生成时间: 2026年07月15日 09:35:45

重点关注论文数量: 0

了解领域论文数量: 5

不相关论文数量: 27

了解领域论文（摘要翻译）

1. 非自治系统的离散时间广义正则变换

Discrete-time generalized canonical transformations for non-autonomous systems

作者: Leonardo Colombo, David Martin de Diego, Riccardo Muradore, Damiano Rigo, Nicola Sansonetto

类别: math.DS, math.NA

发布日期: 2026-07-14

arXiv ID: 2607.12914v1

链接: http://arxiv.org/abs/2607.12914v1

摘要翻译

中文标题: 非自治系统的离散时间广义正则变换

摘要翻译: 当描述其演化的微分方程显含时间时，动力学系统被称为\emph{non-autonomous}。在研究此类系统的各种几何方法中，余辛形式提供了一个自然的框架，将辛几何推广到含时哈密顿系统。然而，在数值离散下保持相关的几何结构仍然是一个具有挑战性的问题：标准积分器，例如显式欧拉格式，通常无法保持余辛体积或底层的泊松结构。 在本文中，我们提出一种基于\emph{generalized canonical transformations}的非自治哈密顿系统离散化的几何方法。该方法在扩展相空间 $T^*(Q \times \mathbb{R})$ 上构造一个辛同胚，其到 $T^*Q \times \mathbb{R}$ 的投影定义了一个保结构的离散流。我们证明，这一构造可保证关键不变量的保持，包括体积形式、泊松括号以及每个时间纤维上的辛结构。

2. 基于能量的物理信息聚类张拉整体结构找形方法

Energy-Based Physics-Informed Form Finding for Clustered Tensegrity Structures

作者: Jing Qin, Muhao Chen

类别: cs.LG, math.NA

发布日期: 2026-07-14

arXiv ID: 2607.12888v1

链接: http://arxiv.org/abs/2607.12888v1

摘要翻译

中文标题: 基于能量的物理信息聚类张拉整体结构找形方法

摘要翻译: 张拉整体结构的找形与物理性质预测是结构力学中的基本反问题，旨在确定平衡构型与内力分布。由于几何与力之间的耦合所导致的强非线性，需要确保结构稳定性，以及边界条件和对称性等约束的施加，这些问题极具挑战性。此外，传统方法往往对噪声和离群值缺乏鲁棒性。本文提出了一种基于能量的聚类张拉整体结构找形与物理性质预测的学习框架。该方法将总势能极小化和本构关系纳入训练目标，能够同时预测平衡节点构型及相关的物理量，包括杆件内力和力密度。通过将基于能量的物理损失直接融入学习过程，该框架提升了物理一致性、鲁棒性和数据效率。在张拉整体结构（包括棱柱和着陆器系统）上的数值实验展示了所提方法的巨大潜力，并证明了其可扩展找形和准确预测结构性质的能力。

3. 基于深度学习的多尺度问题LOD方法代理建模

Deep Learning-based Surrogate Modelling of the LOD Method for Multiscale Problems

作者: Marc Haltmayer, Jaemin Seo, Yuseung Lee, Sungyeop Lee, Jaehoon Jeong, Jae Yong Lee

类别: math.NA, cs.AI, cs.LG

发布日期: 2026-07-14

arXiv ID: 2607.12570v1

链接: http://arxiv.org/abs/2607.12570v1

摘要翻译

中文标题: 基于深度学习的多尺度问题LOD方法代理建模

摘要翻译: 多尺度问题历来难以用传统数值方法处理，因为准确解析细尺度特征常常需要过于精细的离散化。这一挑战在材料科学、流体动力学、气候系统、化学过程和复杂网络等应用中尤为突出。近年来，神经算子模型提供了一种有前景的数据驱动替代方案，但在面对强非均匀或振荡系数时，往往难以达到足够的精度。本文关注具有粗糙和高对比度输入的椭圆偏微分方程的求解。局部正交分解（LOD）方法是求解此类问题的一种成熟数值方法，但其计算代价却相当高。我们研究了常用神经算子架构在这些具有挑战性的多尺度问题上的性能表现，并识别了其在解析细尺度结构方面的关键局限性。为克服这些挑战，我们提出LOD-MSNO（LOD-多尺度神经算子），这是一种混合方法，利用LOD方法作为强多尺度先验，基于其将解表示为问题适配基函数的线性组合，同时通过数据驱动的算子学习来解决其主要计算瓶颈。我们进一步为所提出的系数学习框架提供了理论误差估计。最后，我们证明所提方法对于具有挑战性的多尺度输入，在精度上有潜力超越当前的神经算子基线模型，同时基本保持了神经算子模型的计算效率。

4. 破碎空间加性Schwarz质量逆近似与(块)预处理

Broken-space Additive Schwarz Mass Inverse Approximations and (Block) Preconditioning

