数值分析
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显示 2025年08月07日， 星期四 新的列表
- [1] arXiv:2508.03992 [中文pdf, pdf, html, 其他]
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标题： 无条件能量耗散的Strang分裂方法用于矩阵值Allen-Cahn方程标题： Unconditional energy dissipation of Strang splitting for the matrix-valued Allen-Cahn equation主题： 数值分析 (math.NA)
斯特朗分裂方法的能量耗散特性首次在受限时间步长条件下针对矩阵值阿伦-卡恩（MAC）方程得到了验证 [J. Comput. Phys. 454, 110985, 2022]。在本工作中，我们通过一个改进的稳定性分析框架消除了这一限制，严格证明了斯特朗分裂方法在任意时间步长下无条件地保持能量耗散定律。改进的证明依赖于对修正能量泛函中双井势项的精确估计。利用这一无条件能量耗散特性，我们严格建立了斯特朗分裂方法对于矩阵值阿伦-卡恩方程的全局时间$H^1$-稳定性，保持行列式有界性，并保持二阶时间收敛性。为了验证这些理论结果，我们进行了数值实验，确认了该方法在MAC方程中的能量稳定性以及行列式边界保持性。
The energy dissipation property of the Strang splitting method was first demonstrated for the matrix-valued Allen-Cahn (MAC) equation under restrictive time-step constraints [J. Comput. Phys. 454, 110985, 2022]. In this work, we eliminate this limitation through a refined stability analysis framework, rigorously proving that the Strang splitting method preserves the energy dissipation law unconditionally for arbitrary time steps. The refined proof hinges on a precise estimation of the double-well potential term in the modified energy functional. Leveraging this unconditional energy dissipation property, we rigorously establish that the Strang splitting method achieves global-in-time $H^1$-stability, preserves determinant boundedness, and maintains second-order temporal convergence for the matrix-valued Allen-Cahn equation. To validate these theoretical findings, we conduct numerical experiments confirming the method's energy stability and determinant bound preservation for the MAC equation.
- [2] arXiv:2508.04077 [中文pdf, pdf, html, 其他]
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标题： 通用的稀疏矩阵-矩阵乘法标题： The Ubiquitous Sparse Matrix-Matrix Products主题： 数值分析 (math.NA) ; 分布式、并行与集群计算 (cs.DC) ; 机器学习 (cs.LG) ; 数学软件 (cs.MS) ; 组合数学 (math.CO)
稀疏矩阵与另一个（密集或稀疏）矩阵相乘是一个基础操作，它捕捉了许多数据科学应用中的计算模式，包括但不限于图算法、稀疏连接的神经网络、图神经网络、聚类以及生物测序数据的多对多比较。 在许多应用场景中，矩阵乘法发生在任意的代数半环上，其中标量操作被用户定义的具有特定属性的函数覆盖，或者更一般的异构代数中，甚至输入矩阵的域也可以不同。 在这里，我们提供了一种统一的处理方式，针对稀疏矩阵-矩阵操作及其丰富的应用领域，包括机器学习、计算生物学和化学、图算法以及科学计算。
Multiplication of a sparse matrix with another (dense or sparse) matrix is a fundamental operation that captures the computational patterns of many data science applications, including but not limited to graph algorithms, sparsely connected neural networks, graph neural networks, clustering, and many-to-many comparisons of biological sequencing data. In many application scenarios, the matrix multiplication takes places on an arbitrary algebraic semiring where the scalar operations are overloaded with user-defined functions with certain properties or a more general heterogenous algebra where even the domains of the input matrices can be different. Here, we provide a unifying treatment of the sparse matrix-matrix operation and its rich application space including machine learning, computational biology and chemistry, graph algorithms, and scientific computing.
- [3] arXiv:2508.04082 [中文pdf, pdf, html, 其他]
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标题： 基于POD的全局时间迭代解耦算法在比奥固结模型中的降阶建模标题： POD-based reduced order modeling of global-in-time iterative decoupled algorithms for Biot's consolidation model主题： 数值分析 (math.NA)
本文专注于三场Biot固结模型的高效数值算法。 该方法首先引入了创新的单体和全局时间迭代解耦算法，这些算法结合了后向差分公式进行时间离散化。 在每次迭代中，这些算法涉及在整个时间域上求解一个扩散子问题，随后在同一时间区间上求解一个广义Stokes子问题。 为了加速全局时间迭代过程，我们提出了一种基于本征正交分解的降阶建模方法，旨在降低来自广义Stokes子问题的主要计算成本。 该新方法的有效性通过理论验证和数值实验得到了证实。
This paper focuses on the efficient numerical algorithms of a three-field Biot's consolidation model. The approach begins with the introduction of innovative monolithic and global-in-time iterative decoupled algorithms, which incorporate the backward differentiation formulas for time discretization. In each iteration, these algorithms involve solving a diffusion subproblem over the entire temporal domain, followed by solving a generalized Stokes subproblem over the same time interval. To accelerate the global-in-time iterative process, we present a reduced order modeling approach based on proper orthogonal decomposition, aimed at reducing the primary computational cost from the generalized Stokes subproblem. The effectiveness of this novel method is validated both theoretically and through numerical experiments.
- [4] arXiv:2508.04112 [中文pdf, pdf, html, 其他]
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标题： 高阶PDEs的双曲近似收敛性对于光滑解标题： Convergence of hyperbolic approximations to higher-order PDEs for smooth solutions主题： 数值分析 (math.NA) ; 偏微分方程分析 (math.AP)
我们证明了对于几类高阶偏微分方程的双曲近似方法的收敛性，包括Benjamin-Bona-Mahony方程、Korteweg-de Vries方程、Gardner方程、Kawahara方程和Kuramoto-Sivashinsky方程，前提是极限问题存在光滑解。我们仅要求双曲近似方法的弱（熵）解。由此，我们为这些近似方法提供了坚实的基础，这些方法在文献中已被使用但缺乏严格的收敛分析。我们还给出了支持我们理论结果的数值结果。
We prove the convergence of hyperbolic approximations for several classes of higher-order PDEs, including the Benjamin-Bona-Mahony, Korteweg-de Vries, Gardner, Kawahara, and Kuramoto-Sivashinsky equations, provided a smooth solution of the limiting problem exists. We only require weak (entropy) solutions of the hyperbolic approximations. Thereby, we provide a solid foundation for these approximations, which have been used in the literature without rigorous convergence analysis. We also present numerical results that support our theoretical findings.
- [5] arXiv:2508.04198 [中文pdf, pdf, html, 其他]
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标题： 通过降阶基方法设计多等离子体纳米颗粒宽带吸收器的最佳设计标题： Optimal Design of Broadband Absorbers with Multiple Plasmonic Nanoparticles via Reduced Basis Method主题： 数值分析 (math.NA) ; 优化与控制 (math.OC)
在本文中，我们提出了一种计算框架，用于设计由等离子纳米粒子阵列组成的宽带吸收材料的最优设计。 这一设计问题带来了几个关键挑战：(1) 复杂的多粒子相互作用和高曲率几何结构；(2) 实现宽带频率响应的要求，包括共振区域；(3) 形状导数计算的复杂性；以及 (4) 优化景观的非凸性。 为了系统地解决这些挑战，我们采用了三种顺序策略。 首先，我们引入了一种参数化的积分方程公式，避免了传统的形状导数计算。 其次，我们开发了一种形状自适应的降阶基方法（RBM），该方法利用Neumann-Poincaré算子的本征函数进行正向问题，并利用其伴随形式进行伴随问题，从而解决奇点并加速计算。 第三，我们提出了一种物理信息初始化策略，在弱耦合假设下估计纳米粒子配置，从而提高基于梯度的优化算法的性能。 通过数值实验展示了该方法的计算优势，结果表明在各种几何配置下都能实现准确且高效的設計。 此外，该框架具有灵活性和可扩展性，适用于其他材料系统和边界条件。
In this paper, we propose a computational framework for the optimal design of broadband absorbing materials composed of plasmonic nanoparticle arrays. This design problem poses several key challenges: (1) the complex multi-particle interactions and high-curvature geometries; (2) the requirement to achieve broadband frequency responses, including resonant regimes; (3) the complexity of shape derivative calculations; and (4) the non-convexity of the optimization landscape. To systematically address these challenges, we employ three sequential strategies. First, we introduce a parameterized integral equation formulation that circumvents traditional shape derivative computations. Second, we develop a shape-adaptive reduced basis method (RBM) that utilizes the eigenfunctions of the Neumann-Poincar\'{e} operator for forward problems and their adjoint counterparts for adjoint problems, thereby addressing singularities and accelerating computations. Third, we propose a physics-informed initialization strategy that estimates nanoparticle configurations under weak coupling assumptions, thereby improving the performance of gradient-based optimization algorithms. The method's computational advantages are demonstrated through numerical experiments, which show accurate and efficient designs across various geometric configurations. Furthermore, the framework is flexible and extensible to other material systems and boundary conditions.
