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重点关注论文（完整分析）

1. 超越多项式的自适应有限元空间变分富集

Variational Enrichment of Adaptive Finite Element Spaces Beyond Polynomials

作者: Matthew Francis Dixon

类别: math.NA

发布日期: 2026-08-12

arXiv ID: 2608.11764v1

摘要翻译: 自适应有限元通过细化网格或提高多项式次数来适应，但连贯的非多项式结构无论沿 h 坐标还是 p 坐标都可能仍然代价高昂。我们引入变分富集，其中紧致的问题相关函数补充通常的 h- 和 p- 细化。只要容许函数及其残差值可以被计算，该框架适用于 Galerkin 和极小残差格式；它不局限于某一类 PDE。商-Schur 分析去除冗余增广并度量其变分价值。对于对称强制问题，本文理论还给出精确的能量误差缩减量和一个与参考量无关的上界。

Comment: 无

链接: http://arxiv.org/abs/2608.11764v1

1. 研究对象和背景

本文研究自适应有限元方法中，当解具有超越多项式的紧致结构时，如何高效地扩张逼近空间。传统自适应细化通过减小局部网格尺寸（h-细化）或提高局部多项式阶数（p-细化）来控制误差，但面对诸如输运剖面、界面层、系数适应坐标或振荡相位等相干非多项式特征时，需要大量分片多项式自由度才能逼近。本文提出变分富集（variational enrichment）框架，在 h- 和 p- 细化之外允许加入紧致的问题知情函数（称为 c-富集），这些函数可由短组合、特殊函数、局部 PDE 解或降阶近似等生成，不必预先枚举所有细尺度基函数。问题设定为对称强制椭圆问题，典型形式为 Poisson 方程或 Darcy 方程，能量内积 $a(u,v)=(\nabla u,\nabla v)$，载荷泛函 $F(v)=(q,\nabla v)$；更一般的渗透率场可通过加权内积处理。解空间为 $H_0^1(\Omega)$。自适应有限元通常构造嵌套空间 $V_m = V_1(\mathcal T_m) + \operatorname{span} M_m$，其中 $\mathcal T_m$ 为一致的新顶点二分网格，$M_m$ 为紧致表示的符合物理宏函数集合。本文关注的关键空白在于：缺乏一种共同的、参考自由的变分比较，使得普通 h- 或 p- 行动与来自更丰富（可能非多项式）逼近类的候选函数可以在统一尺度上比较其对于 Galerkin 误差的降低量。仅简单添加后者可能引入冗余方向、拟合不相关特征或导致积分不准确，无法保证经济性和可靠性。已有的相关理论包括：自适应 h- 方法的收敛性与拟最优性 [2,4]；hp 方法残差估计 [12]；单位分解方法允许问题适应的局部函数 [1]；广义多尺度有限元 (GMsFEM) 构造紧致算子适应局部空间 [8]；局部正交分解 (LOD) 的全局修正子极限 [9]；通量重构提供保证的椭圆误差界 [3,11]；兼容有限元与离散微分复形保留结构恒等式 [6]；特征有限元将输运直接纳入演化步 [7]。本文试图将这些思想统一到一个新的细化原则：通过商 (quotienting) 去除已存在内容，用受限残差度量 PDE 相关性，并用精确增益恒等式使不同候选家族可直接比较。这就是变分富集的核心，它为有限元空间提供一种超越多项式阶的近似压缩机制，具有内在保证。

2. 主要定理或主要结果

本文的主要理论结果围绕三个问题组织：可无冗余接纳、可无参考控制误差、是否相对 hp 改善近似。具体结论如下。

(1) 精确保持引理 (Lemma 2.1)：设 $\mathcal T_{m+1}$ 为 $\mathcal T_m$ 的一致新顶点二分加密，且所有已接受富集函数 $M_m$ 作为相同物理函数保留，则空间包含 $V_m \subset V_1(\mathcal T_{m+1}) + \operatorname{span} M_m$ 成立，从而 Galerkin 目标单调不增。这保证了 h-细化不擦除先前非多项式内容，是嵌套性和单调性的基础。

