生成时间: 2026年05月19日 06:40:12

重点关注论文数量: 0

了解领域论文数量: 1

不相关论文数量: 29

了解领域论文（摘要翻译）

1. 离散可压缩正压Navier-Stokes方程的一个不可能定理及其解决

A no-go theorem and its resolution for the discrete compressible barotropic Navier–Stokes equations

作者: Peter Korn

类别: math.AP, math.NA

发布日期: 2026-05-15

arXiv ID: 2605.16554v1

链接: http://arxiv.org/abs/2605.16554v1

摘要翻译

中文标题: 离散可压缩正压Navier-Stokes方程的一个不可能定理及其解决

摘要翻译: 向量不变形式的可压缩正压Navier-Stokes方程组保持了系统的涡量结构，并构成了现代大气和海洋动力核心的基础，但尚未发展出针对该形式的可压缩离散系统的偏微分方程理论。在Delaunay–Voronoi网格上，我们通过离散外微积分证明，每一个具有分部积分相容散度的密度无关质量矩阵都带有一个量级为$\OO(h^2)$的、符号不确定的尖锐能量残差，且任何算子选择都无法消除该残差。这个不可能定理涵盖了A-、B-、C-、D-和准B网格的交错方式。密度加权质量矩阵是唯一的代数补救措施：它在保持向量不变动量方程、Lamb反对称性和拓扑守恒律的同时，恢复了精确的总能量，代价是一个与收敛速率相匹配的$\OO(h^{r_\star})$开尔文缺陷。该残差是导致Hollingsworth不稳定性的原因，这种不稳定性已塑造了向量不变动力核心的设计；密度加权构造在结构上消除了它。对于闭可定向黎曼流形上$d = 2, 3$维的密度加权（DW）格式，我们建立了当$\nu\ge 0$时全局适定性、关于$\nu$一致收敛到光滑解，以及低马赫极限下的渐近保持性；密度无关的残差以$\OO(M^{-1})$发散。通过离散的Arnold能量-Casimir构造，精确的离散守恒强制了围绕三类平衡态的Lyapunov稳定性，从而排除了Hollingsworth不稳定性：围绕静水和恒流层结态的无条件稳定性，以及在离散Charney–Stern判据下围绕剪切斜压态的条件稳定性。只有连续方程本身出现真正的斜压不稳定性时，DW格式才会出现。

不相关论文（基本信息）

1. SURGE: 面向扩散代理的免近似免训练粒子滤波器

SURGE: Approximation-free Training Free Particle Filter for Diffusion Surrogate

作者: Lifu Wei, Yinuo Ren, Naichen Shi, Yiping Lu

类别: stat.ML, cs.LG, math.NA, math.PR, q-fin.MF, stat.CO

发布日期: 2026-05-18

arXiv ID: 2605.18745v1

链接: http://arxiv.org/abs/2605.18745v1

摘要: Diffusion-based generative models increasingly rely on inference-time guidance, adding a drift term or reweighting mixture of experts, to improve sample quality on task-specific objectives. However, most existing techniques require repeated score or gradient evaluations, introducing bias, high computational overhead, or both. We introduce \texttt{URGE}, Unbiased Resampling via Girsanov Estimation, a derivative-free inference-time scaling algorithm that performs path-wise importance reweighting via a Girsanov change of measure. Instead of computing gradient-based particle weights in previous work, \texttt{URGE} attaches a simple multiplicative weight to each simulated trajectory and periodically resamples. No score, no Hessian, and no PDE evaluation is required. We establish an equivalence between path-wise and particle-wise SMC: the Girsanov path weight admits a backward conditional expectation that recovers the previous particle-level weights, guaranteeing that both schemes produce the same unbiased terminal law. Empirically, \texttt{URGE} outperforms existing inference-time guidance baselines on synthetic tests and diffusion-model benchmarks, achieving better generation quality, while being significantly simpler to implement and fully gradient-free.

2. 相对论性等离子体的显式能量守恒粒子网格格式

An explicit, energy-conserving particle-in-cell scheme for relativistic plasmas

作者: Lee Ricketson, Jingwei Hu

类别: physics.plasm-ph, math-ph, math.NA

发布日期: 2026-05-18

arXiv ID: 2605.18542v1

链接: http://arxiv.org/abs/2605.18542v1

摘要: We extend the recently-developed explicit, energy-conserving particle-in-cell (PIC) scheme of [1] to the relativistic Vlasov-Maxwell system. As in the non-relativistic case, the method is built on an optimization problem that is analytically solvable, local to each particle, and designed to enforce exact energy conservation. Although the solution to this optimization problem is not guaranteed to be real, we show that such instances are rare enough for practical simulation parameters to permit dramatic improvements in energy conservation over traditional explicit PIC schemes. We show that, as in the non-relativistic case, the scheme is compatible with popular field-solvers for electromagnetic PIC schemes, including the Yee/FDTD and pseudo-spectral analytic time-domain (PSATD) methods. The scheme is verified on standard relativistic test problems, where its conservation properties are confirmed.

