生成时间: 2026年09月10日 14:10:47
重点关注论文数量: 0
了解领域论文数量: 9
不相关论文数量: 17
了解领域论文（摘要翻译）
1. 一般网格上双曲方程点值与单元平均混合离散的不变域保持
Invariant domain preservation for hybrid point-value and cell-average discretizations of hyperbolic equations on general meshes
作者: Shengrong Ding, Shumo Cui, Remi Abgrall, Kailiang Wu
类别: math.NA, physics.comp-ph
发布日期: 2026-09-09
arXiv ID: 2609.10476v1
链接: http://arxiv.org/abs/2609.10476v1
摘要翻译
中文标题: 一般网格上双曲方程点值与单元平均混合离散的不变域保持
摘要翻译: 本文提出了一个统一的不变域保持 (IDP) 框架，用于双曲守恒律的混合离散，包括 active flux 和 PAMPA 方法；其中单元平均以守恒方式更新，而单元边界点值在一个可能非守恒的算子下演化。主要挑战是在单个局部稳定性条件下，为这两组耦合状态保持可容许性，同时不增加演化自由度，也不依赖更新后修复。对于点值更新，我们引入一种基于 barrier–Legendre 映射的可容许性变换，用于由凹约束描述的凸可容许内部，并证明该逆映射在相关集合上全局定义且 Lipschitz 连续。对于守恒的单元平均更新，我们建立一个结构性障碍定理：仅由可容许边界迹构造的单状态连续物理通量，在内部重构值不受控制时，对任意给定的 CFL 数，都不能提供守恒的 IDP 保证。这指出了必须由额外 IDP 通量机制提供的缺失局部控制。为了显式实现这一机制，我们结合单元平均分解 (CAD)、几何拟线性化和局部先验缩放，在通量求值之前构造可容许的、一般间断的迹状态。在一个显式的基于迹的 CFL 条件下，其中常数由局部 CAD 权重和迹状态波速界确定，耦合混合更新保持所规定的不变域。在三角形、笛卡尔、凸四边形和一般凸多边形网格上发展了具体三阶格式。数值结果展示了光滑解的设计精度阶以及物理可容许性的严格保持。
2. 曲面上平均场规划的梯度增强近端算法
Gradient-Enhanced Proximal Algorithms for Mean Field Planning on Surfaces
作者: Chengrun Jiang
类别: math.NA
发布日期: 2026-09-09
arXiv ID: 2609.10370v1
链接: http://arxiv.org/abs/2609.10370v1
摘要翻译
中文标题: 曲面上平均场规划的梯度增强近端算法
摘要翻译: 曲面上的平均场规划指定初始和终端密度，并在连续性方程约束下最小化一个输运能量。该问题的近端算法反复求解一个时间–空间 Poisson 方程，其时间导数和曲面梯度决定密度与动量修正。我们研究该约束投影的一种梯度增强近似，其中在时间上使用有限差分，在空间上使用曲面有限元，并采用时间多项式保持恢复和空间参数化多项式保持恢复。同一构造被纳入 ISTA、FISTA 和 Douglas–Rachford 分裂。我们区分恢复更新与精确离散投影，并推导残差恒等式以及条件有限迭代扰动界，其中保留了数据、边界和线性求解器误差。现有导数恢复估计在适当的正则性和网格假设下识别出一个高阶贡献；仅凭它们本身并不能建立外层优化迭代的收敛性。我们讨论了球面以及一个更复杂的代数曲面上的可用数值示例，并一并讨论了所记录加密数据的局限。
3. 弱对称和无迹的切向-法向张量有限元：在 Brinkman 方程中的应用
Weakly Symmetric and Traceless Tangential-Normal Tensor Finite Elements: Application to the Brinkman Equations
作者: Xuehai Huang, Xinyue Zhao
类别: math.NA
发布日期: 2026-09-09
arXiv ID: 2609.10331v1
链接: http://arxiv.org/abs/2609.10331v1
摘要翻译
中文标题: 弱对称和无迹的切向-法向张量有限元：在 Brinkman 方程中的应用
摘要翻译: 我们构造了一族在任意空间维数和所有多项式阶数下的弱对称且逐点无迹的切向-法向张量有限元。对称性通过局部单元矩来施加，而唯一全局耦合的应力自由度是切向-法向面矩；不需要任何顶点自由度。在三维及更高维中，最低阶线性富集恢复了离散 Korn 稳定性所需的刚体运动面控制。作为一个主要应用，我们利用物理粘性应力为不可压缩 Brinkman 方程构造了一种分布型混合方法。与散度协调的 BDM 速度和不连续压力耦合，该方法无需稳定化，关于粘性参数一致稳定，精确无散度，且压力鲁棒。我们在自然范数下建立了最优阶误差估计。在适当的参数显式正则性假设下，我们还得到了具有最优 Darcy 逼近阶的参数一致边界层估计。放松切向-法向连续性可得到一个代数等价的应力杂交格式和一个无需稳定化的虚拟元实现。
4. 多组分囊泡的热力学一致模型
