生成时间: 2026年07月30日 12:01:36

重点关注论文数量: 0

了解领域论文数量: 4

不相关论文数量: 19

了解领域论文（摘要翻译）

1. 超导伦敦方程的数值方法

Numerical approach to the London Equation of superconductivity

作者: Nicolás Barnafi, Ignacio Labarca-Figueroa, Carlos Román

类别: math.NA, math-ph, math.AP

发布日期: 2026-07-29

arXiv ID: 2607.27103v1

链接: http://arxiv.org/abs/2607.27103v1

摘要翻译

中文标题: 超导伦敦方程的数值方法

摘要翻译: 本文为求解全空间 $\mathbb{R}^3$ 中第二类超导体的伦敦方程提出一种通用的离散化策略。为了计算磁场 $H_0$，我们将磁位势问题重新表述为传输问题，并通过非标准有限元-边界元耦合方法对其进行离散。该表述同时考虑了有界内部区域和无界外部区域，且无需引入人工截断。随后，我们计算矢量场 $B_0$，它源自超导样品中磁位势的 Helmholtz-Hodge 分解。该场进入等通量问题，后者确定在 Ginzburg–Landau 超导模型中涡旋成核首次在能量上有利的曲线。我们使用 Kikuchi 的混合形式重写 $B_0$ 的方程，在该形式中无散度约束被弱施加，并采用经典的 $H(\operatorname{curl})$-协调有限元离散方法对所得问题进行离散。我们通过收敛性测试验证了离散化策略，并以等通量问题的应用作为总结。对于恒定外加磁场下的球体，唯一的极大化元是与磁场方向对齐的直径。对于恒定外加磁场与长轴对齐的椭球，我们的计算提供了在充分拉长的雪茄状几何体中不同行为的数值证据：非轴向竞争者（让人联想到 U 形涡旋构型）获得了比长轴更大的等通量比。因此，长轴并非极大化元；任意非轴向极大化元通过旋转对称性生成一个连续的等价构型族，这意味着等通量问题中的非唯一性和退化旋转方向的存在。

2. 流函数形式曲面Stokes问题的稳定化Morley有限元方法：基于新几何估计的最优收敛性

Stabilized Morley FEM for surface Stokes in stream-function formulation: Optimal convergence via a new geometric estimate

作者: Shuonan Wu, Hao Zhou

类别: math.NA

发布日期: 2026-07-29

arXiv ID: 2607.26664v1

链接: http://arxiv.org/abs/2607.26664v1

摘要翻译

中文标题: 流函数形式曲面Stokes问题的稳定化Morley有限元方法：基于新几何估计的最优收敛性

摘要翻译: 我们针对闭曲面上的流函数形式的曲面Stokes问题，提出了一种内在稳定化Morley有限元方法。该方法直接建立在曲面的多面体逼近上。引入了一个无参数的跳跃稳定化项以恢复强制性，并建立了关于无迹Hessian矩阵的离散Korn不等式。主要分析贡献是针对分片线性逼近闭曲面的一个法向分离几何估计。该估计表明，尽管直接处理仅显示一阶控制，但一大类依赖于法向的几何相容误差实际上是二阶的。丢失的阶数通过第一法向变分中的积分相消得以恢复。此估计改进了标准的$P_hν$-型估计，并对无迹Hessian矩阵、Stokes型张量Green恒等式以及离散化中出现的余法向流量给出了二阶相容性。结合离散Korn不等式，这些估计在自然的$H^3$正则性下得到了最优阶收敛：在破碎$H^2$范数下为一阶，在破碎$H^1$范数下为二阶。作为直接推论，恢复的切向速度场具有二阶$L^2$估计。提供了数值实验以支持理论结果。

3. 使用分段线性参数有限元的Dziuk方法用于曲线缩短流的稳定化与最优$L^2$收敛性

Stabilization and optimal $L^2$ convergence of Dziuk’s method with piecewise linear parametric finite elements for curve-shortening flow

