生成时间: 2026年10月26日 16:40:46
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了解领域论文数量: 10
不相关论文数量: 16
了解领域论文（摘要翻译）
1. Born 近似的神经网络增强
Neural network enhancement of the Born approximation
作者: J. Ma, A. Desai, T. Lahivaara, P. Monk
类别: math.NA
发布日期: 2026-10-23
arXiv ID: 2610.32764v1
链接: http://arxiv.org/abs/2610.32764v1
摘要翻译
中文标题: Born 近似的神经网络增强
摘要翻译: 我们考察将神经网络与 Born 近似结合，用于求解频率域中可穿透介质的逆散射问题。目标是从单一频率的多基地远场测量中快速确定散射体的折射率。我们提出一个新的神经网络模型，旨在通过锐化重建图像来改进我们此前的结果。此外，我们提出另外两个受修正的 Born 方法启发的网络，它们避免了对正则化 Born 算子求逆的需要。当逆 Born 近似难以计算或使用近场测量时，后两种算法对于未来的三维应用将很重要。计算结果表明，对 Born 近似的新修正可以为逆散射问题提供一种准确而灵活的方法。
2. 基于非线性波动方程与物理信息神经网络的含气泡液体灵敏度引导参数估计
Sensitivity-Guided Parameter Estimation of Bubbly Liquid via a Nonlinear Wave Equation and Physics-InformedNeuralNetworks
作者: Takeru Hasegawa, Naoto Mitsume, Shota Deguchi, Mitsuteru Asai, Tetsuya Kanagawa
类别: physics.flu-dyn, math.NA
发布日期: 2026-10-23
arXiv ID: 2610.32668v1
链接: http://arxiv.org/abs/2610.32668v1
摘要翻译
中文标题: 基于非线性波动方程与物理信息神经网络的含气泡液体灵敏度引导参数估计
摘要翻译: 超声与微气泡之间的相互作用构成了一系列生物医学和工业应用的基础，然而预测介质和气泡的物理参数如何支配非线性波传播仍然很困难。本研究提出了一种参数估计框架，该框架将描述含气泡液体中弱非线性波传播的 Korteweg-de Vries-Burgers（KdVB）方程与物理信息神经网络（PINN）相结合。本文的贡献有三方面。第一，PINN 在正问题中重现了控制方程的参考解，初始时刻的预测误差接近零，这证实初始条件按预期被精确施加。第二，在反演估计之前，引入灵敏度分析，在系数层面和解层面上量化八个物理参数各自如何影响 KdVB 方程，从而可以预先评估每个参数的估计难度。第三，基于这一评估，使用 PINN 对八个参数中的七个进行单参数反演估计。液体黏度、气泡初始半径、液体密度、声速和液体压力的估计误差为 0.8-2.1%，多方指数仅部分收敛，表面张力系数完全无法估计，这与灵敏度分析一致。这些结果建立了一个通用框架，用于从观测波形数据中识别和估计控制含气泡液体中非线性波传播的物理参数。
3. 不可压缩 MHD 的线性双场有限元方法，第一部分：周期域中的交叉螺旋度与 Alfvén 对齐
A linear dual-field finite element method for incompressible MHD, part I: cross-helicity and Alfvénic alignment in periodic domains
作者: Shusheng Li, Siyuan Tong, Qilong Zhai, Qian Zhang
类别: math.NA
发布日期: 2026-10-23
arXiv ID: 2610.32476v1
链接: http://arxiv.org/abs/2610.32476v1
摘要翻译
中文标题: 不可压缩 MHD 的线性双场有限元方法，第一部分：周期域中的交叉螺旋度与 Alfvén 对齐
摘要翻译: 交叉螺旋度与总能量一起决定了 Elsässer 能量，从而量化不可压缩 MHD 中偏离 Alfvén 对齐的程度。在 [Mao and Xi, \emph{J. Comput. Phys.}, 2025] 中双场方法的基础上，我们针对周期域构造了一种线性有限元方法，该格式重现交叉螺旋度平衡。我们通过非负的 Elsässer 能量表示，将离散能量守恒律与离散交叉螺旋度守恒律同 Alfvén 对齐联系起来，并证明：在无外力理想情形下，当任一 Elsässer 能量初值为零时，相应的精确关系得以保持。我们在 Mao-Xi 格式内重新将磁场的两个离散近似分配到整数和半整数时间层，并重新设计 Lorentz 耦合与感应耦合。所得格式采用线性交错更新，并保持能量平衡、强与弱无散度性质以及磁螺旋度和流体螺旋度平衡。数值实验说明了交叉螺旋度平衡在理想情形下保持对齐态以及在弱耗散下对齐动力学中的作用。
4. 用于带 Tresca 摩擦接触力学的任意阶增广拉格朗日离散 de Rham 格式
An Arbitrary-Order Augmented Lagrangian Discrete De Rham Scheme for Contact Mechanics with Tresca Friction
作者: Jérôme Droniou, Raman Kumar, Roland Masson
类别: math.NA
发布日期: 2026-10-23
arXiv ID: 2610.32471v1
链接: http://arxiv.org/abs/2610.32471v1
摘要翻译
中文标题: 用于带 Tresca 摩擦接触力学的任意阶增广拉格朗日离散 de Rham 格式
摘要翻译: 我们设计并分析了一种用于裂隙介质中接触力学的任意阶增广拉格朗日离散 de Rham（DDR）多面体格式。裂隙被表示为基质（一种线弹性介质）中的余维一对象，并且当裂隙壁之间发生接触时采用 Tresca 摩擦模型。法向 Signorini 型/牵引约束与 Tresca 摩擦定律均被重写为投影恒等式，并合并为单个增广拉格朗日双线性形式。位移由任意多项式次数 $k\ge 0,$ 的 DDR 空间离散，并在裂隙面上用泡自由度进行加富；这些泡自由度通过 Lagrange 乘子空间与沿裂隙的位移离散跳跃之间存在离散提升来确保稳定性。我们建立了一个抽象误差估计，给出了位移和 Lagrange 乘子变量的最优阶收敛性。大量数值实验验证了理论分析，并展示了该格式的实用精度和鲁棒性。
