数值分析
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- [1] arXiv:2508.02674 [中文pdf, pdf, html, 其他]
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标题： 球面函数的轨道恢复标题： Orbit recovery for spherical functions评论： 15页主题： 数值分析 (math.NA) ; 信息论 (cs.IT)
轨道恢复是数学和应用科学中的核心问题，具有在结构生物学中的重要应用。 本文关注在 $SO(n)$ 的旋转作用下，对 ${\mathbb R}^{n}$ 和球面 $S^{n-1}$ 上函数的通用轨道进行恢复。 具体而言，我们证明了三次不变量（称为双谱）足以恢复通过限制球面部分的带宽并离散化径向方向得到的 $L^2({\mathbb R}^n)$ 的有限维近似中的函数的通用轨道。 特别地，我们的主要结果明确界定了从三次不变量中恢复所需的径向方向样本数量。 从应用角度来看，最重要的情况是 $SO(3)$，它出现在许多科学领域，并且尤其在诸如冷冻电子断层扫描和冷冻电子显微镜等领先的结构生物学应用中起着核心作用。 我们对 $SO(3)$ 的结果表明，考虑三个球壳（即径向方向的样本）就足以恢复通用轨道，这验证了 Bandeira 等人论文中提出的隐含猜想。 我们的证明技术提供了一种显式的、计算高效的算法，通过依次求解线性方程组来恢复信号。 我们实现了这个算法，并在两个蛋白质结构上展示了它的有效性。
Orbit recovery is a central problem in both mathematics and applied sciences, with important applications to structural biology. This paper focuses on recovering generic orbits of functions on ${\mathbb R}^{n}$ and the sphere $S^{n-1}$ under the rotation action of $SO(n)$. Specifically, we demonstrate that invariants of degree three (called the bispectrum) suffice to recover generic orbits of functions in finite-dimensional approximations of $L^2({\mathbb R}^n)$ obtained by band-limiting the spherical component and discretizing the radial direction. In particular, our main result explicitly bounds the number of samples in the radial direction required for recovery from the degree three invariants. From an application perspective, the most important case is $SO(3)$, which arises in many scientific fields, and in particular, plays a central role in leading structural biology applications such as cryo-electron tomography and cryo-electron microscopy. Our result for $SO(3)$ states that considering three spherical shells (i.e., samples in the radial direction) is sufficient to recover generic orbits, which verifies an implicit conjecture made in a paper of Bandeira et al. Our proof technique provides an explicit, computationally efficient algorithm to recover the signal by successively solving systems of linear equations. We implemented this algorithm and demonstrated its effectiveness on two protein structures.
- [2] arXiv:2508.02675 [中文pdf, pdf, html, 其他]
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标题： 全矢量麦克斯韦方程组与连续角指数标题： Full Vectorial Maxwell Equations with Continuous Angular Indices主题： 数值分析 (math.NA) ; 数学物理 (math-ph) ; 经典物理 (physics.class-ph)
本文提出了一种数学框架，用于在具有连续角指数的圆柱和球形几何中求解麦克斯韦方程组。 我们超越了标准的离散谐波分解，采用广义谱积分进行连续谱表示，捕捉表现出奇异行为但又在几何中心产生有限能量场的电磁解。 对于连续角指数$\ell, m \in \mathbb{R}$，我们在加权Sobolev空间$H^s_{\alpha(\ell,m)}(\Omega)$中研究解的存在性和唯一性，遵循~\cite{adams2003, reed1975}中建立的框架，证明$\ell > -\frac{1}{2}$的有限能量，并通过双正交函数系统构造显式谱核。 该框架涵盖了具有连续方位指数$\nu \in (0,1)$的可分离圆柱模式以及通过矢量旋度运算耦合场分量的不可分离球形模式。 我们给出了奇异场行为的渐近分析，研究了谱近似的收敛速率，并通过Galerkin投影方法和数值谱积分验证了理论框架。
This article presents a mathematical framework for solving Maxwell's equations in cylindrical and spherical geometries with continuous angular indices. We extend beyond standard discrete harmonic decomposition to a continuous spectral representation using generalized spectral integrals, capturing electromagnetic solutions that exhibit singular behavoiur yet yield finite-energy fields at the geometric center. For continuous angular indices $\ell, m \in \mathbb{R}$, we study existence and uniqueness of solutions in weighted Sobolev spaces $H^s_{\alpha(\ell,m)}(\Omega)$ following the framework established in ~\cite{adams2003, reed1975}, prove finite energy for $\ell > -\frac{1}{2}$, and construct explicit spectral kernels via biorthogonal function systems. The framework encompasses both separable cylindrical modes with continuous azimuthal index $\nu \in (0,1)$ and non-separable spherical modes where field components couple through vectorial curl operations. We present asymptotic analysis of singular field behavior, investigate convergence rates for spectral approximations, and validate the theoretical framework through Galerkin projection methods and numerical spectral integration.
- [3] arXiv:2508.02676 [中文pdf, pdf, html, 其他]
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标题： 当表面演化遇到福克-普朗克方程：一种用于均匀参数化的切向速度模型标题： When surface evolution meets Fokker-Planck equation: a novel tangential velocity model for uniform parametrization主题： 数值分析 (math.NA) ; 数学物理 (math-ph)
在模拟曲面几何演化时，一个常见问题是点的意外聚集，这可能导致数值不稳定。 我们为此提出了一种新颖的人工切向速度方法。 人工切向速度由由Fokker-Planck方程控制的表面密度场生成，以指导点的分布。 开发了一种目标分布匹配算法，利用密度函数的表面Kullback-Leibler散度。 数值方法在完全无网格框架内进行，使用移动最小二乘近似，从而消除了网格生成的需要，并允许对非结构化点云数据进行灵活处理。 进行了大量数值实验，以展示所提出方法在各种曲面演化问题中的鲁棒性、准确性和有效性，包括平均曲率流。
A common issue in simulating geometric evolution of surfaces is unexpected clustering of points that may cause numerical instability. We propose a novel artificial tangential velocity method for this matter. The artificial tangential velocity is generated from a surface density field governed by a Fokker-Planck equation to guide the point distribution. A target distribution matching algorithm is developed leveraging the surface Kullback-Leibler divergence of density functions. The numerical method is formulated within a fully meshless framework using the moving least squares approximation, thereby eliminating the need for mesh generation and allowing flexible treatment of unstructured point cloud data. Extensive numerical experiments are conducted to demonstrate the robustness, accuracy, and effectiveness of the proposed approach across a variety of surface evolution problems, including the mean curvature flow.
- [4] arXiv:2508.02677 [中文pdf, pdf, html, 其他]
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标题： 自动化的$h$-自适应有限元近似法求解Falkner-Skan方程标题： Automated $h$-adaptivity for finite element approximations of the Falkner-Skan equation主题： 数值分析 (math.NA)
本文详细介绍了用于求解\textit{法尔克纳-斯坎方程}的$h$自适应有限元方法的开发和应用。后验误差估计控制网格的自适应性，特别是广为人知的\textit{凯利误差估计器}，该方法利用元素之间的梯度跳跃。该方法的实现使得能够准确且高效地解析 Falkner-Skan 流典型的边界层行为。在各种楔形流参数下获得了数值解，包括有利和不利的压力梯度。本研究的重点之一是精确计算皮肤摩擦系数，这是边界层分析中的关键参数，在这一系列多样的流动条件下进行计算。结果被展示并进行了讨论，证明了自适应有限元方法在这一类非线性边界层问题中的鲁棒性和准确性。
This paper details the development and application of an $h$-adaptive finite element method for the numerical solution of the \textit{Falkner-Skan equation}. A posteriori error estimation governs the adaptivity of the mesh, specifically the well-established \textit{Kelly error estimator}, which utilizes the jump in the gradient across elements. The implementation of this method allowed for accurate and efficient resolution of the boundary layer behavior characteristic of Falkner-Skan flows. Numerical solutions were obtained across various wedge flow parameters, encompassing favorable and adverse pressure gradients. A key focus of this study was the precise computation of the skin friction coefficient, a critical parameter in boundary layer analysis, across this diverse range of flow conditions. The results are presented and discussed, demonstrating the robustness and accuracy of the adaptive finite element approach for this class of nonlinear boundary layer problems.
- [5] arXiv:2508.02678 [中文pdf, pdf, html, 其他]
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标题： 关于切片Cramér度量标题： On sliced Cramér metrics评论： 50页，7图。本工作取代arXiv:2101.10867主题： 数值分析 (math.NA)
我们研究了切片Cramér度量族，表明它们对一大类几何变形具有鲁棒性。 我们的核心结果是，函数与其变形之间的切片Cramér距离可能被变形的位移的某些自然度量所限制，再乘以函数的平均混合范数。 这些结果扩展到断层投影之间的切片Cramér距离。 我们还讨论了卷积对切片Cramér度量的影响。 我们将切片Cramér度量的这些性质与Wasserstein距离所满足的类似性质进行了比较。 此外，我们研究了1D和2D中Cramér和切片Cramér距离的计算高效的基于傅里叶的离散化方法，并证明它们对异方差噪声具有鲁棒性。 结果通过数值实验进行了说明。
We study the family of sliced Cram\'er metrics, showing that they are robust to a broad class of geometric deformations. Our central results are that the sliced Cram\'er distance between a function and its deformation may be bounded by certain natural measures of the deformation's displacement, multiplied by the function's mean mixed norm. These results extend to sliced Cram\'er distances between tomographic projections. We also remark on the effect of convolution on the sliced Cram\'er metrics. We compare these properties of sliced Cram\'er metrics to similar properties satisfied by Wasserstein distances. In addition, we study computationally efficient Fourier-based discretizations of the Cram\'er and sliced Cram\'er distances in 1D and 2D, and prove that they are robust to heteroscedastic noise. The results are illustrated by numerical experiments.
