生成时间: 2026年09月23日 14:49:08
重点关注论文数量: 0
了解领域论文数量: 10
不相关论文数量: 20
了解领域论文（摘要翻译）
1. 可压缩 MHD 高阶格式的全离散多熵稳定性：一个弱到强框架
Fully Discrete Multi-Entropy Stability of High-Order Schemes for Compressible MHD: A Weak-to-Strong Framework
作者: Huihui Cao, Kailiang Wu, Caiyou Yuan
类别: math.NA
发布日期: 2026-09-22
arXiv ID: 2609.26791v1
链接: http://arxiv.org/abs/2609.26791v1
摘要翻译
中文标题: 可压缩 MHD 高阶格式的全离散多熵稳定性：一个弱到强框架
摘要翻译: 我们针对一般多面体网格上理想可压缩磁流体力学 (MHD) 方程的任意高阶有限体积与间断 Galerkin (DG) 逼近，建立了一个全离散弱到强 (W2S) 多熵稳定性理论，其中单次数值更新可同时满足对任意给定有限族凸 Harten 熵对的离散熵不等式。由于物理熵仅对正密度和正压力有定义，核心分析困难在于协调离散熵稳定性与正性保持、磁场散度约束以及非守恒 Godunov–Powell 耦合。基于 Wu 和 Shu 的可证明保正 MHD 框架，我们发展了有限体积与 DG 方法共同单元平均演化的弱多熵分析。通过凸分解与相对熵，我们建立了这种弱稳定性在一般多面体网格上成立的两个充分准则。相容磁修正与局部无散度逼近协同抵消非守恒 Godunov–Powell 耦合，并提供对正性保持和熵稳定性都至关重要的离散消去。随后，一个 W2S 提升将正性与熵限制统一为单个逐单元缩放限制器，在严格保持单元平均和局部无散度磁场的同时实现强多熵稳定性。高阶时间精度由变步长强稳定性保持多步方法得到。数值实验验证了理论发现并展示了计算鲁棒性。该框架为在高阶 MHD 逼近中统一物理容许性、磁场散度控制与多熵稳定性提供了严格的全离散基础。
2. 随机神经网络求解椭圆界面问题的误差分析
Error Analysis for Solving Elliptic Interface Problems with Randomized Neural Networks
作者: Jun Hu, Sidi Wu
类别: math.NA
发布日期: 2026-09-22
arXiv ID: 2609.26534v1
链接: http://arxiv.org/abs/2609.26534v1
摘要翻译
中文标题: 随机神经网络求解椭圆界面问题的误差分析
摘要翻译: 本文针对椭圆界面问题，提出了随机神经网络 (RaNNs) 的统一误差分析。我们首先利用标准 $\tanh$ 激活函数推导出 Barron 函数的积分表示，由此得到一个 $C^2$-逼近界，该界对任意支撑在立方体上且密度严格为正的隐参数分布均成立。然后，我们针对同时施加偏微分方程、边界和界面条件的复合损失建立了一个定量的 $\mathcal{O}(n^{-1})$ 泛化界，其中 $n$ 表示训练样本数。将这些结果与 RaNNs 的优化保证相结合，我们得到了在界面间断存在时神经网络解的先验 $L^2$ 误差估计。给出了数值实验以验证理论结果。
3. Richards方程混合形式的线性Robin型区域分解格式
Linear Robin-type Domain Decomposition scheme for Mixed Formulation of Richards’ Equation
作者: Åsmund v. B. Synnevåg, Wietse M. Boon, Florin A. Radu, Sarah E. Gasda
类别: math.NA
发布日期: 2026-09-22
arXiv ID: 2609.26478v1
链接: http://arxiv.org/abs/2609.26478v1
摘要翻译
中文标题: Richards方程混合形式的线性Robin型区域分解格式
摘要翻译: 在这项工作中，我们提出了一种求解Richards方程混合形式的线性Robin型区域分解格式（mLRDD-scheme），该方程描述变饱和多孔介质中的流动。假设多孔介质高度非均质，由多种不同材料构成的多个块体/层组成，我们应用非重叠区域分解来分离各个材料，从而在每个子域内获得近似均匀的条件。Richards方程在时间上通过后向Euler离散，并使用L-scheme线性化。随后推导出一个与混合形式相容的Robin型界面条件，用于耦合各子域。对于空间离散，采用混合有限元，从而确保局部质量守恒，并提供与Robin条件相容的通量变量。所提格式的收敛性得到了严格证明。给出了2D/3D非均质、多区域数值算例，以展示该格式的效率和鲁棒性，并对理论收敛结果进行了数值验证。
4. 非结构网格上连续介质力学的保结构交错半隐式四分裂有限体积格式
A structure-preserving staggered semi-implicit four-split finite volume scheme for continuum mechanics on unstructured meshes
作者: M. Dumbser, W. Boscheri, M. Tavelli, A. Thomann
类别: math.NA
发布日期: 2026-09-22
arXiv ID: 2609.25935v1
链接: http://arxiv.org/abs/2609.25935v1
摘要翻译
中文标题: 非结构网格上连续介质力学的保结构交错半隐式四分裂有限体积格式
摘要翻译: 我们提出了一种新的半隐式保结构 (SP) 有限体积离散，用于连续介质力学的统一一阶双曲模型。该框架为流体和固体提供了统一的数学描述，并在单个双曲偏微分方程系统中包含了输运、粘性效应、热传导和弹性变形。多种物理机制的共存产生了多个特征波速，导致完全显式离散受到严重的时间步长限制。为克服这一困难，我们发展了一种四分裂格式，其中控制方程被分解为对流、温度、力学和压力子系统。对流子系统是唯一在时间上显式推进的子系统，而其余子系统则以顺序方式隐式处理。因此，所提方法的时间步长限制仅依赖于材料速度，而与刻画该模型的声波、剪切波和热波速度无关。该格式旨在非结构网格上保持底层方程的关键结构性质。特别地，三角形上的相容顶点交错空间离散使得：只要控制 PDE 中的松弛源项为线性或不存在，与畸变场和热冲量相关的无旋对合就能够得到满足。与此同时，该格式是渐近保持的，因为它与低 Mach 数极限以及 GPR 模型的刚性松弛极限相容，而这些极限可恢复流体力学中经典的 Navier-Stokes-Fourier 方程。一系列涵盖流体和固体的数值实验展示了所提方法的精度、鲁棒性和多尺度能力。