作者: Genming Bai

类别: math.NA

发布日期: 2026-07-28

arXiv ID: 2607.26290v1

链接: http://arxiv.org/abs/2607.26290v1

摘要翻译

中文标题: 使用分段线性参数有限元的Dziuk方法用于曲线缩短流的稳定化与最优$L^2$收敛性

摘要翻译: 我们提出了一个针对闭合平面曲线的曲线缩短流的全离散Dziuk方法的稳定化版本，其中使用了分段线性参数有限元。通过精心设计的稳定项，我们能够在抛物尺度 $τ\simeq h^2$ 下，展示出Barrett–Garcke–Nürnberg (BGN) 类型的令人惊讶的离散切向稳定性——这一特性在连续层面是隐藏的。结合线性单元反向平均法向量的一个新超逼近结果，这种Dziuk类型的离散切向稳定性产生了最优的 $L^2$ 收敛性。

4. 弹性动力学保结构离散中狄利克雷边界速度的强施加

Strong imposition of Dirichlet boundary velocities in structure-preserving discretizations of elastodynamics

作者: Cristobal Ponce, Hector Ramirez, Yongxin Wu, Ning Liu, Yann Le Gorrec

类别: cs.CE, math.NA, physics.comp-ph

发布日期: 2026-07-28

arXiv ID: 2607.26248v1

链接: http://arxiv.org/abs/2607.26248v1

摘要翻译

中文标题: 弹性动力学保结构离散中狄利克雷边界速度的强施加

摘要翻译: 在端口哈密顿弹性动力学的有限元模型中，边界速度的施加通常依赖于拉格朗日乘子，从而导出微分代数方程。另一种保持常微分方程结构的弱施加方法往往在狄利克雷边界处精度较差。为克服这些局限，本文在连续层面引入加性运动学分解，将位移场和速度场分裂为在边界上为零的相对动态分量和延伸到内部区域的指定提升函数。该分解导出一个分布端口，将边界驱动的作用映射到区域内部。通过将该映射纳入适当的虚功率原理，我们推导出提升的端口哈密顿系统模型，经有限元离散后约化为常微分方程组，其中狄利克雷边界速度被强施加。该框架被用于导出适用于不同端口哈密顿系统几何表示的二场和四场公式。此外，我们表明在特定形函数下，可恢复标准的有限元格式，证明离散模型中的提升框架等价于计算力学实践中经典的代数矩阵分块。所提方法的能量平衡性质和计算性能通过数值模拟得到验证。

不相关论文（基本信息）

1. 从分类到回归：利用果蝇求解方程

From Classification to Regression: Using a Fruitfly to Solve Equations

作者: Shady E. Ahmed, Panos Stinis

类别: cs.LG, math.NA

发布日期: 2026-07-29

arXiv ID: 2607.27196v1

链接: http://arxiv.org/abs/2607.27196v1

摘要: We present a novel approach to regression tasks using classification which is motivated by the mechanism used by fruitflies to sense their environment. Specifically, we formulate a general framework for learning nonlinear input-output relationships by replacing complex global surrogate models with a finite library of representative local patterns. Since scientific data often occupy limited and recurring regions of the input space, we generate predictions by measuring similarities between a query and stored patterns, then combining their associated responses through weighted reconstruction. We apply this approach to nonlinear dynamical systems, data-driven regression, and physics-informed learning using suitable embeddings and similarity measures. For dynamical systems, our offline-online workflow extracts patterns from data or governing equations during the offline phase, while online prediction requires only similarity evaluation and response aggregation. This structure helps us reduce computational and memory demands while providing explicit control over the trade-off among accuracy, storage, and inference cost.

2. 熵守恒的分部求和离散对双曲守恒律光滑解的收敛性

Convergence of entropy-conservative summation-by-parts discretizations to smooth solutions of hyperbolic conservation laws

作者: Hendrik Ranocha

类别: math.NA

发布日期: 2026-07-29

arXiv ID: 2607.27049v1

链接: http://arxiv.org/abs/2607.27049v1

摘要: Although entropy-based summation-by-parts (SBP) discretizations of hyperbolic conservation laws are widely used for their robustness and stability properties, there are very few results on their convergence. We extend a recent convergence analysis of Worku, Del Rey Fernández, and Zingg (2026, DOI: 10.48550/arXiv.2603.18369) in two ways. First, instead of allowing only hyperbolic conservation laws whose fluxes are homogeneous and have globally bounded second derivatives (a restriction essentially to linear or quadratic fluxes), we consider general hyperbolic systems with strictly convex entropy and source terms depending on time and space. Second, instead of requiring a special class of SBP operators, we consider a general framework of diagonal-norm SBP operators on curved meshes, including finite differences, continuous and discontinuous Galerkin methods. Since the error analysis is based on a discrete relative entropy, it is restricted to smooth solutions. To enable a unified treatment of conservation laws, we restrict the analysis to periodic boundary conditions. Numerical results demonstrate that the predicted convergence rates are sharp in general, but can be improved for special cases such as discontinuous Galerkin methods with even polynomial degree and multi-block finite difference methods. An optimal analysis is expected to require more sophisticated arguments specialized to the class of methods instead of the general framework of SBP operators used in this work.

