生成时间: 2026年07月14日 09:32:32
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: 42
了解领域论文（摘要翻译）
1. 物理信息神经网络用于流体动力学中流动测量指标的高分辨率重建
Physics-Informed Neural Networks for the High-Resolution Reconstruction of Flow Measurement Indicators in Fluid Dynamics
作者: Irena Radišić, Raffaele Tirotta, Alberto Zingaro, Stefano Pagani, Luca Dede'
类别: math.NA
发布日期: 2026-07-13
arXiv ID: 2607.11576v1
链接: http://arxiv.org/abs/2607.11576v1
摘要翻译
中文标题: 物理信息神经网络用于流体动力学中流动测量指标的高分辨率重建
摘要翻译: 准确的空间分辨流场测量对于心血管研究和临床实践中血流动力学量的可靠评估至关重要。实验技术，如四维血流磁共振成像（4D flow MRI）、粒子图像测速（PIV）或多普勒超声，常产生稀疏、含噪声或欠分辨的数据，特别是在血管壁附近和复杂流动区域。这限制了分布或导出血流动力学指标（如壁面剪切应力）的保真度以及此类测量的临床效用。为应对这些挑战，我们提出了一个物理信息神经网络（PINN）框架，该框架将不可压缩Navier-Stokes方程与来自实验流场数据的速度测量相结合。通过将物理定律嵌入数据，PINN增强了速度场的重建，能够估计未测量的量（如压力和壁面剪切应力），并提高了血流动力学指标的空间分辨率。我们利用计算机模拟和实验数据展示了该方法的有效性。首先，我们将方法应用于FDA喷嘴基准算例，利用对照粒子图像测速（PIV）测量和计算流体动力学（CFD）模拟。接着，我们利用体外四维血流MRI数据，将方法应用于更复杂的动脉瘤模型中的血流案例。在两种情况下，数据驱动学习与基于物理的正则化之间的协同作用产生的结果比标准CFD或纯数据驱动方法更接近真实观测。我们的发现凸显了PINN在提高欠分辨流场测量的保真度并产生空间分辨血流动力学指标方面的潜力。
2. 基于非标准有限差分离散化的结构保持神经常微分方程
Structure-Preserving Neural ODEs via Nonstandard Finite Difference Discretization
作者: Achraf Zinihi, Matthias Ehrhardt, Moulay Rchid Sidi Ammi
类别: math.NA, physics.comp-ph
发布日期: 2026-07-12
arXiv ID: 2607.10858v1
链接: http://arxiv.org/abs/2607.10858v1
摘要翻译
中文标题: 基于非标准有限差分离散化的结构保持神经常微分方程
摘要翻译: 尽管神经常微分方程（NODE）是学习连续时间动力学的强大框架，它们通常不保持基本的定性性质，例如正性。我们提出了一种基于非标准有限差分（NSFD）离散化的结构保持神经常微分方程框架。学习到的动力学由非负的产生和消耗率参数化，产生一个显式、可微的更新，无缝集成到标准的自动微分流程中。我们证明了所得格式对于任意时间步长无条件保持正性，同时保留一阶相容性。我们概述了一种基于Patankar型离散化的扩展，能够精确保持守恒律。在SIR传染病模型上的数值实验表明，我们的方法生成了物理上有意义的轨迹，在粗离散化下保持稳健，并且在保持学习动力学的定性结构方面优于传统的神经常微分方程。
3. Bernstein-Gelfand-Gelfand (BGG) 构造：代数、几何与分析；第一部分
The Bernstein-Gelfand-Gelfand (BGG) Construction: Algebra, Geometry, and Analysis; Part I
作者: Andreas Cap
类别: math.DG, math.NA
发布日期: 2026-07-12
arXiv ID: 2607.10662v1
链接: http://arxiv.org/abs/2607.10662v1
摘要翻译
中文标题: Bernstein-Gelfand-Gelfand (BGG) 构造：代数、几何与分析；第一部分
摘要翻译: 这些是根据我与Kaibo Hu在维也纳Erwin Schrödinger研究所（ESI）共同教授的短期课程第一部分整理的讲义，该课程是主题项目“微分复形：理论、离散化与应用”的一部分。我们介绍微分形式以及在R^n的开区域上Berstein-Gelfand-Gelfand（BGG）复形的构造，重点阐述半单李群和李代数的表示论在构造中的作用。
4. 波传播的辛哈密顿直接间断伽辽金方法
Symplectic Hamiltonian Direct Discontinuous Galerkin Method for Wave Propagation
作者: Haomiao Li, Yumiao Li, Jiaxin Wang, Tiegang Liu, Kun Wang
类别: math.NA
发布日期: 2026-07-12
arXiv ID: 2607.10652v1
链接: http://arxiv.org/abs/2607.10652v1
摘要翻译
中文标题: 波传播的辛哈密顿直接间断伽辽金方法
摘要翻译: 本文提出了一种辛哈密顿直接间断伽辽金（DDG）方法，用于逼近波传播问题，包括线性和半线性波动方程。在无需辅助变量的 DG 框架下，我们证明数值通量双线性形式的对称性等价于离散哈密顿结构的存在性。由此可得，对称内罚方法和对称 DDG（SDDG）方法等具有离散哈密顿结构，而 Baumann–Oden、DDG 和 BR2 等格式则不具备此性质。利用这一结构，我们将 SDDG 空间离散与辛时间积分器相结合，构造了全离散辛格式。进一步地，我们针对半线性波动方程的 SDDG 方法推导了误差估计，表明位移具有最优收敛阶，而速度具有次优收敛阶。数值实验验证了理论收敛阶，并表明辛哈密顿 DDG 方法在长时间能量守恒和精度方面具有优越性能。
5. 微分神经正切核及其正定性
The Differential Neural Tangent Kernel and Its Positivity
作者: Bangti Jin, Longjun Wu
类别: cs.LG, math.NA, stat.ML
发布日期: 2026-07-11
arXiv ID: 2607.10200v1
链接: http://arxiv.org/abs/2607.10200v1
摘要翻译
中文标题: 微分神经正切核及其正定性
摘要翻译: 神经正切核（NTK）是分析过参数化机制下神经网络训练动力学的有力工具之一。最近，这一理论框架已被扩展至用于求解线性偏微分方程的物理信息神经网络（PINNs），后者是一类非常流行的神经偏微分方程求解器。在分析中，与之相关的NTK的正定性起着基础性作用。然而，由于存在多个微分算子，证明PINNs中NTK的正定性极具挑战性。在本文中，我们提出一个新的理论框架，称为微分神经正切核（DNTK），用于通过NTK的视角分析PINNs，并针对一大类激活函数（包括RePU以及光滑但非多项式的激活函数）和所有线性微分算子，建立了浅层和深层神经网络在无限宽度下DNTK的正定性。这些理论结果为分析训练PINNs的梯度型算法奠定了基础。
不相关论文（基本信息）
1. 先稳定后优化：反馈变换作为最优控制中的预条件子
Stabilize-then-optimize: Feedback transformations as preconditioners in optimal control
作者: Till Preuster, Manuel Schaller, Anton Schiela, Martin Stoll
类别: math.OC, math.NA
发布日期: 2026-07-13
arXiv ID: 2607.11835v1
链接: http://arxiv.org/abs/2607.11835v1
摘要: Many numerical algorithms for optimal control leverage an elimination of the state via the control-to-state map such as condensed approaches or preconditioned conjugate gradient methods for the optimality system. As such, the norm of the control-to-state map directly enters the convergence estimates for these methods, e.g., via the condition number of the associated linear system. In this work we show that using feedback transformations one may reformulate the optimal control problem to decrease the norm of the (feedbacked) control-to-state map, leading to a drastic improvement of the involved condition numbers. We illustrate the abstract approach for ordinary and partial differential equations such as parabolic, hyperbolic or elliptic equations. For each of these problem classes we provide a constructive method to improve solution operator norms via feedbacks. Further, we showcase the efficacy of the method by means of various numerical examples with elliptic, parabolic and hyperbolic partial differential equations.