3. Volterra签名的计算方面

Computational aspects of the Volterra Signature

作者: Paul P. Hager, Fabian N. Harang, Luca Pelizzari, Samy Tindel

类别: math.NA, stat.ML

发布日期: 2026-05-18

arXiv ID: 2605.18406v1

链接: http://arxiv.org/abs/2605.18406v1

摘要: The Volterra signature extends the classical path signature by incorporating general matrix-valued kernel into its iterated integral structure, yielding a flexible notion of memory for time series. Its components can be viewed as successive Picard iterates of linear controlled Volterra equations, making their exact computation of additional mathematical interest. However, the kernel introduces substantial algorithmic challenges. We provide a resolution by first decomposing the Chen-type convolution relation established in [arXiv:2603.04525] into analytic and arithmetic parts, and then introducing several efficient algorithms: a general approximative scheme with quadratic complexity $O(J^2)$ in the number of time steps $J$, an FFT-based acceleration with complexity $O(J\log J)$ for convolution kernels on uniform grids, and an exact recursion with complexity $O(JR^2)$ for kernels admitting a state-space representation of dimension $R$; retaining standard signature complexity in the path dimension and truncation level $N$. We further show that the number of factors in matrix-valued kernels of the form $K(t,s)=\sum_p k_p(t-s)A_p$ do not increase the asymptotic complexity in $J$ and $N$. Finally, we derive a finite-difference predictor–corrector scheme for the associated Volterra signature kernel. All algorithms are implemented in the publicly available JAX-based package “tensordev”.

4. 大规模线性离散不适定问题的随机收敛分析

Stochastic Convergence Analysis for Large-Scale Linear Discrete Ill-posed Problems

作者: Duan-Peng Ling, Wenlong Zhang

类别: math.NA

发布日期: 2026-05-18

arXiv ID: 2605.18259v1

链接: http://arxiv.org/abs/2605.18259v1

摘要: We study weighted Tikhonov regularization for large-scale linear discrete ill-posed problems with random noise. Under a polynomial upper-bound assumption on the generalized eigenvalues of the discrete forward operator, we derive stochastic error bounds for two noise models: expectation bounds for independent zero-mean bounded-variance noise, and high-probability bounds for independent sub-Gaussian noise. The analysis yields an a priori parameter-choice rule and suggests an adaptive strategy suitable for large-scale computation. Numerical experiments support the theory and show that the predicted parameter is nearly optimal and that the adaptive method is effective in practice.

5. 基于高斯混合模型的运动驱动层析成像

Motion-Enabled Tomography via Gaussian Mixture Models

作者: Daniel Burrows, Can Evren Yarman, Ozan Öktem

类别: math.NA

发布日期: 2026-05-18

arXiv ID: 2605.18210v1

链接: http://arxiv.org/abs/2605.18210v1

摘要: Recovering physical properties of objects in motion is a core task across scientific and industrial applications. When the relative motion between the object and the sensing apparatus provides sufficient angular coverage, Computerized Tomography offers a powerful means of reconstruction. For such scenarios, we propose a parametric spatiotemporal model applied to Gaussian Mixture Models (GMM), in which each constituent Gaussian is parameterized by its own angular velocity, projectile motion, and geometry. GMM are a suitable means of reconstruction because they (i) admit accurate approximations in object space and (ii) have a closed form expression under the ray transform; enabling efficient forward predictions and exact gradient computations in data space. By decoupling the reconstruction problem into two sub-inverse problems, we characterize solutions as minimizers of task-specific objective functions that are derived and solved by utilizing the properties of (ii). The resulting algorithm we provide is applicable to objects in Euclidean space of arbitrary dimension. We validate the method on a simulated 2D problem, achieving accurate reconstruction of a 5-Gaussian GMM with intersecting trajectories. This also provides a foundation for further experimentation in settings with noisy data, 3D objects, and non-rigid body dynamics.

6. 一种用于平面等距曲线的正则化方法与双向等距识别

A regularization method for planar offset curves and bi-offset recognition

作者: Rosanna Campagna, Salvatore Mondrone, Tomas Sauer

类别: math.NA

发布日期: 2026-05-18

arXiv ID: 2605.18096v1

链接: http://arxiv.org/abs/2605.18096v1

摘要: Offset curves for planar trajectories are interesting in the generation of tool paths for numerically controlled industrial machines and in trajectory planning methods for autonomous driving systems. Theoretical offset curves may exhibit peculiar singularities, including self-intersections, which limit their use in practical applications. Existing approaches address these issue through geometric filtering techniques to detect and remove undesirable features but the computation of accurate and well-behaved offset curves remains a challenging task. We assume a first stage of functional approximation of trajectories by penalized Hermite spline regression enabling the simultaneous fitting of positions and tangents. The regularization is imposed on the second derivatives, effectively mitigating the jerk effect, which is particularly relevant in motion planning and path smoothing applications. Then, taking into account the geometrical pointwise properties of the resulting curve, we design two offset curves through the simultaneous approximation of function values and derivatives. Then, a mathematical model to obtain the so-called bi-offset as most fitting as with the original generator curve is proposed, also relating the offset range and pointwise curvature values. The adaptive reconstruction of the center line from the external boundaries is a topic of interest and is the main focus of our work. Numerical experiments confirm the reliability of our approach at every stage of the resolution process.