作者: Oliver A. Krzysik, Ben S. Southworth, Golo A. Wimmer

类别: math.NA

发布日期: 2026-07-13

arXiv ID: 2607.12138v1

链接: http://arxiv.org/abs/2607.12138v1

摘要翻译

中文标题: 破碎空间加性Schwarz质量逆近似与(块)预处理

摘要翻译: 有限元质量矩阵求解和近似逆经常出现在显式时间积分以及基于Schur补的块预处理中。质量矩阵的对角近似廉价且广泛使用，但可能是一个较差的近似，特别是对于高阶单元。本文引入了一种破碎空间加性Schwarz（BRAS）质量逆近似，通过在破碎有限元空间上应用精确的单元局部逆质量矩阵并将结果平均回协调空间而构造。该构造使用与标准质量组装相同的单元矩阵和局部-全局映射，具有与协调质量矩阵相同的单元邻接稀疏性图，并且对于任何协调空间、网格几何和多项式基都是对称正定的。我们证明了预处理质量矩阵的谱界，并且在二维和三维单纯形网格上对(H^1)、(H(\operatorname{curl}))和(H(\operatorname{div}))有限元进行了广泛的数值实验，结果表明BRAS相较于对角预处理降低了谱条件数、Krylov迭代次数和求解时间。对于预处理共轭梯度（CG），在所有测试的有限元阶数$p\in[1,4]$的情况下，BRAS相较于对角/雅可比预处理实现了1.1–4.7$\times$的加速。此外，理论和数值实验表明，在块预处理情形下，BRAS可以改善Schur补近似并减少外部求解时间。在混合Poisson和双调和系统上，BRAS相较于标准的基于对角的预处理方法，在求解时间上实现了1.5–3$\times$的加速。

5. 非线性椭圆问题的切空间多尺度流形方法

Tangent-Space Multiscale Manifold Methods for Nonlinear Elliptic Problems

作者: Mats G. Larson, Anna Persson

类别: math.NA

发布日期: 2026-07-13

arXiv ID: 2607.12082v1

链接: http://arxiv.org/abs/2607.12082v1

摘要翻译

中文标题: 非线性椭圆问题的切空间多尺度流形方法

摘要翻译: 针对非线性非均匀椭圆问题，本文引入一种切空间多尺度流形方法。该方法通过粗状态的非线性重构来表示细尺度解。理想重构通过一个约束变分问题消除细尺度，而可计算重构则通过局部非线性子域求解与单位分解的混合来逼近该映射。由于逼近集合是非线性流形，粗方程需借助切多尺度检验函数来建立。我们还提出了一种网络插值变体，其中仅由单位分解混合所使用的受限子域输出及其切作用通过局部学习映射来逼近。对于非均匀单调非线性扩散，我们记录了结构单调性、可微性、子域映射正则性以及条件扰动估计，这些性质将几何稳定性机理与局部化、残差和可选的学习缺陷分离开来。关于局部化缺陷衰减的严格先验理论，以及由此导出的粗网格尺寸收敛速率，将另文分析；本文在条件意义下控制这些缺陷，并通过数值实验验证其衰减。

不相关论文（基本信息）

1. 具有非光滑数据的纳维-斯托克斯方程的后验误差分析

A posteriori error analysis for the Navier-Stokes equations with non-smooth data

作者: María Gabriela Armentano, Mauricio Mendiluce

类别: math.NA

发布日期: 2026-07-14

arXiv ID: 2607.13014v1

链接: http://arxiv.org/abs/2607.13014v1

摘要: We study the stationary Navier-Stokes equations with Dirichlet boundary data in L2, a setting in which the limited regularity of the solution prevents the direct application of standard a posteriori error estimation techniques. To address this issue, we introduce a regularized formulation that yields a well-posed approximation of the original problem and admits a conforming finite element discretization. Using Taylor-Hood P2P1 elements, we construct a residual-based a posteriori error estimator and establish its reliability and efficiency under suitable smallness assumptions on the data. We derive computable upper and lower bounds in an appropriate norm that relate the estimator to the error between the exact solution of the original Navier-Stokes problem and its finite element approximation, showing that the estimator accurately reflects the finite element error. These results provide a rigorous foundation for the analysis and implementation of adaptive finite element methods for incompressible flows with low-regularity Dirichlet boundary data.

2. 隐式数据同化的集合控制流滤波

Ensemble Controlled-Flow Filtering for Implicit Data Assimilation

作者: Zhuoyuan Li, Yue Zhao, Ming Li

类别: stat.ML, cs.LG, math.NA, math.OC

发布日期: 2026-07-14

arXiv ID: 2607.12975v1

链接: http://arxiv.org/abs/2607.12975v1

摘要: Data assimilation estimates the state of a dynamical system from model forecasts and incoming observations. Many observation mechanisms, however, are many-to-one, implicit, non-smooth, or accessible only through simulation, and need not provide the residual structures or likelihood guidance required by existing ensemble filters. We introduce implicit data assimilation, in which the analysis law is defined as an energy tilt of the forecast distribution. We then propose the Ensemble Controlled-flow Filter (EnCF), which realizes this update through a stochastic controlled flow and learns the observation-dependent control by adjoint matching from terminal energy gradients. For simulator-defined observations, EnCF-LF learns a surrogate conditional energy from samples and applies the same controlled-flow solver. We prove ideal exactness, derive a one-step error decomposition, and establish non-accumulation of local errors under filter stability. Numerical results show that Kalman-type filters remain preferable for smooth additive-Gaussian observations, while the proposed methods are better suited to non-Gaussian, many-to-one, multimodal, and implicit observation models.

