生成时间: 2026年05月26日 06:44:02
重点关注论文数量: 0
了解领域论文数量: 8
不相关论文数量: 38
了解领域论文（摘要翻译）
1. 浅水方程协变形式的半隐式二维求解器
A semi-implicit two dimensional solver for a covariant formulation of the shallow water equations
作者: Maurizio Tavelli, Olindo Zanotti
类别: physics.flu-dyn, math.NA
发布日期: 2026-05-25
arXiv ID: 2605.25544v1
链接: http://arxiv.org/abs/2605.25544v1
摘要翻译
中文标题: 浅水方程协变形式的半隐式二维求解器
摘要翻译: 本文将一种灵活的浅水方程协变形式与 Casulli 及其合作者多年来开发的半隐式数值格式结合起来。通过在二维流形上采用正交但非标准正交的坐标基，并以避免引入克里斯托费尔符号的方式编写对称张量的散度，浅水方程与笛卡尔坐标下通常的形式保持了高度相似。通过这种方式，可以对动量方程中的表面高度梯度和连续性方程中的速度采用隐式离散，导出一个稳定的半隐式格式，其稳定性与波速无关。我们已在各种具有挑战性的基准问题上检验了新方法，其中包括水球面上的光滑波传播和动脉分支的变形。两个引人注目的附加特性使得该方法在海洋学应用中尤为强大：首先，得益于我们半隐式方法处理干湿问题的能力，极点处不会出现病态行为；其次，该格式自然具有良好的平衡性，能够以机器精度保持地球整个海洋结构的完美静态。
2. 基于均匀化的原子-连续介质耦合
Atomistic-Continuum Coupling by Homogenization
作者: Aagashram Neelakandan, Karsten Albe, Bernhard Eidel
类别: math.NA
发布日期: 2026-05-24
arXiv ID: 2605.25286v1
链接: http://arxiv.org/abs/2605.25286v1
摘要翻译
中文标题: 基于均匀化的原子-连续介质耦合
摘要翻译: 基于原子间势的经典原子模拟能够高保真地解析晶格失稳、缺陷成核和微观结构演化，但其可处理的系统尺寸仍远低于微米尺度结构分析所需的规模。我们建立了一个双尺度原子-连续介质框架，将微尺度的非线性有限元边值问题与在积分点上定义的周期性分子静力学胞元问题相耦合。尺度过渡通过 Hill-Mandel 能量等价意义下的计算均匀化来表述。不同于在低尺度上预设连续介质本构关系，原子胞元直接由连续介质变形驱动，并以功共轭的形式返回体积平均应力以及有效切线模量。单晶铜的数值算例展现出显著的拉压非对称性、失稳驱动的突变成核、反循环加载下的快速稳定以及悬臂梁弯曲中的局部弹塑性转变。在所有这些强非线性情形中，粗尺度 Newton 求解器保持稳健，并在其最终迭代中恢复到接近二次的收敛速度。因此，该双尺度框架将基于势的原子模拟拓展到了在当前的准静态、非热条件下直接原子模拟无法企及的结构长度尺度。
3. 现代区域分解导览：从 Schwarz 迭代到鲁棒预条件子与高性能计算实现
A Guided Tour of Modern Domain Decomposition: From Schwarz Iterations to Robust Preconditioners and HPC Implementations
作者: Victorita Dolean, Pierre Jolivet, Frédéric Nataf, Pierre-Henri Tournier
类别: math.NA, physics.comp-ph
发布日期: 2026-05-24
arXiv ID: 2605.24982v1
链接: http://arxiv.org/abs/2605.24982v1
摘要翻译
中文标题: 现代区域分解导览：从 Schwarz 迭代到鲁棒预条件子与高性能计算实现
摘要翻译: 区域分解方法为偏微分方程的可扩展数值求解提供了统一框架。该方法起源于 Schwarz 交替法，现已演变为丰富的算法族，兼具局部鲁棒性、全局收敛加速和自然并行性。过去数十年间，区域分解在推动众多应用领域的大规模模拟中发挥了核心作用。本章概述现代区域分解方法，特别强调针对挑战性问题（包括不定问题和高频区域）的可扩展预条件技术。我们重温基本概念——重叠分解、单位分解、加性 Schwarz 和限制性 Schwarz 格式——并阐释其代数解释。进而阐明它们作为 Krylov 子空间求解器中预条件的角色，并讨论为实现可扩展性而引入粗空间校正的必要性。除综述层面外，本章还提炼了过去二十年间出现的关键理论洞见和实用设计原则。特别关注鲁棒粗空间（GenEO，基于 DtN 的方法）和高性能实现。目标是为读者提供领域的一贯概述和一份简洁、面向实践的指南，使其无需遍览全部文献即可理解和应用区域分解方法。
4. 多组分可压缩欧拉流保结构格式的收敛性分析
Convergence analysis of structure-preserving schemes for the multicomponent compressible Euler flows
作者: Jaya Agnihotri, Philipp Öffner
类别: math.NA
发布日期: 2026-05-24
arXiv ID: 2605.24898v1
链接: http://arxiv.org/abs/2605.24898v1
摘要翻译
中文标题: 多组分可压缩欧拉流保结构格式的收敛性分析
摘要翻译: 我们在耗散弱解（DW）解的框架下，对多组分可压缩欧拉系统的有限体积（FV）格式进行了收敛性分析。DW解被引入作为计算流体力学中的广义解框架，近年来得到了广泛关注。它们将著名的Lax等价定理推广到了非线性情形，这意味着如果一个数值格式既相容又稳定，那么它也会收敛。所考虑的FV格式保留了流体混合物的关键物理性质，特别是分密度、压力和温度的正性。利用一致的稳定性界和相容性估计，我们证明了数值解在多组分欧拉系统的DW解框架中收敛。应用相对熵方法和弱强唯一性原理，我们进一步表明，只要经典解存在，近似解就强收敛到该经典解。数值实验证实了理论结果，不仅适用于低阶FV方法，还通过对高阶保结构间断伽辽金格式的扩展数值研究得到了验证。
5. 使用等几何有限元法在曲面上模拟高斯随机场
Simulation of Gaussian random fields on surfaces using the isogeometric finite element method
作者: Helmut Harbrecht, Florian Sonderegger, Remo von Rickenbach
类别: math.NA
发布日期: 2026-05-24
arXiv ID: 2605.24882v1
链接: http://arxiv.org/abs/2605.24882v1
摘要翻译
中文标题: 使用等几何有限元法在曲面上模拟高斯随机场
摘要翻译: 我们关注由 (Whittle-)Matérn 类协方差函数生成的 $\mathbb{R}^3$ 中封闭曲面上随机场的快速模拟。为此，我们利用曲面上的等几何有限元方法结合解的 Balakrishnan 积分表示，求解带有加性白噪声的分数阶随机偏微分方程。通过等几何方法自然蕴含的几何多重网格方法求解底层线性方程组。给出数值结果以演示该方法。
6. IV-Net：一种用于带有随机且高度变化系数的椭圆型偏微分方程的神经网络
IV-Net: A neural network for elliptic PDEs with random and highly varying coefficients
作者: Shan Zhong, George Biros
类别: math.NA, cs.LG
发布日期: 2026-05-24
arXiv ID: 2605.24876v1
链接: http://arxiv.org/abs/2605.24876v1
摘要翻译
中文标题: IV-Net：一种用于带有随机且高度变化系数的椭圆型偏微分方程的神经网络
摘要翻译: 我们提出了一种新颖的神经算子架构，旨在逼近具有高对比度、空间变化系数的线性椭圆型偏微分方程的解。该网络称为迭代V形网（IV-Net），实现了从输入系数和右端项到相应解场的映射。IV-Net的架构受到V循环多重网格求解器的启发，并与其高度相似。IV-Net模型通过定义在物理域中的卷积层进行参数化。对于具有高度非均匀系数的强制问题，与适当正交分解（POD）方法和若干现有神经算子架构相比，所提出的网络展现出更优的性能。对于具有光滑系数的低频振荡亥姆霍兹问题，其性能与傅里叶神经算子相当。我们分析了IV-Net的逼近误差与收敛行为、数据效率，以及其对底层离散化网格的依赖性。此外，我们通过一系列数值实验验证了该架构的实际有效性，包括在不确定性量化、反问题以及感兴趣量预测中的应用。
7. 面向周期性全势电子结构计算的等几何-平面波耦合混合间断伽辽金方法
A Hybrid Discontinuous Galerkin Method with Isogeometric-Planewaves Coupling for Periodic Full-Potential Electronic Structure Calculations
作者: Baowei Lai, Xucheng Meng, Xiaoxu Li
类别: math.NA
发布日期: 2026-05-24
arXiv ID: 2605.24829v1
链接: http://arxiv.org/abs/2605.24829v1
摘要翻译
中文标题: 面向周期性全势电子结构计算的等几何-平面波耦合混合间断伽辽金方法
摘要翻译: 周期性系统的全势电子结构计算在原子核处保留库仑奇性，这导致轨道在核附近呈现尖点行为，而间隙区域保持光滑。这种多尺度正则性促使了一种混合离散框架，该框架将原子补丁中的局部实空间离散与间隙区域中的平面波相结合。本文在原子补丁中使用张量积B样条，并通过对称内罚间断伽辽金公式耦合两个逼近空间，产生等几何-平面波（IGA-PW）方法。为了高效计算受限平面波积分，我们开发了一种结合快速傅里叶变换和切比雪夫修正的策略。我们还构造了一个迹块DG预条件子，以缓解由SIPG罚项引起的条件数恶化，并加速迭代特征求解器。此外，我们针对相应的线性特征值问题建立了{\it a priori}误差估计，证明了在核附近的代数收敛和间隙区域的超代数收敛。数值实验证明了所提方法的准确性和高效性。
8. 活性液晶与固体两相流体动力学模型的高效数值格式
Efficient Numerical Schemes for a Two-Phase Hydrodynamical Model of Active Liquid Crystals and Solids
作者: Xuelong Gu, Guanghua Ji, Qi Wang
类别: math.NA
发布日期: 2026-05-23
arXiv ID: 2605.24758v1
链接: http://arxiv.org/abs/2605.24758v1
摘要翻译
中文标题: 活性液晶与固体两相流体动力学模型的高效数值格式
摘要翻译: 我们针对一个基于Q-张量的新型两相流体动力学模型, 提出若干具有一阶和二阶时间精度的线性、完全解耦数值格式, 该模型描述了活性向列型液晶溶液与各向同性固体基底之间的耦合. 该模型由广义Onsager原理导出, 并包含对总自由能耗散贡献为零的非平凡项. 我们证明了所提出的解耦线性格式在离散层次是热力学相容的. 在被动极限下, SGE-BDF1 和 SGE-PDG 格式无条件能量稳定, 而 SGE-BDF2 格式在标准有界性假设和充分大的稳定化参数下关于修正能量能量稳定. 我们进行了大量数值模拟, 研究活性及其它模型参数如何影响活性向列型流-固相互作用. 最后, 我们分析了所观察到的行为背后的物理机制, 为受限软活性向列型流体的动力学提供了更深入的理解.
不相关论文（基本信息）
1. 用于有限元方法预处理的量子区域分解
Quantum Domain Decomposition for Preconditioning the Finite Element Method
作者: Elise Fressart, Michel Nowak, Nicole Spillane
类别: math.NA, quant-ph
发布日期: 2026-05-25
arXiv ID: 2605.26090v1
链接: http://arxiv.org/abs/2605.26090v1
