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1. 基于简化传输条件的非相对论区域 Klein-Gordon-Schrödinger 方程的一致精确多尺度时间积分器
A Uniformly Accurate Multiscale Time Integrator for the Klein-Gordon-Schrödinger Equations in the Nonrelativistic Regime via Simplified Transmission Conditions
作者: Yue Feng, Caoyi Liu
类别: math.NA
发布日期: 2026-05-13
arXiv ID: 2605.12936v1
摘要翻译: 我们针对具有无量纲参数 $\epsilon\in(0,1]$（其中 $\epsilon$ 与光速成反比）的 Klein-Gordon-Schrödinger (KGS) 方程，提出了一种新颖且简化的多尺度时间积分器傅里叶拟谱（MTI-FP）方法。严格证明了所提出的 MTI-FP 方法在非相对论区域（即 $\epsilon\to0$ 时）在时间上达到一致一阶精度。在该区域，KGS 方程的解表现出波长为 $O(\epsilon^2)$ 的时间振荡，这对经典数值方法提出了苛刻的分辨率要求。一致精确的 MTI-FP 方法建立在两个关键点上：(i) 在每个时间区间上通过频率进行多尺度分解，并采用简化的传输条件；(ii) 时间离散采用指数积分器，空间离散结合傅里叶拟谱方法。利用能量方法和数学归纳法，我们在 $H^1$ 范数下严格建立了两个独立的误差界：$O(h^{m_0-1}+\tau^2/\epsilon^2)$ 和 $O(h^{m_0-1}+\epsilon^2)$，其中 $h$ 为网格尺寸，$\tau$ 为时间步长，$m_0$ 为依赖解的正则性的整数。这些估计表明 MTI-FP 方法在空间上一致且最优收敛，在时间上关于 $\epsilon\in(0,1]$ 以 $O(\tau)$ 一致收敛。此外，通过在每个时间区间上对微观变量进行线性插值与多尺度分解相结合，我们得到了对任意 $t>0$ 都一致精确的数值解。因此，从香农采样理论的角度来看，所提出的 MTI-FP 方法具有时间上的超分辨率性质。提供了大量的数值实验来验证误差估计并展示超分辨率性质。最后，将该方法应用于数值研究 KGS 方程到不同极限模型的收敛速率。
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链接: http://arxiv.org/abs/2605.12936v1
1. 研究对象和背景
本文研究无量纲化的Klein-Gordon-Schrödinger (KGS) 方程组
$$ i\partial_t \psi + \Delta \psi + \lambda \varphi \psi = 0,\qquad \varepsilon^2 \partial_{tt} \varphi - \Delta \varphi + \frac{\mu^2}{\varepsilon^2} \varphi - \lambda |\psi|^2 = 0, $$其中空间变量 $x\in\mathbb{R}^d$ ($d=1,2,3$), 时间 $t>0$, 未知函数 $\psi$ 是复标量核子场, $\varphi$ 是实标量介子场, 无量纲参数 $\varepsilon\in(0,1]$ 与光速成反比, $\mu = m_2/(2m_1)>0$ 是质量比, $\lambda$ 是Yukawa耦合常数. 初始条件为
$$ \psi(x,0)=\psi_0(x),\quad \varphi(x,0)=\varphi_0(x),\quad \partial_t\varphi(x,0)=\frac{1}{\varepsilon^2}\varphi_1(x), $$其中 $\psi_0$ 与 $\varphi_0,\varphi_1$ 均与 $\varepsilon$ 无关. 该方程组作为核子-介子相互作用的平均场模型被广泛研究, 也出现在等离子体物理与非线性光学中.
当 $\varepsilon = O(1)$ 时, 方程处于经典区域, 已有大量解析与数值结果. 但在非相对论极限区域 ($0<\varepsilon\ll1$), 问题呈现出本质困难: 解具有高频时间振荡, 其主导频率为 $\mu/\varepsilon^2$, 波长尺度为 $O(\varepsilon^2)$. 从能量估计可知总能量 $E(t)=O(\varepsilon^{-2})$, 这导致经典的数值方法(如有限差分)需要极其苛刻的网格步长, 例如时间步长需满足 $\tau=O(\varepsilon^3)$ 才能正确分辨振荡[35], 从而计算代价难以承受. Shannon采样理论表明这类方法是时间欠分辨的.
为克服该困难, 近年来已提出多种一致精确的数值方法. 它们均采用 $h=O(1)$, $\tau=O(1)$ 的网格策略, 且对任意 $\varepsilon\in(0,1]$ 均能保持一致的精度. 代表性工作包括: 基于多尺度分解的多尺度时间积分器(MTI)[6], 基于双尺度展开(TSF)的方法[11], 渐近相容的指数型积分器[7], 嵌套Picard迭代(NPI)[10]以及一致精确积分器(UAI)[9]等. 这些方法实现了时间上的超分辨——即只需远大于振荡波长的步长即可获得高精度解.
本文试图在已有MTI框架的基础上进一步推进. 文献[6]的MTI方法采用精心准备的初始条件(well-prepared initial data) 来定义多尺度分解中的微变量, 使得 $\partial_t v(x,0) = -\frac{i}{2\mu}\Delta v(x,0)$, 这导致数值格式较为复杂且对解的正则性要求较高. 本文通过引入简化传输条件, 即直接令 $\partial_t v(x,0)=0$, 构造了一个新的MTI-Fourier伪谱(MTI-FP)方法. 该方法在保持相同的一致一阶时间精度的同时, 显著简化了格式, 降低了计算成本, 并弱化了对解的正则性要求, 同时在空间上达到给定正则性下的最优精度. 此外, 利用多尺度分解中的微变量线性插值, 该方法还可以几乎无代价地得到任意时刻 $t>0$ 的一致精确数值解, 这是其他一致精度方法(如TSF和UAI)所不具备的.
