生成时间: 2026年07月10日 10:39:41

重点关注论文数量: 0

了解领域论文数量: 2

不相关论文数量: 16

了解领域论文（摘要翻译）

1. 高斯空间上的神经算子与谱算子代理模型

Neural and Spectral Operator Surrogates on Gaussian Spaces

作者: Carlo Marcati, Mario Marić, Christoph Schwab, Jakob Zech

类别: math.NA

发布日期: 2026-07-09

arXiv ID: 2607.08492v1

链接: http://arxiv.org/abs/2607.08492v1

摘要翻译

中文标题: 高斯空间上的神经算子与谱算子代理模型

摘要翻译: 我们证明了可分Hilbert空间之间全纯映射的有限参数、谱代理模型和神经代理模型的表示率界。这些代理模型具有编码器-逼近器-解码器架构，采用Karhunen-Loéve编码器和框架解码器。我们证明了两类有限参数代理模型的表示率界：i) 通过Wiener多项式混沌展开的N项截断得到的谱代理模型，以及ii) 通过使用深度前馈神经网络、ReLU和RePU激活函数以及一致有界权重逼近参数映射而得到的神经代理模型。我们在输入空间上高斯测度的协方差特征值满足代数衰减假设的条件下进行研究。我们得到了均方误差的收敛速率，并且还在一阶高斯Sobolev空间中得到了收敛速率，以考虑梯度近似中的误差。

2. 非线性对流-扩散中的热力学结构与组合性

Thermodynamic Structure and Composition in Nonlinear Convection-Diffusion

作者: J. J. Segura

类别: math.NA, cond-mat.stat-mech

发布日期: 2026-07-08

arXiv ID: 2607.07829v1

链接: http://arxiv.org/abs/2607.07829v1

摘要翻译

中文标题: 非线性对流-扩散中的热力学结构与组合性

摘要翻译: 非线性对流-扩散系统在输运现象中扮演着核心角色，包括质量传递、热传递、多孔介质输运以及具有源、交换和界面效应的耦合连续介质过程。对此类系统，关键问题常常不仅在于使用何种控制偏微分方程，更在于模型是否能在输运分析中自然出现的操作下保持热力学一致性：限制到子区域、跨界面耦合、在平衡处线性化以及用于计算的离散化。 本文为开放的非线性对流-扩散系统建立了一个连续介质优先的框架，其中热力学一致性被表述为带有非负体耗散和显式边界及源项贡献的自由能平衡。在此设定下，非线性输运系统被定义为由容许状态场、存储泛函、本构通量分解、源和边界端口构成的结构化对象。我们证明，热力学平衡在精确的保结构变换、限制到子区域、在相容区域分解上的局部到全局重构以及开放子系统的功率守恒互连下得到保持。然后，我们推导出经典的线性对流-扩散模型作为平衡态处的切热力学后代，并证明当数值设计尊重热力学结构时，同样的不变性在弱形式、半离散化和全离散时间步进中得以生存。非线性漂移-扩散和多孔介质对流-扩散被用作显式例子。最终的贡献是一个组合输运框架，在该框架中，第二定律在连续介质建模、子系统耦合、线性近似和计算中始终可见。

不相关论文（基本信息）

1. 沿前向扩散的分数准确性不能保证扩散采样的数值稳定性

Score Accuracy Along the Forward Diffusion Does Not Certify Numerical Stability in Diffusion Sampling