摘要: Even in cases where quantum linear solvers provide significant speedup compared to their classical counterparts, their performance depends on some of the same parameters. In particular, the condition number of the matrix which is to be inverted is a decisive parameter. A well known classical, and now quantum, remedy is to precondition the linear system $A x = b$ by premultiplying it by a matrix $H$ in such a way that the condition number of $HA$ is significantly smaller than the condition number of $A$. In this work, we focus on a family of preconditioners called domain decomposition. First, we prove that it is feasible to apply quantum domain decomposition. We provide upper bounds for the block-encoding parameters of the Poisson problem discretized by the finite element method and preconditioned by the two-level Additive Schwarz preconditioner (one of the most fundamental domain decomposition techniques). From these bounds, we deduce the complexity of the quantum linear system solver. Second, we focus on a particular choice of local solver within the domain decomposition preconditioner by applying recent work by [Deiml and Peterseim, \textit{Math. Comput.}, 2025] on the Bramble–Pasciak–Xu (BPX) preconditioner. Finally, we provide details on how the operators are implemented.
2. 用于变阶时间分数阶波动方程的全离散基于能量的间断伽辽金方法
A Fully Discrete Energy-Based Discontinuous Galerkin Method for Variable-Order Time-Fractional Wave Equations
作者: Lu Zhang
类别: math.NA
发布日期: 2026-05-25
arXiv ID: 2605.26054v1
链接: http://arxiv.org/abs/2605.26054v1
摘要: Variable-order time-fractional wave equations provide a flexible model for wave phenomena with evolving memory effects and anomalous temporal dynamics. Their numerical approximation is challenging because the variable-order fractional derivative generates time-dependent history weights and therefore lacks the standard time-translation-invariant convolution structure of constant-order fractional operators. In this paper, we develop and analyze a fully discrete energy-based discontinuous Galerkin (DG) method for wave equations with a Caputo-type variable-order time-fractional derivative. The equation is reformulated as a reduced first-order-in-time system, discretized in space by an energy-based DG method, and advanced in time using a second-order approximation of the variable-order Caputo derivative at a specially chosen point in each time interval. The main analytical novelty is a cumulative weight-variation estimate for the variable-order memory weights, which requires only that the variable order $α:[0,T] \rightarrow (0,1)$ be Lipschitz continuous. Based on this estimate, we establish energy stability of the fully discrete scheme and derive second-order temporal convergence together with energy-norm spatial error estimates. The analysis gives suboptimal convergence on general affine simplicial or tensor-product meshes and optimal convergence under additional Cartesian and flux assumptions. Numerical experiments in one and two dimensions validate the theoretical findings.
3. 基于黎曼优化的大规模系统非线性降维快速二次流形学习
Fast Quadratic Manifold Learning For Nonlinear Dimensionality Reduction in Large-scale Systems using Riemannian Optimization
作者: Gavin Paxton, Seunghee Cheon, Rudy Geelen, Shane A. McQuarrie
类别: math.NA
发布日期: 2026-05-25
arXiv ID: 2605.26039v1
链接: http://arxiv.org/abs/2605.26039v1
摘要: We present a dynamic subspace approach for efficiently approximating large-scale systems by learning time-continuous trajectories on the Grassmannian manifold. By parameterizing a low-dimensional basis as a geodesic path, the method allows for adaptive tracking of evolving physics. Our approach decouples the geometric drift of the subspace from the intrinsic state evolution. This avoids the typical rank inflation required by static low-dimensional approximation methods to maintain accuracy, effectively breaking the Kolmogorov barrier in transport-dominated phenomena. To ensure scalability for high-dimensional data, the optimization is performed in a reduced feature space, rendering the computational cost independent of the large original state dimension. Numerical results for a 1D transport equation and a large-scale turbulent airfoil wake demonstrate that this dynamic subspace approach achieves higher accuracy than static linear approximations at equivalent ranks, positioning it as a robust and scalable method for the low-rank modeling of complex, non-stationary dynamical systems.
4. 闵可夫斯基空间中标量散射的超双曲面演化
Hyperboloidal evolution for scalar scattering in Minkowski space
作者: Ekrem S Demirboğa, Anıl Zenginoğlu
类别: gr-qc, math.NA
发布日期: 2026-05-25
arXiv ID: 2605.26028v1
链接: http://arxiv.org/abs/2605.26028v1
摘要: We develop a time-domain numerical framework for global scalar wave scattering in Minkowski spacetime. The main contribution is an exact conformal matching of three compactified regions: a past hyperboloidal domain attached to $\mathscr I^-$, a Penrose domain covering a neighborhood of spatial infinity $i^0$, and a future hyperboloidal domain attached to $\mathscr I^+$. The matching surfaces are identical conformal hypersurfaces in the adjacent charts. This yields a global evolution scheme connecting $\mathscr I^-$, the neighborhood of $i^0$, and $\mathscr I^+$ without artificial timelike outer boundaries and without interpolation between scri-fixing gauges. We implement the construction for spherically symmetric scalar waves, including free propagation, localized linear scattering potentials such as the Pöschl–Teller potential, and semilinear wave equations with cubic, quintic, and septic nonlinearities. The numerical experiments demonstrate stable propagation across the matching interfaces, direct extraction of radiation at $\mathscr I^+$, and fourth-order convergence for the free and linear-potential tests. The quintic and septic nonlinear tests exhibit approximately fourth-order convergence and recover the expected late-time tail rates. The cubic case, by contrast, shows only first-order convergence, revealing a limitation of our treatment near compactified boundaries when the conformally rescaled nonlinear source remains non-vanishing. These results validate the conformal matching strategy for long-time simulations, while identifying the boundary regularity issues that must be addressed using a more robust treatment of spatial infinity.