- [6] arXiv:2508.02681 [中文pdf, pdf, html, 其他]
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标题： 用单元神经算子加速均质化问题的共轭梯度求解器标题： Accelerating Conjugate Gradient Solvers for Homogenization Problems with Unitary Neural Operators主题： 数值分析 (math.NA) ; 机器学习 (cs.LG)
快速且可靠的参数偏微分方程（PDE）求解器在许多科学和工程学科中是必需的。例如，对于具有异质微观结构的复合材料和设计材料的需求正在增长。设计这类材料并在实际应用中预测其行为需要解决广泛材料参数和微观结构的均化问题。虽然经典数值求解器提供了由坚实的理论基础支持的可靠和准确的解决方案，但它们的高计算成本和缓慢的收敛性仍然是限制因素。因此，科学机器学习正作为一种有前景的替代方法出现。然而，这些方法通常缺乏保证的精度和物理一致性。这引发了是否可以开发结合数据驱动方法和经典求解器优势的混合方法的问题。为了解决这个问题，我们引入了UNO-CG，这是一种混合求解器，它使用专门设计的机器学习预条件器来加速共轭梯度（CG）求解器，并通过构造确保收敛性。作为预条件器，我们提出了单位神经算子，这是傅里叶神经算子的一种修改。我们的方法可以被解释为数据驱动地发现格林函数，然后用于加速迭代求解器。我们在涉及异质微观结构和数百万自由度的各种均化问题上评估了UNO-CG。我们的结果表明，UNO-CG能够显著减少迭代次数，并且在涉及专家知识的均化问题中与手工设计的预条件器具有竞争力。此外，UNO-CG在各种边界条件下保持强大的性能，其中许多专用求解器不适用，突显了其多样性和鲁棒性。
Rapid and reliable solvers for parametric partial differential equations (PDEs) are needed in many scientific and engineering disciplines. For example, there is a growing demand for composites and architected materials with heterogeneous microstructures. Designing such materials and predicting their behavior in practical applications requires solving homogenization problems for a wide range of material parameters and microstructures. While classical numerical solvers offer reliable and accurate solutions supported by a solid theoretical foundation, their high computational costs and slow convergence remain limiting factors. As a result, scientific machine learning is emerging as a promising alternative. However, such approaches often lack guaranteed accuracy and physical consistency. This raises the question of whether it is possible to develop hybrid approaches that combine the advantages of both data-driven methods and classical solvers. To address this, we introduce UNO-CG, a hybrid solver that accelerates conjugate gradient (CG) solvers using specially designed machine-learned preconditioners, while ensuring convergence by construction. As a preconditioner, we propose Unitary Neural Operators as a modification of Fourier Neural Operators. Our method can be interpreted as a data-driven discovery of Green's functions, which are then used to accelerate iterative solvers. We evaluate UNO-CG on various homogenization problems involving heterogeneous microstructures and millions of degrees of freedom. Our results demonstrate that UNO-CG enables a substantial reduction in the number of iterations and is competitive with handcrafted preconditioners for homogenization problems that involve expert knowledge. Moreover, UNO-CG maintains strong performance across a variety of boundary conditions, where many specialized solvers are not applicable, highlighting its versatility and robustness.
- [7] arXiv:2508.02683 [中文pdf, pdf, html, 其他]
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标题： 双层复合材料的瞬态热分析基于双重可逆包含边界元方法标题： Transient thermal analysis of a bi-layered composites with the dual-reciprocity inclusion-based boundary element method评论： 21页，11图，提交至《国际传热与传质杂志》。在期刊正式出版前，禁止进行任何衍生作品（包括翻译或改编）。数据将在正式出版前保密。主题： 数值分析 (math.NA)
本文提出了一种单域双互易包含边界元方法（DR-iBEM），用于分析在瞬态/谐波热载荷下嵌入椭球异质体的三维完全粘结双层复合材料。 热方程被解释为一个包含时间相关和频率相关非均匀源项的静态问题，这类似于本征场，但通过双互易方法将其转换为边界积分。 利用稳态双材料格林函数，提出了边界积分方程以考虑温度和热通量的连续性条件，从而避免在双材料界面设置任何连续性方程。 引入本征温度梯度和本征热源来分别模拟热导率和比热容的材料不匹配。 DR-iBEM算法特别适用于研究双层复合材料的瞬态和谐波热行为，并通过有限元法（FEM）进行了验证。 与FEM的数值比较证明了其鲁棒性和准确性。 该方法已应用于功能梯度材料作为具有梯度颗粒分布的双材料，其中评估了颗粒尺寸和梯度效应。
This paper proposes a single-domain dual-reciprocity inclusion-based boundary element method (DR-iBEM) for a three-dimensional fully bonded bi-layered composite embedded with ellipsoidal inhomogeneities under transient/harmonic thermal loads. The heat equation is interpreted as a static one containing time- and frequency-dependent nonhomogeneous source terms, which is similar to eigen-fields but is transformed into a boundary integral by the dual-reciprocity method. Using the steady-state bimaterial Green's function, boundary integral equations are proposed to take into account continuity conditions of temperature and heat flux, which avoids setting up any continuity equations at the bimaterial interface. Eigen-temperature-gradients and eigen-heat-source are introduced to simulate the material mismatch in thermal conductivity and heat capacity, respectively. The DR-iBEM algorithm is particularly suitable for investigating the transient and harmonic thermal behaviors of bi-layered composites and is verified by the finite element method (FEM). Numerical comparison with the FEM demonstrates its robustness and accuracy. The method has been applied to a functionally graded material as a bimaterial with graded particle distributions, where particle size and gradation effects are evaluated.
- [8] arXiv:2508.02707 [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)
我们从理论和数值上研究球面上噪声引起的扩散。 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，我们分析球面上带有噪声引起的微分椭圆算子耗散的动力学，并描述其能量和涡旋度衰减特性。 通过使用Zeitlin离散化的保持结构的数值模拟，我们证明适当缩放的输运噪声会引起能量耗散，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球流体模拟中未解析过程的参数化方法。
We investigate transport theoretically and numerically 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.
- [9] arXiv:2508.02714 [中文pdf, pdf, html, 其他]
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标题： 样条浅水矩方程标题： Spline Shallow Water Moment Equations主题： 数值分析 (math.NA) ; 偏微分方程分析 (math.AP) ; 流体动力学 (physics.flu-dyn)
由于底层不可压缩纳维-斯托克斯方程的高维度，需要简化的模型来模拟自由表面流动，这些模型需要完全解析垂直方向上的流动以计算表面高度。另一方面，标准的简化模型，如经典的浅水方程（SWE），假设深度与长度比很小并使用深度平均，无法提供关于垂直速度剖面变化的信息。作为折中方案，最近提出的一种使用勒让德多项式作为垂直速度变化的假设函数的浅流矩方法，展示了所谓的浅水矩方程（SWME）的推导，这些方程结合了低维度和速度剖面建模。然而，到目前为止，只考虑了全局多项式。本文引入了样条浅水矩方程（SSWME），其中分段定义的样条假设函数允许以较低的正则性灵活表示速度剖面。样条基函数的局部支撑为适应性和针对一些典型剖面形状的更大灵活性提供了可能性。我们系统地推导和分析了具有不同数量基函数和不同次数的SSWME模型，在此基础上通过进行双曲正则化，并对高阶SSWME模型层次结构的双曲性进行解析证明，推导出一种正则化的双曲版本。数值模拟显示了新模型的高精度和鲁棒性。
Reduced models for free-surface flows are required due to the high dimensionality of the underlying incompressible Navier-Stokes equations, which need to fully resolve the flow in vertical direction to compute the surface height. On the other hand, standard reduced models, such as the classical Shallow Water Equations (SWE), which assume a small depth-to-length ratio and use depth-averaging, do not provide information about the vertical velocity profile variations. As a compromise, a recently proposed moment approach for shallow flow using Legendre polynomials as ansatz functions for vertical velocity variations showed the derivation of so-called Shallow Water Moment Equations (SWME) that combine low dimensionality with velocity profile modeling. However, only global polynomials are considered so far. This paper introduces Spline Shallow Water Moment Equations (SSWME) where piecewise defined spline ansatz functions allow for a flexible representation of velocity profiles with lower regularity. The local support of the spline basis functions opens up the possibility of adaptability and greater flexibility regarding some typical profile shapes. We systematically derive and analyze hierarchies of SSWME models with different number of basis functions and different degrees, before deriving a regularized hyperbolic version by performing a hyperbolic regularization with analytical proof of hyperbolicity for a hierarchy of high-order SSWME models. Numerical simulations show high accuracy and robustness of the new models.