2. 高维偏微分方程的Inf-Sup神经网络
Inf-Sup Neural Networks for High Dimensional PDEs
作者: Ziren Chen, Hailiang Liu
类别: math.NA, math.AP
发布日期: 2026-07-13
arXiv ID: 2607.11718v1
链接: http://arxiv.org/abs/2607.11718v1
摘要: Solving partial differential equations (PDEs) in high dimensions remains challenging due to the curse of dimensionality. We propose a neural-network-based framework that reformulates PDEs as inf–sup optimization problems through the introduction of a Lagrange multiplier. The primal solution and the associated Lagrange multiplier are parameterized by two networks and are computed via an iterative saddle-point optimization procedure. We prove the theoretical equivalence between the proposed optimization formulation and the original PDE problem, and we derive rigorous error estimates that quantify the total approximation error in terms of the network approximation error, statistical (sampling) error, and optimization error. Numerical experiments demonstrate the accuracy, stability, and efficiency of the proposed method for solving high-dimensional PDEs.
3. 一种用于无约束优化的孪生梯度法
A Twin gradient method for unconstrained optimization
作者: Anna De Magistris, Michiel E. Hochstenbach, Gerardo Toraldo
类别: math.NA, math.OC
发布日期: 2026-07-13
arXiv ID: 2607.11575v1
链接: http://arxiv.org/abs/2607.11575v1
摘要: We propose a new strategy for gradient-based unconstrained optimization, involving two parallel sequences of iterates that cooperate to determine their stepsizes via a \textit{Twin-Step} principle. Rather than minimizing the objective function individually, the algorithm selects steplengths that minimize the Euclidean distance between the two gradient based processes occurring simultaneously at each iteration. The theoretical analysis shows that the convergence of the mutual distance is governed by the angle between the search directions. In particular the effectiveness of the overall process degrades as the directions approach parallelism. To ensure robustness against collinearity, we introduce a hybrid framework, Twin-ABB$_{\min}$, which switches to the Adaptive Barzilai–Borwein method when the geometric cooperation becomes ineffective. Extensive and very promising numerical results evidence that the Twin phase creates favorable initial conditions for subsequent BB-type iterations.
4. Cahn–Hilliard方程的代数不变二次化格式
Algebraic Invariant Quadratization Schemes for Cahn–Hilliard Equations
作者: Fei Xie, Nan Lu, Yajuan Sun
类别: math.NA
发布日期: 2026-07-13
arXiv ID: 2607.11569v1
链接: http://arxiv.org/abs/2607.11569v1
摘要: In this paper, we propose the Algebraic Invariant Quadratization (AIQ) framework for rational-like energy functions by introducing auxiliary variables, which are interpreted as Casimir functions of the extended system. Combining AIQ with symplectic Runge–Kutta (SRK) methods in time and Fourier pseudo-spectral discretization in space, we obtain fully discrete schemes. The resulting schemes are applied to Cahn–Hilliard equations in both the isotropic and anisotropic cases. We analyze the discrete dispersion relation, spinodal instability, coarsening behavior, and missing-orientation phenomena. Numerical comparisons demonstrate the improved performance superiority of the proposed method over the stabilized invariant energy quadratization (S-IEQ) and scalar auxiliary variable (SAV) methods in preserving the original energy evolution and capturing the underlying physical phenomena.
5. 基于截断梯形法则的卷积求积
Convolution quadrature based on a truncated trapezoidal rule
作者: Matteo Ferrari
类别: math.NA
发布日期: 2026-07-13
arXiv ID: 2607.11422v1
链接: http://arxiv.org/abs/2607.11422v1
摘要: We study the truncated trapezoidal rule, a family of A-stable second order multistep methods parametrized by an integer $J \ge 2$, a compromise between BDF2 and the trapezoidal rule. We obtain a closed-form expression for the coefficients that minimize the principal error constant under the A-stability constraint and we derive an explicit formula for the corresponding principal error constant. The latter decreases to the optimal Dahlquist value $1/12$ as $J$ increases, and is strictly smaller than the BDF2 constant for every $J \ge 2$. We apply the truncated trapezoidal rule within the convolution quadrature framework. Its analyticity in a neighbourhood of the closed unit disk yields milder regularity and perturbation requirements than those of the trapezoidal rule, while the error constant can be made arbitrarily close to the optimal one by increasing $J$. Numerical experiments show that convolution quadrature based on the truncated trapezoidal rule remains stable under symbol perturbations, where the trapezoidal rule fails, while achieving a smaller error constant than BDF2.
6. 面向多分量类型多精度浮点运算的无分支融合乘加算法性能评估
Performance evaluation of branch-free fused multiply-add algorithms for multi-component-type multiple-precision floating-point arithmetic
作者: Tomonori Kouya
类别: math.NA, cs.MS
发布日期: 2026-07-13
arXiv ID: 2607.11391v1
链接: http://arxiv.org/abs/2607.11391v1
摘要: High performance in multi-component multiple-precision arithmetic constructed from existing floating-point operations using error-free transformations (EFTs), can be achieved by employing branch-free algorithms that eliminate if-branches. Zhang and Aiken proposed branch-free algorithms for addition and multiplication in double-word(DW), triple-word(TW), and quadruple-word(QW) arithmetic. Among these, we implemented the TW and QW variants, for which substantial performance improvements over existing algorithms can be expected, and demonstrated that they indeed contribute to acceleration. In this paper, we propose new branch-free fused multiply-add operations for DW, TW, and QW arithmetic to achieve further performance improvements, and show through benchmark tests that the proposed operations can enhance performance.
7. 平衡通量估计子驱动的椭圆问题分析感知界面粗化
Analysis-aware interface coarsening for elliptic problems driven by an equilibrated flux estimator
作者: Denise Grappein, Philipp Weder
类别: math.NA
发布日期: 2026-07-13
arXiv ID: 2607.11297v1
链接: http://arxiv.org/abs/2607.11297v1
摘要: We propose an a posteriori error estimator that relies on an equilibrated flux reconstruction to enable analysis-aware interface coarsening decisions in elliptic interface problems. Interface coarsening consists of simplifying an internal interface that separates regions characterized by different physical properties in order to facilitate the meshing process and reduce the total number of degrees of freedom. This process extends the concept of defeaturing, where small features on the boundary of a computational domain are removed. Here, the features lie on an interior interface instead. The focus is on a Laplace problem with a discontinuous diffusion coefficient. The estimator accounts for both the modeling error arising from the interface coarsening and the numerical error from the discrete approximation of the solution to the coarsened-interface problem. It is localized on the mesh elements, and its constants explicitly track the contrast between the diffusion coefficients. The impact of different features can be assessed individually, yielding local contributions that indicate which features to coarsen and which to retain for a given mesh size. The use of an equilibrated flux reconstruction allows us to sharply bound the bulk numerical source of error. We prove the reliability of the estimator and verify it across several numerical examples, including the case where an internal interface is fully removed.