3. 矩阵和张量火车的低精度秩补偿

Low-Precision Rank Compensation for Matrices and Tensor Trains

作者: Ibrahim El Mountasser, Khalide Jbilou

类别: math.NA

发布日期: 2026-07-14

arXiv ID: 2607.12969v1

链接: http://arxiv.org/abs/2607.12969v1

摘要: Lower numerical precision reduces storage and memory traffic but raises the perturbation floor. We study rank compensation: reinvesting saved memory in a larger approximation rank. For matrices, the singular-value error identity yields a directly testable sufficient condition requiring the additional singular component to offset the perturbation from storing the rank-augmented approximation in lower precision. On ten SuiteSparse matrices, all 100 truncation-dominated configurations (50 FP32 and 50 FP16) are certified non-increases and strict accuracy wins, with mean error ratio $0.963$ and storage ratios $58.8\%$ and $29.4\%$ relative to the FP64 baseline. FP16 failures occur only in tail-rank stress tests near the perturbation floor. At the largest resident matrix-application batch, compensated FP32 and FP16 achieve geometric-mean A100 speedups of $1.28\times$ and $2.12\times$; neither accelerates the smallest batch. For Tensor-Train (TT) approximation, we give a conditional a posteriori extension based on the measured truncation gain and rounded-core perturbation. Across three-way and six-way synthetic tests, FP32 and FP16 achieve combined accuracy-memory wins in 10 of 20 and 14 of 20 trials. On public hyperspectral tensors and FROSTT top-active subtensors, the corresponding counts are 44 of 60 and 54 of 60; four FP16 Salinas-A tail-stress cases fail. No certified TT case exceeds the FP64 error beyond numerical tolerance. Reconstruction of six public tensors yields geometric-mean compensated speedups of $1.38\times$ (FP32) and $1.94\times$ (FP16). Timings cover resident downstream kernels, not factorization, transfers, or end-to-end acceleration.

4. 自定义精度OCP MX块浮点格式与算术的仿真

Simulation of Custom-Precision OCP MX Block Floating-Point Formats and Arithmetic

作者: Maliha Islam, Mantas Mikaitis

类别: cs.MS, math.NA

发布日期: 2026-07-14

arXiv ID: 2607.12915v1

链接: http://arxiv.org/abs/2607.12915v1

摘要: We present MXsim v0.1: a MATLAB library for simulating the OCP MX block floating-point arithmetic and formats. The library is built on top of a custom-precision floating-point simulator, CPFloat, which allows custom variants of non-block IEEE 754 floating-point formats to be simulated. MXsim is a MATLAB simulator, and is targeted at mixed-precision algorithm development community to perform experimentation in OCP MX block floating-point arithmetic by using a simple and well-documented interface in MATLAB, which is especially useful when access to the latest NVIDIA and AMD GPUs is not available. Furthermore, due to full customisation of exponents, precisions of input and output formats, and the precision of the accumulator within the dot product operation, MXsim allows experimenting with custom variants of the OCP MX block floating point for research. Finally, since the bit-wise numerical behavior of OCP MX formats is not documented by GPU vendors, and the precision and rounding are not prescribed by the OCP specification itself, MXsim allows experiments with various possible settings in order to determine the effects on mixed-precision numerical algorithms.

5. 求解Fokker-Planck型方程的最小二乘弱Galerkin有限元方法

A Least Squares Weak Galerkin Finite Element Method for Fokker-Planck Type Equations

作者: Chunmei Wang, Shangyou Zhang

类别: math.NA

发布日期: 2026-07-14

arXiv ID: 2607.12803v1

链接: http://arxiv.org/abs/2607.12803v1

摘要: This paper presents a least squares weak Galerkin (LS-WG) finite element method for a class of second order elliptic equations of Fokker-Planck type. To address the numerical challenges arising from non-smooth diffusion tensors, the proposed method utilizes a least-squares formulation that yields a symmetric positive definite (SPD) discrete system. The numerical scheme is designed by employing locally constructed weak second order partial derivatives and the weak divergence commonly used within the weak Galerkin framework. A rigorous theoretical foundation is provided, establishing the uniqueness of the discrete solution and deriving optimal-order error estimates in a discrete energy norm. Finally, extensive numerical experiments are reported to validate the theoretical findings and demonstrate the robustness and performance of the numerical scheme.

6. 多相 Mullins-Sekerka 流的无网格求解器：三重点与九十度边界接触

A Mesh-Free Solver for Multi-Phase Mullins-Sekerka Flow: Triple Junctions and Ninety-Degree Boundary Contact

作者: Tokuhiro Eto

类别: math.NA, math.AP

发布日期: 2026-07-14

arXiv ID: 2607.12759v1

链接: http://arxiv.org/abs/2607.12759v1

摘要: A bulk mesh-free solver for the multi-phase Mullins–Sekerka flow in $\mathbb R^2$ and in a half-plane bounded by a Neumann wall is developed. In the underlying mathematical model, interfaces driven by their curvature are coupled through a harmonic chemical-potential field. We use a charge simulation method, a variant of the method of fundamental solutions: each chemical potential is represented by fundamental solutions centered at charge points off the curve, so no bulk mesh or singular integral is required. It treats curve networks separating several phases at triple junctions, including phases that occupy more than one region; on the half-plane boundary the no-flux condition is imposed exactly by image charges, and mobile contacts stay orthogonal to the wall. The discretization is structure-preserving: between topological events, every bounded phase area is conserved to machine precision at the velocity level by a null-space projection of the discrete area constraints. The junction connectivity is fixed, the only topological event being the disappearance of a region enclosed by a single closed curve and incident to no junction. The proposed scheme is validated against an exact three-concentric-circle solution in terms of convergence, cost, and conditioning.