7. 不适定薛定谔化倒向扩散的 McLachlan 投影约化动力学

McLachlan-projected reduced dynamics for ill-posed Schrödingerized backward diffusion

作者: Jeongbin Jo

类别: math.NA, quant-ph

发布日期: 2026-05-18

arXiv ID: 2605.17996v1

链接: http://arxiv.org/abs/2605.17996v1

摘要: Backward diffusion is a prototype ill-posed evolution: high spatial frequencies grow exponentially in time, so mesh-based time marching without explicit regularization is quickly overwhelmed by noise. Schrödingerization embeds the semidiscrete generator into Hermitian dynamics on an extended space; we ask whether McLachlan projection onto a fixed low-dimensional frame supplies a structured regularizer whose error budget can be read from a projection defect that separates full lifted propagation from the reduced trajectory. We prove uniqueness of the reduced flow, Gram-norm conservation, a lifted–reduced gap bound in terms of that defect, and perturbation estimates that highlight overlap-matrix conditioning when matrix elements are estimated statistically. We also spell out a fair classical baseline – spectral low-pass or Tikhonov filtering on the same semidiscrete model, with bandwidth or ridge strength matched to the information content of the chosen frame – so numerical contrasts isolate the Schrödingerized reduced pipeline rather than an unregularized Crank–Nicolson march that mainly showcases blow-up. A calibrated one-dimensional benchmark pairs a spectrally truncated reference with snapshot-built subspace evolution and finite-shot Qiskit Aer estimation, illustrating how lift, projection, and sampling layers contribute differently to the overall error.

8. 通过路径测度上的序列蒙特卡洛实现扩散模型的简单近似与无导数推理时缩放

Simple Approximation and Derivative Free Inference-Time Scaling for Diffusion Models via Sequential Monte Carlo on Path Measures

作者: Chenyang Wang, Weizhong Wang, Yinuo Ren, Jose Blanchet, Yiping Lu

类别: stat.ML, cs.CV, cs.LG, math.NA, math.PR

发布日期: 2026-05-18

arXiv ID: 2605.17850v1

链接: http://arxiv.org/abs/2605.17850v1

摘要: iffusion-based generative models increasingly rely on inference-time guidance, adding a drift term or reweighting mixture of experts, to improve sample quality on task-specific objectives. However, most existing techniques require repeated score or gradient evaluations, introducing bias, high computational overhead, or both. We introduce \texttt{URGE}, Unbiased Resampling via Girsanov Estimation, a derivative-free inference-time scaling algorithm that performs path-wise importance reweighting via a Girsanov change of measure. Instead of computing gradient-based particle weights in previous work, \texttt{URGE} attaches a simple multiplicative weight to each simulated trajectory and periodically resamples. No score, no Hessian, and no PDE evaluation is required. We establish an equivalence between path-wise and particle-wise SMC: the Girsanov path weight admits a backward conditional expectation that recovers the previous particle-level weights, guaranteeing that both schemes produce the same unbiased terminal law. Empirically, \texttt{URGE} outperforms existing inference-time guidance baselines on synthetic tests and diffusion-model benchmarks, achieving better generation quality, while being significantly simpler to implement and fully gradient-free.

9. 多重网格入门：基本原理

Multigrid Primer: Basic Principles

作者: Stephen F. McCormick, Rasmus Tamstorf

类别: math.NA

发布日期: 2026-05-18

arXiv ID: 2605.17828v1

链接: http://arxiv.org/abs/2605.17828v1

摘要: The goal of this primer is to provide a relatively short exposition of the basics of multigrid methods, simplified by focusing on fundamental concepts in a variational setting. This is done by way of a quadratic energy minimization formulation of symmetric positive definite linear equations arising from the discretization of elliptic partial differential equations. This focus provides an alternate viewpoint to other expositions and, more importantly, it enables a simplification of the development while clarifying the principles that lead to effective algorithms. The development begins with multigrid as an iterative solver exemplified by the so-called V-cycle. It then introduces the full multigrid method as a direct solver in the sense that it is aimed directly at the source of the matrix equations. In this way, full multigrid attempts to achieve discretization-level accuracy at a cost comparable to that of a few matrix multiplies on the finest level.

10. 用自适应埃尔米特谱方法求解弗拉索夫-泊松系统

Solving Vlasov-Poisson system with an adaptive Hermite spectral method

作者: Sihong Shao, Yanli Wang, Jie Wu

类别: math.NA

发布日期: 2026-05-18

arXiv ID: 2605.17820v1

链接: http://arxiv.org/abs/2605.17820v1

摘要: We propose an adaptive Hermite spectral method for the Vlasov-Poisson system based on a recently developed frequency indicator that measures the contribution of the high-order expansion coefficients. Precisely, the symmetrically weighted Hermite basis with a scaling factor is utilized to approximate the distribution function to satisfy the increasing resolution requirement, which, for example, is induced by filamentation. To implement the scaling adjustment, a fast conservative projection operator is constructed in two steps. The first step is to formulate the projection as a constrained optimization problem to preserve key invariants, including mass, momentum, energy, and the $L^2$ norm of the distribution function. The second step is an ODE-based approximation developed to compute the updated expansion coefficients with linear complexity. Numerical experiments with 1D1V and 2D2V settings validate the feasibility and efficiency of this proposed adaptive Hermite method.