2. 主要定理或主要结果
本文针对一维周期边值问题(可张量积推广至高维), 在时间步长 $\tau$ 与空间网格尺寸 $h$ 的设定下, 严格证明了所提MTI-FP方法在 $H^1$ 范数下的两个独立误差界:
$$ \|\psi(\cdot,t_n)-\psi^n\|_{H^1} + \|\varphi(\cdot,t_n)-\varphi^n\|_{H^1} \lesssim h^{m_0-1} + \frac{\tau^2}{\varepsilon^2}, $$$$ \|\psi(\cdot,t_n)-\psi^n\|_{H^1} + \|\varphi(\cdot,t_n)-\varphi^n\|_{H^1} \lesssim h^{m_0-1} + \varepsilon^2, $$其中整数 $m_0$ 依赖于解的正则性(解属于 $H^{m_0}$). 以上界分别通过能量方法与数学归纳法导出. 综合这两个估计即得到对 $\varepsilon\in(0,1]$ 一致有效的时间误差界:
$$ \|\psi(\cdot,t_n)-\psi^n\|_{H^1} + \|\varphi(\cdot,t_n)-\varphi^n\|_{H^1} \lesssim h^{m_0-1} + \tau. $$这表明方法在时间上具有一致的一阶精度, 且空间精度为 $O(h^{m_0-1})$, 即达到给定解正则性下的最优收敛阶(当使用Fourier谱方法时, 若解充分光滑则可获谱精度).
此外, 通过在每个时间区间 $[t_n,t_{n+1}]$ 上对微观变量 $v^n$ 和 $r^n$ 进行线性插值, 可构造出任意时刻 $t\ge 0$ 的数值解, 且该插值不增加额外误差, 从而获得一致精确的全时逼近. 数值上验证了上述误差估计的锐利性, 并展示了时间超分辨性质: 即使在时间步长远大于 $\varepsilon^2$-波长时, 数值解仍保持高精度. 最后, 应用该方法数值研究了KGS方程在不同初始条件下向Schrödinger极限模型以及带波算子的Schrödinger半极限模型的收敛速率, 结果与理论分析吻合.
3. 研究方法、关键技术和核心工具
本文的核心创新在于构造了一种简化的多尺度时间积分器, 其关键技术包括:
- 基于简化传输条件的多尺度频率分解. 对每一时间步 $[t_n, t_{n+1}]$, 将解 $\varphi$ 分裂为
并要求 $v^n$ 满足线性Schrödinger-波算子方程
$$ \varepsilon^2\partial_{ss}v^n + 2i\partial_s v^n - \partial_{xx}v^n = 0, $$而余项 $r^n$ 满足非线性Klein-Gordon方程
$$ \varepsilon^2\partial_{ss}r^n - \partial_{xx}r^n + \frac{1}{\varepsilon^2}r^n - |\psi^n|^2 = 0. $$与文献[6]的关键区别在于传输条件: 此处取初始条件为
$$ v^n(x,0)=\frac12[\varphi^n_0(x)-i\varphi^n_1(x)],\quad \partial_s v^n(x,0)=0,\qquad r^n(x,0)=0,\;\partial_s r^n(x,0)=0, $$即 $v^n$ 的初始导数取为零, 从而避免了准备初始数据所需的额外空间微分, 极大地简化了格式并降低了对初值正则性的要求. 该设定使得微变量 $v^n$ 的线性方程可精确求解, 且其解关于 $\varepsilon$ 一致有界.
- 指数时间离散与Gautschi型求积. 对线性部分采用指数积分器精确求解: $v^n$ 的Fourier系数 $\widehat{(v^n)}_l(s)$ 由 $a_l(s)\widehat{(v^n)}_l(0)$ 给出, 其中 $a_l(s)$ 为显式表达式; 对非线性源项积分, 采用Gautschi型一次近似
并类似处理 $\psi^n$ 方程中的卷积项. 这里 $\omega_l = \frac{1}{\varepsilon^2}\sqrt{1+\varepsilon^2\mu_l^2}$, 系数 $p_l,q_l$ 等均可显式计算出, 从而整个格式是显式的, 且可利用快速Fourier变换(FFT)实现, 每步计算复杂度为 $O(N\log N)$.
误差分析的双重估计策略. 证明的难点在于同时控制时间离散误差与高频振荡在不同 $\varepsilon$ 下产生的放大效应. 作者采用两种独立的技术路径: (i) 能量方法: 较直接地给出 $O(\tau^2/\varepsilon^2)$ 形式的误差估计, 该界在 $\varepsilon$ 很小时不理想, 但结合 $\tau = O(\varepsilon^2)$ 即退化为经典结果; (ii) 数学归纳法: 充分利用分解后 $v^n$ 和 $r^n$ 的一致有界性, 通过精细分析非线性项中的相位振荡与正交性, 建立 $O(\varepsilon^2)$ 形式的误差界, 该界虽对小区间给出更紧估计, 但依赖于解具有更高阶的一致正则性. 将二者结合, 对任意 $0<\varepsilon\le 1$ 取 $\min\{\tau^2/\varepsilon^2,\,\varepsilon^2\}\le \tau$, 即得一致的一阶精度. 该策略避免了对刚性的直接处理, 而是将误差分解为“在 $\varepsilon$ 不小时能量估计有效”与“在 $\varepsilon$ 极小时利用高频平均性质”两部分.
全局时间一致逼近的插值后处理. 在每一时间步计算出 $\varphi^{n+1}$ 后, 可得到离散时刻的数值解. 为获得任意 $t>0$ 处的值, 作者利用多尺度分解的自然结构: 对 $t\in(t_n,t_{n+1})$, 用 $e^{i(t-t_n)/\varepsilon^2}v^{n,1}(x) + e^{-i(t-t_n)/\varepsilon^2}\overline{v^{n,1}}(x) + r^{n,1}(x)$ 进行线性重构, 其中 $v^{n,1}, r^{n,1}$ 是计算得到的 $v^n(\cdot,\tau)$ 和 $r^n(\cdot,\tau)$ 的线性插值. 该重构在原有的 $H^1$-误差基础上仅引入 $O(\tau)$ 的额外误差, 从而保持了整体的一致精度, 并且几乎不增加计算量. 这一性质是MTI方法所独有的, 源自其在每一时间步均显式保留了高频因子 $e^{\pm i t/\varepsilon^2}$.