作者: Yiwei Zhou

类别: stat.ML, cs.LG, math.NA, math.PR

发布日期: 2026-07-09

arXiv ID: 2607.08757v1

链接: http://arxiv.org/abs/2607.08757v1

摘要: Score matching controls average error under the forward marginals, but a discretized reverse-time sampler evaluates the learned score along its own trajectory. We show that small forward-marginal error does not guarantee numerical stability. We construct a single smooth score field with arbitrarily small forward-marginal $L^2$ error. The learned reverse-time process is nonexplosive, has moments of every order, and can be arbitrarily close to the exact reverse-time process in path-space total variation. Yet its Euler–Maruyama discretizations converge in probability while every positive moment diverges. Thus weak convergence can hold even though every Wasserstein distance $W_p$, $p\ge1$, diverges. The same failure can occur within one fixed finite neural architecture. We construct a family of bounded, globally Lipschitz denoisers for which both the forward-marginal error and the path-space total variation distance tend to zero, while their Euler–Maruyama endpoints diverge in every $W_p$. For compactly supported data, we also give a simple positive result. Projecting the learned denoiser onto a known bounded closed convex set containing the support preserves pointwise accuracy, gives grid-uniform moment bounds, and yields Wasserstein convergence under mild local regularity. Experiments with a small fixed DiT-style network show large growth along rare numerical trajectories and its suppression by denoiser projection, while overall trajectory errors remain small.

2. 高阶非标准多步多级方法的若干性质

Some properties of high-order nonstandard multistep multistage methods

作者: Balint M. Takacs

类别: math.NA

发布日期: 2026-07-09

arXiv ID: 2607.08694v1

链接: http://arxiv.org/abs/2607.08694v1

摘要: In this paper, we introduce nonstandard versions of multistep multistage methods. While proving the convergence of these schemes, we also define nonstandard general linear methods. We show that the nonstandard methods can attain the same order as their standard counterparts while preserving certain qualitative properties (e.g., boundedness) for all positive step sizes. These results are also demonstrated by some numerical experiments.

3. 几何约束梯度流的无约束格式

Unconstrained Scheme for Geometrically Constrained Gradient Flows

作者: Sören Bartels, Lucas Bouck, Christian Palus

类别: math.NA

发布日期: 2026-07-09

arXiv ID: 2607.08577v1

链接: http://arxiv.org/abs/2607.08577v1

摘要: In this paper, we study the approximation of gradient flows of harmonic maps, which serve as model problems for applications in micromagnetics, liquid crystals, and nonlinear plate bending. Harmonic maps are vector fields that are critical points of the Dirichlet energy subject to the constraint that the vector field be unit length pointwise. Most existing time-stepping schemes for gradient flows deal with the constraint by linearizing the unit length constraint at every step, which involves solving for the solution increment in the tangent space of the constraint. These schemes lead to robust control over the violation of the constraint, but require solving degenerate saddle point systems at every step that may be difficult to precondition. In this paper, we propose a scheme that first computes the unconstrained increment and then projects this increment pointwise onto the tangent space. With an additional stabilization, this scheme is energy stable under mild step size restrictions and provides robust control of the unit length constraint violation. Our new scheme only requires the solution of decoupled symmetric positive definite systems at every step, which translates to a large increase in computational efficiency. We also propose a computable a posteriori criterion and a variable time-stepping procedure that guarantee the stability of the scheme. We conclude with computational examples demonstrating the efficacy of the scheme, and present a computational extension of the scheme to nonlinear plate bending.

4. 非线性涡流问题瞬态有限元模拟的迭代法

An Iterative Method for Transient Finite Element Simulations of Non-Linear Eddy Current Problems

作者: Karl Hollaus, Haik Silm

类别: math.NA

发布日期: 2026-07-09

arXiv ID: 2607.08432v1

链接: http://arxiv.org/abs/2607.08432v1

摘要: A method is presented to carry out a transient simulation of eddy current problems with nonlinear materials. Coils are voltage-driven. The magnetic field due to currents in coils are considered by their Biot-Savart-fields. The magnetic vector potential is used in the finite element formulation. The time stepping method is based on implicit Euler. The arising nonlinear equation system is split into two parts, the common finite element system and a circuit equation. Each part is solved separately by Newton’s method. Additionally, a line search is used to solve the nonlinear field equations. Inrush currents and average magnetic flux densities through cross sections of laminates of a nonlinear benchmark problem consisting of a laminated iron core inserted in a cylindrical coil are studied. All details of the numerical benchmark are given to evaluate the presented results. Numerical data describing the performance of the presented method are provided.

