生成时间: 2026年07月22日 10:07:51
重点关注论文数量: 0
了解领域论文数量: 3
不相关论文数量: 11
了解领域论文（摘要翻译）
1. 具有道尔蒂-福克尔-普朗克碰撞的弗拉索夫-泊松方程的局部宏观守恒低秩间断伽辽金方法
A Local Macroscopic Conservative Low-Rank Discontinuous Galerkin Method for the Vlasov-Poisson Equation with Dougherty-Fokker-Planck Collisions
作者: Austin Nelson, Wei Guo, Pierson Guthrey
类别: math.NA
发布日期: 2026-07-21
arXiv ID: 2607.19264v1
链接: http://arxiv.org/abs/2607.19264v1
摘要翻译
中文标题: 具有道尔蒂-福克尔-普朗克碰撞的弗拉索夫-泊松方程的局部宏观守恒低秩间断伽辽金方法
摘要翻译: 本文构造了一种低秩、保结构的间断伽辽金方法，用于模拟耦合道尔蒂-福克尔-普朗克碰撞算子的弗拉索夫-泊松系统。当库仑碰撞在稠密或弱碰撞等离子体中发生时，电子被推向低秩稳态。在许多情况下，等离子体很快达到该稳态，这意味着在大多数运行时间内，等离子体主要由数值低秩结构组成。我们构造的新型低秩格式旨在利用这些数值低秩结构，以大幅降低VP-DFP系统模拟所需的存储复杂度。该格式是通过将库仑碰撞纳入系统，对先前建立的局部宏观守恒方法进行的扩展。LoMaC性质在离散层面上保证了宏观质量、动量和能量的局部守恒。本文讨论了该新方法的细节。数值实验表明了该方法的有效性。
2. 守恒律方程间断Galerkin方法的加权逆Lax-Wendroff边界处理
Weighted Inverse Lax-Wendroff Boundary Treatment of Discontinuous Galerkin Methods for Conservation Laws
作者: Yongjie Bi, Yan Jiang, Yong Liu
类别: math.NA
发布日期: 2026-07-21
arXiv ID: 2607.19258v1
链接: http://arxiv.org/abs/2607.19258v1
摘要翻译
中文标题: 守恒律方程间断Galerkin方法的加权逆Lax-Wendroff边界处理
摘要翻译: 本文针对非贴体网格上的间断Galerkin方法,提出一种加权逆Lax-Wendroff边界处理技术,以高效求解复杂几何区域中的双曲守恒律问题。该方法对内部单元采用标准DG格式,而对靠近边界的切割单元,通过逆Lax-Wendroff原理重构高阶近似多项式来施加数值边界条件,从而有效消除通常由小切割单元引起的时间步长限制。特别地,为克服基本ILW格式中数值误差对切割单元几何尺寸的敏感性,我们提高了边界处的重构阶数,使得精度不再依赖于切割单元的大小。此外,方法引入加权最小二乘重构,以减少构造过程中对复杂高阶边界导数的需求。因此,该方法在保持高阶精度的同时,显著提升了多维问题的计算效率。最后,通过线性稳定性分析从理论上验证了所提方法的稳定性,并通过一系列一维和二维标量方程及方程组的数值实验,数值上验证了该格式的有效性和鲁棒性。
3. 椭圆Dirichlet问题的边界适配PINNs: $H^2(Ω)$ 先验误差界及其在平均逃逸时间计算中的应用
Boundary-Adapted PINNs for Elliptic Dirichlet Problems: $H^2(Ω)$ A Priori Error Bounds with Application to Mean Escape Time Computation
作者: Nathanael Tepakbong, Jun Fan, Xiang Zhou, Ding-Xuan Zhou
类别: math.NA, cs.LG, math.ST, stat.ML
发布日期: 2026-07-21
arXiv ID: 2607.19167v1
链接: http://arxiv.org/abs/2607.19167v1
摘要翻译
中文标题: 椭圆Dirichlet问题的边界适配PINNs: $H^2(Ω)$ 先验误差界及其在平均逃逸时间计算中的应用
摘要翻译: 受有限域 $Ω\subseteq\mathbb{R}^d$ 上随机过程平均逃逸时间 (MET) $\tau:Ω\to\mathbb{R}$ 数值计算的启发，我们使用边界强制的物理信息神经网络 (PINNs) 研究椭圆Dirichlet边值问题 (BVPs)，其中通过将网络输出与预先定义的距离到边界近似 $\rho$ 相乘来精确施加Dirichlet边界条件。结合针对修正二次单元 (ReQU) 和双曲正切 (tanh) 网络的逼近论论证和统计学习论证，我们推导出了显式依赖于 $\rho$ 的先验误差界。特别地，我们证明了仅精确边界强加不足以得到 $H^2(Ω)$ 误差界，并且充分且本质上必要的条件是 $\rho$ 为一类在 arXiv:2104.08426 [math.NA] 中所构造的、$\textit{一阶正规化的}$ 光滑距离近似。由此，我们将这一子类的 $\textit{边界适配}$ PINNs 视作求解 Dirichlet 边值问题的合适神经网络 ansatz。数值实验支持理论，表明适当选择 $\rho$ 可提高精度和收敛性，而选择不当的距离函数可能显著恶化解。我们的证明还得到了 ReQU 和 tanh 网络高阶导数的假设空间的新 VC-维数界，以及浅层 ReQU 网络在高阶 Sobolev 范数下的新逼近界，所有这些都具有重要的独立意义。
不相关论文（基本信息）
1. Hadamard流形上的1-Lipschitz神经网络
1-Lipschitz Neural Networks on Hadamard Manifolds
作者: Davide Murari, Marta Ghirardelli, Ben Adcock, Elena Celledoni, Brynjulf Owren, Carola-Bibiane Schönlieb
类别: math.NA, cs.LG
发布日期: 2026-07-21
arXiv ID: 2607.19335v1
链接: http://arxiv.org/abs/2607.19335v1
