生成时间: 2026年08月21日 05:00:01

重点关注论文数量: 0

了解领域论文数量: 3

不相关论文数量: 19

了解领域论文（摘要翻译）

1. 面向高Péclet入口输运的DeepONet代理模型的有理增强Chebyshev截断基

Rationally Enriched Chebyshev Trunk Bases for DeepONet Surrogates of High Péclet Entrance Transport

作者: Mingeun Choi, Satish Kumar

类别: cs.LG, math.NA

发布日期: 2026-08-20

arXiv ID: 2608.19658v1

链接: http://arxiv.org/abs/2608.19658v1

摘要翻译

中文标题: 面向高Péclet入口输运的DeepONet代理模型的有理增强Chebyshev截断基

摘要翻译: 本研究针对奇异摄动和高Péclet输运问题，提出了一种有理增强Chebyshev（REC）截断基，用于深度算子网络（DeepONet）代理模型，此类问题的解剖面以薄局部边界层或壁面层为特征。REC截断基将Chebyshev多项式字典元素与使用自适应Antoulas-Anderson（AAA）算法构造的有理字典元素相结合。在五次独立训练中，将所得的REC截断基DeepONet与普通DeepONet及仅由Chebyshev多项式组成的预定字典的Chebyshev截断基DeepONet，在三个奇异摄动参数（即扩散-对流比）不同的问题上进行了评估：一个奇异摄动标量边值问题（BVP）、一个具有给定壁面温度的热入口问题，以及一个具有吸收壁面的浓度入口问题。在保留的测试剖面上，REC截断基DeepONet在预测标量剖面方面优于普通DeepONet，并与Chebyshev截断基DeepONet相当；当摄动参数介于$1.00\times10^{-4}$与$1.78\times10^{-4}$之间时，其相对于Chebyshev截断基DeepONet的优势最为明显，此时剖面误差指标最多降低$19.5\,\%$。在预测壁面法向温度和浓度剖面时，REC截断基DeepONet相对于普通DeepONet和Chebyshev截断基DeepONet的剖面误差指标分别最多降低$60.2\,\%$和$32.2\,\%$，同时在Péclet数或传质Péclet数从$10^{2}$到$10^{4}$范围内抑制了人为的近壁振荡。

2. 面向抛物型偏微分方程的Evo-GTransNet：具有求积质量正交化的固定特征Galerkin线方法

Evo-GTransNet for Parabolic PDEs: A Fixed-Feature Galerkin Method of Lines with Quadrature-Mass Orthonormalization

作者: Lili Ju, Jin Zhao

类别: math.NA

发布日期: 2026-08-20

arXiv ID: 2608.19615v1

链接: http://arxiv.org/abs/2608.19615v1

摘要翻译

中文标题: 面向抛物型偏微分方程的Evo-GTransNet：具有求积质量正交化的固定特征Galerkin线方法

摘要翻译: 在本文中，我们为抛物型偏微分方程开发了一种进化式广义可迁移神经网络（Evo-GTransNet）求解器，其表述为保留空间固定特征Galerkin线方法。GTransNet提供指定的空间字典，而只有保留的输出系数随时间演化，从而避免时间积分过程中的非线性训练。为了解决严重的质量矩阵病态问题，我们应用加权求积截断奇异值分解（SVD）来选择数值上可分辨的试验空间，然后进行单独的重新缩放，使其基关于装配求积质量内积正交。秩截断修改了逼近空间，而随后的正交化仅改变其坐标表示，并在精确算术中保留离散函数。所得的半离散系数系统具有单位质量矩阵，我们为对称线性抛物问题建立了半离散能量律。对于时间离散采用隐式中点格式，我们进一步证明了方法的压缩性，并导出了一个条件完全离散误差估计，其中误差由保留空间逼近误差和一致性缺陷控制。数值实验使用重复特征样本以及单独的装配和验证求积，证明了二阶时间收敛，并量化了在严重原始质量病态情况下保留空间的准确性。对于本文考虑的高频和多尺度基准问题，GTransNet在相同的标称输出维度下，在测试的固定特征字典中实现了最小的平均验证误差。

3. 移动流形上对流-扩散-反应问题的高阶无网格 Lagrangian-Eulerian RBF-FD 方法

A high-order, meshless, Lagrangian–Eulerian RBF-FD method for advection–diffusion–reaction on moving manifolds