- [10] arXiv:2508.02717 [中文pdf, pdf, 其他]
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标题： DD-DeepONet：用于解决三个应用场景中偏微分方程的域分解和DeepONet标题： DD-DeepONet: Domain decomposition and DeepONet for solving partial differential equations in three application scenarios主题： 数值分析 (math.NA)
在某些实际的工程应用中，需要在短时间内进行偏微分方程（PDEs）的重复求解，这具有迫切的需求。 本文主要考虑三种需要大量重复模拟的场景。 这三种场景是根据几何形状、边界条件（BCs）或参数是否变化来分类的。 我们引入了DD-DeepONet，这是一种具有强可扩展性的框架，其核心概念涉及将复杂的几何结构分解为简单结构，反之亦然。 我们主要研究由矩形和长方体组成的复杂几何结构，这些结构有广泛的实际应用。 同时，对简单几何结构应用拉伸变换，以解决与形状相关的問題。 这项工作在三种场景中解决了几个典型的PDE，包括拉普拉斯方程、泊松方程、纳维-斯托克斯方程和漂移-扩散方程，展示了DD-DeepONet的计算潜力。 实验结果表明，DD-DeepONet降低了训练难度，所需的數據集和每网络的VRAM更少，并加速了求解过程。
In certain practical engineering applications, there is an urgent need to perform repetitive solving of partial differential equations (PDEs) in a short period. This paper primarily considers three scenarios requiring extensive repetitive simulations. These three scenarios are categorized based on whether the geometry, boundary conditions(BCs), or parameters vary. We introduce the DD-DeepONet, a framework with strong scalability, whose core concept involves decomposing complex geometries into simple structures and vice versa. We primarily study complex geometries composed of rectangles and cuboids, which have numerous practical applications. Simultaneously, stretching transformations are applied to simple geometries to solve shape-dependent problems. This work solves several prototypical PDEs in three scenarios, including Laplace, Poission, N-S, and drift-diffusion equations, demonstrating DD-DeepONet's computational potential. Experimental results demonstrate that DD-DeepONet reduces training difficulty, requires smaller datasets andVRAMper network, and accelerates solution acquisition.
- [11] arXiv:2508.02735 [中文pdf, pdf, html, 其他]
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标题： 周期稳态脉冲在短脉冲光纤激光器中的Floquet稳定性标题： Floquet stability of periodically stationary pulses in a short-pulse fiber laser评论： arXiv管理员注释：与arXiv:2508.01133文本重叠期刊参考： 《应用数学杂志》，第84卷，第3期，第961--987页，2024年主题： 数值分析 (math.NA) ; 动力系统 (math.DS)
现代短脉冲光纤激光器的定量建模和设计无法使用平均模型，因为每个往返过程中脉冲参数有较大的变化。 相反，需要使用通过连接激光器各个组件的模型而得到的集中模型。 由于集中模型中的光脉冲是周期性的，因此使用单值化算子来研究其线性稳定性，该算子是关于脉冲的往返算子的线性化。 开发了一种基于梯度的优化方法来发现周期性脉冲。 目标函数梯度的计算涉及往返算子和单值化算子的伴随算子作用的数值计算。 引入了一种新的傅里叶分裂步法来计算非线性、非局部、刚性方程的线性化方程的解，该方程模拟光纤放大器中的光传播。 该方法是通过对非线性方程的分裂步法中的两个求解算子进行线性化得到的。 单值化算子的谱包括本质谱，对于本质谱有一个解析公式，以及特征值。 在$\lambda=1$处有一个重数为二的特征值，这是由于相位和平移不变性引起的。 其余的特征值由单值化算子的矩阵离散化确定。 仿真结果验证了数值方法的准确性，展示了周期稳态脉冲的例子，它们的谱和本征函数，并讨论了它们的稳定性。
The quantitative modeling and design of modern short-pulse fiber lasers cannot be performed with averaged models because of large variations in the pulse parameters within each round trip. Instead, lumped models obtained by concatenating models for the various components of the laser are required. Since the optical pulses in lumped models are periodic, their linear stability is investigated using the monodromy operator, which is the linearization of the roundtrip operator about the pulse. A gradient-based optimization method is developed to discover periodic pulses. The computation of the gradient of the objective function involves numerical computation of the action of both the round trip operator and the adjoint of the monodromy operator. A novel Fourier split-step method is introduced to compute solutions of the linearization of the nonlinear, nonlocal, stiff equation that models optical propagation in the fiber amplifier. This method is derived by linearizing the two solution operators in a split-step method for the nonlinear equation. The spectrum of the monodromy operator consists of the essential spectrum, for which there is an analytical formula, and the eigenvalues. There is a multiplicity two eigenvalue at $\lambda=1$, which is due to phase and translation invariance. The remaining eigenvalues are determined from a matrix discretization of the monodromy operator. Simulation results verify the accuracy of the numerical methods, show examples of periodically stationary pulses, their spectra and eigenfunctions, and discuss their stability.
- [12] arXiv:2508.02797 [中文pdf, pdf, html, 其他]
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标题： 具有静止对流Brinkman-Forchheimer扩展Darcy方程的半变分不等式的混合有限元方法标题： Mixed Finite Element Method for a Hemivariational Inequality of Stationary convective Brinkman-Forchheimer Extended Darcy equations主题： 数值分析 (math.NA)
本文提出了用于从定常对流Brinkman-Forchheimer扩展Darcy（CBFeD）方程中出现的半变分不等式的混合有限元方法的公式和分析。 该模型通过引入阻尼和泵送效应，扩展了不可压缩Navier-Stokes方程。 半变分不等式描述了粘性、不可压缩流体在饱和多孔介质中的流动，受到非光滑、非凸的摩擦型滑动边界条件的约束。 不可压缩性约束通过混合变分公式处理。 我们通过利用底层算子的伪单调性和强制性性质，建立了解的存在性和唯一性，并提供了所提出数值方案的详细误差分析。 在适当的正则性假设下，该方法使用低阶混合有限元对实现了最优收敛率。 该方案使用$\text{P1b/P1}$元素对进行实现，并进行了数值实验以验证理论结果并确认预期的收敛行为。
This paper presents the formulation and analysis of a mixed finite element method for a hemivariational inequality arising from the stationary convective Brinkman-Forchheimer extended Darcy (CBFeD) equations. This model extends the incompressible Navier-Stokes equations by incorporating both damping and pumping effects. The hemivariational inequality describes the flow of a viscous, incompressible fluid through a saturated porous medium, subject to a nonsmooth, nonconvex friction-type slip boundary condition. The incompressibility constraint is handled via a mixed variational formulation. We establish the existence and uniqueness of solutions by utilizing the pseudomonotonicity and coercivity properties of the underlying operators and provide a detailed error analysis of the proposed numerical scheme. Under suitable regularity assumptions, the method achieves optimal convergence rates with low-order mixed finite element pairs. The scheme is implemented using the $\text{P1b/P1}$ element pair, and numerical experiments are presented to validate the theoretical results and confirm the expected convergence behavior.
- [13] arXiv:2508.02861 [中文pdf, pdf, html, 其他]
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标题： 基于H(curl)的斯托克斯问题弱强制无滑移边界条件的近似方法标题： H(curl)-based approximation of the Stokes problem with weakly enforced no-slip boundary conditions评论： 27页，6图主题： 数值分析 (math.NA)
在本工作中，我们展示了如何为不可压缩斯托克斯流的基于H(curl)的公式施加无滑移边界条件，该公式用于纳维-斯托克斯方程和磁流体动力学方程的结构保持离散化。 乍看之下，应用无滑移边界条件似乎很简单：切向部分是H(curl)上的本质边界条件，而法向分量可以通过散度项的分部积分自然地施加。 然而，我们表明这可能导致离散化不适定，并提出了基于Nitsche的有限元方法。 我们分析了离散系统，建立了稳定性并推导了先验误差估计。 数值实验验证了我们的分析，并展示了速度场的最佳收敛率。
In this work, we show how to impose no-slip boundary conditions for an H(curl)-based formulation for incompressible Stokes flow, which is used in structure-preserving discretizations of Navier-Stokes and magnetohydrodynamics equations. At first glance, it seems straightforward to apply no-slip boundary conditions: the tangential part is an essential boundary condition on H(curl) and the normal component can be naturally enforced through integration-by-parts of the divergence term. However, we show that this can lead to an ill-posed discretization and propose a Nitsche-based finite element method instead. We analyze the discrete system, establishing stability and deriving a priori error estimates. Numerical experiments validate our analysis and demonstrate optimal convergence rates for the velocity field.