(2) 一致商稳定性定理 (Theorem 2.2)：考虑逐层生成映射 $S_\ell: E_\ell \to H_0^1(\Omega)$，定义创新部分 $R_\ell = (I-P_{\ell-1}^a)S_\ell$ 及能量 Gram 算子 $D_\ell = R_\ell^* R_\ell$。若在商系数空间上 $\delta I \preceq D_\ell \preceq \Delta I$ 对 $\ell$ 一致成立，且相应块消去映射及其逆的范数有界 $M$ 和 $K$，则对任意层 $m$，有

$$\frac{\delta}{K^2} \|[c]\|^2 \le \|S^{(m)}[c]\|_a^2 \le \Delta M^2 \|[c]\|^2.$$

该结果将抽象 Schur–Riesz 理论转化为序列富集的水平一致稳定性，保证新加入方向可从旧空间中区分，且系数扰动对函数影响有界。

(3) 精确变分增益定理 (Theorem 2.3)：设 $u_m$ 为 $V_m$ 中 Galerkin 解，候选块 $C: E_C \to H_0^1(\Omega)$。先投影掉已表示内容，定义 $Q = (I-P_m^a)C$，$D = Q^* Q$，$b(z) = F(Qz)-a(u_m,Qz)$。在商系数空间上若 $A_C I \preceq D \preceq B_C I$，则因添加该块导致的能量误差平方减少量为

$$\|u-u_m\|_a^2 - \|u-P_{V_m+\operatorname{ran} C}^a u\|_a^2 = \langle b, D^\dagger b\rangle,$$

且该增益介于 $\|b\|^2/B_C$ 和 $\|b\|^2/A_C$ 之间。该恒等式完全可计算，无需知道精确解 $u$。

(4) 可靠商通量上界 (Theorem 3.1)：对 Poisson 形式，定义不可见散度场空间 $N = \{ s \in L^2(\Omega)^d : (s,\nabla v) = 0 \; \forall v \in H_0^1(\Omega) \}$。对任何符合逼近 $u_h$，能量误差满足

$$\|u-u_h\|_a = \inf_{s\in N} \| q - \nabla u_h - s\|_{L^2}.$$

因此任何可计算的 $s_h \in N$ 给出保证上界 $\|u-u_h\|_a \le \| q - \nabla u_h - s_h\|_{L^2}$，无需饱和假设或精确参考解。

(5) 扰动安全排序 (Proposition 3.2)：设计算得到 $\tilde D_C, \tilde b_C$，且满足 $D_C \succeq A_C I$、$\|\tilde D_C - D_C\| \le \varepsilon_{D,C} < A_C$、$\|\tilde b_C - b_C\| \le \varepsilon_{b,C}$。定义 $\tilde G_C = \langle \tilde b_C, \tilde D_C^\dagger \tilde b_C\rangle$，则真实增益 $G_C$ 与 $\tilde G_C$ 的偏差有界于

$$\Xi_C = \frac{2\|\tilde b_C\|\varepsilon_{b,C} + \varepsilon_{b,C}^2}{A_C} + \frac{\|\tilde b_C\|^2 \varepsilon_{D,C}}{A_C(A_C-\varepsilon_{D,C})}.$$

因而 $\tilde G_C - \Xi_C > 0$ 证明确实有正改进；$\tilde G_C - \tilde G_{C'} > \Xi_C + \Xi_{C'}$ 证明 $C$ 的真实增益大于 $C'$。

(6) 条件拟误差收缩 (Proposition 3.3)：若估计量 $\eta_m$ 可靠，满足 $\|u-u_m\|_a^2 \le C_{rel}\eta_m^2$，并且接受的 h-, p- 或 c- 行动满足 $\eta_{m+1}^2 \le q\eta_m^2 + C_{stab}\|u_{m+1}-u_m\|_a^2$，其中 $0 $$E_{m+1}^2 \le \left(1 - \frac{\gamma(1-q)}{C_{rel}+\gamma}\right) E_m^2.$$

该命题指出，稳定创新和精确增益本身只给出单调放大，定量收缩还需要所接受函数族的估计量缩减假设。

数值结果方面，主要应用是 SPE10 非均质渗透率场。在 SPE10 Model 1 的 11 个规定层-载荷情形中，pc 富集在所有情形下同维度优于 p 富集，最高相对能量误差降低 41.6%。具体地，对角线井负载下，23 个 pc 坐标达到误差 0.514，而首个低于该阈值的 h-自适应 P1 和 P2 空间分别需要 79 和 156 个内禀坐标；分阶段 hc 计算用 52 个坐标达到误差 0.432。不匹配坐标（循环移位渗透率场）在所有情形下都劣于 p，证实算子兼容性的必要性。支持性实验包括随机渗透率、高对比度、Allen-Cahn 和输运问题，均支持同一结论。

