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1. 基于量化张量列的多尺度椭圆方程稳定全场模拟
Stable full-field simulation of a multiscale elliptic equation by means of Quantized Tensor Trains
作者: Marc Josien, Anas El Hachimi, Isabelle Ramière
类别: math.NA
发布日期: 2026-05-20
arXiv ID: 2605.21709v1
摘要翻译: 本文针对具有非均匀系数场的线性椭圆方程，设计了一种基于量化张量列（QTT）的新型求解器，该求解器能够处理极细的网格。在维数 $d=2$ 和 $d=3$ 下，它可以实现全场模拟，自由度数（DoFs）比经典求解器高出多达 $20$ 个数量级，并在 $\LL^2$ 范数下精确恢复解及其梯度。为了处理如此庞大的数据量，该求解器关键依赖于 QTT 的指数压缩性质，这显著改进了现有文献的结果。
Comment: 32 pages, 22 figures
链接: http://arxiv.org/abs/2605.21709v1
1. 研究对象和背景
本文考虑具有异质系数场的线性椭圆型偏微分方程的数值求解，方程形式为
$$ -\nabla\cdot (a\nabla u) = \nabla\cdot g, $$定义在周期单位立方体 $Q_{1,\mathrm{per}} = \mathbb{R}^d / \mathbb{Z}^d$ 上，$d = 2$ 或 $3$。其中未知函数 $u$ 是位移场或温度场，$\nabla u$ 是其梯度，系数 $a \in L^\infty(Q_{1,\mathrm{per}})^{d\times d}$ 表示材料的非均匀性质（如热传导系数或弹性张量），满足对称性与一致椭圆性：存在常数 $\Lambda,\lambda > 1$，使得对任意 $x \in Q_{1,\mathrm{per}}$ 有 $\lambda^{-1} \le a(x) \le \Lambda$。源项 $g \in L^2(Q_{1,\mathrm{per}})$ 已知，以散度形式给出，便于后续分析；非散度形式 $f$ 可通过求解 Poisson 方程预处理转化为 $g$。在上述假设下，由 Fredholm 择一定理，方程存在唯一解 $\nabla u \in L^2(Q_{1,\mathrm{per}})$，且满足能量估计 $\|\nabla u\|_2 \le \lambda \|g\|_2$，而 $u$ 仅相差一个常数。
这类问题在材料科学中具有核心地位。当材料内部包含微结构（如夹杂、缺陷或晶界），其特征尺度 $\varepsilon \ll 1$ 远小于宏观尺度，直接的全场有限元模拟需要网格尺寸 $h \ll \varepsilon$，导致自由度数 $(1/h)^d$ 达到天文数字，远超当代超级计算机的内存极限。经典的数值方法如有限元平方 (FE$^2$) 或异质多尺度方法 (HMM) 需要在小尺度和大尺度之间耦合求解，且依赖于尺度分离假设，在真实材料中往往不成立。在计算均匀化框架下，FFT-based 求解器借助快速 Fourier 变换能高效处理周期性问题，但离散尺度仍受限（通常自由度 $< 10^{10}$）且无法直接考虑极细网格下的稳定性。
近年来，张量网络表示，尤其是量化张量列 (Quantized Tensor Train, QTT)，为高维数据压缩提供了可能。QTT 将 $d$ 维空间上的函数 $u(x)$ 沿每个坐标轴的二进分解展开为矩阵乘积态 (MPS)，利用低秩结构实现存储量与 $h$ 的指数级压缩：存储代价仅为 $O(d L r^2)$，其中 $h = 2^{-L}$，$r$ 是最大 TT-秩。已有工作表明，许多解析函数与光滑解可由低秩 QTT 高精度逼近，且基本的线性代数运算（加法、乘法、矩阵作用）均可直接在压缩格式下进行。然而，将 QTT 用于求解椭圆方程面临两个关键障碍。
第一，离散椭圆算子的条件数随网格加密以 $h^{-2}$ 增长，当 $h < 10^{-8}$ 时将超过机器精度，直接迭代求解不可行。Rakhuba [40,41] 通过解析构造一维 Laplace 算子之逆的 MPO，在常系数情形下实现了 $h = 10^{-12}$ 的精细网格，但该方法不能处理非均匀系数 $a$，且无法准确恢复梯度 $\nabla u$。Bachmayr 与 Kazeev [4] 提出了预条件 QTT 有限元方法，理论上可处理异质系数并获得 $H^1$ 收敛，但对常系数 $a=1$ 的二维问题，其 MPO 的 TT-秩至少为 1152，且秩与系数 $a$ 的秩乘积相关，并随维数指数增长，导致三维或稍复杂的非均匀情况即失去实用性。
第二，为了精确获得梯度场并与系数 $a$ 相适应，通常的变分形式将 $a$ 与微分算子乘积作用，在 QTT 框架下导致算子的秩大幅膨胀（乘积的秩近似于秩之积），丧失了压缩优势。此外，现有 QTT 求解器多直接处理原始变量 $u$，但梯度 $\nabla u$ 的精度往往不佳或无法保证。
本文的动机正是针对上述空白：设计一个基于 QTT 的求解器，能够在极细网格上 ($h \ll 10^{-10}$) 对具有高度异质系数场 $a$ 的椭圆方程进行全场模拟，并可靠地输出解 $u$ 及其梯度 $\nabla u$ 的 $L^2$ 逼近，同时保持适中的 TT-秩与无条件稳定性。该求解器不依赖于尺度分离假设，无需事先知道解的解析结构，从用户视角提供了普适性的多尺度线性椭圆方程直接求解工具。
2. 主要定理或主要结果
本文的主要结果包括理论分析与数值实验两部分。
理论分析（第4节）针对离散惩罚泛函
$$ \widehat J_\mu(\hat\psi) = \frac12 \langle \hat\psi, \hat a * \hat\psi \rangle + \langle \hat\psi, \hat g \rangle + \frac{\mu}{2} \langle \hat\psi, \hat p \odot \hat\psi \rangle $$及其对应的 Euler-Lagrange 方程
$$ (\hat a * + \mu \hat p \odot) \hat\psi = -\hat g, \tag{19} $$证明了以下三条性质：