3. 一种用于可酉对角化线性反问题的谱域伪逆构造方法

A Spectral-Domain Pseudo-Inverse Construction Method for Unitary Diagonalizable Linear Inverse Problems

作者: Shengchang Chen

类别: math.NA, physics.geo-ph

发布日期: 2026-07-29

arXiv ID: 2607.26951v1

链接: http://arxiv.org/abs/2607.26951v1

摘要: Linear inverse problems are prevalent in geophysics, signal processing, image restoration, and medical imaging. Mathematically, they can be formulated as $Gm = d$. When $G$ is ill-conditioned or singular, the problem becomes ill-posed and requires regularization methods or generalized inverse methods for a stable solution. However, both types of methods encounter significant computational difficulties for large-scale linear inverse problems. In this paper, we establish a spectral-domain pseudo-inverse construction method for a class of linear inverse problems that can be diagonalized by unitary matrices. The core idea is to construct a stable pseudo-inverse operator directly in the transform domain, starting from the unitary diagonalization structure of the matrix. We first provide the analytic Singular Value Decomposition (SVD) of this class of matrices, clarifying the correspondence between spectral decomposition and SVD. On this basis, we define spectral-domain regularization filtering factors, construct a stable spectral-domain pseudo-inverse operator, and prove its bounded stability as well as its consistency in converging to the Moore–Penrose generalized inverse. This construction is numerically equivalent to zeroth-order Tikhonov regularization and converges to the Moore–Penrose generalized inverse as $α\to 0^+$, but its methodological path differs from both. This paper reveals that the stable generalized inverse of a class of structured matrices can be directly constructed from their spectral decomposition, providing an efficient solution method for large-scale structured inverse problems. The Fourier transform case is a special instance of this method when the unitary matrix is taken as the discrete Fourier transform matrix.

4. 用于量子辅助波形反演的校准压力可观测量Born和Hessian作用量

Calibrated Pressure-Observable Born and Hessian Actions for Quantum-Assisted Waveform Inversion

作者: Guanyu Li, Jiwei Jia, Yu Wang, Yuping Duan

类别: quant-ph, math.NA

发布日期: 2026-07-29

arXiv ID: 2607.26880v1

链接: http://arxiv.org/abs/2607.26880v1

摘要: We construct a pressure-consistent operator-and-readout interface for Born, adjoint, and Gauss–Newton actions in constant-density acoustic full-waveform inversion (FWI) using Schrödingerised propagation. The energy variables $π=c^{-1}\partial_t u$ and $q=\nabla u$ yield an auxiliary-space Hamiltonian, while physical pressure $p=cπ$ depends explicitly on wavespeed. Its derivative $D(cπ)[c_0](δc)=c_0δπ+δc\,π_0$ combines propagated wavefield sensitivity with a direct receiver-calibration term. Duhamel and receiver-row differentiation retain both contributions in the Born map, its adjoint, and the Gauss–Newton normal action. We prove a conditional consistency estimate with a periodic second-order finite-difference specialization and give a resource model for state preparation, normalization, quadrature, and selected-output measurement. A compiled nine-qubit instance realizes structured preparation, product-formula propagation, a derivative-LCU block, and calibrated pressure-overlap measurements. Bernoulli samples from ideal-circuit probabilities drive a four-parameter hybrid inversion. A two-qubit VQLS circuit represents the normalized update direction, while normal-system assembly, line search, and model refresh remain classical. Finite differences, tangent and reverse-adjoint recurrences, autodiff JVP/VJP evaluations, and explicit Jacobians verify the discrete Born, adjoint, and normal actions. Smooth periodic refinement confirms second-order convergence, whereas omitting receiver calibration leaves an order-one Born error and substantially changes the regularized Gauss–Newton direction. All ten predeclared finite-shot runs reduce the initial model error. These results specify the physical-pressure derivative and selected-output measurements needed to connect Schrödingerised propagation to a local FWI update.