A Thermodynamically Consistent Model for Multicomponent Vesicles
作者: Wangbo Luo, Zhonghua Qiao, Yanxiang Zhao
类别: math.NA, math-ph
发布日期: 2026-09-09
arXiv ID: 2609.10168v1
链接: http://arxiv.org/abs/2609.10168v1
摘要翻译
中文标题: 多组分囊泡的热力学一致模型
摘要翻译: 我们建立了一个用于多组分膜的热力学一致扩散界面模型。所提出的模型由耦合自由能泛函导出，该泛函包含依赖蛋白质的弯曲弹性、扩散表面张力、体积惩罚以及与膜相关的 Ohta–Kawasaki 能量。应用 Onsager 变分原理，我们推导了一个耦合的 $L^2$ 梯度流系统及其能量耗散律。随后，我们构造了一个稳定化交替 ETD1 格式，以及一个具有 Strang 型复合的稳定化交替 ETDRK2 格式（交替 Strang-ETDRK2）。据我们所知，所提出的交替 Strang-ETDRK2 格式此前尚未针对耦合相场系统被构造和分析。我们进一步在数值解满足若干正则性假设的条件下，证明了两个格式的离散能量耗散。二维和三维数值实验验证了离散能量耗散律，并展示了所提模型产生的蛋白质分离和膜变形。
5. 多边形 Stokes 与 BGG Hessian 复形的分析性质及其在 Kirchhoff–Love 板中的应用
Analytical properties of polygonal Stokes and BGG Hessian complexes with application to Kirchhoff–Love plates
作者: Daniele A. Di Pietro, Jérôme Droniou, Kaibo Hu, Arax Leroy
类别: math.NA
发布日期: 2026-09-09
arXiv ID: 2609.10150v1
链接: http://arxiv.org/abs/2609.10150v1
摘要翻译
中文标题: 多边形 Stokes 与 BGG Hessian 复形的分析性质及其在 Kirchhoff–Love 板中的应用
摘要翻译: 我们开发并分析了一种新的任意阶方法，用于一般多边形网格上的 Kirchhoff–Love 板问题。该构造依赖于通过 Bernstein–Gelfand–Gelfand 构造得到的离散 Hessian 复形，该复形将一个 Stokes 复形叠加在一个张量化的 de Rham 复形之上。分析的一个关键组成部分是对前者的详细研究。我们具体建立了原始与伴随相容性估计，以及一致 Poincaré 不等式，并利用这些结果推导出所得离散 Hessian 复形的分析性质。随后证明所提方法具有强制性，并关于网格尺寸以 $k+1$ 阶收敛，其中 $k\ge 0$ 是该复形的多项式次数。多边形网格上的数值实验验证了理论收敛速率。
6. 多物种种群超线性非局部扩散系统：适定性与离散链式法则
Superlinear nonlocal diffusion systems for multispecies populations: well-posedness and discrete chain rules
作者: Peter Hirvonen, Ansgar Jüngel, Annamaria Pollino
类别: math.AP, math.NA
发布日期: 2026-09-09
arXiv ID: 2609.09921v1
链接: http://arxiv.org/abs/2609.09921v1
摘要翻译
中文标题: 多物种种群超线性非局部扩散系统：适定性与离散链式法则
摘要翻译: 本文在一个有界区域中分析了一个非局部扩散系统，该系统描述多物种种群的动力学并逼近 Shigesada-Kawasaki-Teramoto (SKT) 模型。该模型由 Andreu-Mazón-Rossi-Toledo 型积分微分方程组构成，其中非局部交叉扩散算子作用于非线性扩散势。我们建立了强解的全局存在性和唯一性。存在性证明依赖于由熵不等式导出的适当先验估计，而该熵不等式的证明需要发展新的离散链式法则不等式。作为副产品，这些不等式为相应的局部 SKT 系统构造保结构有限体积格式提供了基础。解的唯一性通过一种对偶论证建立。最后，一维数值模拟展示了非局部到局部极限以及超线性和次线性扩散势下解的定性行为。
7. 流形上并行区域分解方法的几何收敛性
The geometric convergence of a parallel domain decomposition method on manifolds
作者: Lizhen Qin
类别: math.NA, math.DG
发布日期: 2026-09-09
arXiv ID: 2609.09911v1
链接: http://arxiv.org/abs/2609.09911v1
摘要翻译
中文标题: 流形上并行区域分解方法的几何收敛性
摘要翻译: 本文为流形上椭圆方程的连续区域分解方法建立了一个收敛性理论。该方法起源于 \cite{lions2} 中欧几里得区域的情形，后来由 \cite{qin_wang_wang} 调整并推广到流形上。尽管其收敛性已知，但其收敛是否为几何收敛一直悬而未决。我们证明其几何收敛性，并给出关于收敛速度的各种估计。
