生成时间: 2026年05月18日 06:42:43

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: 14

了解领域论文（摘要翻译）

1. 关于由复形构造复形的 BBG 构造中短与长正合序列的一个注记

A note on short and long exact sequences in the BBG construction of complexes from complexes

作者: Snorre H. Christiansen

类别: math.NA

发布日期: 2026-05-15

arXiv ID: 2605.15933v1

链接: http://arxiv.org/abs/2605.15933v1

摘要翻译

中文标题: 关于由复形构造复形的 BBG 构造中短与长正合序列的一个注记

摘要翻译: 我们首先展示，对偏微分方程数值分析至关重要的某些 Bernstein–Gelfand–Gelfand (BGG) 序列的上同调，如何能通过构造连接上同调群的长正合序列来得到。然后，我们解释如何通过系统地使用复形的短正合序列及其相关联的上同调群长正合序列，将该结果推广到非单射或非满射情形。最后，给出了用谱序列的解释。

2. 磁准静态场的撕裂与互联公式

A Tearing and Interconnecting Formulation for Magneto-Quasi-Statics

作者: Clemens Pechstein

类别: math.NA

发布日期: 2026-05-15

arXiv ID: 2605.15636v1

链接: http://arxiv.org/abs/2605.15636v1

摘要翻译

中文标题: 磁准静态场的撕裂与互联公式

摘要翻译: 本文研究磁准静态场（也称为涡流模型）的撕裂与互联（特殊的非重叠区域分解）公式。仅考虑两个子区域，一个导电，一个绝缘。直接使用树-余树分裂可以消除非导电区域中的某些核分量，但由于界面耦合，大量与界面相关的核函数仍然存在。本文提出的公式通过将空间分裂为梯度场和一个补空间来克服这一问题。在该分裂的一个温和条件下，证明了：(i) 非导电区域中不需要任何梯度部分，因此两个子区域之间没有梯度分量的耦合；(ii) 两个子区域算子均可逆；(iii) 虽然磁矢势在子区域界面上不连续，但相应的磁场全局地属于H(div)。

不相关论文（基本信息）

1. 狼音与大提琴：大型弦乐器中多共振抑制器的建模与设计

The Wolf and the Cello: Modelling and design of multiple resonance suppressors in large string instruments

作者: Simone Cacace, Emiliano Cristiani, Francesca L. Ignoto

类别: math.DS, math.NA, math.OC

发布日期: 2026-05-15

arXiv ID: 2605.16210v1

链接: http://arxiv.org/abs/2605.16210v1

摘要: The wolf note is an acoustic instability that occurs in large bowed string instruments when a strong body resonance interacts with the vibrating string, producing amplitude modulation and loss of tonal control. Various wolf suppressors - tuned mass dampers attached to the string or to the instrument body - are used in practice to mitigate this effect. In this paper, we propose a mathematical model describing the coupled dynamics of a string and a two-dimensional body equipped with one or two wolf suppressors. Both string and body include elastic (second-order) and stiffness (fourth-order) contributions and can be excited either by plucking or bowing. Three performance indicators are introduced: The first one perceives the wolf-tone appearance, the second one quantifies the attenuation of the notes possibly caused by the wolf suppressor, and the third one measures the acoustic fidelity (in terms of spectrum) with respect to the original instrument. The proposed numerical tests give insights about optimal tuning and placement of one or two suppressors, achieving effective wolf-note suppression while preserving as much as possible the global tonal balance.

2. 关于Riesz-Caputo算子下非线性分数阶边值问题的样条配点法收敛性

On the Convergence of a Spline Collocation Method for Nonlinear Fractional Boundary Value Problems with the Riesz-Caputo Operator

作者: Chiara Sorgentone, Enza Pellegrino, Francesca Pitolli

类别: math.NA

发布日期: 2026-05-15

arXiv ID: 2605.16102v1

链接: http://arxiv.org/abs/2605.16102v1

摘要: Fractional boundary value problems are often used to model complex systems and processes characterized by memory effects and anomalous diffusion. In this paper, we consider fractional boundary value problems involving the Riesz-Caputo operator, which is particularly suited for modeling physical phenomena exhibiting symmetric diffusive effects. We provide an integral representation of the solution to prove existence and uniqueness of the fractional differential problem. We introduce a B-spline collocation method to approximate the solution of the problem and provide a convergence analysis, with both theoretical insights and numerical experiments.