5. 稀疏网格与有效低维高维函数的改进采样不等式

Improved Sampling Inequalities for Sparse Grids and High-Dimensional Functions with Effective Low Dimension

作者: Christian Rieger, Holger Wendland

类别: math.NA

发布日期: 2026-07-29

arXiv ID: 2607.26866v1

链接: http://arxiv.org/abs/2607.26866v1

摘要: The approximation of high-dimensional functions is a challenging task due to the often appearing curse of dimensionality. In this paper, we combine sparse grid with anchored projection techniques to derive sampling inequalities for Sobolev functions of a dominating mixed regularity which are effectively low dimensional. To this end, we derive new sampling inequalities for sparse grids and combine these with recently investigated regression processes of non-matching sampling processes.

6. 用于泊松逆问题中分裂方法的松弛梯度步去噪器

A Relaxed Gradient Step Denoiser for Splitting Methods in Poisson Inverse Problems

作者: Alessandro Benfenati

类别: math.NA

发布日期: 2026-07-29

arXiv ID: 2607.26864v1

链接: http://arxiv.org/abs/2607.26864v1

摘要: Plug&Play methods combine classical variational models with learned denoisers and have achieved strong results in imaging inverse problems. Their convergence has been widely studied for Gaussian data, whereas Poisson models require additional care because of the nonquadratic Kullback–Leibler fidelity. This work introduces \gsdsplit, a splitting method built upon \pnpsplit, in which the generic denoising block is replaced by a relaxed Gradient Step Denoiser. The resulting method retains explicit Poisson fidelity updates. The denoiser is defined through the gradient of a learned convex potential parameterized by an Input Convex Neural Network. Its architecture and training promote both blind denoising capability and a smoothness condition sufficient for firm nonexpansiveness. Empirical smoothness estimates are used to select a relaxation parameter compatible with the convergence assumptions of \pnpsplit. Numerical experiments provide evidence that the trained denoiser operates in an FNE-compatible regime on the considered validation data. \gsdsplit also yields stable reconstructions outside the empirically supported range and shows substantially lower sensitivity to the ADMM parameter, while remaining effective for different blur operators and Poisson noise levels.

7. 选主元QR与LU分解的收敛率

Convergence rates for pivoted QR and LU

作者: Marc Aurèle Gilles

类别: math.NA

发布日期: 2026-07-29

arXiv ID: 2607.26863v1

链接: http://arxiv.org/abs/2607.26863v1

摘要: Pivoted QR and pivoted LU decompositions are greedy algorithms used to compute low-rank approximations of matrices from selected columns, or selected rows and columns. Despite their practical robustness, general worst-case bounds comparing their errors with those of the best corresponding low-rank approximations contain exponentially growing factors and do not explain their behavior under modest singular value decay. We prove that under approximate greedy pivoting, their error is controlled by the determinant of a submatrix, which is bounded by the geometric mean of the leading singular values. Using this bound, we establish convergence rates under algebraic and geometric singular value decay. We also extend the LU analysis to functions of two variables. By bounding the determinants of arbitrary sampled submatrices, we obtain algebraic convergence rates under differentiability assumptions and geometric convergence under analyticity.

8. 分数阶哈代常数的 Galerkin 逼近

Galerkin Approximation of the Fractional Hardy Constant

作者: Andreea Dima, Liviu I. Ignat

类别: math.NA, math.AP

发布日期: 2026-07-29

arXiv ID: 2607.26830v1

链接: http://arxiv.org/abs/2607.26830v1

摘要: We establish sharp estimates for the discrete optimal constant of the fractional Hardy Inequality in dimension $N\geq 1$, with fractional exponent $s\in \left(0,\min\left\{1,\frac{N}{2}\right\}\right)$. The convergence rates that we establish take place for the Galerkin approximation with piecewise linear elements, when the computations are carried out in a bounded, convex and smooth domain containing the origin, for which we employ a quasi-uniform and regular mesh.