5. 阻抗边界腔体中的弹性逆散射问题
The inverse elastic scattering problem in the cavity with impedance boundary
作者: Shiyu Xu, Huiling Zheng, Fang Zeng
类别: math.NA, math.AP
发布日期: 2026-09-22
arXiv ID: 2609.25858v1
链接: http://arxiv.org/abs/2609.25858v1
摘要翻译
中文标题: 阻抗边界腔体中的弹性逆散射问题
摘要翻译: 本文研究具有阻抗边界条件的二维腔体中的弹性逆散射问题。我们证明了腔体内部的散射场满足互易关系，并证明了腔体的形状以及边界上的阻抗参数均可由在一条内部曲线上测量的散射场唯一确定。在数值方面，我们首先采用线性采样法重建形状，然后结合位势理论与最小二乘法反演阻抗参数。我们给出了若干数值实验，以验证所提方法的可行性和有效性。
6. Scott-Vogelius 元在 Freudenthal 网格上对于 $k \geq 4$ 是 inf-sup 稳定的
The Scott-Vogelius element is inf-sup stable on Freudenthal meshes for $k \geq 4$
作者: Siqi Ding, Pingbing Ming, Haijun Yu, Quanfan Zhu
类别: math.NA
发布日期: 2026-09-22
arXiv ID: 2609.25690v1
链接: http://arxiv.org/abs/2609.25690v1
摘要翻译
中文标题: Scott-Vogelius 元在 Freudenthal 网格上对于 $k \geq 4$ 是 inf-sup 稳定的
摘要翻译: Scott-Vogelius 元是用于 Stokes 问题的一种经典无散度混合有限元，数十年来一直受到研究关注，但其理论框架仍不完整。在二维情形，Freudenthal 网格及其他正则网格上的 inf-sup 稳定性已被严格建立。在三维情形，Zhang 于 2011 年建立了当 $k\ge6$ 时 Freudenthal 网格上的 inf-sup 稳定性，而数值证据表明 inf-sup 稳定性对 $k=4,5$ 仍然成立。在本文中，我们强化了 Zhang 的方法，并证明 Scott–Vogelius 元在 Freudenthal 网格上对每个速度次数 $k\ge 4$ 都是 inf-sup 稳定的，从而解决了 Farrell、Mitchell 和 Scott 在 2024 年提出的一个猜想。该证明通过对局部小块上的显式构造进行，且不依赖于计算机验证。
7. AKAPINN：用于逼近拟线性偏微分方程解的自适应 Kolmogorov-Arnold 物理信息神经网络
AKAPINN: Adaptive Kolmogorov-Arnold Physics-Informed Neural Networks for approximating solutions to quasilinear partial differential equations
作者: B. Veena S. N. Rao
类别: math.NA
发布日期: 2026-09-22
arXiv ID: 2609.25507v1
链接: http://arxiv.org/abs/2609.25507v1
摘要翻译
中文标题: AKAPINN：用于逼近拟线性偏微分方程解的自适应 Kolmogorov-Arnold 物理信息神经网络
摘要翻译: 本文提出了一项受控比较研究，研究对象是无网格 Kolmogorov–Arnold 物理信息神经网络 (KAN-PINN) 在非线性应变限制型偏微分方程上的应用。在包含精确硬边界 ansatz、稳健的 pseudo-Huber 残差损失以及多阶段 Adam-to-L-BFGS 优化器调度的固定计算流程内，评估了对应于不同材料参数对 $(α, β)$ 的三组不同训练运行。分析揭示了一个基本指标反转：虽然训练损失反映算子特定的残差缩放和本构刚度，但算子无关的相对 $L_2$ 误差表明，中间参数区域实现了更优的解精度和增强的泛化能力。
8. 用于相场梯度流的高阶变尺度能量变分神经网络
High-Order Variable-Scaled Energetic Variational Neural Networks for Phase-Field Gradient Flows
作者: Xiaobo Jing, Leiyi Dong, Jia Zhao
类别: math.NA
发布日期: 2026-09-21
arXiv ID: 2609.25489v1
链接: http://arxiv.org/abs/2609.25489v1
摘要翻译
中文标题: 用于相场梯度流的高阶变尺度能量变分神经网络
摘要翻译: 本文中，我们为相场梯度流发展了一种高阶变尺度能量变分神经网络 (VS-EVNN) 方法。每个离散阶段都使用一个更新状态网络，其空间输入被中心化并除以界面尺度，而采样能量和运动惩罚相对于基线 EVNN 形式保持不变。这种重缩放的动机来自能量中的坐标变换，该变换归一化了变换后泛函的梯度系数。我们将尺度化表示与高阶、多阶段变分外推格式以及启发式自适应时间步进控制器相结合。每一阶段均作为网络最小化问题求解，其中较早阶段在求积网格上充当固定锚点，从而给出这些多阶段变分格式的神经网络实现，包括用于 Cahn–Hilliard 流的迁移率加权、质量守恒版本。该方法在一维和二维 Allen–Cahn 和 Cahn–Hilliard 流上进行了测试，后者在质量守恒度量下。在所报告的优化设置下，尺度化网络比未尺度化 EVNN 基线产生更低的解误差，并且高阶格式在测试示例中提高了精度。
9. 原始 BGN 方法对平均曲率流的收敛性