3. 保端点奇异性的弱奇异积分方程谱逼近理论

Endpoint-singularity-preserving spectral approximation theory for weakly singular integral equations

作者: Mahmoud A. Zaky

类别: math.NA

发布日期: 2026-05-15

arXiv ID: 2605.15825v1

链接: http://arxiv.org/abs/2605.15825v1

摘要: We introduce a fractional approximation framework for functions with limited regularity near the terminal point. The proposed basis is constructed by composing classical Jacobi polynomials with an endpoint algebraic mapping, thereby incorporating the terminal singular structure directly into the approximation space. The main structural properties of the fractional polynomials are established, including orthogonality relations, derivative identities, and a singular Sturm–Liouville eigenvalue formulation. We then introduce the associated weighted Sobolev spaces and prove projection and Gauss-type interpolation error estimates in weighted norms. Inverse inequalities and weighted Sobolev embedding estimates are also derived. The resulting theory provides a rigorous foundation for high-order spectral and collocation approximations of endpoint-singular and weakly regular problems, including terminal value problems, fractional differential equations, and weakly singular Volterra integral equations.

4. 双曲守恒律方程组的自适应人工反扩散方法

Adaptive Artificial Anti-Diffusion Methods for Hyperbolic Systems of Conservation Laws

作者: Shaoshuai Chu, Igor Kliakhandler, Alexander Kurganov

类别: math.NA

发布日期: 2026-05-15

arXiv ID: 2605.15770v1

链接: http://arxiv.org/abs/2605.15770v1

摘要: We introduce new adaptive artificial anti-diffusion (AAAD) methods for one- and two-dimensional hyperbolic systems of conservation laws. The key idea is to reduce the amount of numerical dissipation present in a given numerical method by adding an anti-diffusion (AD) term acting in linearly degenerate fields only. This way, the resolution of contact waves can be improved without risking oscillations, which may be caused if the AD acts in nonlinear fields as well. The AD coefficients are selected adaptively: they are supposed to be proportional to the mesh size near the contact waves to enhance the resolution and to be very small in the smooth parts of the computed solution to ensure a sufficiently high (formal) order of accuracy there. The proposed AAAD methods are realized using either the second-order central-upwind numerical fluxes or their fifth-order extensions implemented within the alternative weighted essentially non-oscillatory (A-WENO) framework. We test the proposed schemes on a series of benchmarks for the one- and two-dimensional Euler equations of gas dynamics and the obtained results demonstrate the robustness and high resolution of the new AAAD methods.

5. 一类用于单流体双温度欧拉非平衡流体动力学的熵稳定有限差分格式

Entropy stable finite difference schemes for One-Fluid Two-Temperature Euler Non-equilibrium Hydrodynamics

作者: Chetan Singh, Harish Kumar

类别: math.NA

发布日期: 2026-05-15

arXiv ID: 2605.15616v1

链接: http://arxiv.org/abs/2605.15616v1

摘要: In this work, we consider the One-Fluid Two-Temperature Euler (OFTT-Euler) equations used for modeling non-equilibrium hydrodynamics. The model comprises a system of nonlinear hyperbolic partial differential equations with non-conservative products. The model decomposed the total pressure into two scalar components: one for electrons and one for ions. Our aim in this work is to design entropy-stable finite difference numerical schemes for the model. This is achieved by reformulating the equations such that the reformulated non-conservative part does not contribute to the entropy. Then, we design higher-order entropy-conservative numerical schemes by using Tadmor’s relation for the conservative part and higher-order central differences for the non-conservative parts. Finally, we design the entropy-dissipation terms using the entropy-scaled right eigenvectors of the conservative part, thereby deriving the entropy inequality for the entire system. We present several test cases in one and two dimensions to demonstrate the accuracy and stability of the proposed schemes.