9. 带加性噪声与非光滑初始数据的半线性随机偏微分方程有限元方法的最优误差估计

Optimal Error Estimates of a Finite Element Method for Semilinear SPDEs with Additive Noise and Nonsmooth Initial Data

作者: Jitendra Nath Naik, Lok Pati Tripathi

类别: math.NA, math.PR

发布日期: 2026-07-29

arXiv ID: 2607.26797v1

链接: http://arxiv.org/abs/2607.26797v1

摘要: This paper presents a strong error analysis of both semidiscrete and fully discrete approximations for semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive noise and subject to nonsmooth initial data. The spatial discretization is based on a standard finite element method, coupled with the linearly implicit Euler scheme in time. Under low-regularity initial conditions, we derive sharp spatial and temporal regularity estimates that isolate the loss of initial regularity into an integrable temporal singularity, allowing us to establish optimal strong error estimates for positive times. Specifically, we prove strong convergence rates of order $O(h^β)$ for the spatially semidiscrete approximation and $O(h^β+ k^{β/2})$ for the fully discrete scheme away from $t = 0$, where the parameter $β\in (0, 2]$ characterizes the spatial regularity of the noise process. Numerical experiments confirm the theoretical convergence rates.

10. 粘性扩散与网络上的哈密顿-雅可比-贝尔曼方程的马尔可夫链逼近

Markov Chain Approximation of Sticky Diffusions and Hamilton-Jacobi-Bellman Equations on Networks

作者: Alessio Basti, Jules Berry, Fabio Camilli

类别: math.NA

发布日期: 2026-07-29

arXiv ID: 2607.26709v1

链接: http://arxiv.org/abs/2607.26709v1

摘要: We propose a discrete Markov-chain approximation of diffusion processes on networks with both Kirchhoff and sticky vertex conditions. Stickiness is modeled by a probabilistic residence mechanism at the vertex, while the motion along the edges follows an Euler-Maruyama-type update at the diffusive scale. We prove that the associated time-interpolated chain converges in distribution to the limiting diffusion in the Skorokhod space using the Ethier-Kurtz framework. Based on this construction, we derive a fully discrete semi-Lagrangian scheme for Hamilton-Jacobi-Bellman equations on networks and establish its convergence using viscosity solution techniques.

11. HORSES3D-GPU: 面向多GPU系统的高阶间断Galerkin求解器

HORSES3D-GPU: A high-order discontinuous Galerkin solver for multi-GPU systems

作者: Gerasimos Ntoukas, Gonzalo Rubio, Abbas Ballout, Stefano Colombo, David Huergo, Eduardo Jané, Albert Jiménez-Ramos, Hatem Kessasra, Himpu Marbona, Oscar Mariño, Rodrigo Salado, Pol Solé-Miró, Andrés M. Rueda-Ramírez, Miguel Chávez-Módena, Eusebio Valero, Esteban Ferrer

类别: math.NA

发布日期: 2026-07-29

arXiv ID: 2607.26674v1

链接: http://arxiv.org/abs/2607.26674v1

摘要: We present the GPU acceleration and large-scale performance assessment of HORSES3D, an open-source high-order discontinuous Galerkin solver for computational fluid dynamics. The solver is ported to NVIDIA GPU architectures using OpenACC directives, preserving the original Fortran code structure while enabling GPU-resident execution of the main computational kernels. The implementation exploits the element-local structure of discontinuous Galerkin spectral element methods by mapping element-level loops to GPU gangs and nodal operations to vector-level parallelism. The GPU version is verified using the method of manufactured solutions and validated on canonical turbulent-flow benchmarks. Its performance is assessed on the MareNostrum 5 accelerated partition using NVIDIA H100 GPUs. Taylor-Green vortex benchmarks show that solver efficiency improves with polynomial order and that near-ideal strong and weak scaling is obtained when the workload exceeds approximately 16,000 to 20,000 elements per GPU. The solver is further evaluated on the High-Lift Common Research Model wing-body configuration, which involves a complex geometry, realistic boundary conditions, and unstructured meshes with up to 20.8 million hexahedral elements. Simulations with polynomial orders up to $P=7$ reach approximately $10.7 \times 10^9$ degrees of freedom and scale efficiently to 2,048 GPUs. The results demonstrate that HORSES3D preserves its performance characteristics for industrially relevant configurations and can exploit modern GPU-based supercomputers for billion-degree-of-freedom high-order CFD simulations.