- [14] arXiv:2508.02925 [中文pdf, pdf, html, 其他]
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标题： 目标导向自适应有限元多级拟蒙特卡洛{M}{C}算法标题： Goal-Oriented Adaptive Finite Element Multilevel Quasi-{M}onte {C}arlo主题： 数值分析 (math.NA)
对具有对数正态扩散系数的偏微分方程得出的兴趣量的有效近似是不确定性量化中的核心挑战。 在本研究中，我们提出了一种多级准蒙特卡罗框架，以近似依赖于具有对数正态扩散系数的线性椭圆偏微分方程解的确定性、实值、有界线性泛函，{由一个49维的高斯随机向量参数化}以及在$\mathbb{R}^d$有界域中的确定性几何奇异性。 我们分析了参数正则性，并基于一系列自适应网格进行多级实现，这些网格在“目标导向的自适应有限元多级蒙特卡罗及其收敛速率”中开发，\emph{计算方法在工程力学中的应用}，402（2022），第115582页。 为了进一步减少方差，我们结合了重要性抽样，并在多级层次结构中引入了零级控制变量。 {引入这样的控制变量可以改变初始网格的最优选择，进一步突出自适应网格的优势。} 数值实验表明，我们的自适应QMC算法在计算成本显著低于标准多级蒙特卡罗方法的情况下实现了预定的精度。
The efficient approximation of quantity of interest derived from PDEs with lognormal diffusivity is a central challenge in uncertainty quantification. In this study, we propose a multilevel quasi-Monte Carlo framework to approximate deterministic, real-valued, bounded linear functionals that depend on the solution of a linear elliptic PDE with a lognormal diffusivity coefficient {parameterized by a 49-dimensional Gaussian random vector} and deterministic geometric singularities in bounded domains of $\mathbb{R}^d$. We analyze the parametric regularity and develop the multilevel implementation based on a sequence of adaptive meshes, developed in "Goal-oriented adaptive finite element multilevel Monte Carlo with convergence rates", \emph{CMAME}, 402 (2022), p. 115582. For further variance reduction, we incorporate importance sampling and introduce a level-0 control variate within the multilevel hierarchy. {Introducing such control variate can alter the optimal choice of initial mesh, further highlighting the advantages of adaptive meshes.} Numerical experiments demonstrate that our adaptive QMC algorithm achieves a prescribed accuracy at substantially lower computational cost than the standard multilevel Monte Carlo method.
- [15] arXiv:2508.03040 [中文pdf, pdf, 其他]
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标题： 随机延迟微分方程的非线性动力学稀疏识别标题： Sparse Identification of Nonlinear Dynamics for Stochastic Delay Differential Equations主题： 数值分析 (math.NA)
首次为随机延迟微分方程提出了从采样轨迹中恢复漂移和扩散动力学的通用框架。 核心依赖于用于非线性动力学稀疏识别的已建立的SINDy算法。 所提出的方法结合了最近提出的高阶漂移和协方差估计，以处理随机问题，并利用扩展库来处理延迟项。 考虑到仅可获得现实数据的情况下，讨论了三种不同的策略。 对不同模型进行了详尽的比较数值研究，这有助于指导选择有效且可能表现更优的方案。
A general framework for recovering drift and diffusion dynamics from sampled trajectories is presented for the first time for stochastic delay differential equations. The core relies on the well-established SINDy algorithm for the sparse identification of nonlinear dynamics. The proposed methodology combines recently proposed high-order estimates of drift and covariance for dealing with stochastic problems with augmented libraries to handle delayed arguments. Three different strategies are discussed in view of exploiting only realistically available data. A thorough comparative numerical investigation is performed on different models, which helps guiding the choice of effective and possibly outperforming schemes.
- [16] arXiv:2508.03049 [中文pdf, pdf, html, 其他]
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标题： 低秩性与平滑性结合子空间：高光谱图像超分辨率的统一张量正则化标题： Low-rankness and Smoothness Meet Subspace: A Unified Tensor Regularization for Hyperspectral Image Super-resolution评论： 13页，71图主题： 数值分析 (math.NA)
高光谱图像超分辨率（HSI-SR）已成为遥感中一个具有挑战性但至关重要的问题。 现有方法主要集中在利用低秩性和局部平滑先验的正则化技术上。 最近，相关总变差已被引入用于张量恢复，将这些先验整合到一个统一的正则化框架中。 然而，直接应用于HSI-SR受到高光谱数据高光谱维度的阻碍。 在本文中，我们提出了一种统一的张量正则化器，称为JLRST，在子空间框架下联合编码低秩性和局部平滑先验。 具体来说，我们在所有三个张量模式上计算聚类系数张量的梯度，以充分利用HSI中的光谱相关性和非局部相似性。 通过在子空间系数上施加先验而不是整个HR-HSI数据，所提出的方法实现了更高的计算效率和准确性。 此外，为了减轻张量核范数（TNN）引入的偏差，我们引入了模式-3对数TNN来处理梯度张量。 开发了一种具有已证明收敛性的交替方向乘子法来求解所提出的模型。 实验结果表明，我们的方法在HSI-SR中显著优于最先进的方法。
Hyperspectral image super-resolution (HSI-SR) has emerged as a challenging yet critical problem in remote sensing. Existing approaches primarily focus on regularization techniques that leverage low-rankness and local smoothness priors. Recently, correlated total variation has been introduced for tensor recovery, integrating these priors into a single regularization framework. Direct application to HSI-SR, however, is hindered by the high spectral dimensionality of hyperspectral data. In this paper, we propose a unified tensor regularizer, called JLRST, which jointly encodes low-rankness and local smoothness priors under a subspace framework. Specifically, we compute the gradients of the clustered coefficient tensors along all three tensor modes to fully exploit spectral correlations and nonlocal similarities in HSI. By enforcing priors on subspace coefficients rather than the entire HR-HSI data, the proposed method achieves improved computational efficiency and accuracy. Furthermore, to mitigate the bias introduced by the tensor nuclear norm (TNN), we introduce the mode-3 logarithmic TNN to process gradient tensors. An alternating direction method of multipliers with proven convergence is developed to solve the proposed model. Experimental results demonstrate that our approach significantly outperforms state-of-the-art methods in HSI-SR.
- [17] arXiv:2508.03325 [中文pdf, pdf, html, 其他]
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标题： 基于随机Koopman正交分解和可解释深度学习的非线性PDE降阶数据驱动双模型标题： Reduced Order Data-driven Twin Models for Nonlinear PDEs by Randomized Koopman Orthogonal Decomposition and Explainable Deep Learning评论： 34页，9图主题： 数值分析 (math.NA)
本研究引入了一种基于现代Koopman算子理论的数据驱动孪生建模框架，通过以较低的复杂度准确捕捉非线性动力学而无需手动参数调整，相较于经典模态分解有了显著进步。 该方法结合了一种新算法与Pareto前沿分析，构建了一个紧凑且高保真的降阶模型，平衡了准确性与效率。 一个可解释的NLARX深度学习框架实现了实时、自适应的校准与预测，而一个关键创新——通过随机正交投影计算正交Koopman模式——确保了最优的数据表示。 这种数据驱动的孪生建模方法是完全自洽的，避免了启发式选择，并通过集成可解释的学习技术增强了可解释性。 所提出的方法在冲击波现象中进行了验证，包括三个复杂度逐渐增加的实验，并对生成的数据驱动孪生模型进行了定性分析。
This study introduces a data-driven twin modeling framework based on modern Koopman operator theory, offering a significant advancement over classical modal decomposition by accurately capturing nonlinear dynamics with reduced complexity and no manual parameter adjustment. The method integrates a novel algorithm with Pareto front analysis to construct a compact, high-fidelity reduced-order model that balances accuracy and efficiency. An explainable NLARX deep learning framework enables real-time, adaptive calibration and prediction, while a key innovation-computing orthogonal Koopman modes via randomized orthogonal projections-ensures optimal data representation. This approach for data-driven twin modeling is fully self-consistent, avoiding heuristic choices and enhancing interpretability through integrated explainable learning techniques. The proposed method is demonstrated on shock wave phenomena using three experiments of increasing complexity, accompanied by a qualitative analysis of the resulting data-driven twin models.