5. 一种大规模非线性系统低秩近似的动态子空间方法
A Dynamic Subspace Approach for Low-rank Approximation of Large-scale Nonlinear Systems
作者: Jack DeChant, Rudy Geelen, Shane A. McQuarrie, Johann Guilleminot
类别: math.NA
发布日期: 2026-05-25
arXiv ID: 2605.26025v1
链接: http://arxiv.org/abs/2605.26025v1
摘要: We present a dynamic subspace approach for efficiently approximating large-scale systems by learning time-continuous trajectories on the Grassmannian manifold. By parameterizing a low-dimensional basis as a geodesic path, the method allows for adaptive tracking of evolving physics. Our approach decouples the geometric drift of the subspace from the intrinsic state evolution. This avoids the typical rank inflation required by static low-dimensional approximation methods to maintain accuracy, effectively breaking the Kolmogorov barrier in transport-dominated phenomena. To ensure scalability for high-dimensional data, the optimization is performed in a reduced feature space, rendering the computational cost independent of the large original state dimension. Numerical results for a 1D transport equation and a large-scale turbulent airfoil wake demonstrate that this dynamic subspace approach achieves higher accuracy than static linear approximations at equivalent ranks, positioning it as a robust and scalable method for the low-rank modeling of complex, non-stationary dynamical systems.
6. 动态反问题：在线正则化理论
Dynamic inverse problems: Online regularisation theory
作者: Jyrki Jauhiainen, Yassine Nabou, Tuomo Valkonen
类别: math.NA, math.OC
发布日期: 2026-05-25
arXiv ID: 2605.26022v1
链接: http://arxiv.org/abs/2605.26022v1
摘要: We develop regularisation theory for dynamic inverse problems, solved using online methods with an infinite time horizon. Using concepts of subregularity to treat nonsmooth regularisers, we prove that time-averaged reconstruction errors converge to zero as noise, algorithmic errors, and regularisation vanish as the horizon grows. We illustrate the theory numerically with a dynamic electrical impedance tomography example.
7. Fuzzy PyTorch：深度学习模型的快速数值变异性评估
Fuzzy PyTorch: Rapid Numerical Variability Evaluation for Deep Learning Models
作者: Inés Gonzalez-Pepe, Hiba Akhaddar, Tristan Glatard, Yohan Chatelain
类别: cs.LG, math.NA
发布日期: 2026-05-25
arXiv ID: 2605.25991v1
链接: http://arxiv.org/abs/2605.25991v1
摘要: We introduce Fuzzy PyTorch, a framework for rapid evaluation of numerical variability in deep learning (DL) models. As DL is increasingly applied to diverse tasks, understanding variability from floating-point arithmetic is essential to ensure robust and reliable performance. Tools assessing such variability must be scalable, efficient, and integrate seamlessly with existing frameworks while minimizing code modifications. Fuzzy PyTorch enables this by integrating stochastic arithmetic into PyTorch through Probabilistic Rounding with Instruction Set Management, a novel library interfacing with Verificarlo, a numerical analysis compiler. The library offers stochastic rounding mode and a novel mode; up-down rounding. Comparative evaluations show Fuzzy PyTorch maintains model performance and achieves runtime reductions of 5x to 60x versus Verrou, a state-of-the-art tool. We further demonstrate scalability by running models from 1 to 341 million parameters, confirming applicability across small and large DL architectures. Overall, Fuzzy PyTorch provides an efficient, scalable, and practical solution for assessing numerical variability in deep learning, enabling researchers and practitioners to quantify and manage floating-point uncertainty without compromising performance or computational efficiency.
8. 双曲动力学松弛的良好平衡格式
Well-Balanced Schemes for Hyperbolic Kinetic Relaxation
作者: León Ávila, Manuel Castro, Victor Michel-Dansac, José M. Gallardo
类别: math.NA
发布日期: 2026-05-25
arXiv ID: 2605.25887v1
链接: http://arxiv.org/abs/2605.25887v1
摘要: This work presents a novel family of well-balanced numerical schemes for hyperbolic systems of balance laws based on the kinetic relaxation approach. The method begins by transforming the original non-linear system into a linearized kinetic system with an increased number of variables. In this framework, non-linearities are shifted to the source term, and the connection to the macroscopic variables is maintained via a projection operator related to the Maxwellian equilibrium states. These relaxed systems are typically solved using splitting techniques, where the evolution is decomposed into two distinct steps: first, the transport stage solving the source-free kinetic system, and second, the relaxation stage governed by an ordinary differential equation where the time derivative is balanced by the kinetic source term. The primary objective is to design schemes that accurately preserve the steady-state solutions of the original macroscopic equations, noting that these do not necessarily coincide with the trivial equilibria of the relaxed system. This well-balanced property is achieved through an appropriate formulation of the kinetic system combined with consistent reconstruction operators. This approach ensures that the balance is maintained throughout both the transport stage and the projection stage, where the treatment of the macroscopic source term is carefully integrated. We demonstrate that this framework can be implemented using both Finite Volume (FV) and Semi-Lagrangian (SL) techniques in the transport step. We develop and analyze these schemes for the Finite Volume case in both explicit and implicit formulations. Results are validated through numerical benchmarks, including the Burgers equation with a source term, 1D Shallow Water Equations (SWE) with bathymetry preserving non-trivial moving water equilibria and the Euler equations under external potentials.
9. 具有粗糙单轨迹信号的ODE的分支签名核求解器
Branched Signature Kernel Solvers for ODEs with rough Single-Trajectory signals
作者: Munawar Ali, Qi Feng, Charlie Pyle, George Xu
类别: math.NA, cs.CE, cs.LG
发布日期: 2026-05-25
arXiv ID: 2605.25826v1
链接: http://arxiv.org/abs/2605.25826v1
摘要: We develop a branched signature kernel solver for linear and nonlinear ordinary differential equations driven by a \emph{single observed trajectory} of a possibly rough forcing signal – a setting that arises naturally in earthquake engineering, finance, biology, and structural health monitoring, where the forcing is observed exactly once and the solver must respect the underlying physical law without recourse to an ensemble of realizations. Two ingredients are new. First, a \emph{count-sampling} construction turns the single observation into a hierarchical family of $N+1$ nested training paths on which the branched signature kernel can be evaluated; this allows the signature kernel machinery, originally designed for multi-realization regression problems, to operate on a single-trajectory observation. Second, a kernel-collocation framework places the ansatz either on the highest-order derivative of the solution (with lower derivatives recovered by integrating the kernel) or on the solution itself (after $m$-fold integration of the ODE). We prove a universal approximation theorem for the branched signature kernel, leveraging the Hairer–Kelly morphism to express branched signature evaluations through geometric signatures of time-extended paths. The offline solver is extended to a streaming Test/Train/Retrain protocol with closed-form online updates in the linear case and scalar Newton steps in the nonlinear case. Numerical experiments on six benchmarks (El-Centro earthquake displacement, the Solow capital-stock model, an fBM-driven second-order ODE, a forced Duffing oscillator, a path-dependent Arias-intensity-degraded oscillator with variable coefficients, and a noisy Kuramoto phase-oscillator system) show that the branched signature-kernel solver delivers accurate, stable predictions across all regimes.
