生成时间: 2026年05月13日 06:07:21

重点关注论文数量: 0

了解领域论文数量: 1

不相关论文数量: 17

了解领域论文（摘要翻译）

1. 一种用于带网格正则化的约束平面几何流的稳定化双SAV参数有限元框架

A stabilized dual-SAV parametric finite element framework for constrained planar geometric flows with mesh regularization

作者: Koya Sakakibara

类别: math.NA

发布日期: 2026-05-12

arXiv ID: 2605.11721v1

链接: http://arxiv.org/abs/2605.11721v1

摘要翻译

中文标题: 一种用于带网格正则化的约束平面几何流的稳定化双SAV参数有限元框架

摘要翻译: 约束几何流的参数有限元离散需要同时处理高阶几何刚度、网格退化和非线性全局约束。本文针对平面闭合曲线，提出了一种稳定化的双SAV（标量辅助变量）参数有限元框架。所提公式分别为物理几何能量和人工网格正则化能量引入了独立的辅助变量。网格正则化仅通过投影掉其法向变化而耦合到切向运动，使得网格重新分布改变参数化而不引入人工法向驱动力。基于这种双能量结构，我们构造了一种带零阶稳定化的半隐式冻结度量格式。该格式导出线性空间响应问题，并且满足关于修正几何能量和网格SAV能量的离散耗散估计。通过代数块缩减处理非线性全局约束：在求解少量对称正定响应问题之后，剩余的非线性系统仅涉及几何辅助变量和拉格朗日乘子。对于K个全局约束，这一缩减后的非线性系统具有K+1维；特别地，同时施加面积和长度约束会导致一个三维非线性系统，且其规模与网格顶点数无关。对曲线缩短、保面积曲线缩短、曲线扩散和Helfrich型流进行的数值实验展示了二阶和四阶算例中的修正能量耗散、几何约束施加以及网格质量的改善。

不相关论文（基本信息）

1. 非散度型二阶椭圆方程的最小二乘弱伽辽金方法

A Least-Squares Weak Galerkin Method for Second-Order Elliptic Equations in Non-Divergence Form

作者: Chunmei Wang, Shangyou Zhang

类别: math.NA

发布日期: 2026-05-12

arXiv ID: 2605.12417v1

链接: http://arxiv.org/abs/2605.12417v1

摘要: This article proposes a novel least-squares weak Galerkin (LS-WG) method for second-order elliptic equations in non-divergence form. The approach leverages a locally defined discrete weak Hessian operator constructed within the weak Galerkin framework. A key feature of the resulting algorithm is that it yields a symmetric and positive definite linear system while remaining applicable to general polygonal and polyhedral meshes. We establish optimal-order error estimates for the approximation in a discrete $H^2$-equivalent norm. Finally, comprehensive numerical experiments are presented to validate the theoretical analysis and demonstrate the efficiency and robustness of the method.

2. 扩展采样方法在逆双调和散射中的一种新实现

Novel implementation of the extended sampling method for inverse biharmonic scattering

作者: Isaac Harris, General Ozochiawaeze

类别: math.AP, math.NA

发布日期: 2026-05-12

arXiv ID: 2605.12367v1

链接: http://arxiv.org/abs/2605.12367v1

摘要: This paper considers an inverse shape problem for recovering an unknown clamped obstacle in two dimensions from far–field measurements generated by a single incident wave or just a few incident waves for the biharmonic (flexural) wave equation. Here we will develop a new extended sampling method (ESM) that is derived using the analysis of the well–known factorization method. We will also consider an ESM using both sound–soft and sound–hard sampling disks to identify sampling points where the reference disk intersects the unknown cavity. The use of a sound–hard sampling disk has not been studied in the literature whereas the sound–soft sampling disk has been used in most recent works. Traditionally the ESM seeks to find the location of the scatterer from limited incident directional data. Here, our method acts more like the factorization method to obtain the location as well as the size (and possibly the shape) of the obstacle. We present numerical experiments with synthetic data that demonstrate how effective this new implementation is with respect to noisy data and illustrate the influence of the reference disk radius on the reconstruction.

