生成时间: 2026年07月13日 10:44:13

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: 12

了解领域论文（摘要翻译）

1. 用于热力学一致系统的广义斜梯度嵌入

Generalized skew-gradient embedding for thermodynamically consistent systems

作者: Xuelong Gu, Qi Wang

类别: math.NA

发布日期: 2026-07-10

arXiv ID: 2607.09617v1

链接: http://arxiv.org/abs/2607.09617v1

摘要翻译

中文标题: 用于热力学一致系统的广义斜梯度嵌入

摘要翻译: 斜梯度嵌入（SGE）框架~\cite{GuWangSGE2025} 通过将热力学一致系统的零能量贡献嵌入到反对称算子中，将其重新表述为广义梯度流。在时间离散格式中，定义此算子的轮廓可在以前的时间层上求值。生成的算子仍保持反对称，因此其对离散能量平衡的贡献为零；这种显式处理通常可解耦多物理场系统。我们证明该算子不唯一：容许的规范构成仿射空间，我们将由此产生的族称为广义斜梯度嵌入（GSGE）。对于任何正定度量，最小二乘会选择唯一的极小 Hilbert–Schmidt 规范，且原始度量可恢复 SGE。该构造还给出正则化近似，非中性残差的修正，以及保持指定不变量的规范。对于秩二规范，我们使用一个必要且充分的雅可比准则。将此准则应用于不可压缩 Navier–Stokes 方程的相容 MAC 离散化，可在半离散层面上给出有限维的二秩 Poisson–GENERIC 公式；隐式中点法则在完全离散层面上保持此二秩 GENERIC 结构，并满足精确的离散能量定律。对于 Cahn–Hilliard–Navier–Stokes 系统，正则化的 GSGE–BDF2 格式保持质量，无条件耗散离散能量，并允许解耦实现。

2. 二维欧拉方程矩阵离散的后向误差分析

Backward error analysis for matrix discretizations of 2-D Euler equations

作者: Eugen Bronasco, Klas Modin

类别: math.NA

发布日期: 2026-07-10

arXiv ID: 2607.09549v1

链接: http://arxiv.org/abs/2607.09549v1

摘要翻译

中文标题: 二维欧拉方程矩阵离散的后向误差分析

摘要翻译: 我们引入 Butcher 级数的 Lie–Poisson 约化形式体系. 相应的森林动量映射可用于描述应用于球面上二维欧拉方程的 Zeitlin 矩阵离散的等谱辛 Runge–Kutta 方法的后向误差分析. 基于此, 我们得到修正哈密顿量守恒的指数小误差界, 这些界在指数长的时间区间上有效. 关键在于, 当不同 $n$ 的时间步长按 $h = \mathcal{O}(n^{-1})$ 缩放时, 误差界和时间区间长度与空间离散参数 $n$ (矩阵大小) 无关. 因此, 我们的结果将有限维哈密顿系统的经典后向误差分析结果推广到了经由矩阵流体动力学离散的二维欧拉方程的无穷维情形.

3. 针对扩展 Fisher-Kolmogorov 方程的线性化保结构混合虚拟元方法

A Linearized and structure-preserving mixed virtual element method for the extended Fisher-Kolmogorov equation

作者: Zhen Guan, Xianxian Cao, Houchao Zhang, Junjun Wang

类别: math.NA

发布日期: 2026-07-10

arXiv ID: 2607.09044v1

链接: http://arxiv.org/abs/2607.09044v1

摘要翻译

中文标题: 针对扩展 Fisher-Kolmogorov 方程的线性化保结构混合虚拟元方法

摘要翻译: 在本文中，基于时间上的蛙跳离散和空间上的混合虚拟元离散，我们提出了一种线性化且保结构的数值算法。本工作的主要贡献在于，我们不仅为全离散数值格式的能量耗散性质提供了严格的证明，而且还通过逆不等式建立了无条件最优收敛性分析。证明的核心在于对 (τ) 和 $h$ 之间关系的分类讨论。最后，提供了两个数值例子以验证理论分析的正确性以及所提格式的能量耗散性质。

4. 一种用于 Vlasov-Maxwell 系统的张量列间断 Galerkin 方法

A Tensor-Train Discontinuous Galerkin Method for the Vlasov-Maxwell System

作者: Rujeko Chinomona, Dibyendu Adak, William J. Barham, Duc P. Truong, Nathan V. Roberts, Kim Ø. Rasmussen, Boian S. Alexandrov