5. CoPiT：一种用于非结构网格气动场预测的条件位置诱导 Transformer
CoPiT: A conditional Position-induced Transformer for aerodynamic field prediction on unstructured meshes
作者: Junfeng Chen, Yongchen Fan, Zhonghua Qiao, Kailiang Wu
类别: physics.comp-ph, cs.LG, math.NA, physics.flu-dyn
发布日期: 2026-10-23
arXiv ID: 2610.32299v1
链接: http://arxiv.org/abs/2610.32299v1
摘要翻译
中文标题: CoPiT：一种用于非结构网格气动场预测的条件位置诱导 Transformer
摘要翻译: 高保真 Reynolds 平均 Navier–Stokes (RANS) 模拟仍然是气动分析的标准工具，但在非结构网格上的每次求解计算代价高昂，使得设计探索、优化和多工况评估等多次查询任务成本很高。神经算子提供了网格灵活的代理模型，用于将几何和运行条件映射到流场；然而，在许多现有架构中，自由来流状态和设计变量等全局参数与逐点输入拼接，使其对中间特征的影响保持隐式。本文提出 CoPiT，一种 \textbf{Co}nditional \textbf{P}osition-induced \textbf{T}ransformer，它以 Position-induced Transformer (PiT) 作为几何骨干，并通过多阶段特征级线性调制 (FiLM) 注入全局条件向量，从而使参数调制显式化，同时保留基于几何的注意力权重。该模型在二维 AirfRANS 翼型场（包括稀缺数据和外推情形）以及三维 BlendedNet++ 翼身融合体表面上进行评估。在 AirfRANS 全数据测试集上，CoPiT 相对于 Transolver 分别将表面损失和体积损失降低约 54% 和 24%，并在 BlendedNet++ 上取得所比较基线中最低的整体归一化测试 MSE。匹配消融实验表明，FiLM 条件化通常优于输入拼接，而更优的放置位置取决于任务。因此，CoPiT 为非结构网格上的参数化气动问题提供了一种实用的算子学习代理模型。
6. 曲基底上具有移动接触线的表面扩散的能量稳定参数化有限元框架
An energy-stable parametric finite element framework for surface diffusion with moving contact lines on curved substrates
作者: Harald Garcke, Robert Nürnberg, Rong Tang
类别: math.NA
发布日期: 2026-10-23
arXiv ID: 2610.32270v1
链接: http://arxiv.org/abs/2610.32270v1
摘要翻译
中文标题: 曲基底上具有移动接触线的表面扩散的能量稳定参数化有限元框架
摘要翻译: 我们针对曲基底上具有移动接触线的固体薄膜表面扩散，发展了能量稳定的参数化有限元方法。通过对薄膜表面和基底贴片同时进行离散，可以引入无条件能量稳定的线性格式。这些格式的一个非线性变体还保证，基底近似的顶点在整个过程中始终保持附着于真实基底。此外，我们的框架能够纳入薄膜表面和基底的各向异性表面能密度，以及不同的切向运动。二维和三维空间中的数值实验说明了所引入方法的实用性。
7. 用于高维 Fokker-Planck 方程的弱对抗张量神经网络
Weak adversarial tensor neural network for high-dimensional Fokker-Planck equations
作者: Sihong Shao, Zhongwei Xu
类别: math.NA
发布日期: 2026-10-23
arXiv ID: 2610.32169v1
链接: http://arxiv.org/abs/2610.32169v1
摘要翻译
中文标题: 用于高维 Fokker-Planck 方程的弱对抗张量神经网络
摘要翻译: 求解高维稳态 Fokker-Planck（SFP）方程仍然具有挑战性。近年来，深度学习方法由于其通用逼近能力和无网格训练而在求解高维 SFP 方程方面展现出巨大潜力。然而，损失计算与归一化中涉及的高维积分若使用 Monte Carlo 积分进行精确逼近，代价可能很高。为解决这一问题，我们提出一种用于求解高维 SFP 方程的弱对抗张量神经网络（WATNN）方法，它将弱对抗（WA）形式与张量神经网络（TNN）相结合。WA 形式避免了对强残差平方所引入的交叉项，从而减少了需要计算的积分数量。当主网络和对抗测试网络均设置为 TNN 时，TNN 求积格式能够以相对较少的固定求积点实现准确且高效的归一化与损失计算，避免了使用 Monte Carlo 积分时产生的统计采样误差。我们在四个高维 SFP 方程上数值验证了该方法的有效性。对于 6D 单峰、6D 多峰、10D 多峰和 20D 多峰问题，WATNN 分别在仅几分钟内达到了 1.50%、1.13%、1.20% 和 7.93% 的平均相对误差。
8. 用神经基方法求解与学习 Navier-Stokes 动力学
Solving and Learning Navier-Stokes Dynamics with the Neural Basis Method
作者: Yuhe Wang, Zhu Wang, Min Wang
类别: physics.flu-dyn, cs.LG, math.NA
发布日期: 2026-10-23
arXiv ID: 2610.32167v1
链接: http://arxiv.org/abs/2610.32167v1
摘要翻译
中文标题: 用神经基方法求解与学习 Navier-Stokes 动力学
摘要翻译: 物理信息机器学习已成为流体动力学预测建模的一个主要方向，它将控制物理与学习表示相结合。然而，现有形式既不能保证小的 PDE 残差损失意味着小的解误差，也不能随着问题规模增大而有效扩展。在此，我们提出神经基方法（NBM），作为处理 Navier–Stokes 动力学的一种新途径。NBM 基于“表示-投影”原则：固定的神经基定义表示空间，而控制方程的投影确定展开系数。同一组基为参数化解流形定义公共坐标，将算子学习重新表述为参数到系数的映射，并通过消除显式空间依赖实现可扩展性。对于不可压缩流，NBM 给出仅速度形式，其残差提供速度误差的后验估计，从而为系数学习建立一个可认证的、无数据的训练目标。理论分析和数值实验证明了 NBM 的准确性、效率、泛化性和可扩展性，并较基线有显著改进。