- 无条件稳定性：离散算子 $\hat A_\mu = \hat a * + \mu \hat p \odot$ 的条件数满足 $$ \kappa(\hat A_\mu) \le \frac{\max(\Lambda, \mu)}{\min(\lambda^{-1}, \mu)}, $$ 仅依赖于系数 $a$ 的上下界 $\lambda^{-1}, \Lambda$ 与惩罚参数 $\mu$，而与离散水平 $L$（即网格尺寸 $h=2^{-L}$）无关。因此无论网格多细，局部问题的条件数均一致有界。
- 先验误差估计：若精确解 $\hat\psi^*$ 满足 $\hat p \odot \hat\psi^* = 0$，则惩罚解 $\hat\psi_\mu$ 满足 $$ \|\hat\psi_\mu - \hat\psi^*\|_{\ell^2} \le \frac{C}{\mu} \|\hat g\|_{\ell^2}, $$ 其中常数 $C$ 仅与 $\lambda$ 有关。增大 $\mu$ 可压制无散分量。
- 后验误差估计：定义残量 $\hat r = \hat a * \hat\psi_\mu + \hat g + \mu \hat p \odot \hat\psi_\mu$，则有可靠且有效的误差界 $$ \frac{1}{\max(\Lambda,\mu)} \|\hat r\|_{\ell^2} \le \|\hat\psi_\mu - \hat\psi^*\|_{\ell^2} \le \frac{1}{\min(\lambda^{-1},\mu)} \|\hat r\|_{\ell^2}. $$ 该估计量可用于自适应调整 $\mu$ 或监控解的质量。
数值实验（第5节）在二维和三维验证了方法的实效：
- 首先研究了 Helmholtz-Leray 投影算子 $\hat p$ 的 QTT 表示：在不同格式（如 x1y1, x1x2_y1y2 等）下，对 $d=2,3$ 以适中的 TT-秩（约 10–100）实现了机器精度。
- 与已有方法的基准对比：使用具有解析解的制造解案例，在二维常系数 ($a=1$) 下，本文方法能以 $r \approx 20$ 的秩获得与 [4] (秩 $>1000$) 及 [27] 可比或更优的 $L^2$ 误差，且无需调整预处理子。
- 三维真实材料模拟：针对具有随机周期微结构且大尺度调制的导热问题，在 $d=3$ 下实现了虚拟自由度 $10^{37}$（对应 $h \approx 2^{-44}$），梯度 $\nabla u$ 的相对 $L^2$ 误差控制在 $10^{-3}$ 以内。在整个过程中，最大 TT-秩不超过 200，显示了极好的压缩效率。
3. 研究方法、关键技术和核心工具
本文的核心思想是将原方程 (2) 重写为关于梯度场 $\psi = \nabla u$ 的极小化问题，并通过惩罚技巧施加无旋约束 $\psi \in L^2_{\mathrm{pot}}(Q_{1,\mathrm{per}})$，从而将微分算子与系数场分离为加法项，既简化了 QTT 算子的构造，又保证了条件数的可控性。具体而言，技术路线包含以下创新点。
1. Helmholtz–Leray 投影与惩罚泛函 引入正交分解 $L^2(Q_{1,\mathrm{per}}) = L^2_{\mathrm{pot}} \oplus L^2_{\mathrm{sol}}$，其中势场子空间 $L^2_{\mathrm{pot}}$ 恰为梯度的集合，无散子空间 $L^2_{\mathrm{sol}}$ 的投影算子 $P$ 即为 Helmholtz–Leray 投影。原方程等价的变分问题为在 $\psi \in L^2_{\mathrm{pot}}$ 约束下极小化
$$ J(\psi) = \frac12 \int \psi \cdot a \psi + \int \psi \cdot g. $$通过增加二次惩罚项 $\frac{\mu}{2} \langle \psi, P\psi \rangle$，得到无约束泛函 $J_\mu(\psi)$。该惩罚精确促使散度部分 $P\psi$ 趋于零，且当 $\mu\to\infty$ 时逼近原始问题。该思想借鉴自 FFT-based 求解器中的散度校正技术，但本文将其提升为核心构造，使得在 QTT 离散后算子的条件数不再与 $h$ 相关。
2. Fourier 空间表达与算子解耦 由于 Helmholtz–Leray 投影 $P$ 在物理空间是奇异卷积，其 Fourier 乘子却十分简单：$\hat p_{ij}(k) = \delta_{ij} - k_i k_j / |k|^2$（在 $k=0$ 处约定为 $\delta_{ij}$）。将泛函变换至 Fourier 空间，利用 Parseval 恒等式，得
$$ \widehat J_\mu(\hat\psi) = \frac12 \langle \hat\psi, \hat a * \hat\psi \rangle + \langle \hat\psi, \hat g \rangle + \frac{\mu}{2} \langle \hat\psi, \hat p \odot \hat\psi \rangle, $$其中 $*$ 为离散卷积，$\odot$ 为 Hadamard (逐点) 乘积。相应的算子分裂为两个 QTT 算子的和：$\hat a *$ 与 $\mu \hat p \odot$。这种加法结构是关键技术关键，因为 TT-秩的和近似等于秩之和，而乘积会使秩急剧增长。作为对比，Bachmayr 与 Kazeev 的算法将微分算子与 $a$ 通过乘法耦合，导致巨大的秩；而此处解耦后，$\hat p \odot$ 的秩直接等于 $\hat p$ 的秩（由于对角结构），$\hat a*$ 的秩通过“魔方张量”构造仅比 $\hat a$ 的秩增加一个小因子（通常为 2 或 4），总体算子的秩保持在可接受范围。
3. QTT 构造与线性系统求解 在均匀网格 $G_L$ 及对偶 Fourier 网格 $\widehat G_L$ 上，利用二进下标将向量场 $\hat\psi$ 表示为具有矢量首核的 MPS（式 (21)）。$\hat p$ 的 MPS 表示通过解析公式 $\hat p_{ij}(k)$ 直接在 Fourier 网格上采样后，经由 TCI (Tensor Cross Interpolation) 算法获得低秩逼近。对于 $\hat a$ 与 $\hat g$，同样采用 TCI 从采样点生成 MPS，无需事先构建满维张量。离散 Fourier 变换 QFT 通过一维 QFT 的 Kronecker 积构造，其秩在不同格式下保持中等。