类别: math.NA

发布日期: 2026-07-09

arXiv ID: 2607.08936v1

链接: http://arxiv.org/abs/2607.08936v1

摘要翻译

中文标题: 一种用于 Vlasov-Maxwell 系统的张量列间断 Galerkin 方法

摘要翻译: 我们提出了一种用于 Vlasov-Maxwell 系统的张量列间断 Galerkin (TT-DG) 格式，该格式将模态 DG 离散与相空间解和离散算子的低秩张量表示相结合。该格式利用 DG 离散的张量积结构，直接在压缩形式下进行求积、微分、非线性迎风通量计算和时间积分。 该方法在若干标准 1D2V Vlasov-Maxwell 基准问题上进行了评估，包括流动 Weibel 不稳定性、弱 Landau 阻尼和双流不稳定性问题。在这些问题中，TT 格式重现了底层全网格 DG 离散的精度和守恒性质，同时大幅减少了内存使用和运行时间。对于弱非线性问题，获得了超过 $10^4$ 的压缩比，并且相对全网格求解器有显著的加速。对于强非线性双流不稳定性问题，尽管相空间细丝化导致可压缩性降低，TT 格式仍然有效。 这些结果表明，张量列表示为降低基于确定性 DG 的动理学等离子体模拟的计算成本提供了一种有效途径，同时保留了底层离散的良好数值性质。

5. 一种乘法Calderón预条件电场积分方程的显式高阶对偶基

An Explicit Higher-Order Dual Basis for a Multiplicatively Calderón Preconditioned Electric Field Integral Equation

作者: Bernd Hofmann, Thomas F. Eibert, Francesco P. Andriulli, Simon B. Adrian

类别: math.NA, physics.comp-ph

发布日期: 2026-07-09

arXiv ID: 2607.08848v1

链接: http://arxiv.org/abs/2607.08848v1

摘要翻译

中文标题: 一种乘法Calderón预条件电场积分方程的显式高阶对偶基

摘要翻译: 对用Rao-Wilton-Glisson (RWG) 函数离散化的电场积分方程(EFIE)进行预条件的最有效方法之一，是采用Buffa-Christiansen (BC) 函数作为RWG基的对偶基的乘法Calderón预条件子。它产生的形式避免了稠密离散化问题和低频崩溃。为了将乘法Calderón预条件子从低阶BC和RWG基推广到高阶，我们利用基于B样条的基函数，建立了第一个显式的高阶对偶基。它可以看作是BC函数到任意多项式次数的推广，并为其他依赖对偶基的方法构成了基本构件。对所得预条件子的数值结果表明，对于标准和实际的理想导电(PEC)散射体，广义极小残量(GMRES)迭代次数低且恒定，与未知量个数和多项式次数无关；这是充分发挥高阶基潜力的关键。

不相关论文（基本信息）

1. 一条通往偏微分方程的量子途径

A Quantum Path to Partial Differential Equations

作者: Xiantao Li

类别: quant-ph, math.NA

发布日期: 2026-07-10

arXiv ID: 2607.09639v1

链接: http://arxiv.org/abs/2607.09639v1

摘要: Partial differential equations are a promising application area for fault-tolerant quantum algorithms, but the subject lies between two communities with different languages: numerical analysis and quantum computation. These lecture notes provide a numerically grounded introduction for readers entering from either field. Block encoding is the organizing principle: once a discretized differential operator is embedded in a unitary, primitives such as quantum singular value transformation, Hamiltonian simulation, linear combinations of unitaries, amplitude amplification, and measurement can be assembled into algorithms for elliptic, hyperbolic, and parabolic PDEs. Each chapter begins with a standard finite difference or finite element discretization and follows the full pipeline from the continuous PDE to quantum encoding, transformation, and extraction of a quantity of interest. Particular attention is paid to the factors governing end-to-end performance, including discretization error, state preparation, normalization, postselection, and measurement cost. A final chapter introduces nonlinear problems through Carleman and Koopman-von Neumann linearizations. The aim is not a comprehensive survey or a claim of universal quantum advantage, but a mathematically transparent entry point and a shared vocabulary for researchers in both communities.

