation is more reliable than the standard soft-boundary PINN for this benchmark, while dense near-boundary evaluation helps reveal errors that may be missed on coarse grids.

6. 一种用于患者特异性斜视手术规划的眼部可微模拟

A Differentiable Simulation of the Eye for Patient-Specific Strabismus Surgery Planning

作者: Even Harsigny, Pablo Alvarez, Michel Duprez, Stéphane Cotin

类别: math.NA

发布日期: 2026-06-08

arXiv ID: 2606.09661v1

链接: http://arxiv.org/abs/2606.09661v1

摘要: Purpose: Up to 4% of adults will develop strabismus in their lifetime. The most common surgical intervention involves adjusting the length of one or more extraocular muscles to correct the angular deviation. This correction depends on surgical expertise and statistical reference tables, which often fail to yield optimal results for patients with atypical eye morphology. Our work proposes a physics-based modeling approach to personalized surgical planning, accounting for patient-specific eye anatomy. Methods: We built a physics-based simulator of the eye and its muscles, incorporating patient-specific geometry and Hill-type muscle biomechanics. We solve an optimization problem to find the surgical dosage that minimizes angular deviation. The model is implemented as a fully differentiable simulation, enabling efficient optimization. We validated the framework by comparing its predictions with standard surgical tables for emmetropic eyes before applying it to anatomically atypical virtual patients. Results: Our model’s predictions for emmetropic eyes were first validated, demonstrating a strong fit with standard surgical tables. More importantly, for high-myopia models, the framework computed a clinically significant increase in the required surgical dosage compared to standard eyes. This computed recession difference is highly relevant as surgical plans are adjusted in 0.5 mm increments. Conclusion: Our results show that our model provides a calibrated surgical plan that, unlike standard tables, also accounts for pathologies involving atypical eye shapes. This patient-specific model represents a step toward personalized surgical planning, with the potential to improve dosage accuracy and surgical outcomes for atypical cases.

7. 贝叶斯反问题中的约束残差、图后验与行列式校正的全空间目标

Constraint residuals, graph posteriors, and determinant-corrected full-space targets in Bayesian inverse problems

作者: Jonathon Cottom, Emilia Olsson

类别: math.ST, cond-mat.stat-mech, math-ph, math.NA

发布日期: 2026-06-08

arXiv ID: 2606.09594v1

链接: http://arxiv.org/abs/2606.09594v1

摘要: Bayesian inverse problems constrained by state equations are often sampled in a full parameter-state space by penalising the residual, rather than in a reduced space where the state is eliminated. We show that these formulations are not automatically equivalent as posterior measures. For finite-dimensional discretisations of equality-constrained inverse problems, assume the state equation (c(θ,u)=0) has a unique solution (u=G(θ)) and nonsingular state Jacobian (\D_u c). The reduced posterior, its graph lift, and the zero-noise residual posterior are then distinct. A local change of variables shows that an uncorrected Gaussian residual penalty converges, after marginalisation over (u), to the reduced density multiplied by (\abs{\det \D_u c(θ,G(θ))}^{-1}). Thus algebraically equivalent residuals can define the same feasible set but different limiting posteriors. We derive determinant corrections for unweighted, weighted, and rescaled residual penalties that have the graph-lifted reduced posterior as their hard-constraint limit. The result separates feasibility from posterior calibration: driving the residual to zero is not sufficient for exact sampling of the graph-lifted reduced posterior unless the sampling or correction step targets the corresponding corrected density.

8. 无度量黎曼优化

Metric-Free Riemannian Optimization

作者: Jonas Püschel

类别: math.OC, math.NA

发布日期: 2026-06-08

arXiv ID: 2606.09465v1

链接: http://arxiv.org/abs/2606.09465v1

摘要: Riemannian optimization provides a powerful framework for constrained optimization by incorporating problem-specific structure directly into the geometry of the search space. In many applications, however, the explicit evaluation or application of the Riemannian metric can be computationally expensive or numerically unstable, limiting the practical efficiency of otherwise well-founded algorithms. Motivated by such settings, this work investigates to what extent classical Riemannian optimization algorithms can be reformulated without explicitly applying the metric. We show that many first-order components of Riemannian optimization only rely on the differential of the objective function and access to the Riemannian gradient, but not on explicit metric application. Based on this observation, we develop metric-free formulations and generalize optimization approaches to Finsler and Banach manifolds. Numerical experiments demonstrate that the proposed metric-free strategies retain the effectiveness of their metric-dependent counterparts while significantly reducing computational overhead. These results highlight that a substantial portion of Riemannian optimization can be carried out independently of explicit metric application, broadening its applicability to problems with expensive or implicitly defined metrics.