11. 可微函数的信息论复杂度

The information-theoretic complexity of differentiable functions

作者: Matthijs Ruijgrok

类别: cs.IT, math.NA, nlin.CG

发布日期: 2026-05-18

arXiv ID: 2605.17801v1

链接: http://arxiv.org/abs/2605.17801v1

摘要: A measure for the complexity of a differentiable function f(x) on an interval is introduced. It is based on approximations of the function by piecewise constant functions. The measure takes into account the quality of the approximation and the number of intervals in the approximating function. This measure, called the V-complexity of f(x), is shown to formalize some intuitions about the simplicity or complexity of f(x). The V-complexity is then compared to another measure of complexity, namely how compressible an approximation of f(x) is. It is hypothesized that V-complexity is equivalent to the compression measure, in the case of the Run Length Encoding and the Lempel Ziv 77 algorithms. V-complexity can be used as an ingredient in the definition of the Effective Complexity (EC) of a Complex System. When the perceived regularities of such a system are described by a differentiable function on an interval, the EC can be defined as the V-complexity of that function. EC is applied to the model of diffusion of cream in a cup of coffee. The perceived regularity of this model is given by the diffusion equation. The V-complexity of the solution of the equation starts at zero, quickly increases to a maximum and then decreases back to zero as the liquid reaches its equilibrium state. It is shown that this is also the result when a cellular automaton approach and the concept of Apparent Complexity is used.

12. Tucker分解的两种交替迭代格式收敛性分析

Convergence Analysis of Two Alternating Iterative Schemes for Tucker Decomposition

作者: Ren-Cang Li, Li Wang, Mei Yang

类别: math.NA

发布日期: 2026-05-18

arXiv ID: 2605.17793v1

链接: http://arxiv.org/abs/2605.17793v1

摘要: The higher-order orthogonal iteration (HOOI) and the alternating subspace iteration (ASI) are two popular numerical methods for computing the Tucker decomposition of a multiple-mode tensor. Xu [Linear and Multilinear Algebra, 66(11):2247–2265, 2018] proposed a variation of HOOI, called the greedy HOOI, which has an extra alignment action between consecutive approximations. Kroonenberg and De Leeuw [Psychometrika, 45(1):69–97, 1980] analyzed the convergence of ASI but their analysis has gaps. These analysis were for a real tensor only. In this paper, we present detailed convergence analysis of the two methods that is applicable to a complex tensor with a real tensor being a special case, and it is shown both methods are globally convergent to stationary points under mild conditions while the objective function monotonically increases. Numerical examples are presented to demonstrate the convergence behavior of the methods.

13. 固定缴费养老金计划最优减额的数值方法

Numerical methods for optimal decumulation of a defined contribution pension plan

作者: Peter A. Forsyth, George Labahn

类别: cs.CE, math.NA

发布日期: 2026-05-18

arXiv ID: 2605.17744v1

链接: http://arxiv.org/abs/2605.17744v1

摘要: The decumulation of a defined contribution (DC) pension plan is well known to be one of the hardest problems in finance. We model this decumulation challenge as an optimal stochastic control problem. The control problem is solved, at each rebalancing date, by alternatively solving a linear partial-integro differential equation (PIDE) followed by an optimization step. We solve the PIDE by using a $δ$-monotone Fourier method, which ensures that monotonicity holds to $O(δ)$. We allow for the use of leverage (i.e. borrowing to invest in stocks), as well as minimum constraints on bond holdings. We pay particular attention to minimizing wrap-around error, an issue which is endemic for Fourier methods and central to the effective use of these methods for optimal control problems. Rather unexpectedly, we find that restricting the portfolio equity fraction to a maximum of 50% does not reduce portfolio efficiency noticeably. This may be a useful strategy for risk-averse retirees.

14. 双尺度神经网络用于线性对流占优方程的最优控制

Two-scale neural networks for optimal control of linear convection-dominated equations

作者: Sijing Liu, Marcus Sarkis, Yi Zhang, Zhongqiang Zhang

类别: math.NA, math.OC

发布日期: 2026-05-18

arXiv ID: 2605.17740v1

链接: http://arxiv.org/abs/2605.17740v1

摘要: We propose a two-scale neural network method for optimal control problems governed by convection-dominated convection-diffusion-reaction equations. Building on two-scale architectures developed for singularly perturbed forward problems, we augment the spatial input with suitably rescaled features that become increasingly important as the diffusion coefficient becomes small. The approach employs separate neural networks for the state and adjoint state variables of the optimality system, reflecting the fact that these quantities develop sharp layers in different parts of the domain due to opposite convection fields. By choosing different center points for the two networks, the architecture naturally aligns with the layer location of each variable. We present two formulations of the method, one based on the first-order optimality conditions and another using penalization of the PDE constraint, and combine them with a successive training strategy that gradually decreases the diffusion coefficient toward its target value. Numerical experiments on benchmark problems illustrate the effectiveness and behavior of the proposed approach.