作者: Matthew Lowery, Grady B. Wright, Varun Shankar

类别: math.NA

发布日期: 2026-08-19

arXiv ID: 2608.19384v1

链接: http://arxiv.org/abs/2608.19384v1

摘要翻译

中文标题: 移动流形上对流-扩散-反应问题的高阶无网格 Lagrangian-Eulerian RBF-FD 方法

摘要翻译: 我们提出了一种用于余维数为1的移动流形 $\mathcal M(t)\subset\mathbb R^3$ 上偏微分方程的高阶径向基函数生成有限差分（RBF-FD）方法。我们的方法基于曲面 RBF-FD 的切平面公式，并结合了 Lagrangian 和 Eulerian 处理方式：流形和物质导数以 Lagrangian 方式演化，而其余曲面微分算子则在瞬时点云上重构，并在必要时通过准解析超黏性公式进行稳定。Lagrangian 标记漂移通过使用移动曲面的紧凑全局参数模型进行自适应重排来处理，该模型还提供精确的法向和基于几何的求积。重排后，我们通过向后半 Lagrangian 追踪和插值重构多步历史，然后恢复 Lagrangian 时间离散化。我们通过三种更新策略利用时间相干性：局部 RBF-FD 权重的缺陷校正、光谱重算之间基于曲率的超黏性系数更新，以及在预条件广义最小残差（GMRES）迭代之前重用不完全 LU（ILU）分解的全局缺陷校正迭代。最后，我们通过基于演化曲面求积的标量投影来强制满足给定的全局质量平衡。数值实验证明了高阶收敛性、无源问题中的舍入级守恒性、稳定的长时间积分，以及所提出的更新策略带来的显著节省。

不相关论文（基本信息）

1. 带奇异噪声的表面生长模型的数值研究

Numerical Study of a Surface Growth Model with Singular Noise

作者: Dirk Blömker, David Buchberger, Johannes Rimmele

类别: math.NA

发布日期: 2026-08-20

arXiv ID: 2608.20159v1

链接: http://arxiv.org/abs/2608.20159v1

摘要: We study a stochastic model for epitaxial thin-film growth driven by spatially rough additive noise in a regime where the noise is singular and regularization via truncation in Fourier space is used to give a meaning to the solution, leading to a vanishing nonlinearity in the limit. In order to study this phenomenon numerically, the nonlinearity is discretized by a spectral Galerkin projection, while time integration is performed with an exponential Euler scheme. For roughness stronger than space-time white noise, we derive strong error estimates in $L^p(Ω;C([0,T];\mathcal H^1))$ that display explicitly the interaction between the spatial cut-off, the time step, and the decay of the nonlinear current. We also quantify the growth of the truncated stochastic convolution and the corresponding vanishing rate of the nonlinearity. Numerical experiments illustrate the transition from persistent hill formation to noise-dominated dynamics as the roughness parameter increases.

2. 非定常可压缩粘性流的隐式BDF2双时间步进保正熵稳定格式

Implicit BDF2 dual time-stepping positivity-preserving entropy-stable schemes for unsteady compressible viscous flows

作者: Mohammed Sayyari, Nail K. Yamaleev

类别: math.NA

发布日期: 2026-08-20

arXiv ID: 2608.20103v1

链接: http://arxiv.org/abs/2608.20103v1

摘要: This paper presents a rigorous extension of the explicit, high-order, positivity-preserving, and entropy-stable spectral collocation schemes developed in Upperman 2023 and Yamaleev 2023 for the 3D compressible Navier-Stokes equations to a time-implicit formulation. The time derivative terms are discretized by using the second-order implicit backward difference formula (BDF2) that is well suited for solving time-variable viscous flows at high Reynolds numbers. The nonlinear system of discrete equations resulting from the BDF2 discretization at each physical timestep is solved using a dual time-stepping (DTS) technique. The BDF2 DTS scheme is entropy-stable and positivity-preserving in the pseudotime and provides unconditional stability properties in the physical time. Numerical results demonstrate the efficiency and accuracy of the positivity-preserving BDF2 DTS scheme as compared with its explicit counterpart are presented for supersonic flows with strong shock waves and contact discontinuities.