9. 残差伪谱揭示用于热带太平洋变率与ENSO预测的物理信息Koopman骨干

Residual Pseudospectra Reveal a Physics-Informed Koopman Backbone for Tropical Pacific Variability and ENSO Prediction

作者: Paula Lorenzo-Sanchez, Matthew J. Colbrook, Antonio Navarra

类别: math.NA, physics.ao-ph

发布日期: 2026-06-08

arXiv ID: 2606.09369v1

链接: http://arxiv.org/abs/2606.09369v1

摘要: Tropical Pacific sea-surface-temperature (SST) variability spans interacting timescales, with the ENSO as its dominant interannual expression. Yet the dynamical structure organizing this variability and underpinning extended-range predictability remains difficult to extract from high-dimensional observations. Koopman operator learning offers spectral coordinates for nonlinear dynamics, yet finite geophysical records often produce dense, sampling-sensitive spectra whose physical content is ambiguous. We show that this apparent redundancy reflects coherent operator-level structure. Combining kernel Extended Dynamic Mode Decomposition with residual minimization and pseudospectral analysis, we use the Koopman eigenvalue relation as a physics-informed consistency test to organize learned spectra. Applied to ERA5 and HadISST tropical Pacific SST anomalies, the residual landscape identifies 19 robust residual-minimum frequencies with coherent spatial modes that persist across products and sampling realizations. Together, these modes define a compact Koopman backbone spanning low-frequency modulation through quasi-biennial components, including ENSO-band variability. The surrounding spectral cloud is structured by integer powers and nonlinear combinations of this backbone, forming a residual-ordered Koopman hierarchy. The backbone reconstructs substantial Nino3.4 variance and enables skillful out-of-sample forecasts, with greatest gains at 8-18-month leads. By embedding dynamical consistency into physics-informed operator learning, the framework turns opaque spectra into robust, interpretable and predictive representations of tropical Pacific variability.

10. 双曲化Cahn-Hilliard方程的正当化与结构和渐近保持离散

Justification and structure- and asymptotic-preserving discretizations of a hyperbolized Cahn-Hilliard equation

作者: Jan Giesselmann, Fabio Leotta, Hendrik Ranocha, Jochen Schuetz

类别: math.NA

发布日期: 2026-06-08

arXiv ID: 2606.09299v1

链接: http://arxiv.org/abs/2606.09299v1

摘要: We study a hyperbolic approximation (“hyperbolization”) of the Cahn-Hilliard (CH) equation, originally proposed by Dhaouadi, Dumbser, and Gavrilyuk (2025, DOI: 10.1098/rspa.2024.0606) and study its convergence towards the CH model in a relaxation limit both via formal asymptotic expansions and, for a slightly modified approximation, via the relative energy framework. Moreover, we develop energy-stable semidiscretizations of the CH equation and of this hyperbolization using upwind summation-by-parts operators in space. Subsequently, we combine them with (additive) implicit-explicit (IMEX) Runge-Kutta methods based on a convex-concave splitting. We show that the resulting method is asymptotic preserving, i.e., it converges in the limit of the relaxation parameter to a stable discretization of the original CH equation. The choice of the necessary parameters is guided by the a priori error estimate based on the relative energy framework.

11. 风险规避型偏微分方程约束优化的多水平随机梯度下降

Multilevel Stochastic Gradient Descent for Risk-Averse PDE-Constrained Optimization

作者: Niklas Baumgarten, Philipp A. Guth, David Schneiderhan, Tommaso Vanzan

类别: math.OC, math.NA

发布日期: 2026-06-08

arXiv ID: 2606.09291v1

链接: http://arxiv.org/abs/2606.09291v1

摘要: We present recent advances in applying and analyzing multilevel stochastic gradient descent algorithms to risk-averse, three-dimensional PDE-constrained optimization problems. The algorithm uses adaptive multilevel Monte Carlo gradient estimates, provides parallel scalability as well as improved convergence rates and computational complexity compared to standard batched stochastic gradient descent methods. We study the method in computationally demanding settings using three-dimensional elliptic diffusion problems and large risk-aversion parameters.

12. 一种基于八叉树的自适应隐式物质点方法

An implicit octree-based adaptive Material Point Method

作者: Robert E. Bird, William M. Coombs, Charles E. Augarde, Ted J. O’Hare

类别: math.NA

发布日期: 2026-06-08

arXiv ID: 2606.09275v1

链接: http://arxiv.org/abs/2606.09275v1

摘要: The Material Point Method provides an effective approach for modelling the large deformations that often arise from contact interactions between rigid structures and surrounding continua. However, solving these problems requires accurate representation of the continuum-structure interface, which necessitates high resolution background mesh and material point discretisations. This requirement, combined with evolving continuum-structure interfaces and the fact that most Material Point Method implementations are dependent on structured meshes, can result in large numerical systems and long run times especially when modelling problems in three-dimensions. Motivated by this issue, this paper provides the first octree-based implicit Material Point Method for efficient solution of large deformation continuum-structure interaction problems. The octree background mesh provides a natural way to automatically adapt both the computational mesh and the material point discretisation based on the position of the interaction between the structure and continuum. The new approach is demonstrated on a number of large deformation benchmark and continuum-structure interaction problems, where up to a 5.5-times speed up and a consequent 21-times CO2 saving is achieved when running on a HPC compared to results obtained using a conforming mesh.