3. 研究方法、关键技术和核心工具

本文面临的核心障碍有三：其一，非多项式富集函数可能具有不同的参数化或构造方式，即使表示相同物理函数也容易产生冗余，导致系数矩阵病态；其二，仅凭新颖性不足以决定是否应接受一个富集方向，必须度量其对于当前 PDE 残差的贡献；其三，所有比较不能依赖精确解，否则失去实用性。作者通过以下技术框架逐层解决。

第一，商-舒尔稳定性与系数商化。作者将逐层生成的映射 $S_\ell$ 视为抽象近似步骤，利用伴随 Schur–Riesz 理论 [10] 的思想，定义创新算子 $R_\ell = (I-P_{\ell-1}^a)S_\ell$，其中 $P_{\ell-1}^a$ 是前一空间上的能量正交投影。$R_\ell$ 只保留真正新的部分，其能量 Gram 算子 $D_\ell = R_\ell^* R_\ell$ 在商空间（模去表示相同函数的零系数方向）上运算。要求商空间上的谱等价 $\delta I \preceq D_\ell \preceq \Delta I$，并控制块消去映射及其逆的范数 $M,K$，则得到水平一致的函数范数上下界 (1)。这一步骤保证了无论后续添加多少稳定层，系数条件数不会恶化，新方向总能从旧空间分离。精确重复函数在商中消失，这正是去除冗余的数学表达。

第二，精确变分增益恒等式的导出。该恒等式将几何稳定性与 PDE 相关性联系。给定当前 Galerkin 解 $u_m$ 和候选块 $C$，先将其投影到当前空间的正交补上，得到 $Q = (I-P_m^a)C$，再定义残差向量 $b(z) = F(Qz) - a(u_m,Qz)$。这里 $D = Q^*Q$ 度量新方向的能量几何，而 $b$ 度量当前残差在该方向上的作用。解商正规方程 $Dz=b$，得最优修正 $z=D^\dagger b$，由 Galerkin 正交性和 Pythagoras 定理，能量误差平方的减少恰为 $\langle b, D^\dagger b\rangle$。该式完全可计算，不涉及未知解 $u$，且可将不同来源的候选函数（h-中点分裂、p-边模态、c-富集函数）放在同一标尺上比较。

第三，参考自由后验误差上界。为了决定何时停止或继续细化，需要全局误差控制。文章利用通量重构思想，将误差表示为与不可见散度场 $N$ 的距离：$\|u-u_h\|_a = \inf_{s\in N} \| q - \nabla u_h - s\|_{L^2}$。这给出一个可靠上界，不需要饱和假设。文中在附录中给出了分片重构的具体算法，用于数值实验。

第四，扰动安全排序。实际计算中装配、积分和代数求解引入误差，作者推导了增益计算的扰动界 $\Xi_C$。关键在于 $A_C$ 作为商 Gram 矩阵的下界：几乎依赖的候选函数会放大不确定性边界，只有残差增益足够强时才被接受。这为算法提供了稳健性。

第五，理论与算法耦合。算法 3.1 采用 Dörfler 标记，生成候选块集合，计算每个块的增益减扰动界，选择增益与成本比最大者进行细化。该算法不预设 c- 候选家族；组合 tent 函数仅用于示意，而实际主实验采用谐波或局部谱函数。此外，为处理因式积分，作者引入有限 filtration $\mathcal G_0 \subset \cdots \subset \mathcal G_L$，在原子 $A$ 上使用高斯规则，并给出严格边界 $2\sum_A \mu(A)\epsilon_A$，其中 $\epsilon_A$ 保证原子上的一致多项式逼近。

最关键的一点是：作者强调分析作用于加入 Galerkin 空间的函数本身，而非其构造公式。因此短组合只有在经济表示有价值方向时才有效；本文方法适用于任何可计算的紧致表示函数，包括特殊函数、局部 PDE 解、系数适应坐标、降阶近似等。这一思想将传统 h-, p- 细化与更丰富的逼近类统一起来，形成近似压缩机制。