7. 基于广义内积的质量守恒鞍点动力学：理论、算法与应用

Mass-Conserving Saddle Dynamics via Generalized Inner Product: Theory, Algorithms, and Applications

作者: Longjing Li, Guanghua Ji, Zhen Xu

类别: math.NA

发布日期: 2026-07-14

arXiv ID: 2607.12715v1

链接: http://arxiv.org/abs/2607.12715v1

摘要: To reveal the effect of the inner product choice, we present a unified formulation of saddle dynamics for the functional F with a mass constraint under different inner products. We establish the equivalence between the index-k saddle points and the linearly stable steady states of the corresponding dynamics. Further, we present the dynamics with discrete H^{-1} and L^2 inner products and numerically verify the convergence orders of both dynamics. Finally, we apply the method to a phase field model with driving force under Neumann and periodic boundary conditions. The results uncover previously unreported saddle points and their connectivity, highlighting how the choice of inner product enriches the solution landscape in conservative systems.

8. 使用指数变换的端点奇异函数的埃尔米特谱逼近

Hermite spectral approximation for functions with endpoint singularities using exponential transforms

作者: Haiyong Wang

类别: math.NA

发布日期: 2026-07-14

arXiv ID: 2607.12648v1

链接: http://arxiv.org/abs/2607.12648v1

摘要: In this paper we introduce Hermite spectral approximation for functions with endpoint singularities using exponential transforms, including single exponential (SE), double exponential (DE) and error function (EF) transforms, and present a comprehensive convergence analysis for these approximations without and with scaling. In the case without scaling, we show that these methods converge at some root-exponential rate. In the case with scaling, we derive optimal scaling factors for each of exponential transforms and show that the convergence rate of Hermite spectral approximation can be significantly improved. Numerical comparisons with sinc method are present and it is shown that Hermite method has comparable or superior accuracy performance when using the same number of terms. Extensions to quadrature and rootfinding algorithm are also discussed.

9. 线性微分-代数方程的围道积分方法与结构扰动

Contour integral methods and structured perturbations for linear differential-algebraic equations

作者: Nicola Guglielmi, Mattia Manucci

类别: math.NA, math.DS

发布日期: 2026-07-14

arXiv ID: 2607.12628v1

链接: http://arxiv.org/abs/2607.12628v1

摘要: We generalize the contour integral methods (CIM) framework to the time integration of linear dynamical systems that are subject to algebraic constraints at all times during their evolution. The proposed approach relies on applying the Laplace transform to the Cauchy problem associated with a linear system of differential-algebraic equations (DAE), and subsequently reconstructing the time-domain solution by approximating the inverse Laplace transform via a suitable quadrature rule. This procedure yields an efficient and accurate alternative to classical Runge-Kutta schemes, which are well known to exhibit order reduction in accuracy when applied to DAE. In the second part of the paper, we address linear parametric DAE and propose an efficient strategy for tuning the integration contour in the CIM framework using suitable structured-unstructured pseudospectral computations. This allows the identification of a single integration profile capable of approximating an entire family of parametric solutions, thereby facilitating the efficient application of model order reduction techniques. Finally, numerical experiments are presented to validate the proposed methodology and support the theoretical findings.

10. 基于张量的降阶建模用于基于优化的反问题

Tensor-Based Reduced-Order Modeling for Optimization-Based Inverse Problems

作者: Sahidul Islam, Andreas Mang, Maxim Olshanskii

类别: math.NA, math.OC

发布日期: 2026-07-14

arXiv ID: 2607.12613v1

链接: http://arxiv.org/abs/2607.12613v1

摘要: We develop a tensor reduced-order modeling (TROM) framework for optimization-based inverse problems governed by parameter-dependent dynamical systems. The approach approximates the parameter-to-observation map directly in tensor-train format, using either TT-SVD or TT-Cross compression, and integrates the resulting representation into a regularized nonlinear least-squares formulation. Beyond accelerating forward evaluations, the low-rank tensor structure is used to reformulate the inverse problem in reduced coordinates, assemble the Gauss–Newton quantities without forming the full observation-space Jacobian, and perform TROM-based objective minimization over the discrete parameter grid. This tensor optimization step can be used either as a stand-alone approximate minimization procedure or as a data-informed initialization for a subsequent Gauss–Newton solve. The method is studied for two inverse problems: an inverse heat-transfer problem in a heterogeneous medium, where the unknown parameters describe the locations of multiple low-conductivity inclusions, and a FitzHugh–Nagumo parameter-estimation problem with a highly nonconvex optimization landscape. Numerical experiments assess the effects of ROM approximation error, measurement noise, regularization, initialization, spatial discretization, and increasing parameter dimension. The results show that TROM can reproduce the behavior of full-order inversion at a substantially reduced online cost. The experiments also demonstrate that reduced-coordinate inversion, tensor-based optimization, and appropriate regularization improve robustness in higher-dimensional, noisy, and strongly nonconvex regimes.

11. 一种用于半无限地基中移动荷载波传播和土-结构相互作用问题的2.5维NURBS-迹无限元方法

A 2.5D NURBS-Trace Infinite-Element Method for Moving-Load Wave Propagation and Soil–Structure Interaction in Semi-Infinite Ground

作者: Yanhui Zhong, Hao Hong, Bei Zhang, Quansheng Zang, Hussein Rappel, Stéphane P. A. Bordas