13. 驯服有限元方法对乘性噪声驱动的超线性随机偏微分方程的时间一致强误差估计

Uniform-in-time Strong Error Estimates of Tamed-FEM to Superlinear SPDEs driven by Multiplicative Noise

作者: Jingjing Cai, Zhihui Liu

类别: math.NA

发布日期: 2026-06-08

arXiv ID: 2606.09173v1

链接: http://arxiv.org/abs/2606.09173v1

摘要: We establish sharp, uniform-in-time strong error estimates for a nonlinearity-explicit tamed finite element method (FEM) applied to a class of superlinear stochastic partial differential equations (SPDEs) driven by multiplicative noise, including the stochastic Allen–Cahn equation with a moderately thick interface. This tamed-FEM was first introduced in [Z. Liu and J. Shen, arXiv:2502.19117] to ensure long-time unconditional stability and to preserve the Lyapunov structure of this class of SPDEs. We further prove that the scheme is exponentially ergodic and derive the convergence rate between the exact invariant measure and its numerical counterpart in the Wasserstein-2 distance. Finally, we present numerical experiments that verify the ergodicity as well as the sharpness and time-independence of the strong convergence rates for this tamed-FEM.

14. 具有记忆的部分耗散粘弹性Timoshenko系统的高阶正则性与全离散傅里叶谱方法

High-Order Regularity and a Fully Discrete Fourier Spectral Method for a Partially Dissipative Viscoelastic Timoshenko System with Memory

作者: Zhenyang Zhong, Hui Liang

类别: math.NA

发布日期: 2026-06-08

arXiv ID: 2606.09007v1

链接: http://arxiv.org/abs/2606.09007v1

摘要: This paper investigates a class of partially dissipative viscoelastic Timoshenko systems with memory, where dissipation is induced by a Volterra-type memory term acting only on the shear variable. The well-posedness of weak and strong solutions is established on finite time intervals, including existence, uniqueness, stability, and higher-order regularity under compatibility conditions consistent with mixed boundary conditions. For the numerical approximation, a Fourier spectral fully discrete scheme is constructed: sine and cosine basis expansions are used in space for unknowns satisfying Dirichlet and Neumann boundary conditions, respectively; in time, a central difference scheme is applied to the second-order derivatives, and the composite trapezoidal rule is used to approximate the memory convolution term. Based on a discrete energy method, the positivity of the constructed discrete energy is proved, and the error estimate for the fully discrete scheme with second-order convergence in time and (q)-th order in space is established for any q \in \mathbb{N}. Numerical experiments are given to verify the theoretical convergence rates and to compare the dynamic responses of the local and nonlocal models, demonstrating that the memory term effectively captures energy dissipation and vibration attenuation behavior in viscoelastic materials.

15. 用无矩阵多尺度 Robin 预条件子加速 GMRES

Accelerating GMRES with Matrix-Free Multiscale Robin Preconditioners

作者: Dilong Zhou, Rafael T. Guiraldello, Felipe Pereira, Fabrício S. Sousa

类别: math.NA

发布日期: 2026-06-07

arXiv ID: 2606.08883v1

链接: http://arxiv.org/abs/2606.08883v1

摘要: We propose a matrix-free right-preconditioning strategy for the Generalized Minimal Residual (GMRES) method based on the Multiscale Robin Coupled Method with oversampling (MRCM-OS) for the numerical solution of elliptic problems arising in subsurface flow. The resulting preconditioner is constructed through local subdomain solves with oversampling and smoothing, and can be applied without explicit assembly of the global operator. After a careful presentation of the new procedure, it is used in extensive numerical experiments. Our results demonstrate that the proposed approach substantially reduces iteration counts across a range of challenging, high-contrast subsurface flow problems. In many cases, convergence is obtained in one or two GMRES iterations when oversampling and smoothing are employed. The results indicate that combining GMRES with multiscale Robin-based operators is a promising direction for the construction of rapidly convergent preconditioning strategies.

16. 带秩1格点与双曲交叉的傅里叶神经算子

Fourier Neural Operators with rank-1 lattice points and hyperbolic cross

作者: Jakob Dilen, Alexander Keller, Frances Y. Kuo, Dirk Nuyens

类别: math.NA, cs.LG

发布日期: 2026-06-07

arXiv ID: 2606.08871v1

链接: http://arxiv.org/abs/2606.08871v1

摘要: The \emph{Fourier neural operator} (FNO) is a neural network architecture that learns mappings between function spaces. Its efficient implementation is based on the multi-dimensional Fourier transform. By deriving general regularity bounds for the FNO with respect to both the spatial and parametric variables, we prove that the generalization error of the FNO can be improved by replacing spatial tensor product grids with purpose-built rank-1 lattice points, and by using a second lattice carefully constructed as training points in the parametric space. We achieve more accurate and efficient approximations from fewer network parameters, fewer spatial points, and fewer training samples. In addition, the architecture is simplified, because the high-dimensional Fourier transform on rank-1 lattices requires only a \emph{one-dimensional fast Fourier transform}, and we can use a \emph{hyperbolic cross} frequency index set with lattice points. We demonstrate the benefits of our \emph{lattice-based hyperbolic-cross FNOs} for an elliptic PDE on the torus.