最终离散算子 $\hat A_\mu = \hat a * + \mu \hat p \odot$ 既是对称正定的（当 $\mu > 0$ 且 $a$ 满足椭圆性），又由于其核心的 TT-秩为加性组合，远低于乘积型构造。求解采用 ALS 算法：依次优化各个核，每次求解一个较小规模的局部对称正定系统。由于 $\kappa(\hat A_\mu)$ 的上界与 $h$ 无关，局部矩阵的条件数也一致有界，保证了 ALS 的快速收敛与数值稳定性。求得 $\hat\psi$ 后，通过 Green 算子 $Q$ (Fourier 乘子 $\hat q(k) = i k / |k|^2$) 获得 $\hat u = -\hat q \cdot \hat\psi$，再经逆 Fourier 变换得到物理空间的 $u$ 与 $\nabla u$。
4. 稳定性与误差分析 无条件稳定性的证明基于谱估计：对任意 $\hat\psi$，
$$ \langle \hat\psi, \hat A_\mu \hat\psi \rangle \ge \min(\lambda^{-1}, \mu) \|\hat\psi\|_{\ell^2}^2,\quad \langle \hat\psi, \hat A_\mu \hat\psi \rangle \le \max(\Lambda, \mu) \|\hat\psi\|_{\ell^2}^2, $$因此 $\kappa(\hat A_\mu) \le \max(\Lambda,\mu) / \min(\lambda^{-1},\mu)$。这里关键利用了 $\hat p$ 的正交投影性质（$\hat p = \hat p^\dagger = \hat p^2$）以及 $a$ 的一致椭圆性。通过调整惩罚参数 $\mu$（典型值 $\sim 1$ 或略大于 1），条件数可保持在较小常数。这一性质使得极细网格上的 ALS 求解成为可能。后验误差估计量由残量 $\|\hat r\|_{\ell^2}$ 的自然界限导出，与 $\kappa(\hat A_\mu)$ 相关联，提供了可靠的质量控制手段。
4. 与之前工作的比较
本文提出的方法与若干已有 QTT 椭圆方程求解器形成鲜明对比，可以在处理对象、精度、秩复杂度、稳定性等多个维度上进行比较。
与 Rakhuba [40,41] 的比较 Rakhuba 的方法针对常系数 Laplace 方程，通过解析构造一维逆 Laplacian 的 MPO 并利用张量积结构避免了直接求逆的 $h^{-2}$ 条件数问题。该方法成功地将网格尺寸推至 $h=10^{-12}$，且计算过程不涉及迭代，效率极高。然而其主要局限在于：
- 只适用于 $a \equiv 1$ 的纯扩散问题，无法处理非均匀系数场；
- 不能获得准确的梯度场 $\nabla u$，仅输出 $u$；
- 若需推广到变系数，可能需要借助 Lippmann-Schwinger 迭代，这会引入 QTT 迭代压缩的困难。
本文的方法放宽了这两项假设：它直接以梯度 $\psi = \nabla u$ 为未知量，并有机地引入系数 $a$，因此既能处理一般异质场，又能精确恢复梯度。在常系数情形下，本文同样可以简化（$\hat a*$ 退化为倍数），且能获得与 Rakhuba 方法相当的 $h$ 尺度，甚至更细（演示中 $d=3$ 可至 $h\approx 2^{-44}$）。代价是需要通过 ALS 求解线性系统，但它得益于无条件稳定性而高效收敛。
与 Bachmayr 与 Kazeev [4] 的比较 Bachmayr 和 Kazeev 提出了首个针对变系数椭圆方程的 QTT 预条件有限元方法，理论上可处理非均匀 $a$ 并实现 $H^1$ 收敛。其核心是构造一个预条件子 $M$，使得 $M A$（其中 $A$ 为刚度矩阵的 QTT 形式）有适中的条件数。然而该方法的根本瓶颈在于：
- 预条件子的 TT-秩与系数 $a$ 的秩倍增，且随维数指数增长。对最简单的常系数二维情形，报告的最小秩为 1152；对于有非平凡结构的 $a$，秩将远超此数，导致存储和计算代价不菲；
- 三维情况因此实际上不可行；
- 尽管理论上有 $H^1$ 收敛，但如何从结果中实际复原 $\nabla u$ 并非直接阐明。
本文的策略通过惩罚形式的 Helmholtz–Leray 投影将算子分离为 $\hat a* + \mu \hat p\odot$，秩仅为两者的线性叠加，而 $\hat p\odot$ 本身秩很小（示例如 10–20），$\hat a*$ 的秩由 $\hat a$ 的秩决定但无指数增长。因而对于同样的二维常系数问题，本文可仅用秩约 20 的表示达到同等精度，并可扩展至三维与复杂微结构场（实验中秩 < 200）。这也使得三维模拟首次在 QTT 框架下变为可行。
与 FFT-based 及 QFT-based 方法的比较 经典的基于 FFT 的多尺度求解器（如 Lippmann-Schwinger 迭代配合固定点法）虽能处理异质系数和获取梯度，但面对极细网格时仍需要存储满维场量，且每次迭代需 Fourier 变换，内存和时间增长迅速。QTT 的出现提供了在压缩域运算的可能性，但早期尝试 (如 Risthaus 与 Schneider [42]，Giacomini 等 [21]) 并未展现出优于传统 FFT 的性能，部分原因在于他们的方法未能全程保持在 QTT 格式内（需解压），或使用了迭代框架，其中每一步的 MPO 乘法与压缩累积了秩与误差。
本文采用的全 QTT 策略在压缩域内完成所有运算（仅 TCI 采样和最终重建时接触物理量），且通过惩罚形式避免了乘性复合导致的秩膨胀，从而在极细网格上显现出压倒性优势（如自由度 $10^{37}$ 级别的模拟）。与 [25] 的理论工作相比，该文主要从均匀化角度证明了在特殊微结构下解和梯度可得到低秩 QTT 表示，但并未给出求解器；本文则贡献了实际可运行的通用工具，并配以详尽的分析。
了解领域论文（摘要翻译）
1. 矩形 Raviart–Thomas 元的精细收敛结构
Refined convergence structures of the rectangular Raviart-Thomas element
作者: Yifan Yue, Hongtao Chen, Shuo Zhang
类别: math.NA
发布日期: 2026-05-21