8. 一般流形上标量值与向量值偏微分方程的内蕴有限元框架
An intrinsic finite element framework for scalar- and vector-valued partial differential equations on general manifolds
作者: Tamara A. Tambyah, Alberto F. Martín, David Lee, Santiago Badia
类别: math.NA
发布日期: 2026-09-09
arXiv ID: 2609.09592v1
链接: http://arxiv.org/abs/2609.09592v1
摘要翻译
中文标题: 一般流形上标量值与向量值偏微分方程的内蕴有限元框架
摘要翻译: 我们提出了一个内蕴有限元框架，用于对由坐标卡图册描述的一般流形上的标量值与向量值偏微分方程进行数值逼近。弱形式及其离散化完全在流形的平坦参数空间中表达，其中精确几何仅通过求积点处计算的度量张量进入。通过纯拓扑的坐标卡间粘合，在多坐标卡图册上获得了整个 de Rham 复形的任意阶协调有限元空间。我们为多坐标卡图册上的向量值空间发展了节点基变换构造，其中节点自由度通过传递映射耦合具有不相容坐标系的坐标卡。坐标卡间连续性在节点处成立，其跳跃为逼近误差量级，并且所得的离散场精确切于流形。对于表面 Stokes 问题，数值上观察到该方法的最优收敛阶；未提供 a priori 误差分析。在二维和三维立方球流形上针对 Hodge Laplacian 问题的数值实验表明，对于足够的求积阶数，该内蕴框架没有几何相容性误差，并且内蕴装配比其外蕴对应方法便宜得多。进一步考虑将地形作为几何扰动的旋转浅水方程，表明质量精确守恒，而能量守恒精确到时间离散误差。
9. 具有相互作用势的收敛反应-漂移-扩散主方程
A Convergent Reaction-Drift-Diffusion Master Equation with Interaction Potentials
作者: Max Heldman, Samuel A. Isaacson
类别: math.NA
发布日期: 2026-09-09
arXiv ID: 2609.09546v1
链接: http://arxiv.org/abs/2609.09546v1
摘要翻译
中文标题: 具有相互作用势的收敛反应-漂移-扩散主方程
摘要翻译: 我们将非结构化网格上的收敛反应-扩散主方程（CRDME）框架推广到包含粒子间两体相互作用势。此类相互作用对于模拟粒子经历静电排斥、体积排斥或范德华吸引的系统至关重要。我们的方法对基于粒子的随机反应-漂移-扩散模型的前向方程的弱形式进行离散，通过结合两种互补技术来获得一个新的 CRDME：一种带质量集中求积的边平均有限元方法，用于包含势相互作用的空间输运项；以及有限体积与质量集中求积近似的混合，用于反应项。这给出了粒子在对偶网格体元之间跳跃并发生反应的空间跳跃过程的转移速率。所得的空间离散 CRDME 在闭区域中保留了连续系统的关键物理性质。在无反应时，它保持吉布斯-玻尔兹曼平衡分布；对于具有可逆反应的系统，它满足漂移-扩散通量和反应通量的细致平衡。与布朗动力学模拟的数值比较验证了我们的方法，并且我们展示了当网格细化时的经验二阶收敛性。
不相关论文（基本信息）
1. 有限元空间的一种拟插值方法
A quasi-interpolation method for finite element spaces
作者: Ohannes Karakashian
类别: math.NA
发布日期: 2026-09-09
arXiv ID: 2609.10374v1
链接: http://arxiv.org/abs/2609.10374v1
摘要: A combination of Taylor polynomial approximation and averaging results in a powerful quasi-interpolation technique that can be used to analyze the approximation properties of finite element spaces. With this particular approach, the notion of affine equivalent families is completely bypassed in the sense that the degrees of freedom are defined locally rather than being exported from a reference element. In particular, domains with curved boundaries can be handled in a natural manner. Finally, the ease of its application to a particular finite element makes it a valuable pedagogical tool.