17. 面向有限元模拟的低方差随机数值线性代数

Low-Variance Randomised Numerical Linear Algebra for Finite Element Simulation

作者: N. Polydorides, Y. Wu, . H. Noori, H. Vandierendonck, R. Woods

类别: math.NA

发布日期: 2026-06-07

arXiv ID: 2606.08817v1

链接: http://arxiv.org/abs/2606.08817v1

摘要: We present a low-variance randomised numerical linear algebra approach for multi-query finite element systems arising from parametric elliptic partial differential equations with applications to digital twins and online model calibration. The method relies on Galerkin subspace projection for reducing the dimensionality, and then combines parameter-oblivious leverage-score Bernoulli sampling with a control variates scheme to yield a reduced-variance forward' sketch and an invertible inverse’ sketch that are then fused to a single efficient regularised estimator. Effectively, this reduces the computational cost in computing the projected system of equations while preserving the structure, stability, and accuracy of the underlying FEM formulation. We derive probabilistic bounds for the sketching error, invertibility, and estimator variance, and then validate the method on large-scale example problems. The results show that when the parameter fields do not vary too sharply, the synergy of control variates together with the sketch fusion can largely offset the loss incurred by the sub-optimal parameter-oblivious sampling. In this regime, our method achieves substantial savings in time, memory, and communication while maintaining accuracy levels that are acceptable for scientific simulation.

18. OptMuon：用于随机优化且具有零噪声最优性的闭环正交化动量方法

OptMuon: Closed-Loop Orthogonalized Momentum Methods for Stochastic Optimization with Zero-Noise Optimality

作者: Ganzhao Yuan

类别: math.OC, cs.LG, math.NA

发布日期: 2026-06-07

arXiv ID: 2606.08783v1

链接: http://arxiv.org/abs/2606.08783v1

摘要: Orthogonalized momentum updates, as used in Muon-style optimizers, have recently shown strong empirical stability in large-scale deep learning. However, existing orthogonalized methods are typically paired with constant or open-loop magnitude rules, and therefore do not explicitly calibrate their update magnitudes from the observed optimization trajectory. Motivated by the closed-loop perspective behind Lipschitz-free and noise-adaptive methods, we propose OptMuon, a family of adaptive momentum orthogonalization methods for stochastic nonconvex optimization. OptMuon combines Muon-style polar-factor directions with a trajectory-dependent AdaGrad-Norm-type coefficient schedule, so that the update magnitude is determined by the observed gradient and momentum history rather than by a prescribed Lipschitz-dependent rule. The schedule does not use the smoothness constant, the variance level, or the bounded-gradient constant in parameter selection, and its running-maximum correction prevents isolated gradient spikes from causing excessive coefficient collapse. Under lower-boundedness, unbiased stochastic gradients with bounded variance, smoothness, and an almost-sure bounded stochastic-gradient condition, we prove two complementary guarantees. OptMuon-A achieves the noise-adaptive rate (\tilde{\mathcal O}(T^{-1/2}+σ^{1/2}T^{-1/4})) under average smoothness, while OptMuon-I achieves (\tilde{\mathcal O}(T^{-1/2}+σ^{1/3}T^{-1/3})) under individual smoothness. In the zero-noise regime, both bounds automatically reduce to a nearly optimal deterministic first-order rate (\tilde{\mathcal O}(T^{-1/2})) without manual hyperparameter retuning. These results show that closed-loop scalar adaptation can be combined with Muon-style momentum orthogonalization while retaining noise adaptivity and zero-noise optimality up to logarithmic factors.

19. 基于有限转置法的倒向随机微分方程数值分析

Numerical Analysis on Backward Stochastic Differential Equations by Finite Transposition Method

作者: Penghui Wang, Yanqing Wang, Xu Zhang

类别: math.PR, math.NA

发布日期: 2026-06-07

arXiv ID: 2606.08731v1

链接: http://arxiv.org/abs/2606.08731v1

摘要: In this paper, we propose a finite transposition method to solve backward stochastic differential equations (BSDEs, for short). Based on the transposition solution theory for BSDEs, our method offers a promising way of efficiently computing solutions, which can be regarded as an analogous method for BSDEs as the classical finite element method for partial differential equations. Our method has the advantage of easily computable conditional expectations.

20. 组合逼近可严格优于叠加逼近

Compositional Approximation Can Strictly Outperform Superpositional Approximation

作者: Dennis Elbrächter, Philipp Petersen

类别: math.NA, cs.LG

发布日期: 2026-06-07

arXiv ID: 2606.08727v1

链接: http://arxiv.org/abs/2606.08727v1

摘要: Many classically studied function classes are known to be approximated optimally by superpositional methods, i.e. with approximants constructed as the linear combination of elements in some dictionary. Here optimality means that the uniform approximation error viewed as a function of the number of parameters used has polynomial decay of the highest order achievable by any parametrized method whose parameters can be encoded as a bit string of length proportional, up to logarithmic factors, to the number of parameters. While compositional methods like neural networks are structurally different, their approximation rates can be made comparable by imposing constraints that ensure such a proportional bit string encoding. In this work we study function classes exhibiting structural properties that limit superpositional approximation rates to be strictly lower than compositional approximation rates. In particular, we construct explicit examples for which there is an arbitrarily large gap.

21. 利用内部测量同时恢复非局部扩散方程的多参数

Simultaneous recovery of multiple parameters in nonlocal diffusion equations from internal measurements

作者: Kai Yu, Zhiyuan Li, Yikan Liu

类别: math.NA, math.AP

发布日期: 2026-06-07

arXiv ID: 2606.08699v1

链接: http://arxiv.org/abs/2606.08699v1

摘要: This paper is devoted to simultaneously recovering multiple parameters from internal measurements for nonlocal diffusion equations. The uniqueness of the inverse problem is established by employing the asymptotic behavior of solutions, analytic continuation, the Laplace transform, and properties of analytic functions. For numerical reconstruction, we apply the Levenberg-Marquardt method to obtain a stable approximate solution of the inverse problem. Numerical examples are provided to demonstrate the efficiency of the proposed algorithm and to validate our theoretical findings.