5. 任意剖分上对流扩散方程的高阶完全通量格式

High-order complete flux schemes for convection-diffusion equations on arbitrary subdivisions

作者: Peng Yang, Wenyu Lei, Liwei Xu

类别: math.NA

发布日期: 2026-07-09

arXiv ID: 2607.08422v1

链接: http://arxiv.org/abs/2607.08422v1

摘要: We develop a complete flux finite volume method for convection-diffusion equations that works on arbitrary meshes in two and three dimensions and for discrete spaces of any polynomial degree. Unlike standard finite volume discretizations, where the numerical flux is directly approximated from the flux definition, we derive the exact normal flux across each control volume edge/face from the underlying PDE. This exact flux splits naturally into a homogeneous part (the classical Scharfetter-Gummel flux) and an inhomogeneous part expressed via a Green’s function that incorporates the tangential flux and the source term. The resulting formulation is exactly equivalent to the continuous equation and, once the discrete space is chosen, yields high-order schemes without any correction or stabilization. For piecewise linear spaces, the scheme achieves optimal second-order accuracy in convection-dominated regimes and can preserve positivity on moderately coarse meshes. For quadratic spaces, standard finite volume methods, based on the Lagrange elements or B-splines, fail to attain optimal $L^2$ convergence unless the control volume mesh is specially designed. The proposed complete flux scheme, however, always achieves optimal $L^2$ convergence independently of the control volume mesh. Numerical experiments in two and three dimensions confirm the robustness and optimal accuracy of the approach.

6. 非线性薛定谔方程的无降阶分裂法

Splitting methods for nonlinear Schrödinger equation without order reduction

作者: Carlos Arranz-Simón, Begoña Cano

类别: math.NA

发布日期: 2026-07-09

arXiv ID: 2607.08387v1

链接: http://arxiv.org/abs/2607.08387v1

摘要: A technique is provided in this paper to integrate nonlinear Schrödinger equation with time-dependent Dirichlet boundary conditions with high-order Yoshida splittings which are based on Strang method. For that, a modification of Strang method is required in which the linear and stiff part of the equation is integrated with a rational-like version of midpoint rule for which the required boundary values can be calculated without resorting to any differentiation of data. Although Yoshida splitting (with real coefficients) cannot be applied to parabolic problems to obtain order higher than two because of stability, the modified Strang method is also applicable to such type of problems and local order $3$ and global order $2$ are also obtained without differentiation of data.

7. 自适应行选择遇见异步：随机Kaczmarz方法

Adaptive Row Selection Meets Asynchrony in Randomized Kaczmarz

作者: Evan Coleman

类别: cs.DC, math.NA

发布日期: 2026-07-09

arXiv ID: 2607.08313v1

链接: http://arxiv.org/abs/2607.08313v1

摘要: Randomized Kaczmarz is a natural fit for large sparse least-squares and tomographic reconstruction, and adaptive row selection can reduce iteration counts. However, deploying adaptive selection on a shared-memory machine means sampling from a residual that lock-free workers are concurrently modifying, often using stale data. We present the first systematic study of this regime: residual-weighted and greedy Kaczmarz under asynchronous execution, measured across 339 runs on a 96-core node with realized (not injected) delays. Four findings carry directly to practice. (i) Stability is governed by a boundary $\ell^*(T)$ between sampling aggressiveness and thread count; below it, more aggressive sampling is strictly better, so one should tune to just inside the cliff. (ii) Threshold-greedy selection (the standard accelerated rule) is unstable at high thread counts, diverging almost immediately. (iii) Under-relaxation buys back the cliff at a predictable cost, giving a usable safety knob. (iv) Consistent-snapshot reads admit a rare, scheduling-dependent divergence that live (inconsistent) reads never exhibited and that is also cheaper, making inconsistent reads the right default. We validate the implementation against published sequential results and outline the distributed two-level sampler these measurements motivate.