2. 量子线性系统求解器的迭代精化分析
An analysis of iterative refinement for quantum linear system solvers
作者: Adrian Harkness, Mohammadhossein Mohammadisiahroudi, Brandon Augustino, Giacomo Nannicini, Tamás Terlaky
类别: quant-ph, math.NA
发布日期: 2026-09-09
arXiv ID: 2609.10250v1
链接: http://arxiv.org/abs/2609.10250v1
摘要: We present and analyze an iterative refinement (IR) framework for improving the precision dependence of algorithms that combine a quantum linear system algorithm (QLSA) with quantum state tomography. Existing QLSAs achieve polylogarithmic dependence on the inverse error tolerance to prepare a quantum state encoding the solution, but extracting a classical description of such a state via tomography typically introduces a polynomial dependence on the target precision. To retain polylogarithmic dependence in the inverse error tolerance throughout the entire process, we develop an IR scheme that solves a sequence of related linear systems to progressively refine the solution while requiring only fixed-precision quantum subroutines, finally obtaining a classical description of a high-precision solution. We analyze the proposed framework under three input models: quantum-read/classical-write RAM (QRAM), linear combinations of unitaries (LCU), and sparse-access oracles. We evaluate the proposed scheme through numerical experiments on both quantum simulators and real quantum hardware. The results demonstrate that iterative refinement efficiently improves the precision of the solution and exhibits robustness to hardware noise.
3. 用于瞬态动力学的直接自启动子步隐式积分器的升阶
Order elevation of directly self-starting sub-step implicit integrators for transient dynamics
作者: Jinze Li, Yaokun Liu, Kewei Chen, Hua Li, Kaiping Yu
类别: math.NA, physics.comp-ph
发布日期: 2026-09-09
arXiv ID: 2609.10240v1
链接: http://arxiv.org/abs/2609.10240v1
摘要: Directly self-starting implicit methods are attractive for transient analysis because they avoid auxiliary starting procedures while retaining the original first- or second-order governing equations. However, most existing formulations usually fix the last sub-step at the end of each time interval, which restricts the attainable order. This study develops a generalized $s$-sub-step implicit framework by releasing this constraint and treating all sub-step locations as design variables. The resulting methods admit a unified Runge–Kutta representation for both first- and second-order transient systems and preserve identical effective matrices over all sub-steps. Accuracy conditions are derived by simultaneously matching the numerical amplification factor and load operator, thereby accounting for both homogeneous and forced responses. For $s=1,~\cdots,~6$, two complementary families are obtained: $s$th-order members with user-controllable high-frequency numerical dissipation and adjustable sub-step locations, and $(s+1)$th-order members obtained by selecting the sub-step locations, with fixed dissipation. The latter reach up to seventh-order accuracy without increasing the number of sub-steps, although some high-order members are $A(α)$-stable with stability angles extremely close to $90^\circ$. Analytical amplitude and phase errors further reveal parity-dependent superconvergence in undamped systems, and appropriate parameter selections can substantially increase either phase or amplitude accuracy beyond the formal order. Numerical benchmarks confirm the predicted convergence orders and the controllable suppression of spurious high-frequency responses.
4. 关于点-回旋曲线距离算法的注记
A Note on the Point-Clothoid Distance Algorithm
作者: Haibin Ye, Hao Ge, Gong Cheng
类别: math.NA, cs.CG
发布日期: 2026-09-09
arXiv ID: 2609.10179v1
链接: http://arxiv.org/abs/2609.10179v1
摘要: Computing the closest point on a clothoid is a recurring task in geometric design, road and railway alignment, and path planning. The efficient algorithm of Frego and Bertolazzi addresses this problem, but its candidate-selection analysis assumes at most one local minimum per search interval. We exhibit admissible configurations with two local minima, raising the question of whether the existing strategy accounts for every possible minimum. Using the geometry of the clothoid evolute, we prove that, for any query point and any proper no-inflection planar clothoid segment with tangent-angle variation at most $2π$, the squared-distance function has at most three stationary points; if all three are local extrema, their order is min-max-min. This establishes the completeness of the original candidate-selection logic beyond the one-minimum premise. It also shows that no interior search is needed when neither endpoint derivative test is active, allowing unnecessary midpoint searches to be omitted while retaining numerical fallback. Numerical experiments demonstrate reductions in iteration count and evaluation time.
