生成时间: 2026年07月16日 09:39:33
重点关注论文数量: 1
了解领域论文数量: 4
不相关论文数量: 18
重点关注论文（完整分析）
1. 可扩展广义多尺度有限元空间构造的误差的一阶下界
An Order-One Lower Bound on the Error of Scalable Generalized Multiscale Finite Element Space Constructions
作者: Changqing Ye
类别: math.NA
发布日期: 2026-07-15
arXiv ID: 2607.13888v1
摘要翻译: Several coefficient-adapted methods provide optimal-order approximation for elliptic equations with rough coefficients. Prominent examples include localized orthogonal decomposition, multiscale spectral GFEM, and constraint energy-minimizing GMsFEM. Their proven accuracy, however, is obtained by allowing the localization radius or the local spectral dimension to grow as the coarse scale \\(H\\) tends to zero. Classical MsFEM has an FEM-like local construction, but its available analysis does not give a coefficient-uniform \\(\\bigO(H)\\) energy estimate over the full bounded-contrast measurable coefficient class. Motivated by this gap, we formalize an FEM-like notion of structural scalability. A chosen spatially local basis has uniformly bounded overlap, hence \\(\\bigO(1)\\) stiffness entries per row, and every anchored local span uses coefficient information from only \\(\\bigO(1)\\) coarse-element layers. We prove that no deterministic construction satisfying fixed bounds on the support radius, coefficient-information radius, and local multiplicity can converge uniformly over the coefficient class. In fact, its worst-case \\(L^2\\)-to-energy Galerkin error remains bounded below by a positive constant independent of \\(H\\) . The lower bound is established using a fixed finite family of smooth periodic coefficients and smooth right-hand sides. The proof combines coefficients that coincide on local patches, a finite-dimensional approximation lower bound for corrector fields, a positive-density mesh argument, and strong periodic corrector convergence. Thus uniform optimal accuracy requires at least one local construction parameter to grow or requires coefficient information beyond the fixed-visibility model.
Comment: 无
链接: http://arxiv.org/abs/2607.13888v1
1. 研究对象和背景
本文研究具有粗糙系数椭圆方程的数值方法。具体考虑 Dirichlet 问题
$$ -\nabla\cdot(\kappa\nabla u) = f \quad \text{在 } \Omega,\qquad u = 0 \quad \text{在 } \partial\Omega, $$其中 $\kappa$ 是可测的且满足 $1\le\kappa\le\rho$，$\rho>1$ 固定。由于系数不规则，标准有限元方法无法得到关于系数一致的 $H^2$ 正则性估计，因此需要构造依赖于系数的多尺度试验空间来恢复最优能量逼近。目前已有多种方法可以达到最优阶精度，如局部正交分解 (LOD)、超局部正交分解 (SLOD)、约束能量极小化广义多尺度有限元 (CEM-GMsFEM) 以及多尺度谱广义有限元 (MS-GFEM)。然而，这些方法的精度证明均要求局部化半径或局部谱维数随粗网格尺度 $H$ 趋于零而增长：LOD 需要 $|\log H|$ 层粗单元，SLOD 的无条件分析至少退化到 LOD 的局部化率，CEM-GMsFEM 的过采样深度为 $\log(\max\kappa/H)$，MS-GFEM 则需 $|\log H|^d$ 的局部谱维数。与之相对，标准低阶有限元基函数具有固定的局部支集和重叠数，每行刚度矩阵仅有 $O(1)$ 个非零元，且每个基函数仅由相邻的 $O(1)$ 个粗单元确定。基于这一对比，本文引入“结构可扩展性”这一 FEM-like 概念：选定的空间局部基具有一致有界的支集半径和局部重数（从而每行刚度矩阵有 $O(1)$ 个非零元），且每个锚定局部空间仅依赖于固定层数粗单元内的系数信息（称为固定可见性）。核心问题是：能否在局部化半径、局部维数和系数信息半径均保持有界的情况下，在整个有界对比度可测系数类上实现一致收敛？本文从反面回答了该问题：任何满足此类固定可见性参数的确定性构造，在最坏情况下的能量误差具有一个与 $H$ 无关的正常数下界，从而无法一致收敛。这一结果填补了既有多尺度方法分析中的理论空白，即证明了若要达到一致最优精度，必须至少有一个局部构造参数随 $H$ 增长，或使用超出固定可见性模型的全局系数信息。
2. 主要定理或主要结果
本文的主要结果是定理 2.3，它以精确的数学语言陈述了在固定可见性条件下的一致误差下界。
设 $d\ge 2$，$\Omega\subset\mathbb{R}^d$ 为有界连通多面体 Lipschitz 区域，固定 $\rho>1$ 以及 $m,k\in\mathbb{N}_0$，$C_{\dim},C_{\mathrm{loc}}>0$。令 $\{\mathcal{T}_H\}_{H\in\mathcal{H}}$ 为满足形状正则和拟一致条件 (2.2) 的网格族。则存在常数 $c_*>0$ 和 $H_*>0$，使得对任意满足 $0 其中 $E_H$ 为由 (2.4) 定义的 $L^2$-到-能量误差。因此， 常数 $c_*$ 和 $H_*$ 仅依赖于 $d,\rho,m,k,C_{\mathrm{loc}},\Omega$ 以及网格正则性参数 $\gamma_{\mathrm{sh}},c_{\mathrm{qu}}$，与 $C_{\dim}$、$H$、具体规则及网格族无关。 定理 7.1 给出了更强的有限族形式：存在有限个光滑周期系数剖面和光滑紧支集右端项构成的固定集合，使得对每个充分小的 $H$ 和每个允许的规则，该集合中至少有一对给出上述下界。下界在 (2.4) 的归一化下是 $\Theta(1)$ 的，因为任何协调试验空间总有平凡上界 $E_H\le C_P$（$C_P$ 为 Poincaré 常数）。因此，定理 2.3 揭示了在固定局部复杂度下，最坏情形的误差完全不趋于零。 证明的核心思想是将固定可见性转化为局部近似空间的维数限制，再构造一族在公共区域内但具有足够多线性独立梯度的系数，最后通过网格正密度论证和精确解逼近得到全域误差下界。 1. 从固定可见性到公共局部空间 (Lemma 3.1)
对任一元素 $K$，定义 $Y_K^\kappa$ 为 $V_H^\kappa$ 在 $K$ 上限制的商空间（模常数）。利用基函数的支集条件：只有锚定元素与 $K$ 的图距离不超过 $m$ 的基函数才能在 $K$ 上非零，而每个锚定元素最多有 $\lfloor C_{\mathrm{loc}}\rfloor$ 个基函数，因此 $\dim Y_K^\kappa\le q$，其中 $q = b_{d,m}\lfloor C_{\mathrm{loc}}\rfloor$，$b_{d,m}=\sum_{j=0}^m (d+1)^j$ 是距离 $m$ 内粗单元个数的通用上界。此外，若两个系数在 $\omega_R(K)$ 上几乎处处相等（$R=2m+k$），则它们在该元素上的限制空间相同，因为可见性条件 (2.3) 保证了所有相关锚定子空间不变。 2. 局部分解中线性无关场的构造 (Lemma 4.2, 第 4 节)