arXiv ID: 2605.21979v1
链接: http://arxiv.org/abs/2605.21979v1
摘要翻译
中文标题: 矩形 Raviart–Thomas 元的精细收敛结构
摘要翻译: 本文充分探究了最低阶矩形 Raviart–Thomas 元在求解 Laplace 特征值问题时的三种精细收敛结构。首先，该格式具有离散特征函数与插值特征函数之间的超逼近性质，从而可以方便地构造后处理，最多可将精度提高一阶。所用核心技巧是对插值项进行积分展开。其次，基于超逼近性质，我们不仅对单特征值、还对于重特征值推导了误差展开式，并给出了严格证明，据此可执行 Richardson 外推。作为附带结果，我们证明了所有特征值均从上方收敛。此外，利用超逼近性质及 Rayleigh 商分析，我们对方形区域上均匀网格的重特征值收敛行为给出了严格证明。第三，我们指出了最低阶矩形 Raviart–Thomas 元与加富旋转双线性元之间的等价性。在文末，设计了一系列数值实验来验证我们的理论。
2. Vlasov-Poisson系统的一种保结构装饰粒子方法
A Structure-Preserving Decorated Particle Method for the Vlasov-Poisson System
作者: Mandela B. Quashie, J. W. Burby, Andrew J. Christlieb, Qi Tang
类别: math.NA, physics.plasm-ph
发布日期: 2026-05-20
arXiv ID: 2605.21794v1
链接: http://arxiv.org/abs/2605.21794v1
摘要翻译
中文标题: Vlasov-Poisson系统的一种保结构装饰粒子方法
摘要翻译: 我们重新审视了Scovel-Weinstein框架（Scovel & Weinstein, CPAM 1994），该框架用于约化Vlasov-Poisson系统同时保持其哈密顿结构。标准的粒子网格（PIC）算法通过具有位置和速度的宏粒子来近似分布函数。相比之下，Scovel-Weinstein装饰粒子引入了额外的形状自由度，同时保持了从连续模型继承的哈密顿结构的有限维约化。虽然原始工作在三十年前就建立了这一结构，但其计算潜力在很大程度上尚未得到探索。我们给出了Scovel-Weinstein模型的一个实用实现，并将其与标准PIC算法进行了比较。数值实验表明，标准PIC中的宏粒子可以被少得多的装饰粒子替代，同时保持相当的精度。这种装饰粒子方法为动理学等离子体模拟提供了一种新的保结构范式。
3. 双调和方程的拟最优多面体有限元方法
Quasi-optimal polytopal finite element methods for biharmonic equation
作者: Ngoc Tien Tran
类别: math.NA
发布日期: 2026-05-20
arXiv ID: 2605.21764v1
链接: http://arxiv.org/abs/2605.21764v1
摘要翻译
中文标题: 双调和方程的拟最优多面体有限元方法
摘要翻译: 本文在一般多面体网格上的极小正则性假设下，建立了双调和方程的弱伽辽金、间断伽辽金和混合高阶有限元方法的拟最优和低阶误差估计。此外，还证明了稳定化项在后验误差估计中是一个有效的贡献。
不相关论文（基本信息）
1. 流形交集上的优化
Optimization over the intersection of manifolds
作者: Yan Yang, Bin Gao, Ya-xiang Yuan
类别: math.OC, cs.LG, math.DG, math.NA
发布日期: 2026-05-21
arXiv ID: 2605.22736v1
链接: http://arxiv.org/abs/2605.22736v1
摘要: Optimization over the intersection of two manifolds arises in a broad range of applications, but is hindered by the coupled geometry of the feasible region. In this paper, we prove that the regularities – clean intersection and intrinsic transversality – are equivalent, which yields a tractable projection onto the tangent space of the intersection. Therefore, we propose a geometric method that employs a retraction on only one manifold and updates the iterate along two orthogonal directions. Specifically, the iterates stay on one manifold, and the two directions are responsible for asymptotically approaching the other manifold and decreasing the objective function, respectively. Under intrinsic transversality, we derive the convergence rate for both the feasibility and optimality measures, and show that every accumulation point is first-order stationary. Numerical experiments on problems stemming from sparse and low-rank optimization, including fitting spherical data, approximating hyperbolic embeddings on real data, and computing compressed modes, demonstrate the effectiveness of the proposed method.
