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重点关注论文（完整分析）

1. 多准轴对称结构声散射的谱方法求解器

A Spectral Solver for Acoustic Scattering by Multiple Quasi-Axisymmetric Structures

作者: Jun Lai, Yuxin Li

类别: math.NA

发布日期: 2026-06-30

arXiv ID: 2606.31380v1

摘要翻译: 声散射在医学成像、地球物理勘探、声学超材料等众多应用中普遍存在。本文针对旋转轴为任意曲线的多准轴对称物体的声散射问题，发展了一种快速高精度算法。该方法基于结合 Gauss-Legendre 求积与梯形法则的 Nyström 离散。为了处理目标点接近或重合于源点时出现的奇异积分，我们将其重新表述为模态 Green 函数及其导数的计算，并利用快速傅里叶变换和卷积高效求值。然后通过体间边界积分相互作用耦合各单散射体离散，构造出多散射求解器。我们还给出了光滑几何散射问题的收敛性分析。数值算例展示了所提方法在求解多达 1000 个准轴对称结构的多散射问题时的效率与精度。

Comment: 无

链接: http://arxiv.org/abs/2606.31380v1

1. 研究对象和背景

声散射问题广泛存在于医学成像、地球物理勘探、无损检测和声学超材料设计等诸多应用中，其中波与多个散射体同时发生作用，准确预测由此产生的集体散射效应对于仿真和优化至关重要。然而，在三维情形下，多散射问题的数值计算面临收敛缓慢和低阶精度的困难。本文旨在为多个准轴对称物体的声散射发展一套高阶边界积分方程方法。

考虑一个嵌入均匀背景介质中的有界障碍物 $\Omega\subset\mathbb{R}^3$，其边界 $\partial\Omega$ 光滑，并受到时谐入射波 $u^{\mathrm{inc}}$ 照射。总场 $u = u^{\mathrm{inc}} + u^{\mathrm{sc}}$ 满足外 Helmholtz 方程与 Dirichlet 边界条件（声软边界）：

$$ \begin{cases} \Delta u + k^2 u = 0, & \text{in } \mathbb{R}^3\setminus\bar{\Omega},\\ u = 0, & \text{on } \partial\Omega, \end{cases} $$

其中 $k>0$ 为波数。散射场 $u^{\mathrm{sc}}$ 还需满足 Sommerfeld 辐射条件

$$ \lim_{r\to\infty} r\left(\frac{\partial u^{\mathrm{sc}}}{\partial r} - i k u^{\mathrm{sc}}\right) = 0,\quad r=|\mathbf{x}|. $$

边界积分方程(BIE)方法可将问题降低一维并自动满足辐射条件，避免了体积方法对无界域的截断，且较少的未知量即可达到高精度。对于光滑表面，常见的离散化策略包括分片配点法、Galerkin 方法和基于修正求积或高阶乘积积分的 Nyström 方法。其中 Nyström 方法兼有实现简单与高阶精度的优点，但需妥善处理奇异性与近奇异性。

障碍物几何形式为所谓“准轴对称结构”：其边界可由参数化

$$ \partial\Omega := \bigl( r(s)\cos\theta + p(s),\, r(s)\sin\theta + q(s),\, z(s) \bigr),\quad s\in[0,T],\;\theta\in[0,2\pi] $$

描述，即半径为 $r(s)$ 的圆形截面沿一条弯曲中心线 $\boldsymbol{\gamma}(s) = (p(s), q(s), z(s))$ 扫掠而成。此类几何在神经螺旋、盘绕波导、光纤等领域出现，其特点是在方位角 $\theta$ 方向上保持周期性，但失去了传统轴对称物体（绕直轴旋转所得）的全局旋转对称性。对于真正的轴对称物体，边界积分方程可利用旋转不变性将表面积分分解为一系列一维方程，通过方位 Fourier 模态解耦，并由模态 Green 函数描述，从而极大地降低了计算量；相关的高阶离散化与 FFT 加速技术已得到充分开发。然而，在准轴对称情形，尽管局部区域仍可延续轴对称处理的思想，但不同弧长位置 $s$ 处的模态相互耦合，且失去全局旋转不变性，不能直接沿用已有的降维策略。因此，为这类几何发展高效的高阶边界积分框架在声散射领域仍属空白，尤其是在多体散射背景下。