4. 与之前工作的比较
本文与已有的KGS方程非相对论区域数值方法作了多方面的比较, 尤其聚焦于同属MTI框架的文献[6].
格式复杂度与计算效率. 文献[6]的MTI方法基于良好准备的初始条件, 即选取 $\partial_t v(x,0) = -\frac{i}{2\mu}\Delta v(x,0)$. 在实际计算中, 这需要在每一时间步对微变量 $v^n$ 的初始导数施加空间微分算子, 不仅增加了计算量, 也对解的正则性提出了更高要求. 本文通过采用齐次初始导数传输条件 $\partial_s v^n(x,0)=0$, 完全避免了该步骤, 使格式大为简化, 数值实现仅需 $O(N\log N)$ 复杂度且易于编程.
正则性要求. 文献[6]的误差分析要求解满足 $\psi,\varphi\in H^{m_0}$, 且需要一定的 $H^{m_0+1}$ 型条件以保证准备初始数据有意义. 本文的误差界中 $m_0$ 仍然表征解的正则性, 但得益于简化的初始条件, 证明只需解与文献[6]同阶的正则性即可获得相同的一阶时间精度, 作者明确声称“正则性要求被弱化”. 从提供的文本看, 具体弱化程度未列出定理严格比较, 但这一声明来自格式本身的简化.
时间步长与空间精度的权衡. 两个方法均得到一致一阶时间精度, 且本文额外证明了空间精度 $O(h^{m_0-1})$ 是最优的 (给定 $H^{m_0}$ 的正则性). 文献[6]的方法是否达到空间最优并未明确提及. 此外, 本文通过双重误差估计 (能量与归纳) 给出了更完整的误差结构图像, 尤其 $O(\varepsilon^2)$ 界单独揭示了在 $\varepsilon\to0$ 时方法的超收敛潜力.
后处理插值能力. 文中指出, 基于多尺度分解的线性插值以获得任意时刻数值解的特性仅适用于MTI方法, 而不能直接应用于TSF或UAI等其他一致精度方法. 这是因为MTI在每一时间步显式重建了 $\varphi$ 的高频振荡因子, 而其他方法通常在每一时间步直接计算低频和高频成分的组合, 丢失了中间的相位信息.
与其他一致精度方法的比较. 与TSF方法[11]相比, MTI方法无需求解复杂的双尺度耦合系统, 而是逐时间步顺序求解线性方程和非线性方程, 实施更简便. 与指数型积分器[7]或UAI[9]相比, 本文格式是显式的, 且利用了Fourier谱方法的快速算法, 在空间精度和计算效率上更具优势. 不过, 由于缺少具体对比实验, 从提供的内容无法确认本文方法在同样粗网格下常数因子是否占优, 或对某些特殊解是否更具稳健性.
局限与未来工作. 文中未讨论可能的局限性, 但考虑到简化传输条件实质上是一种零阶近似, 可以推测当方程参数 $\mu$ 很小或初始相位不匹配时, 可能需要更多时间步才能达到渐近状态. 另外, 误差界中 $O(\tau^2/\varepsilon^2)$ 与 $O(\varepsilon^2)$ 的交错意味着当 $\varepsilon$ 处于中间量级时, 实际误差可能会在两者之间摆动, 但整体仍由 $\tau$ 控制. 文中未提供自适应 $\tau$ 选择策略, 这可以作为未来研究方向.
了解领域论文（摘要翻译）
1. 精确守恒与Onsager阈值：不可压缩Navier–Stokes方程的离散外微积分理论
Exact conservation and the Onsager threshold: a discrete exterior calculus theory for incompressible Navier–Stokes
作者: Peter Korn
类别: math.AP, math.NA
发布日期: 2026-05-13
arXiv ID: 2605.13048v1
链接: http://arxiv.org/abs/2605.13048v1
摘要翻译
中文标题: 精确守恒与Onsager阈值：不可压缩Navier–Stokes方程的离散外微积分理论
摘要翻译: 我们为不可压缩Euler和Navier–Stokes方程建立了一个保持结构的离散化严格理论，该理论基于闭Riemann流形上棱柱Delaunay–Voronoi网格的离散外微积分。核心结果是一个选择原则：离散层面上的精确代数守恒不仅是一个保真性质，而且排除了其他离散化无条件可达的整类弱解。我们在四种情形下证明了这一点。\emph{光滑解情形}：收敛速度为$\mathcal{O}(h^{\min(r_{\rm rec},\,r_\star)}\,|\log h|^{\beta_d})$，关于粘性$\nu\ge 0$一致成立，其中$\beta_3 = 0$，$\beta_2 = 1$；在一般网格上为一阶精度，在具有形心接近性和重构对称性的网格上为二阶精度。\emph{Leray–Hopf弱解情形}：子列的$L^2$极限是粘性系统的弱解。\emph{无粘测度值情形}：极限是守恒的测度值Euler解；若离散解在Onsager阈值$\alpha> 1/3$以上具有一致的$C^{0,\alpha}$界，则其集中缺陷在该阈值以上消失。\emph{耗散情形}：在任何正则性下，没有子列收敛到一个能量耗散的Euler解，这种结构性排除来自于精确的离散能量守恒，并使该格式区别于其他方法。区间$1/3 < \alpha < 1$上能量守恒和无缺陷收敛成立，但唯一性仍是悬而未决的，这孤立了无粘流体动力学的核心公开问题。
不相关论文（基本信息）
1. Elastica++：用于大型相互作用的Cosserat杆组装体的高性能多物理场框架
Elastica++: A high-performance, multiphysics framework for large interacting assemblies of Cosserat rods
作者: Tejaswin Parthasarathy, Seung Hyun Kim, Songyuan Cui, Mattia Gazzola
类别: cs.CE, math.NA, physics.comp-ph
发布日期: 2026-05-13
arXiv ID: 2605.13766v1
链接: http://arxiv.org/abs/2605.13766v1
摘要: Soft, slender structures are ubiquitous in natural and engineered systems, with broad application potential from biomimetic materials to soft robotics. However, there is a notable lack of computational tools that simultaneously preserve high-fidelity continuum rod mechanics, scale to large interacting ensembles, and remain flexible across diverse biophysical settings. Here we introduce Elastica++, an open-source, high-performance implementation of the Cosserat-rod model for large-scale simulations of slender-body dynamics. Elastica++ combines performance-oriented kernels with shared-memory parallelism to sustain teraflop-scale throughput despite complex discretization domains and physical interactions. The framework further interoperates with external numerical solvers, supporting efficient multiphysics workflows. We demonstrate robustness and breadth through case studies spanning passive nest-like metamaterials, collective active-matter dynamics, cilia carpets, soft magnetic microrobots, and schooling swimmers. Elastica++ thus provides a missing foundation for high-throughput studies of emergent behavior in interacting assemblies of elastic slender structures.