8. 低秩指数积分器的自适应Krylov方法
Adaptive Krylov Methods for Low-Rank Exponential Integrators
作者: Rico Weigel, Tom-Christian Riemer, Martin Stoll
类别: math.NA
发布日期: 2026-07-13
arXiv ID: 2607.11293v1
链接: http://arxiv.org/abs/2607.11293v1
摘要: Differential equations arise in numerous applications, particularly within scientific and technical contexts. Systems of stiff, time-dependent ordinary differential equations constitute the focus of this work. Exponential integrators are designed to solve such equations by integrating the linear part exactly, while simultaneously approximating the nonlinear part through a linear combination of $\varphi$-functions. By utilizing an augmented stiffness matrix, state-of-the-art methods like KIOPS and RK2EXPINT solve the linear part and evaluate linear combinations of $\varphi$-functions for the nonlinear part in a single step, effectively reducing the computational effort to a single matrix exponential evaluation. However, these classical approaches assume that the system is represented by matrices and vectors, potentially not utilizing the underlying high-dimensional structure. Tensors address this limitation and offer significant storage efficiency through well-established decompositions like the Tensor Train (TT) format. This work provides a general framework for solving stiff, time-dependent systems directly within the TT format. Specifically, KIOPS-TT and RK2EXPINT-TT are developed as extensions of the original KIOPS and RK2EXPINT algorithms. This involves reformulating the scheme of explicit exponential Runge-Kutta integrators for tensors and augmenting the stiffness tensor to compute linear combinations of $\varphi$-functions acting on tensors through a single evaluation of the exponential function using Krylov subspace methods. Furthermore, it is shown that the underlying theory of the matrix methods remains valid, thereby enabling the transfer of key theorems to the tensor case. Numerical experiments confirm significant speed-ups for KIOPS-TT and RK2EXPINT-TT in low-rank scenarios compared to their classical counterparts.
9. 约束非线性最小二乘的增广拉格朗日算法
An augmented Lagrangian algorithm for constrained nonlinear least-squares
作者: Pierre Borie, Fabian Bastin, Stéphane Dellacherie
类别: math.OC, math.NA
发布日期: 2026-07-13
arXiv ID: 2607.11239v1
链接: http://arxiv.org/abs/2607.11239v1
摘要: We present an algorithm for solving nonlinear least-squares problems subject to a mix of nonlinear and linear constraints. The nonlinear constraints are handled by reformulating the objective as the augmented Lagrangian function while linear constraints are handled directly. Each iteration consists of approximately solving a linearly constrained problem by means of a gradient projection technique. Our approach also involves a structured approximation of the augmented Lagrangian Hessian. We show global convergence of the method and assess the performance through numerical experiments.
10. 一种基于边的二维形状与图像高阶Zernike矩精确计算公式
An Edge-Based Formulation for the Exact Computation of High-Order Zernike Moments of 2D Shapes and Images
作者: Patrice Koehl, Stephan Tillmann
类别: math.NA
发布日期: 2026-07-13
arXiv ID: 2607.11158v1
链接: http://arxiv.org/abs/2607.11158v1
摘要: Zernike moments are widely used rotation-invariant descriptors for shape and image analysis, but their standard computation relies on a pixel-based quadrature that treats each pixel as a point mass located at its center. This approximation introduces spatial aliasing that increases with moment order, degrading image reconstruction and reducing the discriminative power of high-order moments. We present an edge-based formulation that eliminates this source of error by applying Green’s theorem to transform the two-dimensional area integral defining a Zernike moment into a sum of one-dimensional integrals along image boundaries. The resulting framework applies equally to polygonal shapes, binary images, grayscale images, and color images. We derive recurrence relations for the required radial primitives and show that the transformed edge integrands are polynomial functions, allowing their exact evaluation using Clenshaw–Curtis quadrature. The proposed method computes Zernike moments from polygonal image representations without the spatial discretization errors inherent to conventional pixel-based approaches and remains computationally practical for high-order moments. Numerical experiments on image reconstruction, shape analysis, and character classification demonstrate that the proposed formulation matches the accuracy of classical methods at low orders while remaining stable at orders for which pixel-based moments suffer from significant aliasing and numerical degradation.
11. 基于约束 Kolmogorov–Arnold 网络的积分微分方程中记忆与非局部核的神经发现
Neural Discovery of Memory and Nonlocal Kernels in Integro-Differential Equations with Constrained Kolmogorov–Arnold Networks
作者: Aruzhan Tleubek, Salah A Faroughi
类别: cs.LG, math.NA
发布日期: 2026-07-13
arXiv ID: 2607.11110v1
链接: http://arxiv.org/abs/2607.11110v1
摘要: Discovering the memory or nonlocal kernel governing an integro-differential equation (IDE) from sparse and noisy observations is an ill-posed inverse problem. Existing identification methods often rely on problem-specific analytical derivations, specialized observation requirements, or restrictive assumptions about the kernel, limiting their applicability across different classes of IDEs. In this work, we propose a differentiable-solver-based framework for discovering memory and nonlocal kernels directly from spatiotemporal observations. Within the solver, the unknown kernel is represented using a constrained Kolmogorov–Arnold Network (KAN) parameterization, with the physical constraints imposed through two different approaches: a Bernstein-polynomial-based Monotone–Convex KAN (MC-KAN), whose coefficient constraints enforce positivity, monotonic decrease, and convexity by construction, and a Chebyshev-based KAN (Cheb-KAN), in which the same properties are encouraged through soft penalty terms. After training, symbolic regression is applied to the learned kernels to obtain interpretable closed-form representations. We evaluate both methods on benchmarks spanning a one-dimensional Volterra equation, a one-dimensional viscoelastic wave partial integro-differential equation, and a two-dimensional nonlocal reaction-diffusion equation with an anisotropic coupled kernel. For the 1D problems, both methods recover the correct kernel functional form and achieve comparable solution-reconstruction accuracy. In contrast, for the sparse and noisy 2D nonlocal problem, the hard-constrained MC-KAN consistently achieves lower kernel reconstruction errors than the soft-constrained Cheb-KAN. Our results demonstrate that enforcing physically motivated shape constraints by construction provides greater robustness than soft penalties for multidimensional kernel discovery from sparse and noisy observations.
12. 混合精度龙格-库塔方法中稳定修正的性能评估
Performance Evaluation of Stabilized Corrections for Mixed Precision Runge–Kutta Methods
作者: César Herrera, John Driscoll, Sigal Gottlieb, Zachary J. Grant, Tej Sai Kakumanu, Andrew Christlieb
类别: math.NA
发布日期: 2026-07-13
arXiv ID: 2607.10967v1
链接: http://arxiv.org/abs/2607.10967v1
摘要: Mixed precision Runge–Kutta methods reduce the cost of the expensive implicit solves in diagonally implicit Runge–Kutta (DIRK) schemes by evaluating them in low precision, while retaining the accuracy of the scheme for larger time steps. The accuracy lost to the low-precision perturbation can be recovered through inexpensive explicit corrections; however, these corrections have an adverse impact on stability. Recently proposed stabilized corrections remedy this by applying a stabilization matrix to the correction step, but their runtime cost has not previously been quantified. In this work, we present a numerical study of the runtime performance of these stabilized corrections. Using spectral semi-discretizations of two nonlinear partial differential equations, the inviscid Burgers’ equation and the porous medium equation, we compare uncorrected mixed precision DIRK methods against explicitly corrected and stabilized variants across half, single, double, and quadruple precision pairings, for SDIRK methods of orders two through four. We report convergence, runtime, and speedups, and show that the stabilized corrections improve the accuracy of the mixed precision schemes while preserving substantial runtime savings. All experiments were performed on an Intel Xeon Platinum 8480+ CPU with Julia version 1.11.4.
13. 奇异空间：一种用于信号表示的生成扩散框架
The Singularity Space: A Generative Diffusion Framework for Signal Representation
作者: Eli Bar-Yosef, Amir Averbuch, Eli Turkel
类别: cs.LG, cs.AI, eess.SP, math.NA
发布日期: 2026-07-12
arXiv ID: 2607.10930v1
链接: http://arxiv.org/abs/2607.10930v1
摘要: Generative models often represent signals as dense grids of amplitudes, blurring sharp transients that are crucial for the correctness of physical signals. We introduce Singularity Space, a generative framework that represents signals through complex-plane singularities, rooted in the classical pole-residue representation of meromorphic functions. We learn a latent space of physically constrained, per-signal singularity configurations to solve an inverse problem from degraded or partial observations. The framework has three key properties: interpretability, in which each generated singularity configuration corresponds to a set of physical parameters; structural stability, which mitigates Gibbs artifacts at discontinuities; and resolution-free output reconstruction on arbitrary grids without retraining or interpolation. Our framework employs a transformer-based diffusion model that directly predicts samples at complex-plane singularity coordinates, subject to geometric constraints during sampling. As a controlled test case for sharp-feature recovery, we evaluate our framework on 1D Burgers shocks, where each shock is represented by 32 predicted singularities (an $8\times$ reduction versus a 1024-point grid signal). Our framework preserves signal structure ($\text{TV ratio} \approx 1$) under unseen test-time observation noise, achieves a $4.2\times$ lower reconstruction error in zero-shot sub-resolution generalization than a grid-based baseline, and recovers physical parameters to $10^{-4}$ absolute error in-distribution. These results suggest that singularity-based representations may provide a practical foundation for other transient-dominated signals such as speech and biomedical signals, with potential extension to higher-dimensional domains.