3. 球面积分方程 Sloan 迭代的超收敛与混叠饱和

Superconvergence and aliasing saturation in Sloan iteration for spherical integral equations

作者: Hao-Ning Wu

类别: math.NA

发布日期: 2026-08-20

arXiv ID: 2608.20098v1

链接: http://arxiv.org/abs/2608.20098v1

摘要: Sloan iteration raises the convergence order of Galerkin and degenerate-kernel approximations to second-kind integral equations. After quadrature discretization, these become a discrete Galerkin method and a product-integration Nyström method, respectively. How much of this improvement survives quadrature discretization? For zonal integral equations on the sphere, we give a sharp answer by expressing the Sloan error identity in terms of spherical harmonics. In the absence of quadrature, Sloan iteration fully exploits the smoothing of the integral operator. Quadrature can destroy this gain by aliasing unresolved information into low-frequency modes, where further iteration no longer improves the asymptotic rate. This yields a unified tail–aliasing description of these four methods. For positive-weight polynomially exact quadrature and multipliers of exact algebraic order, we derive sharp two-sided worst-case bounds that separate the spectral-tail and aliasing contributions. Their balance yields a depth-dependent recovery–saturation threshold: sufficient overintegration recovers the quadrature-free rate, while below the threshold aliasing determines the sharp order. We also extend the analysis beyond polynomial exactness using Marcinkiewicz–Zygmund stability and Gram-corrected least squares. Numerical experiments illustrate both regimes.

4. 波动方程逆背散射问题的区域导数与形状重构

Domain derivative and shape reconstruction for an inverse backscattering problem for the wave equation

作者: Roland Griesmaier, Marvin Knöller, Eliane Kummer

类别: math.NA

发布日期: 2026-08-20

arXiv ID: 2608.20078v1

链接: http://arxiv.org/abs/2608.20078v1

摘要: We consider an inverse backscattering problem for time-dependent acoustic waves with a compactly supported penetrable scattering obstacle in unbounded three-dimensional free space. Assuming that dynamic backscattering far field data of scattered waves corresponding to a few time-dependent incident plane waves are available, the goal is to reconstruct the shape of the scattering obstacle. We establish Fréchet differentiability of the time-dependent far field pattern with respect to the shape of the scattering obstacle and give a characterization of the associated temporal domain derivative in terms of its Laplace transform. This characterization is then utilized in an efficient implementation of a regularized Gauß-Newton method for the inverse backscattering problem using convolution quadrature and a boundary element method. Numerical examples demonstrate potentials and limitations of the algorithm.

5. 由紧黎曼流形上的嵌套Marcinkiewicz–Zygmund测度构造的近似紧框架小波系统

Nearly tight framelet systems from nested Marcinkiewicz–Zygmund measures on compact Riemannian manifolds

作者: Hao-Ning Wu, Xiaosheng Zhuang

类别: math.NA

发布日期: 2026-08-20

arXiv ID: 2608.20058v1

链接: http://arxiv.org/abs/2608.20058v1

摘要: Tight framelet systems on manifolds provide exact energy preservation and one-pass reconstruction, but their standard semi-discretization relies on polynomial-exact quadrature rules, which are difficult to reconcile with scattered and progressively refined data. We develop nested Marcinkiewicz–Zygmund (MZ) measures on compact Riemannian manifolds as a quantitative alternative. Using dyadic quasi-uniform nested point sets and local partition weights, we establish a sequence of deterministic MZ inequalities for diffusion polynomial spaces. The result has two complementary forms: a fixed-tolerance version, in which the polynomial bandwidth grows dyadically with the level, and a fixed-bandwidth refinement version, in which the MZ tolerance improves as more nested nodes are added. When these measures are used to discretize continuous tight framelets, the MZ tolerance transfers directly to the deviation of the frame bounds from one. Thus quadrature exactness is relaxed with a controlled loss of tightness, and near-tightness improves along the same nested hierarchy. For the fully discrete setting, we develop filter-bank analysis and synthesis procedures, characterize one-pass and canonical reconstruction, and show that the MZ tolerance also controls the conditioning of the frame-operator equation. Fast implementation of filter-bank transforms is also presented. Numerical experiments on the sphere and the flat torus illustrate the resulting multiscale decompositions and reconstruction behavior.