3. 具有齐次Robin条件的腔体形状重构：基于约束耦合复边界方法与ADMM

Cavity shape reconstruction with a homogeneous Robin condition via a constrained coupled complex boundary method with ADMM

作者: Mustapha Essahraoui, El Mehdi Cherrat, Lekbir Afraites, Julius Fergy Tiongson Rabago

类别: math.NA, math.OC

发布日期: 2026-05-12

arXiv ID: 2605.12202v1

链接: http://arxiv.org/abs/2605.12202v1

摘要: We revisit the problem of identifying an unknown portion of a boundary subject to a Robin condition do, based on a pair of Cauchy data on the accessible part of the boundary. It is known that a single measurement may correspond to infinitely many admissible domains. Nonetheless, numerical strategies based on shape optimization have been shown to yield reasonable reconstructions of the unknown boundary. In this study, we propose a new application of the coupled complex boundary method to address this class of inverse boundary identification problems. The overdetermined problem is reformulated as a complex boundary value problem with a complex Robin condition that couples the Cauchy data on the accessible boundary. The reconstruction is achieved by minimizing a cost functional constructed from the imaginary part of the complex-valued solution. To improve stability with respect to noisy data and initialization, we augment the formulation with inequality constraints through prior admissible bounds on the state, leading to a constrained shape optimization problem. The shape derivative of the complex state and the corresponding shape gradient of the cost functional are derived, and the resulting problem is solved using an alternating direction method of multipliers (ADMM) framework. The proposed approach is implemented using the finite element method and validated through various numerical experiments.

4. 一种面向三维计算机断层成像的基于线搜索的随机梯度方法

A Line–Search–Based Stochastic Gradient Method for 3D Computed Tomography

作者: Tatiana A. Bubba, Elena Morotti, Federica Porta, Valeria Ruggiero, Ilaria Trombini

类别: math.NA

发布日期: 2026-05-12

arXiv ID: 2605.12085v1

链接: http://arxiv.org/abs/2605.12085v1

摘要: We introduce FB-LISA, a forward-backward (FB) generalization of a recently proposed line-search-based stochastic gradient algorithm to address the imaging problem of volumetric reconstruction in Computed Tomography, a substantially high demanding problem, which involves orders of magnitude of data, a high computational burden for forward and backprojection, and memory requirements that push current GPU architectures to their limits. Our formulation employs stochastic mini-batches composed of full 2D projections, preserving the physical structure of the acquisition process while enabling significant speed-ups during early iterations. The resulting method demonstrates how concepts traditionally associated with deep learning can be repurposed to accelerate large-scale inverse problems, without relying on training data or learned priors.

5. 分数阶椭圆问题的有限元与箱方法离散：结合数值积分与质量集中

Finite element and box-method discretizations for fractional elliptic problems with quadrature and mass lumping

作者: Kelvin J. R. Almeida-Sousa, David Bolin, Alexandre B. Simas

类别: math.NA

发布日期: 2026-05-12

arXiv ID: 2605.12082v1

链接: http://arxiv.org/abs/2605.12082v1

摘要: We analyze numerical approximation of the fractional elliptic problem $L^βu=f$, ${β>0}$, where $L$ is a second-order self-adjoint elliptic operator with homogeneous Dirichlet or Neumann boundary conditions. The paper develops a unified conforming piecewise linear framework that covers both the standard finite element discretization and the box-method discretization of fractional powers. The key point is that the discrete fractional operator is defined with respect to an admissible inner product on the trial space. This includes, in particular, the standard $L^{2}$ inner product and the quadrature-based mass-lumped inner product, and we also identify a broader family of admissible inner products interpolating between these two realizations. Within this framework, we show that the mass-lumped choice yields the intrinsic fractional box discretization, namely the one obtained by taking fractional powers of the nonfractional box solution operator. For both the finite element and box-method realizations, we establish error estimates under natural consistency assumptions, making explicit the effect of load quadrature in the box case. The analysis applies directly to practical schemes and is supported by numerical experiments in one and two space dimensions.

