生成时间: 2026年08月03日 11:01:15

重点关注论文数量: 1

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重点关注论文（完整分析）

1. I-MCHM：多尺度椭圆问题的界面多重连续介质均匀化方法

I-MCHM: Interface Multicontinuum Homogenization for Multiscale Elliptic Problems

作者: Wing Tat Leung, Zhihang Xu

类别: math.NA

发布日期: 2026-07-31

arXiv ID: 2607.29325v1

摘要翻译: 我们针对具有高对比度的椭圆问题，提出一种界面多重连续介质均匀化方法（I-MCHM），其中子区域可具有不同的微观模式及数目不等的连续介质。多重连续介质模型将微观场升尺度为宏观连续介质量；此处连续介质为高渗透率与低渗透率区域，宏观变量则是细尺度解在尺寸为$H_ε$的观测单元上的相应局部平均值。虽然细尺度解在材料界面上是连续的，但这些宏观连续介质场在界面上未必一致。此外，当相邻子区域保留不同数目的连续介质时，无法定义一对一的粗传输条件。因此，标准的分区多重连续介质均匀化只能确定体方程，而对这些粗变量之间的相互作用不加规定。I-MCHM通过在界面邻域上构造双侧约束的局部基，并在子区域邻接图上用体切割积分和界面片修正组装粗双线性形式，从而填补这一空白，且未预先指定逐点传输律。对平直、弯曲和三重连接几何形状的数值实验，包括连续介质耦合消融和不等连续介质计数的情况，展示了所得升尺度模型的准确性。

Comment: 无

链接: http://arxiv.org/abs/2607.29325v1

1. 研究对象和背景

多尺度椭圆问题广泛存在于复合材料、多孔介质流动等物理建模中。考虑有界区域 $\Omega \subset \mathbb{R}^d$ ($d=2,3$) 上的精细尺度椭圆方程

$$-\nabla\cdot(\kappa_\epsilon(x)\nabla u_\epsilon) = f \quad \text{in } \Omega, \qquad u_\epsilon = 0 \quad \text{on } \partial\Omega,$$

其中 $u_\epsilon$ 为细观解，$f$ 为给定源项，系数 $\kappa_\epsilon$ 可能呈现剧烈振荡和高对比度结构，且 $\epsilon$ 表征微观尺度。直接细网格求解计算代价极高，因此需借助升尺度(homogenization)方法获得宏观有效模型。

经典均匀化与数值均匀化方法常假设局部周期性，推导出单一等效系数；然而，对于同时包含高、低渗透区域且两者相互作用显著的材料，单一宏观场不足以刻画不同物理区域的响应。多连续介质均匀化方法(MCHM) 针对此问题，将细观解分解为若干连续介质分量。每个分量对应一种特征区域（如高渗或低渗区），宏观变量 $U_i$ 定义为该分量在观测单元 $K_{H_\epsilon}$ （尺寸 $H_\epsilon \ge \epsilon$）上的局部平均：

$$U_i(x) := \frac{\int_{K_{H_\epsilon}(x)} u_\epsilon(y) \psi_i(y) dy}{\int_{K_{H_\epsilon}(x)} \psi_i(y) dy},$$

其中 $\psi_i$ 为第 $i$ 个连续介质的指示函数。在粗尺度 $H$ (> $H_\epsilon$) 上视 $U_i$ 为缓慢变化的宏观场，并在代表性体积元(RVE) $\omega(K)$ 上作一阶展开：

$$u_\epsilon(y) \approx \sum_{i=1}^L \left( U_i(x_K) \eta_i^K(y) + \sum_{m=1}^d \partial_m U_i(x_K) \eta_i^{K,(m)}(y) \right), \quad y \in \omega(K).$$

局部基函数 $\eta_i^K$ 通过在 oversampled 区域 $\omega^+(K)$ 上求解约束能量极小问题得到：最小化能量 $\frac12 a_{\epsilon,\omega^+(K)}(\eta,\eta)$，同时满足矩约束

$$\int_{\omega^+(K)} \eta(y) \psi_{j,q}(y) dy = \delta_{ij} \int_{\omega^+(K)} \psi_{j,q}(y) dy,$$

以确保在观测单元上精确重现各连续介质的平均值。类似可得梯度基函数 $\eta_i^{K,(m)}$。这些基函数代入精细双线性形式并取平均，导出有效系数(如 $\alpha_{ij}^{(mn)}, \beta_{ij}^{(m)}, \gamma_{ij}$)，最终得到粗尺度弱形式(12)。该方法不依赖全局周期假设，且有效系数可用于任意粗网格离散。