9. 空间表示随机系数的多面体DG离散的多层蒙特卡洛方法
Multilevel Monte Carlo for Polytopic DG Discretisations with Spatially Represented Random Coefficients
作者: Veronika Chronholm, Matt Evans, Emmanuil H. Georgoulis, Tristan Pryer
类别: math.NA
发布日期: 2026-10-22
arXiv ID: 2610.31861v1
链接: http://arxiv.org/abs/2610.31861v1
摘要翻译
中文标题: 空间表示随机系数的多面体DG离散的多层蒙特卡洛方法
摘要翻译: 我们针对多面体网格上具有随机扩散系数和反应系数的强制椭圆问题，建立了一个多层蒙特卡洛框架，特别强调由空间表示隐场生成的随机系数的逼近以及层间耦合。主要分析贡献是在DG范数下的一种Strang型误差分解，该分解将正演离散误差与逼近随机系数所引入的误差分离。在此基础上，我们研究两种系数生成与多层耦合策略：一种基于定义在固定笛卡尔参考网格上的平稳高斯隐场的谱采样器；另一种是基于SPDE的Matérn采样器，其中各层通过凝聚层级上的投影白噪声载荷进行耦合。两者都纳入同一个系数感知的分析框架，并通过共同的隐随机输入直接耦合相邻层。理论得到了一般Voronoi网格和凝聚多面体网格上的数值实验支持，这些实验展示了所预测的正演误差与系数表示误差、多层修正的衰减、强的层间耦合，以及相对于单层蒙特卡洛所得到的工作量-精度行为。
10. 分段光滑区域上的有限元逼近
Finite element approximation on piecewise smooth domains
作者: Daniele Boffi, Lucia Gastaldi
类别: math.NA
发布日期: 2026-10-22
arXiv ID: 2610.31822v1
链接: http://arxiv.org/abs/2610.31822v1
摘要翻译
中文标题: 分段光滑区域上的有限元逼近
摘要翻译: 有限元逼近理论通常在多边形或多面体区域这类简化几何上描述。虽然回避边界逼近问题使分析更简单，但实际应用可能涉及分段光滑区域，且在某些情况下将经典结果推广到一般边界可能很棘手。本文以系统的方式讨论二维和三维分段曲线区域上标准非等参有限元空间的逼近性质与误差分析。我们详细表明，几何误差为 Dirichlet 和 Neumann 椭圆问题解的 $H^1$ 与 $L^2$ 误差引入一个障碍，且该障碍与问题的正则性以及所用有限元空间的次数无关。
不相关论文（基本信息）
1. 平流方程的 Lax-Wendroff 型格子 Boltzmann 格式的数值分析
Numerical analysis of a Lax-Wendroff type lattice Boltzmann scheme for the advection equation
作者: Thomas Bellotti, Filipa Caetano, François Dubois, Benjamin Graille
类别: math.NA
发布日期: 2026-10-23
arXiv ID: 2610.32810v1
链接: http://arxiv.org/abs/2610.32810v1
摘要: A linear monodimensional lattice Boltzmann scheme with three velocities tackling the advection equation is investigated. We prove that it is second-order accurate and provide necessary and sufficient stability conditions through an explicit formula involving the two relaxation parameters. Relying on existing results for finite difference schemes, we prove convergence for smooth initial data with initialization of the non-conserved moments at equilibrium. Several numerical simulations illustrate these theoretical findings.
2. 学习型优化器调度中的乘性与加性状态耦合
Multiplicative vs. Additive State Coupling in Learned Optimizer Schedules
作者: M. Harper Langston, Pierre-David Letourneau, Dalton Jones, Roland Memisevic, Peter Scott, Mingu Lee, Richard A. Lethin, Christopher Lott, Fatih Porikli, Harris Teague
类别: cs.LG, cs.AI, cs.CL, math.NA, math.OC
发布日期: 2026-10-23
arXiv ID: 2610.32801v1
链接: http://arxiv.org/abs/2610.32801v1