2. 基于变量投影的多快照脉冲反卷积凸性区域刻画

Characterization of the basin of convexity for multi-snapshot spike deconvolution via variable projection

作者: Meghna Kalra, Maxime Ferreira Da Costa, Kiryung Lee

类别: stat.ML, eess.SP, math.NA, math.OC

发布日期: 2026-07-10

arXiv ID: 2607.09593v1

链接: http://arxiv.org/abs/2607.09593v1

摘要: We study the problem of multi-snapshot spike deconvolution, where the goal is to recover the locations of sparse impulses from their noisy convolution with a known point spread function (PSF) across multiple snapshots. We adopt a variable-projection formulation that eliminates the amplitudes in closed form, reducing the task to a nonconvex least-squares problem over the spike locations alone, which we refer to as the variable-projection formulation of spike deconvolution (VarProSD). We provide an explicit characterization of the basin of convexity of the VarProSD objective in terms of key PSF properties, including its power spectral density and smoothness, revealing how sampling bandwidth and spike separation influence the local geometry. Within this basin, we establish that the estimator is consistent in the number of snapshots under stochastic noise, and provide a complementary, sharper error bound under adversarial noise via the local Lipschitz property of the inverse map. We further show local convergence guarantees for gradient descent when initialized within the basin. A central ingredient throughout is the use of Beurling–Selberg extremal approximations, which enable sharp, PSF-agnostic bounds on the conditioning of the structured matrices arising in the optimization landscape. Numerical experiments validate our theoretical findings and demonstrate the effectiveness of modified ESPRIT initialization followed by gradient-based refinement.

3. 基于变量投影的图正则化低秩矩阵补全

Graph-Regularized Low-Rank Matrix Completion by Variable Projection

作者: Benoît Loucheur, P. -A. Absil, Michel Journée

类别: cs.LG, math.NA, math.OC

发布日期: 2026-07-10

arXiv ID: 2607.09546v1

链接: http://arxiv.org/abs/2607.09546v1

摘要: We address the low-rank matrix completion problem by incorporating graph regularization into the existing Riemannian Trust-Region Matrix Completion (RTRMC) framework. The latter uses the geometry of the low-rank constraint to remodel the problem as an unconstrained optimization problem on a single Grassmann manifold. Our approach, named Graph-Regularized RTRMC (GR-RTRMC), exploits the inherent relationships between rows and columns of the matrix. By using these relationships, we aim to improve the accuracy and robustness of matrix completion, particularly in scenarios where the underlying data exhibits strong correlations between rows or columns.

4. 马尔可夫过程优化的凸松弛

Convex Relaxations for the Optimization of Markov Processes

作者: Hongyi Zhang, Yuehaw Khoo, Tianyun Tang

类别: math.OC, math.NA

发布日期: 2026-07-10

arXiv ID: 2607.09423v1

链接: http://arxiv.org/abs/2607.09423v1

摘要: In this paper, we study the problem of optimizing Markov processes that interpolate between two prescribed probability distributions while minimizing a given cost. The main computational challenge is the curse of dimensionality: in high-dimensional state spaces, representing the full distribution is intractable. To address this, we reformulate the problem in terms of sequential couplings and develop convex relaxations based on local marginals and cluster moments. These relaxations exploit locality and sparse interaction structure, provide computable lower bounds, and recover low-order statistics of the intermediate laws. We identify dynamic optimal transport as a special case of our Markov process optimization problem and develop a procedure for recovering the underlying Benamou–Brenier dynamics from the relaxed solution. We also show that the procedure extends to more general Markov processes and illustrate it with a constrained process between Ising models.

5. 用于时间谐和声波定量被动成像的全互相关反演

Full cross-correlation inversion for quantitative passive imaging with time-harmonic acoustic waves

作者: Jean Dutheil, Florian Faucher

类别: math.NA, math.AP

发布日期: 2026-07-10

arXiv ID: 2607.09392v1

链接: http://arxiv.org/abs/2607.09392v1

摘要: We consider the inverse problem for the quantitative reconstruction of physical properties in the context of passive imaging, where ambient wavefields are used to infer a medium. The data are modeled as a superposition of waves generated by stochastic sources. In this work, we focus on time-harmonic acoustic wave propagation and assume that the stochastic sources exciting the medium are zero-mean and spatially uncorrelated. Under these assumptions, the expected value of the cross-correlation between signals recorded at two locations can be related to the deterministic Green’s function and the covariance of the source terms. We follow a first-order formulation of the wave equation, which enables the treatment of correlations between different types of wavefields. A numerical framework is developed for the resulting nonlinear inverse problem. The quantitative reconstruction is carried out using an iterative minimization scheme, in which the gradient of the misfit functional is computed via the adjoint-state method. Numerical experiments in two and three dimensions are performed using synthetic data, and inversions based on the expected value of cross-correlations are compared with those relying on direct wavefield measurements from active-source acquisitions.