Convergence of the original BGN method for mean curvature flow
作者: Yifei Li
类别: math.NA
发布日期: 2026-09-21
arXiv ID: 2609.25402v1
链接: http://arxiv.org/abs/2609.25402v1
摘要翻译
中文标题: 原始 BGN 方法对平均曲率流的收敛性
摘要翻译: 我们证明，曲线平均曲率流的原始 Barrett–Garcke–Nürnberg (BGN) 方法在时间步长 $τ=h^2$ 下以 $h^2$ 阶在流形距离中收敛。关键的新技术是后向误差分析，它源于常微分方程的数值分析。通过结合离散速度在空间和时间上的展开，我们构造了一个近似流，其插值以改进的缺陷满足 BGN 方法，缺陷为 $h^4$ 阶，其中切向部分为 $h^6$ 阶。该构造还确定了 BGN 方法的切向速度，并表明 $τ=O(h^2)$ 是 $τ$ 与 $h$ 之间的正确关系。借助改进的缺陷，我们随后使用演化曲面有限元方法的技术，证明数值解到该近似流的 $H^1$ 超收敛阶为 $h^4$，从而得到流形距离意义下的收敛性。数值实验验证了所预测的缺陷阶。
10. 一种用于流体-多孔弹性结构相互作用的无条件稳定二阶 Robin 分区方法
An unconditionally stable second-order Robin partitioned method for fluid-poroelastic structure interaction
作者: Wenlong He, Thomas Wick, Xiaohe Yue, Jiwei Zhang, Haibiao Zheng
类别: math.NA
发布日期: 2026-09-21
arXiv ID: 2609.25350v1
链接: http://arxiv.org/abs/2609.25350v1
摘要翻译
中文标题: 一种用于流体-多孔弹性结构相互作用的无条件稳定二阶 Robin 分区方法
摘要翻译: 我们研究一个非定常流体-多孔弹性结构相互作用 (FPSI) 问题，其中自由流体区域由不可压缩 Stokes 方程控制，而多孔弹性区域由完全动态 Biot 系统控制。原始物理耦合条件被等价地重新表述为 Robin 型界面条件，并引入一个辅助界面变量作为两个子系统共享的公共 Robin 数据。这种重新表述将耦合的 FPSI 问题分解为流体和多孔弹性子问题，它们分别定义在各自的子区域上。该问题在空间上采用有限元方法 (FEM) 离散，而在时间离散中则采用向后 Euler 格式和结合二阶界面外推的 BDF2 格式。通过利用所得的离散动态关系，我们推导了辅助界面变量的更新公式，并得到全离散的一阶和二阶 Robin 分区格式，其中两个子问题在每个时间步独立求解。两种格式的无条件稳定性均被严格建立。数值实验验证了所提格式的时间收敛阶，并研究了 Robin 参数的影响。一个经典的压力波基准测试验证了所提方法的精度，而补充的非线性移动区域模拟展示了它们对更复杂 FPSI 配置的适用性。
不相关论文（基本信息）
1. 带 $(2,1)$-范数正则化的 Stiefel 流形优化问题的 NEPv 方法
NEPv Approach for Optimization on Stiefel Manifold with the $(2,1)$-norm Regularization
作者: Ren-Cang Li, Li Wang, Lei-Hong Zhang, Zhaojun Bai
类别: math.NA
发布日期: 2026-09-22
arXiv ID: 2609.26675v1
链接: http://arxiv.org/abs/2609.26675v1
摘要: Row-sparse projection provides a useful tool in machine learning (ML) when it comes to, for example, feature selection, aiming to choose most relevant features for various ML objectives. One way to seek a high quality row-sparse projection is to combine an ML objective, such as the ones for PCA, LDA, and OCCA, with the matrix $(2,1)$-norm regularization which is nonsmooth. Such combinations result in challenging optimization problems on the Stiefel manifold that need to be solved efficiently. In this paper, a unifying NEPv framework is established to efficiently deal with optimization on the Stiefel manifold with the $(2,1)$-norm regularization. The effect of the $(2,1)$-norm regularization is also investigated. The wide applicability of the framework is demonstrated through the combinations of common learning objectives in today’s data science applications with the $(2,1)$-norm regularization. Numerical experiments are presented to illustrate the use of the NEPv approach and to gain insights as to what a proper regularizing parameter should have in real-world applications.