22. ND-TNN：高维非局部扩散模型的张量神经网络逼近

ND-TNN: Tensor-Neural-Network Approximation for High-Dimensional Nonlocal Diffusion Models

作者: Ziyue Cai, Zuoqiang Shi

类别: math.NA

发布日期: 2026-06-07

arXiv ID: 2606.08685v1

链接: http://arxiv.org/abs/2606.08685v1

摘要: We study a numerical method, built on the tensor neural network (TNN) architecture introduced in \cite{wang2022tensor}, for solving nonlocal diffusion models in high-dimensional spaces. The tensor-product structure of the TNN ansatz, combined with the separability of the Gaussian kernel, reduces the high-dimensional integrals in the nonlocal energy to products of low-dimensional integrals, which are evaluated by Gauss–Legendre quadrature; nonseparable source and boundary data are handled by a TNN-based preconditioning step. For the Dirichlet boundary condition, we establish the asymptotically compatible $L^2$ error estimate [ |u_{\mathrm{loc}}-u_{δ,p}|{L^2(Ω)} \le C!\left(\frac{\varepsilon_f}{\sqrtδ} +\frac{\varepsilon_g}δ +\frac{\varepsilon_u}{\sqrtδ} +η{\mathrm{opt}}\right) +C\sqrtδ, ] where $\varepsilon_f$, $\varepsilon_g$ and $\varepsilon_u$ are the data and trial-class approximation errors and $η_{\mathrm{opt}}$ is the optimization residual. For the Neumann boundary condition, the $L^2$ estimate is improved to $O(\varepsilon_f+\varepsilon_g/\sqrtδ+\varepsilon_u +η_{\mathrm{opt}}+δ)$, and an $H^1$ gradient estimate is further obtained through a smoothing post-processing step. Numerical experiments on tensor-product domains up to $d=20$ support the theoretical results, and additional tests on two- and three-dimensional $L$-shaped domains demonstrate the practical robustness of the method beyond the smooth-domain setting covered by the analysis.

23. 二维不可压缩Navier-Stokes方程的算子学习：数据稀缺情形下的共形预测方法

Operator learning for the 2D incompressible Navier-Stokes equations: a conformal prediction approach in the data-scarce regime

作者: Weinan Wang, Bowen Gang, Hao Deng

类别: cs.LG, math.AP, math.NA, stat.AP

发布日期: 2026-06-07

arXiv ID: 2606.08654v1

链接: http://arxiv.org/abs/2606.08654v1

摘要: In this paper, we propose a perturbation-based conformal prediction framework for uncertainty quantification in operator learning, with a focus on the 2D Navier–Stokes equations. While neural operators provide fast surrogates for expensive PDE solvers, they do not by themselves provide calibrated uncertainty for spatiotemporal field predictions. Our approach wraps a trained Fourier Neural Operator (FNO) with split conformal prediction and constructs the local uncertainty scale by comparing the predictions of two operators trained on nearly identical datasets: one on the original labels and one on labels perturbed by small Gaussian noise. We consider this procedure in the data-scarce regime, where the total label budget is fixed and methods that require a separate uncertainty network must divide training data between multiple models. On the 2D Navier–Stokes benchmark, the perturbation-based method produces substantially narrower conformal bands than existing methods under matched total data budgets while maintaining the target simultaneous coverage. These results suggest that perturbation sensitivity is a practical and sample-efficient uncertainty proxy for conformalized neural operators.

24. 多尺度傅里叶神经算子用于高振荡介质中的波动反散射

Multiscale Fourier Neural Operator for Inverse Wave Scattering in Highly Oscillatory Media

作者: Zilin You, Zhenli Xu, Wei Cai

类别: math.NA

发布日期: 2026-06-07

arXiv ID: 2606.08448v1

链接: http://arxiv.org/abs/2606.08448v1

摘要: In this paper, we propose an operator learning method based on the multiscale Fourier neural operator (MscaleFNO) for inverse medium problems of Helmholtz equations. The MscaleFNO provides a neural surrogate model with reduced spectral bias for the Helmholtz equations, mapping highly oscillatory medium profiles to scattered wavefields. A plug-and-play inversion using elucidated diffusion model is introduced to regularize the inverse solver based on least squares of data misfits. Numerical results for partial aperture inversion of oscillatory two-dimensional media demonstrate the advantage and effectiveness of MscaleFNO for accurate reconstruction of highly oscillatory medium properties.