在参考周期单元 $Y=(-1/2,1/2)^d$ 中，选取同心立方体 $D_-\Subset D_0\Subset D\Subset Y$，并构造 $q+1$ 个向量场 $e_1,\nabla v_1,\dots,\nabla v_q$，它们在 $D$ 的任意非空开子集上线性无关。具体作法：利用周期 Green 函数 $G$ 的奇性，选取 $Y\setminus D$ 中的点 $z_1,\dots,z_q$，定义 $p_j = \partial_{y_1}G(y-z_j)$，则 $\nabla p_j$ 在 $D$ 内调和且整体线性无关。通过磨光技术将点源替换为支集在 $Y\setminus D$ 内的光滑非负紧支集函数 $b_j$，解周期 Laplace 方程 $(4.3)$ 得到 $v_j$，当磨光参数足够小时，$\nabla v_j$ 在 $L^2(D)$ 中任意接近 $\nabla p_j$，从而保持线性无关性。 3. 构造具有相同局部信息的有限系数族 (第 4 节后续，部分细节未显示)
基于上述 $v_j$，构造一族光滑周期系数 $a_{j,\delta}$，使得它们在 $D$ 上恒等于 $1$（即具有相同的局部系数信息），但相应的单元校正子 $\chi_{a_{j,\delta},1}$ 的梯度在 $D$ 内逼近 $e_1+\nabla v_j$（加上一个小扰动）。通过适当选取参数，这族校正子梯度在 $D$ 的任意开子集上张成 $q+1$ 维空间，从而保证了任何 $q$ 维局部空间都无法同时逼近它们。 4. 网格正密度与局部逼近下界 (引理 4.4 和网格论证)
利用 $D_-\Subset D_0\Subset D$ 且网格拟一致，可以证明：对充分小的 $H$，$\mathcal{T}_H$ 中满足 $K\cap D_-\ne\emptyset$ 且 $\omega_R(K)\subset D$ 的元素 $K$ 至少有 $cH^{-d}$ 个（$c>0$ 依赖于几何）。对于每个这样的元素，限制空间 $Y_K^\kappa$ 的维数 $\le q$，而上述校正子梯度提供 $q+1$ 个线性独立的方向，因此由有限维逼近下界（引理 4.4），存在某个 $j$ 和相应的校正子场，使得该场在 $K$ 上的 $L^2$ 投影到 $Y_K^{a_j}$ 的误差不小于一个统一的正常数。 5. 将周期校正子实现为全域光滑右端项的精确解 (第 6 节)
为了将单元校正子场转换为方程 (1.1) 的真实解，利用紧支集光滑右端项逼近：构造 $f_\varepsilon$，使其对应的解 $u_\varepsilon$ 在 $H^1(\Omega)$ 中强收敛到某个带有适当边界条件的“整体化”校正子。该右端项可以与周期系数一起缩放至物理域，从而对于每一个 $H$，得到相应系数 $\kappa_H$ 和右端项 $f_H$，使得精确解梯度在包含正密度个粗单元区域内表现得接近所需的校正子梯度。将所有贡献累加，并利用 Galerkin 正交性，最终导出全局能量误差的下界 $c_* > 0$。 本文的结果与已有数值均匀化文献中的多尺度方法存在本质区别，主要体现在问题设定、结果性质和理论意义三个层面。 1. 与正向构造性方法的比较
LOD、SLOD、CEM-GMsFEM 和 MS-GFEM 等方法均成功构造了依赖于系数的基函数，并在各自的假设下证明了最优阶 $O(H)$ 能量误差。然而，这些方法的精度证明均依赖于一个随 $H$ 增长的局部参数：LOD 局部化层数为 $O(|\log H|)$，CEM-GMsFEM 过采样深度为 $O(\log(\max\kappa/H))$，而谱 GFEM 的局部谱维数需达到 $O(|\log H|^d)$。本文的结果表明，这种增长不是当前证明技术的偶然产物，而是在固定局部复杂度的 FEM-like 框架中不可避免的。换言之，任何试图将局部半径、局部维数和系数信息半径同时保持有界的确定性构造，必然会在整个有界对比度系数类上出现 $O(1)$ 的误差下界。这为上述方法为何需要增长局部参数提供了理论解释。 2. 与经典 MsFEM 的比较
经典多尺度有限元（MsFEM）具有天然的 FEM-like 局部结构：基函数在单个粗单元上求解局部问题，且仅使用粗单元上的系数信息。其分析在周期情形下给出 $O((\varepsilon/H)^{1/2}+H)$ 的误差，但得不到在整个 $\mathcal{K}_\rho$ 上一致的 $O(H)$ 估计。本文从反面证实：只要坚持这种固定可见性的局部构造，最坏情况的误差不仅不能达到 $O(H)$，甚至根本不会趋于零。这从理论上划清了“严格局部构造”所能达到的精度边界。 3. 与前人下界结果的比较
在数值均匀化领域，已有的下界结果通常聚焦于特定方法的最优局部化半径或谱维数必要性，例如 LOD 的局部化半径必须至少为 $O(|\log H|)$ 等。本文的结果更一般：它不针对特定方法，而是对所有满足固定可见性（即局部支持半径、局部重数和系数信息半径均有界）的确定性空间构造统一适用。证明采用了信息论的框架，构造了一族在局部完全无法区分的系数，它们却能产生任意多个线性独立的局部特征场，从而使任何维数固定的局部空间无法一致逼近。这是一种“不可能性”结果，类似于机器学习中的“没有免费午餐”定理。 4. 未解决的问题与讨论
定理 2.3 针对的是确定性固定可见性规则在 minimax 意义下的下界，即 $\inf_{\mathcal{M}}\sup_{\kappa} E_H$。作者明确指出，若允许构造规则“先获知整个系数再设计局部基”（即 $\sup_{\kappa}\inf_{V_H}$），该问题目前尚无答案。换言之，是否可以通过全局系数信息来设计一个具有固定局部支集的基并达到一致收敛，是一个重要的开放性难题。本文的结果仅限定了在构造必须仅基于局部系数信息时的必然失败，并未否定利用全局信息优化局部基的可能性。此外，随机规则或容忍度依赖的构造也不在本文的覆盖范围内。 Approximation of solutions of parameter-dependent problems by residual neural networks 作者: Ana Carpio 类别: math.NA, cs.LG, math.AP 发布日期: 2026-07-15 arXiv ID: 2607.13574v1 链接: http://arxiv.org/abs/2607.13574v1 中文标题: 基于残差神经网络的参数依赖问题解逼近 摘要翻译: 我们基于梯度流发展了一种收敛的格式来训练具有解析激活函数的神经网络。收敛性由 Lojasiewicz 理论保证。该方法的主要优势在于实现简单。网络的系数通过求解常微分方程组来逼近。我们通过构造参数问题解的残差神经网络逼近来测试该方法。简单常微分方程的解对少量参数的依赖性被正确复现。涉及少量参数依赖的波约束反问题的解可以被合理逼近，甚至在该问题严重不适定的区域中也能做到。 Spectral-Informed Neural Networks Outperform Spectral Methods in High-dimensional PDEs 作者: Tianchi Yu, Ivan Oseledets 类别: math.NA, cs.AI, cs.CE, cs.LG 发布日期: 2026-07-15 arXiv ID: 2607.13566v1 链接: http://arxiv.org/abs/2607.13566v1 中文标题: 谱信息神经网络在高维偏微分方程中优于谱方法 摘要翻译: 对于低维问题 ($d\leq3$)，谱方法可以达到极高的精度。对于中等维度问题 ($4 \leq d \lesssim 10$)，通过稀疏网格或双曲交叉等特定技术，谱方法仍然可行。然而，对于高维问题 ($d\gg 10$)，谱方法受到维数灾难的困扰。物理信息神经网络 (PINNs) 已成为克服这一挑战的有前途的方法，可扩展到高维，但通常精度和效率有限。最近提出的谱信息神经网络 (SINNs) 将谱方法与 PINNs 相结合，直接在谱域中操作以避免空间导数计算并减少内存消耗。在这项工作中，我们引入了改进的 SINNs，它结合了由调和分析启发的系数衰减缩放和基嵌入，以提高高维问题的精度，并实现未知谱系数的精确逼近。对稳态和依赖时间的偏微分方程的数值实验表明，改进的 SINNs 在谱信息不完整的中等维问题上优于稀疏网格谱方法，并且在高维问题上比 PINNs 具有更高的精度。 A linear, fully decoupled, and unconditionally energy-stable SAV-FEM for the Cahn–Hilliard–Navier–Stokes model 作者: Jinting Yang, Nianyu Yi, Peimeng Yin 类别: math.NA 发布日期: 2026-07-14 arXiv ID: 2607.13253v1 链接: http://arxiv.org/abs/2607.13253v1 中文标题: 一种用于Cahn–Hilliard–Navier–Stokes模型的线性、完全解耦且无条件能量稳定的SAV有限元方法 摘要翻译: 本文中，我们采用标量辅助变量(SAV)方法，为Cahn–Hilliard–Navier–Stokes (CHNS)系统构造了一种线性、完全解耦且无条件能量稳定的全离散有限元格式。与现有基于SAV的公式通常引入多个辅助变量或附加技术来处理不同非线性项不同，我们仅引入一个标量辅助变量并采用新颖的辅助变量更新方式，重新表示由Cahn–Hilliard和Navier–Stokes方程产生的所有非线性项，得到了原CHNS系统的等价形式。时间离散采用隐式-显式(IMEX) Euler格式，其中线性项隐式处理，非线性项显式处理，空间离散则采用有限元方法。得到的全离散格式可以高效地分解为两个线性子问题和一个标量二次代数方程，这极大地简化了实现过程。此外，我们证明所提格式满足无条件离散能量耗散律，并建立了其关于若干相关范数的稳定性。还推导了全离散有限元逼近的最优阶$L^2$误差估计。最后，给出了一系列数值实验以验证理论结果并展示所提方法的效率。 Gaussian FSBP operators: Comparison and application to numerical methods for hyperbolic conservation laws 作者: Jan Glaubitz, Henry Haase, Philipp Öffner, Finnja Stafforst 类别: math.NA 发布日期: 2026-07-14 arXiv ID: 2607.13224v1 链接: http://arxiv.org/abs/2607.13224v1 中文标题: 高斯 FSBP 算子：比较及其在双曲守恒律数值方法中的应用 摘要翻译: 函数空间分部求和（FSBP）算子使得基于一般非多项式逼近空间的双曲守恒律的守恒和能量稳定数值方法成为可能。