6. 高效的TV正则化大规模线性反问题：基于SCD半光滑*牛顿法及其在断层成像中的应用

Efficient TV regularization of large-scale linear inverse problems via the SCD semismooth Newton method with applications in tomography*

作者: Helmut Gfrerer, Simon Hubmer, Stefan Kindermann, Jaakko Kultima, Ronny Ramlau, Tanja Tarvainen

类别: math.NA

发布日期: 2026-05-12

arXiv ID: 2605.12041v1

链接: http://arxiv.org/abs/2605.12041v1

摘要: In this paper, we consider the efficient numerical minimization of Tikhonov functionals resulting from total-variation (TV) regularization of linear inverse problems. Since the TV penalty is non-smooth, this is typically done either via smooth approximations, which are inexact, or using non-smooth optimization techniques, which can often be numerically expensive, in particular for large-scale problems. Here, we present a numerically efficient minimization approach based on the recently proposed semismooth* Newton method, which employs a novel concept of graphical derivatives and exhibits locally superlinear convergence. The proposed approach is specifically tailored to TV regularization, suitable for large-scale inverse problems, and supported by strong mathematical convergence guarantees. Furthermore, we demonstrate its performance on two (large-scale) tomographic imaging problems and compare our results to those obtained via other state-of-the-art TV regularization approaches.

7. 多材料拓扑优化的SiMPL方法

The SiMPL Method for Multi-Material Topology Optimization

作者: Peter Gangl, Brendan Keith, Dohyun Kim, Thomas M. Surowiec

类别: math.NA

发布日期: 2026-05-12

arXiv ID: 2605.11994v1

链接: http://arxiv.org/abs/2605.11994v1

摘要: We introduce an efficient and scalable method for density-based multi-material topology optimization, integrating classical mirror descent techniques with point-wise polytopal design constraints. Such constraints arise naturally in this class of problems, wherein the vertices of convex polytopes correspond to distinct design states, only one of which should be occupied at each point in space. The framework generates a descending sequence of iterates by penalizing the design space around the previous iterate with a generalized distance function tailored to the convex geometry of the $n$-dimensional polytope. This distance function, called a Bregman divergence, smooths the optimization landscape, ensuring that each iterate strictly satisfies the point-wise constraints. Subsequently, global constraints (e.g., bounds on the structural mass) can be enforced easily by solving a small, finite-dimensional dual problem. The resulting method is simple to implement and demonstrates robustness and efficiency when combined with an Armijo-type line search algorithm. We validate the method in structural design problems involving the optimal arrangement of both isotropic and anisotropic materials, as well as magnetic flux optimization in electric motors.

8. Parareal 算法中的优化两步粗传播子

Optimized Two-Step Coarse Propagators in Parareal Algorithms

作者: Guanglian Li, Qingle Lin, Kai Zhang, Zhi Zhou

类别: math.NA

发布日期: 2026-05-12

arXiv ID: 2605.11979v1

链接: http://arxiv.org/abs/2605.11979v1

摘要: In this work, we propose a novel framework for accelerating the parareal algorithm, in which the coarse propagator is formulated as a two-step method and optimized with respect to the convergence factor.} We derive a rigorous error estimate for the proposed two-step parareal algorithm, yielding an explicit bound on the linear convergence factor. This estimate is not only of theoretical interest: it provides a quantitative guideline for selecting and designing coarse propagators. Guided by this estimate, we {consider the linear parabolic equation as an illustrative example and }construct an optimized two-step coarse propagator~(O2CP) that delivers very fast convergence in practice. The resulting method attains an optimized convergence factor of approximately $0.0064$, substantially smaller than that of commonly used practical coarse propagators in the classical parareal setting, while keeping the computational cost moderate. Numerical experiments on linear and nonlinear parabolic equations fully support the theoretical analysis and demonstrate rapid convergence of the two-step parareal algorithm equipped with the O2CP.