2. 降低phi-FEM计算成本的多重网格和神经网络方法
A multigrid and neural network approach to reduce the computational cost of phi-FEM
作者: Raphaël Bulle, Michel Duprez, Vanessa Lleras, Killian Vuillemot
类别: math.NA
发布日期: 2026-05-13
arXiv ID: 2605.13718v1
链接: http://arxiv.org/abs/2605.13718v1
摘要: In this work, we present a combination of a multigrid approach and the phi-FEM immersed boundary finite element method to reduce its computational cost while preserving its accuracy. To further reduce the numerical cost of the approach, we also propose the combination of the previous technique with some neural network methods. We illustrate the efficiency of these two approaches with numerical test cases in 2D and 3D.
3. 基于滤波白噪声的二维湍流傅里叶模态随机建模
Stochastic modeling of Fourier modes in two-dimensional turbulence via filtered white noise
作者: Paolo Cifani, Franco Flandoli, Andrea Zanoni
类别: math-ph, math.NA, math.PR, physics.flu-dyn
发布日期: 2026-05-13
arXiv ID: 2605.13671v1
链接: http://arxiv.org/abs/2605.13671v1
摘要: Modeling turbulent flows by a random Fourier decomposition is a classical procedure in order to use simplified models of turbulence in heat transport and other applications. We carefully investigate the Fourier time series of two-dimensional turbulent flows forced at intermediate scales and identify significant statistical structures. In particular, we find the existence of a typical time correlation length, and propose a stochastic model for the Fourier components. Finally, we compute the transport of a passive tracer under purely convective dynamics by means of direct numerical simulation of the turbulent flow and compare it with the effective diffusion produced by the stochastic model.
4. 具有势的粘性 CIR 过程：不变测度与精确采样
Sticky CIR process with potential: invariant measure and exact sampling
作者: Tony Shardlow
类别: math.PR, math.NA
发布日期: 2026-05-13
arXiv ID: 2605.13648v1
链接: http://arxiv.org/abs/2605.13648v1
摘要: We study the sticky Cox-Ingersoll-Ross (CIR) process in one dimension, a diffusion on $[0,\infty)$ with a sticky boundary condition at the origin, arising as the marginal process in a sparse Bayesian inference framework based on Hadamard-Langevin dynamics. For the parameter range $δ\in(1,2)$, in which the origin is accessible but not absorbing, we prove well-posedness of the process and uniqueness of its invariant measure, which is a mixture of a point mass at zero and a weighted gamma-type density on the interior. We derive an explicit Green’s function for the resolvent in terms of confluent hypergeometric functions, and use this to construct an exact sampler for the invariant measure in the zero-potential case. For a non-trivial potential $G$, we establish existence and uniqueness of the tilted invariant measure via a Girsanov change of measure, and develop two sampling algorithms: a Metropolis-Hastings corrected sampler that targets the invariant measure exactly, and an unadjusted Langevin algorithm (ULA) that is cheaper per step but introduces an $O(h)$ bias. Numerical experiments confirm the predicted behaviour: the Metropolis-Hastings sampler achieves the target invariant measure at all step sizes, while the ULA exhibits the expected $O(h)$ bias.
5. 基于蒙特卡洛的Majorization-Minimization超参数估计方法
A Majorization-Minimization with Monte Carlo Approach for Hyperparameter Estimation
作者: Elle Buser, Julianne Chung, Hugo Díaz, Arvind K. Saibaba
类别: math.NA
发布日期: 2026-05-13
arXiv ID: 2605.13620v1
链接: http://arxiv.org/abs/2605.13620v1
摘要: We consider inverse problems with linear forward models and Gaussian priors, but with unknown hyperparameters that may arise from the model, the noise, or the specification of the prior. We model this using a hierarchical Bayes framework resulting in a posterior distribution that is non-Gaussian, in general, and challenging to sample from. Consequently, we use an empirical Bayes framework for estimating the maximum a posteriori estimate of the hyperpameters by considering the marginalized posterior distribution. However, the optimization problem is also computationally challenging due to the need for repeated evaluation of log determinants. To address this issue, we propose a Majorization-Minimization with Monte Carlo approach, which we call M$^{3}$C, for hyperparameter estimation. Specifically, we replace the challenging optimization problem with a sequence of simpler ones by utilizing a majorization function (or majorant) for the log-determinant term, combined with a Monte Carlo estimator to approximate the majorant. We provide theoretical results, showing that under certain assumptions, the M$^{3}$C iterates converge with high probability to a critical point of the original cost function. A variety of numerical examples are provided from seismic tomography, super-resolution imaging, and contaminant source identification.