类别: cs.CE, math.NA

发布日期: 2026-07-14

arXiv ID: 2607.12582v1

链接: http://arxiv.org/abs/2607.12582v1

摘要: For moving-load problems whose geometry and material properties are approximately invariant along the traveling direction, 2.5D analysis retains three displacement components at lower cost than full three-dimensional discretization. We present a 2.5D Non-Uniform Rational B-spline (NURBS)-trace infinite-element method (NBIEM), formulated as a coupled finite/infinite-element scheme, for wave propagation in linear viscoelastic semi-infinite geotechnical media. The bounded near field is discretized by isogeometric analysis, while the exterior is represented by tensor products of the boundary NURBS basis and admissible outgoing or evanescent exponential radial functions. Both subdomains share the same NURBS trace space and control-point degrees of freedom, enforcing displacement continuity without projection or mortar variables. For the selected radial functions, far-field stiffness and mass contributions are evaluated through closed-form radial moments, eliminating finite radial cutoff and radial quadrature. Closed-form half-space solutions verify displacement and stress frequency-response functions in sub-Rayleigh, super-shear but sub-compressional, and super-compressional moving-load regimes. Low-frequency studies assess sensitivity to radial parameters and artificial-boundary placement. Additional tests examine complex-valued response accuracy, phase fidelity, computational cost, and the frequency-dependent working range of the default S-wave-informed exterior realization. Applications to layered media, track–subgrade systems, and buried structures demonstrate the ability to handle heterogeneous materials, multi-patch configurations, curved interfaces, and cover-depth-dependent geotechnical responses. The framework provides a geometrically consistent and computationally efficient treatment of moving-load wave propagation and soil–structure interaction in semi-infinite domains.

12. 由核函数产生的矩阵的连续交叉逼近

Continuous Cross Approximation of Matrices Arising Out of Kernel Functions

作者: Sumit Singh, Sivaram Ambikasaran

类别: math.NA

发布日期: 2026-07-14

arXiv ID: 2607.12540v1

链接: http://arxiv.org/abs/2607.12540v1

摘要: We propose a residual energy-based framework for constructing low-rank approximations of kernel matrices arising from continuous kernel functions. The method operates in a continuous setting and is based on the adaptive selection of pivot nodes, referred to as \emph{optimal nodes}, which are chosen to minimize the residual energy at each step. This leads to a sequence of rank-$1$ updates of the residual kernel and admits a natural interpretation as a continuous analog of Adaptive Cross Approximation (ACA). From a theoretical perspective, we show that the residual kernels remain in the class of compact operators and that the approximation error is exactly characterized by the residual energy. We provide convergence guarantees showing that the method yields monotonic error reduction under an alignment condition and achieves geometric decay under practically motivated assumptions. Extensive numerical experiments demonstrate that the proposed method achieves approximation errors close to those of the truncated singular value decomposition across a range of kernel functions. The method exhibits strong robustness with respect to sampling and maintains stable performance across different discretizations. Furthermore, the close agreement between the continuous residual energy and the discrete approximation error highlights the consistency of the formulation. These results establish the proposed approach as a theoretically grounded, practically effective continuous counterpart to classical cross-approximation techniques.

13. 一种计算矩阵平方根的混合精度算法

A mixed precision algorithm for the matrix square root

作者: Bowen Gao, Daniel Kressner, Meiyue Shao

类别: math.NA

发布日期: 2026-07-14

arXiv ID: 2607.12430v1

链接: http://arxiv.org/abs/2607.12430v1

摘要: Mixed precision algorithms can significantly enhance the performance of linear algebra solvers by leveraging increasingly powerful low precision hardware while recovering working precision accuracy through, for example, iterative refinement. In this paper, we propose a novel mixed precision algorithm for computing matrix square roots. Our algorithm combines a Schur decomposition approach in low precision with iterative refinement performed through an approximate Newton method. We perform a detailed convergence analysis of the approximate Newton method. For the special case of symmetric positive definite matrices, this analysis implies that one can recover full working precision accuracy under mild conditions. Numerical experiments on x86-64 architectures indicate that our algorithm frequently reduces execution time compared with a fixed working-precision Schur algorithm.

14. 具有超线性增长系数的反射随机微分方程的耦合驯服Euler-Peano格式的强1/2阶收敛

Strong order one-half convergence of a coupled tamed Euler–Peano scheme for reflected stochastic differential equations with super-linearly growing coefficients

作者: Ziheng Chen, Caiyun Hua, Meng Cai

类别: math.NA

发布日期: 2026-07-14

arXiv ID: 2607.12415v1

链接: http://arxiv.org/abs/2607.12415v1

摘要: We study strong numerical approximations for reflected stochastic differential equations in possibly unbounded convex domains with super-linearly growing drift and diffusion coefficients. Under a coupled monotonicity condition and polynomial local Lipschitz assumptions, we first establish the well-posedness of the reflected SDE and derive uniform moment bounds for its solution. We then introduce a coupled tamed Euler–Peano scheme, in which the drift and the squared diffusion coefficient are tamed by a common factor and the resulting Euler–Peano path is corrected through the Skorokhod problem. This common taming factor preserves the drift–diffusion coercivity structure and yields uniform moment estimates for the numerical solution. We prove strong convergence of order $1/2$ for both the constrained state process and the boundary regulator, thereby recovering the standard Euler-type strong order in this reflected setting. Numerical experiments for a reflected stochastic Ginzburg–Landau type system illustrate the constraint preservation of the scheme and support the theoretical convergence rate.