6. 基于平均场博弈动力学的走时层析成像

Travel-time tomography from mean field game dynamics

作者: Longqiang Xu, Weishi Yin, Hongyu Liu

类别: math.OC, math.AP, math.NA

发布日期: 2026-05-15

arXiv ID: 2605.15602v1

链接: http://arxiv.org/abs/2605.15602v1

摘要: Travel-time tomography seeks to recover a hidden environment from external measurements generated by propagation through an anomalous region. Standard formulations treat propagation as passive, so the environment influences observations mainly by bending paths or changing travel times. Many collective systems do not operate in that regime: observed arrivals are shaped by strategic motion, congestion, and environmental costs. We formulate this active setting through mean field games, in which the unknown environment enters the running cost through a spatial cost field and observations are read from the resulting population dynamics. This yields three contributions. First, it places propagation, observation generation, and inversion within one PDE-constrained model. Second, it clarifies why the inverse problem differs structurally from passive tomography: kinetic, congestion, and environmental effects are coupled endogenously and appear through space- and time-dependent local weights rather than externally chosen global coefficients. Third, it introduces a two-stage inversion pipeline that combines diffusion-based initialization with full mean field game refinement, and numerical experiments show stable recovery under noise and across multichannel scene families. Taken together, these ingredients establish a foundational framework for a class of inverse problems in which data are generated by optimizing and interacting populations rather than by passive signals. The framework identifies the forward MFG model, admissible observation channels, structural mechanisms, and computational recovery route needed to study active tomography under collective dynamics. Such problems arise, among other settings, in biological transport, vascular flow, and subsurface groundwater dynamics.

7. 不带Lipschitz连续性的大规模单调方程组的谱共轭梯度投影方法

Spectral conjugate gradient projection methods for large-scale monotone equations without Lipschitz continuity

作者: Kabenge Hamiss, Mohammed Alshahrani, Mujahid N. Syed

类别: math.OC, math.NA

发布日期: 2026-05-15

arXiv ID: 2605.15570v1

链接: http://arxiv.org/abs/2605.15570v1

摘要: We introduce two derivative-free projection methods for large-scale systems of nonlinear monotone equations subject to convex constraints. Both methods incorporate an adaptive spectral parameter into established conjugate gradient frameworks: the first generalizes the modified optimal Perry method via an eigenvalue-optimized scaling matrix, and the second generalizes the Hager–Zhang-type conjugate gradient projection method via a spectral Dai–Liao parameter. The resulting search directions satisfy a sufficient descent condition independent of the line search. For the first method, we establish global convergence under monotonicity alone, without requiring Lipschitz continuity of the mapping. For the second, global convergence holds under the standard monotonicity and Lipschitz continuity assumptions. Numerical experiments on 18 test problems across dimensions up to 120{,}000, together with applications to $\ell_1$-regularized signal recovery and regularized logistic regression, confirm the practical effectiveness of the proposed approach.

8. 双周期多层介质中麦克斯韦方程组的基本解法

Method of Fundamental Solutions for Maxwell’s Equations in Bi-Periodic Multilayered Media

作者: Jared Weed, Bowei Wu, Jingfang Huang, Min Hyung Cho

类别: math.NA, physics.comp-ph

发布日期: 2026-05-15

arXiv ID: 2605.15527v1

链接: http://arxiv.org/abs/2605.15527v1

摘要: In this paper, we present an accurate numerical method for the time-harmonic Maxwell’s equations for bi-periodic multilayered media with quasi-periodic incident waves using the Method of Fundamental Solutions in conjunction with a periodization scheme. Following an approach used in acoustic scattering problems, the electric and magnetic fields in each layer are expressed as a sum of near and distant interactions. The near interaction comprises interactions between the unit cell and its nearest neighboring copies, while the distant interaction is approximated by proxy source points placed on spheres surrounding the unit cell. Imposing continuity of tangential components at the layer interface, quasi-periodicity conditions on the walls of the unit cell, and Rayleigh-Bloch expansion for the radiation condition yields a system of equations for the unknown coefficients, which can be solved by Schur complement and a backward-stable solver. The scheme is verified with known solutions and exhibits exponential convergence close to $10^{-14}$ for both single and multiple interfaces. An example with 39 interfaces is presented to demonstrate the solver’s performance. The paper provides promising results for extending this method to a fast and accurate boundary integral equation solver for many cutting-edge applications involving a large number of layers in electromagnetics and optics.