摘要: Controlling the Lipschitz constant of a neural network is a standard way to promote robustness and stability. Most existing constraining strategies are designed for Euclidean spaces. In this work, we construct and analyze a class of 1-Lipschitz neural networks on Hadamard manifolds. Our layers are of gradient-descent type, $1$-Lipschitz, and quasi-$α$-firmly nonexpansive. The core building blocks of the proposed architecture are Busemann functions, and we exploit the properties of Busemann gradient flows to design $1$-Lipschitz geometry-preserving layers. We provide explicit constructions and examples for hyperbolic manifolds and the manifold of symmetric positive definite (SPD) matrices. We test the proposed architecture in two numerical experiments: robust classification on the Poincaré disk and masked-Wishart covariance reconstruction. On the Poincaré disk, the proposed networks yield robust classifiers under hyperbolic perturbations. On the SPD manifold, we train SPD-valued denoisers and adopt them as a Plug-and-Play prior for a masked-Wishart covariance reconstruction problem. We show improved results from the nonexpansive denoiser over static, data-only, and Log-Euclidean denoising baselines, and empirically test its convergence properties.
2. ENO-TV 猜想的解决：奇偶二分
Resolution of the ENO-TV conjecture: a parity dichotomy
作者: Zhuoyun Li, Kailiang Wu
类别: math.AP, math.CO, math.NA, math.RT
发布日期: 2026-07-21
arXiv ID: 2607.19283v1
链接: http://arxiv.org/abs/2607.19283v1
摘要: We resolve the ENO–TV conjecture, a discrete coercivity problem in compactness theory for entropy-stable approximations of hyperbolic conservation laws. For order-$k$ essentially non-oscillatory (ENO) reconstruction from compactly supported cell averages, it asks whether the nonnegative ENO source times the $(k-1)$st power of the amplitude uniformly controls the $(k+1)$st absolute-jump moment. We prove a parity dichotomy: the estimate holds for odd $k\ge3$ and fails for even $k\ge4$; the known second-order case completes the classification. Localization gives a selection-independent finite-difference functional uniformly comparable to the source and reduces the conjecture to discrete interpolation. For odd orders, summation by parts reveals a hidden square; a discrete Gagliardo–Nirenberg inequality yields coercivity. For even orders, Euler-polynomial blocks from the functional’s polynomial kernel yield counterexamples that persist under arbitrarily small perturbations making all affected ENO comparisons strict. We also prove two coercive estimates for every $k\ge2$: control of jumps larger than a fixed fraction of the amplitude and of local blocks modulo sampled polynomials of degree at most $k-2$. Via the Cayley–Sylvester decomposition, we compute the dimensions of homogeneous first-cohomology spaces for the lattice shift on polynomial jump profiles. At fourth order, for a cubic flux and a globally strictly convex entropy, a total-degree-seven component of a reduced entropy-flux mismatch represents a nonzero class on profiles of degree at most two and hence has no translation-invariant finite-stencil $C^7$ local primitive at the zero constant state. Odd-order coercivity persists on globally quasi-uniform meshes, whereas for each $k\ge2$ it fails on a fixed irregular mesh even though every interface contribution remains nonnegative. This failure is due to the mesh geometry.