15. 史前人类迁徙的高效路径重建：基于小波压缩地形数据的自适应Dijkstra算法

Efficient Path Reconstruction in Prehistoric Human Migration: An Adaptive Dijkstra’s Algorithm Based on Wavelet Compression for Topographic Data

作者: Max Brockmann, Lena Perlberg, Angela Kunoth

类别: math.NA

发布日期: 2026-07-14

arXiv ID: 2607.12408v1

链接: http://arxiv.org/abs/2607.12408v1

摘要: The reconstruction of prehistoric migration routes requires the accurate evaluation of effective distances across complex topographies. Traditional Least-Cost Path Analyses (LCPA) using high-resolution elevation models often lead to dense computational grids, making algorithms like Dijkstra’s prohibitively slow for large-scale archaeological modelling. In this paper, we present a novel approach using adaptive wavelet methods to dynamically compress topographic data. By retaining high resolution only in areas of high topographic complexity (e.g., mountain passes) and smoothing homogeneous regions, we significantly accelerate the Dijkstra algorithm without losing essential routing topology. We demonstrate the efficacy of this method using migration scenarios.

16. 面向时间周期问题的傅里叶感知投影周期 Parareal 方法

A Fourier-Aware Projection-Based Periodic Parareal Method for Time-Periodic Problems

作者: Chenyi Tan, Yuncheng Xu, Yehao Zhang, Yangfeng Su

类别: math.NA

发布日期: 2026-07-14

arXiv ID: 2607.12402v1

链接: http://arxiv.org/abs/2607.12402v1

摘要: Time-periodic problems arise when the desired solution is a periodic steady state rather than a transient trajectory. The periodic parareal algorithm with a periodic coarse problem (PP-PC) is a periodicity-preserving parallel-in-time approach for such problems. Projection-based correction can accelerate convergence of both parareal and PP-PC. In this paper, we propose a Fourier-aware construction of projection spaces and a new correction scheme to further accelerate the convergence of projection-based PP-PC. We develop a convergence analysis of projection-based PP-PC with the discrepancy-based correction scheme for general nonlinear time-periodic problems. For an arbitrary orthogonal projection, we derive a local one-step convergence estimate controlled by the unresolved error and explicit nonlinear contributions. A temporal Fourier decomposition bounds the unresolved error by a tail-leak quantity, which is small when dominant error modes are selected and their coefficients are captured by the projection space. For linear problems, the nonlinear contributions vanish, yielding a globally valid one-step tail-leak convergence estimate under weaker assumptions. Experiments on linear and nonlinear problems show that Fourier-aware PP-PC requires fewer outer iterations than Krylov-enhanced PP-PC. For the linear problems, the errors track the tail-leak bound. For the nonlinear problems, the experiments quantify the unresolved-error and explicit nonlinear contributions in the local one-step estimate and show that the evaluated tail-leak estimate follows the observed decay.

17. SinCoTrap：一种适用于任意维周期奇异积分的高阶局部修正梯形法则

SinCoTrap: A High-Order Locally Corrected Trapezoidal Rule for Periodic Singular Integrals in Arbitrary Dimensions

作者: Hengzhun Chen, Yingzhou Li

类别: math.NA

发布日期: 2026-07-14

arXiv ID: 2607.12390v1

链接: http://arxiv.org/abs/2607.12390v1

摘要: We present SinCoTrap (Singularity-Corrected Trapezoidal Rule), a high-order locally corrected trapezoidal method for periodic singular integrals in arbitrary dimension $d$ with kernel $|\boldsymbol{x}|^{-s}$, $0

18. 利用平方指数核的解析特征对的Karhunen-Loève展开进行分层贝叶斯反演

Hierarchical Bayesian inversion using the Karhunen-Loève expansion with analytical eigenpairs of the squared exponential kernel

作者: Tatsuya Shibata, Michael Conrad Koch, Kazunori Fujisawa

类别: stat.ME, math.NA

发布日期: 2026-07-14

arXiv ID: 2607.12387v1

链接: http://arxiv.org/abs/2607.12387v1

摘要: Hierarchical Bayesian inversion with Gaussian random field priors addresses uncertainty in covariance hyperparameters, such as the standard deviation and correlation length. When a Gaussian random field is represented by the Karhunen-Loève (KL) expansion, the basis functions depend on these hyperparameters through an integral eigenvalue problem (IEVP) associated with the covariance kernel. Consequently, the IEVP must be solved repeatedly whenever the hyperparameters are updated, leading to significant computational cost in hierarchical inference. In this paper, we focus on the squared exponential kernel and construct the KL expansion using the analytical solution to a Gaussian-weighted IEVP. This analytical KL expansion offers a computationally efficient alternative to the conventional KL expansion by eliminating the repeated numerical solutions of the IEVP during hyperparameter updates. While the analytical KL expansion is applicable to arbitrary domains and dimensions, it does not have the same mean-square optimality as the conventional KL expansion. To address this limitation, we employ an optimization-based approach that selects the standard deviation of the Gaussian weight function in the IEVP to effectively reduce the truncation error of the KL expansion. Numerical experiments in one- and two-dimensional settings show that this selection strategy provides sufficient accuracy for practical applications. Furthermore, the analytical KL expansion admits closed-form differentiation, enabling efficient posterior sampling via HMC. The proposed framework is applied to Bayesian inversion for a steady Darcy flow model, where the hydraulic conductivity field is successfully estimated using weakly informative hyperpriors.

19. Vlasov-Poisson系统的自适应傅里叶谱方法

An Adaptive Fourier Spectral Method for the Vlasov-Poisson system

作者: Seung Yeon Cho, Giovanni Russo

类别: math.NA

发布日期: 2026-07-14

arXiv ID: 2607.12368v1

链接: http://arxiv.org/abs/2607.12368v1

摘要: Numerical simulation of the Vlasov-Poisson system faces fundamental challenges due to phase-space filamentation. Standard spectral methods rely on artificial filtering to suppress errors, which inadvertently degrades physical structures and accuracy over time. This paper proposes a dynamically adaptive Fourier spectral method combined with high-order time splitting to overcome these limitations. To prevent filamentation-induced aliasing, we dynamically expand the wavenumber domain via adaptive zero-padding whenever spectral tail monitoring detects emerging fine-scale structures. Acting as an exact trigonometric interpolation, it inherently avoids the artificial smoothing of traditional filters, preserving mass and effectively maintaining the $L^2$-norm within the dynamically resolved scales. Numerical experiments on Landau damping and various instability problems validate the robustness of our scheme.