2. 用于走时层析成像中的大规模贝叶斯推断的分布式近端Stein变分梯度下降算法
Distributed Proximal Stein Variational Gradient Descent Algorithm for Large-scale Bayesian Inference in Traveltime Tomography
作者: Akshay Vishwakarma, Kamal Aghazade, Ali Siahkoohi, Ali Gholami
类别: physics.geo-ph, math.NA, physics.comp-ph
发布日期: 2026-09-22
arXiv ID: 2609.26653v1
链接: http://arxiv.org/abs/2609.26653v1
摘要: We present a distributed framework for large-scale Bayesian inverse problems governed by the eikonal equation, with a specific focus on seismic traveltime tomography. Traditional deterministic approaches often fail to provide the uncertainty quantification (UQ) necessary for ill-posed problems, while conventional Bayesian sampling methods such as Markov chain Monte Carlo (MCMC) suffer from the curse of dimensionality and slow convergence in high-dimensional model spaces. The proposed framework addresses these challenges through a three-tier computational strategy. First, we utilize the Fast Marching Method (FMM) to solve the eikonal equation, ensuring high numerical accuracy. Second, we reformulate the global tomographic objective into a decentralized consensus form, allowing the inversion to be decomposed into independent subproblems solved in parallel via the Alternating Direction Method of Multipliers (ADMM). This architecture eliminates the need for the explicit construction of large-scale sensitivity matrices, significantly reducing the memory footprint for 3D surveys. Finally, we integrate Stein Variational Gradient Descent (SVGD) within the ADMM workers to perform approximate posterior sampling. By evolving a set of model particles along a functional gradient direction that balances data-fitting forces with a repulsive kernel-based diversity force, we obtain an ensemble from which posterior summaries are computed. We derive a data-space Gauss-Newton update using the Woodbury matrix identity to further accelerate the particle evolution in large-scale 3D problems. Numerical experiments on complex 2D and 3D models demonstrate that the algorithm achieves stable convergence, produces high-fidelity velocity reconstructions, and provides posterior uncertainty maps.
3. 正则化稀疏随机矩阵的集中性：基于非回溯算子的谱边缘界
Concentration of Regularized Sparse Random Matrices: Spectral Edge Bounds via Nonbacktracking Operators
作者: Hai-Xiao Wang
类别: math.PR, math.CO, math.NA, math.ST
发布日期: 2026-09-22
arXiv ID: 2609.26615v1
链接: http://arxiv.org/abs/2609.26615v1
摘要: In sparse random matrices, spectral outliers (eigenvalues and singular values located away from the bulk) emerge due to degree fluctuations: high degrees inflate the operator norm, while low column degrees reduce the least singular value. As proved by Feige and Ofek (2005) and Le, Levina, and Vershynin (2017), degree regularization enforces concentration at the expected norm scale. However, precise bounds incorporating the cutoffs remain unexplored and challenging since regularization introduces dependencies among entries. For the first time in the literature, we provide variance- and cutoff-dependent bounds for extreme singular values and eigenvalues of regularized inhomogeneous random matrices. In the absence of regularization, our lower bound for the least singular value matches the same leading constant obtained by Brailovskaya and van Handel (2024). Moreover, our error term vanishes under the milder condition $d/\log N\to\infty$, as opposed to their stronger requirement $d/(\log N)^4\to\infty$. A key ingredient is to extend spectral radius bounds for nonbacktracking matrices to the dependent setting. We build on approaches for independent cases established by Benaych-Georges, Bordenave, and Knowles (2020), as well as Dumitriu and Zhu (2024), and carefully handle edges traversed only once. Our proof framework separates deterministic spectral comparisons from probabilistic estimates: once Loewner inequalities and columnwise variance controls are established, the remaining probabilistic analysis boils down to verifying the graph moment conditions formulated in this paper. We hope this framework can be extended to handle general random matrices with more complex dependencies.
4. 关于 Fourier-Bessel 小波的注记
Notes on Fourier-Bessel wavelets
作者: Marcel Venturotti, Georgios Exarchakis
类别: cs.LG, cs.CV, math.NA
发布日期: 2026-09-22
arXiv ID: 2609.26537v1
链接: http://arxiv.org/abs/2609.26537v1
摘要: These notes develop the mathematical foundations and construction of a Fourier-Bessel wavelet family inspired by the disk harmonics of Shaqfa et al.[9]. We begin with the relevant properties of Bessel and modified Bessel functions and introduce the wavelet properties required for the construction. We then derive the Fourier-Bessel disk harmonics as solutions to the Helmholtz equation on the unit disk subject to a Neumann boundary condition. Building on this basis, we construct a wavelet family by applying a Gaussian spatial envelope and introducing a zero-mean correction for the zeroth angular order. We derive the corresponding normalisation constants for $L^2$-based applications and discuss $L^1$-based normalisation for frequency-domain peak consistency. Finally, we derive a closed-form Fourier-domain representation of the resulting wavelets. The main motivation is the approximately linear spacing, which converges to $π$ between consecutive radial eigenvalues. Rather than replacing the conventional dyadic organisation of wavelet families, this construction lays out the foundation to explore whether a more uniform radial frequency allocation can be useful for applications in which broad and balanced frequency coverage is desirable.