摘要: We learn schedules for a continuous penalty relaxation of MAX-CUT by backpropagating through an unrolled classical solver. A state-blind schedule generalizes to instances substantially larger than those used for training. A state-adaptive bilinear schedule also generalizes out of distribution, outperforming its untrained baseline across 15 seeds ($t=4.1$). By contrast, an additive ablation with identical inputs and output heads is statistically indistinguishable from the bilinear model in distribution but performs worse than its own baseline out of distribution ($t=-11.7$). This separation appears without regularization and persists across a $30$ round weight-decay sweep: the bilinear advantage remains significant at every tested value, whereas the additive model never recovers. The results in this setting isolate how solver state enters the schedule, rather than state adaptivity alone, as the architectural distinction associated with out-of-distribution transfer.
3. 使用历史 Ritz 步长的循环梯度法的超线性收敛
Superlinear Convergence of Cyclic Gradient Methods with Historical Ritz Step Sizes
作者: Ran Gu
类别: math.OC, math.NA
发布日期: 2026-10-23
arXiv ID: 2610.32705v1
链接: http://arxiv.org/abs/2610.32705v1
摘要: Simplified spectral models predict dimension-dependent superlinear convergence for gradient methods, but the corresponding behavior of actual iterations remains only partially understood. Historical methods introduce an additional difficulty: the gradients defining the step sizes differ from those receiving them. We study this offset for cyclic reuse of historical Ritz step sizes on strictly convex quadratics. Within the existing limited memory steepest descent with retards (LMSDR) family, a rolling cyclic Ritz (RCR) rule reuses $m$ historical Ritz values $\ell$ times per cycle, ending with a complete sweep. At $q=m+1$ active distinct eigenvalues, an exact delayed recurrence gives almost-everywhere doubly exponential convergence with an explicit rate and exceptional weights. Nonempty open basins also exist for $m+1
4. 带正权的随机节点：Richter-Tchakaloff 规则与最优正求积
Random nodes with positive weights: Richter-Tchakaloff rules and optimal positive quadrature
作者: Sebastian Neumayer, Martin Schäfer, Tino Ullrich
类别: math.NA
发布日期: 2026-10-23
arXiv ID: 2610.32699v1
链接: http://arxiv.org/abs/2610.32699v1
摘要: We show that independent random points support exact quadrature rules with positive weights. For any $n$-dimensional space $V$ of integrable functions a constant multiple of $n\log n$ points sampled from a suitable density yields, with high probability, positive weights which integrate the functions in the space exactly. The weights stay close to the natural importance sampling weights according to the Lewis density with respect to $V$. They are the result of a weight adaptation procedure which requires an explicit $L_1$-norm discretization result that is proved on the way. As an application, we study numerical integration in reproducing kernel Hilbert spaces. Combining the random rules with a subsampling step, we show that positive quadrature rules with constantly oversampled nodes achieve the best possible worst-case error bound in terms of the singular numbers of the embedding into $L_2$.