6. 基于分层超体积和幅度指标的非光滑集合梯度上升逼近帕累托前沿
Nonsmooth Set-Gradient Ascent to the Pareto Front via Layered Hypervolume and Magnitude Indicators
作者: Michael T. M. Emmerich
类别: math.OC, cs.NE, math.NA
发布日期: 2026-05-13
arXiv ID: 2605.13468v1
链接: http://arxiv.org/abs/2605.13468v1
摘要: A nonsmooth set-gradient ascent method is developed for moving finite approximation sets toward the Pareto front in multiobjective optimization. The method optimizes layered set indicators: a base indicator is evaluated on successive nondomination layers, and the layer values are combined with rapidly decreasing weights. This gives ascent directions to nondominated and dominated points while preventing deeper layers from compensating for deterioration of the first front. Two base indicators are treated: the hypervolume indicator and the magnitude indicator of the dominated set, whose expansion over coordinate projections contains extent, projected-area, and volume terms. The scalar objectives are nonsmooth because nondomination layers change combinatorially and the active orthogonal-union geometry changes piecewise. On fixed strata, where layer assignments and active geometry remain unchanged, the indicators are piecewise smooth and chamberwise continuous. For the magnitude indicator, an exact gradient formula is derived as a linear combination of hypervolume gradients of projected shadow sets. Thus, for fixed objective dimension, magnitude gradients have the same asymptotic time complexity as hypervolume gradients. Lexicographic layer aggregation is related to a unary infinitesimal encoding. For finite-$ε$ surrogates, the main nonsmoothness mechanisms are isolated and chamberwise Lipschitz continuity on bounded sets is proved; a two-point counterexample shows that hard-layer scalarization is not globally continuous across layer switches. The theory motivates a projected finite-difference implementation with repulsion and recovery from stagnation. Numerical examples and reproducible code cover two- and three-objective settings, including objective-space tests, curved fronts, a supersphere benchmark, and traces comparing layered magnitude and hypervolume ascent.
7. 再论 CUR 扰动分析：局部切空间展开
Revisiting CUR Perturbation Analysis: A Local Tangent-Space Expansion
作者: Longxiu Huang
类别: math.NA, cs.IT
发布日期: 2026-05-13
arXiv ID: 2605.13437v1
链接: http://arxiv.org/abs/2605.13437v1
摘要: CUR decompositions approximate a matrix using selected columns, rows, and their intersection. Classical CUR theory provides exactness results for low-rank matrices and perturbation bounds controlled by the size of the noise. In this work we develop a local perturbation expansion for a fixed-index rank-truncated CUR map near an admissible rank-(r) matrix. We show that the Fréchet derivative of the rank-truncated CUR map is a sampling-induced oblique tangent-space projector determined by the selected rows and columns. Consequently, the local recovery error for an underlying low-rank matrix is governed not by the full perturbation norm alone, but by the image of the perturbation under this sampling-induced tangent projector. In particular, perturbations that are invisible to the selected rows and columns are removed to first order. We compare this behavior with the classical local expansion of the rank-(r) SVD truncation. SVD removes orthogonal-normal perturbations to first order, whereas rank-truncated CUR removes perturbations in the kernel of the sampling-induced oblique tangent projector. Numerical experiments illustrate these regimes and confirm the predicted first- and second-order local rates.
8. 分数阶Sobolev常数的伽辽金逼近
Galerkin Approximation of the Fractional Sobolev Constant
作者: Andreea Dima, Liviu I. Ignat
类别: math.NA, math.AP, math.CA
发布日期: 2026-05-13
arXiv ID: 2605.13347v1
链接: http://arxiv.org/abs/2605.13347v1
摘要: We establish sharp estimates for the discrete optimal constant of the fractional Sobolev inequality in dimension $N\geq 1$, with fractional exponent $s\in (0,\min\{1,N/2\})$. The convergence rates that we establish take place for the Galerkin approximation with piecewise linear elements, when the computations are carried out in the unit ball, for which we employ a quasi-uniform and regular mesh.
9. 面向交互式物理模拟的分层Transformer预条件
Hierarchical Transformer Preconditioning for Interactive Physics Simulation
作者: Carl Osborne, Minghao Guo, Crystal Owens, Wojciech Matusik
类别: cs.GR, cs.DC, cs.LG, math.NA
发布日期: 2026-05-13
arXiv ID: 2605.13343v1
链接: http://arxiv.org/abs/2605.13343v1
摘要: Neural preconditioners for real-time physics simulation offer promising data-driven priors, but they often fail to capture long-range couplings efficiently because they inherit local message passing or sparse-operator access patterns. We introduce the Hierarchical Transformer Preconditioner, a neural preconditioner anchored to a weak-admissibility H-matrix partition. The partition provides a multiscale structural prior (dense diagonal leaves plus coarsening off-diagonal tiles) that enables full-graph approximate-inverse computation with O(N) scaling at fixed block sizes. The network models the inverse through low-rank far-field factors and uses highway connections (axial buffers plus a global summary token) to propagate context across transformer depth. At each PCG iteration, preconditioner application reduces to batched dense GEMMs with regular memory access. The key training contribution is a cosine-Hutchinson probe objective that learns the action of MA on convergence-critical spectral subspaces, optimizing angular alignment of MAz with z rather than forcing eigenvalue clusters to a prescribed location. This removes unnecessary spectral-placement constraints from SAI-style objectives and improves conditioning on irregular spectra. Because both inference and apply are dense, dependency-free tensor programs, the full solve loop is captured as a single CUDA Graph. On stiff multiphase Poisson systems (up to 100:1 density contrast, N = 1,024-16,384), the solver runs from ~143 to ~21 fps. At N = 8,192, it reaches 17.9 ms/frame, with 2.2x speedup over GPU Jacobi, ~28x over GPU IC/DILU (AMGX multicolor_dilu), and 2.7x over neural SPAI retrained per scale on the same benchmark.