- [18] arXiv:2508.03326 [中文pdf, pdf, html, 其他]
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标题： 通过包含红细胞压积依赖流变学的物理信息神经网络估算血流动力学参数标题： Estimation of Hemodynamic Parameters via Physics Informed Neural Networks including Hematocrit Dependent Rheology主题： 数值分析 (math.NA)
物理信息神经网络（PINNs）在解决逆问题方面显示出显著的潜力，特别是在观测数据有限和稀疏的情况下，前提是已知相关的物理方程。 我们使用PINNs从合成的4D流动磁共振成像（MRI）数据中估计平滑的速度和压力场。 我们在一个现实的主动脉模型中分析了五个非牛顿动态3D血液流动案例，涵盖了从贫血到红细胞增多症条件下的不同血细胞比容值。 为了提高状态估计结果，我们考虑了PINNs的各种设计和训练技术，包括自适应损失平衡、课程训练和一个现实的测量算子。 关于血液流变学，PINN方法在峰值收缩期条件下能够准确地全局和局部估计粘度。 它还为舒张阶段提供了清晰的模式识别。 关于质量守恒，PINN估计有效地再现了通过主动脉不同分支的流动分叉，展示了壁面处无滑移条件的出色表示，并在整个压力场中以低于5%的相对误差准确估计压力下降。 我们将压力下降估计与最先进的虚拟功能相对压力（vWERP）估计器进行对比，观察到我们的结果在准确性和时间分辨率方面都优于vWERP。 此外，我们发现通过PINN计算速度场，然后将其集成到vWERP框架中，可以得到时间超采样和高阶近似，具有临床可接受的准确性。
Physics-Informed Neural Networks (PINNs) show significant potential for solving inverse problems, especially when observations are limited and sparse, provided that the relevant physical equations are known. We use PINNs to estimate smooth velocity and pressure fields from synthetic 4D flow Magnetic Resonance Imaging (MRI) data. We analyze five non-Newtonian dynamic 3D blood flow cases within a realistic aortic model, covering a range of hematocrit values from anemic to polycythemic conditions. To enhance state estimation results, we consider various design and training techniques for PINNs, including adaptive loss balancing, curriculum training, and a realistic measurement operator. Regarding blood rheology, the PINN approach accurately estimates viscosity globally and locally under peak systolic conditions. It also provides a clear pattern recognition for diastolic stages. Regarding mass conservation, PINN estimations effectively reproduce the bifurcation of flow through the different branches of the aorta, demonstrate an excellent representation of the non-slip conditions at the walls, and accurately estimate pressure drops with relative errors below the 5% in the whole pressure field. We test our pressure drop estimations against the state of the art Virtual Work Energy Relative Pressure (vWERP) estimator, and we observe how our results outperform vWERP in terms of both accuracy and time resolution. Additionally, we find that the best results are achieved by computing the velocity field using the PINN, which is then integrated into the vWERP framework, leading to time super-sampled and high-order approximations, with a clinically admissible accuracy.
- [19] arXiv:2508.03390 [中文pdf, pdf, html, 其他]
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标题： 三维乘性噪声随机麦克斯韦方程的两种算子分裂方法标题： Two operator splitting methods for three-dimensional stochastic Maxwell equations with multiplicative noise主题： 数值分析 (math.NA)
在本文中，我们开发了两种能量保持的分裂方法，用于求解由乘性噪声驱动的三维随机麦克斯韦方程。我们使用算子分裂方法将随机麦克斯韦方程分解为简单的二维子系统，并通过空间紧致差分方法和时间离散化中的中点法则结合，构建了两种随机分裂方法，即分裂方法I和分裂方法II，以及随机部分的精确酉解析解。理论证明表明，这两种方法严格保持离散能量守恒定律。最后，数值实验充分验证了方法的能量守恒性，并表明两种分裂方法的时间收敛阶为一阶。
In this paper, we develop two energy-preserving splitting methods for solving three-dimensional stochastic Maxwell equations driven by multiplicative noise. We use operator splitting methods to decouple stochastic Maxwell equations into simple one-dimensional subsystems and construct two stochastic splitting methods, Splitting Method I and Splitting Method II, through a combination of spatial compact difference methods and the midpoint rule in time discretization for the deterministic parts, and exact unitary analytical solutions for the stochastic parts. Theoretical proofs show that both methods strictly preserve the discrete energy conservation law. Finally, numerical experiments fully verify the energy conservation of the methods and demonstrate that the temporal convergence order of the two splitting methods is first-order.
- [20] arXiv:2508.03421 [中文pdf, pdf, 其他]
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标题： 基于伴随方法的物理信息神经网络矩阵预处理框架标题： A matrix preconditioning framework for physics-informed neural networks based on adjoint method评论： 16页，8图主题： 数值分析 (math.NA) ; 流体动力学 (physics.flu-dyn)
物理信息神经网络（PINNs）最近成为解决涉及偏微分方程（PDEs）的正问题和反问题的一种流行方法。 与全连接神经网络相比，基于卷积神经网络的PINNs在硬性执行边界条件和减少偏导数的计算成本方面具有优势。 然而，后者在某些情况下仍存在收敛缓慢甚至失败的问题。 在本研究中，我们提出了一种矩阵预处理方法来改善后者的收敛性。 具体而言，我们将自动微分与矩阵着色相结合，以计算PDE系统的雅可比矩阵，该矩阵通过不完全LU分解用于构建预处理器。 随后，我们使用预处理器对损失函数中的PDE残差进行缩放，以降低雅可比矩阵的条件数，这是提高PINNs收敛性的关键。 为了克服预处理过程中自动微分与三角求解之间的不兼容性，我们还设计了一个基于伴随方法的框架，以计算损失函数相对于网络参数的梯度。 通过数值实验，我们验证了所提出的方法成功且高效地解决了多尺度问题和高雷诺数问题，在这两种问题中，PINNs无法获得满意的结果。
Physics-informed neural networks (PINNs) have recently emerged as a popular approach for solving forward and inverse problems involving partial differential equations (PDEs). Compared to fully connected neural networks, PINNs based on convolutional neural networks offer advantages in the hard enforcement of boundary conditions and in reducing the computational cost of partial derivatives. However, the latter still struggles with slow convergence and even failure in some scenarios. In this study, we propose a matrix preconditioning method to improve the convergence of the latter. Specifically, we combine automatic differentiation with matrix coloring to compute the Jacobian matrix of the PDE system, which is used to construct the preconditioner via incomplete LU factorization. We subsequently use the preconditioner to scale the PDE residual in the loss function in order to reduce the condition number of the Jacobian matrix, which is key to improving the convergence of PINNs. To overcome the incompatibility between automatic differentiation and triangular solves in the preconditioning, we also design a framework based on the adjoint method to compute the gradients of the loss function with respect to the network parameters. By numerical experiments, we validate that the proposed method successfully and efficiently solves the multi-scale problem and the high Reynolds number problem, in both of which PINNs fail to obtain satisfactory results.
- [21] arXiv:2508.03439 [中文pdf, pdf, html, 其他]
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标题： 多维无压欧拉型模型的数值研究，包含非局部相互作用和趋化性用于集体细胞迁移标题： Numerical study on a multi-dimensional pressureless Euler-type model with non-local interactions and chemotaxis for collective cell migration主题： 数值分析 (math.NA)
在本文中，我们提出对集体细胞迁移的宏观模型进行数值研究，重点研究一种多维无压欧拉型模型，该模型包含非局部相互作用并与趋化性耦合，严格从微观动力学推导而来。不同的机械相互作用被研究，包括吸引-排斥效应。此外，该模型被扩展到不同相互作用细胞群体的情况。最后，通过特定设置中的参数估计分析评估了此类宏观模型的有效性及其与微观动力学的一致性。
In this paper we propose a numerical study of macroscopic models for collective cell migration, focusing on a multi-dimensional pressureless Euler-type model with non-local interactions coupled with chemotaxis, rigorously derived from microscopic dynamics. Different mechanical interactions are investigated, including attraction-repulsion effects. Moreover, the model is extended to the case of different populations of interacting cells. The validity of such macroscopic model and its agreement with the microscopic dynamics is finally assessed through a parameter estimation analysis in a specific setting.
- [22] arXiv:2508.03455 [中文pdf, pdf, html, 其他]
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标题： 半拉格朗日格式在扩散守恒定律中的误差估计标题： Error Estimates of Semi-Lagrangian Schemes for Diffusive Conservation Laws评论： 25页，3图主题： 数值分析 (math.NA)
我们给出了使用高阶插值算子的完全半拉格朗日格式在求解一维非线性扩散守恒律的初值问题中的误差估计，包括伯格斯方程。 我们对插值算子施加了一些假设，这些假设被样条插值和埃尔米特插值所满足。 我们建立了空间网格尺寸$ h $和时间步长$ \Delta t $的$ O(\Delta t + h^{2 s} / \Delta t) $在$ L^2 $范数下和$ O(\Delta t + h^{s} / (\Delta t)^{1/2} + h^{2s} / \Delta t) $在$ H^s $范数下的收敛率，其中使用了次数为$ (2s - 1) $的样条或埃尔米特插值算子。 数值结果与理论分析一致。
We present error estimates of the fully semi-Lagrangian scheme with high-order interpolation operators, solving the initial value problems for the one-dimensional nonlinear diffusive conservation laws, including the Burgers equations. We impose certain assumptions on the interpolation operator, which are satisfied by both spline and Hermite interpolations. We establish the convergence rates of $ O(\Delta t + h^{2 s} / \Delta t) $ in the $ L^2 $-norm and $ O(\Delta t + h^{s} / (\Delta t)^{1/2} + h^{2s} / \Delta t) $ in the $ H^s $-norm for the spatial mesh size $ h $ and the temporal step size $ \Delta t $, where the spline or Hermite interpolation operator of degree $ (2s - 1) $ is employed. The numerical results are in agreement with the theoretical analysis.