15. 超越精确求积的超插值：一种谱乘子方法

Hyperinterpolation beyond exact cubature: a spectral multiplier approach

作者: Hao-Ning Wu

类别: math.NA

发布日期: 2026-05-18

arXiv ID: 2605.17739v1

链接: http://arxiv.org/abs/2605.17739v1

摘要: We study hyperinterpolation and its spectral multiplier variants on the sphere under weak cubature assumptions formulated through Sobolev discrepancy estimates. In contrast with classical hyperinterpolation theory, our framework does not require exact polynomial cubature formulas or Marcinkiewicz–Zygmund inequalities. The main idea is to interpret the discretization error as the action of a spectral multiplier operator on the cubature discrepancy measure. This viewpoint separates approximation properties of the underlying spectral operator from geometric properties of the sampling measure, leading to stable Sobolev approximation estimates under weak cubature assumptions. The resulting theory applies to a broad class of spectral approximation operators, including sharp spectral projections, compactly supported smooth filters, Bessel potential operators, and heat kernel operators. For sufficiently localized spectral multipliers, we additionally obtain uniform $L^\infty$-stability of the corresponding discrete approximation operators. The results establish a direct connection between hyperinterpolation, Sobolev discrepancy, and quasi-Monte Carlo (QMC) designs, showing that stable approximation from scattered data can be achieved without exact polynomial reproduction.

16. 求解非Hermitian四元数线性系统的保结构四元数共轭梯度型方法

Structure preserving quaternion conjugate gradient-type methods for solving non-Hermitian quaternion linear systems

作者: Baohua Huang, Tao Li, Wen Li

类别: math.NA

发布日期: 2026-05-18

arXiv ID: 2605.17732v1

链接: http://arxiv.org/abs/2605.17732v1

摘要: In this paper, we consider the non-Hermitian quaternion linear systems arising from color image restoration and three-dimensional signal filtering problems. For exploring to solve such systems, we present two innovative structure-preserving conjugate gradient-type methods, QNHERLQ and QNHERQR, which are based on the unitary equivalence transformations of the non-Hermitian quaternion matrices to tridiagonal forms, called quaternion Saunders-Simon-Yip tridiagonalization procedure. The proposed tridiagonalization procedure for non-Hermitian quaternion matrices is closely related to the quaternion Lanczos process for Hermitian matrices, and is very different from the quaternion Lanczos biorthogonalization process for non-Hermitian matrices. The convergence of QNHERLQ and QNHERQR is discussed, which depends on the singular values of the coefficient matrix. Also we show that both algorithms have the finite termination property and constant costs per iteration step. Numerical results illustrate that the proposed algorithms are with the robustness and effectiveness compared with QGMRES and QQMR.

17. 带警察响应的住宅入室盗窃模型中的犯罪热点动力学

Crime hotspot dynamics in residential burglary models with police response

作者: Baoli Hao, Kamrun Mily, Annalisa Quaini, Ming Zhong

类别: math.DS, math.NA

发布日期: 2026-05-18

arXiv ID: 2605.17709v1

链接: http://arxiv.org/abs/2605.17709v1

摘要: We develop and analyze mathematical models for residential burglary that incorporates police deployment through a delayed feedback mechanism. Motivated by empirical observations from publicly available crime and policing data, we extend a well-known agent-based model by introducing a dynamic police response driven by crime information that becomes available only after a finite delay. Taking the mean-field limit, we derive a coupled continuum system consisting of three partial differential equations and one ordinary differential equation describing the interactions among criminal density, environmental attractiveness, delayed crime signal, and police deployment. Linear stability analysis of homogeneous steady states reveals that response delays can destabilize otherwise stable equilibria through Hopf bifurcations. As a result, the model predicts sustained temporal oscillations and dynamically evolving crime hotspots. Numerical simulations of both the agent-based and continuum models confirm the theoretical analysis and uncover rich spatio-temporal behaviors, including moving, splitting, and merging hotspots. Through a parametric study, we investigate the roles of police density, crime information delay, and neighborhood effects in controlling stability, hotspot size, and oscillatory behavior. Our results indicate that timely access to crime data plays a more important role than police density in stabilizing crime levels.

18. 一种具有精确线性收敛速率预测的改进安德森加速方法及其在不可压缩流动中的应用

A modified Anderson acceleration with sharp linear convergence rate predictions and application to incompressible flows

作者: Yunhui He, Leo Rebholz

类别: math.NA

发布日期: 2026-05-17

arXiv ID: 2605.17664v1

链接: http://arxiv.org/abs/2605.17664v1

摘要: In this work, we extend a modified Anderson acceleration proposed in [Y. He, arXiv:2603.25983, 2026] to accelerate the Picard iteration for the Navier-Stokes equations. In this variant of Anderson acceleration, named AAg, the nonlinear residual–rather than the standard fixed-point iteration residual–is used to define the associated least-squares problem. We establish a convergence analysis for this method with any depth that shows how AAg accelerates convergence through the gain of the optimization problem, and obtain a sharp prediction of its linear convergence rate (a feature that is not part of the known theory for classical Anderson acceleration). Additionally, motivated by this sharp convergence prediction, we introduce an adaptive strategy that automatically selects the depth parameter. Results of several numerical experiments are given that illustrate the new theory and also demonstrate the effectiveness of the proposed adaptive approach. Comparisons of AAg to usual AA and nonlinear GMRES are also provided.