9. 用于工业聚合物混合的流体体积浸入边界法

A Volume of Fluid Immersed Boundary Method for Industrial Polymer Mixing

作者: Emilia Capuano, Daniele Cerroni, Holger Marschall, Giorgio Negrini, Nicola Parolini, Marco Verani

类别: physics.flu-dyn, math.NA

发布日期: 2026-05-12

arXiv ID: 2605.11896v1

链接: http://arxiv.org/abs/2605.11896v1

摘要: This work develops advanced numerical methods for free-surface simulations of polymer mixing processes, integrating a Volume of Fluid (VOF) interface-capturing approach with a non-conforming Immersed Boundary (IB) method to model two-phase flows of highly viscous polymer melts and air within partially filled rotating mixing devices, implemented within the Finite Volume OpenFOAM library. To overcome severe numerical instabilities arising from the strong viscosity contrast between polymer melts and air, a block-coupled scheme providing fully implicit viscous diffusion treatment is integrated into the VOF-IB framework, relaxing time-step stability constraints and substantially reducing computational cost with respect to standard segregated solvers. The resulting BC-VOF-IB solver is applied to industrially relevant geometries of single- and twin-screw extruders, yielding physically consistent predictions of velocity and pressure fields under partial filling conditions. While further developments, most notably the inclusion of thermal effects, remain necessary, the proposed framework represents a meaningful step toward bridging academic CFD research and the practical demands of industrial polymer processing.

10. 最长边高线二分算法的动力学

Dynamics of the Longest-Edge Altitude Bisection Algorithm

作者: Jérôme Michaud, Sergey Korotov

类别: math.NA

发布日期: 2026-05-12

arXiv ID: 2605.11842v1

链接: http://arxiv.org/abs/2605.11842v1

摘要: We study a longest-edge based refinement scheme for triangulations, termed the longest-edge altitude bisection (LEAB), in which each triangle is subdivided by dropping the altitude from the vertex opposite to its longest edge. Using the normalized shape space of triangles introduced by Perdomo and Plaza in: Properties of triangulations obtained by the longest-edge bisection. \emph{Cent. Eur. J. Math.}, 12(12) (2014), 1796-1810, we show that LEAB admits a simple geometric description: the normalized left and right children of a triangle in focus are obtained by intersecting the geodesic of right triangles with rays issued from the endpoints of the longest edge and explicit formulas for the mappings are derived. This characterization implies an interesting observation that the associated refinement dynamics collapse the entire shape space onto the right-triangle geodesic in a single step and that every point on this geodesic is fixed. Two-sided bounds for the contraction of the mesh size (discretization parameter) are derived. Also, applications and limitations of the method are briefly discussed.

11. 参数形状设计中降维的参数模型嵌入的非线性扩展

A nonlinear extension of parametric model embedding for dimensionality reduction in parametric shape design

作者: Andrea Serani, Giorgio Palma, Matteo Diez

类别: cs.CE, cs.LG, math.NA

发布日期: 2026-05-12

arXiv ID: 2605.11759v1

链接: http://arxiv.org/abs/2605.11759v1

摘要: Dimensionality reduction is essential in simulation-based shape design, where high-dimensional parameterizations hinder optimization, surrogate modeling, and systematic design-space exploration. Parametric Model Embedding (PME) addresses this issue by constructing reduced variables from geometric information while preserving an explicit backmapping to the original design parameters. However, PME is intrinsically linear and may become inefficient when the sampled design space is governed by nonlinear geometric variability. This paper introduces a nonlinear extension of PME, denoted NLPME. The proposed framework preserves the defining principle of PME – geometry-driven latent variables and parameter-mediated reconstruction – while replacing the linear reduced subspace with a nonlinear latent representation. Geometry is not reconstructed directly from the latent variables; instead, the latent representation is decoded into admissible design parameters, and the corresponding geometry is recovered through a forward parametric map. The method is assessed on a bio-inspired autonomous underwater glider with a 32-dimensional parametric shape description and a CAD-based geometry-generation process. NLPME reaches a 5% reconstruction-error threshold with (N=5) latent variables, compared with (N=8) for linear PME, and a 1% threshold with (N=9), compared with (N=15) for PME. Comparison with a deep autoencoder shows that most of the nonlinear compression gain can be retained while preserving an explicit backmapping to the original design variables. The results establish NLPME as a compact, admissible, and engineering-compatible nonlinear reduced representation for parametric shape design spaces.