- [23] arXiv:2508.03512 [中文pdf, pdf, html, 其他]
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标题： 梁格栅到微极连续体的均质化率标题： Homogenization rates of beam lattices to micropolar continua主题： 数值分析 (math.NA)
当具有梁模化边界的机械晶格的尺寸趋近于零时，它会经历均质化过程，转化为连续体模型，该模型表现出偏离经典柯西弹性理论的异常力学特性，称为微极弹性。 通常，在工程界中，均质化过程是定性的，缺乏定量的均质化误差估计。 在本文中，我们严格分析了从梁晶格到连续体的均质化过程。 我们的方法始于一个定义在具有周期性边界条件的三角晶格上的工程力学问题。 通过应用傅里叶变换，我们将问题简化为频率域中的一系列方程。 当晶格尺寸趋近于零时，这将产生一个具有周期性边界条件的偏微分方程形式的均质化模型。 如果频率域中的外部条件仅在低频模式下非零，则此过程可以很容易地得到验证。 然而，通过数值实验，我们发现除了低频范围之外，由于梁中的额外旋转自由度，梁晶格的均质化与经典周期性均质化理论不同。 我们分析中的关键技巧是通过一种称为舒尔补的线性代数运算来解耦位移场和旋转场。 通过专门的分析，我们在晶格和连续体模型中建立了舒尔补的强制性，这使我们能够推导出均质化误差的收敛速率估计。 数值实验验证了均质化速率估计的最优性。
As the size of a mechanical lattice with beam-modeled edges approaches zero, it undergoes homogenization into a continuum model, which exhibits unusual mechanical properties that deviate from classical Cauchy elasticity, named micropolar elasticity. Typically, the homogenization process is qualitative in the engineering community, lacking quantitative homogenization error estimates. In this paper, we rigorously analyze the homogenization process of a beam lattice to a continuum. Our approach is initiated from an engineered mechanical problem defined on a triangular lattice with periodic boundary conditions. By applying Fourier transformations, we reduce the problem to a series of equations in the frequency domain. As the lattice size approaches zero, this yields a homogenized model in the form of a partial differential equation with periodic boundary conditions. This process can be easily justified if the external conditions in the frequency domain are nonzero only at low-frequency modes. However, through numerical experiments, we discover that beyond the low-frequency regime, the homogenization of the beam lattice differs from classical periodic homogenization theory due to the additional rotational degrees of freedom in the beams. A crucial technique in our analysis is the decoupling of displacement and rotation fields, achieved through a linear algebraic manipulation known as the Schur complement. Through dedicated analysis, we establish the coercivity of the Schur complements in both lattice and continuum models, which enables us to derive convergence rate estimates for homogenization errors. Numerical experiments validate the optimality of the homogenization rate estimates.
新提交 (展示 23 之 23 条目 )
- [24] arXiv:2508.02673 (交叉列表自 cs.CE) [中文pdf, pdf, html, 其他]
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标题： 定量系统分析中决策图的数值误差标题： Numerical Errors in Quantitative System Analysis With Decision Diagrams主题： 计算工程、金融与科学 (cs.CE) ; 数值分析 (math.NA) ; 量子物理 (quant-ph)
决策图（DDs）是一种强大的数据结构，用于解决状态空间爆炸问题，不仅适用于离散系统，也适用于概率和量子系统。 虽然在概率和量子领域中使用的许多DDs使用浮点数，但这并非没有挑战。 浮点计算容易产生微小的舍入误差，这可能会影响结果的正确性和DD压缩的有效性。 在本文中，我们研究了与多终端二进制决策图（MTBDDs）进行矩阵-向量乘法的数值稳定性，即算法对小数值误差的鲁棒性。 矩阵-向量乘法特别重要，因为它用于计算概率和量子系统的后继状态。 我们证明，在某些条件下，MTBDD矩阵-向量乘法算法可以实现数值稳定，尽管在许多MTBDD的实际实现中这些条件并未满足。 此外，我们提供了一个关于量子电路模拟中数值误差的案例研究，该研究表明实际中的数值误差程度在不同实例之间差异很大。
Decision diagrams (DDs) are a powerful data structure that is used to tackle the state-space explosion problem, not only for discrete systems, but for probabilistic and quantum systems as well. While many of the DDs used in the probabilistic and quantum domains make use of floating-point numbers, this is not without challenges. Floating-point computations are subject to small rounding errors, which can affect both the correctness of the result and the effectiveness of the DD's compression. In this paper, we investigate the numerical stability, i.e. the robustness of an algorithm to small numerical errors, of matrix-vector multiplication with multi-terminal binary decision diagrams (MTBDDs). Matrix-vector multiplication is of particular interest because it is the function that computes successor states for both probabilistic and quantum systems. We prove that the MTBDD matrix-vector multiplication algorithm can be made numerically stable under certain conditions, although in many practical implementations of MTBDDs these conditions are not met. Additionally, we provide a case study of the numerical errors in the simulation of quantum circuits, which shows that the extent of numerical errors in practice varies greatly between instances.
- [25] arXiv:2508.02763 (交叉列表自 math.ST) [中文pdf, pdf, html, 其他]
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标题： 基于顺序蒙特卡罗的多模态分布多项式复杂度采样标题： Polynomial complexity sampling from multimodal distributions using Sequential Monte Carlo评论： 58页，5图主题： 统计理论 (math.ST) ; 数值分析 (math.NA) ; 概率 (math.PR) ; 计算 (stat.CO)
我们研究一种顺序蒙特卡罗算法，用于在低温下从具有非凸能量函数的吉布斯测度中采样。 我们使用实际且流行的几何退火调度，并在每个温度级别使用朗之万扩散。 朗之万扩散只需要运行足够长的时间以确保能量谷内的局部混合，这比全局混合所需的時間要短得多。 我们的主要结果表明，蒙特卡罗估计量的收敛性具有时间复杂度，大约与逆温度的四次方成比例，以及允许误差的平方。 我们还在一个说明性的模型场景中研究了该算法，其中可以给出更明确的估计。
We study a sequential Monte Carlo algorithm to sample from the Gibbs measure with a non-convex energy function at a low temperature. We use the practical and popular geometric annealing schedule, and use a Langevin diffusion at each temperature level. The Langevin diffusion only needs to run for a time that is long enough to ensure local mixing within energy valleys, which is much shorter than the time required for global mixing. Our main result shows convergence of Monte Carlo estimators with time complexity that, approximately, scales like the forth power of the inverse temperature, and the square of the inverse allowed error. We also study this algorithm in an illustrative model scenario where more explicit estimates can be given.
- [26] arXiv:2508.02882 (交叉列表自 cs.LG) [中文pdf, pdf, html, 其他]
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标题： 具有正交雅可比矩阵的神经网络标题： Neural Networks with Orthogonal Jacobian主题： 机器学习 (cs.LG) ; 数值分析 (math.NA)
非常深的神经网络通过提取丰富的分层特征实现了最先进的性能。 然而，通过反向传播训练它们通常受到梯度消失或爆炸的阻碍。 现有的解决方法，如正交或方差保持初始化和残差结构，允许更稳定的梯度传播并训练更深的模型。 在本工作中，我们引入了一个统一的数学框架，描述了一类广泛的非线性前馈和残差网络，其输入到输出的雅可比矩阵几乎处处正好正交。 这种约束使得这些网络能够实现完美的动态等距性，并且即使非常深也能高效训练。 我们的公式不仅恢复了标准架构作为特例，还产生了新的设计，这些设计在不依赖传统跳跃连接的情况下匹配残差网络的可训练性。 我们提供了实验证据，证明初始时刻的完美雅可比正交性足以稳定训练并实现具有竞争力的性能。 我们将这种策略与那些被正则化以保持雅可比正交性的网络进行了比较，并获得了相当的结果。 我们进一步将分析扩展到一类可以由具有正交雅可比的网络很好近似的网络，并介绍了雅可比代表部分等距的网络。 这些广义模型随后被证明保持了有利的可训练性特性。
Very deep neural networks achieve state-of-the-art performance by extracting rich, hierarchical features. Yet, training them via backpropagation is often hindered by vanishing or exploding gradients. Existing remedies, such as orthogonal or variance-preserving initialisation and residual architectures, allow for a more stable gradient propagation and the training of deeper models. In this work, we introduce a unified mathematical framework that describes a broad class of nonlinear feedforward and residual networks, whose input-to-output Jacobian matrices are exactly orthogonal almost everywhere. Such a constraint forces the resulting networks to achieve perfect dynamical isometry and train efficiently despite being very deep. Our formulation not only recovers standard architectures as particular cases but also yields new designs that match the trainability of residual networks without relying on conventional skip connections. We provide experimental evidence that perfect Jacobian orthogonality at initialisation is sufficient to stabilise training and achieve competitive performance. We compare this strategy to networks regularised to maintain the Jacobian orthogonality and obtain comparable results. We further extend our analysis to a class of networks well-approximated by those with orthogonal Jacobians and introduce networks with Jacobians representing partial isometries. These generalized models are then showed to maintain the favourable trainability properties.
- [27] arXiv:2508.02928 (交叉列表自 q-bio.QM) [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 mode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，该模型捕捉传染病传播的时空动态。 作为半线性抛物型偏微分方程（PDEs）系统表述，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离散化设计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，这些特征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.