- [6] arXiv:2508.04356 [中文pdf, pdf, html, 其他]
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标题： 单块多级重叠 Schwarz 求解器用于流体问题标题： Monolithic Multi-level Overlapping Schwarz Solvers for Fluid Problems评论： GAMM95主题： 数值分析 (math.NA)
加性重叠Schwarz方法是用于求解偏微分方程的域分解类型迭代方法。通过添加粗粒度层次可以实现这些方法的数值和并行可扩展性。受迭代子结构启发的成功粗空间是广义Dryja-Smith-Widlund（GDSW）空间。在https://doi.org/10.1137/18M1184047中，基于GDSW方法，引入了用于鞍点问题的双层单块重叠Schwarz预条件器。我们展示了使用双层和三层单块重叠Schwarz预条件器对单位立方体上的Poiseuille流示例和复杂挤压模具几何结构求解不可压缩流体问题的并行结果，最多达到32768个MPI进程。这些结果是通过结合Fast and Robust Overlapping Schwarz（FROSch）库https://doi.org/10.1007/978-3-030-56750-7_19中实现的加性重叠Schwarz求解器，该库是Trilinos包ShyLU https://doi.org/10.1109/IPDPS.2012.64的一部分，以及FEATFLOW库http://www.featflow.de，通过可扩展接口高效耦合这两个库实现的。这项工作是StroemungsRaum项目的一部分，该项目名为面向计算流体力学模拟的新型exascale架构，具有异构硬件组件，由德国联邦研究、技术和航天部BMFTR（前身为BMBF）资助，作为新方法和技术用于exascale计算（SCALEXA）计划的一部分。
Additive overlapping Schwarz Methods are iterative methods of the domain decomposition type for the solution of partial differential equations. Numerical and parallel scalability of these methods can be achieved by adding coarse levels. A successful coarse space, inspired by iterative substructuring, is the generalized Dryja-Smith-Widlund (GDSW) space. In https://doi.org/10.1137/18M1184047, based on the GDSW approach, two-level monolithic overlapping Schwarz preconditioners for saddle point problems were introduced. We present parallel results up to 32768 MPI ranks for the solution of incompressible fluid problems for a Poiseuille flow example on the unit cube and a complex extrusion die geometry using a two- and a three-level monolithic overlapping Schwarz preconditioner. These results are achieved through the combination of the additive overlapping Schwarz solvers implemented in the Fast and Robust Overlapping Schwarz (FROSch) library https://doi.org/10.1007/978-3-030-56750-7_19, which is part of the Trilinos package ShyLU https://doi.org/10.1109/IPDPS.2012.64, and the FEATFLOW library http://www.featflow.de using a scalable interface for the efficient coupling of the two libraries. This work is part of the project StroemungsRaum - Novel Exascale-Architectures with Heterogeneous Hardware Components for Computational Fluid Dynamics Simulations, funded by the German Bundesministerium fur Forschung, Technologie und Raumfahrt BMFTR (formerly BMBF) as part of the program on New Methods and Technologies for Exascale Computing (SCALEXA).
- [7] arXiv:2508.04360 [中文pdf, pdf, 其他]
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标题： 热力学一致磁两相流模型的推导与数值模拟用于磁性药物靶向标题： Derivation and Numerical Simulation of a Thermodynamically Consistent Magneto Two-Phase Flow Model for Magnetic Drug Targeting主题： 数值分析 (math.NA)
在本文中，我们推导了一个新颖且全面的热力学一致模型，用于描述超顺磁性氧化铁纳米颗粒（SPIONs）、载液和磁场之间在磁性药物靶向（MDT）中的复杂相互作用，即通过外部磁场靶向传递磁功能化药物载体的过程。 该模型包括一个用于SPIONs的对流-扩散方程，一个修正的Navier-Stokes系统用于载液-纳米颗粒混合物的平均速度，以及一个准静态Maxwell系统用于磁变量。 所推导的模型通过考虑载液和磁场对SPIONs动力学的响应，扩展了之前的MDT模型，从而提供了一个全面的工具，用于预测和优化MDT过程。 在引入用于模型数值模拟的半隐式有限元方案后，对完全耦合模型进行了仿真，并与一种简化模型的结果进行了比较，该简化模型中忽略了载液流动和磁场对SPION动力学的响应。 此外，还研究了MDT相对于实验参数（如磁体位置）的敏感性。
In this paper, we derive a novel and comprehensive thermodynamically consistent model for the complex interactions between superparamagnetic iron oxide nanoparticles (SPIONs), a carrier fluid, and a magnetic field, as they occur in Magnetic Drug Targeting (MDT), the targeted delivery of magnetically functionalized drug carriers by external magnetic fields. It consists of a convection-diffusion equation for SPIONs, a modified Navier-Stokes system for the averaged velocity of the carrier fluid-nanoparticle mixture and a quasi-stationary Maxwell system for the magnetic variables. The derived model extends previous models for MDT by taking into account the response of the carrier fluid and of the magnetic field to the dynamics of the SPIONs, and thus provides a comprehensive tool for the prediction and optimization of MDT processes. After introducing a semi-implicit finite element scheme for the numerical simulation of the model, simulation results for the fully coupled model are performed and compared with results from a reduced version of the model, where the response of the carrier flow and of the magnetic field to the SPION dynamics is neglected. Furthermore, the sensitivity of MDT with respect to experimental parameters, such as magnet positioning, is investigated.
- [8] arXiv:2508.04392 [中文pdf, pdf, html, 其他]
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标题： 具有C2光滑性的近似柯尔莫哥洛夫-阿诺德叠加的显式构造标题： Explicit Construction of Approximate Kolmogorov-Arnold Superpositions with C2-Smoothness评论： 21页，17图主题： 数值分析 (math.NA)
我们显式构造了Kolmogorov-Arnold叠加的一个近似版本，该版本由C2内函数和外函数组成，能够很好地近似任意alpha-Holder连续函数。 内函数是通过对一个分段C2、严格递增的函数应用适当的平移和缩放生成的，而外函数则是通过使用新设计的形状函数进行分段C2插值逐行构建的。 这种Kolmogorov-Arnold叠加的新变体克服了固有单变量函数的极端和病态行为，但保留了Kolmogorov精确表示策略的本质，这是Sprecher在Neural Networks 144, 2021中积极追求的目标。
We explicitly construct an approximate version of the Kolmogorov-Arnold superpositions, which is composed of C2 inner and outer functions, and can approximate an arbitrary alpha-Holder continuous function well. The inner functions are generated by applying suitable translations and dilations to a piecewise C2, strictly increasing function, while the outer functions are constructed row-wise through piecewise C2 interpolation using newly designed shape functions. This novel variant of Kolmogorov-Arnold superpositions overcomes the wild and pathological behaviors of the inherent single variable functions, but retains the essence of Kolmogorov strategy of exact representation, an objective that Sprecher, Neural Networks 144, 2021, has actively pursued.