8. 使用伴随方程追踪绝对/对流不稳定性边界

Tracking the boundary between absolute/convective instability using adjoint equations

作者: Yue Xiao, Hui Li, Zijing Ding

类别: physics.flu-dyn, math.NA

发布日期: 2026-07-09

arXiv ID: 2607.08305v1

链接: http://arxiv.org/abs/2607.08305v1

摘要: Determining absolute/convective instability boundaries conventionally requires repeated saddle searches in the complex-wavenumber plane and a subsequent scan of the physical parameter space to locate zero absolute growth. Such nested calculations become costly and sensitive to modal branch association for large non-normal eigenvalue problems. This work develops a direct continuation method for neutral stationary-saddle boundaries of frequency-affine generalised eigenvalue problems. The zero-group-velocity condition is expressed as an adjoint solvability residual and solved together with the direct and adjoint eigenproblems, complex gauge constraints and the neutral-growth condition. The resulting one-dimensional solution manifold in the combined state–parameter space is tracked by scaled pseudo-arclength continuation, allowing parameter folds to be crossed without switching the physical continuation variable. The formulation recovers the analytical Ginzburg–Landau boundary and, for a Gaussian-wake Orr–Sommerfeld problem, agrees with separately formulated finite-difference saddle corrections to approximately $10^{-8}$ in relative critical Reynolds number. Compared with nested complex-wavenumber and parameter-plane saddle scanning, the scanning calculations require $14.0$–$30.6$ times the wall time of the direct adjoint continuation, with the cost increasing as the reconstructed boundary is refined. Application to a coupled Oldroyd–B free-surface film reveals genuine folds of the neutral-saddle manifold and a re-entrant CI–AI–CI boundary geometry for the selected saddle family. The results show that adjoint-augmented pseudo-arclength continuation can replace nested saddle searches and parameter-plane reconstruction by direct and computationally efficient tracking of the neutral boundary itself.

9. 局部拟一致多面体上变系数对流-扩散-反应问题的三维虚单元方法

3D Virtual Element Method for Advection-Diffusion-Reaction Problems with Variable Coefficients on Locally Quasi-Uniform Polytopes

作者: M. Trezzi

类别: math.NA

发布日期: 2026-07-09

arXiv ID: 2607.08237v1

链接: http://arxiv.org/abs/2607.08237v1

摘要: In this paper, we propose and analyze a Continuous Interior Penalty (CIP) stabilized Virtual Element Method (VEM) for three-dimensional advection-diffusion-reaction equations on general polyhedral meshes. While CIP-VEM schemes have been recently explored in a two-dimensional setting, their analysis heavily relies on global mesh quasi-uniformity and constant physical parameters. We overcome these limitations by introducing a novel three-dimensional variant of the Oswald-type quasi-interpolant. This allows us to establish robust, uniform error estimates in the hyperbolic limit under a realistic local quasi-uniformity assumption and variable coefficients. Finally, we provide a comprehensive set of three-dimensional numerical experiments to validate the theoretical convergence rates and demonstrate the absence of non-physical oscillations.

10. 算法 XXXX：有限元自由度变换矩阵的计算

Algorithm XXXX: Computation of finite element degree-of-freedom transformation matrices

作者: Matthew W. Scroggs, Garth N. Wells

类别: math.NA

发布日期: 2026-07-09

arXiv ID: 2607.08172v1

链接: http://arxiv.org/abs/2607.08172v1

摘要: The arithmetic intensity of algorithms for computing finite element operators increases with increasing polynomial degree. This has made high degree methods particularly attractive on modern CPU and GPU architectures, since on these architectures performance at low degree is limited (severely) by the available memory bandwidth and only a very small fraction of the floating point capacity of the processor is used. Higher degree methods can exploit a significantly greater fraction of the available compute power of modern architectures. However, whilst stable methods for computing high-degree finite element bases are well-established, there is no universal and automated algorithm for the efficient construction of the degree-of-freedom map for arbitrary degree elements. We address this with a new algorithm that can be used in computing degree-of-freedom maps for an arbitrary Ciarlet-type finite element using only the element’s definition and properties of the reference cell, and without requiring a specific implementation for each element. This method is implemented in the library Basix, a component of the FEniCSx libraries. As well as allowing vast simplifications of parts of a codebase, the algorithm allows for new elements to be implemented with ease and has allowed us to support user-defined custom elements that a user can create at runtime without requiring the user to input any information about transformations required to construct a degree-of-freedom map.