近期工作表明，与主要关注等距网格的现有构造相比，使用广义高斯求积法能显著减少所需的网格点数。在本文中，我们比较了用广义高斯求积法构造的开型和闭型 FSBP 算子，并将其应用于数值求解双曲守恒律。此外，为了支持开节点分布，我们扩展了 FSBP 框架，引入了函数空间精确外推算子，并将其应用于求解双曲守恒律的数值格式中。我们的数值实验包括一维线性平流、无黏 Burgers 方程和气体动力学的可压缩 Euler 方程。我们观察到，在数值格式中应用 FSBP 算子能够提高效率和精度。值得注意的是，与其他关于高斯 FSBP 算子的近期工作相比，我们在更具挑战性的时间依赖情形中展示了这些优势。 Adaptivity in Local Kernel Based Methods for Approximating Solutions to the Poisson Equation 作者: Jonah A. Reeger, Anders R. Johnson, Shelby W. Woodrum 类别: math.NA 发布日期: 2026-07-15 arXiv ID: 2607.14057v1 链接: http://arxiv.org/abs/2607.14057v1 摘要: Expanding on the recent development of adaptive local kernel methods for approximating the action of linear operators, a local estimate of the error and an adaptive procedure for approximating solutions to the Poisson equation is developed. The error estimate is used in the midst of the adaptive procedure to determine locations in the solution domain where decreasing the spacing between nodes can decrease the error in the solution. The approach described here is essentially “meshless”, with only local Delaunay triangulations leveraged as a convenience to provide information on the local geometry of the node set. Once this information is utilized, it is discarded. The experiments performed show close agreement between the error estimate and actual absolute forward error in the approximate solution along with a comparison to existing indicators for refinement. The combination of the adaptive procedure and error estimate provides an automated technique to resolve localized features of a solution without the, often intractable, expense of uniform refinement across the entire solution domain. Revisiting column subset selection through the lens of submodularity 作者: Ilse C. F. Ipsen, Arvind K. Saibaba 类别: math.NA 发布日期: 2026-07-15 arXiv ID: 2607.13823v1 链接: http://arxiv.org/abs/2607.13823v1 摘要: The problem is to select k columns with maximal volume from a real mxn matrix X. We show that the logarithm of the volume is a submodular function on columns of X, and for full column-rank matrices X with sufficiently large singular values, it is a non-negative non-decreasing function. As a consequence, traditional Businger-Golub QR with column pivoting is a greedy algorithm, with a relative error of at most 37 percent. In contrast, Gu-Eisenstat strong rank-revealing QR is a 1-interchange algorithm, with a relative error of at most 50 percent. The higher accuracy, under this metric, of the simple QR with column pivoting confirms its well known effectiveness in practice. For general matrices, we show that Businger-Golub QR is a greedy algorithm, and Gu-Eisenstat QR a 1-interchange algorithm for maximizing the trace of the upper triangular matrix in a QR factorization of X, with Businger-Golub again having a better error bound. This is extended to finding kxk submatrices of maximal volume in symmetric positive-definite matrices. Microstructure-Conditioned Surrogate Models for Graded Multiscale Optimization of Mycelium Composites 作者: J. Storm, I. B. C. M. Rocha, S. Schyck, K. Masania, F. P. van der Meer 类别: cs.LG, cond-mat.mtrl-sci, math.NA, physics.comp-ph 发布日期: 2026-07-15 arXiv ID: 2607.13688v1 链接: http://arxiv.org/abs/2607.13688v1 摘要: Emerging sustainable materials increasingly rely on engineered hierarchy and microstructure to achieve control of their properties and mechanical behavior. Optimizing these materials with controllable microstructures requires efficient multiscale simulations. Data-driven surrogate models for the microscale can accelerate multiscale simulations, but require large amounts of data even for a fixed microstructure. When a range of microstructures is considered, as is the case in multiscale optimization, even more data is needed to train a surrogate. To overcome this challenge, we condition a hybrid physics-data surrogate on microstructural variables using a hypernetwork. This approach enables accurate predictions of multiscale mechanical behavior for a mycelium-woodchip composite material, even when trained on small datasets. The conditioned surrogate makes multiscale simulations of functionally graded structures tractable, and we validate it against a full FE^2 simulation. We optimize a graded multiscale disk, and reduce