19. 基于LCHS的带边界条件的对流-扩散方程的量子线路

Quantum circuits for the advection-diffusion equation with boundary conditions based on LCHS

作者: Leyu Chen, Tiegang Liu, Liang Xu, and Kun Wang

类别: math.NA, physics.comp-ph

发布日期: 2026-05-17

arXiv ID: 2605.17542v1

链接: http://arxiv.org/abs/2605.17542v1

摘要: This paper proposes a systematic and explicit quantum circuit framework for solving advection-diffusion equations with boundary conditions, based on the Linear Combination of Hamiltonian Simulations (LCHS) method. By employing the Finite Volume Method (FVM) combined with various flux construction schemes, we elaborate the design of quantum circuits tailored explicitly for Robin boundary conditions (including Dirichlet and Neumann boundary conditions as special cases) and periodic boundary conditions. In contrast to prior works on quantum simulation of advection-diffusion equations, we present a detailed error analysis for the linear combination of unitaries (LCU) induced by the constructed quantum circuits. A comprehensive gate complexity analysis demonstrates the quantum advantages over classical computing in high-dimensional scenarios. We simulate the proposed circuits on a fault-tolerant emulator, and numerical results validate the effectiveness of the proposed framework across homogeneous, inhomogeneous, and high-dimensional cases. The proposed framework is compatible with numerous spatial discretization methods and numerical schemes, extends naturally to other linear PDEs, and establishes a practical foundation for solving large-scale PDE problems on future fault-tolerant quantum computers.

20. WinQ: 加速语言模型在鞍点附近的量化感知训练

WinQ: Accelerating Quantization-Aware Training of Language Models Around Saddle Points

作者: Dongyue Li, Zechun Liu, Kai Yi, Zhenshuo Zhang, Changsheng Zhao, Raghuraman Krishnamoorthi, Harshit Khaitan, Hongyang R. Zhang, Steven Li

类别: cs.LG, math.NA

发布日期: 2026-05-17

arXiv ID: 2605.17471v1

链接: http://arxiv.org/abs/2605.17471v1

摘要: Quantization-aware training (QAT) is widely adopted to quantize language models by training full-precision weights using gradients from the quantized model. The main bottleneck is its slow convergence and early performance plateau, particularly below 4-bit-widths. While this problem has been observed in prior work, its precise cause remains unclear. In this paper, we analyze the convergence of QAT by estimating the spectrum of the loss-surface Hessians. We find that the weights converge to flat regions around saddle points, where a large fraction of the Hessian eigenvalues are both positive and negative. During training, an increasing fraction of Hessian eigenvalues concentrates around zero, whose magnitude decreases. At lower bit-widths, the magnitude of eigenvalues in the Hessian spectrum is significantly smaller. To mitigate these issues, we propose an algorithm called WinQ to accelerate QAT, which involves: (1) periodically resetting weights to the linear interpolation of full-precision and quantized weights, reducing the distance to the quantization grid and increasing eigenvalue magnitude, and (2) computing gradients of noise-injected weights to regularize the Hessian. Extensive experiments show that WinQ accelerates QAT by up to 4 times across various quantization methods and models. Under the same training cost, WinQ improves state-of-the-art sub-4-bit quantization by up to 8.8%. These results are consistent across 16 settings with different language models, quantization methods, and bit widths.

21. 使用封闭与算子分裂求解线性速率常微分方程层次结构（如主方程）

Solving linear-rate ODE hierarchies (like master equations) using closures and operator splitting

作者: Joshua C Chang

类别: math.NA, math.DS, math.PR, q-bio.QM, stat.CO

发布日期: 2026-05-16

arXiv ID: 2605.17186v1

链接: http://arxiv.org/abs/2605.17186v1

摘要: Countably infinite systems of linear ODEs arise as forward equations for many continuous-time Markov processes. The standard recipe – truncate to a finite cap N and exponentiate – pays cubic cost in N and a time-growing boundary-feedback bias. We identify a structural condition on the rates, L_{n+r,n} = alpha_r n + beta_r (“linear-rate”), under which the generating function G(z,t) = sum_n x_n(t) z^n satisfies a first-order linear PDE in z, and the method of characteristics yields a composition-multiplier representation G(z,t) = K_t(z) G(Phi_t(z), 0). The Taylor coefficients of Phi_t and K_t on any output window {0,…,N} are determined exactly by a closed lower-triangular polynomial ODE on R^{2(N+1)}, independent of any coefficients above N. Truncation enters only through the support M_0 of the initial law, set independently of N. For binary birth-death the closure collapses to the geometric tail p_n(t) = p_1(t) rho(t)^{n-1} with rho(t) = lambda(1 - e^{-(mu-lambda)t})/(mu - lambda e^{-(mu-lambda)t}). The linear-rate class spans Markov branching with immigration, multi-type branching, matrix-valued telegraph and G/R elongation, and signed or non-stochastic hierarchies. When the generator decomposes as L = A + B with A linear-rate and B non-affine (Schlogl bistable, predator-prey, lattice reaction-diffusion), we pair the closure with Strang splitting on B; Richardson extrapolation lifts the time order to Delta-t^4 at ~3x wall clock. On the Schlogl problem at V=500, N=8,000, the split runs 6.3x faster than dense Pade and 20x faster than sparse Krylov expv. For the stationary regime, a closure-Strang power iteration extends the same machinery to multi-dimensional product-state-space generators where sparse LU hits OOM/OOT or boundary-projection bias at usable caps. Numerical experiments locate where each route wins and where it is dominated by standard tools.