4. 与之前工作的比较

本文与既有有限元文献的关系可从以下几个维度比较。

(1) 与传统 h-/p- 自适应方法。经典理论讨论了自适应 h- 方法的收敛性与拟最优性 [2,4]，以及 hp 残差估计 [12]。本文在 h-, p- 之外引入 c- 富集动作，并提供统一的参考自由增益公式 (2) 来比较三者。传统 h-/p- 方法通过局部网格尺寸或多项式阶数捕捉结构，在相干非多项式特征处可能效率低下；本文方法可通过紧致算子适应函数直接表示这些特征，从而在相同维度下降低能量误差（如 SPE10 上 41.6% 的改进）。但定量收敛率仍依赖于额外的估计量缩减假设，这是本文相对于传统自适应理论的一个未完全闭合之处。

(2) 与单位分解和广义多尺度有限元。单位分解方法已经允许问题适应的局部函数 [1]，GMsFEM 构造紧致算子适应局部空间 [8]，LOD 则考虑全局修正子极限 [9]。这些工作提供了丰富的候选函数生成方式，但缺乏在统一变分尺度上评估和选择这些富集方向的机制。本文的贡献正是补充这一点：商-舒尔稳定性去除冗余，精确增益恒等式度量 PDE 价值，使来自不同构造的候选函数可以直接比较。此外，本文的理论不要求候选函数来自特定公式，因此可涵盖单位分解或 GMsFEM 类型的函数。

(3) 与后验误差估计和通量重构。通量重构提供保证椭圆误差界 [3,11]，但通常需要某种平衡或饱和假设。本文利用商通量恒等式 $\|u-u_h\|_a = \inf_{s\in N} \| q - \nabla u_h - s\|_{L^2}$ 给出一个不依赖饱和假设的可靠上界，并通过分片重构实现。这为自适应终止提供更严格的基础。

(4) 与 Schur–Riesz 理论。本文的直接理论源头是伴随文献 [10] 的连续 Schur–Riesz 定理，该文在抽象 Hilbert 空间逼近中建立商系数稳定性。本文将其转化为 PDE 细化方法，通过耦合稳定创新与变分残差，得到精确增益和参考自由误差界。没有这个转化，Schur–Riesz 理论只保证空间几何的良性，而本文证明它还提供了可计算的实际增益。

(5) 与离散微分复形和特征有限元。兼容有限元通过离散微分复形保留结构恒等式 [6]，特征有限元将输运直接纳入演化步 [7]。本文不修改离散复合体本身，而是允许加入诸如输运剖面等物理结构，作为额外的富集函数。这为处理含有运输或界面等非多项式特征的问题提供了另一种思路。

代价与局限：本文的主要理论结果限于对称强制问题；最小残差扩展和条件复杂度分析在附录中。数值证据表明，hc 只在紧凑函数与 PDE 算子对齐时有效；如果函数冗余或不匹配，则无法改善甚至倒退。作者明确指出，主要贡献是近似压缩具有内在保证，而非普遍更快的线性求解器；均衡 CEM-多重网格虽然保持网格独立收敛并减少迭代次数，但不减少墙钟时间。此外，组合 tent 函数本身为分段仿射，仍需大量方向才能表示弯曲或各向异性结构，且不含 PDE 算子信息，因此不适合作为通用富集家族。这些诚实的局限性使得本文的适用范围清晰：适用于已知和可计算的、与算子兼容的紧致函数能够显著压缩近似空间的场景，如非均质 Darcy 流。

了解领域论文（摘要翻译）

1. 一种用于间断 Galerkin 方法完整离散 $W^{1,\infty}$ 分析的界面格林函数框架

An Interface Green’s Function Framework for Complete Discrete $W^{1,\infty}$ Analysis of Discontinuous Galerkin Methods

作者: Haitao Leng, Weifeng Qiu

类别: math.NA

发布日期: 2026-08-12

arXiv ID: 2608.11559v1

链接: http://arxiv.org/abs/2608.11559v1

摘要翻译

中文标题: 一种用于间断 Galerkin 方法完整离散 $W^{1,\infty}$ 分析的界面格林函数框架

摘要翻译: 在过去的二十年间，间断 Galerkin（DG）方法的逐点误差分析受到了相当大的关注。现有分析针对若干 DG 方法在 $L^{\infty}$ 范数和破碎 $W^{1,\infty}$ 半范数下建立了最优误差估计。然而，由于破碎 $W^{1,\infty}$ 半范数不能控制跨网格界面的间断，针对间断逼近的完整离散 $W^{1,\infty}$ 理论一直未能建立。