25. 自定义精度的浮点自动调优

Floating-point autotuning with customized precisions

作者: Xinye Chen, Thibault Hilaire, Fabienne Jézéquel

类别: cs.MS, math.NA

发布日期: 2026-06-06

arXiv ID: 2606.08339v1

链接: http://arxiv.org/abs/2606.08339v1

摘要: Reduced-precision arithmetic offers significant opportunities to improve performance, memory usage, and energy efficiency in numerical applications, provided that numerical accuracy is preserved. This work investigates automated precision tuning through customized floating-point formats with user-defined exponent and significand sizes, enabling the emulation of emerging low-precision formats and the exploration of non-standard precision configurations within a unified mixed-precision framework. The proposed methodology, implemented in the PROMISE precision autotuning tool, combines numerical validation with a systematic search to generate program variants that satisfy user-defined accuracy requirements. To address the computational cost of this exploration, a containerized benchmarking framework supports parallel execution across multiple algorithms and parameter configurations. The approach is evaluated on a suite of numerical programs, including linear solvers and applications from the Rodinia benchmark. Results show that a substantial proportion of variables can be safely reduced to lower precision while preserving accuracy, indicating that standard double precision is often over-provisioned. These findings highlight the potential of automated precision tuning to derive efficient mixed-precision configurations tailored to application-specific accuracy requirements.

26. 用于刚性高维常微分方程的稳定且可扩展的概率数值求解器

Stable and Scalable Probabilistic Numerical Solvers for Stiff and High-Dimensional ODEs

作者: Nathanael Bosch

类别: math.NA, cs.LG, stat.ML

发布日期: 2026-06-06

arXiv ID: 2606.08203v1

链接: http://arxiv.org/abs/2606.08203v1

摘要: Filtering-based probabilistic numerical solvers for ordinary differential equations (ODEs) have been established as a flexible and efficient simulation framework with built-in numerical uncertainty quantification. However, problems that are both stiff and high-dimensional remain a challenge, as current methods are either stable and have cubic cost in the ODE dimension, or scale linearly at the expense of stability. In this paper, we close this gap and develop probabilistic ODE solvers that are both stable and scalable. We propose two complementary strategies. First, we develop a matrix-free update step that uses Jacobian-vector products, iterative linear solvers, and stochastic covariance estimation to enable linear scaling, all while retaining stability. Second, we propose iterative re-linearization to further improve stability without sacrificing scalability, turning probabilistic ODE solvers into fully implicit methods. We evaluate the proposed approaches on a range of stiff and high-dimensional problems and demonstrate improved stability and scalability over established probabilistic solvers.

27. 四元数最大体积子矩阵选择及其在多通道成像与视觉数据中的应用

Quaternion Maximum-Volume Submatrix Selection with Applications to Multichannel Imaging and Visual Data

作者: Vsevolod Kliushev, Junjun Pan, Valentin Leplat

类别: math.NA

发布日期: 2026-06-06

arXiv ID: 2606.08175v1

链接: http://arxiv.org/abs/2606.08175v1

摘要: Low-rank approximation based on selected rows and columns is a useful alternative to singular value decompositions when the goal is an interpretable and compact matrix representation. A standard way to choose these rows and columns is the maximum-volume principle: it selects submatrices with large volume, which usually leads to stable interpolation coefficients and accurate CUR-type approximations. In this paper, we study this idea for quaternion matrices. This setting is natural for color images, three-dimensional motion data, and multi-channel signals, but requires care because quaternion multiplication is noncommutative. We define quaternion maximum-volume submatrix selection using quaternion singular values and the Study determinant. We then derive quaternion rank-one update formulas and use them to build two selection procedures: a greedy square-core method for row and column replacement, and a rectangular method that enlarges a selected row set until the interpolation coefficients are controlled. We prove that successful row and column swaps increase the quaternion volume of the selected square core when the exact quaternion inverse is used. We also connect the stopping criterion with quasi-dominance, prove an exact quaternion CUR identity in the full-rank case, and derive an interpolation stability bound. For the rectangular case, we derive an append-row pseudoinverse update and show how it gives a natural right preconditioner for overdetermined quaternion least-squares problems. Finally, we illustrate the methods on three applications: quaternion CUR approximation of RGB images, RectMaxVol-based preconditioning for ill-conditioned quaternion least-squares systems, and row selection in quaternion motion-capture data. The experiments show that the proposed quaternion MaxVol and RectMaxVol methods provide stable and efficient selection routines.

28. AttentionCap: 基于Transformer的电容矩阵学习面向全芯片提取

AttentionCap: Transformer Based Capacitance Matrix Learning Toward Full-Chip Extraction

作者: Jiechen Huang, Hector R. Rodriguez, Dingcheng Yang, Zuochang Ye, Yibo Lin, Wenjian Yu

类别: cs.LG, cs.AR, math.NA

发布日期: 2026-06-06

arXiv ID: 2606.08161v1

链接: http://arxiv.org/abs/2606.08161v1

摘要: As capacitance extraction accuracy of rule-based pattern matching becomes difficult to sustain at advanced nodes, a growing trend emerges to develop deep-learning-based 2D capacitance models. However, existing MLP- and CNN-based methods constrain their input to fixed metal-layer combinations in a specific process node, limiting their usability in practice. Recognizing the inherent similarity between capacitance matrix and the prevailing attention mechanism, we propose AttentionCap, a customized Transformer for capacitance matrix learning, with a Gram representation framework, a physics-aligned symmetric-attention output layer, and a novel normalized Laplacian loss. We also introduce a process-node embedding to enable multi-node learning. Trained on synthetic data, AttentionCap attains 0.67%/3.99% self/coupling-capacitance error on unseen real designs under a multi-layer and multi-node setting, surpassing the CNN-Cap baseline with 4.6$\times$/5.7$\times$ lower self/coupling error and 192$\times$ faster inference speed. A pretrained AttentionCap accurately transfers to an unseen node with only 5K samples and 4K finetuning steps. With sufficient accuracy on unseen real designs and strong transferability to new process nodes, AttentionCap offers highly practical value for modern EDA workflows. Code and data are available at https://github.com/THU-numbda/AttentionCap.