22. Kaczmarz方法中的数值不稳定性与迭代细化稳定化

Numerical Instabilities in the Kaczmarz Method and Stabilization by Iterative Refinement

作者: Michał Dereziński, Ethan N. Epperly, Deanna Needell, Alexander Xue

类别: math.NA

发布日期: 2026-05-16

arXiv ID: 2605.17185v1

链接: http://arxiv.org/abs/2605.17185v1

摘要: The randomized Kaczmarz method and its accelerated variants are a powerful class of iterative methods for solving large-scale linear systems, offering guaranteed convergence with low per-iteration cost. However, their numerical stability remains poorly understood. In this work, we investigate the stability properties of both classical and accelerated randomized Kaczmarz methods, with an emphasis on how error propagates across iterations and interacts with acceleration. We show that both classical and accelerated randomized Kaczmarz fail to be forward stable. To address this issue, we propose the integration of iterative refinement into randomized Kaczmarz frameworks. We demonstrate that refinement can effectively control error accumulation and recover high-accuracy solutions, even when the system is ill-conditioned. Numerical experiments corroborate our theoretical findings and illustrate the practical benefits of combining refinement with both classical and accelerated Kaczmarz methods.

23. 有理逼近与本征高斯过程

Rational approximation and intrinsic Gaussian processes

作者: Christopher Beattie, David Higdon, Leanna House, Colby Stakun-Pickering, Jared Clark

类别: math.NA

发布日期: 2026-05-16

arXiv ID: 2605.17168v1

链接: http://arxiv.org/abs/2605.17168v1

摘要: Gaussian processes (GPs) defined through intrinsic random fields provide a flexible framework for modeling spatial phenomena, and have been advocated in a variety of applications over the past several decades. Nevertheless, their adoption has lagged behind traditional, covariance-based approaches, in part because the intrinsic formulation has lacked an accompanying toolkit of computational methods and dependence specifications that facilitate fitting and prediction. We develop here a systematic framework for modeling intrinsic GPs and introduce practical algorithms and dependence/variogram models for modeling, inference and computation that parallel those of traditional, stationary GPs. We explore a close connection between intrinsic GP models and rational approximation, which clarifies the underlying problem structure. Numerical examples illustrate how the new tools can be deployed in practice, highlighting the advantages of intrinsic-field modeling in terms of robustness, interpretability, and computational efficiency.

24. 基于Q-和D-条件的少级数高阶显式龙格-库塔方法：一般理论与高效递归构造

Low Stage High Order Explicit Runge–Kutta Methods via Q- and D-Conditions: General Theory and Efficient Recursive Construction

作者: Junyuan He, Jizu Huang

类别: math.NA

发布日期: 2026-05-16

arXiv ID: 2605.16995v1

链接: http://arxiv.org/abs/2605.16995v1

摘要: Constructing explicit Runge–Kutta (ERK) methods with as few stages as possible for a given order is a classical problem in numerical analysis. In this work, we introduce a $Q$/$D$-space framework of sufficient order conditions for ERK methods. This framework generalizes Butcher’s classical simplifying assumptions by reformulating them in terms of simplified $Q$- and $D$-spaces defined through their residual vectors. It yields sufficient conditions which, together with $B(p)$, ensure order $p$. It also leads to a recursive construction procedure for ERK methods of arbitrary even order, in which the Butcher coefficients are obtained from two structured linear systems. For every even order $p\ge 4$, the construction produces ERK methods with stage number $s(p)=(p^2-2p+8)/4$. This stage count has the same leading term as that of the classical Gragg families, while improving the linear term. The free parameters retained by the construction further provide a systematic framework for designing methods with enhanced stability and short-time accuracy.

25. 块Krylov子空间与正交矩阵多项式：一种结构对应及其在酉矩阵中的应用

Block Krylov subspaces and orthogonal matrix polynomials: a structural correspondence with applications to unitary matrices

作者: Michele Rinelli, Raf Vandebril

类别: math.NA

发布日期: 2026-05-16

arXiv ID: 2605.16954v1

链接: http://arxiv.org/abs/2605.16954v1

摘要: We study the connection between block Krylov subspaces and matrix orthogonal functions. Under a no-deflation assumption, we show that polynomial block Krylov subspaces are isometrically isomorphic to spaces of matrix polynomials of bounded degree, providing a unified framework for the analysis and construction of orthonormal bases and recurrence relations. The same correspondence holds for rational block Krylov subspaces and matrix-valued rational functions, and in the extended Krylov setting this leads naturally to Laurent matrix polynomials. When the matrix $A$ is normal, we prove that the induced inner product admits a representation in terms of a discrete spectral matrix measure, extending a classical result for Hermitian matrices. In the unitary case, where the measure is supported on the unit circle, this connection allows us to transfer the Szegő recurrence for orthogonal matrix polynomials and the CMV framework for Laurent matrix polynomials to the block Krylov setting, yielding efficient procedures for the orthogonalization of polynomial and extended block Krylov subspaces.