交叉提交 (展示 4 之 4 条目 )
- [28] arXiv:2404.16978 (替换) [中文pdf, pdf, html, 其他]
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标题： 一种三场多尺度方法标题： A three-field Multiscale Method主题： 数值分析 (math.NA)
\emph{一种三场区域分解方法}是 F. Brezzi 和 L. D. Marini 的一篇开创性论文的标题，该论文介绍了用于椭圆偏微分方程的三场公式。 基于此，我们提出了多尺度混合混合方法（MH$^2$M）用于达西模型，这是一种多尺度有限元方法，经过一系列形式操作后，得到一个仅依赖于解的迹的对称正定公式。 我们展示了有限元空间族的稳定性与收敛性结果，并建立了与其他多尺度有限元方法之间的关系。
\emph{A Three-Field Domain Decomposition Method} is the title of a seminal paper by F. Brezzi and L. D. Marini which introduces a three-field formulation for elliptic partial differential equations. Based on that, we propose the Multiscale-Hybrid-Hybrid Method (MH$^2$M) for the Darcy model, a multiscale finite element method that yields, after a series of formal manipulations, a symmetric positive definite formulation that depends only on the trace of the solution. We show stability and convergence results for a family of finite element spaces and establish relationships with other multiscale finite element methods.
- [29] arXiv:2405.16400 (替换) [中文pdf, pdf, html, 其他]
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标题： 函数混合光滑性的加权采样恢复标题： Weighted sampling recovery of functions with mixed smoothness评论： arXiv管理员备注：与arXiv:2309.04994文本重叠主题： 数值分析 (math.NA)
我们研究稀疏网格线性采样算法及其在从$\mathbb{R}^d$的一组$n$的采样值中近似恢复具有混合光滑性的函数时的最优性，在两种不同情况下：(i) 要恢复的函数属于混合光滑性的加权 Sobolev 空间$W^r_{p,w}(\mathbb{R}^d)$，近似误差由加权 Lebesgue 空间$L_{q,w}(\mathbb{R}^d)$的范数来衡量，以及 (ii) 要恢复的函数属于具有测度$W^r_p(\mathbb{R}^d; \mu_w)$的混合光滑性的 Sobolev 空间，近似误差由具有测度$L_q(\mathbb{R}^d; \mu_w)$的 Lebesgue 空间范数来衡量。 这里，函数$w$是张量积形式的弗洛伊德型权函数，是在情形 (i) 中的权函数，而$\mu_w$是情形 (ii) 中测度的密度函数。在线性采样算法的最优性方面，根据相关的采样$n$-宽度进行了研究。我们构造了稀疏网格线性采样算法，这些算法在情形 (i) 和 (ii) 中完全不同，并给出了相应采样$n$-宽度的上界。我们证明了在一维情况下，这些算法实现了采样宽度的正确收敛速度。在高维情形 (ii)（$d\ge 2$）中，我们也通过一种非构造性方法实现了$n$-宽度的正确收敛速度，对于$1\le q \le 2 \le p \le \infty$。
We study sparse-grid linear sampling algorithms and their optimality for approximate recovery of functions with mixed smoothness on $\mathbb{R}^d$ from a set of $n$ their sampled values in two different settings: (i) functions to be recovered are in weighted Sobolev spaces $W^r_{p,w}(\mathbb{R}^d)$ of mixed smoothness and the approximation error is measured by the norm of the weighted Lebesgue space $L_{q,w}(\mathbb{R}^d)$, and (ii) functions to be recovered are in Sobolev spaces with measure $W^r_p(\mathbb{R}^d; \mu_w)$ of mixed smoothness and the approximation error is measured by the norm of the Lebesgue space with measure $L_q(\mathbb{R}^d; \mu_w)$. Here, the function $w$, a tensor-product Freud-type weight is the weight in the setting (i), and the density function of the measure $\mu_w$ in the setting (ii). The optimality of linear sampling algorithms is investigated in terms of the relevant sampling $n$-widths. We construct sparse-grid linear sampling algorithms which are completely different for the settings (i) and (ii) and which give upper bounds of the corresponding sampling $n$-widths. We prove that in the one-dimensional case, these algorithms realize the right convergence rate of the sampling widths. In the setting (ii) for the high dimensional case ($d\ge 2$), we also achieve the right convergence rate of the sampling $n$-widths for $1\le q \le 2 \le p \le \infty$ through a non-constructive method.
- [30] arXiv:2410.04915 (替换) [中文pdf, pdf, html, 其他]
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标题： 基于有限差分的剪切柔性几何精确梁单元标题： Shear-flexible geometrically exact beam element based on finite differences评论： 42页，18图期刊参考： 计算机方法在应用力学中的工程，436，117671 (2025)主题： 数值分析 (math.NA)
所提出的二维几何精确梁单元通过包括剪切变形的影响，以及沿梁作用的分布力和力矩的影响，扩展了我们之前的工作。 一般的基于柔度的公式利用运动学方程结合倒置截面方程和平衡方程的积分形式。 得到的一组三个一阶微分方程通过有限差分进行离散化，并使用射击法将边界值问题转换为初始值问题。 由于控制方程的特殊结构，该方案即使使用中心差分近似一阶导数，仍保持显式，从而实现高精度。 采用的方法的主要优点是，可以通过在元素级别上细化有限差分所使用的计算网格来高效地减少误差，同时保持全局自由度的数量较低。 通过直接处理元素级别的全局中线坐标和截面相对于全局轴的倾斜角度作为主要未知量，也提高了效率，从而避免了局部坐标和全局坐标之间的转换。 介绍了两种截面方程的公式，分别称为Reissner模型和Ziegler模型，并进行了比较。 特别是研究了轴向载荷梁/柱的稳定性，并讨论了与Haringx和Engesser稳定理论的联系。 两种方法都在一系列数值示例中进行了测试，这些示例说明了（i）当空间离散化细化时具有二次收敛的高精度，（ii）容易对元素上的可变刚度进行建模（如刚性节点偏移），（iii）对屈曲和后屈曲行为进行高效准确的表征。
The proposed two-dimensional geometrically exact beam element extends our previous work by including the effects of shear distortion, and also of distributed forces and moments acting along the beam. The general flexibility-based formulation exploits the kinematic equations combined with the inverted sectional equations and the integrated form of equilibrium equations. The resulting set of three first-order differential equations is discretized by finite differences and the boundary value problem is converted into an initial value problem using the shooting method. Due to the special structure of the governing equations, the scheme remains explicit even though the first derivatives are approximated by central differences, leading to high accuracy. The main advantage of the adopted approach is that the error can be efficiently reduced by refining the computational grid used for finite differences at the element level while keeping the number of global degrees of freedom low. The efficiency is also increased by dealing directly with the global centerline coordinates and sectional inclination with respect to global axes as the primary unknowns at the element level, thereby avoiding transformations between local and global coordinates. Two formulations of the sectional equations, referred to as the Reissner and Ziegler models, are presented and compared. In particular, stability of an axially loaded beam/column is investigated and the connections to the Haringx and Engesser stability theories are discussed. Both approaches are tested in a series of numerical examples, which illustrate (i) high accuracy with quadratic convergence when the spatial discretization is refined, (ii) easy modeling of variable stiffness along the element (such as rigid joint offsets), (iii) efficient and accurate characterization of the buckling and post-buckling behavior.
- [31] arXiv:2412.05598 (替换) [中文pdf, pdf, html, 其他]
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标题： 通过等分布进行变量步长有限差分的实用方法标题： A practical recipe for variable-step finite differences via equidistribution评论： 6页，2图主题： 数值分析 (math.NA) ; 计算物理 (physics.comp-ph)
我们描述了一个简短且可重复的工作流程，用于在由正权函数g确定的非均匀网格上应用有限差分。该网格通过等分布获得，通过累积积分$S(x)=\int_a^x\!1/g(s)\,ds$及其逆变换将统一的计算坐标$\xi\in[0,1]$映射到物理空间，在多维情况下则通过相应的变扩散（调和）映射及张量$P=(1/g)I$进行映射。然后，我们在不均匀间距上使用标准的三点中心模板计算一阶和二阶导数。我们收集了公式，说明了对g的温和约束（正性、有界性、可积性），并提供了一个小型参考实现。最后，我们在均匀网格和变网格上求解二维定态薛定谔方程，展示了解决局部本征函数的预期改进，而无需增加矩阵大小。我们希望这篇笔记作为一种操作指南，而不是一种新方法，将广泛使用的想法整合成一个单一的、即用型配方，不声称具有新颖性。薛定谔方程，实现了在不增加计算成本的情况下改进本征函数分辨率。
We describe a short, reproducible workflow for applying finite differences on nonuniform grids determined by a positive weight function g. The grid is obtained by equidistribution, mapping uniform computational coordinates $\xi\in[0,1]$ to physical space by the cumulative integral $S(x)=\int_a^x\!1/g(s)\,ds$ and its inverse, and in multiple dimensions by the corresponding variable-diffusion (harmonic) mapping with tensor $P=(1/g)I$. We then use the standard three-point central stencils on uneven spacing for first and second derivatives. We collect the formulas, state the mild constraints on g (positivity, boundedness, integrability), and provide a small reference implementation. Finally, we solve the 2D time-independent Schr\"odinger equation for a harmonic oscillator on uniform vs. variable meshes, showing the expected improvement in resolving localized eigenfunctions without increasing matrix size. We intend this note as a how-to reference rather than a new method, consolidating widely used ideas into a single, ready-to-use recipe, claiming no novelty.ent Schr\"odinger equation for a harmonic oscillator, achieving improved eigenfunction resolution without increased computational cost.