the peak stress by 42% compared to one with a random microstructure. Then, we go one step further, conditioning the network directly on manufacturing variables that can have a complex influence on the microstructure. This is a practical route to engineer the microscale for desired macroscale behavior. This contribution highlights the benefits of microarchitectured structures and demonstrates how conditioned surrogate models enable their multiscale optimization, which will accelerate the development and design of future sustainable materials and structures. Error Analysis of a Fully-Discrete Implicit $θ$-Scheme with WG-FEM for Parabolic Singularly Perturbed Boundary Turning Point Problems 作者: Aayushman Raina, Srinivasan Natesan 类别: math.NA 发布日期: 2026-07-15 arXiv ID: 2607.13611v1 链接: http://arxiv.org/abs/2607.13611v1 摘要: In this article, we introduce a weak Galerkin finite element method (WG-FEM) for a class of parabolic singularly perturbed boundary turning point problems (SPBTPPs). The proposed numerical scheme employs an implicit $θ$-scheme for temporal discretization over a uniform mesh and applies WG-FEM spatial discretization on a layer-adapted Shishkin mesh. Rigorous stability estimates are established for both the semi-discrete and fully-discrete formulations. Furthermore, we derive error estimates in the energy norm and prove that the convergence of the scheme is uniform with respect to the perturbation parameter. Numerical tests are conducted to verify the theoretical findings and illustrate the efficiency of the proposed method. In addition, the theoretical framework developed in this work lays the foundation for future extensions to higher-dimensional problems using ADI-type operator splitting WG-FEM schemes. Filon Methods for Highly Oscillatory Controlled Quantum Systems 作者: Spencer Lee, Daniel Appelö 类别: quant-ph, math.NA 发布日期: 2026-07-15 arXiv ID: 2607.13580v1 链接: http://arxiv.org/abs/2607.13580v1 摘要: Fast and accurate classical simulation of quantum systems is a central challenge in the design and control of quantum computers, but the highly oscillatory dynamics of these systems severely limit the efficiency of standard numerical methods. To address this, we adapt Filon quadrature for oscillatory integrals into two numerical methods, called Filon and Controlled Filon, for solving linear systems of ODEs with highly oscillatory solutions. We tailor both methods for efficient implementation in controlled quantum systems, and the Controlled Filon method additionally accounts for the oscillatory structure of the control pulses. We show by numerical experiments that these methods significantly reduce the computational cost of accurately simulating systems of superconducting transmon qubits by decreasing the number of timesteps needed to reach a given level of precision, with only a modest increase in the cost per timestep. For a realistic simulation of the dynamics of a CNOT gate, the Controlled Filon method is the most efficient method tested at every target accuracy, outperforming the best Hermite method by up to 6x and the Hermite method of the same order by up to 500x. Adaptive space-time BEM for the heat equation with Neumann boundary conditions 作者: Gregor Gantner, Helmut Harbrecht, Franz Nowakowsky 类别: math.NA 发布日期: 2026-07-15 arXiv ID: 2607.13578v1 链接: http://arxiv.org/abs/2607.13578v1 摘要: We consider the space-time boundary element method (BEM) for the heat equation with prescribed initial and Neumann data. We propose a weighted-residual a posteriori error estimator that is an upper bound for the unknown BEM error. The possibly locally refined meshes are assumed to be parabolically scaled prismatic, i.e., their elements are tensor-products $J\times K$ of elements in time $J$ and space $K$ with $|J| \eqsim \text{diam}(K)^2$. In the considered numerical experiments on two-dimensional domains in space, an adaptive algorithm steered by the derived estimator yields significantly faster convergence compared to uniform refinement, achieving near-optimal rates even in the presence of strong singularities. A Spectrally Damped Tensor Randomized Kaczmarz Method for Doubly Noisy Tensor Systems 作者: Nahyun Lee, Jiyoung Choi 类别: math.NA 发布日期: 2026-07-15 arXiv ID: 2607.13552v1 链接: http://arxiv.org/abs/2607.13552v1 摘要: Tensor randomized Kaczmarz (TRK) methods are efficient row-action solvers for tensor