12. 从有限元法与逆解视角看先进脑电源模型

Advanced EEG Source Models from the Perspective of FEM and Inverse Solutions

作者: Santtu Söderholm, Joonas Lahtinen, Sampsa Pursiainen

类别: math.NA

发布日期: 2026-07-29

arXiv ID: 2607.26556v1

链接: http://arxiv.org/abs/2607.26556v1

摘要: In this study, we compare forward solutions computed with finite element methods in Zeffiro Interface and DUNEuro, using the different source models they provided. We compared two of the source models from DUNEuro, called Whitney basis and Local subtraction, and the divergence-conforming model of Zeffiro Interface. For source estimation, we applied sparsity-promoting standardized hierarchical adaptive L1 regression (SHAL1R), standardized Kalman filtering (SKF), classical sLORETA, and dipole scanning. Analyses include Earth Mover’s Distance, depth bias scatter plots, and qualitative assessments of amplitude distribution and focality. Preliminary experiments with source interpolation for each method revealed that Local subtraction closely matches expectations for the local behavior of the lead field at various depths. The main results reveal that the success of an inverse method depends strongly on the compatibility between its assumptions about the focality of the source and the chosen source model, with point-source models performing best when paired with methods designed for such sources, i.e., sparsity-promoting methods and methods that scan for a single source. Moreover, source models that admit patch sources with a wide distribution are more sensitive to additional noise.

13. 基于集合卡尔曼滤波的常微分方程离散误差均值贝叶斯推断

Bayesian Inference of Discretization Error Means in ODEs via Ensemble Kalman Filtering

作者: Shoji Toyota, Yuto Miyatake

类别: math.NA, stat.ME

发布日期: 2026-07-29

arXiv ID: 2607.26552v1

链接: http://arxiv.org/abs/2607.26552v1

摘要: We propose a Bayesian framework to quantify discretization errors in numerical solutions of ODE models based on observational data. The discretization error is modeled as a random variable, and its mean-referred to as the discretization error mean-is inferred from the observations. By introducing a Markov prior on the temporal evolution of the discretization error mean, we formulate the problem as a state-space model with a linear Gaussian observation process, which enables efficient inference via the Ensemble Kalman Filter. We also propose a specific form of a Markov prior motivated by classical discretization error analysis, in which global errors accumulate from local errors. The proposed prior depends on the step size of a numerical solver, and we establish its convergence rate in probability as the step size tends to zero. Numerical experiments on the pendulum system and the FitzHugh-Nagumo model demonstrate the effectiveness of the proposed approach.

14. 一个新的择一定理导出不一致情形下动态字符串平均优越化保证问题的充分条件

A new theorem of alternatives leading to sufficient conditions for the superiorization guarantee question of Dynamic String-Averaging in the inconsistent case

作者: Kay Barshad, Yair Censor

类别: math.OC, math.FA, math.NA

发布日期: 2026-07-29

arXiv ID: 2607.26489v1

链接: http://arxiv.org/abs/2607.26489v1

摘要: We study the Superiorization Methodology (SM) in the context of the General Dynamic String-Averaging (GDSA) method in the inconsistent case (that is, where the input operators don’t have a common fixed point) which primarily aims at achieving convex feasibility while simultaneously reducing an objective function. In many scientific and real-world problems modeled as constrained minimization tasks, striving for the exact constrained optimum can be costly in terms of time, energy, and resources. Therefore, applying the SM can offer a practical and efficient alternative. In particular, we present a new “theorem of alternatives” for the superiorization method which leads to investigation of theoretical conditions under which the superiorized version of the GDSA algorithm converges to a “superior” feasible point, i.e., one with an objective function value that is smaller or equal to that produced by the unperturbed feasibility-seeking algorithm. While this question has only been partially addressed in the existing literature, we present new sufficient conditions that guarantee that the SM attains such a superior outcome.