3. Nyström 误差超出 $M$-矩阵：一个最小对角占优障碍
Nyström Error Beyond $M$-Matrices: A Minimal Diagonally Dominant Obstruction
作者: Matthew J. Colbrook
类别: math.NA
发布日期: 2026-07-21
arXiv ID: 2607.19282v1
链接: http://arxiv.org/abs/2607.19282v1
摘要: We study the nuclear-norm error of a column-selected Nyström approximation to $K=(L+γI)^{-1}$, where $L$ is symmetric diagonally dominant and $γ>0$. Our central question is whether this error has diminishing returns. A Schur-complement identity reduces the question to traces of inverses of principal submatrices. Existing $M$-matrix results settle the case in which $L$ is a symmetric diagonally dominant $M$-matrix (SDDM). However, diagonal dominance alone is not enough: failure occurs already in dimension three. We construct an exact one-parameter SDD family and determine its sharp failure interval. A $2\times2$ identity proves that dimension three is minimal within the SDD class. We then show that failure persists under strict diagonal dominance; with a nonempty selected base set, dimension four is minimal. Finally, we prove invariance under signature switching, derive a three-dimensional formula showing how a signed triangle causes failure, and give an example in which greedy column selection misses the optimal pair. Together, these findings complete the answer to Problem 4.6 in a recent Simons workshop report.
4. 乘性噪声驱动的超线性随机偏微分方程的一种非线性显式全离散格式的时间一致弱误差和遍历误差估计
Uniform-in-Time Weak and Ergodic Error Estimates of a Nonlinearity-Explicit Full Discretization for Superlinear SPDEs Driven by Multiplicative Noise
作者: Jingjing Cai, Zhihui Liu, Xiaoming Wu
类别: math.NA
发布日期: 2026-07-21
arXiv ID: 2607.19250v1
链接: http://arxiv.org/abs/2607.19250v1
摘要: For a class of superlinear SPDEs driven by multiplicative noise, we prove an (essentially) sharp uniform-in-time (UIT) weak convergence rate for the nonlinearity-explicit Galerkin tamed Euler method (GTEM). Under standard monotonicity assumptions, the proof combines Malliavin calculus with regularity theory for the associated backward Kolmogorov equation (BKE), leading to UIT moment, Hölder, and Malliavin estimates, along with regularity estimates for the BKE solution. These estimates, together with a weak error decomposition and Malliavin integration by parts (IBP) formula, then yield a UIT weak convergence rate $τ^ρ+λ_N^{-ρ}$ for any $ρ\in (0,1)$. Consequently, we obtain a sharp ergodic error estimate between the exact and numerical invariant measures. Numerical experiments support the theory.
5. Langevin型SPDE的数值方法：隐式Milstein方法与多层蒙特卡洛技术
Numerical methods for Langevin-type SPDE: an implicit Milstein approach and multilevel Monte Carlo techniques
作者: Sascha Portaro, Carlos Vázquez
类别: math.NA, math.PR
发布日期: 2026-07-21
arXiv ID: 2607.19188v1
链接: http://arxiv.org/abs/2607.19188v1
摘要: In this work, we investigate the numerical approximation of degenerate Langevin-type stochastic partial differential equations (SPDEs) in two spatial dimensions. These SPDEs arise in stochastic dynamics and mathematical finance, among other applications. In order to handle the mixed deterministic-stochastic structure of the equation and the degeneracy of the differential operator, we propose a semi-implicit Milstein finite difference scheme for the numerical solution. Through the Fourier analysis of the mean-square stability and convergence, we derive explicit conditions on the coefficients under which the scheme is stable, jointly with explicit convergence rates in terms of the discretization parameters. We further embed the proposed scheme within a Multilevel Monte Carlo (MLMC) framework to reduce the computational cost associated with SPDE simulations, and we derive its theoretical computational complexity. Numerical experiments confirm theoretical convergence rates and show that the MLMC strategy achieves an accuracy comparable to standard Monte Carlo at a fraction of the computational cost, reducing the complexity from $\mathcal{O}(\varepsilon^{-5})$ to $\mathcal{O}(\varepsilon^{-3})$ for a target root-mean-square error $\varepsilon$. These results show that combining semi-implicit Milstein schemes with MLMC techniques provides an effective approach for the numerical simulation of Langevin-type SPDEs.