10. 分析适用的 $G^1$ 等几何多片离散化的鲁棒逼近误差估计
Robust approximation error estimates for analysis-suitable $G^1$ isogeometric multi-patch discretizations
作者: Fatima Hasanova, Stefan Takacs, Thomas Takacs
类别: math.NA
发布日期: 2026-05-13
arXiv ID: 2605.13270v1
链接: http://arxiv.org/abs/2605.13270v1
摘要: We prove $p$-robust approximation error estimates for $H^2$-conforming isogeometric discretizations over planar multi-patch domains. Possible applications are fourth order boundary value problems, like the biharmonic equation or Kirchhoff-Love plates. Using Isogeometric Analysis, such conforming discretizations can be constructed effortlessly for the single-patch case. In order to obtain a globally $H^2$-conforming discretization in the multi-patch case, the functions must be $C^1$-smooth across the interfaces between the patches. To obtain optimal approximation properties, those $C^1$-smooth spaces must also reproduce splines of sufficiently high degree for traces and transversal derivatives at all patch interfaces. Such constructions are based on some assumptions on the geometry. We restrict ourselves to the class of analysis-suitable $G^1$ (AS-$G^1$) multi-patch domains, which is the subset of $C^0$-matching multi-patch domains that allows the definition of spline spaces that yield the necessary reproduction properties without the need to locally increase the degree. While approximation error estimates have been established for single-patch and $C^0$ isogeometric multi-patch spaces, corresponding results for the $C^1$ multi-patch setting have been missing. The resulting bounds on the approximation error depend on the geometry parameterization and on the Sobolev regularity of the target function, but are independent of the spline degree $p$.
11. 关于不适定问题的自适应随机重球法的后验停止准则
On a posteriori stopping rules of adaptive stochastic heavy ball method for ill-posed problems
作者: Ruixue Gu, Qinian Jin
类别: math.NA
发布日期: 2026-05-13
arXiv ID: 2605.13144v1
链接: http://arxiv.org/abs/2605.13144v1
摘要: In this paper we develop a stochastic heavy ball method for solving ill-posed inverse problems. The method updates the iterate using only a randomly selected equation at each iteration step while incorporating a momentum term into the process. To facilitate fast convergence, we propose an adaptive strategy for selecting the step size and the momentum coefficient. Inspired by the spirit of the discrepancy principle, we introduce an {\it a posteriori} stopping rule for our adaptive stochastic heavy ball method. This rule avoids the need to compute residuals of all equations in the system at every iteration or at fixed frequency intervals, thereby enhancing computational efficiency and practicality. Additionally, convex penalty functions are employed to capture the specific features of the desired solutions. Under suitable conditions, we establish almost sure convergence as well as convergence in expectation. Extensive numerical experiments are conducted to evaluate the performance of the proposed method, demonstrating its efficiency and promising potential for solving large-scale ill-posed problems.
12. 球面上的多项式插值—回归
Polynomial interpolation–regression on the sphere
作者: Francesco Dell’Accio, Federico Nudo, Teresa E. Pérez, Miguel A. Piñar
类别: math.NA
发布日期: 2026-05-13
arXiv ID: 2605.13124v1
链接: http://arxiv.org/abs/2605.13124v1
摘要: We introduce an interpolation–regression operator for polynomial approximation on the unit sphere $\mathbb{S}^2$ from discrete samples. The approximant is a spherical polynomial of degree $r$ which interpolates the data on a prescribed subset of nodes and uses the remaining sampling nodes to minimize the residual in a least squares sense. Under natural rank assumptions on the associated Vandermonde matrices, the approximant is unique and is characterized by an orthogonality condition with respect to the discrete inner product on the sampling set. We then focus on the case in which the sampling and interpolation nodes are antipodally symmetric. In this setting, when the polynomial is expressed in real spherical harmonics, the constrained problem can be decomposed into independent even and odd components. In the same framework, we prove equivariance under the antipodal map and, more generally, under orthogonal transformations preserving the node sets. We also consider spherical designs. In this case, the normal matrix is a scalar matrix. Consequently, the spectral condition number of the associated KKT matrix can be written explicitly. Numerical experiments in both antipodal and non-antipodal settings illustrate the effectiveness of the proposed method.
13. 面向一维守恒律的超网络条件化WENO5守恒型CNN
Hypernetwork-Conditioned WENO5 Conservative-Form CNNs for One-Dimensional Conservation Laws
作者: Yongsheng Chen, Wei Guo, Xinghui Zhong
类别: math.NA
发布日期: 2026-05-13
arXiv ID: 2605.13106v1
链接: http://arxiv.org/abs/2605.13106v1
摘要: We study a conservative data-driven discretization for one-dimensional hyperbolic conservation laws based on the classical fifth-order WENO finite-volume scheme and a hypernetwork architecture. In the proposed Hyper–WENO5 Conservative-Form Convolutional Neural Network (Hyper–CFCNN), a lightweight target network predicts the nonlinear WENO weights on each stencil, while a hypernetwork generates the target-network parameters from problem metadata, including the mesh spacing, mesh layout, and coarse descriptors of the initial condition. The construction preserves the standard polynomial reconstruction and conservative flux-difference update of WENO, which enables adaptation across problem instances and spatial resolutions without retraining. We also consider an unknown-flux variant, Hyper–CFCNN–F, in which a compact FluxNet is used in place of the analytical flux inside the numerical flux function while retaining a conservative update form. To improve long-time prediction quality, training uses a multi-step recurrent loss that penalizes error accumulation over successive time advances. Numerical experiments on one-dimensional test problems, including single- and multi-shock Burgers equations, the shallow-water system, and the Shu–Osher Euler example, show that Hyper–CFCNN attains accuracy comparable to classical WENO5, achieves near machine-precision conservation in the known-flux setting on fine meshes, and generalizes to unseen spatial resolutions and initial conditions without retraining. The flux-learning variant remains stable on meshes outside the training set and exhibits bounded conservation drift. These results show that hypernetwork-conditioned conservative WENO discretizations provide an effective framework for adaptive high-order learning of nonlinear conservation laws with either known or unknown fluxes.