14. 基于分裂的自监督稀疏视角CT重建中的设计选择
Design Choices in Splitting-Based Self-Supervised Sparse-View CT Reconstruction
作者: Nadja Gruber, Lukas Neumann, Ander Biguri, Gyeongha Hwang, Markus Haltmeier, Johannes Schwab
类别: cs.CV, math.NA
发布日期: 2026-07-12
arXiv ID: 2607.10898v1
链接: http://arxiv.org/abs/2607.10898v1
摘要: Self-supervised data splitting has emerged as a promising paradigm for sparse-view CT reconstruction, enabling training from incomplete measurements without fully sampled ground truth. However, the influence of key design choices, including partitioning strategy, preprocessing, and inference, remains insufficiently understood. In this work, we introduce a unified framework that decomposes splitting-based reconstruction into these three components, enabling controlled comparison of existing methods and two incremental extensions: multi-partition splitting and an alternative inference strategy. Experiments on simulated LoDoPaB-CT data under independent and correlated noise, together with validation on the real-world 2DeteCT dataset, show that the optimal partitioning strategy strongly depends on the measurement noise structure. Lattice-based splitting performs favorably under independent noise, whereas angular masking is more robust under correlated noise and real measured data. Multi-partition splitting consistently improves over pure projection-wise splitting in several settings. Complementary perceptual and structural metrics, including LPIPS and HaarPSI, reveal differences between masking strategies that are less apparent from PSNR and SSIM alone. These results provide practical guidelines for designing self-supervised sparse-view CT reconstruction methods and highlight the limitations of common independence assumptions in realistic imaging environments.
15. 图像去模糊中凸优化方法的多层预处理策略
Multilevel Preconditioning Strategies for Convex Optimization Methods in Image Deblurring
作者: Stefano Aleotti, Claudia Binda, Marco Donatelli, Rolf Krause
类别: math.NA
发布日期: 2026-07-12
arXiv ID: 2607.10864v1
链接: http://arxiv.org/abs/2607.10864v1
摘要: Proximal gradient methods are widely used in imaging, and their speed of convergence can be accelerated by incorporating variable metrics and/or extrapolation steps. Recent works have shown that preconditioning strategies can significantly enhance this acceleration, in particular, for image deblurring problems. In parallel, a multilevel framework has been introduced to speed up inertial and inexact forward-backward schemes for image restoration problems. In this paper, we combine preconditioning and multilevel strategies to design a robust and consistent acceleration framework for both standard and inexact forward-backward schemes applied to regularized convex optimization problems. Numerical experiments in image deblurring confirm that our approach yields a substantial improvement in convergence speed compared to standard methods.
16. 用最小二乘弱伽辽金方法求解斯托克斯方程
Solving the Stokes Equations via a Least Squares Weak Galerkin Method
作者: Chunmei Wang, Shangyou Zhang
类别: math.NA
发布日期: 2026-07-12
arXiv ID: 2607.10831v1
链接: http://arxiv.org/abs/2607.10831v1
摘要: We present a least-squares weak Galerkin (LS-WG) finite element method for solving the Stokes equations on arbitrary polygonal and polyhedral meshes. By utilizing discrete weak derivatives on discontinuous polynomial spaces, the proposed framework naturally accommodates complex domain geometries and general partitions. Crucially, this least-squares formulation bypasses the traditional inf-sup (LBB) compatibility condition, transforming the standard indefinite saddle-point problem into an inherently symmetric and positive definite (SPD) discrete linear system. We establish the well-posedness of the numerical scheme and rigorously derive optimal-order error estimates in a custom discrete energy norm. Specifically, we prove convergence rates of $\mathcal{O}(h^k)$ for the discrete projection error and $\mathcal{O}(h^{k-1})$ for the global approximation error when employing polynomials of degree $k \ge 1$ for the velocity field and $k-1$ for the pressure. Extensive numerical experiments confirm these theoretical convergence rates, demonstrating the method’s robustness, geometric flexibility, and overall efficiency.
17. 一个用于双曲守恒律的紧致简单HWENO格式与ADER时间离散 II：三角网格
A compact simple HWENO scheme with ADER time discretization for hyperbolic conservation laws II: triangular meshes
作者: Dongmi Luo, Zhuang Zhao, Jianxian Qiu, Yibing Chen
类别: math.NA
发布日期: 2026-07-12
arXiv ID: 2607.10809v1
链接: http://arxiv.org/abs/2607.10809v1
摘要: A compact and high order HWENO scheme using ADER (Arbitrary high order using DERivatives) time discretization is developed for hyperbolic conservation laws on the triangular mesh, which is the extension of the work on the structured mesh (Luo et. al. (2024) \cite{luo2023}). The Lax-Wendroff procedure is employed to convert time derivatives to spatial derivatives. Thanks to this, the cell averages of the derivatives of the solution can be obtained by the time accurate solution as Gaussian points along the cell interfaces through the Green-Gauss theorem instead of by the evolution solution directly in the conventional HWENO methods. Comparing with the existing Runge-Kutta HWENO (RK-HWENO) method on the unstructured mesh (Zhao et. al. (2025) \cite{zhao2025}), the new method has the following advantages. Firstly, the RK-HWENO method must solve the additional equations for reconstructions and time advancing, which is avoided for the new method. Secondly, the HWENO reconstruction in the new method is performed once per time step and is different from the RK-HWENO method, in which the reconstruction is performed several times every time step. Because of these advantages the new method is more efficient than the RK-HWENO method with smaller numerical errors and less computational costs. Besides, comparing with the existing ADER-WENO methods \cite{dumbser20071,dumbser20072} under the same order of accuracy, the stencil of the new method is more compact since the both the function and its first derivative values are used in the reconstruction of the HWENO schemes. Numerical examples demonstrate that the new method can achieve the high order for smooth solutions both in space and time, keep non-oscillatory near discontinuities.
18. 分层贝叶斯求积
Hierarchical Bayesian Quadrature
作者: Tim Weiland, Toni Karvonen, Philipp Hennig
类别: cs.LG, math.NA
发布日期: 2026-07-12
arXiv ID: 2607.10793v1
链接: http://arxiv.org/abs/2607.10793v1
摘要: Numerical integration is a cornerstone of various scientific computing applications, such as engineering simulations and model evidence computations in probabilistic machine learning. Bayesian Quadrature uses Gaussian process surrogates that explicitly encode structural assumptions about the integrand to obtain integral estimates with quantified uncertainty. These surrogates are predominantly based on stationary covariance functions, which results in model misspecification for integrands exhibiting nonstationary behavior. We tackle this issue through an adaptively growing, tree-based partition of the integration domain into local stationary models. Our method recombines the local integral estimates through a hierarchy of GP conditioning that reintroduces cross-subdomain correlations, while model selection criteria control the tree growth to avoid unnecessary partitioning. The resulting algorithm is simple, requires no MCMC, and adapts its evaluation budget to local integrand complexity. On benchmark integration problems and a model evidence computation for an epidemiological model, Hierarchical Bayesian Quadrature achieves substantial gains over standard Bayesian Quadrature on nonstationary integrands while matching its performance on stationary ones.