6. 面向实时应用的张量层析X射线散射数据快速重建
Fast reconstruction of tensor tomographic X-ray scattering data for real-time applications
作者: André M. Antunes, Daniël M. Pelt, K. Joost Batenburg
类别: eess.IV, math.NA, physics.app-ph, physics.ins-det
发布日期: 2026-07-21
arXiv ID: 2607.19124v1
链接: http://arxiv.org/abs/2607.19124v1
摘要: X-ray scattering tensor tomography reveals nanoscale structural orientation in 3D, but its reliance on slow iterative reconstruction limits real-time use. We introduce an extension of a direct reconstruction approach for parallel-beam geometries that computes algebraic filters approximating multiple iterative updates with a single filtering and back-projection step. By explicitly separating the tomographic projector from a view-dependent mixing operator, the method generalizes across tensor representations and modes of (small-angle) X-ray scattering measurement. Simulated and experimental results show that the approach approximates iterative reconstructions while reducing computation time by over an order of magnitude, realizing a reconstruction of a 53x53x53x28 tensor volume in 1 second on commercially-available hardware. The resulting speed and stability enable high-throughput analysis and open the door to real-time scattering-based imaging.
7. 抛物问题的时空张量积有限元方法
Space-time tensor-product finite element methods for parabolic problems
作者: Richard Löscher, Michael Reichelt, Olaf Steinbach
类别: math.NA
发布日期: 2026-07-21
arXiv ID: 2607.19107v1
链接: http://arxiv.org/abs/2607.19107v1
摘要: We study space-time Galerkin–Petrov formulations for parabolic evolution problems and their relation to classical implicit time-stepping schemes. Although such schemes are stable in the usual time-stepping sense, their interpretation as space-time operator equations may lead to conditional stability, with constants depending on the relation between temporal and spatial mesh sizes. We revisit this phenomenon for the continuous Galerkin method of Aziz and Monk, which yields the Crank–Nicolson scheme in the lowest-order case, and provide a detailed space-time error analysis for solutions of both high and low regularity. In particular, the space-time framework allows us to analyze the deteriorated behaviour of classical time-stepping methods for nonsmooth initial data. By applying integration by parts in time, we derive an adjoint space-time formulation that incorporates the initial condition in a natural variational way. In the lowest-order case, this formulation leads to a Rannacher-type smoothing of the initial data. The theoretical results are complemented by numerical experiments.
8. 随机化单对角隐式Runge-Kutta方法的误差界与稳定性分析
Error Bound and Stability Analysis for a Randomized Singly Diagonally Implicit Runge-Kutta Method
作者: Monika Eisenmann, Marvin Jans, Raphael Kruse, Helmut Podhaisky
类别: math.NA
发布日期: 2026-07-21
arXiv ID: 2607.18928v1
链接: http://arxiv.org/abs/2607.18928v1
摘要: A randomized Singly Diagonally Implicit Runge-Kutta (SDIRK) method, based on the randomized trapezoidal rule as the underlying quadrature scheme, is proposed. Every realization of the scheme is an algebraically stable SDIRK method of at least second order. The main result is the proof that the randomized scheme converges with order 2.5 in the root mean square sense under low regularity assumptions. Numerical experiments illustrate the robustness of the new scheme when applied to nonsmooth problems.
9. 浮点归约树的二阶矩理论
A Second-Moment Theory for Floating-Point Reduction Trees
作者: Piyush Sao, Narasinga Miniskar, Pedro Valero-Lara, Keita Teranishi, Sudip Seal
类别: math.NA, cs.DC, cs.IT, cs.MS
发布日期: 2026-07-21
arXiv ID: 2607.18758v1
链接: http://arxiv.org/abs/2607.18758v1
摘要: Summation error depends on partial-sum order, which standard worst-case bounds omit. To capture this dependence, we derive an exact mean-square error (MSE) recurrence for a binary reduction tree T under conditionally unbiased rounding. With unit roundoff u, the constant-nu model sets the local variance at pre-rounding value x to nu u^2 x^2. Its leading tree-dependent cost for the input vector p is p^T K_T p, where the common-ancestor kernel K_T counts the internal ancestors shared by each pair of leaves. For i.i.d. inputs of mean mu and variance tau^2, this expected cost is tau^2 Lambda_1(T) + mu^2 Lambda_2(T), where Lambda_1 is total leaf depth and Lambda_2 sums squared internal-subtree sizes; Lambda_1 governs centered inputs, while Lambda_2 captures nonzero means. We use these statistics to characterize optimal tree topologies and schedules. Balanced and sequential trees attain the centered extrema. For k inputs, optimal two-stage sequential blocking yields root-mean-square (RMS) error scaling as k^{3/4}. For fixed-stage hierarchies, geometric schedules are optimal for centered inputs, whereas the optimal noncentered stage exponents halve successively. For independent centered inputs with unequal variances, Huffman coding minimizes variance-weighted depth over free leaf assignments. We extend the kernel to matrix multiplication through operand Gram matrices. We then test the approximation under round-to-nearest using exact residuals. Across binary64, binary32, and software-emulated binary16 and bfloat16, the model recovers the ordering among tree topologies; K_T tracks AR(1) partial-sum costs. For GEMM, independently calibrated predictions differ from measurements by at most 3% on the tested grid. A reduction tree extracted from an array library predicts the measured RMS scaling. However, stagnation and bias in positive low-precision sums limit the model’s applicability.