linear systems under the t-product framework. We study their behavior under a doubly noisy perturbation model. In this model, both the system tensor and the right-hand side tensor are corrupted. We first analyze standard TRK and derive an expected error recursion with two terms. One term is contractive, and the other is a persistent perturbation term. This explains the noise-limited and semi-convergent behavior that can occur when the observed tensor system is inconsistent. We then introduce a spectrally damped tensor randomized Kaczmarz method (SD-TRK). We prove an expected error recursion for SD-TRK that separates error propagation from noise injection. The bound makes explicit a speed-robustness trade-off. We also give an FFT-based implementation that applies the damped update slice-wise in the Fourier domain. This implementation allows frequency-dependent damping parameters in practice. Numerical experiments on synthetic tensor systems illustrate the stabilization behavior of SD-TRK relative to standard TRK in noisy and ill-conditioned settings. We also include a two-pass image reconstruction comparison under the same noisy reconstruction pipeline. Computing Strong Rank-Revealing Factorizations for Matrices with Orthonormal Rows 作者: Anil Damle 类别: math.NA 发布日期: 2026-07-15 arXiv ID: 2607.13532v1 链接: http://arxiv.org/abs/2607.13532v1 摘要: We show that a pivoting strategy due to Stewart (based on work by Bischof) computes a strong rank-revealing factorization when applied to a matrix with orthonormal rows. When paired with the classical column selection algorithm of Golub, Klema, and Stewart (GKS) it helps achieve rank-$k$ approximation accuracy bounds and basis conditioning as good as those from applying a strong rank-revealing factorization directly to A. We then extend this framework in two directions: (1) providing analysis of GKS when only approximations of right singular vectors are available and (2) providing a randomized variant of the pivoting strategy for matrices with orthonormal rows that achieves the same theoretical guarantees but can return the desired subset two orders of magnitude faster than the deterministic variant. Reconstruct the ambient noise source from the multi-frequency sparse correlation data 作者: Hao Gu, Hongxia Guo, Xiang Xu, Yue Zhao 类别: math.NA 发布日期: 2026-07-15 arXiv ID: 2607.13509v1 链接: http://arxiv.org/abs/2607.13509v1 摘要: In this paper, we develop a novel multi-frequency factorization method to reconstruct the spatial support of the ambient noise source. The proposed method only requires sparse correlation data and has low computational cost. Numerical experiments in two and three dimensions are presented to demonstrate the effectiveness of the proposed method. Inverse scattering for three-dimensional random obstacles with multi-frequency data 作者: Zhiqi Sun, Yiwen Lin 类别: math.NA 发布日期: 2026-07-15 arXiv ID: 2607.13473v1 链接: http://arxiv.org/abs/2607.13473v1 摘要: In many practical scenarios the shapes of scatterers exhibit uncertain geometric variations arising from diverse physical or environmental factors. For inverse scattering problems which are inherently ill-posed, the presence of such geometric uncertainties may have a non-negligible impact on the recovery process. With the aim of recovering both obstacle geometry and statistics of the shape uncertainties, in this paper we study an inverse acoustic scattering problem for three-dimensional smooth star-shaped obstacles with random isotropic fluctuations. We propose an efficient Monte Carlo-based multi-frequency recursive linearization algorithm in which the far-field operator is linearized with respect to the geometry parameters and frequency continuation is employed to recover the unknown geometry from coarse to fine scales. Based on the reconstructed samples, we further estimate the reference geometry and key statistics of the shape fluctuation field including Karhunen–Loève eigenvalues, covariance hyper-parameters for Gaussian perturbations and covariance structure, representative marginal distributions for non-Gaussian perturbations. We also prove that the probability law of the far-field data uniquely determines the radial function in distribution which implies uniqueness of the reference shape and related statistics. Numerical experiments demonstrate the effectiveness of the proposed method in recovering both