9. 浮力驱动热交换器的一个模型，及其对最优设计的启示

A Model of a Buoyancy-Driven Heat Exchanger, with Implications for Optimal Design

作者: Sylvie Bronsard, Charles S. Peskin

类别: math.NA

发布日期: 2026-05-14

arXiv ID: 2605.15409v1

链接: http://arxiv.org/abs/2605.15409v1

摘要: In this paper, we introduce a model for a buoyancy-driven, air-to-air heat exchanger. This model, derived from first principles, features a conservative boundary condition at inflow based on the compressible Bernoulli equation, and a dissipative boundary condition at outflow based on pressure continuity. We solve for the steady-state behavior numerically and asymptotically, with excellent agreement between the two, and we study the tradeoff between the efficiency and air flow predicted by the model.

10. 基于能量的摊销贝叶斯推断

Amortized Energy-Based Bayesian Inference

作者: Hojjat Kaveh, Ricardo Baptista, Andrew M. Stuart

类别: math.NA, cs.AI

发布日期: 2026-05-14

arXiv ID: 2605.15407v1

链接: http://arxiv.org/abs/2605.15407v1

摘要: We consider amortized Bayesian inference for nonlinear inverse problems in settings where only samples from the joint distribution of parameters and observations are available. Classical methods such as Markov chain Monte Carlo require solving a new inference problem for each observation, which can be computationally prohibitive when inference must be repeated many times. We propose a transport-based approach that learns an observation-dependent map pushing forward a reference measure to approximate the posterior distribution. The map is trained by minimizing an averaged energy-distance objective between the true posterior and the learned pushforward. This formulation is likelihood-free, requiring only joint samples, and avoids density evaluation, invertibility constraints, and Jacobian determinant computations. For function-space inverse problems with Gaussian priors, we parameterize the transport map as the identity plus a perturbation in the Cameron-Martin space of the prior, preserving absolute continuity with respect to the prior. In infinite-dimensional settings, the map is represented using neural operators. We illustrate the method on a finite-dimensional nonlinear inverse problem and two PDE-constrained inverse problems arising in porous medium flow and seismic inversion. The results show that the learned transport captures posterior structure, including multimodality and dominant modes, while enabling fast posterior sampling for new observations.

11. 盈亏复杂度：神经偏微分方程求解器的新视角

Breakeven complexity: A new perspective on neural partial differential equation solvers

作者: Yijing Zhang, Nicholas Roberts, Tanya Marwah, Mikhail Khodak

类别: cs.LG, cs.AI, math.NA, physics.comp-ph

发布日期: 2026-05-14

arXiv ID: 2605.15399v1

链接: http://arxiv.org/abs/2605.15399v1

摘要: Neural surrogate solvers of partial differential equations (PDEs) promise dramatic speedups over numerical methods, especially in scenarios requiring many solves. However, current accuracy-based evaluations do not fully consider two central issues: (1) neural solvers incur substantial up-front costs for data generation, training, and tuning; and (2) classical solvers can also generate low-fidelity solutions at a sufficiently low simulation cost. To explicitly account for these realities and fully incorporate end-to-end costs, we propose an evaluation framework centered on breakeven complexity, a metric that counts the forward solves before a learned solver is cost-effective relative to an error-equivalent traditional solver. To evaluate this measure, we apply scaling laws to determine how much training budget to allocate to data generation and discuss how to achieve smooth error-matching in diverse settings. We evaluate the breakeven complexity of multiple neural PDE solvers on three PDEs on 2D periodic domains from APEBench and a novel benchmark of flows past multiple obstacles generated by the GPU-native PyFR code. Among other findings, our results suggest that neural PDE solvers become more effective as problems get harder in terms of cost, dimension, rollout, physics regime (e.g. higher Reynolds number), etc.

12. 具有刚性Fokker-Planck碰撞的动理学方程的隐式动态张量列车近似

Implicit Dynamical Tensor Train Approximation for Kinetic Equations with Stiff Fokker–Planck Collisions