6. 全局延拓作为传统延拓与分岔分析的补充

Global continuation as a complement to traditional continuation and bifurcation analysis

作者: George Datseris, Andreas Morr, Muhammed Fadera, Juergen Kurths

类别: nlin.CD, math.DS, math.NA, physics.app-ph

发布日期: 2026-07-10

arXiv ID: 2607.09332v1

链接: http://arxiv.org/abs/2607.09332v1

摘要: Multistable dynamical systems are ever-prevalent, used to model for example ecosystems, power grids, climate elements, neurons, and more. When perturbed, such systems may ``tip’’ from one state of operation to another, often with abrupt, irreversible, and high-impact consequences in each context. Traditionally, these systems are analysed via bifurcation diagrams, the result of a process we refer to as \emph{local continuation}, as it only captures the linear (local) system response to infinitesimal perturbations. Local continuation requires substantial expertise, constant interventions, and may yield inaccurate assessment of the system’s response to large perturbations that is crucial for tipping analysis. To address some inherent challenges of local continuation and to provide fundamentally new information during a continuation, this paper introduces \emph{global continuation} as a complement suitable for the study of multistability, critical transitions and real-world-oriented applications. Global continuation finds and continues in parallel (practically) all system attractors and their response to finite perturbations by synthesising information from the whole state space, while placing a focus on the qualities or observables of a dynamical system that the practitioner cares about in context. Global continuation does not require deep expertise and is effortless to use and troubleshoot, making it attractive to applied scientists from different disciplines. We highlight several unique advantages that allow global continuation to complement the status quo and exemplify them through a plethora of representative examples. Global continuation is also implemented as open source software in DynamicalSystems.jl, enhancing its accessibility.

7. 用于Ripa模型的五阶良好平衡路径守恒A-WENO格式

Fifth-Order Well-Balanced Path-Conservative A-WENO Scheme for the Ripa Model

作者: Yan-Ping Qiu, Zhen Gao, Alexander Kurganov, Bao-Shan Wang, Xiao Wen

类别: math.NA

发布日期: 2026-07-10

arXiv ID: 2607.09293v1

链接: http://arxiv.org/abs/2607.09293v1

摘要: In this work, we introduce a fifth-order well-balanced (WB) path-conservative A-WENO scheme with the central-upwind numerical fluxes (PCCU-5) for the Ripa model. The proposed scheme is capable of exactly preserving a variety of steady states, including still-water, moving-water, isobaric, and constant water height ones. This goal is achieved with the help of a flux globalization technique: The source terms are incorporated into the fluxes, resulting in a quasi-conservative system, for which central-upwind numerical fluxes are computed using the path-conservative integration. The proposed A-WENO scheme utilizes a WENO interpolation of the equilibrium variables rather than the conservative ones to ensure the WB property. In addition, we perform the WENO interpolation of the local characteristic equilibrium variables to mitigate numerical oscillations near discontinuities. We perform a series of numerical experiments, which demonstrate that the proposed fifth-order WB PCCU-5 scheme achieves high resolution and clearly outperforms its second-order counterpart. Our numerical results also demonstrate the importance of the local characteristic projection for significantly reducing (eliminating) numerical oscillations near discontinuities.

8. 具有混合边界条件的拟线性椭圆问题的最优有限元误差估计与牛顿收敛性

Optimal finite element error estimates and Newton convergence for a quasilinear elliptic problem with mixed boundary conditions

作者: Mihai Bucataru

类别: math.NA

发布日期: 2026-07-10

arXiv ID: 2607.09181v1

链接: http://arxiv.org/abs/2607.09181v1

摘要: The article studies finite element approximations of a quasilinear elliptic heat-conduction problem with inhomogeneous mixed boundary conditions. The conductivity tensor is matrix-valued, anisotropic, possibly nonsymmetric, and dependent on both position and temperature, rendering the problem nonlinear, nonmonotone, and nonpotential. The nonlinear algebraic system arising from the Galerkin discretization is solved using Newton’s method, with a posteriori guarantees provided by a computable Newton-Kantorovich criterion and a mesh-dependent stopping rule that ensures that the algebraic error is asymptotically negligible relative to the discretization error. Since the discrete solution need not be unique, we prove that every discrete solution satisfies the optimal convergence rates. The analysis combines mixed-boundary elliptic regularity with an Aubin-Nitsche duality argument adapted to the quasilinear setting. Numerical experiments in two and three dimensions confirm the predicted convergence rates and demonstrate the feasibility of the proposed criterion.