15. 当正SIMP密度下界不足时：无矩阵三维拓扑优化中的求解器可容许性与受保护下界选择

When a positive SIMP density floor is not enough: solver admissibility and guarded floor selection in matrix-free 3D topology optimization

作者: Shaoliang Yang, Jun Wang, Yunsheng Wang

类别: cs.CE, math.NA

发布日期: 2026-07-29

arXiv ID: 2607.26382v1

链接: http://arxiv.org/abs/2607.26382v1

摘要: In a matrix-free geometric-multigrid FGMRES solver for three-dimensional SIMP topology optimization, a reported converged solve is not always a converged solve. On four of 102 held-out states, the projected residual used for stopping falls below $10^{-6}$ while a recomputed true residual is 1.35 to 49.5 times the tolerance; in an unguarded optimization trajectory, 22 of 40 state solves reach the iteration cap without raising an error. We formulate floor selection as a verified control problem: probe the frozen state at the original floor, use two residual features to choose the first attempted floor, and accept no solution until $||f-Ku||/||f||\le10^{-6}$ is recomputed. The two-feature rule matches 98 of 102 reference classifications; the residual guard detects the four missed escalations, and all 102 selected solves satisfy the tolerance. Relative to always using a $10^{-3}$ floor, the policy preserves the original operator on 24 admissible states and avoids mean compliance and gradient changes of 31.0% and 0.340 on those severe random states, and 0.48% and 0.008 on seven optimized designs, at 2.5 times the mean wall time. In a 12-state subset of the held-out states, eight still require escalation at the conventional floor $10^{-6}$. In a nine-state control with the preconditioner’s adaptive components disabled, every failure is visible and no false acceptance occurs, tying the stopping-estimate drift to the iterate-dependent preconditioner. The recomputed residual is the correctness safeguard; the probe and floor ladder govern an implementation-specific cost-fidelity tradeoff.

16. 一种用于移动区域上含Robin边界条件的对流扩散方程的笛卡尔网格方法

A Cartesian Grid Method for Advection-Diffusion Equations with Robin Boundary Conditions on Moving Domains

作者: Han Zhou, Yoichiro Mori, Lingxing Yao

类别: math.NA, math-ph

发布日期: 2026-07-28

arXiv ID: 2607.26341v1

链接: http://arxiv.org/abs/2607.26341v1

摘要: We develop a Cartesian grid method for advection–diffusion equations with Robin boundary conditions on moving domains. The moving-domain problem is reformulated as an interface problem on a box, with an unknown density introduced on the moving interface to enforce the Robin condition. The bulk equation is discretized by a cell-centered finite-difference scheme on the Cartesian grid, while interface corrections are obtained from local problems in a narrow band around the interface. The resulting method requires only modest computational geometry, avoids remeshing and cut cells, and is compatible with geometric multigrid and matrix-free GMRES. The GMRES iteration count is essentially independent of the mesh size, and the computational cost scales linearly with the number of bulk degrees of freedom. For the one-dimensional scheme, first-order convergence in time and second-order convergence in space are proved. Numerical examples in one and two dimensions, including manufactured solutions and an active transport problem without an exact solution, demonstrate the accuracy and efficiency of the method.

17. 去噪增长复杂性：数据几何与扩散采样的可认证调度

Denoising growth complexity: Data geometry and certified schedules for diffusion sampling

作者: Martin J. Wainwright

类别: math.ST, cs.LG, math.NA, stat.ML

发布日期: 2026-07-28

arXiv ID: 2607.26285v1

链接: http://arxiv.org/abs/2607.26285v1

摘要: Two central challenges in diffusion-based sampling are the theoretical one of understanding their remarkable effectiveness even in high-dimensional settings, and the practical one of designing algorithms with certified performance guarantees. We show that these questions are intimately connected via the \emph{denoising growth complexity} ($\mathsf{DGC}$). It is a geometric measure defined by a log-time weighted integral of the derivative of the denoising mean-squared error along the Gaussian heat flow. We show how the $\mathsf{DGC}$ increments lead to a simple and explicit bound on the KL error of an Euler scheme applied to the stochastic innovations representation. The bound is local along the path: each step is controlled by the corresponding $\mathsf{DGC}$ increment and its relative stepsize. This structure allows us to derive KL sampling guarantees for optimized stepsize schedules, both in a simpler single-block setting and in a more refined $K$-block setting. The $\mathsf{DGC}$ function has a natural martingale structure, which we exploit to develop fully data-certified versions of these algorithms. It also admits information-theoretic upper bounds in terms of covariance, rate distortion, metric entropy, and the Poincar’e constant, thereby recovering and sharpening a range of existing diffusion-sampling guarantees, as well as giving new results. In log heat-time, the fine partition limit is governed by an integral involving the square root of the $\mathsf{DGC}$ density, whereas a single-block schedule depends on its ordinary integral. This comparison precisely characterizes when adaptation to data geometry yields substantial computational gains, including logarithmic-to-constant separations for simple Gaussian mixture models.