5. 基于六元排列的四面体多形式最长边二分法
Multiform Longest Edge Bisection of Tetrahedra via Sextuple Permutations
作者: Agustin Trujillo, Jose Pablo Suarez, Tania Moreno-García
类别: cs.CG, math.NA
发布日期: 2026-09-22
arXiv ID: 2609.26522v1
链接: http://arxiv.org/abs/2609.26522v1
摘要: We introduce a new formulation of the Longest Edge Bisection (LEB) of tetrahedra entirely in sextuple space R6, where tetrahedra are represented by the squares of their edge lengths. This representation renders the LEB refinement equations fully linear and eliminates the need for coordinate-based data structures. A central difficulty in three-dimensional LEB arises when a tetrahedron possesses multiple longest edges, making the refinement rule intrinsically multivalued. We formalize this phenomenon through the notion of Multiform Longest Edge Bisection (MLEB), which systematically explores all admissible longest-edge choices. To encode this multivalued behavior, we introduce the concept of bisection patterns, defined as sequences of sextuple permutations governing the refinement process. We prove that the set of sextuples sharing a common LEB pattern forms a convex region in R6. For structurally significant families of tetrahedra, including the R1+ family and the Liu-Joe family, we show that the infinite refinement tree collapses into a finite directed graph with eight states. Remarkably, both families are governed by the same graph, differing only in their initial state. This directed-graph formulation provides a unified combinatorial description of the refinement process and offers an efficient computational framework for deep iterative LEB analysis.
6. 基于 FFT 加速 Toeplitz 算子的快速 Karhunen-Loève 展开
Fast Karhunen-Loève Expansions via FFT-Accelerated Toeplitz Operators
作者: Nils Wildt, Wolfgang Nowak
类别: math.NA
发布日期: 2026-09-22
arXiv ID: 2609.26459v1
链接: http://arxiv.org/abs/2609.26459v1
摘要: Gaussian random fields are a versatile tool used in the fields of stochastic PDEs, uncertainty quantification, and geostatistical simulation. One way to obtain them is to use a truncated Karhunen–Loève expansion (KLE). Computing the expansion requires the leading eigenpairs of an $N \times N$ covariance matrix, where $N$ is the total number of grid cells. These are usually computed with a Krylov eigensolver, which relies on the covariance operator only within matrix–vector products. Stored densely, the matrix takes $\mathcal{O}(N^{2})$ memory and each product $\mathcal{O}(N^{2})$ time. For a stationary kernel on an equispaced grid, the covariance matrix becomes (block-) Toeplitz and the product evaluates in $\mathcal{O}(N\log N)$ time using FFT-based circulant embedding, without the need to assemble the dense matrix. In a matched single-threaded comparison, the median speedup of the eigensolve grows from $18 \times$ at $N = 4096$ to $183 \times$ at $N = 2^{15}$. This makes it possible to compute discretized fields that would otherwise be infeasible to compute in the standard formulation. We show that the same construction carries over to non-separable kernels as well as $d$ dimensions, using block-Toeplitz matrices. We extend it to piecewise-constant fields on arbitrary domains given as subsets of a tensor grid. Computational savings grow with problem size, and storage drops from $\mathcal{O}(N^{2})$ to $\mathcal{O}(2^{d}N)$.
7. 基于块三角联合漂移的一步生成式代理模型
One-Step Generative Surrogate Models via Block-Triangular Joint Drifting
作者: Nicholas Geissler, Shreya Jha, Ricardo Baptista, Benjamin Peherstorfer
类别: cs.LG, math.NA
发布日期: 2026-09-22
arXiv ID: 2609.26435v1
链接: http://arxiv.org/abs/2609.26435v1
摘要: Drifting provides a direct route to one-step generative models, but applying it directly to stochastic transition modeling requires multiple samples of the next state conditioned on the same current state. Standard trajectory data, however, typically provide only one realized next state for each observed current state and therefore do not provide an empirical approximation of the corresponding conditional distribution over possible next states. We introduce block-triangular joint drifting, which instead applies a projected drift field to the empirically accessible joint distribution of consecutive states. Importantly, the block-triangular architecture preserves the current-state marginal while making its second component a direct sampler of the conditional distribution of possible next states. The resulting surrogate generates stochastic trajectories with one model evaluation per time step, without auxiliary generative steps between time steps. Numerical experiments demonstrate accurate marginal and trajectory-dependent statistics and favorable accuracy-cost tradeoffs compared with deterministic, diffusion-, flow-, and distillation-based generative surrogate models.
8. SuperPCA：子空间分析与高维 PCA 的高效算法
SuperPCA: subspace analysis and an efficient algorithm for high-dimensional PCA
作者: Irina-Beatrice Haas, Maike Meier, Yuji Nakatsukasa, Taejun Park
类别: stat.ML, cs.LG, math.NA, stat.CO
发布日期: 2026-09-22
arXiv ID: 2609.26406v1
链接: http://arxiv.org/abs/2609.26406v1
摘要: Principal component analysis (PCA) is a fundamental tool to reduce the dimensionality of the data in many applications. PCA finds a few signal directions that contain most of the variability of the data by computing the eigenvectors of the sample covariance matrix. In this work, we focus on the spiked covariance model, in which the data vectors are defined by a few orthogonal signals plus an isotropic Gaussian noise, and our goal is to estimate one or more of the leading signals. Our main theoretical finding is that the subspace spanned by several leading eigenvectors of the sample covariance matrix contains significant information about the desired signals long before the individual eigenvectors converge to the population principal components. To prove this, we derive a posteriori bounds for the angle between the subspace spanned by the desired population signals and the subspace obtained from the sample using perturbation theory for singular vectors. This leads to a new algorithm, SuperPCA (SUbsPace subsamplER PCA), which capitalizes on an approximate eigenspace of the sample covariance matrix to find the leading signals far more efficiently and accurately than classical PCA in the high-dimensional, multi-signal setting. SuperPCA exploits only a small number of subsampled coordinates of the data, which can lead to tremendous savings in data acquisition cost, especially when the signals are approximately sparse. For the same number of measurements, SuperPCA can offer a factor $10$ improvement in accuracy compared to the classical PCA method.