6. 面向任意形状先进变刚度板的综合 p-VEM 框架

A Comprehensive p-VEM Framework for Advanced Variable Stiffness Plates with Arbitrary Shapes

作者: Paola Pia Foligno, Daniele Boffi, Fabio Credali, Riccardo Vescovini

类别: math.NA

发布日期: 2026-08-20

arXiv ID: 2608.20034v1

链接: http://arxiv.org/abs/2608.20034v1

摘要: This paper presents a comprehensive, high-order (p-version) Virtual Element Method (VEM) framework for the structural analysis of innovative variable stiffness plates. VEM is particularly suited for complex configurations due to its ability to handle arbitrary polygonal meshes, including curved edges. However, its mathematical formulation may hinder its spread in the engineering community. This work illustrates a formulation with an accessible implementation using well-known FEM notation and integrating at the same time a set of new advanced capabilities. Specifically, both standard stabilized and advanced self-stabilized strategies are adopted. To further improve the robustness of VEM in the presence of variable coefficients, polynomial projections taking into account the coefficients are employed. This approach is referred to as Variable Coefficients-VEM approach (VC-VEM). This unified framework is applied to linear static, free-vibration, and buckling analyses, and validated against analytical solutions and numerical benchmarks. In particular, plates with cutouts and problems featuring high-gradient solutions are investigated, demonstrating that the proposed comprehensive approach provides a flexible and ready-to-implement tool for advanced structural design.

7. 应用谱方法建模线性滤波器：贝塞尔、帕普利斯和勒让德滤波器

Applying the Spectral Method for Modeling Linear Filters: Bessel, Papoulis, and Legendre Filters

作者: Konstantin A. Rybakov

类别: eess.SP, eess.SY, math.NA

发布日期: 2026-08-20

arXiv ID: 2608.19840v1

链接: http://arxiv.org/abs/2608.19840v1

摘要: This paper proposes a new technique for computer simulation of linear filters. It allows simulating continuous-time linear filters based on the spectral method for analyzing linear control systems. Applying the spectral method implies that the input and output signals are represented by ordered sets of expansion coefficients in a chosen basis, and the filter itself is specified by a two-dimensional nonstationary transfer function. The described technique is tested on Bessel, Papoulis, and Legendre filters of various orders. For each filter, the corresponding two-dimensional nonstationary transfer function, i.e., the matrix of the linear transformation relating expansion coefficients of the input and output signals, is obtained.

8. T网格上最高阶光滑样条空间的最优维数界

Sharp Dimension Bounds for Spline Spaces over T-meshes with Highest Order of Smoothness

作者: Bingru Huang, Falai Chen

类别: math.NA

发布日期: 2026-08-20

arXiv ID: 2608.19839v1

链接: http://arxiv.org/abs/2608.19839v1

摘要: The dimension of a polynomial spline space of bi-degree $(d_1,d_2)$ over a T-mesh $\mathscr{T}$ with the highest order of smoothness $(d_1-1,d_2-1)$ depends on both mesh topology and geometric configurations. Under the assumption that the T-connected components of the T-mesh $\mathscr{T}$ contain no vanishable T $l$-edges, we develop explicit upper and lower bounds of the dimension of the polynomial spline space. By introducing a decoupling technique within the completely non-diagonalizable component (CNDC) of the T-mesh $\mathscr{T}$, we separate tightly coupled multi-vertex constraints and transform global conformality conditions into localized linear equations along each interior large edge. Based on the decoupling technique, a new dimension formula of the polynomial spline space is then presented, and from which sharp upper and lower bounds of the dimension are obtained. The bounds are sharp in the sense that different geometric realizations of T-meshes with the same topology can attain the lower and upper bounds for the dimension of the polynomial spline space. We further prove that the new formula is consistent with Mourrain’s homological dimension formula, and a sharper lower bound is obtained by our method.