然而，上述标准 MCHM 假设全域具有统一的连续介质数量，且无材料界面。实际工程问题常涉及多个具有不同微观结构的子域 $\Omega_k$ ($k=1,\dots,N_\Omega$)，它们经由界面 $\Gamma_{k\ell} = \partial\Omega_k \cap \partial\Omega_\ell$ 连接。尽管细观解满足连续条件 $[\! [u_\epsilon]\! ]_{\Gamma_{k\ell}} = 0$ 和 $[\! [\kappa_\epsilon \nabla u_\epsilon \cdot \mathbf{n}]\! ]_{\Gamma_{k\ell}} = 0$，但各子域独立定义的宏观平均 $U_j^{(k)}$ 在界面处一般并不连续：$U_j^{(k)}|_{\Gamma_{k\ell}} \neq U_\ell^{(\ell)}|_{\Gamma_{k\ell}}$。更严重的是，若相邻子域包含不同数量的连续介质 ($L_k \neq L_\ell$)，则不存在一对一的传输映射，无法通过显式条件耦合两侧宏观变量。因此，单纯在每子域内应用 MCHM 只能确定子域内部的体方程，跨界面相互作用缺失，导致全局宏观系统不完备。

本文提出的界面多连续介质均匀化方法(I-MCHM) 旨在填补这一空白。它在界面邻域上构造双侧约束局部基函数，允许两侧宏观变量完全独立且数量无需相等，并通过对体切割单元和界面段能量的变分组装，自动产生对称的跨界面耦合项，无需预设任何点式传输定律。

2. 主要定理或主要结果

本文主要理论结果如下：

  1. 双侧约束基函数的构造：对每个界面邻域 $E$ (与界面相交的粗块)，在 oversampled RVE $\omega^+(E)$ 上定义鞍点问题，同时施加来自界面两侧子域的矩约束，生成一组基函数 $\eta_{i,s}^E$ 和 $\eta_{i,s}^{E,(m)}$ ( $i$ 指标遍历两侧各自的连续介质，$s=1,2$ 标记子域)。这些基函数能独立地捕获指定子域、指定连续介质的局部平均值和一阶导数矩，从而支持对不等数量宏观场的完备描述。

  2. 对称粗双线性形式：基于体-界面分解，全局粗双线性形式表达为体切割单元积分与界面段校正项之和：

    $$a_{\text{coarse}}(U, V) = \sum_{K \in \mathcal{T}_H^{\text{bulk}}} a_{\text{bulk},K}(R_K U, R_K V) + \sum_{E \in \mathcal{T}_H^{\text{interface}}} \Bigg[ \frac{1}{|\omega(E)|} \int_{\omega(E)} \kappa_\epsilon \nabla R_E U \cdot \nabla R_E V \,dy - \sum_{t=1,2} c_t \frac{1}{|\omega(E_t)|} \int_{\omega(E_t)} \kappa_\epsilon \nabla R_{E_t} U \cdot \nabla R_{E_t} V \,dy \Bigg],$$

    其中 $R_E U$ 为利用双侧基在 $\omega(E)$ 上构造的重建，$R_{E_t} U$ 为在单侧切割区域 $\omega(E_t) = \omega(E) \cap \Omega_t$ 上的重建。该表示通过减去重复的体贡献保证了能量一致性。整个形式对称正定，且组装无需预设点式传输条件。

  3. 离散适定性：在重构能量范数 $\|\cdot\|_{\mathcal{R}}$ 下，证明了粗问题具有强制性：存在常数 $c>0$ 使得

    $$a_{\text{coarse}}(U, U) \ge c \|U\|_{\mathcal{R}}^2 \quad \forall U,$$

    从而确保了解的适定性。证明基于约束能量最小化基函数的局部衰减性质(由 oversampling 和 CEM/NLMC 理论保证)以及体-界面分解下的范数等价。

  4. 数值验证：针对直界面、弯曲界面、三叉界面以及不等连续介质数量(包括部分连续介质人为消融)的多种配置，数值实验表明 I-MCHM 能精确捕捉细观解的宏观平均行为，即使在极端界面不匹配情况下也表现出良好的精度。

3. 研究方法、关键技术和核心工具

I-MCHM 的方法创新集中于三个关键技术点：

一、双侧约束局部基函数的鞍点构造。 核心障碍在于：标准 MCHM 的基函数仅满足单一子域的矩约束，无法同时兼顾界面两侧的宏观变量。本文的策略是直接在界面邻域 $E$ 的 oversampled RVE $\omega^+(E)$ 上构造基函数，并施行同时涉及两侧的约束条件。具体地，对每个子域 $t$ 和连续介质指标 $i$，定义 $\eta_{i,t}^E$ 为以下能量极小问题的解： \begin{align*} \eta_{i,t}^E = \operatorname{argmin}{\eta \in H_0^1(\omega^+(E))} \frac12 a{\epsilon,\omega^+(E)}(\eta,\eta) \quad \text{s.t.} \quad \int_{\omega^+(E)} \eta(y) \psi_{j,q}^{(s)}(y) dy = \delta_{is}\delta_{jt} \int_{\omega^+(E)} \psi_{j,q}^{(s)}(y) dy, \ \forall j, q, s, \end{align*} 其中 $\psi_{j,q}^{(s)}$ 为第 $s$ 子域内第 $j$ 连续介质在观测单元 $q$ 上的特征函数。约束强制基函数仅与指定子域 $t$ 的连续介质 $i$ 耦合，并压制其他子域和其他介质的矩。通过引入拉格朗日乘子，将该问题转化为鞍点系统，进而使用标准有限元或谱方法求解。一阶基函数类似，但约束条件中的权函数变为 $(y_m - x_{E,m})$。这一双侧约束构造无需全局契合条件，且天然允许 $L_1 \neq L_2$。