5. 对称多凸能量及其包络的通用神经网络逼近
Universal neural network approximation of symmetric polyconvex energies and envelopes
作者: Florian Gammer
类别: math.NA, cs.CE, cs.LG
发布日期: 2026-10-23
arXiv ID: 2610.32598v1
链接: http://arxiv.org/abs/2610.32598v1
摘要: Symmetric polyconvexity provides a sufficient condition for weak lower semicontinuity in small-strain elasticity and a tractable lower bound for the relaxation of non-(quasi)convex energies. Since classical algorithms for computing such functions can be computationally expensive, neural network approximations have recently emerged as an alternative. A central difficulty with establishing such neural network based approaches for symmetric polyconvexity is the additional monotonicity property. As we show, the corresponding weight restrictions on ReLU-based input-convex networks do not yield universal approximation of the lifted representers. Here, we introduce symmetric polyconvex neural networks (SPCNNs) for constitutive modeling and approximation of symmetric polyconvex envelopes. The deep architecture combines convexity with the monotonicity required of a representer in the space of minors. We prove a universal approximation theorem for symmetric polyconvex energies, without requiring isotropy, and quantify the number of affine pieces attainable by the lifted network and derive reduced architectures for isotropic energies. Numerical examples illustrate stress-based regression and approximation of the symmetric polyconvex envelope of a double-well potential. Symmetric polyconvexity is guaranteed by construction, while the accuracy of the fitted energies and envelopes is assessed numerically.
6. 随机调和映射热流的概率强解的数值逼近
Numerical approximation of probabilistically strong solutions to the stochastic harmonic map heat flow
作者: Robert Lasarzik, Stefan Metzger, Maximilian E. V. Reiter
类别: math.NA
发布日期: 2026-10-23
arXiv ID: 2610.32588v1
链接: http://arxiv.org/abs/2610.32588v1
摘要: We study the stochastic harmonic map heat flow into the sphere driven by a Stratonovich noise term that formally preserves the spherical constraint by construction. Starting from a fully discrete, implementable numerical scheme, we prove its convergence to the novel solution concept of probabilistically strong, analytically energy-variational solutions. This yields the first existence result for probabilistically strong solutions in the three-dimensional setting on the prescribed stochastic basis. In addition, we establish weak-strong uniqueness and illustrate our theoretical findings with numerical experiments.
7. 具衰退记忆的抛物型方程 BDF3 方法的长时间稳定性与收敛性
Long-time stability and convergence of a BDF3 method for a parabolic equation with fading memory
作者: Wenlin Qiu
类别: math.NA
发布日期: 2026-10-23
arXiv ID: 2610.32364v1
链接: http://arxiv.org/abs/2610.32364v1
摘要: We analyze the long-time stability and convergence of a third-order backward differentiation formula (BDF3) for parabolic equations with exponentially fading memory. The BDF3 scalar resolvent has oscillatory coefficients, so the monotonicity argument used for lower-order formulas in [Math. Comp. 62 (1994) 477-496] does not apply. We instead control memory feedback by the absolute mass of its discrete Green sequence and establish the rigorous stability-constant bounds $23/20<Λ_3<579/500$. A causal third-order memory quadrature with nonnegative assembled weights then yields exponentially weighted stability without a parabolic step-size restriction. With third-order starting values, we prove a uniform-in-time third-order temporal error estimate and extend it to conforming finite elements. Numerical tests confirm the predicted convergence orders and the long-time behavior.