2. 基于$\operatorname{prox}$的半光滑牛顿法求解TV最小化问题
A $\operatorname{prox}$-Based Semi-Smooth Newton Method for TV-Minimization
作者: Sören Bartels, Alex Kaltenbach
类别: math.NA, math.OC
发布日期: 2026-05-21
arXiv ID: 2605.22728v1
链接: http://arxiv.org/abs/2605.22728v1
摘要: In this paper, we devise a $\operatorname{prox}$-based semi-smooth Newton method for the non-differentiable TV-minimization problem. To this end, the primal-dual optimality conditions are reformulated as a nonlinear operator equation with Newton-(type-)differentiable structure. We investigate the question of well-posedness of the resulting semi-smooth Newton scheme in the infinite-dimensional setting and identify structural properties of the associated Newton-type derivatives. For a conforming finite element discretization, we prove that the resulting semi-smooth Newton method is globally well-posed and locally super-linearly convergent. The approach extends to a large class of convex minimization problems, coincides with established semi-smooth Newton methods for obstacle problems, satisfies a primal-dual invariance, and, under suitable additional assumptions, is well-posed in the infinite-dimensional setting. Numerical experiments indicate a robust practical performance of the proposed method, including reliable reduction of the discrete primal-dual gap estimator to machine precision, robustness with respect to the choice of proximity parameters, an improved convergence basin compared to a canonical primal semi-smooth Newton method, and effective performance even for quadratically graded meshes using only a mesh-independent initialization criterion.
3. 多个神经算子实现多任务学习的近最优速率
Multiple Neural Operators Achieve Near-Optimal Rates for Multi-Task Learning
作者: Adrien Weihs, Hayden Schaeffer
类别: cs.LG, math.NA, stat.ML
发布日期: 2026-05-21
arXiv ID: 2605.22724v1
链接: http://arxiv.org/abs/2605.22724v1
摘要: We study the approximation and statistical complexity of learning collections of operators in a shared multi-task setting, with a focus on the Multiple Neural Operators (MNO) architecture. For broad classes of Lipschitz multiple operator maps, we derive near-optimal upper bounds for approximation and statistical generalization. On the lower-bound side, we establish a curse of parametric complexity and prove corresponding minimax rates. Together, these results show that shared representations across tasks do not increase the overall cost: multi-task operator learning follows the same scaling laws as single operator learning. We also compare MNO with a multi-task extension of DeepONet based on concatenated task inputs and show that, from a worst-case approximation-complexity perspective, both architectures satisfy essentially the same asymptotic rates.
4. 关于正则轨道基下耦合簇振幅的正则性与插值
On the Regularity and Interpolation of Coupled Cluster Amplitudes in Canonical Orbital Basis
作者: Jonas Beck, Benjamin Stamm
类别: math.NA, physics.chem-ph
发布日期: 2026-05-21
arXiv ID: 2605.22584v1
链接: http://arxiv.org/abs/2605.22584v1
摘要: Arguably the most widely used approaches for obtaining highly accurate molecular ground-state energies are coupled cluster methods. Despite introducing two layers of approximation, a linear and a nonlinear one, coupled cluster methods remain computationally intensive, with the complexity scaling as $O(poly(N))$, where $N$ is the number of electrons. Moreover, this method must be applied over a large set of different nuclear coordinates in order to study certain chemical phenomena. Therefore, in this work, we investigate the regularity of single-reference coupled cluster amplitudes with respect to nuclear coordinate displacements, with the aim of enabling interpolation or extrapolation approaches that rely on only a limited number of reference geometries. We show that, in theory, under certain non-degeneracy assumptions on the Hartree-Fock level of theory, and the coupled cluster level of theory the amplitudes behave real analytic. Furthermore, we analyze the artifacts that arise in practical calculations that use canonical orbitals, which hinder this high degree of regularity, and suggest strategies to mitigate these issues. Finally, we validate our findings through numerical experiments by interpolating the amplitudes and comparing the performance of the interpolants with that of the exact amplitudes.