本文在凸多面体区域上建立了用于 DG 方法完整离散 $W^{1,\infty}$ 分析的界面格林函数框架。所提出的框架引入了新的界面格林函数，用以表示间断逼近跨网格界面的跳跃。其核心分析要素是这些格林函数的一个新的局部能量估计，该估计通过泰勒展开论证得到，并利用了相邻离散 delta 函数之间的相互抵消。该估计与现有格林函数分析中的估计有本质区别，并且能够在不引入额外对数因子的情况下推导界面估计。

2. 球面有限元网格最大角条件的精确刻画

Exact characterisation of maximum-angle conditions for spherical finite element meshes

作者: Hiroki Ishizaka

类别: math.NA

发布日期: 2026-08-11

arXiv ID: 2608.11353v1

链接: http://arxiv.org/abs/2608.11353v1

摘要翻译

中文标题: 球面有限元网格最大角条件的精确刻画

摘要翻译: 最大角条件是标准的有限元网格假设，它允许使用那些被最小角或形状正则性假设排除的各向异性三角形。然而，对于精确球面三角形，弦仿射核心的角度与内蕴球面角不一定一致，而径向几何引入了曲率尺度的畸变。我们通过一个由三个顶点向量计算的无量纲量，给出内蕴球面最大角条件的代数刻画。该量的一个一致正下界等价于一致球面最大角界，且既不需要球面角也不需要球面面积的计算。我们确定了将弦外接圆半径与径向畸变联系起来的支撑平面几何，并推导了内蕴球面与弦半正则性参数之间的尖锐比较。特别地，一个与局部性无关的比较以尖锐常数 $2/\sqrt3$ 成立，并具有一个被刻画的等式情形。一个基于面积的参数被证明小于球面半正则性参数，在局部平坦极限下具有尖锐常数一。对于球面有限元网格，顶点准则以显式常数蕴含一致的弦半正则性，而相对细化使得径向畸变因子一致收敛到一。因此，内蕴准则与相对细化一起，提供了各向异性有限元分析中使用的几何控制，而不施加最小角或形状正则性条件。

不相关论文（基本信息）

1. 通过极分解将反对称特征值问题的规模减半

Halving the size of skew-symmetric eigenvalue problems via the polar decomposition

作者: Daniel Kressner, Simon Mataigne

类别: math.NA

发布日期: 2026-08-12

arXiv ID: 2608.12153v1

链接: http://arxiv.org/abs/2608.12153v1

摘要: This paper introduces a novel algorithm for computing eigenvalues and eigenvectors of a dense real skew-symmetric matrix $A$. Its main ingredient is the computation of a polar factor of $A$ that is both skew-symmetric and orthogonal. This polar factor is then used to transform the original problem into a Hermitian eigenvalue problem of half the size, which can be solved accurately and efficiently with standard software such as LAPACK. Numerical experiments demonstrate the stability of the method and show that its running time is competitive with existing approaches for skew-symmetric eigenvalue problems. Finally, we show that the same principle can be used to reduce an orthogonal eigenvalue problem to a unitary eigenvalue problem of half the size.

2. 量子行走与量子退火模拟的数值方法

Numerical methods for the simulation of quantum walks and quantum annealing

作者: Asa Hopkins, Viv Kendon

类别: math.NA, quant-ph

发布日期: 2026-08-12

arXiv ID: 2608.12058v1

链接: http://arxiv.org/abs/2608.12058v1

摘要: It is known that Chebyshev based polynomial approximation gives a near-optimal rate of convergence for calculating a function of a Hermitian matrix. However, previous work has not discussed the option of true minimax approximation, nor the specifics of writing a high performance implementation with a rigorous analysis of errors. This work provides such an analysis and an open-source implementation of three approximation methods in C++.

3. 面向三维稀疏多频电磁源重建的模型一致结构化汉克尔矩阵补全

Model-Consistent Structured-Hankel Matrix Completion for 3D Sparse Multi-frequency Electromagnetic Source Reconstruction