11. 通过多个代数不变量获得对称正定张量的尖锐谱界

Sharp Spectral Bounds for Symmetric Positive Definite Tensors via Multiple Algebraic Invariants

作者: Hemant Sharma, Snigdhashree Nayak, Ankit Singh

类别: math.OC, math.NA

发布日期: 2026-07-09

arXiv ID: 2607.08113v1

链接: http://arxiv.org/abs/2607.08113v1

摘要: We extend the trace–determinant framework of Nayak, Sharma, and Mishra~\cite{nayak2026} for bounding the H-eigenvalues of symmetric positive definite tensors. First, we replace the Arithmetic–Geometric Mean (AM–GM) relaxation underlying previous bounds by the exact solution of the associated constrained optimization problem, yielding sharp upper and lower bounds that are attained on the admissible spectral variety. Second, we incorporate higher-order power sums as additional spectral invariants and prove a structural theorem showing that any extremizer over a $K$-invariant feasibility region has at most $K$ distinct spectral values. This reduces the problem to a finite collection of low-dimensional polynomial systems and yields a hierarchy of increasingly tight bounds. For the four-invariant case $(T,S,p_3,D)$, we develop a complete theory including solution-count estimates, a multistart Newton algorithm, and sharpness conditions. We also derive closed-form bounds in small dimensions, establish perturbation estimates, and obtain refined Lyapunov region-of-attraction bounds. Numerical experiments for dimensions up to $d=100$ show that the sharp three-invariant bound reduces the median relative overestimation gap from $53\%$ to $6\%$ while maintaining low computational cost. The framework is validated on tensors with real H-spectrum.

12. 基于偏微分方程的贝叶斯反问题的多类型传感器放置

Multi-type Sensor Placement for PDE-based Bayesian Inverse Problems

作者: Steven Maio, Alen Alexanderian, Karina Koval, Ruanui Nicholson

类别: math.NA

发布日期: 2026-07-09

arXiv ID: 2607.08074v1

链接: http://arxiv.org/abs/2607.08074v1

摘要: We address optimal placement of multi-type sensors for Bayesian inverse problems governed by partial differential equations (PDEs). The proposed framework allows for sensors with different accuracies and observation types. We formulate the optimal experimental design (OED) problem as a knapsack-constrained binary optimization problem for maximizing expected information gain (EIG). To approximately solve the resulting optimization problems, we propose a stochastic cost-benefit greedy algorithm, which admits theoretical guarantees for monotone submodular set functions. Specifically, these guarantees apply in the case of linear Gaussian inverse problems with uncorrelated measurement errors, where the EIG admits a convenient closed-form expression. For nonlinear inverse problems, we develop a non-intrusive approach that uses the Bayesian approximation error framework to define an observation model with an error-corrected global linear model. We show that the corresponding approximate EIG is a lower bound for the exact EIG and thus provides a principled surrogate objective for the OED problem. The effectiveness of the proposed methods is demonstrated in two model inverse problems governed by PDEs.

13. 面向反弹性介质散射的非解耦时域直接采样方法

A Non-Decoupled Time-Domain Direct Sampling Method for Inverse Elastic Medium Scattering

作者: Lefu Cai, Hongjie Li, Xianchao Wang

类别: math.NA, math.AP

发布日期: 2026-07-09

arXiv ID: 2607.08067v1

链接: http://arxiv.org/abs/2607.08067v1

摘要: This work is concerned with an inverse medium problem for elastic waves, in which unknown inhomogeneities are reconstructed from time-resolved boundary measurements. We propose a novel time-domain direct sampling method for locating scatterers from a single incident source, without imposing specific assumptions on the temporal profile of the excitation. In particular, the imaging functional introduces a time-shifted correlation strategy that replaces the traditional $P$-$S$ wave decomposition with a travel-time alignment mechanism, thereby enabling direct imaging from the coupled elastic wave field. To analyze the proposed time-domain imaging functional, we employ Parseval’s identity for the Fourier–Laplace transform and reformulate the functional in the frequency domain. By exploiting properties of modified Bessel functions, we characterize the asymptotic behavior of the imaging functional and show that it attains its maximum at the target location, which enables reliable identification of the scatterer. Rigorous theoretical justifications are provided to substantiate the effectiveness of the proposed method. Numerical experiments are also presented to demonstrate its performance and applicability.