5. 用于抑制显式等几何分析中裁剪诱导的高频异常值的边界层级约束细化
Boundary-Level-Constrained Refinement for Suppressing Trimming-Induced High-Frequency Outliers in Explicit Isogeometric Analysis
作者: Christoph Hollweck, Lukas Leidinger, Stefan Hartmann, Marcus Wagner, Roland Wüchner
类别: math.NA
发布日期: 2026-09-09
arXiv ID: 2609.10161v1
链接: http://arxiv.org/abs/2609.10161v1
摘要: In explicit dynamics, the critical time step is governed by the maximum eigenfrequency of the semi-discrete system. In Isogeometric Analysis (IGA), open knot vectors introduce a characteristic spectral boundary effect that can lead to high-frequency outliers and restrict the admissible time step. In the row-sum-lumped setting considered here, extending the computational patch and trimming away the exterior boundary functions mitigates this time-step penalty, but does not necessarily prevent boundary-adjacent basis functions from governing the maximum eigenfrequency. We show that reduced-support basis functions adjacent to the trimming boundary can remain critical spectral contributors even for knot-exact trimming, i.e., in the absence of small cut cells. Hence, small cut cells are not required for trimming-induced time-step penalties. Knot-exact and arbitrary trimming represent different degrees of support reduction within the same underlying mechanism. To control this effect, we propose the Boundary-Level-Constrained Refinement (BLCR) strategy, a local refinement constraint for LR- and THB-splines. BLCR constrains the refinement level of basis functions whose support intersects the trimming boundary relative to that of the refined interior. In all configurations investigated in this work, this constraint ensures that the maximum eigenfrequency is governed by untrimmed interior basis functions rather than by refined trimmed functions. Consequently, the trimming-induced high-frequency outliers governing the critical time step are suppressed, yielding a larger admissible time step than globally refined trimmed B-splines with the same interior resolution.[…]
6. 加筋铝制翼盒壁板中的连接非线性及其对降阶基的要求
Joint nonlinearity in a stiffened aluminium wingbox panel and what it requires of a reduced basis
作者: Nikolaos D. Tantaroudas, Evangelos Papatheou
类别: cs.CE, eess.SY, math-ph, math.NA
发布日期: 2026-09-09
arXiv ID: 2609.10088v1
链接: http://arxiv.org/abs/2609.10088v1
摘要: A shell finite-element model of a stiffened aluminium wingbox panel, a laboratory structure shared by several experimental studies, is built with its plate-to-stiffener joints at the $78$ physical fastener positions and calibrated against measured modal data, a six-parameter sensitivity update of the substructure moduli within $\pm20\%$ reproducing the first five measured modes to $1.1$–$4.3\%$ with modal-assurance values of $0.88$–$1.00$. Making the fasteners nonlinear, at equal mesh, mass and damping, shows that joint nonlinearity enters the response strongly asymmetrically. Hardening is almost invisible, at most $+1\%$ in frequency, whereas softening or slipping the joints moves the first three resonances by $-2.1$, $-4.9$ and $-9.2\%$, and friction removes up to $80\%$ of the resonant peak before it recovers. Newmark integration of all $18{,}804$ degrees of freedom and a harmonic balance condensed exactly onto the joints and continued in arclength agree to $5.5\times10^{-4}$. A projection-based nonlinear model order reduction then locates the criterion that a jointed structure imposes on a reduced basis. The binding quantity is not the linear response, which eleven vectors reproduce to better than $0.01\%$, but the receptance of the structure between the joints, of which $126$ global eigenvectors carry about $2\%$, and without which the reduced model overpredicts the hardening shift of the fundamental by a factor of thirty and returns a value beyond the rigid-joint limit of the panel.
7. 线性代数中计算不确定性的随机 Lanczos 求积
Stochastic Lanczos Quadrature for Computational Uncertainty in Linear Algebra
作者: Hassan Fifen, Issa Karambal, Philipp Hennig
类别: math.NA
发布日期: 2026-09-09
arXiv ID: 2609.10080v1
链接: http://arxiv.org/abs/2609.10080v1
摘要: Applying a matrix function to a vector is a common operation in large-scale scientific computing, such as Bayesian inference. When the matrix is symmetric positive definite and the function is its inverse, the Lanczos algorithm gives a matrix-free approximation from matrix-vector products alone. Stopping after a fixed number of steps keeps the approximation on the explored Krylov subspace but assigns nothing to its orthogonal complement, which holds the smallest eigenvalues and therefore the largest contributions to the inverse. This truncation is usually treated as a silent numerical error rather than as measurable uncertainty. We show that a single Lanczos run yields both a matrix-free approximation of the inverse and a Gaussian distribution over the exact result, whose covariance measures what the truncation discarded and concentrates as the number of steps grows. The covariance restores the complement in two parts: a rank-one coupling at the truncation boundary, given exactly by the Lanczos residual, and an isotropic term on the remaining bulk, whose size we estimate by Projected Stochastic Lanczos Quadrature, reusing one Krylov basis across all probes. We prove consistency, with bias decaying exponentially in the quadrature depth and variance at the Monte Carlo rate
8. McKean-Vlasov 随机微分方程的单集合多水平格式的收敛性
Convergence of a single-ensemble multilevel scheme for McKean-Vlasov SDEs
作者: Arne Bouillon, Giovanni Samaey
类别: math.NA, math.PR
发布日期: 2026-09-09
arXiv ID: 2609.10000v1
链接: http://arxiv.org/abs/2609.10000v1
摘要: Numerically solving McKean-Vlasov stochastic differential equations is computationally challenging due to the compounding costs of discretizing in time and in the distribution of the solution. Multilevel ideas have been proposed to provide speed-ups. In this work, we study the multilevel Monte Carlo method proposed by Ricketson (2015) for equations whose drift and diffusion terms depend on the law of the solution $X_t$ through the expectation $E[R(X_t)]$. The scheme follows the single-ensemble paradigm, where particles interact across levels at each timestep. While cross-level feedback makes this scheme attractive in practice, the correlations it introduces have so far confined its cost-error analysis to a model problem with linear drift, deterministic diffusion, and $R$ the identity. We use additional coarse particles to enforce geometrically decaying coupling errors towards the coarser levels. This allows us to prove our main contribution, an $L^p$-error of $O(ε)$ at cost $O(ε^{-2-δ})$ for any $p\ge2$ and $δ>0$ (with a constant that grows as $δ\to0$), assuming only global Lipschitz bounds on the drift, the diffusion, and $R$. An exploratory experiment is consistent with the derived rates and suggests that in practice the constant does not grow significantly for small $δ$. Our methodology and proof strategy may also be useful for other single-ensemble multilevel schemes, such as multilevel ensemble Kalman filters.