作者: Shujaat Khan, Abdul Wahab

类别: math-ph, math.NA, math.OC

发布日期: 2026-08-12

arXiv ID: 2608.12044v1

链接: http://arxiv.org/abs/2608.12044v1

摘要: Reconstruction of 3D electromagnetic currents from sparse, multi-frequency, far-field radiation data is severely ill-posed because sparse sampling masks specific current Fourier modes and the missing spectrum enlarges the null space of the forward operator. Although structured-Hankel completion can recover missing spectrum by exploiting the finite rate of innovations structure of a compactly supported geometrically sparse source, it treats current components independently without guaranteeing compatibility with the model. This often renders non-physical solutions that violate model consistency. To resolve this, we propose a training-free Maxwell-model-consistent 3D structured-Hankel framework (MC-Hankel) that alternates joint low-rank Hankel completion with an exact, closed-form projection onto the true Maxwell synthesis subspace while enforcing conjugate symmetry and noise-aware data fidelity. Across tests on two source models, three sparse sampling rates (30%, 40%, and 50% of a Nyquist-sampled grid), and clean versus noisy (10 dB SNR additive white Gaussian noise) conditions, MC-Hankel consistently outperforms sub-sampled Fourier inversion, scale-normalized $\ell_1$-compressed sensing baseline, and the standard joint 3D annihilating filter-based low-rank Hankel matrix completion approach (ALOHA). Across all considered source-noise-sampling configurations, MC-Hankel improves mean peak SNR (PSNR) by 0.65-4.22 dB and tri-planer structural similarity (3D SSIM) by $0.0266$-$0.0703$, reduces relative full-volume $\ell_2$ reconstruction error by $7.4\%$-$38.9\%$, and brings non-physical model residuals down to machine precision over standard joint 3D ALOHA with merely 17% computational overhead. Exact signed-rank tests are underpowered for five paired trials, so statistical conclusions are reported together with paired bootstrap intervals, effect sizes, and small-sample limitations.

4. 一种用于具有斜 Robin 边界条件的 Hamilton–Jacobi–Bellman 方程的保正性期望格式

A Positivity-Preserving Expectation Scheme for Hamilton–Jacobi–Bellman Equations with Oblique Robin Boundary Conditions

作者: Haoran Xu, Xingye Yue

类别: math.NA

发布日期: 2026-08-12

arXiv ID: 2608.11936v1

链接: http://arxiv.org/abs/2608.11936v1

摘要: For anisotropic diffusion with mixed derivatives, standard compact coordinate-aligned stencils and some local finite-volume or finite-element constructions can lose nonnegative coefficients unless suitable coefficient or mesh conditions are imposed. Here nonnegative coefficients are generated directly from conditional expectation rather than from an algebraic stencil decomposition. We construct a positivity-preserving scheme for possibly degenerate Hamilton–Jacobi–Bellman equations with controlled oblique Robin boundary conditions. At interior nodes the construction stems from a one-step reflected Feynman–Kac identity: an $m$-dimensional Rademacher vector generates $P=2^m$ equally probable weak-Euler branches, an exterior branch is mirrored about its oblique projection, and twice its overshoot serves as the discrete boundary local time $D$. The Robin coefficient enters only through the attenuation factors $e^{-κD}$ and $De^{-κD/2}g$, leaving the equal branch probabilities unchanged. Boundary nodes use a separate same-level closure over a spatial offset $\ell_h\asymp h$, which may be implicit; uniform obliqueness and $\mathbb P_1$ interpolation give a mesh-independent positive weight on interior nodes, so fixed linear Robin data yield a sparse nonsingular $M$-matrix system and the general control set yields a monotone contraction. Positivity is preserved for nonnegative data without a diagonal-dominance condition and without any CFL-type relation between $Δt$ and $h$.

5. 多元三角多项式的正求积与移动采样

Positive quadrature and mobile sampling of multivariate trigonometric polynomials

作者: Stefan Kunis

类别: math.NA

发布日期: 2026-08-12

arXiv ID: 2608.11915v1

链接: http://arxiv.org/abs/2608.11915v1

摘要: The geometric properties of quadrature points in one and multiple dimensions are a classical topic in numerical analysis. Recently, generalized quadrature methods, where discrete points and weights are replaced by integration along curves, have attracted growing interest. This paper focuses on positive generalized quadratures that are exact for multivariate trigonometric polynomials. We derive upper bounds on the covering radius of such quadratures using sign-localized test functions, a technique that also extends naturally to algebraic polynomials on intervals, spherical polynomials, and hyperbolic cross trigonometric polynomials. For lower bounds on the length of quadrature curves, we compare two recent approaches in the context of multivariate trigonometric polynomials. The second part of the paper studies a well-known, geometrically simple curve that underlies rank-1 Korobov lattice rules for periodic functions. We prove that these curves are quasi-optimal with respect to multiple optimality criteria and demonstrate its adaptability to the 2-sphere.