- [9] arXiv:2508.04484 [中文pdf, pdf, html, 其他]
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标题： 一种用于非均匀介质中质子传输的高阶确定性动力低秩方法标题： A high-order deterministic dynamical low-rank method for proton transport in heterogeneous media主题： 数值分析 (math.NA) ; 计算物理 (physics.comp-ph) ; 医学物理 (physics.med-ph)
质子治疗中的剂量计算需要快速且准确地求解大量不同能量和方向的（铅笔）束的高维输运方程。 在足够分辨率下确定性地求解这个输运问题却可能非常昂贵，尤其是由于质子的高度前向散射。 我们提出使用一种模型降阶方法，即动态低秩近似（DLRA），它在低秩矩阵流形上以（伪）时间演化解。 为此，我们比较了线性玻尔兹曼方程及其福克-普朗克近似的碰撞-未碰撞分解。 我们使用光线追踪器处理未碰撞部分，并将高阶相空间离散化和材料混合模型与DLRA结合用于碰撞方程。 我们的方法在显著较低的秩下再现了全秩参考代码的结果，从而降低了计算成本和内存，并进一步使在更高分辨率下的计算成为可能。 在更高分辨率下，我们在均匀和非均匀材料中都实现了与TOPAS MC相当的良好准确性。 最后，我们证明了与单个束相比，多个不同角度的束源可以以很小的成本增加进行计算。
Dose calculations in proton therapy require the fast and accurate solution of a high-dimensional transport equation for a large number of (pencil) beams with different energies and directions. Deterministically solving this transport problem at a sufficient resolution can however be prohibitively expensive, especially due to highly forward peaked scattering of the protons. We propose using a model order reduction approach, the dynamical low-rank approximation (DLRA), which evolves the solution on the manifold of low-rank matrices in (pseudo-)time. For this, we compare a collided-uncollided split of the linear Boltzmann equation and its Fokker-Planck approximation. We treat the uncollided part using a ray-tracer and combine high-order phase space discretizations and a mixture model for materials with DLRA for the collided equation. Our method reproduces the results of a full-rank reference code at significantly lower rank, and thus computational cost and memory, and further makes computations feasible at much higher resolutions. At higher resolutions, we also achieve good accuracy with respect to TOPAS MC in homogeneous as well as heterogeneous materials. Finally, we demonstrate that several beam sources with different angles can be computed with little cost increase compared to individual beams.
- [10] arXiv:2508.04560 [中文pdf, pdf, html, 其他]
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标题： 用对称无迹张量离散化线性化的爱因斯坦-比安基系统标题： Discretizing linearized Einstein-Bianchi system by symmetric and traceless tensors评论： 28页主题： 数值分析 (math.NA) ; 广义相对论与量子宇宙学 (gr-qc)
爱因斯坦-比安基系统使用对称和迹为零的张量来重新表述爱因斯坦的原始场方程。 然而，同时保持这些代数约束对于数值方法仍然是一个挑战。 本文提出了一种新的公式，将线性化的爱因斯坦-比安基系统（接近平凡的闵可夫斯基度规）视为与共形海森复形相关的霍奇波动方程。 为了离散这个方程，在一般的三维四面体网格上构造了一个符合有限元共形海森复形，该复形能够同时保持对称性和迹为零性，并证明了其精确性。
The Einstein-Bianchi system uses symmetric and traceless tensors to reformulate Einstein's original field equations. However, preserving these algebraic constraints simultaneously remains a challenge for numerical methods. This paper proposes a new formulation that treats the linearized Einstein-Bianchi system (near the trivial Minkowski metric) as the Hodge wave equation associated with the conformal Hessian complex. To discretize this equation, a conforming finite element conformal Hessian complex that preserves symmetry and traceless-ness simultaneously is constructed on general three-dimensional tetrahedral grids, and its exactness is proven.
- [11] arXiv:2508.04582 [中文pdf, pdf, html, 其他]
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标题： $h$-三角函数B样条标题： $h$-Trigonometric B-splines主题： 数值分析 (math.NA)
我们引入指数函数、正弦函数和余弦函数的离散模拟。 然后，使用非多项式差商的离散三角函数版本，我们定义了三角函数B样条的离散模拟。 我们推导出一个两项递推关系，一个离散导数的两项公式，以及这些离散三角函数B样条的Marsden恒等式的两个变体。 由于经典指数函数、正弦函数和余弦函数是它们的离散模拟的极限情况，我们得出结论，许多经典多项式B样条的标准结果自然地扩展到三角函数B样条和离散三角函数B样条。
We introduce discrete analogues of the exponential, sine, and cosine functions. Then using a discrete trigonometric version of a non-polynomial divided difference, we define discrete analogues of the trigonometric B-splines. We derive a two-term recurrence relation, a two-term formula for the discrete derivative, and two variants of the Marsden identity for these discrete trigonometric B-splines. Since the classical exponential, sine, and cosine functions are limiting cases of their discrete analogues, we conclude that many of the standard results for classical polynomial B-splines extend naturally both to trigonometric B-splines and to discrete trigonometric B-splines.
新提交 (展示 11 之 11 条目 )
- [12] arXiv:2508.03755 (交叉列表自 cs.LG) [中文pdf, pdf, html, 其他]
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标题： LRTuckerRep：多维数据补全的低秩Tucker表示模型标题： LRTuckerRep: Low-rank Tucker Representation Model for Multi-dimensional Data Completion主题： 机器学习 (cs.LG) ; 计算机视觉与模式识别 (cs.CV) ; 数值分析 (math.NA)
多维数据补全是计算科学中的一个关键问题，特别是在计算机视觉、信号处理和科学计算等领域。 现有方法通常利用全局低秩近似或局部平滑正则化，但每种方法都有显著的局限性：低秩方法计算成本高且可能破坏数据的内在结构，而基于平滑性的方法通常需要大量手动参数调整且泛化能力较差。 在本文中，我们提出了一种新颖的低秩Tucker表示（LRTuckerRep）模型，在Tucker分解中统一了全局和局部先验建模。 具体而言，LRTuckerRep通过因子矩阵上的自适应加权核范数和稀疏的Tucker核心来编码低秩性，同时通过因子空间上的无参数拉普拉斯正则化来捕捉平滑性。 为了高效求解 resulting 的非凸优化问题，我们开发了两种具有可证明收敛保证的迭代算法。 在多维图像修复和交通数据插补的大量实验表明，与基线方法相比，LRTuckerRep在高缺失率下实现了更高的补全精度和鲁棒性。
Multi-dimensional data completion is a critical problem in computational sciences, particularly in domains such as computer vision, signal processing, and scientific computing. Existing methods typically leverage either global low-rank approximations or local smoothness regularization, but each suffers from notable limitations: low-rank methods are computationally expensive and may disrupt intrinsic data structures, while smoothness-based approaches often require extensive manual parameter tuning and exhibit poor generalization. In this paper, we propose a novel Low-Rank Tucker Representation (LRTuckerRep) model that unifies global and local prior modeling within a Tucker decomposition. Specifically, LRTuckerRep encodes low rankness through a self-adaptive weighted nuclear norm on the factor matrices and a sparse Tucker core, while capturing smoothness via a parameter-free Laplacian-based regularization on the factor spaces. To efficiently solve the resulting nonconvex optimization problem, we develop two iterative algorithms with provable convergence guarantees. Extensive experiments on multi-dimensional image inpainting and traffic data imputation demonstrate that LRTuckerRep achieves superior completion accuracy and robustness under high missing rates compared to baselines.