本文的目标正是填补这一空白：首先为单个准轴对称散射体构建一个利用方位角周期性、结合三角插值与模态 Green 函数的高阶 Nyström 离散化方法，然后将它系统性地扩展至多散射问题，并借助 FFT 和块结构复用技术实现快速求解。

2. 主要定理或主要结果

本文的主要成果包括以下方面。

  1. 针对单个准轴对称障碍物的声散射 Dirichlet 边值问题，建立了基于双层位势（亦可处理单层位势）的边界积分方程，并给出了一个谱精度的 Nyström 离散化方案。该方案在方位角方向采用 $2N$ 点等距梯形法则，在母线方向采用分片 Gauss-Legendre 求积，对自作用面板和相邻面板的奇异或近奇异积分，通过三角插值与模态 Green 函数将其转化为反离散 Fourier 变换，并利用广义 Gauss 求积处理外层弱奇异性。最终形成一个规模为 $N_{\mathrm{tot}}=N_p\times N_{\mathrm{order}}\times(2N)$ 的线性系统。

  2. 在多散射体情形（所有物体全等且形状相同），将总体散射场表示为各物体双层位势之和，导出块结构线性系统

$$ \frac12 \mathbf{I}\boldsymbol{\mu} + \mathbf{D}\boldsymbol{\mu} = -\mathbf{u}^{\mathrm{inc}}, $$

其中 $\mathbf{D}$ 为 $N_{\mathrm{tot}}\times N_{\mathrm{tot}}$ 的系数矩阵，由 $N_{\mathrm{geo}}\times N_{\mathrm{geo}}$ 个子块 $\mathbf{D}_{t',t}$ 组成，对角线块为自作用矩阵，非对角线块表示不同物体间的相互影响。当物体相同且位于规则网格时，这些互作用块仅依赖于两物体中心的位移向量，因而具备多级块 Toeplitz 结构。此时，需要计算的不同非对角块个数从 $\mathcal{O}(N_{\mathrm{geo}}^2)$ 降至 $\mathcal{O}(N_{\mathrm{geo}})$；自作用块只需求解一次并复用。对非规则排列（如随机旋转和平移），自作用矩阵仍可复用，但非对角块必须逐一计算，可通过并行计算抵消其开销损失。

  1. 对于光滑几何，作者在第 5 节给出了该 Nyström 方法的收敛性分析，但本文提供的 PDF 摘录未包含该部分的具体定理表述与误差估计。数值实验部分（第 6 节）展示了涉及最多 1000 个准轴对称散射体的计算结果，表明该方法在保持高阶精度的同时具有较高的计算效率。读者可从后续完整论文中获取收敛性的严格刻画。

  2. 数值结果另外显示，通过对物体施加随机旋转和位置扰动，算法仍能有效求解，证明了其应对一般构型的鲁棒性。

3. 研究方法、关键技术和核心工具

本工作的核心技术贡献体现在单散射体离散化的数值实现、模态 Green 函数的推广计算以及多体相互作用的有效组装策略三个层面。

单散射体 Nyström 离散化与奇异积分处理

对单个准轴对称表面 $\partial\Omega$ 上的积分方程（以双层位势为例）

$$ \frac12 \mu(\mathbf{x}) + \int_{\partial\Omega} \frac{\partial G_k(\mathbf{x},\mathbf{y})}{\partial n(\mathbf{y})} \mu(\mathbf{y}) \,\mathrm{d}s(\mathbf{y}) = -u^{\mathrm{inc}}(\mathbf{x}),\quad \mathbf{x}\in\partial\Omega, $$

在母线方向分 $N_P$ 个面板、每面板布置 $N_{\mathrm{order}}$ 个 Gauss-Legendre 节点，方位角方向等距取 $2N$ 个点。当目标点 $\mathbf{x}$ 位于面板 $\gamma_p$ 且源点 $\mathbf{y}$ 位于同一面板或其相邻面板 $\gamma_{p-1},\gamma_{p+1}$ 时，积分核表现出弱奇异性（双层核含有 $1/\rho$ 型奇异性）。处理这一困难的关键步骤是：