18. 路径的简洁 $(\varepsilon,r)$-表示

Concise $(\varepsilon,r)$-representations of a path

作者: Emilio Ferrucci, Oliver Perrée, Terry Lyons

类别: math.NA, math.PR

发布日期: 2026-07-28

arXiv ID: 2607.26281v1

链接: http://arxiv.org/abs/2607.26281v1

摘要: Paths $X \colon [0,T] \to \mathbb R^d$ are traditionally stored in finite memory as time series. Recent research has underscored the benefits of instead representing them as collections of iterated integrals $\{\int_{0 < u_1 < \ldots < u_n < T} \mathrm{d} X_{u_1} \otimes \cdots \otimes \mathrm{d} X_{u_n}\}_{n = 0}^N$. These two encodings can be viewed as the extrema on a two-parameter spectrum of representations of the path as degree-$N$ signatures on $m$ intervals in a partition of $[0,T]$. We ask the question of which such representation takes up the least amount of memory, measured as number of real values needed to store the truncated log-signature, subject to the constraint of it being able to approximate solutions to linear controlled differential equations (CDEs) $\mathrm{d} Y = AY \mathrm{d} X$ with $|A| \leq r$ at accuracy at least $\varepsilon$. Estimating the error in terms of the length of $X$, we find that the optimal representation generally lies strictly in between the two naive choices $N = 1$ or $m = 1$, and derive its asymptotics as $r \to \infty$ and $\varepsilon \to 0^+$. Similar considerations can be made when estimating the error in terms of the $p$-variation norm of $X$: in this regime we prove an error bound of the degree-$N$ Euler scheme for linear CDEs with decay in both $m$ and (factorially) in $N$ with the other arbitrarily fixed. We conclude by setting up the analogous problem for SDEs, with the error measured in $L^2$, and derive a similar $L^2$-Euler error estimate for Itô SDEs with drift. We include an empirical study of the optimisation problem, which we demonstrate for toy examples of $p$-rough paths and for fractional Brownian motion.

19. 守恒Vlasov模拟中正交约束下的非负低秩矩阵修正

Nonnegative Low-Rank Matrix Correction under an Orthogonality Constraint in Conservative Vlasov Simulations

作者: Yue Wu, Stephen Becker, Jingmei Qiu, Xiangxiong Zhang

类别: math.NA, math.OC

发布日期: 2026-07-28

arXiv ID: 2607.26272v1

链接: http://arxiv.org/abs/2607.26272v1

摘要: In low-rank numerical methods for Vlasov dynamics, the SVD-type truncation procedure may introduce negative entries into the numerical solution. Such negative values are unphysical because the solution is a probability distribution function. We design optimization-based post-processing algorithms to recover nonnegativity while preserving the macroscopic quantities (density, momentum, and energy) pointwise. The preservation of the macroscopic quantities is written as an orthogonality constraint on the correction term. For a convex formulation based on squared nuclear norm minimization, we show that the proximal operator with the orthogonality constraint is characterized by an implicit singular value thresholding equation, and the threshold can be computed efficiently by bisection. Based on this result, we develop five algorithms for the convex formulation: Douglas–Rachford splitting, restarted dual FISTA, restarted dual accelerated gradient descent, dual PR+ conjugate gradient, and dual L-BFGS. We also consider a non-convex formulation with an explicit rank constraint and develop a tangent-space accelerated alternating projection algorithm that only requires a (2r \times 2r) SVD per iteration. Numerical results for a Landau damping test case show that the proposed algorithms give comparable correction quality. Among them, the tangent-space accelerated alternating projection is the most cost-efficient, increasingly so as the problem size grows. We further demonstrate the correction as a positivity limiter inside a time-dependent conservative low-rank Vlasov solver, where it removes the negativity introduced by the SVD-type truncation while preserving the conserved mass, momentum, and energy.