- [13] arXiv:2508.03857 (交叉列表自 cs.DS) [中文pdf, pdf, html, 其他]
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标题： 一种用于精确3x3矩阵乘法的60加法，秩23方案标题： A 60-Addition, Rank-23 Scheme for Exact 3x3 Matrix Multiplication主题： 数据结构与算法 (cs.DS) ; 计算复杂性 (cs.CC) ; 数值分析 (math.NA)
我们将一般（非交换）3x3矩阵乘法的加法成本从之前记录的61（Schwartz-Vaknin，2023）和62（Martensson-Wagner，2025）降低到60，而无需改变基。据我们所知，这代表了新的最先进水平。
We reduce the additive cost of general (non-commutative) 3x3 matrix multiplication from the previous records of 61 (Schwartz-Vaknin, 2023) and 62 (Martensson-Wagner, 2025) to 60 without a change of basis. To our knowledge, this represents a new state-of-the-art.
- [14] arXiv:2508.03926 (交叉列表自 cs.LG) [中文pdf, pdf, html, 其他]
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标题： 通过机器学习的下一代无方程多尺度人群动力学建模标题： Next Generation Equation-Free Multiscale Modelling of Crowd Dynamics via Machine LearningHector Vargas Alvarez, Dimitrios G. Patsatzis, Lucia Russo, Ioannis Kevrekidis, Constantinos Siettos评论： 29页（附录9页），9图（附录3图）主题： 机器学习 (cs.LG) ; 动力系统 (math.DS) ; 数值分析 (math.NA)
在人群动力学中弥合微观和宏观建模尺度构成了系统数值分析、优化和控制的重要且开放的挑战。我们提出了一种结合流形和机器学习的方法，从高保真基于代理的模拟中学习隐空间中出现的人群动力学的离散演化算子。该框架建立在我们之前关于下一代无方程算法的研究基础上，用于学习高维和多尺度系统的替代模型。我们的方法是一个四阶段的方法，在高维空间中显式地保持重构动力学的质量。在第一步中，我们使用核密度估计（KDE）从离散微观数据（行人位置）推导出连续宏观场（密度）。在第二步中，基于流形学习，我们构建了一个从宏观环境空间到由对应密度分布的POD参数化的潜在空间的映射。第三步涉及在潜在空间中使用机器学习技术（特别是LSTMs网络和MVARs）学习降阶替代模型（ROMs）。最后，我们根据宏观密度轮廓在高维空间中重构人群动力学。我们证明了通过SVD进行的密度分布的POD重构保持了质量。通过这种“嵌入->在潜在空间中学习->重新提升到环境空间”的流程，我们创建了一个不可用宏观PDE的密度演化的有效解算子。在我们的示例中，我们使用社会力模型在带有障碍物的走廊中生成数据，并施加周期性边界条件。数值结果表明了高精度、鲁棒性和泛化能力，从而允许从基于代理的模拟中快速准确地对人群动力学进行建模/仿真。
Bridging the microscopic and the macroscopic modelling scales in crowd dynamics constitutes an important, open challenge for systematic numerical analysis, optimization, and control. We propose a combined manifold and machine learning approach to learn the discrete evolution operator for the emergent crowd dynamics in latent spaces from high-fidelity agent-based simulations. The proposed framework builds upon our previous works on next-generation Equation-free algorithms on learning surrogate models for high-dimensional and multiscale systems. Our approach is a four-stage one, explicitly conserving the mass of the reconstructed dynamics in the high-dimensional space. In the first step, we derive continuous macroscopic fields (densities) from discrete microscopic data (pedestrians' positions) using KDE. In the second step, based on manifold learning, we construct a map from the macroscopic ambient space into the latent space parametrized by a few coordinates based on POD of the corresponding density distribution. The third step involves learning reduced-order surrogate ROMs in the latent space using machine learning techniques, particularly LSTMs networks and MVARs. Finally, we reconstruct the crowd dynamics in the high-dimensional space in terms of macroscopic density profiles. We demonstrate that the POD reconstruction of the density distribution via SVD conserves the mass. With this "embed->learn in latent space->lift back to the ambient space" pipeline, we create an effective solution operator of the unavailable macroscopic PDE for the density evolution. For our illustrations, we use the Social Force Model to generate data in a corridor with an obstacle, imposing periodic boundary conditions. The numerical results demonstrate high accuracy, robustness, and generalizability, thus allowing for fast and accurate modelling/simulation of crowd dynamics from agent-based simulations.
- [15] arXiv:2508.04020 (交叉列表自 math.AP) [中文pdf, pdf, html, 其他]
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标题： 受控领导者-跟随者系统的微观-宏观和宏观-宏观极限标题： Micro-macro and macro-macro limits for controlled leader-follower systems评论： 41页，6图主题： 偏微分方程分析 (math.AP) ; 数值分析 (math.NA) ; 优化与控制 (math.OC)
我们研究一个受反馈控制的交互粒子领导者-跟随者系统，并通过两步传递过程推导其平均场极限：首先将其转化为耦合领导者粒子与跟随者流体的微观-宏观系统，然后进一步转化为完全连续的宏观-宏观系统。 对于每种极限过程，我们基于调制能量方法和Wasserstein距离建立了定量稳定性与收敛性估计。 这些结果为受控多智能体系统的分层约简提供了严格的理论基础。 给出了数值模拟，包括超出所考虑解析类别的相互作用势的例子，以展示动态行为并支持理论结果。
We study a leader-follower system of interacting particles subject to feedback control and derive its mean-field limits through a two-step passage: first to a micro-macro system coupling leader particles with a follower fluid, and then to a fully continuum macro-macro system. For each limiting procedure, we establish quantitative stability and convergence estimates based on modulated energy methods and Wasserstein distances. These results provide a rigorous foundation for the hierarchical reduction of controlled multi-agent systems. Numerical simulations are presented, including examples with interaction potentials beyond the analytical class considered, to demonstrate the dynamics and support the theoretical results.
- [16] arXiv:2508.04234 (交叉列表自 cs.CV) [中文pdf, pdf, html, 其他]
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标题： 一种用于合成孔径雷达图像分类的机器学习方法标题： A machine learning approach for image classification in synthetic aperture RADAR评论： 22页主题： 计算机视觉与模式识别 (cs.CV) ; 数值分析 (math.NA)
我们考虑合成孔径雷达（SAR）中的问题，即通过卷积神经网络（CNNs）来识别和分类地面上的物体。 具体而言，我们采用单次散射近似来使用模拟的SAR数据以及从这些数据重建的图像对物体的形状进行分类，并比较这些方法的成功率。 然后，我们识别来自卫星Sentinel-1的真实SAR图像中的冰类型。 在两个实验中，我们实现了有前景的高分类准确率（$\geq$75%）。 我们的结果证明了CNN在用于几何和环境分类任务时的有效性。 我们的研究还探讨了在不同天线高度下获取SAR数据对我们成功分类物体的能力的影响。
We consider the problem in Synthetic Aperture RADAR (SAR) of identifying and classifying objects located on the ground by means of Convolutional Neural Networks (CNNs). Specifically, we adopt a single scattering approximation to classify the shape of the object using both simulated SAR data and reconstructed images from this data, and we compare the success of these approaches. We then identify ice types in real SAR imagery from the satellite Sentinel-1. In both experiments we achieve a promising high classification accuracy ($\geq$75\%). Our results demonstrate the effectiveness of CNNs in using SAR data for both geometric and environmental classification tasks. Our investigation also explores the effect of SAR data acquisition at different antenna heights on our ability to classify objects successfully.
- [17] arXiv:2508.04444 (交叉列表自 cs.LG) [中文pdf, pdf, html, 其他]
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标题： 矩阵自由的两到无穷范数和一到两范数估计标题： Matrix-Free Two-to-Infinity and One-to-Two Norms Estimation主题： 机器学习 (cs.LG) ; 数值分析 (math.NA) ; 机器学习 (stat.ML)
在本文中，我们提出新的随机算法，在无需矩阵的条件下估计矩阵的二到无穷范数和一到二范数，仅使用矩阵-向量乘法。 我们的方法基于对Hutchinson的对角线估计器及其Hutch++版本的适当修改。 我们为这两种修改提供了Oracle复杂度界限。 我们进一步展示了我们的算法在图像分类任务中的深度神经网络训练中的基于雅可比的正则化方面的实际效用。 我们还证明了我们的方法可以用于减轻推荐系统领域对抗攻击的影响。
In this paper, we propose new randomized algorithms for estimating the two-to-infinity and one-to-two norms in a matrix-free setting, using only matrix-vector multiplications. Our methods are based on appropriate modifications of Hutchinson's diagonal estimator and its Hutch++ version. We provide oracle complexity bounds for both modifications. We further illustrate the practical utility of our algorithms for Jacobian-based regularization in deep neural network training on image classification tasks. We also demonstrate that our methodology can be applied to mitigate the effect of adversarial attacks in the domain of recommender systems.