9. 确定性及随机浅层Sigmoid网络的Sobolev最优逼近

Optimal Sobolev Approximation by Deterministic and Random Shallow Sigmoidal Networks

作者: Zhaohui Fu, Yangshuai Wang

类别: math.NA

发布日期: 2026-08-20

arXiv ID: 2608.19797v1

链接: http://arxiv.org/abs/2608.19797v1

摘要: Shallow networks with prescribed or randomly sampled hidden parameters are widely used as numerical trial spaces, yet their optimal Sobolev approximation power with standard smooth sigmoidal activations in general dimension remains unresolved. We establish the corresponding optimal rates for a class of smooth sigmoidal activations with Schwartz-class derivative decay, including $\tanh$, the logistic sigmoid, and the error function $erf$. We first construct deterministic direction–offset dictionaries with $M$ features such that every $u\in H^k(Ω)$ can be approximated with error of order $M^{-(k-m)/d}$ in $H^m(Ω)$ for all $0\le m\le k$. This rate is optimal in the sense of Kolmogorov widths for Sobolev balls. We further prove that dictionaries obtained by independent parameter sampling from any prescribed density bounded away from zero attain the same approximation exponent with high probability, up to logarithmic oversampling. The analysis develops a sigmoidal ridge representation and combines it with deterministic or probabilistic quadrature in direction–offset space while retaining polynomial control of the output coefficients. Numerical experiments across a broad range of dimensions, target regularities, and Sobolev error norms recover the predicted algebraic rates for both deterministic and random feature dictionaries.

10. 学习分段光滑动力系统

Learning piecewise-smooth dynamical systems

作者: Davide Murari, Erik Jansson, Chris Budd OBE, Carola-Bibiane Schönlieb

类别: math.DS, cs.LG, math.NA

发布日期: 2026-08-20

arXiv ID: 2608.19785v1

链接: http://arxiv.org/abs/2608.19785v1

摘要: Discovering dynamical systems from trajectory data is a central problem in applied mathematics and engineering. Whilst recent advances in machine learning have led to strong progress in data-driven system identification, much less attention has been given to systems with discontinuous dynamics. These systems are nevertheless highly relevant in applications, including climate dynamics and mechanical systems with friction. In this work, we consider the problem of identifying piecewise-smooth dynamical systems directly from trajectory data. Compared with the smooth setting, this requires recovering the governing equations and detecting the switching hyperplanes that separate different dynamical regimes and characterising their behaviour, such as sliding motion. We present a modular framework for discovering such systems by first estimating switching hyperplanes from data and then learning smooth dynamics within each region using geometry-constrained neural networks. The geometry-learning phase is studied from a statistical perspective, analysing the identifiability of the discontinuities and the robustness of the procedure. We also introduce a novel neural network architecture with a prescribed discontinuity set, and provide a theoretical analysis of its approximation properties. The approach is tested on low-dimensional benchmark problems, including dry-friction oscillators and the PP04 climate model for the ice ages.

11. 用于加速稀疏网格粒子网格方法中粒子-网格耦合的精确分层算法

Exact hierarchical algorithms for accelerating particle–mesh coupling in sparse-grid particle-in-cell methods

作者: Clément Guillet

类别: math.NA

发布日期: 2026-08-20

arXiv ID: 2608.19702v1

链接: http://arxiv.org/abs/2608.19702v1

摘要: In this paper, we propose two hierarchical algorithms for charge deposition and electric-field interpolation that apply to both the sparse-grid combination technique (SGCT-PIC) and hierarchical sparse-grid (HSG-PIC) particle-in-cell methods. The two algorithms are inspired by the fast multipole method (FMM) and exploit clusters of particles associated with a directed acyclic graph (DAG) of particle-populated boxes to reduce the number of particle–mesh interactions. The particle–mesh interactions are governed by piecewise-polynomial kernels, so that the associated multipole expansions are exact, requiring neither truncation nor approximation, and are valid in both near- and far-field regions, thereby eliminating the need for multipole-to-local translations. The arithmetic complexity of the charge deposition and field interpolation steps is reduced from $Ø(p^d n^{d-1}N)$ to $Ø(p^d(N+M))$, where $M=2^{dn}$ denotes the number of full-grid mesh nodes and is typically no larger than the particle population in the considered regime, $M\lesssim N$. Numerical experiments in two-dimensional configurations demonstrate charge-deposition speedups of $8.2\times$–$66.9\times$ for SGCT-PIC and $3.1\times$–$18.8\times$ for HSG-PIC, and field-interpolation speedups of $4.1\times$–$62.6\times$ and $4.2\times$–$13.7\times$, respectively, depending on the particle-per-cell ratio, while preserving the exact particle–mesh interactions. The speedups increase with the particle-per-cell ratio, reflecting the reduced dependence of the hierarchical algorithms on the number of particles and their increasing advantage for large particle populations.