9. 计算同伦路径奇异端点的螺旋终局
Spiral Endgames for Computing Singular Endpoints of Homotopy Paths
作者: Emma R. Cobian, Jonathan D. Hauenstein, Charles W. Wampler
类别: math.NA
发布日期: 2026-09-22
arXiv ID: 2609.26370v1
链接: http://arxiv.org/abs/2609.26370v1
摘要: We develop a new endgame for numerically computing singular solutions to complex analytic systems by means of a homotopy path whose endpoint is the solution to be computed. When the solution is singular, prediction-correction path tracking may fail at the endpoint. Instead of tracking the path all the way to its end, endgames sample nonsingular points on the incoming path and approximate the endpoint by interpolating the samples with a truncated Puiseux series. By collecting sample points along a logarithmic spiral in the complex plane, the proposed spiral endgame balances between the superior numerical conditioning of a circular sample (known as the Cauchy endgame) and the update efficiency of a linear sample (known as the power series endgame), these being tied together as two extremes of the new approach. The numerical conditioning of the methods are analyzed and compared, and computational examples illustrate the effectiveness of the new spiral sampling method compared with the circular sampling method of the Cauchy endgame.
10. 病例特异性非线性颈动脉力学的流固耦合建模与出口边界条件的作用
FSI modeling of case-specific nonlinear carotid artery mechanics and the role of outlet boundary conditions
作者: Tristan Probst, Anna Hundertmark, Ashkan Shiravand, Giorgio Cattaneo
类别: math.AP, math.NA
发布日期: 2026-09-22
arXiv ID: 2609.26038v1
链接: http://arxiv.org/abs/2609.26038v1
摘要: Compliant arterial wall mechanics, contributing to the Windkessel effect in the carotid artery (CA), have significant impact on hemodynamic patterns and surface shear indicators for cardiovascular diseases, e.g. atherosclerosis. In this study, we extend the linear elastic Fluid-Structure Interaction (FSI) framework to account for nonlinear strain-dependent wall behavior. The model incorporates a Young’s modulus generalized from tensile tests on silicone phantoms to capture the nonlinear stress-strain relation of arterial tissue. The final computational model, using a resistance-type boundary condition and in vitro measured stress-strain relation is validated against both, in vitro assessed silicon CA phantom as well as published clinical data on the flow splitting to the daughter branches in CA bifurcations. To better reflect physiological conditions, our model is subsequently extended to incorporate prestress of patient-specific geometries, and clinically measured stress-strain relations, followed by validation against clinical CA data. The present study demonstrates the feasibility of strain-dependent Young’s (elastic) modulus as a means to enhance the capacity of the linear elastic framework to accurately represent the physiologically nonlinear mechanics of arterial walls, striking a balance between implementation effort and physiological fidelity. This approach yields realistic strains, volumetric inflation behavior and nonlinear pressure-volume relationships. Furthermore, the study reveals the importance of proper outlet boundary conditions and the shortcomings of resistance-type boundary condition in patient-specific modeling, leading to non-physiological pressure profiles and non-physiological temporal flow splitting.
11. 一种用于具有高维参数空间的参数化系统模型降阶的迭代活跃子空间方法
An Iterative Active Subspace Approach for Model Order Reduction of Parametric Systems with High-Dimensional Parameter Spaces
作者: Chenzi Wang, Peizhi Yu, Lihong Feng, Peter Benner, Wenshuai Lu, Zheng You
类别: math.NA
发布日期: 2026-09-22
arXiv ID: 2609.26009v1
链接: http://arxiv.org/abs/2609.26009v1
摘要: The increasing complexity in design and manufacturing has driven the need for advanced techniques for fast modeling problems with large-dimensional parameter spaces. Avoiding high-fidelity finite element models while achieving fast and accurate simulations in such contexts is challenging. Parametric model order reduction (pMOR) has drawn significant attention in recent years. Nevertheless, the curse of dimensionality in parameter spaces has severely limited its effectiveness. The active subspace (AS) approach has been successfully applied to pMOR for systems with many parameters. However, the balance between accuracy and compactness of the reduced model remains problematic for such systems with high-dimensional parameter spaces. It often results in models that are either not small enough or not accurate enough. In this paper, we propose an iterative active subspace (IAS) approach for parametric model order reduction, which, to some extent, addresses the trade-off between accuracy and reduced model size and achieves substantial computational gains compared to the original active subspace method.
12. 抽象框架下第二类变分不等式的后验误差估计
A posteriori error estimates for variational inequalities of the second kind in an abstract framework
作者: Lothar Banz, Miriam Schönauer, Andreas Schröder
类别: math.NA
发布日期: 2026-09-22
arXiv ID: 2609.26001v1
链接: http://arxiv.org/abs/2609.26001v1
摘要: In this paper, an abstract framework is introduced for the derivation of reliable a posteriori error estimators for variational inequalities of the second kind. Under certain conditions (local) efficiency of the error estimators can also be derived. The framework is based on the estimation of a residuum which results from an equivalent mixed formulation and from the replacement of the Lagrange multiplier by its (possibly post-processed) discrete counterpart. The approach is applied to an idealized frictional contact problem and a Mosolov type model problem in an $hp$-finite element discretization setup. Several numerical experiments demonstrate the broad applicability of the theoretical findings.