  • 将密度与 Jacobi 因子的乘积在 Gauss-Legendre 节点和等距 $\theta$ 节点上分别进行多项式插值和三角插值，使得奇异积分归结为形如

    $$ \sum_{l,j} \mu(\mathbf{y}_{pjl})|J(\mathbf{y}_{pjl})| \int_{\gamma_p} P_l(s) \int_0^{2\pi} \frac{\partial G_k}{\partial n} L_j(\theta)\,\mathrm{d}\theta\,\mathrm{d}s $$

    的内外层积分。

  • 利用三角基函数 $L_j(\theta)$ 与 Fourier 模态的关系，内层积分可以通过模态 Green 函数的 Fourier 反变换得到：

    $$ \int_0^{2\pi} \frac{\partial G_k}{\partial n} L_j(\theta)\,\mathrm{d}\theta = \mathcal{F}^{-1}\bigl[\widehat{\partial_n G}_m^{(k)}\bigr]_j, $$

    其中模态 Green 函数及其导数定义为

    $$ \widehat{\partial_n G}_m^{(k)} = \int_0^{2\pi} \frac{\partial G_k(\mathbf{x},\mathbf{y})}{\partial n(\mathbf{y})} e^{-im\theta}\,\mathrm{d}\theta,\quad m = -N,\dots,N-1. $$

模态 Green 函数的快速计算与 FFT 加速

对上述模态量的高效计算是算法成败的关键。注意到在准轴对称参数化下，源-目标距离可写为

$$ \rho^2 = \tilde{r}_t^2 + r^2 + c^2 - 2r\tilde{r}_t\cos\phi,\quad \phi = \theta - \alpha, $$

其中 $\tilde{r}_t = \sqrt{(x_t-p(s))^2+(y_t-q(s))^2}$ 和 $\alpha = \arg((x_t-p)+i(y_t-q))$ 均为弧长参数 $t$ 和 $s$ 的函数。尽管 $\alpha$ 随源点位置变化，但角度依赖始终以 $\cos(\theta-\alpha)$ 形式出现。与传统轴对称情形相比，唯一的区别是相位 $\alpha$ 并非恒为零，这并不妨碍 Fourier 模态的局部定义。因此，可以沿用轴对称情形下基于 Legendre 函数和卷积的模态 Green 函数计算技术：

  • 静力情形 $k=0$：此时 Green 函数及其导数可表为半奇数阶 Legendre 函数 $Q_{m-1/2}(\chi)$ 及其导数，其中 $\chi = \frac{\tilde{r}_t^2+r^2+c^2}{2r\tilde{r}_t}$。利用递推公式 (27)–(28) 可从低阶 $Q_{-1/2}$ 和 $Q_{1/2}$（由完全椭圆积分表示）高效生成高阶值。

  • 动力情形 $k\neq 0$：将 $e^{ik\rho}$ 分解为实部与虚部，并注意到 $\sin(k\rho)/(4\pi\rho)$ 在 $\rho=0$ 处可去奇异，因而可直接在等距网格上用 FFT 计算其 Fourier 系数；而 $\cos(k\rho)/(4\pi\rho)$ 的模态则通过卷积

    $$ \int_0^{2\pi} \frac{\cos(k\rho)}{4\pi\rho} e^{-im\theta}\,\mathrm{d}\theta = \frac1{2\pi} \sum_{n} \widehat{\cos(k\rho)}_n \,\widehat{(1/4\pi\rho)}_{m-n} $$

    得到，由于 $\widehat{\cos(k\rho)}_n$ 快速衰减，该卷积可截断至少量项。类似方法适用于法向导数的模态量。

所有模态量计算均利用 FFT 进行加速，使得每个目标点附近的模态生成成本保持在对数线性量级。

外层弱奇异积分与广义 Gauss 求积

获得内层积分 $G_j(t,s)$ 后，对 $s$ 积分时仍面临弱奇异性。本文采用预计算的广义 Gauss 求积法则（由 Bremer 的非线性优化程序生成）：自面板上使用 16 个目标依赖节点，相邻面板使用 48 个目标无关节点，从而以极低节点数达到高阶精度，并保证线性系统的稀疏性。