10. 一种适用于超定最小二乘问题的多层概略-求解方法
A multilevel sketch-and-solve method for overdetermined least squares problems
作者: Irina-Beatrice Haas, Michael B. Giles, Yuji Nakatsukasa
类别: math.NA, stat.CO
发布日期: 2026-05-25
arXiv ID: 2605.25809v1
链接: http://arxiv.org/abs/2605.25809v1
摘要: Sketch-and-solve (SAS) is a very successful method to efficiently estimate the solution of heavily overdetermined large linear least squares problems. It uses random sketching to reduce the size of the problem, hence reducing the computational cost. Several authors have shown that averaging several solutions from SAS further improves the accuracy, which is measured by the residual associated to the approximate solution. Going further, we combine solutions from sketch-and-solve in a multilevel manner, such that the approximate solution is a combination of SAS samples obtained from small sketches and more accurate correction terms obtained from larger sketches. We first consider the variance of the estimator, which depends on the variance of the coarse samples and the correction terms. We show that the variance of the correction terms on each level follows a trend and decreases faster than the variance of the simple SAS estimator. However, we then show that the overall computational cost of our multilevel framework is slightly higher than that of the simple average estimator, so a naive application of multilevel methods appears unattractive for least squares problems.
11. 带相关噪声的高维非线性滤波问题的张量列车方法
Tensor train methods for high-dimensional nonlinear filtering problems with correlated noise
作者: Yuhua Meng, Stephen S. -T. Yau, Zhiwen Zhang
类别: math.NA
发布日期: 2026-05-25
arXiv ID: 2605.25677v1
链接: http://arxiv.org/abs/2605.25677v1
摘要: Nonlinear filtering with correlated noise leads to a Duncan-Mortensen-Zakai (DMZ) equation in the form of a stochastic partial differential equation (SPDE). Unlike the independent noise case, the presence of correlation prevents the classical invertible transformation that reduces the DMZ equation to a deterministic partial differential equation, requiring a direct numerical treatment of the SPDE. This paper develops a tensor train (TT) based framework for solving medium- to high-dimensional DMZ equations with correlated noise. Spatial discretization transforms the SPDE into a high-dimensional stochastic differential system, which is efficiently compressed using TT approximation. A semi-implicit Milstein scheme is employed for temporal integration to ensure stability and accuracy. Under suitable regularity assumptions, we establish a convergence analysis of the proposed method. In particular, the spatial error is controlled by both the mesh size and the prescribed TT approximation accuracy. In the temporal direction, the convergence is proved by estimating stochastic integrals involving drifted observations, without invoking a change-of-measure argument. Numerical experiments demonstrate that the proposed method achieves stable and accurate performance for cubic sensor problems. In challenging multi-modal settings, where particle filter and extended Kalman filter deteriorate, the proposed method maintains accuracy and effectively captures the posterior distribution.
12. 用于对角占优对称矩阵的雅可比型特征值求解器
A Jacobi-Type Eigensolver for Diagonally Dominant Symmetric Matrices
作者: Luca Gemignani
类别: math.NA
发布日期: 2026-05-25
arXiv ID: 2605.25613v1
链接: http://arxiv.org/abs/2605.25613v1
摘要: This paper presents a Jacobi-type iteration for computing a given specified eigenpair of a symmetric matrix. For a certain class of diagonally dominant matrices, the procedure is shown to converge at a linear rate depending on how the matrix is significantly dominated. The cost per iteration is generally quadratic. Therefore, the proposed procedure can compute an approximation of the desired eigenpair in quadratic time.
13. 弯曲水平集区域椭圆型偏微分方程的相容CutPINNs
Consistent CutPINNs for Elliptic PDEs on Curved Level-Set Domains
作者: Maneesh Kumar Singh
类别: math.NA
发布日期: 2026-05-25
arXiv ID: 2605.25562v1
链接: http://arxiv.org/abs/2605.25562v1
摘要: We propose \emph{Consistent CutPINN}, a framework for partial differential equations posed on bounded curved domains defined implicitly by a $\C^2$ level-set function, $Ω= \{\varphi < 0\}$. In this paper we develop the framework for second-order elliptic problems in two dimensions. The standard PINN loss penalises the boundary mismatch in $L^2(\partialΩ)$, but $L^2(\partialΩ)$ does not control the $H^{1/2}(\partialΩ)$ trace norm that appears in the $H^1(Ω)$ energy estimate. The consistent PINN framework of Bonito et al.~\cite{bonito2025} fixes this on the unit cube $(0,1)^d$ via a Kuhn–Tucker simplicial decomposition of the flat boundary faces, but the construction relies on the affine structure of the faces and does not carry over to smooth curved boundaries. We address this gap. Specifically, (i) we introduce a discrete $H^{1/2}(\partialΩ)$ surrogate built directly from collocation points on a $\C^2$ curve, (ii) we prove a \textit{Chord-arc} norm equivalence between this surrogate and the continuous trace norm, (iii) we establish an \emph{a priori} $H^1$ error bound on cut domains, and (iv) we derive convergence rates under Besov regularity using optimal recovery theory. Numerical experiments on a disk and a non-convex flower domain confirm that the consistent loss is much more accurate than the standard PINN loss and far more robust to cut-cell configurations.
14. 欧几里得空间中非光滑凸优化的镜面梯度法：一种次梯度选择策略
Specular gradient methods for nonsmooth convex optimization in Euclidean spaces: a subgradient selection strategy
作者: Kiyuob Jung
类别: math.OC, math.NA
发布日期: 2026-05-25
arXiv ID: 2605.25490v1
链接: http://arxiv.org/abs/2605.25490v1
摘要: This paper deals with nonsmooth convex optimization problems in Euclidean spaces. We identify special elements of the subdifferential of a convex function, called specular gradients. Based on this observation, we propose three numerical methods that use specular gradients in subgradient methods. We prove the convergence of the proposed methods under suitable step sizes. Numerical experiments demonstrate that the proposed methods are capable of minimizing non-differentiable functions that classical methods fail to minimize.
15. 用于含时偏微分方程的无自回归神经算子
Autoregression-Free Neural Operators for Time-Dependent PDEs
作者: Jiaquan Zhang, Caiyan Qin, Haoyu Bian, Libin Cai, Yi Lu, Chaoning Zhang, Wei Dong, Yuanfang Guo, Yang Yang, Hen Tao Shen
类别: cs.LG, cs.AI, math.NA
发布日期: 2026-05-25
arXiv ID: 2605.25413v1
链接: http://arxiv.org/abs/2605.25413v1
摘要: Neural operators learn mappings from function-dependent inputs to solutions, providing an effective framework for solving partial differential equations (PDEs). For time-dependent PDEs, existing methods typically perform long-horizon prediction through autoregressive rollout directly in high-dimensional physical field spaces, where each predicted state is recursively fed back as the input for the next step. Although effective for short-term prediction, this autoregressive rollout and the lack of continuous-time modeling lead to progressive error accumulation over long-horizon rollouts. In this work, we propose Autoregression-Free Neural Operators (AFNO), which map the time evolution of PDEs into a latent space and model continuous-time vector fields within it. AFNO uses flow matching to learn the latent vector field, thereby enabling continuous evolution over extended horizons, avoiding autoregressive rollout and capturing dynamics under varying parameter configurations through explicit conditioning on physical parameters. Theoretical analysis and extensive experiments on six PDEs demonstrate that AFNO improves long-horizon prediction stability and consistently reduces rollout errors compared with the baselines.
16. 一种具有宿主内病毒载量和抗体动态的媒介传播疾病结构化模型
A structured model of vector-borne disease with within-host viral load and antibody dynamics
作者: Paulo Amorim, Maria Soledad Aronna, Debora de Oliveira Medeiros
类别: math.AP, math.NA
发布日期: 2026-05-25
arXiv ID: 2605.25370v1
链接: http://arxiv.org/abs/2605.25370v1
摘要: We present an epidemiological model for vector-borne diseases that includes within-host viral load and antibody dynamics using structured transport equations. By incorporating the internal dynamics into the infected and recovered host compartments, the formulation introduces nonlinearities and nonlocalities. We establish analytical properties, including well-posedness and mass conservation, and characterize its characteristic curves. Furthermore, we derive a simplified Uniform Host Response (UHR) model featuring delay-type terms. For both the full and UHR frameworks, the basic reproduction number is determined and shown to serve as a threshold for the existence of an endemic equilibrium, and is related to the linear stability of the disease-free state. Finally, numerical experiments, parameterized specifically for Dengue fever, demonstrate how within-host mechanisms influence population-level epidemiological outcomes.