29. 软化塑性中的应变局部化，无需修改标准本构模型：一种可变形Cosserat方法

Strain localization in softening plasticity without modifying standard constitutive models: a deformable Cosserat approach

作者: Andrea Panteghini, M. B. Rubin

类别: math.NA

发布日期: 2026-06-06

arXiv ID: 2606.08095v1

链接: http://arxiv.org/abs/2606.08095v1

摘要: This paper presents a formulation for strain localization in softening plasticity based on a deformable Cosserat model. The approach enables the direct use of standard elastoplastic constitutive models formulated for a classical Cauchy continuum, without modifying the stress update algorithm or consistent tangent operator. A key feature of the framework is the strict separation of dissipative and energetic mechanisms: all dissipation is confined to the macro-continuum, while the micro-continuum contributes only through linear elastic terms associated with the director field. As a result, the constitutive structure of the elastoplastic model is preserved, and existing models can be employed as black-box components. The internal length scale arises naturally from the micro-continuum and governs the development, interaction and selection of localization patterns, rather than acting as a diffusive parameter. The formulation is easy to implement within standard finite element frameworks, requiring only additional linear contributions to the residual and tangent operators. The performance of the approach is assessed through benchmark problems involving shallow foundations on soil, a demanding test due to complex and unstable localization mechanisms. Both Tresca and Matsuoka-Nakai plasticity models are considered, including cases with highly unstable post-peak responses. Numerical results show convergence of load-displacement responses, dissipated energy and shear-band patterns upon mesh refinement, even in the presence of nonlinear interacting localization processes. These findings demonstrate a robust and physically consistent approach for the analysis of strain localization in softening plasticity.

30. 基于分裂变分量子算法的非线性狄拉克方程数值解

Numerical solution of the nonlinear Dirac equation by a splitting variational quantum algorithm

作者: Qian Zuo, Ying He, Xiaofei Zhao

类别: quant-ph, math.NA

发布日期: 2026-06-06

arXiv ID: 2606.08053v1

链接: http://arxiv.org/abs/2606.08053v1

摘要: In this work, we propose an operator-splitting variational quantum algorithm, termed Dirac-sVQA, for simulating the nonlinear Dirac equation (NLDE). The main difficulty arises from the state-dependent nonlinear interaction, its time-discrete update depends explicitly on the intermediate spinor state and, in general, cannot be implemented as a fixed state-independent unitary circuit. To address this difficulty, we decompose the NLDE evolution into a structured linear Dirac substep and a nonlinear variational correction. The linear substep is implemented by a spinor-Fourier Dirac propagator on a joint position-spin register, preserving the spin-momentum coupling and mass-induced spin evolution of the Dirac operator. The nonlinear correction is reformulated as a measurement-based variational update through a small set of overlap, self-channel, and cross-channel observables. We provide the corresponding quantum circuits and derive measurement-aware resource and complexity estimates. Numerical experiments in several nonlinear regimes show that Dirac-sVQA accurately captures both the total density and the componentwise spinor dynamics, agrees well with classical Fourier pseudospectral splitting solutions, and exhibits stable error behavior over time. These results provide numerical evidence for the feasibility of operator-splitting variational quantum simulation for nonlinear relativistic wave equations.

31. 带约束的极大似然估计与最小二乘问题中的拉格朗日乘子

Lagrange multipliers in Maximum likelihood estimations and Least squares problems with Constraints

作者: Takeshi Fukasawa

类别: econ.EM, math.NA, stat.CO

发布日期: 2026-06-06

arXiv ID: 2606.07984v1

链接: http://arxiv.org/abs/2606.07984v1

摘要: This study investigates a statistical property of Lagrange multipliers in constrained Maximum Likelihood Estimation (MLE) and Least Squares (LS) problems from the perspective of numerical optimization. Building on large-sample theory, we show that the associated Lagrange multipliers converge to zero as the sample size increases, provided the distribution is correctly specified in MLE or the residuals are normally distributed in LS. Although this asymptotic behavior has long been recognized in statistics, it has received little explicit attention in numerical optimization and has rarely been exploited in algorithmic design. Importantly, the insight extends beyond classical low-dimensional settings: even in modern high-dimensional applications, such as deep learning, where the number of parameters may exceed the sample size, the same reasoning applies provided the generalization performance is good. This observation has two main implications. First, many constrained optimization algorithms, including the Augmented Lagrangian Method, Sequential Quadratic Programming, and Interior Point methods, require initial values for the multipliers, and choosing zero is statistically justified. Numerical experiments for constrained regressions and dynamic discrete choice model estimations support this implication by showing that initializing multipliers at zero usually lead to stable and efficient performance. Second, penalty-based approaches that convert constrained problems into unconstrained ones can perform well when the true multipliers are small. This helps explain why penalty-based methods often perform well in practice.