交叉提交 (展示 6 之 6 条目 )
- [18] arXiv:2409.18490 (替换) [中文pdf, pdf, html, 其他]
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标题： 分数Korteweg-de Vries方程的零色散极限的谱Galerkin方法标题： Spectral Galerkin method for the zero dispersion limit of the fractional Korteweg-de Vries equation主题： 数值分析 (math.NA)
我们提出了一种全离散的Crank-Nicolson傅里叶谱伽辽金（FSG）方案，用于近似分数Korteweg-de Vries（KdV）方程的解，该方程涉及指数为$\alpha \in [1,2]$的分数拉普拉斯算子和一个阶为$\varepsilon^2$的小色散系数。 当$\varepsilon \to 0$趋于极限时的解被称为零色散极限。 我们证明了半离散的FSG方案保持前三个积分不变量，从而保持结构，并且全离散的FSG方案是$L^2$-保守的，确保稳定性。 使用紧致性论证，我们构造性地证明了近似解在$C([0,T]; H_p^{1+\alpha}(\mathbb{R}))$中收敛于分数KdV方程的唯一解，对于周期初始数据在$H_p^{1+\alpha}(\mathbb{R})$中。 所设计的方案对于初始数据在$H_p^r,$ $r \geq 1+\alpha$ 中具有谱精度，对于解析初始数据具有指数精度。 此外，我们证明从全离散 FSG 方案得到的零色散极限的近似解在$\varepsilon \to 0$时收敛到 Hopf 方程在$L^2$中的解，直至梯度爆破时间$t_c$。 超过$t_c$后，数值研究显示近似解收敛到渐近解，该解在振荡区域中由 Whitham 平均方程弱描述，对于$\alpha = 2$。 提供了数值结果来展示方案的收敛性并验证理论结果。
We present a fully discrete Crank-Nicolson Fourier-spectral-Galerkin (FSG) scheme for approximating solutions of the fractional Korteweg-de Vries (KdV) equation, which involves a fractional Laplacian with exponent $\alpha \in [1,2]$ and a small dispersion coefficient of order $\varepsilon^2$. The solution in the limit as $\varepsilon \to 0$ is known as the zero dispersion limit. We demonstrate that the semi-discrete FSG scheme conserves the first three integral invariants, thereby structure preserving, and that the fully discrete FSG scheme is $L^2$-conservative, ensuring stability. Using a compactness argument, we constructively prove the convergence of the approximate solution to the unique solution of the fractional KdV equation in $C([0,T]; H_p^{1+\alpha}(\mathbb{R}))$ for the periodic initial data in $H_p^{1+\alpha}(\mathbb{R})$. The devised scheme achieves spectral accuracy for the initial data in $H_p^r,$ $r \geq 1+\alpha$ and exponential accuracy for the analytic initial data. Additionally, we establish that the approximation of the zero dispersion limit obtained from the fully discrete FSG scheme converges to the solution of the Hopf equation in $L^2$ as $\varepsilon \to 0$, up to the gradient catastrophe time $t_c$. Beyond $t_c$, numerical investigations reveal that the approximation converges to the asymptotic solution, which is weakly described by the Whitham's averaged equation within the oscillatory zone for $\alpha = 2$. Numerical results are provided to demonstrate the convergence of the scheme and to validate the theoretical findings.
- [19] arXiv:2412.16902 (替换) [中文pdf, pdf, html, 其他]
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标题： 对对数薛定谔方程的指数波积分傅里叶谱方法的最优误差界标题： Optimal error bounds on an exponential wave integrator Fourier spectral method for the logarithmic Schrödinger equation评论： 21页，10图主题： 数值分析 (math.NA)
我们在$H^2$-解的假设下，证明了对数薛定谔方程（LogSE）的指数波积分傅里叶谱（EWI-FS）方法的几乎最优误差界，该解在理论上是保证的。在满足由对数非线性奇异性影响的数值格式稳定性所需的CFL类型时间步长限制$\tau |\ln \tau| \leq h^2/|\ln h|$的条件下，建立了$L^2$-范数的误差界，其阶数为$O(\tau |\ln \tau|^2 + h^2 |\ln h|)$，其中$\tau$是时间步长，$h$是网格大小。与文献中对数薛定谔方程的误差估计相比，我们的误差界在相同正则性假设下显著提高了收敛速度，或者在获得相同收敛速度时大大降低了正则性要求。此外，我们的结果可以直接应用于具有低正则性$L^\infty$-势的LogSE，而这是现有误差估计不允许的。 证明中采用两个主要要素：(i) 一个$H^2$-条件的$L^2$-稳定性估计，这是使用能量方法建立的，以避免对数非线性的奇异性，以及 (ii) 使用逆不等式进行数学归纳法来控制数值解的$H^2$-范数。 数值结果被报告以确认我们的误差估计并展示所施加的时间步长限制的必要性。 我们还将 EWI-FS 方法应用于研究一维中的孤子碰撞和二维中的涡旋偶极子动力学。
We prove a nearly optimal error bound on the exponential wave integrator Fourier spectral (EWI-FS) method for the logarithmic Schr\"odinger equation (LogSE) under the assumption of $H^2$-solution, which is theoretically guaranteed. Subject to a CFL-type time step size restriction $\tau |\ln \tau| \leq h^2/|\ln h|$ for obtaining the stability of the numerical scheme affected by the singularity of the logarithmic nonlinearity, an $L^2$-norm error bound of order $O(\tau |\ln \tau|^2 + h^2 |\ln h|)$ is established, where $\tau$ is the time step size and $h$ is the mesh size. Compared to the error estimates of the LogSE in the literature, our error bound either greatly improves the convergence rate under the same regularity assumptions or significantly weakens the regularity requirement to obtain the same convergence rate. Moreover, our result can be directly applied to the LogSE with low regularity $L^\infty$-potential, which is not allowed in the existing error estimates. Two main ingredients are adopted in the proof: (i) an $H^2$-conditional $L^2$-stability estimate, which is established using the energy method to avoid singularity of the logarithmic nonlinearity, and (ii) mathematical induction with inverse inequalities to control the $H^2$-norm of the numerical solution. Numerical results are reported to confirm our error estimates and demonstrate the necessity of the time step size restriction imposed. We also apply the EWI-FS method to investigate soliton collisions in one dimension and vortex dipole dynamics in two dimensions.