多体散射的组装策略与块结构复用

对于 $N_{\mathrm{geo}}$ 个全同物体的多体系统，将其离散化为块线性系统 (36)。对角线上的自作用块 $\mathbf{D}_{t,t}$ 完全相同，只需计算并存储一个代表。在规则排列情形，利用位移不变性 $\mathbf{D}_{t',t} = \mathbf{D}_{\mathbf{d}}$ ( $\mathbf{d} = \mathbf{c}_{t'} - \mathbf{c}_t$ )，可将需要显式计算的互作用块数量从 $N_{\mathrm{geo}}(N_{\mathrm{geo}}-1)$ 降至 $(2P-1)(2Q-1)(2R-1)-1$，并通过内存拷贝完成其余块的填充，组装成本由 $\mathcal{O}(N_{\mathrm{geo}}^2)$ 降至 $\mathcal{O}(N_{\mathrm{geo}})$。当物体受到随机旋转和扰动时，自作用块仍然相同，非对角块则各自独立计算，但可并行处理以控制时间开销。

4. 与之前工作的比较

本文贡献与既有工作的比较可从以下几个方面展开。

与轴对称声散射方法的比较

轴对称物体的声散射 BI 方法已相当成熟，其核心是利用旋转不变性将方程 Fourier 模态完全解耦，在生成曲线上一维求解。经典工作如 Young 等 (2004)，Helsing 和 Karlsson (2014)，Lai (2020) 等均发展了基于模态 Green 函数与 FFT 加速的高阶 Nyström 格式。本文所处理的准轴对称几何可视为轴对称自然的推广：旋转轴由直线变为任意光滑曲线。在这种推广下，方位角方向仍保持周期性，但 $\theta$ 方向的 Fourier 系数之间不再独立，因此不能直接对每个模态独立求解。然而，局部来看积分核的角度依赖仍然可用 $\cos(\theta-\alpha)$ 刻画，从而允许在离散化阶段沿用模态 Green 函数的框架。因此，本文成功地将轴对称模态技术这一有力工具移植到了更一般的准轴对称情形。与之前用于 Stokes 问题的类似思想（[21]，采用预计算求积处理轴对称型模态核）相比，本文首次在声学 Helmholtz 方程中建立了完整的准轴对称 Nyström 求解器，明确了从参数化、奇异积分正则化到 FFT 加速的全套处理流程，并给出了相应的收敛性分析。

与通用三维多散射求解器的比较

现有三维多体声散射算法（如 [8, 16, 17, 19] 等）通常不以特定几何结构为前提，因而在每一个物体表面都需要相当多的离散点来捕获振荡核和表面形状。本文的方法专门针对准轴对称物体，充分利用方位角方向的平移不变性（更准确地说，是回转方向上的卷积结构），使用均匀的角向网格和 FFT，使单物体的未知数仅约为传统边界元方法的 $1/N$ （假设 $2N$ 方位角节点）。同时，互作用计算由于物体形状相同、排列规则时出现的 Toeplitz 结构，进一步大幅度压缩了组装开销。原文数值实验显示，即使物体数达 1000 个，算法依然可行，而一般三维全矩阵的边界元法在该尺寸下几乎不可能完成。但需指出，文中并未提供与其它多散射求解器的直接计算时间或精度对比数据，因此比较仅停留在理论效率论证层面。