5. 神经流算子能逼近任意算子：抽象框架与万能逼近
Neural Flow Operators can Approximate any Operator: Abstract Frameworks and Universal Approcimations
作者: Shuang Chen, Juncai He, Xue-Cheng Tai
类别: cs.LG, math.NA
发布日期: 2026-05-21
arXiv ID: 2605.22557v1
链接: http://arxiv.org/abs/2605.22557v1
摘要: We introduce an abstract neural flow framework for neural networks and neural operators. The framework contains two continuous-depth models, namely neural flows with composition and separation structures, and covers both finite-dimensional function approximation and infinite-dimensional operator approximation. We prove well-posedness and universal approximation properties for the corresponding neural flows, including, to the best of our knowledge, the first universal approximation result for flow-based models between infinite-dimensional spaces. We also obtain universal approximation results for convolutional neural flow models. Through suitable time discretizations, the composition structure recovers ResNet-type architectures, while the separation structure, via a splitting-based discretization, yields plain architectures. This gives a unified flow-based route to both residual and plain architectures for neural networks and neural operators with fully connected or convolutional linear layers.
6. 椭圆-抛物问题的解耦Runge-Kutta格式
Decoupling Runge-Kutta schemes for elliptic-parabolic problems
作者: Robert Altmann, Abdullah Mujahid, Benjamin Unger
类别: math.NA
发布日期: 2026-05-21
arXiv ID: 2605.22485v1
链接: http://arxiv.org/abs/2605.22485v1
摘要: We study the construction and convergence of semi-explicit and iterative decoupling schemes for an elliptic-parabolic problem using higher-order Runge-Kutta methods. For the semi-explicit schemes, which are constructed using a nearby delay system with $k$ time delays, we establish the convergence of $k$th-order Runge-Kutta methods under a weak coupling condition. We develop the convergence analysis by adapting the Fourier stability and perturbation techniques of [Lubich, Ostermann, Math. Comp., 64(210):601–627, 1995]. The key tool is the generating function framework, in which the Runge-Kutta discretization is encoded through an operator-valued function. Stability estimates are then obtained via Parseval’s identity on the unit circle. We further present convergence results for iterative (fixed-stress and undrained-split) higher-order Runge-Kutta schemes. Here, a spectral decomposition of the Schur complement operator is central. Finally, we provide numerical examples to verify the proven convergence results.
7. 具有极小多项式外推加速的基于Schwarz模的矩阵分裂用于美式期权定价中产生的线性互补问题
Schwarz Modulus Based Matrix Splittings with Minimal Polynomial Extrapolation Acceleration for linear complementarity problems arising from American option pricing
作者: Martin J. Gande, Si-Wei Liao, Liu-Di Lu
类别: math.NA
发布日期: 2026-05-21
arXiv ID: 2605.22315v1
链接: http://arxiv.org/abs/2605.22315v1
摘要: Pricing American options is more complicated than pricing European options, because they can be exercised at any time, and one thus needs to solve a linear complementarity problem instead of simply doing time stepping for computing European options. We introduce a new Schwarz modulus-based splitting method for solving such linear complementarity problems, and further accelerate them using Modified Polynomial Extrapolation, a non-linear vector sequence acceleration technique, which is very much related to Krylov methods in the linear case. Numerical experiments on a model problem show that our new solver can have close to an order of magnitude lower iteration counts than the classically used modulus-based matrix splitting technique.
8. 从偏微分方程约束优化到能控性问题：基于时域分解的方法
From PDEs constrained optimization to controllability problems via time domain decomposition
作者: Pierre-Henri Cocquet, Liu-Di Lu
类别: math.NA, math.OC
发布日期: 2026-05-21
arXiv ID: 2605.22312v1
链接: http://arxiv.org/abs/2605.22312v1
摘要: This paper focuses on the application of time domain decomposition to solve partial differential equations constrained optimization problems and controllability problems. After clarifying the link between these two types of problems, we show that applying time domain decomposition to both problems leads to the same convergence behavior. Our numerical experiments also confirm these theoretical findings.
9. 随机化柔性LSQR和LSMR及其在反问题中的应用
Randomized Flexible LSQR and LSMR with applications to inverse problems
作者: Alberto Bucci, Silvia Gazzola, Leonardo Robol
类别: math.NA
发布日期: 2026-05-21
arXiv ID: 2605.22281v1
链接: http://arxiv.org/abs/2605.22281v1
摘要: LSQR and LSMR are iterative methods, based on the Golub-Kahan bidiagonalization algorithm, widely used for large-scale linear least squares problems. FLSQR and FLSMR are flexible variants of LSQR and LSMR, respectively, based on a flexible Golub-Kahan (Arnoldi-like) factorization algorithm, which naturally allow modifications of the solution approximation subspace and/or handling inexact matrix-vector multiplications with the (transpose of the) coefficient matrix, thereby enabling to enforce prior information into the computed solution. The goal of this paper is to introduce sFLSQR and sFLSMR, i.e., sketched variants of FLSQR and FLSMR, respectively, where randomization becomes particularly effective, as it allows to recover short recurrences for the solution approximation. In particular, this paper explores applications to large-scale inverse problems, showing the ability of the new randomized solvers to alleviate computational bottlenecks while preserving reconstruction quality. A theoretical analysis of sFLSQR and sFLSMR is provided, and their performance is validated through numerical experiments.
10. 变系数二阶问题的边界元法
BEM for variable coefficient second-order problems
作者: Benedikt Gräßle, Stefan A. Sauter
类别: math.NA
发布日期: 2026-05-21
arXiv ID: 2605.22159v1
链接: http://arxiv.org/abs/2605.22159v1
摘要: A novel boundary element method (BEM) removes the classical dependence on explicit fundamental solutions and extends quasi-optimal BEM discretisations to strongly elliptic operators with variable coefficients. The approach constructs a computable approximation of the boundary operator from a Galerkin discretisation of the underlying elliptic differential operator in a one-time preprocessing step, for instance by conforming finite elements. The resulting algebraic formulation retains the dimension reduction intrinsic to boundary integral methods and is compatible with standard data-sparse matrix compression techniques.