12. 学习确定性及随机受迫哈密顿系统

Learning Deterministic and Stochastic Forced Hamiltonian Systems

作者: Benedikt Brantner, Tomasz Tyranowski

类别: math.NA, cs.LG, math.SG

发布日期: 2026-08-20

arXiv ID: 2608.19688v1

链接: http://arxiv.org/abs/2608.19688v1

摘要: We develop a geometric framework for learning deterministic and stochastic forced Hamiltonian systems with neural networks. Motivated by the Lagrange-d’Alembert principle and the theory of variational integrators, we introduce the notion of a Lagrange-d’Alembert map and establish a $C^r$ convergence theorem for first-order one-step methods. Building on these results, we propose Generalized Forced Hamiltonian Neural Networks (GFHNNs), a class of structure-preserving neural networks obtained by concatenating Lagrange-d’Alembert-Euler maps, and prove a universal approximation theorem for this architecture. We further extend the framework to parameter-dependent systems, leading to Parametric Generalized Forced Hamiltonian Neural Networks (PGFHNNs). By interpreting the multiple Stratonovich integrals appearing in the Stratonovich-Taylor expansion as parameters, the same framework can be applied to stochastic forced Hamiltonian systems whenever information about the underlying Wiener process is available. Our numerical experiments demonstrate that the proposed geometric architectures provide significantly improved long-time stability and accuracy compared to non-geometric residual neural networks, while requiring substantially less training data to achieve a comparable level of performance.

13. 面向混合离子Vlasov-Fokker-Planck与流体电子系统的自适应守恒低秩IMEX求解器

An adaptive and conservative low-rank IMEX solver for the hybrid ion Vlasov-Fokker-Planck and fluid electron system

作者: Joseph Nakao, Dylan T. Jacobs, William Taitano

类别: math.NA

发布日期: 2026-08-20

arXiv ID: 2608.19603v1

链接: http://arxiv.org/abs/2608.19603v1

摘要: We present a macroscopically conservative, rank-adaptive method for solving a hybrid Vlasov-Fokker-Planck (VFP) equation in which the ions are treated kinetically, and the electrons are treated as a fluid. Solving this system poses several coupled computational challenges: the curse of dimensionality, conservation of macroscopic quantities, stiffness arising from the multiscale nature of the model, and structure preservation. To address these challenges, we combine several established methods, each targeting a specific difficulty, into a unified framework for solving this nonlinear system. Specifically, we employ the recent Reduced Augmentation Implicit Low-rank (RAIL) method for rank adaptivity in velocity space to combat the curse of dimensionality, high-order implicit-explicit time stepping to handle the multiscale stiffness, and the Local Macroscopic Conservative (LoMaC) procedure for conservative truncation of the solution. The RAIL and LoMaC methods are extended from Cartesian to cylindrical coordinates. The resulting framework accommodates implicit rank-adaptive time integration while remaining conservative and leveraging structure-preserving discretizations. We verify the scheme on a suite of test problems and simulate a standing shock to demonstrate the importance of conserving the macroscopic quantities.

14. 正规Procrustes问题：一种黎曼优化方法

The Normal Procrustes Problem: A Riemannian Optimization Approach

作者: Kyle Bierly

类别: math.NA, math.OC

发布日期: 2026-08-20

arXiv ID: 2608.19513v1

链接: http://arxiv.org/abs/2608.19513v1

摘要: For given $m \times n$ data matrices $X, Y$, we investigate the Normal Procrustes Problem—the least squares optimization problem that aims to minimize $\|AX-Y\|_F^2$, where $A$ is constrained to be a normal $m \times m$ matrix. As far as the author of this article is aware, no other method that attempts to solve the Normal Procrustes Problem exists in the literature; we thus propose what is, to our knowledge, the first such method. We, furthermore, adapt our approach to address the Real Normal Procrustes Problem, where $A$ must be real. In our treatment of these problems, we first reduce our complex and real objective functions to be purely optimizable over the Riemannian manifolds of the unitary and real orthogonal matrices, respectively. This reduction enables us to apply techniques in Riemannian manifold optimization to approximate solutions to both. The Closest Normal Matrix and Real Closest Normal Matrix Problems are both special cases of their respective Procrustes Problems and have been previously studied in the literature. Our approach thus recovers a novel Riemannian optimization method for approximating solutions to both these problems. We further numerically test the performance of our method across all such problems (including against previously developed algorithms on the Closest Normal Matrix Problems) and obtain competitive residuals and favorable scaling in wall-clock time.