二、体-界面分解与对称粗形式组装。 获得所有局部基后，需将子域内部与界面的能量耦合为一个全局对称格式。作者采用体-界面分解策略：将每个界面粗块 $E$ 视为由两个切割单侧区域 $E_t = E \cap \Omega_t$ 拼接而成。在 $E$ 上，利用双侧基重建 $R_E U$，得到全界面邻域能量密度；而在切割区域 $E_t$ 上，若仅保留该侧宏观变量并利用单侧基（即标准 MCHM 基）重建，则可算得体贡献。界面段校正项定义为全界面能量减去两侧体贡献的加权和，从而精准抽取出跨界面耦合部分。最终全局双线性形式为所有内部块体贡献与所有界面段校正项之和，保持了整体对称性。由于校正项直接源于局部变分问题，它自动蕴含了物理上正确的传输行为，无需额外假设。

三、重构能量范数与离散稳定性。 为分析全局粗问题的适定性，定义宏观场 $U$ 的重构能量范数

$$\|U\|_{\mathcal{R}} := \left( \sum_{K\in\mathcal{T}_H^{\text{bulk}}} a_{\text{bulk},K}(R_K U, R_K U) + \sum_{E\in\mathcal{T}_H^{\text{interface}}} a_{\epsilon,\omega(E)}(R_E U, R_E U) \right)^{1/2}.$$

稳定性证明的关键是建立该范数与标准 Sobolev 范数的关系，并证明粗双线性形式的强制性。这依赖于CEM-GMsFEM 和 NLMC 理论[24] 中约束能量最小化基函数的指数衰减性质：在 oversampled 区域边界，基函数能量按 $\sim e^{-\beta \text{sos}}$ 衰减，因此局部重建的能量贡献主要集中在 RVE 内部，全局求和时不同邻域的相互作用迅速衰减。结合体-界面分解的精确能量等式，可完成强制定理。具体论证中，需利用局部稳定分解的有限覆盖和局部范数等价，最终导出全局强制性。

4. 与之前工作的比较

I-MCHM 与已有工作的主要差异体现在以下几个方面：

  1. 与标准 MCHM 的比较。标准多连续介质均匀化[11,12] 仅适用于全域具有统一微观结构和相同数量连续介质的情形。当应用于分区具有不同介质的区域时，只能分别在每个子域独立生成宏观方程，但无法提供界面处的连接条件。I-MCHM 直接将 MCHM 推广至多子域界面问题，通过双侧基函数产生界面耦合项，并允许相邻子域的连续介质数量任意。这实质上突破了原方法对一致连续介质数的限制。

  2. 与多尺度有限元界面方法的比较。针对高对比度界面的多尺度有限元构造[20,21] 常常通过求解局部细观问题并施加某种平均或传输条件来定义跨越界面的粗基函数，其本质依赖于界面两侧粗变量之间预设的映射关系。当 $L_k \neq L_\ell$ 时，这种映射无从定义，导致方法失效。I-MCHM 完全放弃显式传输映射，转而从双侧独立约束的变分过程中自动提取耦合信息，因此能处理界面两侧连续介质数不等的复杂情形。

  3. 与基于 CEM-GMsFEM 和 NLMC 的升尺度方法的比较。约束能量最小化广义多尺度有限元 (CEM-GMsFEM) 和非局部多连续介质方法 (NLMC) [10,22] 同样利用约束能量极小化构造具有局部衰减性质的基函数，并已成功应用于多孔介质高对比度问题。但这些方法通常针对单一区域，且跨粗网格的耦合通过 Robin 边界条件或预定义的传输算子实现，缺乏对子域界面的直接处理。I-MCHM 继承了 CEM 框架的优点（如 oversampling 和指数衰减），但通过引入双侧约束，将该框架从单区域推广到多子域界面问题，并使得跨界面耦合由能量极值原理内生决定。