- [32] arXiv:2503.10194 (替换) [中文pdf, pdf, html, 其他]
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标题： 通过自适应有理逼近和草图技术对散射问题中的共振行为进行代理建模标题： Surrogate modeling of resonant behavior in scattering problems through adaptive rational approximation and sketching主题： 数值分析 (math.NA) ; 数学物理 (math-ph)
本文描述了用于识别散射问题中（几乎）共振行为的新算法。 我们的方法依赖于有理逼近，旨在构建我们所谓的“场增强”的代理模型，该模型定义为散射问题解算子的范数，我们通过边界积分方程来表达。 为了给我们的技术提供理论基础，我们首先推导出将场增强与定义散射问题的算子的谱性质联系起来的结果。 然后利用这些结果来证明在代理建模任务中有理逼近的使用是合理的。 我们提出的一些方法以“标准”方式应用有理逼近，直接为解算子构建有理逼近，或者为了计算效率，为它的随机“压缩”版本构建有理逼近。 我们的其他“混合”方法更具创新性，结合了有理逼近辅助的根查找与径向基函数的逼近。 我们方法的三个关键特征是：(i) 它们对用于离散散射问题的策略是无偏的，(ii) 它们不需要涉及非实波数的任何计算，(iii) 它们可以通过自适应采样策略调整到不同的设置中。 我们进行了一些涉及二维散射体的数值实验，以比较我们的方法。 在我们的测试中，我们的两种方法（一种标准，一种混合）表现最佳，根据是否强调准确性或效率，其中一种或另一种更优。
This paper describes novel algorithms for the identification of (almost-)resonant behavior in scattering problems. Our methods, relying on rational approximation, aim at building surrogate models of what we call "field amplification", defined as the norm of the solution operator of the scattering problem, which we express through boundary-integral equations. To provide our techniques with theoretical foundations, we first derive results linking the field amplification to the spectral properties of the operator that defines the scattering problem. Such results are then used to justify the use of rational approximation in the surrogate-modeling task. Some of our proposed methods apply rational approximation in a "standard" way, building a rational approximant for either the solution operator directly or, in the interest of computational efficiency, for a randomly "sketched" version of it. Our other "hybrid" approaches are more innovative, combining rational-approximation-assisted root-finding with approximation using radial basis functions. Three key features of our methods are that (i) they are agnostic of the strategy used to discretize the scattering problem, (ii) they do not require any computations involving non-real wavenumbers, and (iii) they can adjust to different settings through the use of adaptive sampling strategies. We carry out some numerical experiments involving 2D scatterers to compare our approaches. In our tests, two of our approaches (one standard, one hybrid) emerge as the best performers, with one or the other being preferable, depending on whether emphasis is placed on accuracy or efficiency.
- [33] arXiv:2505.13929 (替换) [中文pdf, pdf, html, 其他]
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标题： 非线性梯度流模型数值近似的方法误差估计标题： Error estimates for numerical approximations of a nonlinear gradient flow model主题： 数值分析 (math.NA)
我们对非线性梯度流进行数值分析，该问题可以看作是抛物型极小曲面问题或正则化总变分流，使用梯度离散化方法（GDM）。 GDM 是一个统一的收敛分析框架，涵盖了符合和不符合的数值方法，例如符合和不符合的有限元、两点通量近似等。 在本文中，提出了该模型的一个完全离散的隐式格式，证明了该格式解的存在性和唯一性，分析了该格式的稳定性和一致性，并建立了误差估计。 还提供了基于符合和不符合 P1 有限元的数值结果。
We perform numerical analysis of a nonlinear gradient flow, which can be regarded as a parabolic minimal surface problem or a regularised total variation flow, using the gradient discretisation method (GDM). GDM is a unified convergence analysis framework that covers conforming and nonconforming numerical methods, for instance, conforming and nonconforming finite element, two-point flux approximation, etc.. In this paper, a fully discretised implicit scheme of the model is proposed, the existence and uniqueness of the solution to the scheme is proved, the stability and consistency of the scheme are analysed, and error estimates are established. Numerical results based on the conforming and nonconforming P1 finite elements are also provided.
- [34] arXiv:2507.13640 (替换) [中文pdf, pdf, 其他]
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标题： 多项式空间中的插值标题： Interpolation in Polynomial Spaces of p-Degree主题： 数值分析 (math.NA)
我们最近引入了快速牛顿变换（FNT），这是一种分层算法，用于在任意向下闭合的多项式空间中进行多变量牛顿插值，该空间的空间维数为$m$。 在此，我们分析了 FNT 在特定的向下闭合集$A_{m,n,p}$的上下文中，该集定义为所有满足$\ell^p$范数小于$n$的多重指标，其中$p \in [0,\infty]$。 FNT 在所诱导的向下闭合多项式空间$\Pi_{m,n,p}$上的时间复杂度为$\mathcal{O}(|A_{m,n,p}|mn)$。 我们证明，与张量积空间 $\Pi_{m,n,\infty}$相比， $\Pi_{m,n,p}$的选择将时间复杂度降低了 $\rho_{m,n,p}$倍，在 $m \lesssim n^p$时随着空间维度呈超指数级衰减。我们展示了 FNT 的效率，通过在敏感性分析中计算活动得分来说明。
We recently introduced the Fast Newton Transform (FNT), an hierarchical algorithm for performing multivariate Newton interpolation in arbitrary downward closed polynomial spaces of spatial dimension $m$. Here, we analyze the FNT in the context of a specific family of downward closed sets $A_{m,n,p}$, defined as all multi-indices with $\ell^p$ norm less than $n$ with $p \in [0,\infty]$. The FNT performs with time complexity $\mathcal{O}(|A_{m,n,p}|mn)$ on the induced downward closed polynomial spaces $\Pi_{m,n,p}$. We show that the $\Pi_{m,n,p}$ choice compared to the tensor product spaces $\Pi_{m,n,\infty}$, reduces time complexity by a factor of $\rho_{m,n,p}$, decaying super exponentially with spatial dimension when $m \lesssim n^p$. We showcase the efficiency of the FNT by computing activity scores in sensitivity analysis.
- [35] arXiv:2412.13527 (替换) [中文pdf, pdf, html, 其他]
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标题： Lyapunov分析用于单调前向后向加速算法标题： Lyapunov Analysis For Monotonically Forward-Backward Accelerated Algorithms评论： 20页，4幅图和1张表主题： 优化与控制 (math.OC) ; 数值分析 (math.NA) ; 机器学习 (stat.ML)
Nesterov加速梯度方法（NAG）在凸优化中比梯度下降具有更快的收敛速度，但在函数值上缺乏单调性。 为了解决这个问题，Beck和Teboulle[2009b]提出了一种单调变体M-NAG，并将其扩展到近端设置中作为M-FISTA，用于复合问题如Lasso。 然而，在强凸性下建立M-NAG和M-FISTA的线性收敛仍然是一个开放问题。 在本文中，我们通过隐式速度相位表示分析M-NAG，并表明为了完全恢复NAG迭代，需要额外的假设，要么是位置更新，要么是相位耦合关系。 M-NAG的本质在于控制一个辅助序列以强制非增加。 我们进一步证明，仅M-NAG更新就足以构建一个保证线性收敛的李雅普诺夫函数，而无需依赖完整的NAG迭代。 通过修改混合序列以包含前向索引的梯度，我们开发了一个新的李雅普诺夫函数，消除了动能项，使得能够直接扩展到M-NAG。 所需的起始索引仅取决于动量参数，而不依赖于问题常数。 最后，利用新开发的近端不等式，我们将结果扩展到M-FISTA，建立了其线性收敛性，并加深了对单调加速方法理论的理解。
Nesterov's accelerated gradient method (NAG) achieves faster convergence than gradient descent for convex optimization but lacks monotonicity in function values. To address this, Beck and Teboulle [2009b] proposed a monotonic variant, M-NAG, and extended it to the proximal setting as M-FISTA for composite problems such as Lasso. However, establishing the linear convergence of M-NAG and M-FISTA under strong convexity remains an open problem. In this paper, we analyze M-NAG via the implicit-velocity phase representation and show that an additional assumption, either the position update or the phase-coupling relation, is necessary to fully recover the NAG iterates. The essence of M-NAG lies in controlling an auxiliary sequence to enforce non-increase. We further demonstrate that the M-NAG update alone is sufficient to construct a Lyapunov function guaranteeing linear convergence, without relying on full NAG iterates. By modifying the mixed sequence to incorporate forward-indexed gradients, we develop a new Lyapunov function that removes the kinetic energy term, enabling a direct extension to M-NAG. The required starting index depends only on the momentum parameter and not on problem constants. Finally, leveraging newly developed proximal inequalities, we extend our results to M-FISTA, establishing its linear convergence and deepening the theoretical understanding of monotonic accelerated methods.