8. 含移动井的两相 CO$_2$ 封存的区域分解降阶模型
A domain-decomposition reduced-order model for two-phase CO$_2$ sequestration with moving wells
作者: Xin He, Xianda Shen, Lei Zhang
类别: math.NA
发布日期: 2026-10-23
arXiv ID: 2610.32340v1
链接: http://arxiv.org/abs/2610.32340v1
摘要: We present a domain-decomposition reduced-order model for two-phase geological CO$_2$ storage with fixed and moving wells. The approach builds on a nonoverlapping Schur formulation of a conservative RT0–P0 discretization solved by a sequential pressure–transport algorithm. The distinctive feature of the formulation is that local reduced models are coupled through physically meaningful interface variables: pressure traces for the pressure solve and inflow saturation traces for the transport solve. To address moving wells, we introduce a well-aware strategy that combines source-response enrichment, HAPOD-based far-field compression, and hybrid full-order/reduced-order treatment of well and near-well subdomains. We also formulate conservation-oriented hyper-reduction rules where interface fluxes, boundary terms and source terms are protected. Numerical results are presented to demonstrate accuracy, computational efficiency, robustness of the overall methodology.
9. 面向乳腺成像全波形反演的即插即用流匹配探索
An exploration of plug-and-play flow matching for full-waveform inversion on breast imaging
作者: Cristina Garcia-Cardona, Siting Liu, Fatma Terzioglu, Shanyin Tong, Lu Zhang
类别: eess.IV, math.NA
发布日期: 2026-10-23
arXiv ID: 2610.32185v1
链接: http://arxiv.org/abs/2610.32185v1
摘要: Full-waveform inversion (FWI) reconstructs sound-speed fields from measured waveforms, by solving an optimization problem that minimizes the discrepancy between measured waveforms and those predicted by a numerical wave-propagation model. However, choosing suitable regularization for this optimization is challenging because it must stabilize the inversion while preserving relevant structures, prompting us to investigate whether priors learned from representative sound-speed images can improve reconstruction. In this work, we adapt plug-and-play flow matching (PnP-Flow) to FWI by alternating waveform-misfit updates with updates from a pretrained flow prior. We explore the effectiveness of PnP-Flow reconstruction by comparing it with total-variation (TV) regularization using noiseless transmission-only plane-wave data. Separate flow priors are trained on two circular-inclusion model families with constant and spatially varying backgrounds. We evaluate the method both on these in-distribution circle models and under substantial distribution shift by transferring the Gaussian-circle prior and reconstruction settings to breast phantom patches without retraining or retuning. Using the same number of waveform-misfit updates, the proposed PnP-Flow framework generally improves circle reconstruction accuracy and recovers breast lesion contrast more accurately than TV baselines.
10. 混合整数线性规划的分支-过滤方法
A branch-and-filter method for mixed-integer linear programming
作者: Nikos Dimou
类别: math.OC, math.NA
发布日期: 2026-10-23
arXiv ID: 2610.32165v1
链接: http://arxiv.org/abs/2610.32165v1
摘要: We develop a new tree-search framework for mixed-integer linear programming problems, which incorporates a filter mechanism as a node-acceptance device. The introduced branch-and-filter algorithm combines proper branching actions with a filter on integrality violation and an incumbent-dependent objective coordinate. Inspired by classical filter methods in nonconvex constrained optimization, the proposed method moves to a restoration phase whenever sufficient progress cannot be provided by the filter and funnel mechanisms, the latter of which is anchored at global relaxation lower bounds. Under mild assumptions, the introduced method either certifies infeasibility in finitely many iterations, or converges to global optimality. Moreover, we establish a total outer iteration complexity bound of order $\mathcal{O}(\varepsilon^{-1})$ to certification of an $\varepsilon$-optimal mixed-integer solution through fixed-integer linear programming recovery. We finally perform several numerical experiments to study the performance of branch-and-filter against benchmark solvers, under a branch-and-bound, a branch-and-cut, and a novel cell-deletion realization of the proposed restoration oracle.