15. 超越线性子空间：非线性矩匹配与二次流形的交汇

Beyond linear subspaces: Nonlinear moment matching meets quadratic manifolds

作者: Reetish Padhi, Serkan Gugercin

类别: math.OC, math.NA

发布日期: 2026-08-19

arXiv ID: 2608.19486v1

链接: http://arxiv.org/abs/2608.19486v1

摘要: Quadratic manifold-based model order reduction offers a viable pathway to circumvent the limitations of linear subspaces for linear control systems characterized by slow Kolmogorov $n$-width decay. However, a system-theoretic framework for constructing such quadratic approximations remains absent from the literature. This paper presents a system-agnostic, optimization-free framework for the direct construction of quadratic projection matrices. We prove that the synthesized reduced-order model matches the nonlinear moments of the full-order system and preserves its exact center manifold mapping, thereby ensuring asymptotic tracking of steady-state outputs under specific input classes. Numerical results on transport-dominated benchmark problems, namely, the one-dimensional damped wave and advection equations, show that the proposed framework achieves high-fidelity trajectory reconstruction within a significantly reduced-dimensional state space, yielding substantial online computational savings.

16. 弱相互作用费米子动力学的有效经典模拟

Efficient Classical Simulation of Weakly Interacting Fermion Dynamics

作者: Chu Zhao, Iman Marvian, Yu Tong

类别: quant-ph, math-ph, math.NA

发布日期: 2026-08-19

arXiv ID: 2608.19448v1

链接: http://arxiv.org/abs/2608.19448v1

摘要: We consider the task of simulating the real-time dynamics of weakly interacting fermionic systems. In particular, we focus on computing the expectation value of a local observable $A$ at time $t$. By analyzing the convergence of the perturbative expansion in the interaction strength $λ$ for the Heisenberg-picture observable, we propose a polynomial-time algorithm for estimating this expectation value in the weakly interacting regime $λ|t|^{2D+1}=\mathcal{O}(1)$, when the Hamiltonian is geometrically local on a $D$-dimensional lattice. Importantly, this condition is independent of the system size. If the goal is instead to approximate the time-evolved observable in normalized Frobenius norm, we extend the convergence regime to $λ|t|=\mathcal{O}(1)$ with quasi-polynomial runtime. When the non-interacting part exhibits Anderson localization, our polynomial-time algorithm can be extended up to $λ|t|=\mathcal{O}(1)$, modulo polylogarithmic factors. Our algorithm brings together ideas from continuous-time QMC, diagrammatic QMC, and Majorana Propagation, but with a new Heisenberg-picture operator-growth analysis that makes the sampling complexity rigorously controllable. This leads to provably efficient classical algorithms in regimes where the interaction is weak enough that the sampling variance remains bounded independently of system size. Together, these results identify broad regimes in which weak interactions, locality, and localization can be leveraged to make real-time fermionic dynamics classically tractable.

17. 无序交通的二阶连续模型

Second-Order Continuum Model for Disordered Traffic

作者: Shashank Rajput, Venkatesan Kanagaraj, Martin Treiber, Gowri Asaithambi, Ostap Okhrin