13. 多光谱计算机断层成像的增强衰减建模
Enhanced attenuation modelling for multispectral computed tomography
作者: Jürgen Jeschke, Dirk Lorenz
类别: math.NA
发布日期: 2026-09-22
arXiv ID: 2609.25926v1
链接: http://arxiv.org/abs/2609.25926v1
摘要: We revisit the modelling of the material decomposition problem in multispectral computed tomography and propose to leverage the known absorption spectra of different materials. With this we both derive new uniqueness conditions for the material decomposition and also offer a three-step reconstruction model. Our uniqueness criterion only requires certain characteristics of the bin sensitivity functions and the material attenuation spectra and hence, can be computed a-priori before measurements are taken.
14. AURA：用于训练复值神经网络的角度更新速率自适应
AURA: Angular Update Rate Adaptation for training complex-valued neural networks
作者: Enrico Ballini, Allan Peter Engsig-Karup, Tito Andriollo
类别: cs.LG, math.NA
发布日期: 2026-09-22
arXiv ID: 2609.25914v1
链接: http://arxiv.org/abs/2609.25914v1
摘要: Complex-valued neural networks (CVNNs) are increasingly adopted for complex-valued data; however, they are often trained with first-order optimizers inherited from the real-valued case. The efficiency of these methods depends largely on the step size, and their step-size rules ignore the angular information available in the complex plane. We address step-size adaptation in the complex domain by introducing AURA (Angular Update Rate Adaptation), a per-parameter step-size adaptation that can be added on top of any first-order optimizer, and removed from it, without altering its update direction. AURA measures the agreement between consecutive updates of each complex parameter, in length, alignment, and sense of rotation, and enlarges the step when they are consistent and reduces it when they are not. It requires no additional gradient evaluations and only inexpensive vector operations per step. We combine AURA with Adam and Muon and compare the resulting methods with well-known first-order optimizers on four test cases of increasing complexity, ranging from the approximation of scalar complex functions to physics-informed training. Fully connected neural networks are used throughout this work. All hyperparameters other than the step size are held fixed across test cases; for one case, we also tune the hyperparameters of each optimizer under the same budget. Our empirical tests show that AURA improves the convergence of its base optimizer in most cases with a small per-step overhead, and we identify the conditions under which it fails to do so.
15. 一种由积分微分方程描述的电路的多速率瞬态仿真技术
A Multirate Transient Simulation Technique for Circuits described by Integro-Differential Equations
作者: Hans Georg Brachtendorf, Kai Bittner
类别: math.NA
发布日期: 2026-09-22
arXiv ID: 2609.25854v1
链接: http://arxiv.org/abs/2609.25854v1
摘要: The multirate partial differential equation (PDE) technique is considered as one of the most powerful methods for the simulation of radio frequency (RF) circuits. The method reformulates the underlying ordinary differential algebraic equation (DAE) by a PDE. The DAE solution is obtained along a characteristic curve of the PDE. Numerical solution techniques comprise (envelope modulated) Harmonic Balance (HB), multistep finite difference (FD) and spline/wavelet techniques. All techniques have disadvantages when mixed lumped/distributed devices occur and the waveforms exhibit pulse shaped behavior. Furthermore, communication systems are modeled and simulated by the Equivalent Complex Baseband (ECB) approach. Signals and impulse responses are described by their baseband counterparts, employing the Hilbert transformation. A coupled system/circuit simulation requires therefore novel numerical techniques for the solution of integro-differential equations.
16. 稀疏网格上高维函数的基于各向异性核的多水平插值
Anisotropic Kernel-based Multilevel Interpolation of High-Dimensional Functions on Sparse Grids
作者: Rüdiger Kempf
类别: math.NA
发布日期: 2026-09-22
arXiv ID: 2609.25828v1
链接: http://arxiv.org/abs/2609.25828v1
摘要: The tensor product multilevel method (TPML) approximates high-dimensional functions from scattered data on general bounded domains by combining Smolyak’s sparse grid construction with the kernel-based multilevel interpolation, coupling the kernel scale to the fill distance at each level so that condition numbers stay uniformly bounded. Its original formulation depended only implicitly on the target function, which precluded numerical evaluation. Using a recent nodal representation of the low-dimensional multilevel operators, we derive an explicit, computable representation of the high-dimensional interpolant. On this basis we prove error estimates in mixed-regularity Sobolev and continuity norms, extending the existing $ L_2 $-theory, and we characterise the full range of anisotropy weights that attain the optimal rate against the number of degrees of freedom. Three explicit weight strategies follow, equilibrating the accuracy, the degrees of freedom, or the cost-benefit ratio. Numerical experiments in up to ten dimensions illustrate the theory.