精度与适用范围

与已有的准轴对称相关工作相比（如上述 Stokes 问题的高阶 Nyström 方法），本文明确将框架建立在 Helmholtz 方程上，并处理了实波数 $k$ 下的振荡核，同时结合了广义 Gauss 求积与模态 Green 函数的递推和 FFT 加速，保证了整个管道的高阶精度。此外，作者提供了光滑几何的收敛性分析（第 5 节），这在同类工作中尚不多见，但具体收敛速率及误差界未能从当前摘录内容中获知。在几何假设方面，本文假定边界光滑，但指出方法可推广至非光滑情形；入射波为平面波，但显式仅处理 Dirichlet 边界条件，尽管推广到其他边界条件（如 Neumann 或阻抗条件）可通过结合单、双层位势实现，文末也提到可通过组合位势克服谐振频率问题。物体的全同假设在一定程度上限制了非均匀多散射场景的适用性，但实验表明，即使对物体施加随机旋转与平移，自作用矩阵仍可复用，互作用矩阵需要非规则计算，算法仍具实效。

总体而言，本文推进了准轴对称声散射的边界积分方法，将轴对称情形的高效技术系统性地推广到弯曲旋转轴几何，并首次构建了能处理数百至上千个物体的多散射求解器，填补了该领域的空白。

了解领域论文（摘要翻译）

1. 特征向量降阶加速 WaveHoltz 方法的数值研究

Numerical Study of Eigenvector Deflation to Accelerate the WaveHoltz Method

作者: Daniel Appelo, William D. Henshaw, Zhichao Peng

类别: math.NA

发布日期: 2026-06-30

arXiv ID: 2606.31842v1

链接: http://arxiv.org/abs/2606.31842v1

摘要翻译

中文标题: 特征向量降阶加速 WaveHoltz 方法的数值研究

摘要翻译: 我们对特征向量降阶作为加速求解 Helmholtz 方程的 WaveHoltz 方法的手段进行了数值研究。在能量守恒（Dirichlet 或 Neumann）边界条件下，WaveHoltz 不动点迭代在高频时收敛缓慢，在 $d$ 维空间中大约需要 $\mathcal{O}(ω^{2d})$ 次迭代。我们证明，将特征值最接近驱动频率的特征向量降阶可显著减少迭代次数，并考察了两种引入特征向量的方式：直接特征向量降阶 (DEVD)，其将右端项和迭代向量投影到降阶集上；以及增广 Krylov 特征向量降阶 (AUKED)，采用降阶共轭梯度法 (DCG)、增广 GMRES (AGMRES) 和增广（循环）BICGSTAB (ABICGSTAB)。所需的特征对可用 EigenWave 方法高效计算，并且在二维情况下，我们展示了当降阶向量数目随 $ω$ 二次增长时，渐近收敛速率基本保持不变。由于结构网格上的特征向量自然表示为矩阵，我们进一步应用基于 SVD 的压缩来减少其存储。在采用分部求和算子离散的单一曲线网格以及重叠网格上的数值实验表明了该方法的鲁棒性和高效性，在考虑特征向量预计算代价的情况下，仅需处理少至两个右端项，降阶求解器的总开销即可与未降阶求解器持平。

不相关论文（基本信息）

1. 基于GQL的任意维狭义相对论流体力学保物理约束高阶有限差分格式

GQL-Based Physical-Constraint-Preserving High-Order Finite Difference Schemes for Special Relativistic Hydrodynamics in Arbitrary Dimensions

作者: Linfeng Xu, Shengrong Ding, Kailiang Wu

类别: math.NA, astro-ph.IM, physics.comp-ph, physics.flu-dyn

发布日期: 2026-06-30

arXiv ID: 2606.31992v1

链接: http://arxiv.org/abs/2606.31992v1

摘要: High-order accurate simulations of special relativistic hydrodynamics (RHD) are prone to numerical breakdown if intrinsic physical constraints (positive rest-mass density/pressure and subluminal velocity) are violated near strong discontinuities. In this work, we develop a robust and efficient physical-constraint-preserving (PCP) flux-limiting framework for high-order schemes, using finite-difference WENO as a representative example. By leveraging the geometric quasilinearization (GQL) representation, which equivalently reformulates the nonlinear RHD constraints into a family of linear inequalities, we integrate a Zalesak-type Flux-Corrected Transport (FCT) update into a scalar-style limiter that acts directly on conservative variables. A critical innovation is the explicit, non-iterative determination of limiting parameters via a rational stereographic parameterization of the GQL normal vector. This technique transforms the required worst-case minimization over auxiliary variables into a generalized Rayleigh-quotient formulation, allowing the optimal parameters to be obtained by solving small symmetric eigenvalue problems ($2\times2$ in 1D; $(d+1)\times(d+1)$ in $d$ dimensions). Relaxed variants are further introduced to reduce computational costs in multidimensions while retaining the PCP guarantee. Extensive numerical benchmarks ranging from 1D to 3D, including ultra-relativistic Riemann problems and astrophysical jets, demonstrate that the proposed method robustly enforces physical admissibility, sharply resolves discontinuities, and maintains design-order accuracy for smooth solutions.