17. 计算马赫数2000天体物理喷流的弱-强唯一性
Computing weak-strong uniqueness of a Mach 2000 astrophysical jet
作者: Stephan Simonis, Gauthier Wissocq
类别: math.NA, cs.MS, physics.comp-ph, physics.flu-dyn
发布日期: 2026-05-24
arXiv ID: 2605.25282v1
链接: http://arxiv.org/abs/2605.25282v1
摘要: The simulation of extreme Mach astrophysical flows is traditionally viewed through the lens of deterministic positivity-preserving schemes. However, due to phenomena such as Kelvin–Helmholtz instabilities and shock anomalies, the multi-dimensional Euler equations admit a plethora of non-unique entropy solutions in turbulent regimes. For the first time, we computationally explore the weak-strong uniqueness of a Mach 2000 jet by defining the statistical solution as the pushforward of a probability measure through a vectorial lattice Boltzmann method (VLBM) operator. Utilizing highly optimized CUDA kernels, we compute an ensemble of 1000 Monte Carlo samples across a sequence of unprecedentedly refined spatial grids of up to 3.2 million cells, and subsequently post-process the empirical measures via memory-mapped CPU streaming. We contrast the strong sample-wise $L^1$ error divergence with the convergence of the probability measure in the 1-point Wasserstein distance via empirical Cauchy rates. Our mathematical results demonstrate that while individual flow realizations physically diverge due to chaotic shear-layer instabilities, the macroscopic statistical solution converges to a well-defined limit measure at a rate of 0.5. Conclusively, we provide the first numerical verification of statistical solution stability in the extreme compressible regime.
18. 采样分布作为学习型反问题中的正则化
Sampling Distributions as Regularization in Learned Inverse Problems
作者: Sandra R. Babyale, Jodi Mead
类别: math.NA
发布日期: 2026-05-24
arXiv ID: 2605.25177v1
链接: http://arxiv.org/abs/2605.25177v1
摘要: Neural networks have emerged as effective tools for solving ill-posed inverse problems. In many scientific applications, however, observational training data are insufficient, and learned inverse operators must instead be trained on synthetic data generated from the forward model. This requires specifying unknown parameters in the forward model and solving the model to generate synthetic observations. Typically, the unknown parameters are sampled from a prescribed probability distribution. Here, we show that this sampling strategy is not a neutral preprocessing step, but instead defines an implicit regularization operator. This result follows from the fact that the learned inverse operator minimizes empirical risk together with the classical result that conditional expectation minimizes mean-square error. We present theoretical results for the implicit regularization operator in both infinite- and finite-data settings, including Physics Informed Neural Networks (PINNs). These results are demonstrated numerically on three inverse problems of increasing complexity: a 1D linear Fredholm integral equation, a 1D nonlinear subsurface interface inversion, and a 2D nonlinear cross-well seismic traveltime tomography problem. Across all three problems, three distinct sources of regularization are identified in the learned operator: prior sampling, architectural, and physics-informed regularization. A mismatched sampling distribution is shown to degrade reconstruction quality in ways that neither more expressive architectures nor augmented physics residuals can fully correct. The results demonstrate that the sampling distribution should be chosen with the same care as a classical regularization functional and provide a practical framework for implementing more sophisticated regularization operators using neural networks.
19. Arnoldi增强的流形值数据多元Hermite插值
Arnoldi-Enhanced Multivariate Hermite Interpolation of Manifold-Valued Data
作者: Yuxuan Li, Qiang Niu, Wubin Zhou
类别: math.NA
发布日期: 2026-05-24
arXiv ID: 2605.25176v1
链接: http://arxiv.org/abs/2605.25176v1
摘要: This paper presents a robust enhancement of the Tangent space Hermite Interpolation (THI) method for manifold-valued data by integrating the multivariate Arnoldi process. To circumvent the inherent numerical instability of multivariate confluent Vandermonde matrices, we use a $G$-Arnoldi-based recurrence to construct a discrete orthogonal polynomial basis directly on the tangent space. The method generates better numerical conditioning for high-order approximations. We analyze the convergence rates for both $C^0$ and $C^1$ errors in the multivariate setting. When only function values are used, the $C^0$ approximation error decays as $\mathcal{O}\left(\sqrt{M} n^{-m}\right)$. For the $C^1$ error without derivative data, the rate becomes $\mathcal{O}\left(\sqrt{M} h^{-1} n^{-m}\right)$, where $h$ is the fill distance of the sampling set. When derivative data are additionally available, the $C^1$ error is $\mathcal{O}\left(\sqrt{M} n^{-(m-1)}\right)$. In all cases, $n$ is the polynomial degree, $m$ denotes the regularity of the target function, and $M$ is the number of sampling points. Importantly, as $n$ increases, the required number of points $M$ must also increase. This reveals the interplay among approximation order, sampling density ($M$), fill distance ($h$), dimension ($d$), and the regularity ($m$) of the target function. Extensive numerical experiments conducted on the special orthogonal group $SO(3)$ and the unit sphere $S^2$ show that the Arnoldi-enhanced THI method outperforms the Kriging-based approaches in terms of both computational efficiency and accuracy.
20. 基于逆算子的拟正交方法求解 Schr{ö}dinger 特征值问题
A quasi-orthogonal method based on the inverse operator for Schr{ö}dinger eigenvalue problems
作者: Shengyue Wang, Aihui Zhou
类别: math.NA
发布日期: 2026-05-24
arXiv ID: 2605.25149v1
链接: http://arxiv.org/abs/2605.25149v1
摘要: Computing many eigenpairs of the Schr{ö}dinger operator presents a computational bottleneck in large-scale quantum simulations due to the global communication overhead of explicit orthogonalization. To address this issue, we propose a quasi-orthogonal evolution model utilizing inverse operators and develop a corresponding discrete numerical scheme. Instead of forcing explicit orthogonalization, the proposed framework confines the numerical approximations within a quasi-Stiefel set, ensuring the iterates maintain full column rank without requiring $\left\langle U, U \right\rangle=I_N$. Moreover, the method naturally absorbs orthogonality errors and asymptotically converges to the exact eigenfunctions, even when initialized with non-orthogonal random data. The scheme guarantees monotonic dissipation of the target energy functional, with exponential convergence rates rigorously established for the discrete energy, gradient, and eigenfunction approximations. Furthermore, infinite-dimensional analysis proves that the admissible time step size is independent of the spatial discretization. This property overcomes the mesh-dependent stability constraints typical of conventional explicit or semi-implicit schemes, permitting larger time increments to accelerate global convergence. Numerical experiments validate the theoretical findings.
21. 利用规范自由度学习随机系统的非梯度种群动力学
Leveraging Gauge Freedom for Learning Non-Gradient Population Dynamics of Stochastic Systems
作者: Jules Berman, Tobias Blickhan, Benjamin Peherstorfer
类别: cs.LG, cs.AI, math.NA
发布日期: 2026-05-24
arXiv ID: 2605.25107v1
链接: http://arxiv.org/abs/2605.25107v1
摘要: Existing work on population dynamics inference often focuses on flows arising from vector fields that are the gradients of scalar potentials. Among all admissible flows that are compatible with the population dynamics, gradient flows are optimal in a specific sense: they minimize kinetic energy. The selection of fields based on different criteria corresponds to a gauge freedom when determining population dynamics, which we leverage in this work. We propose Non-Gradient Inference Flows (NGIF), an algorithm to infer non-gradient population dynamics using a weak formulation of the continuity equation. This allows us to parameterize general vector fields and choose other selection criteria beyond minimal kinetic energy. We demonstrate on a variety of low- and high-dimensional physics problems that this more general approach improves distributional accuracy over gradient-restricted baselines and better captures non-potential transport.