9. 指数 Runge–Kutta 方法的阶数降低：非交换算子的四阶格式
Order Reduction of Exponential Runge–Kutta Methods: Fourth-Order Schemes for Non-Commuting Operators
作者: Thi Tam Dang, Pablo Alexei Gazca-Orozco, Trung Hau Hoang
类别: math.NA
发布日期: 2026-09-09
arXiv ID: 2609.09932v1
链接: http://arxiv.org/abs/2609.09932v1
摘要: This paper extends the convergence analysis of explicit exponential Runge–Kutta methods for linear parabolic problems $u'(t) + Au(t) = Bu(t)$, where $A$ generates an analytic semigroup and $B$ is relatively bounded with respect to $A$, from the third-order case to fourth-order schemes. By establishing the global error recursion relation and extending the defect-based analytical framework, we identify the terms responsible for stiff order reduction when $A$ and $B$ do not commute. Numerical experiments are performed on a non-commuting advection-diffusion problem to validate the theoretical results. Numerical tests using the classical four-stage, fourth-order schemes of Krogstad and Strehmel & Weiner exhibit an observed convergence order of approximately 2.75, which matches the theoretical prediction from the convergence analysis.
10. 针对倾斜绝对值的依赖于初始化的 BFGS 试探速率
Initialization-dependent BFGS trial rates for tilted absolute values
作者: Qiuyu Chen
类别: math.NA, math.OC
发布日期: 2026-09-09
arXiv ID: 2609.09872v1
链接: http://arxiv.org/abs/2609.09872v1
摘要: Lewis and Overton conjectured in 2008 that, for BFGS applied to the one-dimensional tilted absolute value with their Armijo–Wolfe line search, every nonterminating execution converges at a trial-normalized rate determined only by the tilt parameter and independent of the initial data. We prove that this initialization-independent rate assertion is false. In fact, for every tilt parameter in an explicit open interval, we construct two nonterminating executions with different sharp trial-normalized convergence rates. The proof reduces the line search to a scale-free state consisting of the iterate sign and the zero-crossing stepsize, and verifies two periodic state cycles by elementary rational inequalities. This yields an explicit family of counterexamples and shows that the asymptotic trial-normalized behavior of nonsmooth BFGS can depend essentially on the initialization even for this canonical one-dimensional model.
11. 一种实用的面向中等规模黑箱全局优化的 DIRECT 型算法
A practical DIRECT-type algorithm for medium-scale black-box global optimization
作者: Linas Stripinis, Remigijus Paulavičius
类别: cs.LG, math.NA
发布日期: 2026-09-09
arXiv ID: 2609.09796v1
链接: http://arxiv.org/abs/2609.09796v1
摘要: The DIRECT algorithm is a deterministic global optimization method known for its versatility and balanced exploration-exploitation strategy. However, DIRECT-type algorithms are primarily effective for low-dimensional problems and often exhibit slow convergence as dimensionality increases, limiting their applicability to more complex optimization tasks. To address this limitation, this paper introduces X-DTC-GL, a novel DIRECT-type algorithm that incorporates dynamic partitioning and hybridization techniques. The dynamic partitioning approach adaptively refines the search space based on local one-dimensional surrogate models, enabling rapid subdivision of promising hyper-rectangles. The hybridization strategy selectively employs a hill-climbing method to exploit promising regions identified by the surrogate models. Extensive experiments on four diverse benchmark suites demonstrate that X-DTC-GL significantly outperforms existing DIRECT-type baselines, achieving improvements of ~12% in solvability and ~27% in solution quality. Performance-profile analyses indicate the fastest convergence on up to ~40% of instances, the best runtime performance on ~17% of problems, and competitive overall execution times. By improving performance within the partition-based framework, these advances strengthen the algorithm’s competitiveness in state-of-the-art black-box optimization.