  4. 与均匀化分析中界面跳跃处理的比较。文献[23] 从经典均匀化角度出发，推导了带界面跳跃的宏观模型，其传输校正依赖于细观界面层的详细分析。I-MCHM 不同于此类基于两尺度展开的分析方法，它采用数值均匀化范式，直接在离散层面通过解局部约束问题来摄取界面行为，因此更易于处理任意复杂界面几何和非周期性微观结构。

  5. 适用范围与代价。I-MCHM 放宽了子域间连续介质一致性和传输映射存在的假设，显著拓展了多连续介质均匀化的应用范围，可处理包含直、弯、多叉界面及连续介质消融等复杂构型。代价在于界面邻域需要求解额外的鞍点系统以构造双侧基，且全局组装需计算切割单元和校正项，计算量高于各子域独立 MCHM。不过，数值实验表明这些增加的计算开销换来了模型精度的大幅提升，特别是当相邻子域介质结构差异显著时，I-MCHM 是唯一能保持合理的宏观描述的方案。

了解领域论文（摘要翻译）

1. 偶数几何的奇怪行为：超收敛几何相容误差的一种解释

Odd behaviour of even geometries: an explanation for superconvergent geometric consistency errors

作者: Hanne Hardering, Simon Praetorius, Gentian Zavalani

类别: math.NA

发布日期: 2026-07-31

arXiv ID: 2607.29466v1

链接: http://arxiv.org/abs/2607.29466v1

摘要翻译

中文标题: 偶数几何的奇怪行为：超收敛几何相容误差的一种解释

摘要翻译: 表面有限元方法中使用的分片多项式曲面近似，若其多项式阶数为偶数，通常表现得比其标准逼近性质所暗示的更好。我们通过在某些细分过程中自然出现的适当结构化网格上主要插值误差的抵消来解释这种超收敛。这种抵消改进了函数、导数和几何量的加权积分估计。应用包括曲面法向量、Weingarten 映射和高斯曲率的估计。数值实验再现了预测的依赖于奇偶性的行为，并支持了对超收敛几何相容误差所提出的解释，而相应的逐点误差保持其标准阶。

不相关论文（基本信息）

1. 冻结，然后选择：面向稀疏观测偏微分方程发现的结构化场适配器与稳定性验证的弱选择

Freeze, Then Select: Structured Field Adapters and Stability-Validated Weak Selection for PDE Discovery from Sparse Observations

作者: Juncheng Zhong, Chenghuang Shen, Jianfeng Liu, Zhengdong Xiao, Longjiu Luo, Qianrong Wang, Wenjun Xu, Wenlian Lu

类别: cs.LG, math.NA

发布日期: 2026-07-31

arXiv ID: 2607.29665v1

链接: http://arxiv.org/abs/2607.29665v1

摘要: PDE discovery from sparse observations requires reconstructing a continuous field and selecting the correct differential terms. Our analysis of optimization paths in coupled neural PDE discovery reveals three behaviors: the exact support can persist to the end of training, appear only transiently, or fail to emerge. To decouple equation selection from neural optimization, we develop a freeze-then-select method combining a structured field adapter with Stability-Validated Weak Selection (SVWS). Trained from observations without a PDE residual, the adapter factorizes the field into learned spatial features and temporal coefficients represented by cubic splines. After freezing the field, SVWS identifies recurrent terms across independent weak-form systems, refits candidate supports, and selects the final equation on held-out weak-form systems. Beyond fixed libraries, we apply the same principle to expressions generated by genetic programming and recover the power-law form of an unknown nonlinear diffusion function from sparse, noisy observations. Across all six sparse MDBench regimes, our method attains the highest exact support recovery rate, with its clearest gains over classical and neural baselines on challenging Kuramoto-Sivashinsky dynamics.

2. 多保真度蒙特卡洛样本量估计的递归舍入

Recursive rounding of sample size estimation for multi-fidelity Monte Carlo

作者: Jiaxing Liang

类别: math.NA

发布日期: 2026-07-31

arXiv ID: 2607.29607v1

链接: http://arxiv.org/abs/2607.29607v1

摘要: In multifidelity Monte Carlo (MFMC), optimal sample allocations are typically derived from a continuous relaxation of a variance minimization problem, with integer solutions obtained through post hoc rounding. Such rounding procedures may fail to fully exploit the available cost budget, particularly under tight cost budgets or when model costs vary significantly. In this work, we reformulate the MFMC allocation problem as a variance-constrained cost minimization problem that is equivalent to the standard budget-constrained formulation at the continuous level. This reformulation admits a recursive structure that enables the construction of an integer allocation strategy based on Bellman’s principle of optimality. Since the resulting MFMC allocation has a mathematical structure similar to the optimal multilevel Monte Carlo (MLMC) allocation, the proposed strategy naturally extends to MLMC. The resulting algorithm constructs integer-valued sample allocations that more closely follow the continuous variance–cost tradeoff while using the prescribed variance tolerance more efficiently. Numerical experiments demonstrate that the proposed approach satisfies the prescribed variance tolerance with less computational overhead than standard rounding strategies.