19. Cahn-Hilliard 时间步的四场辅助重构：分析与协调有限元离散化
A four-field auxiliary reformulation of a Cahn-Hilliard time step: Analysis and conforming finite element discretization
作者: Marvin Fritz
类别: math.NA, math.AP
发布日期: 2026-07-12
arXiv ID: 2607.10724v1
链接: http://arxiv.org/abs/2607.10724v1
摘要: We propose a four-field auxiliary reformulation of a time-discrete Cahn-Hilliard step. The construction is motivated by the scalar trace structure underlying two-dimensional Rafetseder-Zulehner decompositions and represents the scalar quantity $-Δc$ in the form $2p+div\,u$ through an auxiliary scalar field $p$ and an auxiliary vector field $u$. The resulting auxiliary problem is a mixed second-order Stokes/elasticity-type system, while the evolution equation for the phase field retains its standard mass-conserving gradient-flow structure. We derive a continuous four-field formulation that is equivalent to the classical convex-splitting mixed time step. We also state a conforming finite element discretization and prove one-step spatial estimates for the phase-field variables together with a stable auxiliary block estimate containing an explicit weak-Laplacian recovery defect. The numerical experiments verify the expected phase-field convergence in a manufactured setting, mass conservation, energy decay, and projected consistency of the auxiliary block.
20. JD-V：一种面向大型Hermitian特征值问题的Jacobi–Davidson方法新变体
JD-V: A new variant of Jacobi–Davidson method for large Hermitian eigenproblems
作者: Jinzhi Huang
类别: math.NA
发布日期: 2026-07-12
arXiv ID: 2607.10635v1
链接: http://arxiv.org/abs/2607.10635v1
摘要: A novel variant of the Jacobi-Davidson (JD) type method for Hermitian eigenvalue problems, designated as JD-V, is proposed based on a newly designed correction equation, whose solution is shown to be nearly as effective as that of the standard correction equation for subspace expansion. Rigorous convergence analysis of MINRES for solving these equations reveals that the inner iterations of JD-V are significantly more efficient than those of the standard JD method when highly clustered eigenvalues are of interest. A thick-restart JD-V algorithm with deflation and purgation is developed to compute several eigenpairs of a a large-scale Hermitian matrix. Numerical experiments confirm the theoretical results and demonstrate the considerable superiority of JD-V over standard JD in overall efficiency.
21. 可调节深度和宽度的ReLU神经网络对解析函数的逼近
Approximation of Analytic Functions by ReLU Neural Networks with Adjustable Depth and Width
作者: Yanming Lai, Defeng Sun, Yang Wang
类别: stat.ML, cs.IT, cs.LG, math.NA
发布日期: 2026-07-12
arXiv ID: 2607.10589v1
链接: http://arxiv.org/abs/2607.10589v1
摘要: In contrast to most studies on neural network approximation theory that characterize results through a single parameter, such as the total number of network parameters, \cite{shen2020deep} pioneered the characterization of approximation rates as a joint function of the width parameter $N$ and the depth parameter $L$, thereby granting greater architectural flexibility. Existing works using the $(N,L)$-characterization focus on function classes with finite smoothness $s$, establishing a typical approximation rate of $\mathcal{O}\left(N^{-2s/d}L^{-2s/d}\right)$ with $d$ denoting the input dimension, which indicates that network depth and width play symmetric roles for these classes. In contrast, this paper establishes upper bounds for the approximation of analytic functions, which possess infinite smoothness, via ReLU networks under the $(N,L)$-characterization. Specifically, we derive approximation rates of $\mathcal{O}\left(N^{-C L^τ}\right)$, where $C>0$ is some constant and $τ>0$ is a parameter influenced by the relation between $L$ and $N$. In particular, $τ=1$ if $N$ scales roughly as $L^d$. Our findings reveal that depth plays a more critical role than width in the context of analytic function approximation. The main technical difficulty of obtaining such upper bounds lies in the trade-off between the smoothness parameters and the approximation accuracy. To overcome this difficulty, we employ refined constructions of several ReLU networks to approximate power functions, multivariate multiplication, and polynomials, which may be of independent interest.
22. 内参数标定扰动下椎骨 DRR 的投影域灵敏度分析
Projection-Domain Sensitivity Analysis of Vertebral DRRs Under Intrinsic Calibration Perturbation
作者: Lin Li, Chaochao Zhou, Benjamin Aubert, Junlin Guo, Junchao Zhu
类别: eess.IV, cs.CV, cs.LG, math.NA, physics.comp-ph
发布日期: 2026-07-12
arXiv ID: 2607.10551v1
链接: http://arxiv.org/abs/2607.10551v1
摘要: Accurate geometric calibration is essential for fluoroscopy-guided spinal imaging, digitally reconstructed radiograph (DRR) generation, and 2D–3D vertebral registration. Although calibration quality is typically evaluated using reconstruction-based metrics such as reprojection error, its influence on projection-domain consistency remains poorly understood. This study presents a synthetic framework for evaluating how intrinsic calibration perturbations affect vertebral fluoroscopic projections and downstream registration performance. CT-derived vertebral models and controlled cone-beam imaging geometry were used to generate DRRs with both ground-truth and perturbed intrinsic calibration parameters while maintaining identical anatomy and acquisition pose. Projection-domain changes were quantified using anatomical landmark displacement, contour distance, silhouette overlap, image similarity, and landmark-based 2D–3D registration accuracy in anterior–posterior (AP) and lateral (LAT) views. Results show that even small intrinsic calibration perturbations produce measurable changes in vertebral projection geometry, contour morphology, landmark localization, and DRR appearance. Sensitivity is strongly view dependent, with LAT projections exhibiting substantially greater deformation and anatomical displacement than AP projections. These projection inconsistencies also degrade downstream 2D–3D registration, particularly rotational alignment accuracy. The findings demonstrate that projection-domain consistency complements conventional reconstruction-based calibration metrics and provides a practical framework for assessing calibration robustness. This approach may improve the reliability of DRR generation and fluoroscopy-guided vertebral registration in image-guided spinal applications.
23. 用于非各向同性分数阶拉普拉斯方程、亥姆霍兹方程及汤川方程的立方体行走蒙特卡洛模拟
Walk-on-Cubes Monte Carlo Simulation for nonisotropic fractional Laplace, Helmholtz, and Yukawa equations
作者: Antti Rasila, Tommi Sottinen, Yaotong Yuan
类别: math.NA, math.AP, math.PR
发布日期: 2026-07-12
arXiv ID: 2607.10528v1
链接: http://arxiv.org/abs/2607.10528v1
摘要: We study the nonisotropic fractional analogs of Laplace, Helmholtz and Yukawa equations. We provide a Duffin correspondence for the Yukawa equation and a Feynman–Kac reconstruction for the Helmholtz equation. The foundation of our analysis is the fact that the nonisotropic fractional Laplace equation is related to a symmetric $α$-stable Lévy process with independent identically distributed components. By using this relation we provide a Walk-on-Cubes algorithm that simulates the solutions of Helmholtz and Yukawa equations.
24. 稀疏傅里叶和的最优外推界
Optimal Extrapolation Bounds for Sparse Fourier Sums
作者: Ruizhe Zhang
类别: cs.DS, math.CA, math.NA
发布日期: 2026-07-11
arXiv ID: 2607.10501v1
链接: http://arxiv.org/abs/2607.10501v1
摘要: We prove an optimal extrapolation theorem for $k$-sparse Fourier sums over arbitrary real frequencies, without any separation assumption, bounding how large such a sum can be just outside an interval on which its energy is observed. For every $g(t)=\sum_{j=1}^k v_j e^{iλ_jt}$ with $λ_j\in\mathbb R$ and every $x\ge1$,
$$ |g(x)|\le k^{O(1)}\exp(O(k\mathop{\mathrm{arcosh}} x))\|g\|_{L^2[-1,1]} . $$In the endpoint regime, this refines to the explicit bound
$$ |g(1+δ)|\le O(k)\exp(O(k\sqrtδ))\|g\|_{L^2[-1,1]}, \qquad 0\leδ\le1 . $$This improves on the $\exp(O(k^2\log k\cdotδ))$ growth estimate of Chen and Price (ICALP 2019), and the exponential scaling is optimal up to constants and polynomial factors in $k$. As an algorithmic consequence, we improve the cluster-center resolution of Chen–Price’s clustered-frequency recovery algorithm by a factor of $k$, while preserving its sample complexity up to logarithmic factors. We also obtain exterior leverage-score and transfer bounds for sparse Fourier feature spaces, converting in-domain active-regression guarantees into essentially sharp prediction guarantees just outside the sampling interval.