12. 广义Klein-Gordon-Zakharov方程组的新型能量守恒Crank-Nicolson有限元方法

A novel energy-conservation Crank-Nicolson finite element method for generalized Klein-Gordon-Zakharov equations

作者: Xuemiao Xu, Maosheng Jiang, Jiansong Zhang, Jiang Zhu

类别: math.NA

发布日期: 2026-05-12

arXiv ID: 2605.11686v1

链接: http://arxiv.org/abs/2605.11686v1

摘要: This article focuses on an energy-conservation Galerkin finite element method (FEM) for the generalized Klein-Gordon-Zakharov (KGZ) equations. This method combines the bilinear finite element method for spatial discretization with the Crank-Nicolson (CN) scheme for temporal discretization, thereby guaranteeing exact conservation of the discrete energy functional. A rigorous theoretical analysis is devoted to deriving error bounds for the fast-time-scale electronic field $u$ and the ion density deviation $\varphi$. By systematically integrating interpolation estimates, Ritz projection, and a postprocessing technique, the superclose error estimates and global superconvergence are established for $u$ in the $H^1$-norm, even under weakened regularity assumptions on the exact solution. Concurrently, we prove $H^1$-norm superconvergence for the auxiliary variable $φ$ ($-Δφ= \varphi_t$) and optimal-order $L^2$-norm error estimates for the auxiliary variable $p$ ($p=u_t$) and $\varphi$. Numerical examples are provided to confirm theoretical results.

13. 低正则性下波映射方程的一个分裂格式

A splitting scheme for the wave maps equation at low regularity

作者: Katie Marsden, Frédéric Rousset, Katharina Schratz

类别: math.NA

发布日期: 2026-05-12

arXiv ID: 2605.11507v1

链接: http://arxiv.org/abs/2605.11507v1

摘要: We prove convergence of a filtered Lie splitting scheme for the wave maps equation with low regularity initial data in dimension 3. The convergence analysis is performed in discrete Bourgain spaces, as has proved fruitful for the low regularity analysis of the equation in the continuous setting. An important difficulty here is that the analysis of wave maps at low regularity requires the use of the null structure of the system, this structure thus has to be preserved at the discrete level to get an effective stable low regularity scheme. Since the null structure involves time derivatives, the scheme has to be designed carefully. The presence of time derivatives in the nonlinearity then constitutes the most significant source of numerical error. Nonetheless, we are able to prove convergence of the scheme for all subcritical initial data in $H^s$, $s>d/2$.

14. 通过逆多项式重构消除分数阶傅里叶级数中的吉布斯现象

Resolving the Gibbs Phenomenon in Fractional Fourier Series via Inverse Polynomial Reconstruction

作者: Faiza Afzal, Xu Xiao

类别: math.NA

发布日期: 2026-05-12

arXiv ID: 2605.11452v1

链接: http://arxiv.org/abs/2605.11452v1

摘要: The fractional Fourier series generalizes the classical Fourier series by introducing a rotation angle $α$ in the time-frequency plane, but inherits the Gibbs phenomenon for piecewise smooth functions. Unlike the classical setting, the chirp modulation factor renders the fractional partial sum complex-valued, corrupting both real and imaginary components simultaneously and making direct adaptation of classical remedies insufficient. The Inverse Polynomial Reconstruction Method (IPRM) resolves the Gibbs phenomenon by enforcing that the Fourier coefficients of a Gegenbauer polynomial expansion match the given spectral data, rather than projecting the corrupted partial sum onto a polynomial basis. This paper extends the IPRM to fractional Fourier series for the first time. The fractional transformation matrix is derived and its conditioning is shown to be governed by an $α$-independent Gram matrix, which reveals the dependence on the Gegenbauer parameter $λ$ and the polynomial degree $m$, while being entirely insensitive to the transform angle. An $L^{\infty}$ error estimate is established, guaranteeing exponential convergence for analytic functions. Numerical experiments on piecewise analytic test functions demonstrate complete elimination of the Gibbs phenomenon and confirm the theoretical predictions.