14. 直边三角形拟合网格上有限元对光滑边界自然边界条件的离场施加

Off-site enforcement of natural conditions on smooth boundaries for finite elements upon fitted straight-edged triangular meshes

作者: Vitoriano Ruas

类别: math.NA

发布日期: 2026-07-08

arXiv ID: 2607.07963v1

链接: http://arxiv.org/abs/2607.07963v1

摘要: A few decades ago some possible remedies to an inaccurate enforcement of Neumann or Robin conditions prescribed on the boundary of a smooth domain, owing to the approximation of a curved domain by the union of straight-edged triangles or tetrahedra in a fitted mesh, were addressed in the literature. By that time authors such as Barrett and Elliott (1988) advocated the use of elements with a single curved edge or face fitting the true boundary not only at two or three vertexes, but also at additional points on those curves or curved surfaces, so as to define a polynomial surface of a certain type compatible with the theoretical approximation order of the method in use. In this work we adopt a different approach, whose main feature is the use of a fitted mesh consisting of straight-edged elements only. The recovery of lost accuracy due to the domain’s approximation by a polytope is achieved by means of the addition of terms to the bilinear form, which account for natural boundary conditions of the same type to be prescribed on the approximating boundary, though much closer to the true ones. This technique is applied here to the case of triangular Lagrange finite elements, for which we give a rigorous reliability study in the solution of reaction-diffusion equations. Numerical experimentation is supplied in support of the theoretical results.

15. 具有任意三维旋转的旋转非线性薛定谔方程的高阶时间分裂的容许离散线性传播子

Admissible Discrete Linear Propagators for High-Order Time Splittings of Rotational Nonlinear Schrödinger Equations with Arbitrary Three-Dimensional Rotation

作者: Tianqi Zhang, Fei Xue

类别: math.NA

发布日期: 2026-07-08

arXiv ID: 2607.07923v1

链接: http://arxiv.org/abs/2607.07923v1

摘要: We study robust high-order time splittings for nonlinear Schrödinger equations whose linear part is defined by the Laplacian and an arbitrary three-dimensional rotation operator. After Fourier pseudospectral discretization, a continuous exact factorization of the linear flow need not yield a method self-adjoint fixed-grid propagator. For the original stage-wise explicit exact integrator, we identify a quadratic even term in the local logarithm and show that its visibility is state-dependent, so the observed temporal order of accuracy can depend on the initial data. We then formulate fixed-grid admissibility for discrete linear propagators and construct two admissible propagators for arbitrary three-dimensional rotation: a symmetrized explicit exact integrator and a palindromic generalized shear propagator. Both are unitary, first-order consistent, method self-adjoint, and have odd local logarithms. Numerical experiments verify the predicted defect mechanism and demonstrate recovery of the designed second-, fourth-, and sixth-order behavior with the admissible propagators.

16. 基于树的二次双线性系统解表示及其在模型降阶中的结果

Tree-based solution representations for quadratic bilinear systems and their consequences in model order reduction

作者: Martin Redmann

类别: math.OC, math.NA, math.PR

发布日期: 2026-07-08

arXiv ID: 2607.07841v1

链接: http://arxiv.org/abs/2607.07841v1

摘要: We investigate quadratic bilinear systems by developing novel tree-based representations of their solutions. The proposed framework decomposes the solution into a sequence of coupled bilinear subsystems whose components admit explicit expansions indexed by full binary trees. These representations yield sufficient conditions for the existence of global solutions and lead to new output bounds in terms of reachability Gramians. Motivated by these estimates, we introduce time-limited and infinite-horizon reachability and observability Gramians, establish sufficient conditions for their existence, and characterize them through nonlinear matrix equations. The associated Gramians are employed to identify dominant state-spaces and to derive exact reduced-order models obtained by removing Gramian kernels. Building on these results, we develop a balanced truncation method for quadratic bilinear systems and prove an error bound for the reduced-order approximation. The proposed framework provides a unified connection between tree-based solution representations, nonlinear Gramian theory, and balanced truncation for quadratic bilinear systems, closing several theoretical gaps in the analysis of Gramian-based model reduction for this class of systems.