3. 一种基于阈值化的算子分裂法用于曲率正则化曲面重建

A Thresholding Based Operator-Splitting Method for Curvature-Regularized Surface Reconstruction

作者: Yang Hu, Hao Liu, Dong Wang, Tieyong Zeng

类别: math.NA

发布日期: 2026-07-31

arXiv ID: 2607.29584v1

链接: http://arxiv.org/abs/2607.29584v1

摘要: Surface reconstruction from point clouds is a fundamental problem in computational geometry with broad applications in computer graphics, medical imaging, and manufacturing. In this paper, we propose an efficient method for solving a curvature-regularized surface reconstruction model. To avoid the computational cost associated with reinitialization in traditional level set methods, we represent the reconstructed surface by an indicator function. By introducing an auxiliary variable, we reformulate the original optimization problem as the computation of the steady-state solution of an initial value problem. We then develop an operator-splitting method to decompose the resulting problem into two tractable subproblems, one of which can be efficiently solved by iterative thresholding. The proposed approach combines the advantages of curvature regularization, operator splitting, and threshold dynamics. Numerical experiments show that the proposed method is computationally efficient and can accurately reconstruct sharp corners and concave features.

4. 带噪声系数及维纳过程信息的标量SDE的随机化Milstein格式的误差

Error of randomized Milstein scheme for scalar SDEs with noisy information about coefficients and Wiener process

作者: Paweł M. Morkisz, Paweł Przybyłowicz, Martyna Wiącek

类别: math.NA, math.PR

发布日期: 2026-07-31

arXiv ID: 2607.29578v1

链接: http://arxiv.org/abs/2607.29578v1

摘要: We investigate the strong approximation of scalar stochastic differential equations when the available standard information about the drift coefficient, the diffusion coefficient, the derivative of the diffusion coefficient, and the observed Wiener path is corrupted by noise. The precision of the drift, diffusion-information, and Wiener-path observations is described by three nonnegative parameters $δ_1,δ_2,δ_3$, where $δ_2$ controls both the noisy diffusion coefficient and the separate noisy derivative oracle required in the Milstein correction. We analyze a randomized Milstein scheme based only on this noisy information and prove, for $r\geq 2$, that its $L^r$-error is bounded by $C(n^{-\min\{γ_1+1/2,γ_2\}}+δ_1+δ_2+δ_3)$, where $n$ is the number of time steps and $γ_1,γ_2$ are the temporal Hölder exponents of the coefficients. We also prove a matching minimax lower bound in the randomized standard-information model considered in the paper. In particular, the Wiener-path contribution proportional to $δ_3$ is unavoidable, and the noisy randomized Milstein scheme is minimax order-optimal.

5. iSMART: 一种求解基于鞅的偏微分方程的迭代采样与回归技术

iSMART: An Iterative Sampling-and-Regression Technique for Solving Martingale-Based PDEs

作者: Tiejun Li, Xiaoguang Li, Fugui Ma

类别: math.NA

发布日期: 2026-07-31

arXiv ID: 2607.29470v1

链接: http://arxiv.org/abs/2607.29470v1

摘要: We propose the {\bf i}terative {\bf S}a{\bf M}pling-{\bf A}nd-{\bf R}egression {\bf T}echnique (iSMART) for high-dimensional martingale-based partial differential equations (PDEs) in this paper. By leveraging the $L^2$-projection property of conditional expectation and adopting the stop-gradient technique, iSMART reformulates the continuous martingale condition derived from PDEs into a sequence of tractable sampling-regression problems within an iterative framework. This approach relies solely on standard SDE path simulation and plain squared-error loss minimization, completely bypassing the need for adversarial optimization or nested expectation estimation in previous methods. iSMART accommodates linear, semi-linear, and fully nonlinear martingale-based PDEs within a unified iterative procedure. In particular, for fully nonlinear Hamilton-Jacobi-Bellman (HJB) equations, a freezing-and-compensating technique is introduced to strategically shift a portion of the nonlinearity into the SDE drift, thereby improving the convergence behavior of the iterations. Numerous numerical experiments on linear reaction-diffusion equations with sharp gradients, semilinear Burgers-type equations, and fully nonlinear HJB equations demonstrate the accuracy, efficiency, and robustness of the proposed approach in various high dimensions.