the scatterer shapes and the associated statistical information under Gaussian and non-Gaussian random variations. Residual-Christoffel Sampling for Random Feature Collocation of Linear PDEs 作者: Jiale Linghu, Yangshuai Wang 类别: math.NA, physics.comp-ph 发布日期: 2026-07-15 arXiv ID: 2607.13382v1 链接: http://arxiv.org/abs/2607.13382v1 摘要: Random feature collocation fixes a randomly generated trial space and determines its coefficients from a linear least-squares system. Stability then depends on whether the sampled residual equations represent the geometry induced by the differential operator. We construct an operator-aware discretization in which the operator-applied features determine both the collocation measure and a coefficient whitening map. The randomized scheme combines a residual-Christoffel density with inverse-density weights, while a deterministic scalar-row alternative maximizes successive regularized log-determinant increments. Conditional on the realized trial space, the sampled whitened interior Gram is a spectral approximation to the reference Gram on the retained residual space, with sample complexity linear in the retained dimension up to a logarithmic factor. For uniformly analytic residual kernels, the associated operator has stretched-exponentially decaying eigenvalues and ridge effective dimension that is polylogarithmic in the inverse ridge scale. Experiments on scalar and vector equations, varied geometries, and one to three spatial dimensions show that residual-space sampling and whitening produce numerically full-rank transformed systems with substantially smaller condition numbers and iteration counts. The deterministic construction attains the lowest errors at the smallest scalar sample sizes. Residual-space geometry therefore yields a principled design for stable strong-form random feature collocation. Hybrid GKB Methods for X-Ray Tomography Problems with Unmatched Back Projector 作者: Abdulmajeed Alsubhi, Rosemary Renaut 类别: math.NA 发布日期: 2026-07-15 arXiv ID: 2607.13381v1 链接: http://arxiv.org/abs/2607.13381v1 摘要: X ray Computed Tomography (CT) is a widely used imaging modality in medical, industrial, and scientific applications. In practical CT implementations, forward, A, and back projection, B, operators are often constructed using different discretization schemes to improve computational efficiency on available software and hardware platforms. Consequently, the resulting projector pair is generally unmatched, meaning that the back projector is not the exact adjoint of the forward projector. This mismatch alters the mathematical properties of the reconstruction problem and can affect the convergence behavior of iterative reconstruction methods. Previous studies have proposed AB and BA Golub Kahan bidiagonalization (GKB) methods, as well as GMRES and hybrid GMRES methods, for solving CT reconstruction problems with unmatched projector pairs, demonstrating reduced semiconvergence effects compared with conventional approaches. In this work, we develop hybrid variants of the AB- and BA-GKB algorithms by incorporating regularization within the projection process. The singular value decomposition for the operator on the projected space is used to efficiently reconstruct the solution, and to automatically select the regularization parameter using either the L-curve or generalized cross-validation. Numerical experiments on several CT reconstruction problems demonstrate the effectiveness of the proposed hybrid methods in improving robustness against semiconvergence. Data driven non-equilibrium moist phase exchanges for atmospheric convection within a discontinuous Galerkin model of the compressible Euler equations 作者: David Lee, Kieran Ricardo, Junwei Lyu 类别: math.NA, physics.flu-dyn 发布日期: 2026-07-15 arXiv ID: 2607.13360v1 链接: http://arxiv.org/abs/2607.13360v1 摘要: A neural network is trained to learn the mass exchanges between vapour, liquid and ice phases in atmospheric convection. The network is trained on convection resolving output from a regional configuration of the LFRic model with a multi-moment microphysics parameterisation (CASIM). The loss function for learning these phase exchanges is formulated under the assumptions of thermal and mechanical equilibrium (same temperature and pressure for all phases), and mechanical dis-equilibrium (different Gibbs free energies for all