12. 满足变差型条件的函数的一致切比雪夫逼近：图上时变信号的重构
Uniform Chebyshev approximations of functions satisfying a variation-type condition: reconstruction of time-varying signals on graphs
作者: Davide Bianchi, Sandra Saliani, Dimitrios Vavitsas
类别: math.NA, math.FA
发布日期: 2026-09-09
arXiv ID: 2609.09792v1
链接: http://arxiv.org/abs/2609.09792v1
摘要: We prove uniform approximation theorems for Chebyshev expansions of continuous functions of two variables. Under a variation-type condition on the square $[-1,1]^2$, a continuous function admits a uniformly convergent Chebyshev expansion in the first variable whose coefficients are continuous functions of the second variable. These results are applied to the spectral graph wavelet transform of time-varying signals on finite weighted graphs: the scaling and wavelet kernels are approximated by expansions with time-varying coefficients whose degrees do not depend on time, the composition of the approximate transform with its adjoint admits the same explicit coefficient formulas as in the time-independent case, and reconstruction by the pseudoinverse is stable, with explicit bounds in terms of the uniform kernel errors. Numerical experiments on a sensor network confirm the convergence and stability estimates. In a denoising problem, soft thresholding of the graph wavelet coefficients with a time-varying transform parameter improves over its time-independent, fixed-parameter counterpart.
13. 带控制与状态约束的稀疏最优控制中 BDDC 预处理的函数空间框架
A Function-Space Framework for BDDC Preconditioning in Control- and State-Constrained Sparse Optimal Control
作者: Harbir Antil, Keegan L. A. Kirk, Umarkhon Rakhimov
类别: math.OC, math.NA
发布日期: 2026-09-08
arXiv ID: 2609.09541v1
链接: http://arxiv.org/abs/2609.09541v1
摘要: We develop a Balancing Domain Decomposition by Constraints (BDDC) preconditioner for the interface systems arising from active-set semi-smooth Newton linearizations of elliptic optimal control problems with box-constrained controls, $L^1$-sparsity, and Moreau–Yosida regularized state constraints. The domain decomposition and BDDC construction are formulated directly at the infinite-dimensional level. Eliminating the interior variables yields a Schur complement for the state and adjoint traces assembled from local subdomain operators. We prove a global–local equivalence result, establish well-posedness of the local and partially assembled problems under verifiable conditions on the primal constraints. The BDDC operator admits a two-level additive Schwarz representation with independent local solves and a finite-dimensional coarse solve. We implement the preconditioner using conforming finite elements and test it for distributed and Neumann boundary control. For a fixed ratio of subdomain diameter $H$ to mesh size $h$, the GMRES iteration counts are nearly independent of mesh refinement and grow only moderately as this ratio increases. Enriching the coarse space substantially improves convergence and reduces sensitivity to the control regularization and sparsity weights. For the state-constrained problem, scaling the Moreau–Yosida penalty parameter proportionally to $h^2$ yields nearly constant GMRES iteration counts as the mesh is refined.
14. 构造性 Tchakaloff 结果与基于随机最小二乘的良态求积
Constructive Tchakaloff results and well-conditioned quadrature through randomized least squares
作者: Filip Bělík, Akil Narayan, John Turnage
类别: math.NA
发布日期: 2026-09-08
arXiv ID: 2609.09506v1
链接: http://arxiv.org/abs/2609.09506v1
摘要: We consider using randomized least squares to construct quadrature rules exact on a subspace of functions. Using new conditions that we call relative admissibility and reference weight concentration, we establish both that the quadrature weights from such a procedure concentrate close to their asymptotic values with prescribed and arbitrarily large probability, and this in turn provides useful finite-sample probabilistic bounds on the stability of the resulting quadrature rules for very general classes of possibly complex-valued functions. Our analysis both significantly generalizes the existing analysis of randomized least squares quadrature construction, and provides new bounds on stability for these rules. These results specialize to existence results for positive quadrature rules, when such rules can be theoretically expected. Because our procedures are formally algorithmic, our analysis is a substantive advance toward computationally constructive generalized Tchakaloff theorems.