14. U-HNO: 用于非稳态偏微分方程动力学的带有稀疏点自适应路由的 U 形混合神经算子
U-HNO: A U-shaped Hybrid Neural Operator with Sparse-Point Adaptive Routing for Non-stationary PDE Dynamics
作者: Yingzhe Ma, Xiao Yang, Yuxin Xie, Zihan Xiong, Jinliang Liu
类别: cs.LG, math.NA
发布日期: 2026-05-13
arXiv ID: 2605.12965v1
链接: http://arxiv.org/abs/2605.12965v1
摘要: Solutions to many partial differential equations (PDEs) display coexisting smooth global transport and localized sharp features within a single trajectory: shock fronts, thin interfaces, and concentrated high-frequency content sit on top of slowly varying backgrounds. This poses a challenge for neural operators: Fourier-based architectures mix nonlocal interactions efficiently but tend to under-resolve localized non-smooth features, whereas spatially local architectures recover fine detail at the cost of long-range propagation and rollout stability. Existing hybrid operators paper over this tension with a fixed, spatially uniform fusion that forces the same trade-off everywhere. We propose U-HNO, a U-shaped hybrid neural operator whose central design is Sparse-Point Adaptive Routing (SPAR): at every spatial location, a per-pixel hard mask selects whether the global Fourier branch or the local multi-scale Gaussian branch should dominate, and the sparsity ratio is a function of the local contrast of the routing signal, so smooth and shock-aligned regions receive different mixtures of global and local computation. SPAR is embedded in a hierarchical encoder-bottleneck-decoder backbone with skip connections so that the dual branches and the gate operate at every resolution. Training combines pointwise supervision with a finite-difference H^1 gradient term and a band-wise spectral consistency regularizer. Across benchmarks spanning 1D Burgers, Kuramoto-Sivashinsky, KdV, 2D advection, Allen-Cahn, Navier-Stokes, Darcy flow, and 3D transonic compressible Navier-Stokes from PDEBench, U-HNO achieves state-of-the-art rollout accuracy on the majority of tasks in both relative L^2 and H^1 metrics, with the largest gains on problems dominated by sharp localized features. Ablations show that removing any single component substantially degrades rollout error.
15. 一种用于张量块对角化的 NPDo 方法
An NPDo Approach for Tensor Block-Diagonalization
作者: Ren-Cang Li, Li Wang, Mei Yang
类别: math.NA
发布日期: 2026-05-13
arXiv ID: 2605.12932v1
链接: http://arxiv.org/abs/2605.12932v1
摘要: This paper is concerned with Partial Tensor Block-Diagonalization of a multiway tensor by orthonormal matrices so that the extracted block-diagonal part optimally represents the tensor. The basic idea is to maximize the block-diagonal part via the tensor’s mode-multiplications by orthonormal matrices. For that reason, it will be referred to Principal Tensor Block-Diagonalization (PTBD), which contains the Tucker decomposition (TD) of a tensor as a special case with just one block. Also as a special case is the approximate dominant tensor SVD in which each block-size is 1-by-1. An NPDo approach is proposed to optimize the block-diagonal part for computing \ptbd. It is shown the NPDo approach combined with Gauss-Seidel-type updating is globally convergent to a stationary point while the objective increases monotonically. Numerical experiments are presented to illustrate the efficiency of the NPDo approach.
16. 基于输运声学增量的可压缩欧拉方程组全离散Active Flux方法
Fully Discrete Active Flux Method based on Transported Acoustic Increments for the Compressible Euler Equations
作者: Karthik Duraisamy
类别: math.NA, physics.flu-dyn
发布日期: 2026-05-13
arXiv ID: 2605.12915v1
链接: http://arxiv.org/abs/2605.12915v1
摘要: A fully discrete Active Flux method is proposed for the 2D compressible Euler equations. The method builds on the evolution-operator formulation proposed by Roe in which conservative cell averages are updated by unsplit flux quadrature while primitive point values are evolved by acoustic and advective subsolvers. The proposed method reconstructs the acoustic increment as a cellwise Q2 field and evaluates this field at the convective foot of the target point. For constant frozen coefficients, the resulting point update reduces to the transported composition, eliminating the additive split defect and yielding the exact unsplit frozen evolution when the acoustic and advective generators commute. The resulting method preserves the exact locally linearized acoustic evolution operator of Barsukow (2025), the compact stencil, and the conservative one-stage average update. Numerical experiments probe several facets of the numerical method. A mixed Fourier wave packet isolates the split error and shows third-order point accuracy for the transported update, compared with second-order behavior for the additive update. Isentropic vortex convection confirms third-order convergence for the full nonlinear scheme, reduced error constants, and an enlarged empirical CFL range. Nonlinear Gaussian acoustic pulse evolution demonstrates preservation of radial symmetry and near-third-order decay of the symmetry error. Low-Mach shear layer tests show coherent vorticity evolution, ultra-low entropy dissipation, and absence of the coarse-grid secondary vortices seen in displayed DG/CG comparisons. Finally, a compressible under-resolved Kelvin-Helmholtz test demonstrates robust no-limiter evolution to late time with consistent entropy dissipation. Fourier diagnostics of the vertical-edge point operator support the observed improvements in acoustic phase and amplification behavior.
17. 厄米特特征值问题中一种带有可靠伪Ritz值去除方法的改进CJ–SS–RR方法
A refined CJ–SS–RR method with a reliable removal approach of spurious Ritz values for the Hermitian eigenvalue problem
作者: Zhongxiao Jia, Tianhang Liu
类别: math.NA
发布日期: 2026-05-13
arXiv ID: 2605.12846v1
链接: http://arxiv.org/abs/2605.12846v1
摘要: Under the hypothesis that the deviations of the desired eigenvectors of the matrix $A$ from the underlying subspace tend to zero, the Ritz vectors may not converge and have poor or little accuracy. This phenomenon is not unusual and particularly occurs when the associated Ritz values are close, which is independent of the eigenvalue distribution of $A$. For the (block) SS–RR methods, there are possibly {\em more} Ritz values that converge to the same desired eigenvalue(s) counting multiplicity in the region of interest, meaning that some of the Ritz values must be spurious and the corresponding residual norms of the Ritz pairs may not be small. Consequently, the (block) SS–RR methods including the CJ–SS–RR method cannot base on the corresponding residual norms to effectively identify if the Ritz values in the region are genuine or spurious. This paper proposes refined SS–RR, abbreviated as SS–RRR, methods based on the refined Rayleigh–Ritz projection that compute the eigenpairs of large matrices with the eigenvalues located in the given region. We present a new approach to accurately implement the RRR methods more efficiently than ever before for a general subspace.Exploiting the unconditional convergence of the refined Ritz vectors when the subspace is sufficiently accurate, we propose a tune-free removal approach to effectively remove spurious Ritz values with a rigorous theory supported, and develop a restarted CJ–SS–RRR algorithm. Numerical experiments show that the restarted CJ–SS–RRR algorithm is more efficient and effective than the restarted CJ–SS–RR algorithm.