- [20] arXiv:2503.19483 (替换) [中文pdf, pdf, html, 其他]
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标题： 基于统一积分准则的实用超单元积分方法（EHEIM）用于高效超降阶FE$^2$仿真标题： Empirical Hyper Element Integration Method (EHEIM) with Unified Integration Criteria for Efficient Hyper Reduced FE$^2$ Simulations期刊参考： 材料力学 210 (2025) 105444主题： 数值分析 (math.NA) ; 计算物理 (physics.comp-ph)
数值均质化方法通过有限元法（FEM）用于机械多尺度建模，是一种优雅的方法，如果对低尺度组分的行为有很好的理解。 然而，所谓的FE$^2$方法的计算成本非常高，因此降阶方法是必不可少的。 虽然使用本征正交分解（POD）构建微观节点位移的降阶基已成为一种标准技术，但对投影节点力的计算工作量进行降阶，即所谓的超降阶，是一个额外的挑战，文献中已提出了不同的策略。 经验立方方法（ECM）已被证明非常稳健，在结果优化问题中使用总容积守恒作为约束，而其他贡献中提出了基于能量的标准。 本文提出了一种统一的积分标准概念，包括上述标准等。 这些标准既与基于高斯点的超降阶方案一起使用，也与基于单元的超降阶方案一起使用，后者保留了与常见模块化有限元框架的完全兼容性。 这些方法与之前提出的聚类训练策略和整体求解器相结合。 数值示例经验表明，附加标准在给定模式数的情况下提高了准确性。 反之，为了达到给定的精度水平，所需的模式数更少，从而计算成本更低。
Numerical homogenization for mechanical multiscale modeling by means of the finite element method (FEM) is an elegant way of obtaining structure-property relations, if the behavior of the constituents of the lower scale is well understood. However, the computational costs of this so-called FE$^2$ method are so high that reduction methods are essential. While the construction of a reduced basis for the microscopic nodal displacements using proper orthogonal decomposition (POD) has become a standard technique, the reduction of the computational effort for the projected nodal forces, the so-called hyper reduction, is an additional challenge, for which different strategies have been proposed in the literature. The empirical cubature method (ECM), which has been proven to be very robust, implemented the conservation of the total volume is used as a constraint in the resulting optimization problem, while energy-based criteria have been proposed in other contributions. The present contribution presents a unified integration criteria concept, involving the aforementioned criteria, among others. These criteria are used both with a Gauss point-based as well as with an element-based hyper reduction scheme, the latter retaining full compatibility with the common modular finite element framework. The methods are combined with a previously proposed clustered training strategy and a monolithic solver. Numerical examples empirically demonstrate that the additional criteria improve the accuracy for a given number of modes. Vice verse, less modes and thus lower computational costs are required to reach a given level of accuracy.
- [21] arXiv:2504.11926 (替换) [中文pdf, pdf, html, 其他]
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标题： 肿瘤生长的Eyles-King-Styles模型有限元收敛性标题： Convergence of finite elements for the Eyles-King-Styles model of tumour growth主题： 数值分析 (math.NA)
本文提出了对肿瘤生长原始Eyles-King-Styles模型应用的演化曲面有限元方法（ESFEM）的收敛性分析。 该模型包括体内的泊松方程、表面上的强制平均曲率流以及体和表面之间的耦合速度定律。 由于非平凡的体-表面耦合，所有先前的分析都需要一个额外的正则化项。 通过引入$H^{1/2}(\Gamma)$能量估计理论，我们建立了一个本质上新的理论框架，解决了固有的体-表面耦合问题。 基于此框架，我们提供了无需正则化的原始模型的第一个严格的收敛证明。
This paper presents a convergence analysis of the evolving surface finite element method (ESFEM) applied to the original Eyles-King-Styles model of tumour growth. The model consists of a Poisson equation in the bulk, a forced mean curvature flow on the surface, and a coupled velocity law between bulk and surface. Due to the non-trivial bulk-surface coupling, all previous analyses required an additional regularization term. By introducing a $H^{1/2}(\Gamma)$ energy estimates theory, we develop an essentially new theoretical framework that addresses the intrinsic bulk-surface coupling. Based on this framework, we provide the first rigorous convergence proof for the original model without regularization.
- [22] arXiv:2504.13809 (替换) [中文pdf, pdf, html, 其他]
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标题： 一种使用求积展开的边界积分方程快速直接求解器标题： A Fast Direct Solver for Boundary Integral Equations Using Quadrature By Expansion评论： 31页，12张图；修复了TeX格式（v2）；在引言中突出显示了贡献（v3）主题： 数值分析 (math.NA)
我们构建并分析了一个分层直接求解器，用于求解使用求积展开（QBX）方法离散化边界积分方程产生的线性系统。 我们的方案基于现有的分层半分离（HSS）矩阵算子理论，这些算子包含低秩的非对角子矩阵。 我们使用基于代理的远场相互作用近似和插值分解（ID）来构建压缩的HSS算子，这些算子被用作原始系统的快速直接求解器。 我们描述了对标准HSS框架的一些修改，以实现与QBX离散化方法族的兼容性。 我们建立了一个基于QBX介导的代理相互作用多重展开和ID的标准估计的直接求解器误差模型。 基于这些理论结果，我们开发了一种自动设置方案参数的方法，该方法基于用户提供的误差容限。 该求解器无缝地适用于二维和三维问题，并实现了最先进的渐近缩放性能。 我们通过数值实验结束，这些实验支持了对直接求解器误差和计算成本的理论预期。
We construct and analyze a hierarchical direct solver for linear systems arising from the discretization of boundary integral equations using the Quadrature by Expansion (QBX) method. Our scheme builds on the existing theory of Hierarchical Semi-Separable (HSS) matrix operators that contain low-rank off-diagonal submatrices. We use proxy-based approximations of the far-field interactions and the Interpolative Decomposition (ID) to construct compressed HSS operators that are used as fast direct solvers for the original system. We describe a number of modifications to the standard HSS framework that enable compatibility with the QBX family of discretization methods. We establish an error model for the direct solver that is based on a multipole expansion of the QBX-mediated proxy interactions and standard estimates for the ID. Based on these theoretical results, we develop an automatic approach for setting scheme parameters based on user-provided error tolerances. The resulting solver seamlessly generalizes across two- and tree-dimensional problems and achieves state-of-the-art asymptotic scaling. We conclude with numerical experiments that support the theoretical expectations for the error and computational cost of the direct solver.
- [23] arXiv:2505.11762 (替换) [中文pdf, pdf, html, 其他]
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标题： 求解薛定谔方程的参数化Wasserstein哈密顿流方法标题： A parameterized Wasserstein Hamiltonian flow approach for solving the Schrödinger equation主题： 数值分析 (math.NA)
在本文中，我们提出了一种新方法来计算时间依赖的薛定谔方程（TDSE）的解。 利用前推映射和Wasserstein哈密顿流，我们将TDSE重新表述为前推映射的哈密顿系统。 这种新公式可以被视为Wasserstein空间中的生成模型，这是概率密度函数的流形。 然后我们通过降阶模型（如神经网络）对前推映射进行参数化。 这通过将密度流形上的Wasserstein度量拉回到参数空间，从而在参数空间中引入了一种新的度量，进一步导致了降阶模型参数的常微分方程（ODE）系统。 利用深度学习中的计算技术，如神经ODE，我们设计了一个算法，在参数化的前推映射空间中求解TDSE，该算法提供了一种潜在可扩展到高维问题的替代方法。 给出了几个数值例子来展示该算法的性能。
In this paper, we propose a new method to compute the solution of time-dependent Schr\"odinger equation (TDSE). Using push-forward maps and Wasserstein Hamiltonian flow, we reformulate the TDSE as a Hamiltonian system in terms of push-forward maps. The new formulation can be viewed as a generative model in the Wasserstein space, which is a manifold of probability density functions. Then we parameterize the push-forward maps by reduce-order models such as neural networks. This induces a new metric in the parameter space by pulling back the Wasserstein metric on density manifold, which further results in a system of ordinary differential equations (ODEs) for the parameters of the reduce-order model. Leveraging the computational techniques from deep learning, such as Neural ODE, we design an algorithm to solve the TDSE in the parameterized push-forward map space, which provides an alternative approach with the potential to scale up to high-dimensional problems. Several numerical examples are presented to demonstrate the performance of this algorithm.