15. 从速度测量恢复压力与应力的基于应力的估计子
A Stress-Based Estimator for Pressure and Stress Recovery from Velocity Measurements
作者: Nicolás Barnafi, Benjamín Cisternas, Ernesto Castillo, Felipe Galarce
类别: math.NA
发布日期: 2026-09-08
arXiv ID: 2609.09484v1
链接: http://arxiv.org/abs/2609.09484v1
摘要: Non-invasive pressure field estimation from velocity measurements is a longstanding engineering problem. We propose, analyze, and test a pressure-recovery method that computes a full stress field from velocity measurements, and leaves the pressure estimation as a cheap post-processing step. The method relies on a stress-velocity first order formulation of the Navier-Stokes equations, and we show that the formulation accounts for deviations from incompressibility in the measured velocity field by construction. In addition, we theoretically establish the convergence of the finite element (FE) approximation scheme, the stability of the stress recovery with respect to finite-resolution velocity measurements, and then validate this theory numerically. Our results show that the proposed estimator is robust in convective flow regimes and remains accurate at reduced spatial resolution, improving upon state-of-the-art pressure-recovery strategies.
16. 用于低秩量子态层析的块张量列车 Burer-Monteiro 框架
A Block Tensor Train Burer-Monteiro Framework for Low-Rank Quantum State Tomography
作者: Shakir Showkat Sofi, Charlotte Vermeylen, Fatemeh Mohammadi, Lieven De Lathauwer
类别: quant-ph, cs.LG, eess.SP, math.NA, math.OC
发布日期: 2026-09-08
arXiv ID: 2609.09457v1
链接: http://arxiv.org/abs/2609.09457v1
摘要: Quantum state tomography is a fundamental technique for estimating the state of a quantum system from measured data and plays a crucial role in evaluating the performance of quantum devices. However, standard estimation methods become computationally prohibitive as the system size increases due to the exponential growth of the density matrix, describing a quantum state, with the number of qubits. We propose a low-rank tensor-network framework for mixed-state quantum state tomography based on a block tensor train (Block-TT) factorization. Specifically, the density matrix is represented as the contraction of a Block-TT with its Hermitian transpose, yielding a TT analogue of the Burer-Monteiro factorization. This parameterization guarantees Hermiticity and positive semidefiniteness by construction while compressing the number of optimization variables from exponential to linear in the number of qubits. Building on this representation, we develop single-site and two-site density matrix renormalization group (DMRG) algorithms for estimating quantum states from compressed measurements. The resulting methods operate directly on the compressed parameterization, support adaptive rank refinement, and exploit efficient tensor-network contractions for expectation-value evaluation. The framework is applicable to a broad class of low-rank quantum states, including pure states, nearly pure states, and ground states that admit accurate tensor-network approximations. Numerical experiments demonstrate accurate state reconstruction from limited measurements together with substantial reductions in memory requirements and computational cost compared with conventional low-rank tomography methods.
17. 基于迭代重加权最小二乘的固定费用网络流支撑集发现
Support Discovery With Iteratively Reweighted Least Squares for Fixed-Charge Network Flow
作者: Sindura Saraswathi, Christian Kümmerle
类别: math.OC, cs.AI, cs.LG, math.NA
发布日期: 2026-09-08
arXiv ID: 2609.09295v1
链接: http://arxiv.org/abs/2609.09295v1
摘要: The fixed-charge network flow problem (FCNFP) couples continuous flow allocation with discrete arc-activation decisions, making it a canonical but computationally challenging model for a variety of network design and resource allocation problems. Exact mixed-integer linear programming formulations capture the fixed-charge structure faithfully, but often become difficult to solve on large networks. We propose a scalable continuous-optimization algorithm for large-scale single-commodity FCNFP based on an iteratively reweighted least-squares (IRLS) framework. The method replaces the discontinuous fixed-charge and linear arc cost objective with a smooth nonconvex Lasry–Lions surrogate and solves a sequence of weighted quadratic flow subproblems. Each subproblem is solved by a warm-started dual semismooth Newton method whose Newton systems have weighted graph-Laplacian structure, enabling the use of modern Laplacian solvers. To further improve the discovered arc supports of the challenging underlying combinatorial problem, we also develop an algorithmic variant that incorporates objective-driven perturbation restarts and an anchor-union restricted search that jointly leverages supports discovered by IRLS and by complementary FCNFP heuristics. Computational experiments on 410 benchmark, synthetic, and large-scale instances show that our method obtains the best objective quality among the evaluated scalable FCNFP algorithms, with a mean gap of $1.316\%$ to a time-limited MILP reference and a win-or-tie rate of $90.0\%$ among the non-MILP methods. The results indicate that combining smooth continuous optimization with support-level search is an effective strategy for producing high-quality feasible solutions to large-scale FCNFP.