11. 面向蒙日-安培方程的带有注意力特征扩展的物理信息神经网络
Physics-Informed Neural Networks with Attention Feature Expansion for Monge-Ampère Equations
作者: Anxiao Yu, Bangmin Wu, Zhengbang Zha, Xinlong Feng, Dongwoo Sheen
类别: math.NA
发布日期: 2026-05-21
arXiv ID: 2605.22115v1
链接: http://arxiv.org/abs/2605.22115v1
摘要: The Monge-Ampère equation is a fundamental fully nonlinear elliptic partial differential equation that finds extensive applications across multiple disciplines. This study proposes a novel physics-informed neural network integrated with attention feature expansion (PINN-AFE) for its numerical solution. A multi-head attention enhanced feature pool is constructed to enable adaptive nonlinear feature representation, and input convex neural networks are adopted to impose strict convexity of solutions with rigorous theoretical guarantees. Meanwhile, a dynamically weighted loss function combined with hybrid optimization is formulated to accelerate training convergence. Comprehensive numerical experiments validate the accuracy and computational efficiency of the developed framework. The PINN-AFE paradigm is further extended to image processing tasks, delivering high-quality and physically consistent results in both image enhancement and medical image registration scenarios.
12. 比例延迟微分方程的稳定性分析
Stability Analysis of Pantograph Delay Differential Equations
作者: Sachin Bhalekar
类别: math.DS, math.NA
发布日期: 2026-05-21
arXiv ID: 2605.22019v1
链接: http://arxiv.org/abs/2605.22019v1
摘要: This article investigates the stability of pantograph delay differential equations, in which the delayed argument is proportional to the present time. We derive analytic criteria that partition the parameter plane into unstable, asymptotically stable, and delay-dependent stability regions. The theoretical results are supported by numerical simulations that illustrate the sharpness of the stability boundaries. We also formulate a proportional-delay analogue of the Mackey–Glass chaotic delay differential equation and examine the resulting dynamical behaviour.
13. 图上瓦瑟斯坦空间中哈密顿-雅可比方程的单调格式的一阶收敛
First-Order Convergence of Monotone Schemes for Hamilton–Jacobi Equations on the Wasserstein Space on Graphs
作者: Jianbo Cui, Tonghe Dang
类别: math.NA
发布日期: 2026-05-21
arXiv ID: 2605.22016v1
链接: http://arxiv.org/abs/2605.22016v1
摘要: We prove first-order convergence of semi-discrete monotone finite difference schemes for Hamilton–Jacobi equations on the Wasserstein space over a finite graph. A central challenge is the boundary degeneracy of the Wasserstein simplex, which prevents the direct use of the standard $L^1$ adjoint method and limits doubling-of-variables arguments to the suboptimal rate $\mathcal O(h^{\frac 12})$ \cite{CDM25}. We address this issue by introducing a weighted $L^1$ framework with a boundary-vanishing weight and by analyzing the corresponding weighted adjoint equation for the linearized operator of the scheme, featuring a new geometric drift term. Our proof relies on uniform bounds for the weighted adjoint variable and the mesh-parameter derivative of the numerical solution. These estimates are derived from discrete gradient and semi-concavity bounds, obtained through a bootstrap argument for two classes of monotone Hamiltonians.
14. 基于近端梯度的低秩张量分解用于状态依赖Riccati方程
Proximal Gradient-based Low Rank Tensor Decomposition for State Dependent Riccati Equation
作者: Jiahua Jiang, Carmeliza Navasca
类别: math.OC, math.NA
发布日期: 2026-05-21
arXiv ID: 2605.21885v1
链接: http://arxiv.org/abs/2605.21885v1
摘要: We address the optimal control problems arising from partial differential equations with large discrete dimensional control systems. To obtain reduced order models, we find basis elements from the canonical polyadic (CP) decomposition. Tensor datasets are from snapshots of the large models. Our method to reduce the control system is to use dimensionality reduction approaches through sparse optimization and flexible hybrid methods is to obtain low rank CP tensor basis elements. The reduced optimal control problem leads to reduced state-dependent Riccati Equations which can be solved efficiently.
15. 一种蝴蝶加速的流形调和变换
A Butterfly-Accelerated Manifold Harmonic Transform
作者: Paul G. Beckman, Samuel F. Potter, Michael O’Neil
类别: math.NA
发布日期: 2026-05-20
arXiv ID: 2605.21828v1
链接: http://arxiv.org/abs/2605.21828v1
摘要: The eigenfunctions of the Laplacian are a natural basis of functions for many tasks in computational mathematics. On the circle and sphere, the eigenfunctions are given by complex periodic exponentials and spherical harmonics, respectively, and much work has been done to develop fast algorithms for analyzing and synthesizing data in these bases. In this work, we generalize these special-case transforms to Laplace-Beltrami eigenfunctions of arbitrary surfaces, referred to as manifold harmonics. The resulting fast algorithm for computing linear combinations of the manifold harmonics is based on a butterfly factorization, which hierarchically compresses the transform matrix by constructing nested low-rank approximations of carefully selected submatrices. Several numerical examples are provided which demonstrate the speedups and reduction in memory requirements achieved by our algorithm for a variety of geometries, discretizations, and applications. In addition, a detailed analysis of the algorithm is given in the case that the underlying manifold is the flat periodic square.