22. 随机神经网络对非线性偏微分方程的表达能力
Random Neural Network Expressivity for Non-Linear Partial Differential Equations
作者: Muhammed Ali Mehmood, Lukas Gonon
类别: math.NA, cs.LG
发布日期: 2026-05-24
arXiv ID: 2605.25057v1
链接: http://arxiv.org/abs/2605.25057v1
摘要: Neural networks with randomly generated hidden weights (RaNNs) have been extensively studied, both as a standalone learning method and as an initialization for fully trainable deep learning methods. In this work, we study RaNN expressivity for learning solutions to non-linear partial differential equations (PDEs). Despite their widespread use in practical applications, a rigorous theoretical understanding of the approximation properties of RaNNs in this context remains limited. Here, we derive error bounds for RaNN approximations to time-dependent Sobolev functions and obtain a dimension-free approximation rate $\frac{1}{2}$ for sufficiently regular functions. We apply our results to two important classes of non-linear PDEs: Porous Medium Equations and Compressible Navier-Stokes Equations, showing that RaNNs are capable of efficiently approximating solutions to these complex, non-linear PDEs. Our theoretical analysis is supported by numerical experiments, showing that the obtained convergence rates extend beyond the considered setting.
23. 随机化共轭梯度最小二乘方法
Randomized conjugate gradient least squares
作者: Yun Zeng, Jian-Feng Cai, Deren Han, Jiaxin Xie
类别: math.NA
发布日期: 2026-05-24
arXiv ID: 2605.25034v1
链接: http://arxiv.org/abs/2605.25034v1
摘要: We develop a novel randomized conjugate gradient least squares (RCGLS) method for solving least-squares problems, in which iterative sketching is employed at each step to reduce the dimension and hence the computational cost. In particular, we propose a new perspective on the classical CGLS method, where the next descent direction is determined via a constraint correction problem associated with the gradient. Based on this insight, we replace the gradient with a randomized coordinate gradient that naturally satisfies the variance reduction property, leading directly to the proposed RCGLS method. We prove that RCGLS converges linearly in expectation, with a better convergence bound compared to the randomized coordinate descent method. Furthermore, we investigate an implementation of the method that avoids full-dimensional vector operations, which are the major bottleneck of vanilla RCGLS for sparse matrices and render it impractical. We also show how to apply the RCGLS method to solve the ridge regression problem, yielding a lightweight, parallelizable, and accelerated method for such problems. Numerical experiments are provided to confirm our results.
24. 高维下子采样OLS与快速CUR的随机斜投影去偏
Debiasing Random Oblique Projections for Subsampled OLS and Fast CUR in High Dimensions
作者: Chengmei Niu, Sachin Garg, Michał Dereziński, Zhenyu Liao
类别: math.NA, stat.ML
发布日期: 2026-05-24
arXiv ID: 2605.24955v1
链接: http://arxiv.org/abs/2605.24955v1
摘要: Random sampling is a fundamental tool in modern machine learning and numerical linear algebra for reducing the computational cost of large-scale matrix problems. Existing analyses, however, rely primarily on subspace embedding guarantees, which do not precisely characterize the statistical bias of nonlinear random oblique projections induced by sampling, which arises ubiquitously in subsampled least squares and fast low-rank approximation methods. Because (pseudo)inversion is nonlinear, these random oblique projections can be systematically biased even when the underlying sketch is unbiased, thereby introducing hidden bias into downstream least squares and low-rank approximation solutions. In this work, we develop a unified non-asymptotic theory for random oblique projections in high dimensions. We show that standard random sampling schemes generally induce a systematic statistical bias overlooked by classical subspace embedding-style analyses, and we propose a principled debiasing framework to correct it. We illustrate the power of the theory through two canonical applications. For subsampled least squares, we obtain sharp bias–variance characterizations, reveal previously unrecognized statistical suboptimality in widely used sampling schemes, and identify when debiasing yields provable improvements. For fast CUR decomposition, we develop a debiased approach with improved approximation accuracy. Numerical experiments further validate our theoretical findings.
25. 具有局部Lipschitz漂移的随机微分方程的驯服Euler格式与随机Euler格式的误差估计及其在非对数凹抽样与优化中的应用
Error estimates for tamed Euler and Randomized Euler schemes for SDEs with locally Lipschitz drift with applications to non-logconcave sampling and optimization
作者: Iosif Lytras, Angelos Ntousis
类别: math.PR, math.NA, math.OC, stat.CO, stat.ML
发布日期: 2026-05-24
arXiv ID: 2605.24937v1
链接: http://arxiv.org/abs/2605.24937v1
摘要: In this paper, we study the numerical discretization of stochastic differential equations with locally Lipschitz, super-linearly growing drift, and the resulting implications for sampling from non-log-concave distributions satisfying a logarithmic Sobolev inequality. In this regime, the classical Euler–Maruyama scheme underlying the unadjusted Langevin algorithm (ULA) is known to be unstable. We analyze the KL-accelerated tamed unadjusted Langevin algorithm (kTULA) and introduce a new tamed randomized midpoint scheme, termed tRLMC. Building on the shifted-composition approach of \cite{chewi2024local}, we develop two new local-error frameworks that yield finite-time, non-asymptotic error estimates against the underlying SDE – in KL divergence for kTULA, and in total variation for tRLMC – valid for general locally Lipschitz drift. Specializing these frameworks to the sampling problem under a logarithmic Sobolev inequality, we obtain a near-optimal $\widetilde{O}(\varepsilon^{-1/2})$ iteration complexity for kTULA in KL divergence, with corresponding guarantees in total variation and Wasserstein distance. We further establish, for the first time, a non-asymptotic guarantee in total variation for a tamed randomized Langevin scheme under super-linear drift growth, together with the corresponding Wasserstein-distance bound, both with $\widetilde{O}(\varepsilon^{-1})$ complexity for tRLMC. As a consequence, both schemes yield non-asymptotic bounds for a non-convex excess-risk optimization problem.
26. 基于太阳能相控阵的商业航空无线电力传输可降低美国能源成本与碳排放
Solar phased arrays-based wireless power transfer for commercial airlines can reduce energy costs and carbon emissions in the United States
作者: Tianyi Wang, Yiming Xu, Jiseop Byeon, Junfeng Jiao, Javad Mohammadi, Kara Kockelman, Christian Claudel, Alexandre Bayen
类别: eess.SY, cs.ET, math.NA
发布日期: 2026-05-24
arXiv ID: 2605.24875v1
链接: http://arxiv.org/abs/2605.24875v1
摘要: Decarbonizing aviation remains challenging because energy-dense jet fuels dominate beyond short-range operations, while batteries impose severe range and payload penalties. Here we evaluate a new infrastructure pathway in which utility-scale solar farms equipped with solar phased arrays wirelessly beam microwave power to hybrid-electric aircraft during cruise. Integrating 143,152 U.S. flight trajectories, 5,712 solar farms and wireless power transfer models, we quantify the spatial, temporal, and operational potential of this concept at continental scale. We find that benefits are highly concentrated in solar-rich, traffic-dense states and are dominated by short- and medium-range flights, accounting for nearly all delivered energy and cost savings. Schedule optimization and higher cruise altitudes further increase value by improving alignment between aircraft demand and beaming availability. Market penetration analysis reveals non-linear scaling between solar farm and flight adoption. These results show that wireless power beaming is best understood as a corridor-specific strategy complementing other aviation decarbonization pathways.
27. 面向指定指标特征空间反射的移位矩阵符号反射子
Shifted Matrix-Sign Reflectors for Prescribed-Index Eigenspace Reflection
作者: Jin Zhao
类别: math.NA
发布日期: 2026-05-24
arXiv ID: 2605.24840v1
链接: http://arxiv.org/abs/2605.24840v1
摘要: Spectral projectors and the reflectors derived from them are basic objects in numerical linear algebra. This paper studies the prescribed-index reflector I-2P_k, where P_k is the spectral projector associated with the first k eigenvectors of a symmetric matrix. If a shift s lies in the target spectral gap, then this reflector is exactly the shifted matrix sign sign(H-sI). The exact identity is elementary, but its algorithmic consequences are not: all admissible shifts give the same exact reflector, while finite-step sign filters can have very different errors. We analyze odd sign-preserving spectral filters, prove local inheritance and discrete stability for the induced reflector iterations, derive a gap-dependent Newton-Schulz operator bound, and give deterministic admissibility certificates for inexact and reused shifts. The analysis identifies the shifted spectral margin as the quantity controlling finite-step reflector accuracy and explains why the midpoint shift is the natural default. Numerical experiments separate the matrix-function issues from the outer saddle-search dynamics: controlled spectra verify the margin predictions, low-dimensional tests distinguish shifted signs from raw signs, target-index scans probe non-small k, and Allen-Cahn and dense timing tests identify the regimes in which full-matrix sign filters are useful and the stiff regimes in which stronger sign engines are needed.