6. 具有最优性保证的递归蝴蝶分解

A recursive butterfly factorization with optimality guarantees

作者: David Persson, Paul G. Beckman, Tyler Chen, Diana Halikias, Christopher Musco

类别: math.NA

发布日期: 2026-07-31

arXiv ID: 2607.29361v1

链接: http://arxiv.org/abs/2607.29361v1

摘要: We formalize a recursive format for representing a butterfly matrix. This new format naturally leads to a simple recursive algorithm for computing a quasi-optimal butterfly approximation to an arbitrary $N \times N$ matrix $A$. When the entries of $A$ are explicitly available, we show that the algorithm computes a butterfly matrix $B$ in $O(N^2)$ operations with approximation error $\|A - B\|_F$ at most a $O(\sqrt{\log(N)})$ factor away from that of the best possible approximation by a butterfly matrix. We also develop a matrix-free variant of the method, which uses $\widetilde{O}(\sqrt{N})$ matrix-vector products and $\widetilde{O}(N)$ working memory and, with high probability, returns a butterfly approximation with Frobenius norm error within a $O(N^{1/4})$-factor of the optimal error. We show that the algorithm is a reformulation of the hybrid butterfly factorization approach presented in [Liu et. al.; SISC, 43 (2021)]. Our paper therefore provides the first theoretical quasi-optimality guarantee for that algorithm.

7. 计算分裂四元数矩阵的Moore–Penrose逆的迭代方法及其应用

Iterative Methods for Computing the Moore–Penrose Inverse of Split-Quaternion Matrices with Applications

作者: Salman Ahmadi-Asl, Valentin Leplat, Mohammad S. Alkousa

类别: math.NA

发布日期: 2026-07-31

arXiv ID: 2607.29270v1

链接: http://arxiv.org/abs/2607.29270v1

摘要: We study iterative methods for computing the Moore–Penrose inverse of split-quaternion matrices. We first establish a consistent framework based on a (2\times2) real representation and the associated (i)-conjugate transpose. This representation gives a direct definition of the Moore–Penrose inverse and clarifies the treatment of nonzero zero divisors. We then analyze Newton–Schulz iterations for rectangular and rank-deficient matrices. Using a thin singular value decomposition of the real representative, we derive the convergence conditions, the evolution of the projector residuals, and an exact relation between the residual and the error. The resulting analysis applies both to an embedded real implementation and to a native split-quaternion iteration, which are shown to be equivalent under the representation. We also propose a low-degree polynomial initialization inspired by Souriau’s inverse recursion. The polynomial is obtained by a least-squares approximation of the inverse Gram factor and is combined with a spectral acceptance test and a safe fallback initialization. Finally, we apply the proposed methods to cross and CUR approximations of split-quaternion matrices. We characterize the optimal middle factor for fixed sampled rows and columns and give conditions under which the cheaper cross factor gives an exact reconstruction. Numerical experiments illustrate the accuracy of the methods and the practical benefit of the polynomial initialization.

8. 固定先验期望改进在Matérn和平方向指数RKHS中的简单遗憾率与极小化极大最优性

Simple-regret rates and minimax optimality of fixed-prior expected improvement in Matérn and squared-exponential RKHSs

作者: Emmanuel Vazquez, Sébastien Petit

类别: stat.ML, cs.LG, math.NA, math.ST

发布日期: 2026-07-31

arXiv ID: 2607.29245v1

链接: http://arxiv.org/abs/2607.29245v1

摘要: We study the expected improvement (EI) policy for minimizing a deterministic objective function $f$ on a nonempty compact set $\mathcal X \subset\mathbb R^d$. We assume that $f$ belongs to the RKHS $\mathcal H_k$ of a continuous positive-semidefinite kernel $k$ on $\mathcal X$. Function values are observed exactly, and EI is computed from a fixed zero-mean Gaussian-process model with covariance $σ^2k$. After an initial design, the policy queries a point whose EI is at least a fixed positive fraction of its maximum. We identify the normalized posterior standard deviation at a candidate point $x$ with the norm of the corresponding innovation in the canonical feature space, namely the component of $k(x,\cdot)$ orthogonal to the span of the preceding evaluation representers. Sequential separation radii bound the ranked innovation norms along arbitrary query sequences. We estimate these radii using Gram determinants and Kolmogorov widths for subspaces of different dimensions, then combine the estimates with a one-step regret inequality to obtain finite-budget bounds for simple regret. After $N$ post-initial queries, simple regret is $O(N^{-ν/d})$ for isotropic Matérn kernels of smoothness $ν>0$. For the isotropic squared-exponential kernel, simple regret is $O(\exp[-c_1\min\{N, N^{1/d}\log(eN)\}])$ for some $c_1>0$. With exact EI maximization, it is $O(\exp[-c_2N^{1/d} \log(eN)])$ for some $c_2>0$. For every fixed $B\geq0$, these bounds are uniform over the RKHS ball of radius $B$. If $\mathcal X$ has nonempty interior and $B>0$, then, among deterministic methods whose final recommendation may be any point of $\mathcal X$, the exact EI policy is minimax-rate optimal over the RKHS ball of radius $B$ for Matérn kernels and minimax-rate optimal up to constants in the exponent for squared-exponential kernels.