20. 参数动力系统中解流形的残差认证自适应跟踪

Residual-Certified Adaptive Tracking of Solution Manifolds in Parametric Dynamical Systems

作者: Yiran Xing, Yuandi Xu, Sulei Hu

类别: math.NA, cs.CE

发布日期: 2026-07-14

arXiv ID: 2607.12351v1

链接: http://arxiv.org/abs/2607.12351v1

摘要: This paper presents a residual-certified adaptive method for tracking local solution manifolds in parametric dynamical systems. The method combines local POD reduction, full physical residual checks, state-distance snapshot forgetting, high-fidelity resampling, and a lightweight physics-informed neural correction. Instead of learning one global parameter-to-state map, the algorithm maintains the currently active local branch and updates it when the residual indicates loss of validity. The analysis explains why residual thresholds are meaningful on regular branches through local residual-error control, and why stricter local updates are needed near folds or other degenerate neighborhoods. Numerical studies on Ostwald ripening, a particle population-balance model, and the Bratu equation test the approach across low-dimensional dynamics, nonlinear nonlocal residual compensation, and near-fold model failure. The results show that residual-certified local model management can concentrate high-fidelity computation in difficult parameter regions while preserving an interpretable link between surrogate prediction, physical consistency, and active-branch tracking.

21. 一种带回溯线搜索的自适应内点法用于凸约束优化

An adaptive interior-point method with backtracking line search for convex constrained optimization

作者: Fadi Hamad, Oliver Hinder

类别: math.OC, math.NA

发布日期: 2026-07-14

arXiv ID: 2607.12318v1

链接: http://arxiv.org/abs/2607.12318v1

摘要: Interior-point methods (IPMs) are widely adopted due to their high practical efficiency in solving linear, convex, and nonconvex optimization problems. For convex optimization, this performance is theoretically well-supported: there are strong complexity guarantees for self-concordant barrier setups \cite{nesterov1994interior}, which cover linear and conic optimization. On the other hand, unconstrained convex optimization is well-studied. However, there is limited analysis of constrained convex optimization methods without the self-concordance assumption. We develop and analyze a regularized Newton method with line search applied to the log barrier function in the setting that the objective and constraints are thrice differentiable and have Lipschitz continuous first and second derivatives. Starting from a strictly feasible point, our method finds an $ε$-approximately optimal solution in $\tilde{O}(ε^{-2/3})$ iterations.

22. 多类数据分类的扩散方法

Diffusive method for multiple class data classification

作者: Toyohiko Aiki, Honoka Yano, Takeshi Ohtsuka

类别: math.NA

发布日期: 2026-07-14

arXiv ID: 2607.12313v1

链接: http://arxiv.org/abs/2607.12313v1

摘要: A simple algorithm for multiple class data classification is proposed, which is based on the infinit speed of propagation and the strong maximum principle for the heat equation. In this method, solutions of the heat equation whose initial data are the characteristic function of the training dataset are considered. The classification is established by the sign of the difference of two solutions in a very short time interval. A numerical algorithm for discrete training data in high-dimensional feature space is proposed. As an application of the proposed method, numerical results on the classification of handwritten digits database are presented in this paper.

23. 一种用于含裂缝的混合维孔隙弹性模型的时间解耦格式

A time-decoupling scheme for mixed-dimensional poroelastic models with fractures

作者: Dongchun Tang, Linshuang He, Shuyu Sun, Minfu Feng

类别: math.NA

发布日期: 2026-07-14

arXiv ID: 2607.12309v1

链接: http://arxiv.org/abs/2607.12309v1

摘要: We propose a locking-free decoupling method for a mixed-dimensional poroelasticity model with fractures. By introducing the total pressure, the fractured Biot system is reformulated as a four-field formulation involving the displacement, total pressure, matrix pressure, and fracture pressure. We establish an energy dissipation law for the continuous model, which shows its consistency with the second law of thermodynamics. Based on this formulation, a time-decoupled scheme is developed. At the initial time step, a fully coupled scheme is employed, while for subsequent time steps, the flow problem is solved first, followed by the mechanics problem. A stabilization term is incorporated into the mechanical equation to help reduce the restrictions imposed on the model parameters in the stability analysis. Energy stability is established for the semi-discrete scheme. For the spatial discretization, the displacement and total pressure are approximated by the Taylor–Hood element, while Lagrange finite elements are used for the matrix and fracture pressures. A fully discrete decoupled scheme is then constructed. Energy stability and error estimates are derived for the fully discrete scheme, and the method is shown to be locking-free. Numerical experiments are presented to support the theoretical results.