28. 弹性静力学中的弦核表示
String kernel representations in elastostatics
作者: Jeremy G. Hoskins, Alan E. Lindsay, Manas Rachh
类别: math.NA
发布日期: 2026-05-24
arXiv ID: 2605.24827v1
链接: http://arxiv.org/abs/2605.24827v1
摘要: In this paper we present a new boundary integral equation formulation for the solution of the elastostatic traction boundary value problem in two and three dimensions. The approach relies on the introduction of new layer potentials, called string kernels, which are based on modifications of the Boussinesq-Cerruti family of half-space solutions. We prove that the resulting integral equations are second-kind integral equations, and show that they are well-behaved in the incompressible limit. We illustrate the performance of the method with several numerical examples.
29. Cimmino反射算法的尖锐收敛率与最优权重
Sharp Convergence Rates and Optimal Weights for Cimmino’s Reflection Algorithm
作者: Hemant Sharma
类别: math.NA
发布日期: 2026-05-23
arXiv ID: 2605.24692v1
链接: http://arxiv.org/abs/2605.24692v1
摘要: In this paper, Cimmino’s classical reflection algorithm for solving the $n\times n$ nonsingular linear system $A\bx=\bb$ is analysed through the lens of spectral theory. Reformulating the weighted iteration as $\e^{(ν+1)}=M_w\,\e^{(ν)}$, where $M_w = I - A^\top D_w A$, the error is shown to contract by the spectral radius $\sprad(M_w)$ at every step, with a sharp, asymptotically tight bound. For $n=2$, a closed-form expression for the contraction factor is derived, [ \sprad(M_w) ;=; |1-μ|
- \tfrac{1}{2}\sqrt{(w_1-w_2)^2 + 4w_1w_2\cos^2!θ}, ] where $μ=(w_1+w_2)/2$ and $θ$ denotes the angle between the hyperplane normals. A central result of this paper is that the standard unit weights $w_1^*=w_2^*=1$ are \emph{globally optimal} over all positive weight pairs, uniquely achieving the minimum contraction factor $\sprad^*=|\cosθ|$ – a quantity determined solely by the geometry of the hyperplane normals. The inter-normal angle $θ$ thus emerges as the single diagnostic parameter governing both convergence speed and weight selection. Extensions to a single-step convergence criterion at $θ=π/2$ and to an exact spectral rate for general~$n$ are also established.
30. 基于个性化PageRank寻找Koopman不变子空间
Finding Koopman Invariant Subspaces via Personalized PageRank
作者: Hyukpyo Hong, Qin Li, Matthew J. Colbrook, Hanbaek Lyu
类别: math.DS, math.NA, stat.ML
发布日期: 2026-05-23
arXiv ID: 2605.24666v1
链接: http://arxiv.org/abs/2605.24666v1
摘要: Selecting a finite dictionary of observables whose span is Koopman-invariant is a central challenge in data-driven Koopman operator approximation. We address this problem by exploiting zero-block structure in Extended Dynamic Mode Decomposition (EDMD) matrices. We show that any sub-dictionary whose span is Koopman-invariant induces an exact zero block in the EDMD matrix, even for finite data. We then show that such blocks can be detected by applying PageRank to a row-normalized EDMD matrix constructed from a large initial dictionary. The theory extends to approximately invariant subspaces and yields stronger guarantees for personalized PageRank (PPR) when the seed observables lie inside the target block and reach all observables in that block. Combining EDMD concentration bounds with PageRank perturbation theory gives end-to-end detection guarantees with $O(1/\sqrt{M})$ finite-sample scaling and explicit constants. More generally, without assuming an invariant subspace exists, high PPR mass on a sub-dictionary controls discounted multi-step leakage from the seed observables. Numerical experiments on the Duffing oscillator, Van der Pol oscillator, Lorenz system, and a three-well Ramachandran potential suggest that the method identifies compact, interpretable dictionaries with accurate predictions.
31. WINO：一种用于可变区域超弹性的弱形式物理信息神经算子
WINO: A Weak-Form Physics Informed Neural Operator for Hyperelasticity on Variable Domains
作者: Bokai Zhu, Qinghui Zhang, Timon Rabczuk
类别: math.NA, cs.LG
发布日期: 2026-05-23
arXiv ID: 2605.24651v1
链接: http://arxiv.org/abs/2605.24651v1
摘要: We propose a Weak-form Physics-Informed Neural Operator (WINO), a data-free framework that combines the efficiency of neural operators with the geometric flexibility of the $\varphi$-finite element method ($\varphi$-FEM). $\varphi$-FEM is an unfitted method that accommodates geometric variations without body-fitted meshes, where the domain geometry is represented by the level-set function $\varphi$. To impose the boundary conditions, Dirichlet problems adopt the $\varphi$-FEM lifting so only the homogeneous displacement contribution is learned, whereas traction-driven Neumann problems additionally predict the auxiliary fields necessary for the unfitted weak formulation. Parameters are trained by minimizing squared weak-form residuals aligned with $\varphi$-FEM together with squared penalties on the cut-cell auxiliary equations, which removes the need for large paired datasets of converged reference solutions. After training, WINO outputs can seed the nonlinear $\varphi$-FEM solvers as neural operator warm starts (NOWS), which reduce iteration counts relative to traditional cold-started solvers. Numerical benchmarks show that WINO achieves high accuracy below 0.04 across all benchmarks, while reducing total computational time by 50–80% compared with purely data-driven methods.
32. 巴拿赫空间中优化的多层蒙特卡洛方法
Optimized multilevel Monte Carlo methods in Banach spaces
作者: Kristin Kirchner, Fabio Nobile, Christoph Schwab, Tommaso Vanzan
类别: math.NA, math.PR
发布日期: 2026-05-23
arXiv ID: 2605.24620v1
链接: http://arxiv.org/abs/2605.24620v1
摘要: We present a theoretical and numerical analysis of Monte Carlo methods for the estimation of statistical moments of random variables $X:Ω\rightarrow E$ taking values in a Banach space $E$. For practical computation, we consider finite-dimensional approximation subspaces ${(E_\ell)_{\ell\in\mathbb{N}}\subset E}$ of increasing dimension. We develop a refined error analysis that explicitly accounts for a dependence of the Rademacher type constants on the dimension of $E_\ell$, leading to novel complexity results for single- and multilevel Monte Carlo methods to estimate the mean and injective moments of arbitrary order, which are, in certain cases, sharper than those derived in [Kirchner, Schwab, J. Funct. Anal, 2024]. Moreover, we show that, in favorable cases, the resulting error-vs.-work bounds are independent of the Rademacher type of $E$. We then focus on $L^p(S)$-valued random variables for a $σ$-finite measure space satisfying certain approximation properties, and prove that for a random variable $X\in L^q(Ω;L^p(S))\cap L^p(S;L^q(Ω))$, with $q\in (1,\infty)$ and $p\in [1,\infty)$, the $L^q$-convergence rate of a Monte Carlo estimator is determined exclusively by the integrability parameter $\min\{q,2\}$, with no dependence on the Rademacher type $\min\{p,2\}$ of $L^p(S)$. We further investigate the impact of measuring the (multilevel) Monte Carlo error in the $L^q(Ω;L^p(S))$-norm while $X$ possesses additional regularity, $X\in L^{\tilde{q}}(Ω;L^p(S))\cap L^p(S;L^{\tilde{q}}(Ω))$ with $\tilde{q}\in [q,\infty)$. This analysis reveals an interplay between the sampling error and the strong approximation error, and leads to optimized error-vs.-work bounds for both single- and multilevel Monte Carlo methods. Numerical experiments confirm the sharpness of the analyses presented.