phases). The network outputs determine the exchanges of vapour, liquid and ice so as to conserve mass, and the resulting change in entropy is determined from the network outputs so as to conserve energy. The neural network is implemented in a thermodynamically consistent manner within a 2D vertical slice discontinuous Galerkin model of a moist, non-hydrostatic atmosphere in order to simulate the formation of three-phase clouds for convection at sub-km resolution. The results are compared to those from a physics based representation of three-phase moist processes at thermodynamic equilibrium. A Weighted Integral-Regularized Finite Difference Scheme for the Tempered Fractional Laplacian 作者: Mingyi Wang, Lisen Ding, Dongling Wang 类别: math.NA 发布日期: 2026-07-14 arXiv ID: 2607.13317v1 链接: http://arxiv.org/abs/2607.13317v1 摘要: The intrinsic singularity of the tempered fractional Laplacian (TFL) remains a major challenge in developing numerical methods that are simultaneously accurate, efficient, and easy to implement. We develop a weighted integral-regularized finite difference (WIRFD) method that regularizes the singular integrand via a multidimensional Taylor expansion incorporating a smooth window function. The resulting integral is decomposed into a regularized term, which is discretized by a punctured trapezoidal rule, and a directly evaluated correction term. For the multidimensional TFL operator, we derive an $O(h^{4-α})$ truncation error bound in the $l^{\infty}$-norm for $α\in(0,2)$ and $u\in C^s(\mathbb{R}^d)$ with $s\geq 8$ by introducing a smooth auxiliary function together with the aliasing formula. For the one-dimensional TFL equation, we establish stability in both the $l^2$- and $l^{\infty}$-norms and optimal $O(h^{4-α})$ convergence for $α\in[1,2)$ based on the strict diagonal dominance of the discrete matrix and a lower bound for its minimum eigenvalue. The Toeplitz structure of the discrete matrix enables FFT-based matrix-vector multiplication, and the resulting linear systems are solved efficiently by a preconditioned conjugate gradient (PCG) method. Numerical experiments corroborate the theoretical results, demonstrating the accuracy, efficiency, and robustness of the proposed method. Duality Framework for Flux Constrained Flow in Porous Media: Analysis and Numerics 作者: Harbir Antil, Keegan L. A. Kirk, Felipe Pérez 类别: math.NA, math.AP, math.OC 发布日期: 2026-07-14 arXiv ID: 2607.13256v1 链接: http://arxiv.org/abs/2607.13256v1 摘要: We introduce and analyze Darcy flow through a saturated porous medium subject to bilateral constraints on the normal flux across a portion of the boundary. The problem is posed as the maximization of a velocity-based dual concave energy over a convex subset of $H(\mathrm{div};Ω)$; Fenchel duality identifies a pressure-based predual formulation, yields strong duality, and provides convex optimality conditions with a complementarity structure on the constrained boundary. The primal–dual gap satisfies an a posteriori error identity, free of generic constants, valid for arbitrary admissible approximations. The duality structure is inherited by a Raviart–Thomas/Crouzeix–Raviart discretization, from which we derive a discrete error identity and a priori error decay rates under fractional regularity assumptions on the solution and the flux bounds. Numerical experiments, including adaptive refinement driven by localized primal–dual gap indicators, support the theory. MCMC Methods for Parameter Inference in Structurally Nonidentifiable Models 作者: Xuyuan Wang, Donglin Han, Michael Y. Li 类别: stat.ME, math.NA 发布日期: 2026-07-14 arXiv ID: 2607.13243v1 链接: http://arxiv.org/abs/2607.13243v1 摘要: We consider the problem of parameter inference for ordinary differential equation (ODE) models with structural non-identifiability. Such models arise in a wide range of scientific fields, including control theory, systems biology, and public health. Structural non-identifiability occurs when distinct parameter values provide identical model outputs, resulting in lower-dimensional manifolds of observationally equivalent solutions in the parameter space. This poses challenges for Bayesian inference and Markov chain Monte Carlo (MCMC) methods, often leading to poor mixing and slow convergence. We develop two MCMC methods that use information from structural identifiability analysis. The first, Identifiability-Aware