类别: physics.soc-ph, math.NA

发布日期: 2026-08-19

arXiv ID: 2608.19440v1

链接: http://arxiv.org/abs/2608.19440v1

摘要: In this study, a two-dimensional second-order macroscopic traffic flow model is proposed to capture the complex dynamics of disordered traffic by explicitly incorporating coupled longitudinal and lateral interactions. The framework extends the classical continuity equation to two spatial dimensions and introduces acceleration equations consisting of self-driven, interaction-induced, and road-boundary-related components. The macroscopic longitudinal acceleration is formulated based on the Full Velocity Difference Model (FVDM), while lateral dynamics are governed by interaction principles consistent with the Optimal Velocity Model (OVM). The explicit inclusion of road-boundary effects enables the representation of vehicle confinement and space-sharing behavior that are essential in lane-free and weakly lane-disciplined traffic systems. The model is examined through a series of numerical experiments in which longitudinal and lateral dynamics are analysed individually as well as simultaneously within a coupled two-dimensional setting under a range of initial conditions, including transitions between free-flow, medium congestion, and heavy congestion, and different lateral density configurations. The simulations demonstrate the model’s ability to reproduce key traffic features such as shockwave propagation, lateral dispersion, vehicle rearrangement, and stable density evolution across the road width. The numerical scheme, based on upwind and Lax-Friedrichs discretisations, ensures stable solutions of the coupled partial differential equations. Overall, the proposed framework provides a robust macroscopic description of disordered traffic and offers a consistent basis for analysing two-dimensional vehicular flow dynamics.

18. 二维共轭传热的隐式-伴随有限体积拓扑优化

Implicit-adjoint finite-volume topology optimization of two-dimensional conjugate heat transfer

作者: Sam Yang

类别: math.NA

发布日期: 2026-08-19

arXiv ID: 2608.19426v1

链接: http://arxiv.org/abs/2608.19426v1

摘要: Designing compact, high-efficiency thermal architectures requires resolving the competing demands of solid conduction, fluid convection, and flow resistance within highly constrained physical envelopes. Density-based topology optimization provides a systematic framework for synthesizing these coupled layouts, yet the reproducibility and numerical stability of the resulting designs depend critically on the underlying discrete solvers and adjoint sensitivity mechanics. In this work, we present a transparent, self-contained two-dimensional finite-volume formulation on a staggered Marker-and-Cell grid for conjugate heat transfer governed by design-dependent energy transport coupled to Stokes–Brinkman or Darcy flow at fixed solid volume. To prevent spurious artificial thermal sources in porous, weakly compressible Brinkman domains, the discrete advection operator is constructed to satisfy the identity $\mathbf{u}\cdot\nabla T=\nabla\cdot(\mathbf{u}T)-T(\nabla\cdot\mathbf{u})$ cellwise, ensuring that uniform temperature fields remain exact discrete nullspaces even under inexact continuity satisfaction. Reverse-mode derivatives are evaluated via the implicit function theorem rather than unrolled iterative loops, yielding exact discrete adjoints with bounded memory requirements. The discrete operators are systematically validated through the method of manufactured solutions and directional Taylor remainder tests. Four representative thermofluid design benchmarks are optimized using a projected-gradient scheme with $β$-continuation, wherein candidate iterates are accepted and published only upon satisfying rigorous, predeclared gates on residual convergence, mass conservation, volume feasibility, and numerical finiteness. The resulting formulation provides an inspectable, deterministic reference stack for verifiable conjugate thermofluidic topology optimization.

19. 利用层次Tucker分解求解高维扩散方程的高阶秩自适应隐式算法

A High-Order Rank-Adaptive Implicit Algorithm for Solving High Dimensional Diffusion Equations using the Hierarchical Tucker Decomposition

作者: Paolo Bosques-Paulet

类别: math.NA

发布日期: 2026-08-19

arXiv ID: 2608.19368v1

链接: http://arxiv.org/abs/2608.19368v1

摘要: This paper presents a high-order rank-adaptive implicit integrator for the tensor solution of high-dimensional diffusion equations. We extend the 3D version of this method from the Tucker decomposition to higher dimensions using the hierarchical Tucker (HT) decomposition, since the storage complexity for the Tucker decomposition increases exponentially with the number of dimensions $d>3$. The HT format avoids this issue by decomposing the solution according to a binary tree consisting of bases for each dimension and core tensors which connect the bases. Spectral methods are considered for spatial discretization, and diagonally implicit Runge-Kutta methods are considered for time discretization. At each stage of the Runge-Kutta method, the bases computed at the previous stages are augmented to predict the upcoming basis and construct projection subspaces. By projecting onto these enriched subspaces, the bases and cores can be updated in a sequential manner going up the tree from leaf-to-root. Unlike the 3D Tucker method which has a single core tensor, the HT method also updates the intermediate core tensors. Numerical experiments demonstrate that the method observes high-order accuracy, and test how well the integrator captures the solution rank for various sets of time-dependent diffusion coefficients.