26. 张量空间中由迹泛函和Wasserstein距离导出的张量的谱界

Spectral Bounds for Tensors Derived from Trace Functionals and Wasserstein Distance in Tensor Spaces

作者: Hemant Sharma, Nachiketa Mishra

类别: math.NA

发布日期: 2026-05-16

arXiv ID: 2605.16930v1

链接: http://arxiv.org/abs/2605.16930v1

摘要: This article introduces a trace-based metric on the space of positive semi-definite (PSD) tensors, offering a geometric perspective that connects their algebraic structure to their intrinsic geometric properties. It defines the Bures-Wasserstein distance on tensor spaces, establishing clear measurements between tensors. Moreover, the study derives trace-based eigenvalue bounds for PSD tensors and analyzes how these bounds depend on the PSD condition. The behavior of these bounds is further explored when the PSD requirement is relaxed, with illustrative examples provided to support the theoretical findings. In addition, a detailed complexity analysis is carried out for the methods proposed in this study.

27. 带有非守恒项的熵稳定间断伽辽金欧拉求解器的GPU性能

GPU Performance of an Entropy-Stable Discontinuous Galerkin Euler Solver with Non-Conservative Terms

作者: Henry Waterhouse, Maciej Waruszewski, Lucas C. Wilcox, Francis X. Giraldo

类别: math.NA

发布日期: 2026-05-15

arXiv ID: 2605.16684v1

链接: http://arxiv.org/abs/2605.16684v1

摘要: The entropy-stable discontinuous Galerkin method for compressible Euler equations with buoyancy is implemented on graphics processing unit (GPU) hardware. We measure the performance of the solver on three-dimensional problems: the rising thermal bubble and the baroclinic instability in a channel. On NVIDIA A100 hardware, the solver achieves nearly 70% of 64-bit floating-point peak performance for the most computationally expensive kernel (volume terms) and significantly reduces the computational overhead typically incurred by two point entropy-stable fluxes in the volume terms. We also present impressive strong and weak scaling performance of the solver and compare to a highly-optimized central processing unit (CPU) code showing that the GPU kernels are a factor of $10\times$ faster and better than $13\times$ more energy efficient than the CPU code. We also show that the solver achieves the expected $2\times$ speedup when run at 32-bit floating-point peak performance. We discuss the different modifications that we implemented to reach the final form of the GPU implementation and measure the performance gain of each of the implementation strategies ranging from reduction in complex operations and memory traffic as well as load balancing. We also extend symmetry-based flux savings to the non-symmetric gravity term, preserving nearly the full factor-of-two speedup achieved for the symmetric flux.

28. fPINN-DeepONet：一种面向多项时间分数阶混合扩散-波动方程的物理信息算子学习框架

fPINN-DeepONet: A Physics-Informed Operator Learning Framework for Multi-term Time-fractional Mixed Diffusion-wave Equations

作者: Binghang Lu, Zhaopeng Hao, Christian Moya, Guang Lin

类别: math.NA, cs.LG

发布日期: 2026-05-15

arXiv ID: 2605.16594v1

链接: http://arxiv.org/abs/2605.16594v1

摘要: In this paper, we develop a physics-informed deep operator learning framework for solving multi-term time-fractional mixed diffusion-wave equations (TFMDWEs). We begin by deriving an $L_2$ approximation, which achieves first-order accuracy for the Caputo fractional derivative of order $β\in (1,2)$. Building upon this foundation, we propose the fPINN-DeepONet framework, a novel approach that integrates operator learning with the $L_2$ approximation to efficiently solve fractional partial differential equations (FPDEs). Our framework is successfully applied to both fixed and variable fractional-order PDEs, demonstrating the framework’s versatility and broad applicability. To evaluate the performance of the proposed model, we conduct a series of numerical experiments that involve dynamically varying fractional orders in both space and time, as well as scenarios with noisy data. These results highlight the accuracy, robustness, and efficiency of the fPINN-DeepONet framework.

29. 一种面向非均质多孔介质的并行自适应无网格方法

A Parallel and Adaptive Mesh-Free method for Heterogeneous Porous Media

作者: Kapil Chawla, Sanghyun Lee, Yeonjong Shin

类别: math.NA

发布日期: 2026-05-15

arXiv ID: 2605.16564v1

链接: http://arxiv.org/abs/2605.16564v1

摘要: Material properties such as permeability fields in heterogeneous porous media are often represented as discontinuous, piecewise constant data tied to a given spatial discretization. Such representations are inherently mesh-dependent, requiring interpolation or projection whenever they are transferred to a different discretization. In this work, we develop \emph{Parallel and Adaptive Mesh-Free Approximation (PAM)}, a mesh-independent framework that approximates discontinuous data by a continuous, closed-form function. The resulting approximation can be evaluated consistently across different geometries and numerical discretizations, while preserving sharp interface features. The proposed PAM framework employs radial basis functions (RBFs) to construct continuous approximations of discontinuous data. To accurately capture discontinuities, we incorporate Shepard-normalization, which stabilizes the approximation near sharp interfaces. The coefficients of the RBF expansion are determined via sparse regression, enabling automatic selection of the most relevant basis functions and promoting robust representations. In addition, we develop a novel adaptive refinement approach which further enriches the approximation in regions of rapid spatial variation. We provide a theoretical analysis showing that the proposed normalized RBF framework achieves arbitrarily small $L^1$ error in approximating discontinuous step functions. To enhance computational efficiency, the domain is partitioned into subdomains, and the reconstruction problem is solved independently on each subdomain in parallel. Numerical experiments demonstrate the accuracy, adaptivity, and scalability of the proposed method, including applications to challenging heterogeneous permeability fields.