15. 基于Carleman估计与压缩映射原理的非线性薛定谔方程初始数据反演

Inverse initial data for nonlinear Schrödinger equation via Carleman estimates and the contraction principle

作者: Navaraj Neupane, Loc Nguyen

类别: math.NA

发布日期: 2026-05-12

arXiv ID: 2605.11409v1

链接: http://arxiv.org/abs/2605.11409v1

摘要: We study an inverse initial-data problem for a nonlinear Schrödinger equation in which the initial wave field is reconstructed from lateral measurements. Our approach combines a Legendre-polynomial-exponential-time dimensional reduction with a Carleman-based contraction principle. First, we expand the solution in a weighted Legendre basis in time and truncate the expansion to obtain a coupled nonlinear elliptic system for the spatial coefficients. Next, we solve this reduced system by constructing a contraction map on a suitable admissible set. This contraction map admits a unique fixed point, which is the limit of the corresponding Picard iteration. We also establish a stability estimate showing that this fixed point remains close to the exact reduced solution in the noisy-data case. Finally, we present numerical experiments in two space dimensions for several different geometries and nonlinear exponents. The numerical results show that the proposed method accurately reconstructs the main features of the initial wave field and remains stable even when the boundary data contain noise.

16. Muon 并不那么特殊：随机或反转频谱同样有效

Muon is Not That Special: Random or Inverted Spectra Work Just as Well

作者: Zakhar Shumaylov, Nathaël Da Costa, Peter Zaika, Bálint Mucsányi, Alex Massucco, Yoav Gelberg, Carola-Bibiane Schönlieb, Yarin Gal, Philipp Hennig

类别: cs.LG, cs.AI, math.NA, math.OC, stat.ML

发布日期: 2026-05-11

arXiv ID: 2605.11181v1

链接: http://arxiv.org/abs/2605.11181v1

摘要: The recent empirical success of the Muon optimizer has renewed interest in non-Euclidean optimization, typically justified by similarities with second-order methods, and linear minimization oracle (LMO) theory. In this paper, we challenge this geometric narrative through three contributions, demonstrating that precise geometric structure is not the key factor affecting optimization performance. First, we introduce Freon, a family of optimizers based on Schatten (quasi-)norms, powered by a novel, provably optimal QDWH-based iterative approximation. Freon naturally interpolates between SGD and Muon, while smoothly extrapolating into the quasi-norm regime. Empirically, the best-performing Schatten parameters for GPT-2 lie strictly within the quasi-norm regime, and thus cannot be represented by any unitarily invariant LMO. Second, noting that Freon performs well across a wide range of exponents, we introduce Kaon, an absurd optimizer that replaces singular values with random noise. Despite lacking any coherent geometric structure, Kaon matches Muon’s performance and retains classical convergence guarantees, proving that strict adherence to a precise geometry is practically irrelevant. Third, having shown that geometry is not the primary driver of performance, we demonstrate it is instead controlled by two local quantities: alignment and descent potential. Ultimately, each optimizer must tune its step size around these two quantities. While their dynamics are difficult to predict a-priori, evaluating them within a stochastic random feature model yields a precise insight: Muon succeeds not by tracking an ideal global geometry, but by guaranteeing step-size optimality.

17. 一种用于铁电向列平衡态有限元近似的能量递减算法

An energy-decreasing algorithm for the finite element approximation of ferronematic equilibrium states

作者: Alexandre Ern, Ruma R. Maity

类别: math.NA

发布日期: 2026-05-11

arXiv ID: 2605.11160v1

链接: http://arxiv.org/abs/2605.11160v1

摘要: We develop an energy-decreasing algorithm for the finite element approximation of two-dimensional ferronematic equilibrium states. The problem is formulated as the minimization of the harmonic energy of two two-dimensional vector fields, both with prescribed length, together with an additional nonlinear relation on the orientation of the two vectors. The finite element setting is based on piecewise continuous finite elements on a weakly acute triangulation. The computational realization of the energy-decreasing algorithm employs a decomposition-coordination framework and a Uzawa-like iteration. Numerical experiments are presented to illustrate the computational performances of the algorithm.