25. 比较样条弯曲单元上的代数求积规则
Comparing algebraic cubature rules on spline curved elements
作者: A. Sommariva, M. Vianello
类别: math.NA
发布日期: 2026-07-11
arXiv ID: 2607.10485v1
链接: http://arxiv.org/abs/2607.10485v1
摘要: We compare four methods for the construction of algebraic cubature rules on planar elements, whose boundary is tracked by splines. The methods, that we have developed over the last two decades, are based on Green theorem together with some cornerstones of polynomial approximation theory: Gaussian quadrature,Tchakaloff theorem, discretized Chebyshev expansion (hyperinterpolation), Fekete-like interpolation. We discuss their advantages and drawbacks in view of the application to curved polytopal element methods. We have also made freely available at a single site the corresponding open-source Matlab codes.
26. 通过多参数特征值问题计算波导中的临界速度
Computing critical velocities in waveguides via multiparameter eigenvalue problems
作者: Hauke Gravenkamp, Xiang Liu
类别: math.NA
发布日期: 2026-07-11
arXiv ID: 2607.10462v1
链接: http://arxiv.org/abs/2607.10462v1
摘要: In waveguide dynamics and moving-load problems (e.g., high-speed trains), critical velocities indicate the onset of strong vibration amplification. In systems that are invariant in the direction of motion, these velocities can be identified from dispersion relations as points where the phase and group velocities of a propagating mode coincide. Finding such points indirectly by tracing dispersion curves can be cumbersome and potentially unreliable for multimodal systems with complex branch interactions. We present a direct method for computing critical velocities in such scenarios, specifically in the context of semi-analytical methods. Starting from a polynomial parameter-dependent eigenvalue problem for the wavenumber-frequency relation, incorporating the additional condition of equal phase and group velocities yields a singular polynomial multiparameter eigenvalue problem that can be linearized and solved using established algorithms. The proposed approach enables the simultaneous computation of all critical points without requiring the tracing of dispersion curves. Its performance is demonstrated by several benchmark problems, confirming the accurate and robust identification of critical velocities.
27. 关于有理Bézier曲线一阶导数的界
On the First Derivative Bounds for Rational Bézier Curves
作者: Mao Shi
类别: math.NA
发布日期: 2026-07-11
arXiv ID: 2607.10425v1
链接: http://arxiv.org/abs/2607.10425v1
摘要: In this paper we investigate sharp upper bounds for the first derivative of rational Bézier curves. A long-standing conjecture posited that the linear bound $\|\mathbf{R}'(t)\| \le nΩ\max\|Δ_i\|$ holds for all degrees. We prove that the bound is indeed valid for $n \leq 6$, thus resolving the last open low-degree case. The problem is reformulated as maximizing a variance-like function over a compact box. Using a block argument we show that optima can only appear on one-dimensional faces, reducing the task to a finite family of polynomial inequalities, which are verified exactly via real quantifier elimination. A notable practical feature is that the bound can be evaluated in linear time with respect to the degree, making it attractive for real-time geometric processing. The same structural analysis illustrates the failure for $n=7$ and outlines how the true worst-case constant can be computed.
28. 基于预解式有理逼近的非线性特征值求解器的收敛性分析
Convergence analysis of a nonlinear eigensolver based on rational approximation of the resolvent
作者: Nian Shao, Yuji Nakatsukasa
类别: math.NA
发布日期: 2026-07-11
arXiv ID: 2607.10377v1
链接: http://arxiv.org/abs/2607.10377v1
摘要: Given a holomorphic matrix-valued function, the poles of its sketched resolvent are generically its eigenvalues. Once a good rational approximation of the sketched resolvent is obtained, the poles of this rational approximation typically lie close to those eigenvalues, thus providing a flexible framework for solving both linear and nonlinear eigenvalue problems. However, the accuracy of the computed eigenvalues is limited and remains poorly understood. This paper analyzes the convergence of this approach and demonstrates the effectiveness of two techniques to improve accuracy: block probing and zooming in. We also establish the backward and forward stability of polefinding for a barycentric rational form via a generalized eigenproblem. Numerical experiments demonstrate the sharpness of our theoretical results.
29. M1辐射传输的双曲型神经闭包
A Hyperbolic Neural Closure for M1 Radiation Transfer
作者: Bongseok Kim, Jiahao Zhang, Johannes Krotz, Dinshaw Balsara, Ryan McClarren, Guang Lin
类别: cs.LG, cs.AI, cs.CE, math.NA
发布日期: 2026-07-11
arXiv ID: 2607.10364v1
链接: http://arxiv.org/abs/2607.10364v1
摘要: In radiation transfer simulations, an M1 method achieves substantial computational savings by replacing the full angular transport equation with a low-order moment system. Because this reduced system is not closed, a closure model is required to represent the unknown higher-order moments using lower-order moments. While machine learning (ML)-based closures can improve accuracy beyond classical analytic closures, unconstrained learned closures may produce non-real characteristic speeds and consequently cause numerical solver breakdown. To guarantee real eigenvalues of the Jacobian associated with ML closures, we propose a hyperbolic neural closure for the M1 radiative transfer system. Rather than directly predicting closure terms, we parameterize the Jacobian through two neural networks: (i) a symmetric matrix network and (ii) a strictly convex entropy network whose Hessian defines a positive definite symmetrizer. These components are combined to yield a Jacobian that is similar to a symmetric matrix, thereby ensuring real eigenvalues. The closure is then reconstructed by numerical integration of the learned Jacobian field along a prescribed integration path. Numerical experiments show that the proposed closure not only achieves higher closure accuracy than classical analytic closures, but also improves solution accuracy and remains stable in discontinuous Galerkin simulations for radiative transfer problems.
30. 素描-重启：矩阵函数求积重启中的随机化素描
Sketch-and-Restart: Randomized Sketching in Quadrature-Based Restarting for Matrix Functions
作者: Stefan Güttel, Jingyu Liu, Lauri Nyman
类别: math.NA
发布日期: 2026-07-11
arXiv ID: 2607.10354v1
链接: http://arxiv.org/abs/2607.10354v1
摘要: We develop a sketch-and-restart framework for computing the action of a matrix function on a vector, $f(A) b$, where $A$ is large, sparse, and non-Hermitian. The framework combines quadrature-based restarting with Arnoldi-like decompositions generated by sketched or truncated Arnoldi processes. Within this framework, we develop two classes of restarted algorithms. The first uses a fixed Krylov subspace dimension and is based either on the sketched Arnoldi process or on a new sketched harmonic Arnoldi process proposed in this work. The second class chooses the Krylov subspace dimension adaptively by running the truncated Arnoldi process until the condition number of the generated basis, estimated from its sketch, exceeds a prescribed threshold. We also establish the convergence of the restarted sketched harmonic Arnoldi method for Stieltjes functions under the assumption that $A$ is positive real. Numerical experiments demonstrate the effectiveness of the proposed framework, including the computational savings achieved through sketching, the storage reduction enabled by adaptive truncation, and the acceleration obtained from thick restarting.