9. 基于节点扫掠协同设计的认证七阶双导数Hermite延迟校正

Certified Seventh-Order Two-Derivative Hermite Deferred Correction via Node-Sweep Co-Design

作者: Zhixin Huo

类别: math.NA

发布日期: 2026-07-31

arXiv ID: 2607.29217v1

链接: http://arxiv.org/abs/2607.29217v1

摘要: Two-derivative Hermite deferred correction combines high collocation order with sequential single-state solves, but the stopped method depends jointly on the nodes and the correction sweep. We co-design these ingredients for diffusion-dominated semilinear problems. For three subintervals, an H4 predictor, and two corrections, the complete order-seven B-series defect has rank one in the 48-dimensional rooted-tree space: two corrections apply two unary graftings to the one-directional order-five predictor defect. Hence one scalar chain coefficient controls every nonlinear principal-error condition. Rational nodes and an isolated algebraic correction parameter cancel this coefficient; coprimality with the order-eight chain polynomial proves classical order exactly seven. A complementary design, Accuracy-P40, retains generic sixth order but reduces the complete principal-error norm to $9.8\%$ of the LGL–L3 value while satisfying $J_{\mathrm{stiff}}<0.40$. We also distinguish convergence of repeated corrections from absolute stability after a fixed number of sweeps: two corrections have finite negative-real stability intervals, whereas a third correction restores far-stiff output damping for the new designs. High-precision nonlinear order tests verify sixth versus seventh order, and Allen–Cahn and Cahn–Hilliard calculations show that Accuracy-P40 reduces correction and Krylov work.

10. 打乱数字网的分离性质及相关随机点集

Separation properties of scrambled digital nets and related random point sets

作者: Kosuke Suzuki

类别: math.NA

发布日期: 2026-07-31

arXiv ID: 2607.29063v1

链接: http://arxiv.org/abs/2607.29063v1

摘要: We study how standard randomization procedures affect the local geometry of quasi-Monte Carlo point sets, as measured by their minimum distance and mesh ratio. Although probabilistic selection within structured lattice families can produce quasi-uniform point sets, randomizing an existing low-discrepancy construction need not preserve quasi-uniformity. We first determine sharp probabilistic orders for Monte Carlo, jittered, and Latin hypercube sampling, whose mesh ratios diverge as positive powers of $N$. The orders are $Θ_{\mathbb{P}}(N^{1/d}(\log N)^{1/d})$ for Monte Carlo sampling, $Θ_{\mathbb{P}}(N^{1/(d+1)})$ for jittered sampling, and, for $d\ge 2$, $Θ_{\mathbb{P}}(N^{1/d}(\log N)^{1/d})$ for Latin hypercube sampling. We also obtain a Weibull limit law for the minimum distance of jittered samples. For full Owen scrambling, every family of fixed-$t$ nets has minimum distance $O_{\mathbb{P}}(N^{-3/(2d)})$, and its mesh ratio is therefore $Ω_{\mathbb{P}}(N^{1/(2d)})$. Under uniform coincidence and common-prefix conditions, these bounds are sharp up to logarithmic factors. Moreover, a single full Owen scrambling of any $(t,d)$-sequence is almost surely non-quasi-uniform. By contrast, for matrix and linear scrambling of binary digital nets with fixed $t$ in dimension $d\ge 2$, the mesh ratio is $O_{\mathbb{P}}(\log N)$, whereas it is $Θ_{\mathbb{P}}(\log N)$ in the separate balanced-prefix affine-tail model. The model also yields the exact probabilistic order for one-dimensional binary digital $(0,m,1)$-nets under matrix or linear scrambling. These results demonstrate that the geometric effect of randomization is governed by whether it introduces local independence or shared algebraic randomness.

11. 从逐点不变量稳定恢复矩阵规范类

Stable Recovery of Matrix Gauge Classes from Pointwise Invariants

作者: Dexuan Zhou, Huajie Chen, Bernie Hsu, Christoph Ortner

类别: math.NA

发布日期: 2026-07-31

arXiv ID: 2607.29021v1

链接: http://arxiv.org/abs/2607.29021v1

摘要: A parameterized matrix family $x\mapsto H(x)$ on a configuration domain is determined by its physical content only up to a constant orthogonal change of basis. This gauge ambiguity is intrinsic to data-driven Hamiltonian models, such as tight-binding parameterizations, reduced-order electronic structure methods, or excited-state models. It raises a basic inverse problem: what observations of $H(x)$ suffice to identify the family up to this gauge? The pointwise spectrum is incomplete already for linear families on $\mathbb{R}$. Here, we prove that, under natural non-degeneracy and connectivity assumptions, augmenting the spectrum with loop products of the coupling matrices in the instantaneous eigenframe yields a complete invariant and that inversion is stable. We support the theory with numerical experiments.