- [24] arXiv:2508.01238 (替换) [中文pdf, pdf, html, 其他]
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标题： 三维有限元共形复形标题： Finite element conformal complexes in three dimensions评论： 35页主题： 数值分析 (math.NA)
本文将Bernstein-Gelfand-Gelfand（BGG）框架扩展到构造三维空间中涉及共形张量（即对称且无迹张量）的有限元共形Hessian复形和共形弹性复形。 这些复形包含高阶微分算子，包括线性化Cotton-York算子，并需要具有非平凡光滑性和迹条件的共形张量空间。 引入了离散BGG框架的新应用，结合气泡空间的几何分解和约简操作，用于局部气泡有限元复形。 这比基于全局BGG的方法得到更简单且更易处理的构造，并导致气泡共形复形。 在这些气泡共形复形和相关的面气泡复形基础上，系统地开发了具有不同光滑度的有限元共形Hessian复形和共形弹性复形。 所得复形支持在相对论、Cosserat弹性力学和流体力学应用中的稳定且保持结构的数值方法。
This paper extends the Bernstein-Gelfand-Gelfand (BGG) framework to the construction of finite element conformal Hessian complexes and conformal elasticity complexes in three dimensions involving conformal tensors (i.e., symmetric and traceless tensors). These complexes incorporate higher-order differential operators, including the linearized Cotton-York operator, and require conformal tensor spaces with nontrivial smoothness and trace conditions. A novel application of the discrete BGG framework, combined with the geometric decomposition of bubble spaces and a reduction operation, to local bubble finite element complexes is introduced. This yields simpler and more tractable constructions than global BGG-based approaches, and leads to the bubble conformal complexes. Building on these bubble conformal complexes and the associated face bubble complexes, finite element conformal Hessian complexes and conformal elasticity complexes with varying degrees of smoothness are systematically developed. The resulting complexes support stable and structure-preserving numerical methods for applications in relativity, Cosserat elasticity, and fluid mechanics.
- [25] arXiv:2508.02707 (替换) [中文pdf, pdf, html, 其他]
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标题： 传输噪声在$\mathbb{S}^2$上的扩散行为标题： Diffusive behavior of transport noise on $\mathbb{S}^2$评论： 13页，2张图。欢迎所有评论！主题： 数值分析 (math.NA) ; 概率 (math.PR) ; 流体动力学 (physics.flu-dyn)
我们从理论上和数值上研究球面上由输运噪声引起的扩散。 在环面之前的分析表明，适当选择的欧拉方程中的输运噪声会导致类似于纳维-斯托克斯方程的扩散行为。 在这里，我们分析球面上由噪声引起的微分椭圆算子耗散的动力学，并表征其能量和涡旋度衰减特性。 通过使用Zeitlin离散化的结构保持数值模拟，我们证明适当缩放的输运噪声会引发能量耗散，同时保持涡旋度和共伴随轨道。 所提出的分析为输运噪声的进一步理论研究奠定了基础，并支持将输运噪声模型校准为地理流体模拟中未解析过程的参数化方法。
We investigate theoretically and numerically transport noise-induced diffusion in flows on the sphere. Previous analysis on the torus demonstrated that suitably chosen transport noise in the Euler equations leads to diffusive behavior resembling the Navier--Stokes equations. Here, we analyze dynamics on the sphere with noise-induced differential elliptic operator dissipation and characterize their energy and enstrophy decay properties. Through structure-preserving numerical simulations with the Zeitlin discretization, we demonstrate that appropriately scaled transport noise induces energy dissipation while preserving enstrophy and coadjoint orbits. The presented analysis lays a groundwork for further theoretical investigation of transport noise and supports the calibration of transport noise models as a parametrization for unresolved processes in geophysical fluid simulations.
- [26] arXiv:2306.07886 (替换) [中文pdf, pdf, html, 其他]
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标题： 对称性与对称张量分解问题的临界点标题： Symmetry & Critical Points for Symmetric Tensor Decomposition Problems主题： 优化与控制 (math.OC) ; 机器学习 (cs.LG) ; 代数几何 (math.AG) ; 数值分析 (math.NA) ; 机器学习 (stat.ML)
我们考虑与实对称张量分解为秩一项之和相关的非凸优化问题。 利用丰富的对称结构，构造了由问题维度的Puiseux级数表示的无限族临界点，从而得到了目标函数值和Hessian谱的精确解析估计。 这些结果使得能够对局部优化方法的各种障碍进行解析表征，特别是揭示了在对称性、结构和解析性质方面有所不同的鞍点和极小值的复杂阵列。 一个显著的现象是，对于所有考虑的临界点，Hessian的指标随着目标函数值的增加而增加。
We consider the nonconvex optimization problem associated with the decomposition of a real symmetric tensor into a sum of rank-one terms. Use is made of the rich symmetry structure to construct infinite families of critical points represented by Puiseux series in the problem dimension, and so obtain precise analytic estimates on the objective function value and the Hessian spectrum. The results enable an analytic characterization of various obstructions to local optimization methods, revealing, in particular, a complex array of saddles and minima that differ in their symmetry, structure, and analytic properties. A notable phenomenon, observed for all critical points considered, concerns the index of the Hessian increasing with the objective function value.
- [27] arXiv:2410.10040 (替换) [中文pdf, pdf, 其他]
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标题： 漂移-扩散方程的饱和标题： Drift-diffusion equations with saturation评论： 52页，7图主题： 偏微分方程分析 (math.AP) ; 数值分析 (math.NA)
我们关注一个非线性连续性方程族，用于描述非负密度$\rho$的演化，其具有连续且紧支撑的非线性迁移率$\mathrm{m}(\rho)$，该迁移率不一定为凹函数。 速度场是包含内部和约束能量项的自由能变化的负梯度。 具有紧支撑迁移率的问题通常被称为饱和问题，因为密度的值被限制在一个最大值以下。 利用一组近似问题，我们证明了$C_0$-半群的$L^1$压缩映射的存在性。 我们研究了该问题的$\omega$极限，其最相关的性质以及在长时间行为中自由边界的出现。 这个问题具有形式上的梯度流结构，我们讨论了对应自由能在与概率密度的$L^\infty$约束梯度流的初始数据集相关的自然拓扑下的局部/全局极小值。 此外，我们分析了一个保持结构的隐式有限体积格式，并讨论了其收敛性和长时间行为。
We focus on a family of nonlinear continuity equations for the evolution of a non-negative density $\rho$ with a continuous and compactly supported nonlinear mobility $\mathrm{m}(\rho)$ not necessarily concave. The velocity field is the negative gradient of the variation of a free energy including internal and confinement energy terms. Problems with compactly supported mobility are often called saturation problems since the values of the density are constrained below a maximal value. Taking advantage of a family of approximating problems, we show the existence of $C_0$-semigroups of $L^1$ contractions. We study the $\omega$-limit of the problem, its most relevant properties, and the appearance of free boundaries in the long-time behaviour. This problem has a formal gradient-flow structure, and we discuss the local/global minimisers of the corresponding free energy in the natural topology related to the set of initial data for the $L^\infty$-constrained gradient flow of probability densities. Furthermore, we analyse a structure preserving implicit finite-volume scheme and discuss its convergence and long-time behaviour.
- [28] arXiv:2410.21111 (替换) [中文pdf, pdf, 其他]
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标题： LAMA：稀疏视图CT的稳定双域深度重建标题： LAMA: Stable Dual-Domain Deep Reconstruction For Sparse-View CT评论： arXiv:2507.22316 旨在取代本文期刊参考： 《数学成像与视觉》67，30（2025）主题： 计算机视觉与模式识别 (cs.CV) ; 机器学习 (cs.LG) ; 数值分析 (math.NA)
反问题出现在许多应用中，尤其是在断层成像中。 我们开发了一种学习交替最小化算法（LAMA），通过结合数据驱动和经典技术的两块优化来解决这些问题，并具有已证明的收敛性。 LAMA 是由一个变分模型自然产生的，在数据和图像域中具有可学习的正则化项，参数化为使用特定领域数据训练的神经网络的复合函数。 我们允许这些正则化项是非凸和非光滑的，以有效地从数据中提取特征。 我们使用 Nesterov 的平滑技术以及残差学习架构来最小化整体目标函数。 实证表明， LAMA 降低了网络复杂度，提高了内存效率，并增强了重建精度、稳定性和可解释性。 大量实验表明，LAMA 在常用的计算机断层扫描基准数据集上显著优于最先进方法。
Inverse problems arise in many applications, especially tomographic imaging. We develop a Learned Alternating Minimization Algorithm (LAMA) to solve such problems via two-block optimization by synergizing data-driven and classical techniques with proven convergence. LAMA is naturally induced by a variational model with learnable regularizers in both data and image domains, parameterized as composite functions of neural networks trained with domain-specific data. We allow these regularizers to be nonconvex and nonsmooth to extract features from data effectively. We minimize the overall objective function using Nesterov's smoothing technique and residual learning architecture. It is demonstrated that LAMA reduces network complexity, improves memory efficiency, and enhances reconstruction accuracy, stability, and interpretability. Extensive experiments show that LAMA significantly outperforms state-of-the-art methods on popular benchmark datasets for Computed Tomography.