作者: Geshuo Wang, Jingwei Hu

类别: math.NA, math.AP

发布日期: 2026-05-14

arXiv ID: 2605.15382v1

链接: http://arxiv.org/abs/2605.15382v1

摘要: Low-rank methods for kinetic equations have attracted increasing attention due to their effectiveness in reducing the high dimensionality of phase space. In our previous work [G. Wang & J. Hu, J. Comput. Phys. 558 (2026) 114884], we developed a dynamical low-rank method based on the projector-splitting integrator in tensor-train (TT) format, in which explicit time integration is employed in all substeps. As a result, the method is subject to severe stability constraints in the strongly collisional regimes. In this paper, we consider kinetic equations with the (nonlinear) Fokker–Planck collision operator and develop a dynamical low-rank method that employs implicit or implicit-explicit (IMEX) discretizations in appropriate substeps to overcome stiffness. In these implicit substeps, the resulting equations can be formulated as matrix or tensor Sylvester equations, for which we propose efficient direct solvers by exploiting their underlying structure. The overall computational cost of the proposed method scales linearly with respect to the number of grid points in a single velocity dimension, comparable to that of a fully explicit low-rank scheme. We demonstrate the accuracy and efficiency of the proposed method on several representative kinetic test problems.

13. PDE驱动高维稀有事件估计的提议引导贪心代理精化

Proposal-Guided Greedy Surrogate Refinement for PDE-Driven High-Dimensional Rare-Event Estimation

作者: Zhiwei Gao, George Karniadakis

类别: math.NA, stat.ML

发布日期: 2026-05-14

arXiv ID: 2605.15356v1

链接: http://arxiv.org/abs/2605.15356v1

摘要: Accurate surrogate construction for PDE-driven high-dimensional rare-event simulation is challenging when performance evaluations are expensive. Since a globally accurate surrogate may require many high-fidelity evaluations, adaptive importance sampling provides a natural localization tool: its evolving proposal distribution progressively identifies the failure-relevant region. Motivated by this observation, we propose a surrogate-assisted adaptive importance sampling framework that refines the surrogate locally along the evolving proposal, rather than over the entire input space. The surrogate combines an encoder with a neural network, providing a low-dimensional latent representation for both prediction and sample selection. At each adaptive iteration, candidates drawn from the current proposal are selected by a greedy latent-space rule balancing proximity to the estimated failure boundary and sample diversity. The selected samples are evaluated by the high-fidelity model and used to refine the surrogate, which then guides the subsequent cross-entropy-type adaptive proposal update. We establish one-step proposal stability bounds under local surrogate errors, together with surrogate-induced misclassification and finite-sample estimation error bounds. Numerical experiments on multimodal benchmarks and PDE-driven rare-event problems up to 100 dimensions show that the proposed method achieves accuracy comparable to true-model adaptive importance sampling while requiring substantially fewer high-fidelity evaluations.

14. 非线性算子及其导数的通用逼近

Universal Approximation of Nonlinear Operators and Their Derivatives

作者: Filippo de Feo

类别: cs.LG, cs.AI, math.FA, math.NA, math.OC

发布日期: 2026-05-14

arXiv ID: 2605.15285v1

链接: http://arxiv.org/abs/2605.15285v1

摘要: Derivative-Informed Operator Learning (DIOL), i.e. learning a (nonlinear) operator and its derivatives, is an open research frontier at the foundations of the influential field of Operator Learning (OL). In particular, Universal Approximation Theorems (UATs) of nonlinear operators and their derivatives are foundational open questions and delicate problems in nonlinear functional analysis. In this manuscript, we prove the first UATs of non-linear $k$-times differentiable operators between Banach spaces and their derivatives, uniformly on compact sets and in weighted Sobolev norms for general finite input measures, via OL architectures. Our results are the first complete generalizations of the corresponding influential classical results in [Hornik, 1991] to infinite-dimensional settings and OL. We discuss several open areas where DIOL and our UATs find applications: high-order accuracy in OL, fast constrained optimization in Banach spaces (e.g. optimal control of PDEs, inverse problems) and numerical methods for infinite-dimensional PDEs (e.g. HJB PDEs on Banach spaces from optimal control of PDEs, SPDEs, path-dependent systems, partially observed systems, mean-field control). We parameterize nonlinear operators via Encoder-Decoder Architectures, renowned classes in OL due to their generality, including classical architectures, such as DeepONets, Deep-H-ONets, PCA-Nets. Our results are based on four key features that allow us to prove UATs in full generality: (i) Approximation Properties of Banach spaces. (ii) $k$-times continuous differentiability in the sense of Bastiani (weaker than $k$-times continuous Fréchet differentiability). (iii) Natural compact-open topologies for UA; indeed, we show that UA in standard compact-open topologies induced by operator norms is violated even for Fréchet derivatives. (iv) Construction of novel weighted Sobolev spaces for the UA.