Geometric MCMC, constructs proposals that move within and between non-identifiable manifolds. The second, Identifiability-Aware Pseudo-Marginal MCMC, performs inference on the space of identifiable parameter combinations and reconstructs full parameter values. We show that both methods target the correct posterior distribution and are ergodic under standard conditions. Numerical examples demonstrate improved sampling efficiency and convergence compared with standard MCMC methods. A Generative Model-Free Form Deformation Approach for the Generation of Mesh Motions with Applications to PDE 作者: Gugliemo Padula, Artem Sinitsa, Gianluigi Rozza 类别: math.NA 发布日期: 2026-07-14 arXiv ID: 2607.13202v1 链接: http://arxiv.org/abs/2607.13202v1 摘要: We introduce a topology-agnostic framework for matching deformations of three-dimensional shapes with non-isomorphic mesh graphs by modelling the deformation as the flow of an Ordinary Differential Equation (ODE). The velocity field is parameterised by a time-dependent Free Form Deformation (FFD), expressed through displacements of a coarse control lattice, yielding a smooth and low-dimensional representation that decouples the deformation model from the discretisation of the source and target surfaces. Under mild regularity assumptions, we prove that the induced ODE map is a universal approximator (in the sup norm) for mappings between genus-0 surfaces, providing a theoretical expressivity guarantee. To further compress the representation and enable probabilistic inference, we couple the ODE–FFD model with a flow-based generative approach in the TarFlow framework, learning a compact latent parametrisation over time series of FFD maps. The resulting method supports efficient sampling and optimisation of plausible deformation trajectories while preserving mesh quality, and it enables scalable reduced-order modelling. Experiments on deforming-body flow benchmarks demonstrate improved accuracy and computational efficiency of reduced-order models constructed from the learned latent dynamics. Tensor-Network Finite Elements for Analytic Operator Equations 作者: Abhijatmedhi Chotrattanapituk, Michael J. Landry, Chu-Liang Fu, Mingda Li 类别: math.NA, cond-mat.other, hep-lat, physics.comp-ph 发布日期: 2026-07-14 arXiv ID: 2607.13129v1 链接: http://arxiv.org/abs/2607.13129v1 摘要: Operator equations (OEs) underpin quantitative modeling across science and engineering. Finite-element (FE) methods discretize continuous OEs into finite-dimensional algebraic systems, whereas tensor networks (TNs) provide flexible variational representations of correlated discrete systems. Here, we develop a framework that connects FE with TN for analytic OEs. The power of this method comes from its ability to convert highly non-linear partial differential equations into linear matrix equations. In particular, we show that FE discretization induces a hierarchy of multilinear interaction tensors, through which differential, integral, nonlinear, memory, and delay equations can be expressed within a common algebraic structure. The resulting systems are reformulated as weighted-residual optimization problems over TN degrees of freedom. Matrix-product-state calculations for one-dimensional linear and nonlinear diffusion reproduce conventional solutions with controlled error while preserving continuity and Neumann boundary conditions. The framework provides a common variational language for analytic OEs and establishes a direct connection between FE numerical formalism and TN variational algorithms, offering a general foundation for TN-based and quantum-inspired approaches to solving OEs.3. 研究方法、关键技术和核心工具
4. 与之前工作的比较
了解领域论文（摘要翻译）
1. 基于残差神经网络的参数依赖问题解逼近
摘要翻译
2. 谱信息神经网络在高维偏微分方程中优于谱方法
摘要翻译
3. 一种用于Cahn–Hilliard–Navier–Stokes模型的线性、完全解耦且无条件能量稳定的SAV有限元方法
摘要翻译
4. 高斯 FSBP 算子：比较及其在双曲守恒律数值方法中的应用
摘要翻译
不相关论文（基本信息）
1. 局部核基方法逼近泊松方程解的自适应性
2. 从次模性视角重新审视列子集选择
3. 用于菌丝体复合材料梯度多尺度优化的微观结构条件化代理模型
4. 全离散隐式 $θ$-格式与 WG-FEM 对于抛物型奇异摄动边界转向点问题的误差分析
5. 高振荡受控量子系统的Filon方法
6. 具有Neumann边界条件的热方程的自适应时空边界元法
7. 用于双噪声张量系统的谱阻尼张量随机化Kaczmarz方法
8. 计算具有标准正交行的矩阵的强秩揭示分解
9. 从多频稀疏相关数据重构环境噪声源
10. 基于多频数据的三维随机障碍物逆散射
11. 用于线性偏微分方程随机特征配置的残差-Christoffel采样
12. 用于反投影器不匹配的X射线层析成像问题的混合GKB方法
13. 可压缩欧拉方程间断伽辽金模型中大气对流的数据驱动非平衡湿相位交换
14. 一个用于调节分数阶拉普拉斯算子的加权积分正则化有限差分格式
15. 多孔介质中通量约束流动的对偶框架：分析与数值方法
16. 用于结构不可识别模型中参数推断的MCMC方法
17. 一种生成式无模型形状变形方法用于网格运动生成及其在偏微分方程中的应用
18. 用于解析算子方程的张量网络有限元方法