31. 计算伪逆的直接方法
Direct Methods for Calculating PseudoInverses
作者: Jeff Knisley
类别: math.NA
发布日期: 2026-07-11
arXiv ID: 2607.10302v1
链接: http://arxiv.org/abs/2607.10302v1
摘要: The Moore-Penrose pseudoinverse of a matrix can be defined and calculated using its singular value decomposition. There are also direct methods for computing matrix pseudo-inverses (those that avoid eigenvalue computations), but these are often rank-revealing, poorly conditioned, or otherwise limited in practice. In this paper, we demonstrate that direct methods can overcome these limitations. In particular, we reinterpret several existing direct methods and introduce new variations that are appropriate for large scale or sparse multilinear regression applications.
32. 关于“HLL型MHD黎曼求解器中中间状态的能量一致性”中HLLD型格式的数值耗散
Regarding the numerical diffusion of the HLLD-type scheme in “On energy consistency of intermediate states in HLL-type MHD Riemann solvers”
作者: Fan Zhang
类别: math.NA
发布日期: 2026-07-11
arXiv ID: 2607.10293v1
链接: http://arxiv.org/abs/2607.10293v1
摘要: A simple modification to the HLLD-ec scheme is introduced to remove a density oscillation that appears in the Dai-Woodward shock tube problem with a stronger longitudinal magnetic field. The original HLLD-ec scheme was introduced in “On energy consistency of intermediate states in HLL-type MHD Riemann solvers, Journal of Computational Physics, 553 (2026), 114724”, and the authors were not aware of this issue.
33. 退化扩散随机控制的神经反馈近似：误差估计与数值分析
Neural feedback approximation for stochastic control with degenerate diffusions: error estimates and numerical analysis
作者: Olivier Bokanowski, Jean-François Chassagneux, Marco Scaratti, Xavier Warin
类别: math.OC, math.NA
发布日期: 2026-07-11
arXiv ID: 2607.10254v1
链接: http://arxiv.org/abs/2607.10254v1
摘要: We study finite-horizon stochastic optimal control problems and approximate the resulting time-discrete formulation by a direct policy-learning problem over neural-network feedback maps. We prove a quantitative convergence estimate, in an averaged sense, for the error between the time-discrete value and the value induced by an approximately optimized neural policy. The bound separates the approximation of near-optimal feedback policies, the localization of stochastic trajectories on compact sets, and the optimization tolerance in training. The analysis does not require transition-density assumptions and covers possibly degenerate diffusions and deterministic controlled dynamics in a unified framework. Numerical experiments are provided for a degenerate stochastic radial target problem, a Hamilton–Jacobi–Bellman benchmark, and a gas storage problem, illustrating the approach and separating the main error sources: time discretization, restriction to piecewise-constant policies, neural-network approximation, and Monte Carlo evaluation.
34. PnP-IPA：一种可证收敛的即插即用非精确邻近算法用于非凸成像问题
PnP-IPA: A Provably Convergent Plug-and-Play Inexact Proximal Algorithm for Nonconvex Imaging Problems
作者: Cristiano Parenti, Silvia Bonettini, Marco Prato
类别: math.NA, math.OC
发布日期: 2026-07-11
arXiv ID: 2607.10223v1
链接: http://arxiv.org/abs/2607.10223v1
摘要: Plug-and-Play (PnP) methods have emerged as a highly effective paradigm for solving imaging inverse problems by replacing traditional proximity operators of regularization terms with highly expressive deep denoisers. While empirically successful, establishing rigorous convergence guarantees for PnP algorithms remains a major challenge. Existing provable approaches based on the Gradient-Step (GS) denoiser suffer from theoretical and practical limitations, such as restrictive bounds on the regularization parameter, rigid step-size rules, and the inability to handle nonconvex data-fidelity terms. In this paper, we introduce PnP-IPA (Plug-and-Play Inexact Proximal Algorithm), a novel optimization scheme that overcomes these bottlenecks. We propose a new splitting strategy that evaluates the proximal operator of the scaled implicit regularizer inexactly. To enable adaptive step-size selection without exact objective evaluations, we design a novel surrogate merit function that successfully drives an Armijo-like backtracking line-search. Relying on the Kurdyka-Lojasiewicz property, we establish global convergence to a stationary point of the nonconvex objective without imposing any assumption on the regularization parameter. Extensive numerical experiments on image deblurring under both Gaussian and Cauchy noise demonstrate the practical advantages of PnP-IPA. By effectively lifting previous theoretical constraints, our method allows for optimal parameter tuning, yielding state-of-the-art restoration quality and robust convergence even in nonconvex regimes.
35. 关于多元解析函数的切比雪夫多项式逼近的指数收敛性
On exponential convergence of Chebyshev polynomial approximation for multivariate analytic functions
作者: Xinyu Wang, Haiyong Wang
类别: math.NA
发布日期: 2026-07-11
arXiv ID: 2607.10209v1
链接: http://arxiv.org/abs/2607.10209v1
摘要: This paper presents a new analysis of the Chebyshev projection for multivariate analytic functions, drawing on pluripotential theory. It is proved that in any downward closed convex polynomial space, the Chebyshev projection achieves the same exponential convergence rate as the best polynomial approximation. This result enables a precise quantification of the exponential convergence rate of the Chebyshev projection. The analysis is then extended to several related topics, including tensorized Chebyshev interpolation, tensor product Gauss–Legendre quadrature, Padua interpolation and cubature, and Chebyshev-Galerkin method, with the corresponding exponential convergence rate established in each case. Supporting numerical experiments are provided to validate the theoretical results.
36. 基于临近交替线性化最小化的耦合张量-矩阵恢复及其在劳动力技能与小型企业健康估计中的应用
Coupled Tensor-Matrix Recovery via Proximal Alternating Linearized Minimization, with an Application to Workforce Skill and Small-Business Health Estimation
作者: Analee Miranda
类别: math.OC, cs.CY, math.NA
发布日期: 2026-07-11
arXiv ID: 2607.10163v1
链接: http://arxiv.org/abs/2607.10163v1
摘要: We study recovery of a low-rank tensor $\mathcal{T}$ and a low-rank matrix $M$ from sparse, noisy observations. $\mathcal{T}$ and $M$ share one mode. We relax tensor rank using the nuclear norm of the mode-1 unfolding. This unfolding carries the coupling. It also has an exact proximal operator. We couple $\mathcal{T}$ and $M$ through a learned linear operator $G$. We prove a minimizer exists for the ridge-stabilized penalized objective. We prove that a proximal alternating linearized minimization (PALM) scheme converges to a critical point, for the algorithm as implemented, by verifying the hypotheses of a known nonconvex block-coordinate convergence theorem against our objective and identifying which conditions come from this problem’s structure. For the matrix-only sub-problem, we state a proven sampling bound from matrix completion theory. For the coupled problem, we prove a sample-complexity result for a sequential sub-case: a separately-known coupling operator recovers $M$ from $\mathcal{T}$’s recovery accuracy alone, with no observations of $M$ needed. For the fully joint, alternately-estimated case, we state a conjecture and test it empirically, including a low-density regime where coupling does not help. We report multi-seed synthetic experiments with mean and standard deviation across sampling densities, an asymmetric-density experiment, and convergence curves, and we explain why recovery error stays high at low density. We apply the framework to workforce-skill and small-business-health estimation. Every application-specific choice is a proposed design, not a validated result; we have not run the framework on deployed data.
37. 带奇异初值的半线性次扩散方程超越 $L^\infty$ 框架的正则性与高阶时间步进
Regularity and high-order time stepping for semilinear subdiffusion equations with singular initial data beyond the $L^\infty$ framework
作者: Runjie Zhang, Dongling Wang
类别: math.NA
发布日期: 2026-07-11
arXiv ID: 2607.10100v1
链接: http://arxiv.org/abs/2607.10100v1
摘要: This paper aims to analyze a numerical scheme for semilinear subdiffusion problems with singular initial data beyond the $L^\infty$ framework. The main difficulty lies in the stronger singular behavior of the nonlinear term compared with previous analyses. Since the singular initial datum is too rough to guarantee a uniform $L^\infty$ bound for the solution, the usual Lipschitz framework in the base space is no longer sufficient. The analysis must instead be carried out in weaker fractional Sobolev-type spaces, where nonlinear composition is more delicate and the term $f(u(t))$ may exhibit an amplified singularity relative to that of $u(t)$. To overcome this difficulty, we exploit the smoothing properties of the subdiffusion solution operators and formulate suitable nonlinear assumptions in fractional operator spaces. These smoothing estimates allow part of the singularity to be transferred from the nonlinear term to the solution operators, where it can be controlled. Under these assumptions, we establish well-posedness and regularity results for the mild solution and derive a pointwise-in-time error estimate for the exponential convolution quadrature method. Numerical experiments confirm the predicted convergence rates.