33. 用于大规模动态矩阵的增量奇异值分解：精确性、子空间稳定性、刷新策略与基于金融因子的风险模型
Incremental SVD for Large-Scale Dynamic Matrices: Accuracy, Subspace Stability, Refresh Strategies, and Financial Factor-Based Risk Models
作者: Stilyan Staykov
类别: math.NA, q-fin.RM
发布日期: 2026-05-23
arXiv ID: 2605.24514v1
链接: http://arxiv.org/abs/2605.24514v1
摘要: Return panels, covariances, and large feature matrices evolve one observation or one entry at a time, yet downstream models require an up-to-date low-rank factorization $A_t \approx U_t Σ_t V_t^\top$ on every tick – a regime where full SVD is prohibitive and existing alternatives sacrifice either singular vectors, singular values, or long-horizon stability. We present a practical, metric-driven study of Brand-style incremental SVD, built around a unified engine that handles row appends, column appends, rank-1 entry updates, and metrics tracking within a single framework, with two core contributions. For rank-1 entry updates, we derive an explicit projection-based rule $U'Σ'(V')^\top = P_U(\widehat{A} + δ\,e_ie_j^\top)P_V$ that keeps rank fixed while discarding only the out-of-subspace remainder in a quantifiable way, turning Brand’s rank-suppression heuristic into an operational scheme. We then treat refresh scheduling as a first-class design axis, systematically comparing periodic, error-threshold, angle-threshold, and adaptive-rank policies on the accuracy-latency frontier. A unified framework tracks error ratios, principal angles, explained variance, and per-update runtime on long synthetic streams and a multi-asset ETF factor model for covariance and portfolio-risk estimation. With a sensible rank and refresh cadence, incremental SVD matches full-SVD accuracy within a few percent at a fraction of the cost, scaling to high-frequency regimes where batch SVDs are infeasible.
34. 在各向异性力不确定性下的保几何轻推弹性带与二聚体方法
Geometry-Preserving Nudged Elastic Band and Dimer Methods under Anisotropic Force Uncertainty
作者: Yifan Yu, Yangshuai Wang
类别: math.NA, physics.comp-ph
发布日期: 2026-05-23
arXiv ID: 2605.24401v1
链接: http://arxiv.org/abs/2605.24401v1
摘要: The nudged elastic band (NEB) and Dimer methods are standard tools for computing minimum-energy paths and index-one saddle points in atomistic transition problems. They are increasingly driven by surrogate or learned force models, whose force errors are often anisotropic and spatially varying near transition states and defect cores, where saddle-search iterations are most sensitive. We introduce uncertainty-aware NEB and Dimer methods (UA-NEB, UA-Dimer) that use covariance as an optimizer-level reliability metric while preserving the mean-potential saddle-search equations: an oblique normal projection for NEB and covariance-weighted rotation and translation for Dimer. Both algorithms fit Robbins–Monro recursions; under a local Lyapunov stability hypothesis, verified explicitly for a canonical UA-NEB setting and stated as a hypothesis for UA-Dimer, the stochastic iterations converge almost surely within the corresponding local stability neighborhood. In the analytic benchmark, UA-NEB reduces mean barrier error by $21\%$ relative to stochastic NEB and UA-Dimer reduces the reflected-gradient residual by $22\%$; in the 127-atom tungsten-vacancy benchmark, full UA-NEB reduces mean barrier error by $56\%$ relative to stochastic NEB and by $23\%$ relative to diagonal covariance weighting. These results show that anisotropic uncertainty is most useful when embedded in the constrained geometry of the optimizer rather than collapsed into a scalar acquisition or trust criterion.
35. 逼近具有对数势的空间分数阶Allen-Cahn方程的结构保持稳定化格式的谱分析与基于正弦变换的预条件处理
Spectral analysis and sine transform based preconditioning for a structure preserving stabilized scheme approximating the space-fractional Allen Cahn equation with logarithmic potential
作者: Danyal Ahmad, Stefano Serra-Capizzano, Muhammad Sohaib, Cristina Tablino-Possio
类别: math.NA
发布日期: 2026-05-23
arXiv ID: 2605.24387v1
链接: http://arxiv.org/abs/2605.24387v1
摘要: We consider an initial boundary value problem of the space fractional Allen-Cahn equation with logarithmic Flory-Huggins potential. As an approximation technique, first-order weighted and shifted Grunwald difference formulae of the left and right fractional derivatives are used. The main focus of the present work is to study the spectral features of the underlying matrices and matrix sequences and to design proper preconditioners based on the spectral information. Then a computational and spectral analysis of the resulting preconditioned matrix sequences is performed. Numerical evidence and a short list of open problems complete the current study.
36. 提高指数拟合格式在分段均匀网格上的精度
Improving the Accuracy of the Exponentially Fitted Scheme on Piecewise Uniform Meshes
作者: Relja Vulanović
类别: math.NA
发布日期: 2026-05-23
arXiv ID: 2605.24336v1
链接: http://arxiv.org/abs/2605.24336v1
摘要: A linear one-dimensional singularly perturbed convection-diffusion problem is solved numerically after its solution is decomposed as $u_0+w$, where $u_0$, the corresponding reduced solution, is treated as a function known exactly or approximately. The component $w$ is then calculated using the exponentially fitted Allen-Southwell-Il’in (ASI) scheme on the Shishkin mesh and its asymptotic version. We prove that this numerical method is highly accurate, with errors that diminish when the discretization parameter increases, and, in some cases, even when the perturbation parameter decreases. This is a theoretical confirmation of earlier numerical results showing that the ASI scheme outperforms the general class of Samarskii-type schemes to which it belongs. Even higher accuracy is proved when $u_0$ is linear, in which case, the decomposition is not needed. New numerical experiments are provided to illustrate all this.
37. 代理点方法的精度分析及其在某些托普利茨矩阵中的应用
Accuracy Analysis of the Proxy Point Method with Applications to Some Toeplitz Matrices
作者: Mikhail Lepilov, Jianlin Xia
类别: math.NA
发布日期: 2026-05-22
arXiv ID: 2605.24231v1
链接: http://arxiv.org/abs/2605.24231v1
摘要: For some kernel matrices, low-rank approximations can be quickly obtained via analytic techniques. One important class of analytic methods that has received attention in recent years is based on the use of proxy points. Accuracy analysis for various proxy point methods has often been heuristic in nature, other than for certain special kernels. For more general cases, the methods lack an explicit number or location of proxy points required to yield a particular approximation accuracy. In this work, we carry out new analysis of a proxy point method that is applicable to general complex-analytic kernels. An intuitive way of choosing proxy points is used to show explicit error bounds. Such bounds decay exponentially with regard to the number of proxy points. This also leads to convenient estimates of numerical ranks of relevant kernel matrices. To showcase the utility of this new analysis, we apply it to design a new sublinear-time hierarchically semiseparable approximation method for certain Toeplitz matrices, including ones that frequently arise from real-world applications. This allows, for example, inversion of such matrices with lower computational complexity compared with existing direct methods. Some extensions of these ideas are also discussed.
38. 极化诱导的梁弯曲：数学模型、离散化与算法
Polarization-Induced Beam Bending: Mathematical Model, Discretization, and Algorithm
作者: Harbir Antil, Rainald Löhner, Sarswati Shah
类别: physics.optics, math.AP, math.NA
发布日期: 2026-05-22
arXiv ID: 2605.24151v1
链接: http://arxiv.org/abs/2605.24151v1
摘要: We study a reduced hydrodynamic formulation of paraxial vector beam propagation in which the beam intensity, optical phase, and spatially-dependent polarization are coupled through a nonlinear dispersive system. While prior analytical work derived a solution for the beam path valid for short propagation distances, a fully resolved numerical treatment of the model over long ranges has not previously been available. Here we present a conservative numerical scheme for the coupled system, combining a finite-volume discretization of the intensity equation with monotone Hamilton–Jacobi (H-J) solvers for the phase dynamics and upwind transport of polarization. The method preserves the nonnegativity of the intensity and remains stable under long-distance propagation. We perform large-scale simulations over propagation distances of tens of meters, while resolving millimeter-scale transverse structure. The numerical results reproduce the analytically predicted and experimentally observed quadratic beam bending at short distances and reveal systematic deviations beyond the asymptotic regime. These deviations arise from nonlinear phase accumulation and dispersive effects captured by the full model but are neglected in the short-distance approximation.