17. 基于离散正弦变换离散化的线性二阶微分算子谱的一致逼近
Uniform approximation of spectra of linear second order differential operators via discrete sine transform based discretisation
作者: Oliver Křenek, Vít Průša, Rebecca Tozzi, Martin Vejvoda
类别: math.NA, physics.comp-ph
发布日期: 2026-09-22
arXiv ID: 2609.25796v1
链接: http://arxiv.org/abs/2609.25796v1
摘要: We study various discretisation schemes for the regular Sturm–Liouville operator on a bounded interval and for the Laplace operator on an arbitrary planar domain, in both cases subject to zero Dirichlet boundary conditions. The objective is to identify a discretisation scheme such that the corresponding discretised operator—a matrix of size $N \times N$—produces $N$ eigenvalues that approximate as closely as possible the first $N$ eigenvalues of the corresponding operator at the continuous level. By means of numerical experiments we examine several conventional discretisation schemes, and we corroborate the known fact that the conventional discretisations fail to achieve the objective, with the failure attributable to the poor approximation of high-index eigenvalues, and, as a result, to the non-uniform spectrum approximation. In contrast, the newly proposed discrete sine transform based discretisation scheme is designed in such a way that it replicates the asymptotic behaviour of high-index eigenvalues, thereby providing the sought uniform spectrum approximation. Of equal significance is the fact that the proposed discrete sine transform based scheme can be, unlike many Fourier transform based methods, applied to non-rectangular domains.
18. 关于一个与特征值扰动相关的猜想
On a Conjecture Related to Eigenvalue Perturbations
作者: Lei-Hong Zhang, Ren-Cang Li
类别: math.NA
发布日期: 2026-09-22
arXiv ID: 2609.25552v1
链接: http://arxiv.org/abs/2609.25552v1
摘要: In connection with eigenvalue perturbation bounds, Li [Linear Algebra Appl., 278 (1998), pp. 317-326] formulated three minimization problems involving Hadamard products. The third asks how to relate
$$ \min_{W\ \mathrm{nonsingular}}\|W\circ G\|_\mathrm{ui}\,\|W^{-1}\circ G^\mathrm{T}\|_\mathrm{ui} \quad\mbox{and}\quad \min_{P\ \mathrm{permutation}}\|P\circ G\|_\mathrm{ui}^2, $$where $\|\cdot\|_\mathrm{ui}$ denotes a unitarily invariant norm and $\circ$ denotes the Hadamard product, i.e., entrywise multiplication. The first quantity never exceeds the second, and a reverse inequality with a constant independent of the matrix size was conjectured. Romeo and Tilli [Linear Algebra Appl., 326 (2001), pp. 161-172] showed that if such a bound holds for every $G$, then $\|\cdot\|_\mathrm{ui}$ must be uniformly equivalent to the Frobenius norm $\|\cdot\|_\mathrm{F}$. We prove that the two quantities coincide when $\|\cdot\|_\mathrm{ui}=\|\cdot\|_\mathrm{F}$: for every complex matrix $G$,
$$ \min_{W\ \mathrm{nonsingular}} \|W\circ G\|_\mathrm{F}\,\|W^{-1}\circ G^\mathrm{T}\|_\mathrm{F} =\min_{P\ \mathrm{permutation}}\|P\circ G\|_\mathrm{F}^2. $$The associated conjecture therefore holds with the optimal constant $c=1$.
19. 基于 CP 参数化核的三维盲反卷积
Three-dimensional blind deconvolution by CP-parameterized kernels
作者: Fatoumata Sanogo, Stefan Kindermann
类别: math.NA
发布日期: 2026-09-21
arXiv ID: 2609.25394v1
链接: http://arxiv.org/abs/2609.25394v1
摘要: We consider the problem of reconstructing a kernel $k$ and image $u$ in a blind deconvolution problem from convolutional data $g = k*u$. We particularly focus on the 3D case, i.e., on volumetric images. The problem is approached by imposing a semiparametric CP decomposition for the kernel and a variational TV penalty for the image. We also discuss a motion blur ambiguity effect, i.e., a nonuniqueness issue of the kernel in the blind deconvolution problem that may appear in certain videos of moving objects. The blind deconvolution algorithm uses an alternating minimization approach with TV regularization for $u$ and a CP decomposition step for the kernel. These are implemented with positivity projection, kernel normalization, and\textbackslash or causal support projection steps. Several numerical examples for 3D MRI images, hyperspectral images, and a grayscale test video show the effectiveness of the method.
20. 基于采样的随机模型预测控制中退出泛函的高阶近似
Higher-Order Approximation of Exit Functionals in Sampling-Based Stochastic Model Predictive Control
作者: Sashank Modali, Takashi Tanaka
类别: eess.SY, cs.RO, math.NA, math.OC
发布日期: 2026-09-21
arXiv ID: 2609.25257v1
链接: http://arxiv.org/abs/2609.25257v1
摘要: Safety evaluation in sampling-based stochastic model predictive control often requires numerical estimation of exit functionals. The approximation of first-exit times and exit indicators is therefore a key numerical bottleneck, and discretization error in these quantities directly affects the resulting controller. This paper studies how existing higher-order methods for strong approximation of exit times can be brought into safe control. Two cases are highlighted. For general noncommutative dynamics, an adaptive order-1 Milstein discretization is used together with Lévy-area simulation via Wiktorsson’s method. For commutative dynamics, an adaptive order-1.5 construction achieves a stronger exit-time rate. Under a local anti-concentration condition on the exit-time law, we show that strong exit-time approximation transfers to strong approximation of the failure indicator. The methods are then studied in the context of chance-constrained path integral control, which provides an exact continuous-time representation of safety through exit events. Numerical experiments compare the two cases in terms of strong exit-time error, failure-indicator error, and closed-loop constraint satisfaction, showing improvement over Euler-Maruyama and thereby enabling existing and future techniques whose applicability depends on improved strong approximation.