38. 基于本征广义分解的参数化稀薄气体流动的渐近保持降阶方法
An asymptotic-preserving reduced-order method for parametrised rarefied gas flow by proper generalised decomposition
作者: Luowei Yin, Wei Su
类别: math.NA, physics.flu-dyn
发布日期: 2026-07-11
arXiv ID: 2607.10085v1
链接: http://arxiv.org/abs/2607.10085v1
摘要: Modelling rarefied gas flow using the Boltzmann equation is vital in many areas. Due to the high dimensionality and coexistence of multiple characteristic scales, conventional solution strategies to this equation incur prohibitively high computational costs and are inadequate for rapid response in engineering design simulations. Based on proper generalised decomposition (PGD), we propose an \textit{a priori}, asymptotic-preserving reduced-order method to solve the high-dimensional, parametrised Shakhov kinetic model equation. The method reduces the original problem to a few low-dimensional problems by formulating separated representations for the low-rank solution, thereby mitigating the curse of dimensionality. To capture the hydrodynamic asymptotics, we incorporated solutions of some synthetic equations into the PGD algorithm. This treatment allows the PGD solver to automatically reduce to a macroscopic solver for the Navier-Stokes equations, whose solution naturally exhibits low-rank structure. By treating the rarefaction parameter as an additional coordinate, a parametrised solution can be computed once and for all over the entire range of rarefaction, enabling fast multiple queries to any points in the parameter space. Numerical examples are presented to demonstrate the capability of the method to simulate rarefied gas flow with certain accuracy and a significant reduction in computational costs.
39. 自适应字典的非线性逼近
Nonlinear approximation with adaptive dictionaries
作者: V. Temlyakov
类别: math.NA, math.FA
发布日期: 2026-07-11
arXiv ID: 2607.10052v1
链接: http://arxiv.org/abs/2607.10052v1
摘要: It is well known that the study of the Kolmogorov widths of a function class, which is the image of the unit ball of the $L_q$ space of an integral operator $J_K$ with the kernel $K$, is closely connected with the study of sparse approximations of the kernel $K$ with respect to the classical bilinear dictionary. Recently, it was discovered that if instead of the Kolmogorov widths we study the errors of optimal linear sampling recovery of the same classes, then we need to study sparse approximations of the kernel $K$ with respect to an adaptive dictionary, which is determined by the kernel $K$. In this paper we study this important problem of nonlinear approximation with respect to an adaptive dictionary. Also, in this paper we continue to develop the following general approach, which is related to the above nonlinear approximation problem. We study asymptotic behavior of the errors of sampling recovery not for an individual smoothness class, how it is usually done, but for the collection of classes, which are defined by integral operators with kernels coming from a given class of functions. Earlier, such approach was realized for the Kolmogorov widths and very recently for the entropy numbers.
40. 弹塑性遗传律的最优历史编码：精确输入与本构近似
Optimal history encoding for elastic-plastic hereditary laws: Sharp input and constitutive approximation
作者: Michael Ortiz, Pablo Pedregal
类别: math.NA
发布日期: 2026-07-10
arXiv ID: 2607.09974v1
链接: http://arxiv.org/abs/2607.09974v1
摘要: We formulate rate-independent elastic-ideally-plastic response directly in hereditary form and study its approximation by finite history surrogates. At the material-point level, the constitutive law is the vector play operator generated by metric projection onto a closed convex elastic domain in stress space. Starting from the closest-point return mapping for step inputs, we pass to absolutely continuous driving histories, for which the constitutive law admits a differential form: the stress remains in the elastic domain and the difference of rates belongs almost everywhere to the normal cone. In this $W^{1,1}$ setting, the hereditary law is causal, contracts variation, and satisfies a $BV$-to-$L^\infty$ stability estimate. We then approximate histories by right-continuous step surrogates with at most $N$ constant pieces. For absolutely continuous inputs, we prove a sharp minimax theorem for input approximation in $L^\infty$, normalized by the $BV$ norm: the optimal encoder is given by equal-variation sampling. For constitutive approximation, the correct vector-valued minimax statement is obtained by allowing the encoder to be material-law aware: it may compress the exact stress history $\mathcal P(π)$ rather than only the driving history $π$. The resulting stress-aware encoder, followed by the same discrete hereditary decoder, gives the sharp value $(2N)^{-1}$ under the natural nondegeneracy assumption $0\in\operatorname{int}C$. The scalar complementary-variable case is also recorded: there the input equal-variation encoder is sharp because the scalar stop/play operator is $L^\infty$-nonexpansive in the complementary variable. The results identify cumulative variation as the natural variable for sampling and compressing both driving and constitutive histories.
41. 带加性噪声的三维纳维-斯托克斯方程的误差分析
Error analysis for 3D Navier–Stokes equations with additive noise
作者: Soumya Ranjan Behera, Dominic Breit, Abhishek Chaudhary
类别: math.NA, math.AP, math.PR
发布日期: 2026-07-10
arXiv ID: 2607.09926v1
链接: http://arxiv.org/abs/2607.09926v1
摘要: We consider the three-dimensional stochastic Navier–Stokes equations with additive stochastic forcing. We study temporal discretizations based on a semi-implicit Euler as well as a Crank-Nicolson scheme. We prove that locally in time the former converges with rate 1 while the latter converges with rate 3/2. These theoretical results are confirmed by extensive numerical simulations. To the best of our knowledge this is the first time that numerical simulations for the three-dimensional stochastic Navier–Stokes equations have been performed.
42. Galinstan中强迫对流绕过等热通量圆柱的流动与传热特性：雷诺数最高至600
Flow and Heat Transfer Characteristics of Forced Convection Past an Isoflux Circular Cylinder in Galinstan for Reynolds Numbers up to 600
作者: Dipjyoti Nath
类别: physics.flu-dyn, math.NA
发布日期: 2026-07-10
arXiv ID: 2607.09922v1
链接: http://arxiv.org/abs/2607.09922v1
摘要: A numerical investigation of steady forced convection heat transfer from an isoflux circular cylinder immersed in the liquid metal Galinstan is presented. The governing streamfunction, vorticity, and energy equations are solved using a fourth-order compact finite difference scheme in cylindrical coordinates (FOCS–CC) coupled with a stable pseudo-time iteration (PTI) technique. The influence of the Reynolds number ($1 \leq Re \leq 600$) on the flow and heat transfer characteristics is systematically investigated for Galinstan with a Prandtl number of $Pr=0.025$. The performance and accuracy of the proposed scheme are first established through grid independence studies and validation against previously published numerical results for the average Nusselt number and total drag coefficient over a range of Reynolds numbers. Excellent agreement with the available literature confirms the reliability and robustness of the present formulation. The effects of Reynolds number ($1\leq Re\leq600$) on the flow and thermal fields are examined through streamline patterns, isotherm distributions, and local Nusselt number variations. The results reveal that increasing the Reynolds number promotes flow separation, enlarges the wake region, intensifies downstream thermal transport, and significantly enhances convective heat transfer from the cylinder surface. Furthermore, a new empirical correlation for the average Nusselt number is proposed for Galinstan fluid over the Reynolds number range $1\leq Re\leq600$, exhibiting excellent agreement with the numerical data with a coefficient of determination of $R^2=0.99939$.