9. 群不变谱嵌入

Group Invariant Spectral Embedding

作者: Yeari Vigder, Paulina Hoyos, David Thong, Joakim andén, Joe Kileel, Amit Moscovich

类别: cs.LG, math.NA, math.ST

发布日期: 2026-07-09

arXiv ID: 2607.08987v1

链接: http://arxiv.org/abs/2607.08987v1

摘要: Spectral embedding methods are widely used for dimensionality reduction and clustering of high-dimensional datasets with intrinsic low-dimensional structures. Although many datasets of practical interest exhibit invariance under symmetries such as rotations, standard spectral embedding methods do not account for this, treating symmetry-related data points as unrelated. Our approach to this problem is to incorporate the symmetries directly into the affinity kernels used for spectral embedding. We analyze the case of a Riemannian data manifold $M$ with symmetries given by a compact Lie group~$G$ and prove that, under suitable conditions, graph Laplacians constructed from three types of invariant kernels converge pointwise to explicit second-order differential operators on the quotient space $M/G$. Our analysis implies improved convergence rates, as the effective dimension drops according to the dimension of the group. We validate our approach on datasets with $\mathrm{SO}(2)$ or $\mathrm{SO}(3)$ symmetry, and show that $G$-invariant spectral embedding recovers the intrinsic geometry of the data, in contrast to standard spectral embedding, which fails to do so even in the limit of infinite data.

10. 再论多维外推的静态偏微分方程方法

Another look at the static PDE approach to multidimensional extrapolation

作者: Hwi Lee

类别: math.NA

发布日期: 2026-07-09

arXiv ID: 2607.08980v1

链接: http://arxiv.org/abs/2607.08980v1

摘要: We present a variant of the static PDE based extrapolation method (2014 Aslam, SIAM J. Sci. Comp) for extending smooth fields across interfaces implicitly defined as zero level sets. Our approach introduces the idea of relaxed upwinding finite differences as a key modification within the fast sweeping method. Unlike existing approaches, we require function values and their normal derivatives only on the first grid layer inside the domain adjacent to the boundary, thereby improving computational efficiency and flexibility. To further improve accuracy, we introduce a simple boundary reconstruction technique that significantly reduces the numerical error in the extrapolated solution near the boundary. Numerical experiments indicate enhanced performance of our approach across a range of domain geometries, including those with high curvature features.

11. 利用改进的广义高斯求积算法构造与优化一般函数空间的分部求和算子

Construction and Optimization of Summation-by-Parts Operators for General Function Spaces Using an Improved Generalized Gaussian Quadrature Algorithm

作者: Alex Bercik, Lisa Patrascu, David W. Zingg

类别: math.NA

发布日期: 2026-07-09

arXiv ID: 2607.08934v1

链接: http://arxiv.org/abs/2607.08934v1

摘要: We construct optimized summation-by-parts (SBP) operators for general function spaces with provably minimal degrees of freedom on open, closed, and half-open nodal distributions. These operators rely on generalized Gaussian quadrature rules, for which we present an improved algorithm that is flexible, efficient, and provably convergent. In cases where free parameters are available, we further introduce two operator optimization strategies. We test our operators on a handful of numerical examples that contain large or unbounded gradients, in which some a priori knowledge of the solution has been assumed to select an appropriate basis. The novel operators are found to outperform standard polynomial operators by several orders of magnitude in solution accuracy relative to degrees of freedom. Furthermore, our novel operators significantly outperform function-space SBP operators with equispaced nodal distributions, which require significantly more nodes for the same operator basis. Finally, we demonstrate that the operator optimization procedures are critical to achieving accurate and efficient discretizations, as the standard SBP construction procedure can lead to nullspace-inconsistent and poorly-conditioned operators.

12. 色散浅水矩方程的分裂格式

A Splitting Scheme for Dispersive Shallow Moment Equations

作者: Ullika Scholz, Robin Paar, Manuel Torrilhon

类别: physics.flu-dyn, math.AP, math.NA

发布日期: 2026-07-09

arXiv ID: 2607.08870v1

链接: http://arxiv.org/abs/2607.08870v1

摘要: The well-known Shallow Water Equations (SWE) are used for modeling incompressible free-surface flows whenever the shallowness allows for a vertical-averaging; i.e., vertical effects are negligible in comparison to horizontal ones. But vertical averaging comes with the price of losing information along the vertical axis. Moment models for shallow flow contain information on the vertical velocity and pressure profile despite being dimensionally reduced. A class of these models incorporating a non-hydrostatic pressure have been introduced before as Dispersive Shallow Moment Models (DSM). However, no method for solving the non-stationary equations has been presented yet, mainly because it was unclear how to compute the pressure equation in the form of the divergence-free constraint. We rewrite the pressure equations of the DSM models in the form of a Poisson-like problem to enable their solution with a projection-type splitting scheme. For the linear equations, we present the calculations for the generalized model and discuss the non-linear case. We state the first two linear models and the corresponding nonlinear counterparts. Finally, we introduce a hybrid Finite-Volume Finite-Difference method and discuss the non-stationary numerical results for an experiment with periodic boundary and uneven bottom topography.