6. 基于切比雪夫多项式的快速八阶Pade格式用于直接Zakharov-Shabat问题

Fast eight-order Pade schemes based on Chebyshev polynomials for direct Zakharov-Shabat problem

作者: Sergey Medvedev, Igor Chekhovskoy, Irina Vaseva, Mikhail Fedoruk

类别: math.NA

发布日期: 2026-08-12

arXiv ID: 2608.11892v1

链接: http://arxiv.org/abs/2608.11892v1

摘要: In this work, we construct fast eighth-order Pade schemes for the direct Zakharov-Shabat scattering problem. The schemes are based on an eighth-order exponential integrator obtained from the Magnus expansion. A direct extension of the conventional fast Pade representation to the eighth-order case leads to insufficiently accurate fast variants in the considered tests. To overcome this difficulty, we reformulate the spectral dependence on a finite real interval using the Joukowski mapping and represent the local numerators and denominators in the Chebyshev polynomial basis. This makes it possible to construct the global transition matrix by a fast product tree while retaining a compact polynomial representation. Numerical experiments for chirped hyperbolic secant potentials with both signs of dispersion show that the proposed Chebyshev-based fast schemes substantially improve the accuracy of the corresponding direct fast variants and can be used for efficient computation of the continuous nonlinear spectrum.

7. 一个一致椭圆约化、一个阶匹配准则，以及用于格点三玻色子三聚体强耦合 Birman-Schwinger 分析的精度基准

A uniform elliptic reduction, an order-matching criterion, and precision benchmarks for the strong-coupling Birman-Schwinger analysis of the lattice three-boson trimer

作者: Mikhail Dolgopolov

类别: math-ph, math.NA, math.SP, quant-ph

发布日期: 2026-08-12

arXiv ID: 2608.11848v1

链接: http://arxiv.org/abs/2608.11848v1

摘要: We study the strong-coupling Birman-Schwinger analysis of the three-boson lattice Schroedinger operator at the exceptional quasimomentum K = pi. First, we provide an exact closed-form benchmark for the fiber Fredholm determinant at a distinguished flat momentum, valid for every quasimomentum K, obtained via a uniform elliptic reduction of the two-dimensional lattice integral. Second, we formulate and prove a general order-matching criterion. This criterion decides whether a leading-order Fredholm determinant asymptotic, with relative accuracy of order 1/mu, suffices to determine the constant-order additive correction in the strong-coupling energy, or if the next-order refinement is required. Applying the criterion at K = pi, we derive the complete strong-coupling ground-state asymptotics, including the coefficient of the order 1/mu term. The results are cross-validated against the exact closed-form benchmark and two independent high-precision numerical schemes. Furthermore, we determine the corresponding spectral gap and prove an exact Fredholm-determinant underestimation factor equal to 2. Additionally, we independently confirm that the previously established K = 0 constant is correctly derived and requires no analogous refinement. Finally, we compare the orders of asymptotic precision controlled across recent lattice few-body models and draw a structural parallel with the parity classification of topological band structures.

8. 黎曼 $Ξ$ 函数的围道汉克尔动力学与指示场

Contour Hankel dynamics and indicator fields for the Riemann $Ξ$-function

作者: Zhiliang Deng, Xiaomei Yang, Huazhong Lü

类别: math.NA

发布日期: 2026-08-12

arXiv ID: 2608.11520v1

链接: http://arxiv.org/abs/2608.11520v1

摘要: We develop a moving-contour Hankel framework for encoding local zero configurations of the Riemann $Ξ$-function. Weighted contour integrals of the logarithmic derivative $Ξ'/Ξ$, expressed in a holomorphic coordinate associated with the contour, are identified with the power moments of a finite atomic measure supported at the coordinate images of the enclosed zeros. This representation yields exact zero-free and rank criteria and, for conjugation-compatible contour-coordinate pairs, an inertia formula: once the matrix order is at least the number of distinct coordinate nodes, the negative index equals the number of distinct nonreal conjugate pairs. Consequently, the Riemann hypothesis admits a local finite-dimensional Hankel-positivity formulation, although establishing this positivity independently of the zero set remains unresolved. As the contour moves, the Hankel matrix evolves by a continuous congruence flow between zero crossings and undergoes finite-rank jumps at crossing events. An isolated zero produces a signed rank-one jump, whereas a nonreal conjugate pair produces a rank-two indefinite event in a real-axis circular scan that meets the pair. Removing the continuous coordinate drift yields a piecewise-constant matrix process from which crossing coordinates, multiplicities, and zero locations can be recovered. Numerical experiments validate the contour quadrature, indicator fields, continuous flow, crossing signatures, and recovery procedure.