11. 协方差信息引导的集合得分滤波：将局部化协方差几何嵌入基于得分的输运
Covariance-Informed Ensemble Score Filtering: Embedding Localized Covariance Geometry into Score-Based Transport
作者: Siming Liang, Peter Jan van Leeuwen, Hristo G. Chipilski, Zezhong Zhang, Feng Bao, Guannan Zhang
类别: math.NA, physics.ao-ph, stat.ML
发布日期: 2026-10-23
arXiv ID: 2610.32148v1
链接: http://arxiv.org/abs/2610.32148v1
摘要: Background-error covariance models are central to atmospheric data assimilation because they spread partial observational information across space, variables, and dynamically coupled scales. Ensemble and variational methods exploit this covariance geometry efficiently, but their standard analysis formulations are largely built around Gaussian or locally quadratic approximations. Score-based ensemble filtering provides a framework for nonlinear distributional transport, but existing Ensemble Score Filter (EnSF) formulations lack an intrinsic covariance-mediated pathway for propagating likelihood information from observed to unobserved state components. We introduce a covariance-informed ensemble score filter (CI-EnSF) that embeds localized forecast-error covariance geometry directly into the forward diffusion and reverse transport. The proposed forward diffusion is covariance-shaped in standardized state space and becomes isotropic under the corresponding control-variable transformation. This covariance structure continuously propagates observational information from observed to unobserved state components throughout the reverse transport analysis trajectory. Experiments with Lorenz-96 and surface quasi-geostrophic (SQG) dynamics show a clear regime dependence. For linear observations, a well-tuned LETKF is more accurate, while CI-EnSF maintains stable cycling and effective covariance-mediated information propagation. Under strongly nonlinear observations and under climatological initialization, CI-EnSF achieves lower analysis error and greater cycling robustness than Gaussian ensemble benchmarks such as the LETKF, with improvements extending across spatial scales in the nonlinear SQG experiment. These results suggest that the principal benefit of CI-EnSF arises from combining nonlinear distributional transport with the mature covariance geometry used in conventional data assimilation.
12. 任意维矩形网格上用于双调和方程的分片 $P_2$ 有限元空间
Piecewise $P_2$ finite element space on rectangular meshes in arbitrary dimensions for the biharmonic equation
作者: Rui Ma, Yang Xing, Shuo Zhang
类别: math.NA
发布日期: 2026-10-23
arXiv ID: 2610.32048v1
链接: http://arxiv.org/abs/2610.32048v1
摘要: This paper develops a piecewise $P_2$ finite element space on rectangular meshes in arbitrary dimensions for the biharmonic equation. The discrete space imposes continuity of vertex sums on $(d-2)$-faces and of averaged normal derivatives across interior $(d-1)$-faces. A local algebraic characterization of $P_2$ is first established through the unisolvence of an enriched polynomial space and the resulting compatibility conditions. For each interior element, a $3^d$-element patch is then introduced, and the associated homogeneous patch space is shown to be one-dimensional. The corresponding patch functions, extended by zero, form a basis of the global homogeneous discrete space. A local quadratic interpolation operator is further constructed in arbitrary dimensions. Its approximation properties, together with a consistency estimate derived from the weak continuity conditions, lead to the optimal broken $H^2$ error estimate for the biharmonic problem.
13. 任意剖分上二次非节点样条插值的最优收敛性
Optimal Convergence of Quadratic Not-a-Knot Spline Interpolation on Arbitrary Partitions
作者: Aussie Greene
类别: math.NA
发布日期: 2026-10-23
arXiv ID: 2610.32044v1
链接: http://arxiv.org/abs/2610.32044v1
摘要: The not-a-knot condition provides a simple spline interpolation scheme that requires no derivative data at the endpoints. Whereas the classical not-a-knot scheme using odd-degree splines selects knots that coincide with the data points, this approach does not extend to the even-degree case. In this paper we establish the well-posedness of the not-a-knot interpolation problem for even-degree splines, where knots are selected at midpoints of data values. We then show that the quadratic case, introduced in [M. J. D. Powell, Approximation Theory and Methods, Cambridge University Press, 1981], achieves $\mathcal{O}(h^3)$ convergence for $C^3$ functions on arbitrary partitions without mesh-ratio restrictions. The proof derives a linear system defining the interpolating spline, normalizes the system to bound the matrix norm of its inverse using a Neumann-series argument, and establishes global error bounds.