32. 一种用于改进型狄利克雷特征值最小化器的新的水平集公式

A New Level Set Formulation for Improved Dirichlet Eigenvalue Minimizers

作者: Atharv Thakur

类别: math.NA, math.AP, math.OC

发布日期: 2026-06-06

arXiv ID: 2606.07979v1

链接: http://arxiv.org/abs/2606.07979v1

摘要: This paper makes several improvements to existing level set based approaches to computing shape optimizers for the Dirichlet eigenvalues subject to a volume constraint. The most notable changes in formulation include an overhaul of the classical level set construction and root-finding procedures as well the use of a regularized approximation to the standard objective function. Our resulting computational minimizers are either comparable to or improvements on the best known minimizers from the literature. We conclude with a survey of subproblems within the field that may benefit from numerical experiments; these include the existence of cusps on the boundary, the end-behavior of eigenfunction weights in the p-parameterized problem, and the nature of Weyl asymptotics as they relate to the Pólya conjecture.

33. 用于无穷维主方程的测度一致算子学习方法

A Measure-Consistent Operator Learning Method for Infinite-Dimensional Master Equations

作者: Chenyao Wang, Hongyu Liu, Hui Liang

类别: math.NA

发布日期: 2026-06-06

arXiv ID: 2606.07976v1

链接: http://arxiv.org/abs/2606.07976v1

摘要: Master equations in mean field game theory characterize feedback value functions that depend on time, state (space), and the population distribution. Their numerical approximation is challenging because the unknown is defined on a space of probability measures and the equation involves intrinsic measure derivatives and nonlocal population terms. This paper proposes a measure-consistent operator learning method (MCOL) for infinite-dimensional master equations. The population distribution is represented by an empirical measure and encoded through a symmetric pooling structure, so that the network input is built directly from the particles representing the measure. The same particles are used in the empirical quadrature of the nonlocal residual terms, avoiding additional quadrature grids or auxiliary integration points. A key feature is that the intrinsic derivative appearing in the residual is induced by the same measure-dependent representation that defines the approximation of the value function. Consequently, the value function, its measure derivative, and the empirical residual are tied to a common measure representation, leading to a structurally coupled value-derivative approximation. We also introduce an error decomposition separating neural approximation error from empirical discretization error. Numerical experiments on several master equations show that MCOL accurately approximates the value function, intrinsic measure derivatives, and feedback quantities, and remains robust under changes in the input measures.

34. 关于子空间上外心反射方法的精确线性收敛速率

On the sharp linear convergence rate of the circumcentered–reflection method on subspaces

作者: Yunier Bello-Cruz

类别: math.OC, math.NA

发布日期: 2026-06-05

arXiv ID: 2606.07888v1

链接: http://arxiv.org/abs/2606.07888v1

摘要: For two subspaces $U,V\subseteq\RR^n$, the circumcentered–reflection method (CRM) of Behling, Bello-Cruz, and Santos~\cite{BBS2018} computes the projection onto $U\cap V$ using only the reflections across $U$ and $V$, with known linear-convergence rate $c_F$, the cosine of the Friedrichs angle. We prove that, when CRM is initialized in $V$, it contracts at the strictly smaller rate $ρ_V=(\sin^2θ_p-\sin^2θ_F)/(\sin^2θ_p+\sin^2θ_F)$, where $θ_F\in(0,π/2]$ is the Friedrichs angle and $θ_p\in[θ_F,π/2]$ the largest principal angle between $U$ and $V$. The bound is sharp, attained on an explicit ray in $V$, and optimal among parameter-free single-step iterations. The constant itself is not new: Bauschke, Bello-Cruz, Nghia, Phan, and Wang~\cite{BBNPW2016} identified it as the optimal rate of the relaxed alternating-projection family and of their adaptive linesearch map $B_T$. Our contribution is that the parameter-free geometric circumcenter attains it as well, via Kantorovich’s inequality applied to a single self-adjoint operator on $V$. Restricted to $V$, CRM coincides pointwise with the linesearch maps $A_T$ and $B_T$ from the Gubin–Polyak–Raik framework~\cite{GPR1967}. We further prove $ρ_V

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33. 用于无穷维主方程的测度一致算子学习方法#

34. 关于子空间上外心反射方法的精确线性收敛速率#

重点关注论文（完整分析）

1. 多边形网格上椭圆界面问题的混合扩展虚拟元方法

1. 研究对象和背景

2. 主要定理或主要结果

3. 研究方法、关键技术和核心工具

4. 与之前工作的比较

2. 针对周期障碍物阵列散射的一致高精度 PML-BIE 方法：二维情形

1. 研究对象和背景

2. 主要定理或主要结果

3. 研究方法、关键技术和核心工具

4. 与之前工作的比较

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1. $p$-Stokes 问题的压力鲁棒拟最优间断 Galerkin 离散

摘要翻译

2. 含动力学边界条件的波动方程的二阶体–表面分裂方法

摘要翻译

3. 随机 Schrödinger–KdV 方程的新型保物理性质方法

摘要翻译

4. 变阶四面体空间的稳定三角形投影与一致IPDG预处理

摘要翻译

5. 一种用于曲面演化的二阶结构保持参数有限元方法

摘要翻译

6. 曲面Stokes问题的无散度Scott-Vogelius有限元方法

摘要翻译

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