16. 跨退火贝叶斯后验的似然信息降维
Likelihood-informed dimension reduction across tempered Bayesian posteriors
作者: Arne Bouillon, Oliver R. A. Dunbar
类别: stat.CO, math.NA
发布日期: 2026-05-20
arXiv ID: 2605.21717v1
链接: http://arxiv.org/abs/2605.21717v1
摘要: Scientific computer simulations cannot represent all scales in realistic applications. To bridge this model-data gap, parameters are injected into models and constrained with noisy data using Bayesian inversion. To reduce the number of simulator evaluations, which can be 10^5 or more, modern approaches employ dimension reduction in conjunction with emulation of the forward map (that contains the simulator). Due to scarcity of model evaluations and data, this dimension reduction becomes very important for posterior sampling performance. Recent work on likelihood-informed subspaces (LIS) truncates to informative directions by optimizing bounds on information loss, and though mathematically well-adapted to sampling, they are often restrictive in practice. In this work, we provably generalize this methodology to facilitate application to $α$-tempered (i.e., annealed, power-posterior) distributions for $α$ in [0,1]. We provide theory to build partially-informed spaces termed $α$-LIS. We show how $α$ < 1 can often produce near-optimal spaces. In addition, we focus on applying $α$-LIS to practical cases, where the available data is severely limited and noisy. We propose and test extensions for utilizing data from the entire sequence of distributions $α$_0 < … < $α$_k, and use simple approximations of model gradients so that our approach can be used for emulation of forward maps for chaotic or stochastic systems where derivatives are unavailable or uninformative due to noise. In experiments, our accumulated approach is much more robust to these challenging circumstances than the theoretically optimal $α$ = 1.
17. 肿瘤生长Cahn-Hilliard-Darcy模型的保结构迎风DG格式
Structure-preserving upwind DG scheme for a Cahn-Hilliard-Darcy model of tumor growth
作者: Daniel Acosta-Soba, Francisco Guillén-González, J. Rafael Rodríguez-Galván
类别: math.NA
发布日期: 2026-05-20
arXiv ID: 2605.21716v1
链接: http://arxiv.org/abs/2605.21716v1
摘要: In this work, we develop a structure-preserving numerical scheme for a Cahn-Hilliard-Darcy model that describes tumor growth in a fluid-saturated porous medium. First, we derive a physically consistent model from the general framework proposed in [29] that guarantees mass conservation and pointwise bounds on the phase-field and nutrient variables, with a decreasing energy law. The resulting model couples the evolution of tumor cells via a Cahn-Hilliard equation with a diffusion equation for the nutrients thro chemotactic interactions and extends the model in [1] by introducing the effect of a surrounding fluid described by Darcy’s law. Subsequently, we propose a fully discrete scheme that combines an upwind discontinuous Galerkin method in space and a convex splitting strategy in time, which inherits the fundamental properties of the continuous model: mass conservation, pointwise bounds and discrete energy law. Our theoretical analysis is accompanied by numerical experiments that demonstrate the robustness of the proposed scheme and show the influence of the surrounding fluid on the tumor evolution.
18. 经典风险模型中破产概率的龙格-库塔数值方法
Runge–Kutta numerical methods for ruin probabilities in classical risk model
作者: George Kanakoudis, Lazaros Kanellopoulos
类别: math.NA
发布日期: 2026-05-20
arXiv ID: 2605.21693v1
链接: http://arxiv.org/abs/2605.21693v1
摘要: In this paper, we study Runge–Kutta methods for the computation of ruin probabilities in the classical risk model through the associated Volterra integro-differential equation. The proposed framework combines fourth-order one-step and two-step Runge–Kutta schemes with numerical quadrature formulas to approximate the convolution term. In particular, the convolution term is approximated using Newton–Cotes and Gaussian quadrature formulas, including Simpson’s 1/3 rule and Pareto-adapted Gauss–Jacobi quadrature. An equivalent reformulation of the Volterra equation as a system of ordinary differential equations is also considered. Implementations for Gamma and Pareto claim-size distributions are developed. Numerical results are presented to illustrate the effectiveness of the proposed methods.
19. M-矩阵平方根的逐分量精确计算
Component-wise accurate computation of the square root of an M-matrix
作者: Dario A. Bini, Bruno Iannazzo, Beatrice Meini, Jie Meng
类别: math.NA
发布日期: 2026-05-20
arXiv ID: 2605.21679v1
链接: http://arxiv.org/abs/2605.21679v1
摘要: Component-wise accurate algorithms for computing the principal square root of an M-matrix are designed in terms of triplet representations. A triplet representation of an M-matrix $A$ is the triple $(P, {\bf u},{\bf v})$, where the matrix $P$ is such that $p_{ij}=-a_{ij}$ for $i\ne j$, $p_{ii}=0$, and ${\bf u}>0$, ${\bf v}\ge 0$ are two vectors such that $A{\bf u}={\bf v}$. It is shown that if $A$ is an M-matrix representable by a triplet, then its principal square root exists and is an M-matrix represented by a triplet as well. New versions of the Cyclic Reduction and the Incremental Newton iterations are provided in terms of triplets, to compute the principal matrix square root of $A$. It is shown that these algorithms are component-wise numerically stable independently of the singularity of $A$ and of its condition number. Numerical experiments are shown to confirm the component-wise stability.