- [29] arXiv:2502.13675 (替换) [中文pdf, pdf, 其他]
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标题： 一种有限单元法的CFL条件标题： A CFL condition for the finite cell method评论： 19页，9图，4表主题： 计算工程、金融与科学 (cs.CE) ; 数值分析 (math.NA)
浸入边界有限元方法使用户能够绕过可能令人头疼的边界拟合网格生成任务。 当与显式时间积分结合使用时，物理域中支持很少的切割单元会导致临界时间步长显著减小，这对浸入波传播模拟构成了重大挑战。 有限单元法通过在虚构域中定义问题的弱形式来稳定切割单元，但该弱形式被一个小值$\alpha$缩放。 本文研究了有限单元法对显式时间积分临界时间步长的影响。 从一个解析的一自由度模型开始，我们系统地研究了$\alpha$稳定化对最大特征值的影响，从而对临界时间步长的影响，针对角切割和薄片切割。 分析通过一个具有一个单元且多项式次数增加的示例的数值研究得到补充，确认即使切割比例趋于零，临界时间步长也不会低于某个极限。 这个下限由$\alpha$的选择控制。 在高维情况下，发现薄片切割比角切割更具破坏性，因此决定了最小临界时间步长。 增加多项式次数对此退化影响很小。 基于这些观察，我们推导了一个作为$\alpha$函数的最小临界时间步长的估计值，并以此为有限单元法提出了一种修改后的CFL条件。 该条件的有效性在一个二维穿孔板示例中得到了验证。
Immersed boundary finite element methods allow the user to bypass the potentially troublesome task of boundary-conforming mesh generation. When combined with explicit time integration, poorly cut elements with little support in the physical domain lead to a severely reduced critical time step size, posing a major challenge for immersed wave propagation simulations. The finite cell method stabilizes cut elements by defining the weak form of the problem also in the fictitious domain, but scaled by a small value $\alpha$. This paper investigates the effect of the finite cell method on the critical time step size for explicit time integration. Starting with an analytical one-degree-of-freedom model, we systematically study the influence of $\alpha$-stabilization on the maximum eigenvalue, and thus on the critical time step size, for corner and sliver cuts. The analysis is complemented by a numerical study of an example with one element and increasing polynomial degree, confirming that the critical time step size does not decrease below a certain limit, even as the cut fraction tends to zero. This lower bound is controlled by the choice of $\alpha$. In higher dimensions, sliver cuts are found to be more detrimental than corner cuts, thus determining the minimum critical time step size. Increasing the polynomial degree has only little effect on this degradation. Based on these observations, we derive an estimate of the minimum critical time step size as a function of $\alpha$, which we use to propose a modified CFL condition for the finite cell method. The validity of this condition is demonstrated on a two-dimensional perforated plate example.
- [30] arXiv:2504.15110 (替换) [中文pdf, pdf, html, 其他]
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标题： Besov范数中的逼近率以及带有残差连接的Kolmogorov-Arnold网络的样本复杂度标题： Approximation Rates in Besov Norms and Sample-Complexity of Kolmogorov-Arnold Networks with Residual Connections主题： 机器学习 (cs.LG) ; 神经与进化计算 (cs.NE) ; 泛函分析 (math.FA) ; 数值分析 (math.NA) ; 机器学习 (stat.ML)
受Kolmogorov-Arnold叠加定理的启发，Kolmogorov-Arnold网络（KANs）最近作为大多数深度学习框架的改进骨干出现，通过允许可训练的样条激活函数，比其多层感知器（MLP）前身具有更高的适应性。在本文中，我们通过表明KAN架构可以以最优逼近率在有界开域或甚至分形域$\mathcal{X}$在$\mathbb{R}^d$上最优逼近任何Besov函数$B^{s}_{p,q}(\mathcal{X})$，相对于任何较弱的Besov范数$B^{\alpha}_{p,q}(\mathcal{X})$；其中$\alpha < s$。我们通过限制相关Res-KAN类的伪维数来补充我们的逼近结果。作为后者的应用，我们直接推导出当从$N$独立同分布的无噪声样本中学习Besov正则性的函数时，残差KAN模型的样本复杂度的无维度估计，表明KAN可以学习它们能够逼近的平滑映射。
Inspired by the Kolmogorov-Arnold superposition theorem, Kolmogorov-Arnold Networks (KANs) have recently emerged as an improved backbone for most deep learning frameworks, promising more adaptivity than their multilayer perceptron (MLP) predecessor by allowing for trainable spline-based activation functions. In this paper, we probe the theoretical foundations of the KAN architecture by showing that it can optimally approximate any Besov function in $B^{s}_{p,q}(\mathcal{X})$ on a bounded open, or even fractal, domain $\mathcal{X}$ in $\mathbb{R}^d$ at the optimal approximation rate with respect to any weaker Besov norm $B^{\alpha}_{p,q}(\mathcal{X})$; where $\alpha < s$. We complement our approximation result with a statistical guarantee by bounding the pseudodimension of the relevant class of Res-KANs. As an application of the latter, we directly deduce a dimension-free estimate on the sample complexity of a residual KAN model when learning a function of Besov regularity from $N$ i.i.d. noiseless samples, showing that KANs can learn the smooth maps which they can approximate.
- [31] arXiv:2508.02928 (替换) [中文pdf, pdf, html, 其他]
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标题： 用于SEIQR流行病学PDE模型的非标准有限差分格式标题： A nonstandard finite difference scheme for an SEIQR epidemiological PDE model主题： 定量方法 (q-bio.QM) ; 动力系统 (math.DS) ; 数值分析 (math.NA)
本文介绍了一种非标准有限差分（NSFD）方法，用于反应扩散SEIQR流行病学模型，该模型能够捕捉传染病传播的时空动态。 作为一类半线性抛物型偏微分方程（PDEs）系统，该模型通过引入空间扩散来考虑人口流动和空间异质性，从而扩展了经典的分 compartment 模型。 所提出的NSFD离散化方法旨在保持连续模型的基本定性特征，如正性、有界性和稳定性，这些特性通常会被标准有限差分方法破坏。 我们严格分析了模型的适定性，为PDE系统构建了一个结构保持的NSFD方案，并研究了其收敛性和局部截断误差。 数值模拟验证了理论结果，并展示了该方案在保持生物上一致的动力学方面的有效性。
This paper introduces a nonstandard finite difference (NSFD) approach to a reaction-diffusion SEIQR epidemiological model, which captures the spatiotemporal dynamics of infectious disease transmission. Formulated as a system of semilinear parabolic partial differential equations (PDEs), the model extends classical compartmental models by incorporating spatial diffusion to account for population movement and spatial heterogeneity. The proposed NSFD discretization is designed to preserve the continuous model's essential qualitative features, such as positivity, boundedness, and stability, which are often compromised by standard finite difference methods. We rigorously analyze the model's well-posedness, construct a structure-preserving NSFD scheme for the PDE system, and study its convergence and local truncation error. Numerical simulations validate the theoretical findings and demonstrate the scheme's effectiveness in preserving biologically consistent dynamics.