14. 用于间断 Galerkin 谱元高效预处理的新型无矩阵近似单元逆
A Novel Matrix-Free Approximate Elemental Inverse for Efficient Preconditioning of Discontinuous Galerkin Spectral Elements
作者: Boyang Xia, David Moxey
类别: math.NA
发布日期: 2026-10-22
arXiv ID: 2610.31982v1
链接: http://arxiv.org/abs/2610.31982v1
摘要: Efficient matrix-free preconditioning remains a challenge for discontinuous Galerkin spectral element discretisations of elliptic problems, particularly for general element shapes with collapsed-coordinate bases, such as prisms, tetrahedra and pyramids. Multigrid provides a promising framework, but its efficiency depends strongly on the effectiveness and computational cost of the smoother. For modal bases, point Jacobi can provide insufficient smoothing, whereas applying block Jacobi through stored dense inverses incurs substantial storage and computational costs. We propose a matrix-free approximate elemental inverse, an approximation to the inverse of each elemental diagonal block, constructed from inverse modal-to-nodal transforms and an element-dependent sparse matrix. Its application exploits generalised sum-factorisation and partial sum-factorisation in collapsed coordinates, avoiding the explicit storage and multiplication of large dense matrices. The method is evaluated as both a standalone preconditioner and a multigrid smoother on hexahedral, prismatic, tetrahedral and pyramidal elements. Although the proposed approximation generally increases iteration counts compared to exact block Jacobi, the cost per iteration is far lower. This trade-off yields substantial reductions in total solution time for hexahedral and prismatic elements, while achieving comparable performance for tetrahedral and pyramidal elements. This study can also serve as a practical guide for tuning the convergence of the discontinuous Galerkin interior-penalty Poisson solver.
15. 极化集合卡尔曼反演：一种用于多模态反问题的无导数方法及其非渐近分析
Polarized Ensemble Kalman Inversion: A Derivative-Free Method for Multimodal Inverse Problems with Non-asymptotic Analysis
作者: Huchen Yang, Jin-Long Wu
类别: math.NA, math.OC
发布日期: 2026-10-22
arXiv ID: 2610.31907v1
链接: http://arxiv.org/abs/2610.31907v1
摘要: Inverse problems often admit multiple solutions consistent with the same noisy, indirect observations, especially with sparse data and/or strongly nonlinear forward models. Ensemble Kalman inversion (EKI) provides a robust, derivative-free framework to solve inverse problems, but its global empirical covariances couple all ensemble members across distinct solution modes, limiting its effectiveness for multimodal problems. We introduce polarized ensemble Kalman inversion (PEKI), which constructs distance-weighted local covariances for each ensemble member. This localization weakens contributions from distant, data-consistent regions and enables different clusters of ensemble members to evolve toward distinct solution modes. For the practical algorithm with finite ensemble size and polarization radius, we also establish non-asymptotic bounds and perform numerical studies, showing that cross-cluster effects decay exponentially with cluster separation relative to the polarization radius. For linear forward maps, we further establish particle-wise residual contraction and derive an exact cluster-mean decomposition that separates cross-cluster leakage, within-cluster gain variation, and observation perturbations. These results further motivate a one-step procedure for choosing the polarization radius by reusing stored forward-model evaluations. We studied the performance of PEKI on several numerical experiments, including low-dimensional nonconvex benchmarks, a problem equipped with many-mode landscape, and a high-dimensional inverse problem governed by Navier–Stokes equations. The results confirm the theoretical exponential decay of cross-cluster effects and show that PEKI can serve as a robust and derivative-free method for inverse problems with multiple solution modes.
16. 近静力平衡大气流动的高阶模拟
High-order Simulation of Nearly Hydrostatic Atmospheric Flows
作者: Shoyon Panday, Stéphane Gaudreault, Carlos Pereira, Philipp Birken
类别: physics.flu-dyn, math.NA
发布日期: 2026-10-22
arXiv ID: 2610.31836v1
链接: http://arxiv.org/abs/2610.31836v1
摘要: Accurate simulation of atmospheric flows is challenging. Low-Mach-number dynamics often need to be resolved in the presence of a dominant, nearly hydrostatic background state. Conventional upwind numerical fluxes can introduce excessive acoustic-speed-based dissipation in the low-Mach regime. Discretization errors in the balance law can also generate spurious motion. This work investigates these two issues within a high-order Direct Flux Reconstruction (DFR) framework coupled with a third-order exponential time integrator for the 3D compressible Euler equations with gravity. Low-Mach-number accuracy is addressed using a modified AUSM$^+$-up flux, denoted AUSM$^+$-upH. Preservation of the background equilibrium is improved through a deviation-based formulation. The resulting approach is assessed using a hierarchy of idealized atmospheric test cases. The results show that AUSM$^+$-upH preserves low-Mach flow structures more effectively than the more dissipative Rusanov flux. However, reduced numerical dissipation also makes AUSM$^+$-upH more sensitive to residual errors in the discrete hydrostatic balance. The deviation formulation substantially suppresses these errors by several orders of magnitude, with the strongest improvement observed for flows over orography. Overall, AUSM$^+$-upH exhibits greater robustness when combined with the deviation formulation across the atmospheric test cases considered.