24. 亚纯延拓中可见极点簇的围道计数指示场

Contour-count indicator fields for visible pole clusters in meromorphic continuation

作者: Zhiliang Deng, Xiaomei Yang

类别: math.NA

发布日期: 2026-07-14

arXiv ID: 2607.12256v1

链接: http://arxiv.org/abs/2607.12256v1

摘要: We develop a contour-count indicator method for visible pole clusters in outward meromorphic continuation from circular boundary data. The method starts from determinant characteristics built from positive Fourier coefficients. In the pure finite-pole model, the correct determinant characteristic factors into a polynomial whose zeros are the reciprocals of the exterior poles. In the presence of a holomorphic background, finite sampling, and noise, roots of individual determinants are unstable and are used only as local evidence. We aggregate this evidence into a scalar indicator field on the reciprocal pole plane: at each sampling point, the field records the fraction of determinant orders and shifts for which a small contour centered at that point encloses exactly one empirical determinant zero. The resulting field plays the role of a sampling-type imaging functional for pole visibility. Fixed superlevel sets give visible-pole clusters, while zero-dimensional persistent homology is used only as a threshold-robust post-processing step. We prove deterministic results linking pure-pole contour counts, Rouché stability, indicator-field contrast, fixed-threshold component recovery, and persistence-gap stability. These results explain why isolated poles with sufficient residue and separation generate stable high-value components, whereas weak, close, boundary-near, or noise-dominated poles may give low, short-lived, or merged components. The framework is a cluster-certification and imaging method, not an unconditional all-pole recovery procedure.

25. 一种用于混合连续–弹道气体动力学的对数高斯尺度空间限制器

A Log-Gaussian Scale-Space Limiter for Hybrid Continuum–Ballistic Gas Dynamics

作者: Bjørn Wu

类别: math.NA, physics.flu-dyn

发布日期: 2026-07-13

arXiv ID: 2607.12218v1

链接: http://arxiv.org/abs/2607.12218v1

摘要: We propose a log-Gaussian scale-space limiter for hybrid continuum-ballistic gas dynamics. The method defines complementary continuum and ballistic weights as Gaussian cumulative probabilities in logarithmic Knudsen-number space. This construction gives a smooth numerical transition between Navier-Stokes-Fourier and kinetic/free-molecular fluxes and suppresses asymptotically invalid correction branches in the continuum and free-molecular limits. The limiter is incorporated into a conservative finite-volume interface flux by blending a Navier-Stokes-Fourier flux with a half-range Maxwellian kinetic flux. Numerical tests using one-dimensional DVM/BGK Fourier and Couette benchmarks show that rarefaction corrections improve macroscopic profiles relative to NSF. A parameter scan over the transition center and width gives a DVM/BGK-calibrated pair K0=0.03 and sigma=2.5, reducing the combined mean profile error by about 40% for the tested planar wall-bounded flows. The paper is intended as a proof-of-concept numerical method and calibration study.

26. 汤川势的分裂分析

Splitting Analysis for Yukawa Potential

作者: Di Fang, Jiaqi Zhang

类别: math.NA, quant-ph

发布日期: 2026-07-13

arXiv ID: 2607.12153v1

链接: http://arxiv.org/abs/2607.12153v1

摘要: Splitting methods are among the most classical and fundamental tools for the simulation of quantum dynamics, and their importance has grown further with the rise of quantum computing. In this work, we analyze the Schrödinger equation with Yukawa potential, a physically relevant and widely used model potential. It may be viewed as a Coulomb interaction with exponential decay at spatial infinity, preserving the Coulomb singularity at the origin while removing the long-range Coulomb tail. We prove that the operator splitting for this unbounded Hamiltonian achieves a global $1/4$-order convergence rate in the time step for many-body Yukawa interactions, with explicit polynomial dependence on the number of particles. The result holds for all initial wavefunctions in $H^2(\mathbb R^{3N})$, the natural domain of the Hamiltonian, and our numerical experiments are consistent with the theoretical estimates. To identify the sharp obstruction behind this rate, we prove a short-time lower bound in the one-body setting of order $t^{5/4}$ for the one-step error, which rules out any uniform global estimate of order better than $1/4$ in general. This agreement with the optimal $1/4$ rate in the Coulomb case is particularly interesting, as Yukawa potential is short-ranged compared to Coulomb potential. For the many-body upper bound, one of the new technical ingredients is the explicit polynomial-in-system-size Sobolev estimates of many-body Yukawa systems. These estimates are crucial for obtaining fully a priori bounds that depend only on the norms of the initial states, rather than on the solution at time $t$. For the one-body lower bound, we leverage a new analysis argument based on Fourier analysis and Kato smoothing.

27. 基于水平集网格演化方法的二维开曲线贴体追踪

Body-fitted tracking of 2d open curves with a level set based mesh evolution method

作者: Charles Dapogny

类别: math.OC, math.NA

发布日期: 2026-07-13

arXiv ID: 2607.12116v1

链接: http://arxiv.org/abs/2607.12116v1

摘要: This article describes a novel numerical algorithm for tracking the motion of a collection of open curves in two space dimensions. The proposed strategy combines two complementary representations of these curves at each iteration of the evolution process: on the one hand, they are meshed explicitly, as a sub-collection of the entities of a mesh of the total computational domain. Concurrently, using a variant of the Level Set Method, they are captured implicitly as algebraic combinations of the negative, zero and positive subsets of two auxiliary scalar functions, defined on the whole ambient space. This coupling of representations allows to perform accurate geometric or mechanical computations on these curves, while leaving room for large evolution of their shape. After the description of its main numerical ingredients, several applications examples of this methodology are proposed, where it is used to simulate the motion of open physical discontinuities, such as a vortex sheet roll-up, and to optimize the shape of open-ended curves, e.g. with respect to their anisotropic length, with the aim to improve the trajectory of a laser acting on a powder bed in the context of additive manufacturing, or to reconstruct fracture sets.