18. 球面游走与Array-RQMC
Walk on spheres and Array-RQMC
作者: Valerie N. P. Ho, Art B. Owen
类别: math.NA, stat.CO
发布日期: 2026-05-13
arXiv ID: 2605.12844v1
链接: http://arxiv.org/abs/2605.12844v1
摘要: We use Array-RQMC sampling in a walk on spheres (WOS) algorithm for Dirichlet boundary value problems. On a collection of problems, we find that Array-RQMC-WOS reduces the Monte Carlo variance by factors ranging from $57$-fold to $2290$-fold at $n=2^{17}$ trajectories. The variance is known to be $o(1/n)$ but attains empirical rates between $n^{-1.4}$ and $n^{-1.8}$ in our examples. A simpler RQMC-WOS algorithm studied in Ho and Owen (2026) has more theoretical support but only reduced variance by 1.8 to 10.7-fold on the same set of examples. In order to explain this improvement, we introduce a column-wise mean dimension of the RQMC error based on Sobol’ indices. It matches the usual mean dimension for Monte Carlo and the mean dimension of a dual lattice error for randomized lattices. We find for a gasket example from Crane et al.\ (2025) that the mean dimension of Array-RQMC-WOS errors is much higher than an analogous Array-MC-WOS algorithm has.
19. 导数自由优化中二次模型的精度与关系
Accuracy and Relationships of Quadratic Models in Derivative-free Optimization
作者: Yiwen Chen, Warren Hare, Lindon Roberts
类别: math.OC, math.NA
发布日期: 2026-05-12
arXiv ID: 2605.12819v1
链接: http://arxiv.org/abs/2605.12819v1
摘要: We study three quadratic models in model-based derivative-free optimization: the minimum norm (MN), minimum Frobenius norm (MFN), and quadratic generalized simplex derivative (QS) models. Despite their widespread use, their approximation accuracy and relationships have not been systematically explored. We establish fully linear error bounds for all three models, removing the uniformly bounded model Hessian assumption required in existing MN analyses and deriving the first such results for the QS model. We further analyze Hessian approximation accuracy via directional error bounds, showing that all three models achieve fully quadratic accuracy along sample directions under a mild condition on the sample set. This reveals a form of directional fully quadratic accuracy not captured by existing theory. Finally, we characterize the relationships among these models, identifying conditions under which they coincide and clarifying their structural connections.
20. 厄米特矩阵函数双线性形式的快速稳定梯度逼近
Fast and Stable Gradient Approximation for Bilinear Forms of Hermitian Matrix Functions
作者: Navjot Singh, Kipton Barros, Xiaoye Sherry Li
类别: math.NA
发布日期: 2026-05-12
arXiv ID: 2605.12801v1
链接: http://arxiv.org/abs/2605.12801v1
摘要: Objectives involving bilinear forms $u^\top f(A(θ))v$ for Hermitian $A$ arise widely in scientific computing and probabilistic machine learning. For large matrices, Lanczos efficiently approximates these quantities, but differentiating them with respect to $θ$ is challenging. Existing approaches either backpropagate through the Lanczos recurrence, requiring reorthogonalization for stability, or apply Arnoldi to an augmented block matrix of twice the original size. Both introduce extra computation and orthogonalization costs that can limit performance on modern hardware. We propose a forward-only gradient approximation that reuses the Lanczos pass and adds very minimal overhead in most cases. We prove that its error is proportional to the Lanczos residual norm, the same quantity controlling the forward approximation. Whereas a traditional adjoint-based calculation would be unstable without reorthogonalization, the new method appears unconditionally stable in our tests. It is also faster than existing state-of-the-art approaches.
21. 基于非对称格林核的逼近及其在分数阶微分方程中的应用
Approximations with Non-Symmetric Green’s Kernels and their Application to Fractional Differential Equations
作者: Nick Fisher
类别: math.NA
发布日期: 2026-05-12
arXiv ID: 2605.12707v1
链接: http://arxiv.org/abs/2605.12707v1
摘要: Several kernel-based methods for the numerical solution of fractional differential equations have been developed in the recent past; however, these techniques exclusively relied on the use of radial basis function approximations. In the present work, we consider the non-symmetric Green’s kernel perspective on fractional order spline interpolation and its application to a kernel Galerkin method for the numerical solution of certain fractional order differential equation. Unfortunately, the reliance on a non-symmetric kernel requires that our theoretical analysis of the kernel interpolants must take place outside the familiar setting of reproducing kernel Hilbert spaces. Nevertheless, we are able to prove that the proposed kernel interpolants obtain optimal order convergence rates in a reproducing kernel Banach space.
22. UFO: 一种面向广义算子学习的无域统一算子框架
UFO: A Domain-Unification-Free Operator Framework for Generalized Operator Learning
作者: Hanli Qiao, George Em Karniadakis, Muhammad Muniruzzaman
类别: cs.LG, math.NA
发布日期: 2026-05-12
arXiv ID: 2605.12700v1
链接: http://arxiv.org/abs/2605.12700v1
摘要: Neural operators have become an effective framework for learning mappings between function spaces, yet most existing architectures realize operators within a single representational domain, such as physical, spectral, or latent space. In this work, we introduce UFO (Domain-Unification-Free Operator), a cross-domain neural operator framework that realizes operators through adaptive, jointly conditioned interactions among representations defined on distinct domains. UFO enables discretization decoupling: the input function can be observed at resolutions or locations different from those used during training, while the solution can be queried at arbitrary output resolutions. Across four complementary benchmarks covering discontinuous inputs, irregular sampling with spectral mismatch, nonlinear dynamics, and stochastic high-frequency fields, UFO delivers accurate, robust, and physically coherent predictions under distribution shifts. These results establish cross-domain, phase-modulated realization as a powerful framework for discretization-decoupled neural operator learning.