12. 一种用于线性算子方程模型降阶的伪时间数据驱动框架

A Pseudo-time Data-Driven Framework for Model Reduction of Linear Operator Equations

作者: Zhentong Wei, Tingen Xiong, Wenlong Zhang, Zhiwen Zhang

类别: math.NA

发布日期: 2026-07-31

arXiv ID: 2607.28972v1

链接: http://arxiv.org/abs/2607.28972v1

摘要: This paper proposes a novel Pseudo-time Proper Orthogonal Decomposition (POD) framework to enable model reduction for stationary problems lacking temporal snapshot data. By recasting static operator equations into a pseudo-dynamic evolution form, we artificially generate temporal data while preserving the system’s intrinsic spectral properties. Mathematically, we rigorously prove the exponential convergence of the pseudo-time trajectory to the exact stationary solution. Furthermore, the approximation properties of the generated POD basis functions are rigorously established. Finally, the universality and accuracy of the framework are validated both theoretically and numerically across two representative settings: elliptic inverse source problems and Fredholm integral equations of the first kind.

13. 基于SPH(2)的广义垂向坐标变换用于高效自由表面流模拟

A generalized vertical coordinate transformation based on SPH(2) for efficient free surface flow simulations

作者: Shujiro Fujioka, Kumpei Tsuji, Naoto Mitsume, Mitsuteru Asai

类别: math.NA

发布日期: 2026-07-30

arXiv ID: 2607.28872v1

链接: http://arxiv.org/abs/2607.28872v1

摘要: We propose three new particle methods that improve computational efficiency by introducing a generalized Vertical Coordinate Transformation (VCT) for free surface flow problems with complex bottom boundaries. The first method is a bottom boundary-fitted particle method (BF-SPH). The BF-SPH is simply an arrangement of the body-fitted-coordinate system in the finite difference method to the particle method. The BF-SPH can accurately impose the bottom boundary conditions, while a simple procedure is performed by transforming the complex bottom into a flat one. The second method is the bottom boundary-fitted ellipsoidal particle method (BFE-SPH), which combines the BF-SPH with the ellipsoidal particle model proposed by Shibata et al. The BFE-SPH can speed up the particle simulation by choosing a reasonable aspect ratio of ellipsoidal particles. The last method is the $σ$-SPH method, which automatically selects the aspect ratios of ellipsoidal particles concerning water depth using the $σ$-coordinate system. The $σ$-coordinate is often employed in numerical simulations of oceanographic fields, such as in the Princeton Ocean Model. However, this is the first attempt to apply the $σ$-coordinate to a particle method. Vertical resolution is required from offshore to the coastal region in oceanographic problems such as tsunamis, especially when conducting detailed analysis using a 3-D particle method. Using the $σ$-coordinate allows for a stepwise transition to a naturally efficient coordinate system by referencing water depth. In this paper, we have shown that the above three methods can be generalized as Vertical Coordinate Transformations (VCTs), and the VCTs are successfully achieved by employing SPH(2) with the second-order accuracy of the second-order derivatives, including cross derivatives.

14. 切割单元有限体积算子中的缺陷子空间与局部不稳定性

Defect Subspaces and Localized Instabilities in Cut-Cell Finite-Volume Operators

作者: Justo E. Karell

类别: math.NA

发布日期: 2026-07-30

arXiv ID: 2607.28808v1

链接: http://arxiv.org/abs/2607.28808v1

摘要: Cut-cell instability in explicit finite-volume methods arises when an embedded boundary clips a background cell to a small volume fraction~$α$, forcing the local CFL number to $λ_α= λ/α\gg 1$. This paper characterizes the spectral structure of this instability and derives a minimum SRD blending parameter per cut cell computable from local mesh geometry. The update operator has exactly $m$ unstable eigenvalues, one per cut cell, with dominant eigenvalue $μ_u = 1 - λ/α+ O(1)$ and eigenvector $v_u = e_c + O(α)$. The growth rate is governed by the defect matrix $Γ= (Δt/h)(a_{c+1/2,c} - a_{c-1/2,c})$, yielding the criterion [ s_{\mathrm{stab}} = \frac{λ_α- 2}{λ_α- 1} + O(α), ] which is evaluated from $α$ and $λ$ at each cut cell at mesh generation time with no time-stepping and no global matrix assembly. The criterion prevents the class of crash that would otherwise require rerunning a production simulation at significant cost. For merge-left state redistribution, $|\cos_α(D,v_u)| = 1 - O(α)$, proving SRD acts in the unstable eigendirection. We also characterize which linearized explicit finite-volume operators on scalar periodic 1D cut-cell meshes admit this structure, covering schemes from first-order Godunov to high-order MUSCL with bounded flux coefficients, and prove $\|P_{\cU_α} - P_\cC\| = O(α)$ for $m$ non-adjacent cut cells.