9. 由矩阵谱分解导出的新型正交多小波滤波器

New Orthogonal Multiwavelet Filters Derived by Matrix Spectral Factorization

作者: Vasil Kolev, Todor Cooklev, Fritz Keinert

类别: cs.CV, cs.DB, math.NA, stat.AP

发布日期: 2026-08-12

arXiv ID: 2608.11518v1

链接: http://arxiv.org/abs/2608.11518v1

摘要: The paper considers the construction of two new orthogonal multiwavelets with supercompact support by using the Fast Bauer’s method for matrix spectral factorization on the matrix product filter of the orthogonal CL multiwavelet filter. The new multiwavelets possess orthogonality, symmetry/antisymmetry, and one of them provides better coding and smoothness than other supercompact multiwavelets. The performance of the new multiwavelet filters in subband-based edge detection, grayscale and color image compression and 1D and 2D signal denoising is compared with the GHM, SA4, CL, Integer Haar and Alpert multifilters. The comparative analysis shows that new multiwavelets can provides better human visual measures, SSIM and MS-SSIM in image compression and denoising applications.

10. 超越强凸性的有限批次粒子平均场变分推断稳定性

Stability of Finite-Batch Particle Mean-Field Variational Inference Beyond Strong Convexity

作者: Vinh Nguyen, Truong Vu

类别: math.NA, math.OC, math.PR, stat.ML

发布日期: 2026-08-11

arXiv ID: 2608.11486v1

链接: http://arxiv.org/abs/2608.11486v1

摘要: We study the implementable finite-batch particle algorithm for mean-field variational inference as a fully discrete stochastic approximation of the projected Wasserstein dynamics. The target potential is globally smooth but need not be strongly convex. The departure from contractivity is quantified by the curvature defect [ \mathfrak d_α(x,y) = \bigl[α|x-y|^2- \langle\nabla V(x)-\nabla V(y),x-y\rangle\bigr]_+, ] which is the additive loss in the one-step Euler contraction estimate. We prove a non-asymptotic Wasserstein stability bound that separates initialization, product-empirical approximation, finite-batch drift error, time discretization, and the defects accumulated along the coupled trajectories. Under the uniform bound $\mathfrak d_α\leqβ$, the particle iterates remain within $O(\sqrt{β/α})$ of any MFVI minimizer, up to explicit errors in the particle number, batch size, and step size. The proof uses a stationary comparison array whose population law is an MFVI minimizer but whose particle-level law is a random product empirical measure, and it controls the resulting projected-drift discrepancy explicitly. We also give coordinatewise defect estimates and structural conditions for dimension-independent projected-drift sensitivity, construct an arbitrary-dimensional smooth nonconvex benchmark with a closed-form MFVI minimizer, and explain why polynomially growing drifts require a modification of the untamed explicit scheme.

11. 线性成本的任意次数多调和样条插值

Linear-cost Polyharmonic Spline Interpolation of Arbitrary Degree

作者: Christopher J. Geoga, Michael O’Neil

类别: math.NA, stat.CO

发布日期: 2026-08-11

arXiv ID: 2608.11462v1

链接: http://arxiv.org/abs/2608.11462v1

摘要: We introduce a simple and performant approach for rapidly and accurately performing polyharmonic spline (PHS) interpolation using a combination of the fast multipole method (FMM) from computational electrostatics and the Vecchia approximation from the Gaussian process and sparse approximate inverse literatures. Using basic properties about Hadamard products and low-rank matrices, we demonstrate that an FMM with two kernels, the logarithmic and distance kernels, results in fast PHS interpolation for all orders. Furthermore, we demonstrate the exceptional performance of sparse inverse approximation methods with the Matérn covariance model for preconditioning. Combined with careful management of disallowed subspaces, we describe a procedure for obtaining prediction weights using preconditioned conjugate gradient that converges in less than $15$ iterations, even for problem sizes with over one million points. As a result, thin-plate spline interpolation—a particularly popular method that does not require parameter tuning—that matches the fully dense $\mathcal{O}(n^3)$ computation in accuracy can be done at the cost of approximately $50-60$ FMMs. A high-performance software library for odd-order PHS interpolation in two dimensions is made available as a companion to this work.