10. 量化矩阵乘法的收缩-规范预处理
Contraction-Gauge Preconditioning for Quantized Matrix Multiplication
作者: Piyush Sao, Narasinga Miniskar, Pedro Valero-Lara, Keita Teranishi, Sudip Seal
类别: cs.LG, cs.IT, math.NA
发布日期: 2026-07-21
arXiv ID: 2607.18745v1
链接: http://arxiv.org/abs/2607.18745v1
摘要: We study low-precision computation of C=AB with both factors quantized. We derive an exact finite-dimensional identity for the expected squared product error under independent, zero-mean entrywise errors with known variance fields; it holds exactly for non-overloading subtractive dither and for independent stochastic rounding, and we empirically assess deterministic round-to-nearest (RTN). Using the product-preserving equivalence AB=(AT)(T^{-1}B), we formulate contraction-gauge preconditioning: jointly choosing a factor representation and its sharing pattern before quantization. Preconditioning can reduce product error but may require extra transformed, quantized copies of the opposite operand: a shared transform needs one copy, a block-specific transform up to one per block. Within the bounded family of positive diagonal gauges (folds), a geometric program computes a globally optimal shared fold and a linear program decides whether the identity fold is already optimal. For other families we derive computable selection statistics – tail index for scaling, profile spread for partitioning, coherence and weighted-Gram energy for rotations, slice-energy covariance for hierarchy depth – with upper bounds for ranking heuristic candidates. Across twelve linear products from a trained three-block image classifier, median within-product rank correlations between dither-model predictions and deterministic-RTN errors are 0.937 at 8 bits and 0.918 at 4 bits. The GP fold cuts held-out product error over the identity fold by 18.0% (8-bit) and 20.5% (4-bit) in geometric mean, beats a SmoothQuant-style grid baseline at both precisions and on ten of twelve products, and lowers composed logit MSE by 15.4% and 26.4%. We thus provide exact stochastic product-error accounting, certified selection within the diagonal family, and a common objective for evaluating reusable transform candidates under RTN.
11. 电子磁流体动力学中具有分裂分数阶耗散的电流片形成
Current-Sheet Formation in Electron Magnetohydrodynamics with Split Fractional Dissipation
作者: Ruimeng Hu, Qirui Peng, Xu Yang
类别: physics.plasm-ph, math.NA
发布日期: 2026-07-21
arXiv ID: 2607.18648v1
链接: http://arxiv.org/abs/2607.18648v1
摘要: Thin current sheets are central small-scale structures in electron magnetohydrodynamics (EMHD), closely associated with energy dissipation and fast magnetic reconnection at electron scales. We study their formation numerically in a $2\frac{1}{2}$-dimensional EMHD system on a periodic domain with split fractional dissipation, where the magnetic potential and the vertical magnetic component are damped separately. The local theory is governed by a symmetric combined damping balance, but the numerical onset of small-scale growth need not follow this symmetry. A scaling analysis identifies the out-of-plane current as the primary concentration observable, since it is regularized only through the magnetic-potential equation. Using a validated Fourier pseudospectral exponential time-differencing solver with resolution-controlled diagnostics, we find a clear decay/concentration dichotomy. The onset boundary is markedly asymmetric: current-sheet formation appears to be controlled mainly by damping of the magnetic potential, rather than by the combined damping strength. The analyticity strip collapses to the grid scale, the concentration sharpens under grid refinement, and the observed growth is consistent with an energy-critical self-similar rate, with exponent near three. These experiments indicate that magnetic-potential damping is the apparent binding constraint for current-sheet concentration, refining the symmetric sum picture.