2. 重叠网格上中心间断Galerkin方法的隐藏精度与超收敛分析

Hidden Accuracy and Superconvergence Analysis of Central Discontinuous Galerkin Methods on Overlapping Meshes

作者: Manting Peng, Kailiang Wu

类别: math.NA

发布日期: 2026-06-30

arXiv ID: 2606.31931v1

链接: http://arxiv.org/abs/2606.31931v1

摘要: This paper establishes the first rigorous superconvergence theory for semidiscrete and fully discrete central discontinuous Galerkin (CDG) methods for linear hyperbolic equations on overlapping meshes. While the optimal $L^2$ convergence of $\mathbb{Q}^k$ CDG schemes was established on uniform Cartesian meshes by Liu, Shu, and Zhang [ SIAM J. Numer. Anal.}, 56 (2018), pp. 520–541], their observed $\mathcal{O}(h^{k+2})$ pointwise superconvergence has remained unproven, due to the loss of standard single-mesh Galerkin orthogonality inherent in the CDG overlapping structure. To overcome this fundamental barrier, we introduce a projection-correction framework that identifies a hidden superconvergent mechanism: an asymptotic weak residual cancellation in one dimension, and a high-order cancellation-by-aggregation (HOCA) mechanism in multiple dimensions. This HOCA approach overcomes the analytical challenge posed by coupled primal-dual directional residuals, recovering critical error cancellation properties absent from the standard variational formulation. Consequently, we provide the rigorous proof of the conjectured $\mathcal{O}(h^{k+2})$ pointwise superconvergence in the discrete $\ell^{\infty}$ norm across all superconvergent points. Furthermore, we reveal that under a systematically corrected initialization, this framework yields a previously undiscovered, stronger cell-average superconvergence estimate of order $\mathcal{O}(h^{\min\{2k+1,k+3\}})$. The theory is extended to fully discrete explicit Runge–Kutta CDG schemes, where stagewise corrected errors are constructed to preserve spatial superconvergence up to temporal truncation errors, yielding a stable reconstruction-based postprocessing estimate. Numerical experiments in one and two spatial dimensions confirm the sharpness of the theoretical rates.

3. 面向鲁棒内聚力模型模拟的界面感知神经网络牛顿预处理

Interface-Aware Neural Newton Preconditioning for Robust Cohesive Zone Model Simulations

作者: Zhangyong Liang, Huanhuan Gao

类别: cs.LG, math.NA

发布日期: 2026-06-30

arXiv ID: 2606.31921v1

链接: http://arxiv.org/abs/2606.31921v1

摘要: Cohesive Zone Models (CZMs) are widely used to simulate interface fracture, delamination, adhesive failure, and fiber–matrix debonding in aerospace composite structures. In implicit quasi-static finite element analyses, cohesive softening may introduce negative interface tangents, solution jumps, and Newton-basin mismatch, so the previous converged state can become a poor initial guess for the next increment. This may lead to stagnation, wrong-branch convergence, or repeated step cuts. Existing remedies, including viscous regularization, path following, dynamic relaxation, and manual Newton–Raphson (NR) modification, either alter the effective response, increase cost, or rely on hand-crafted interface rules. This work proposes an Interface-Aware Neural Newton Preconditioner (IA-NNP) for difficult CZM increments. IA-NNP recasts manual NR modification as rule-based interface lifting and generalizes it into a learned, state-dependent interface correction. The method acts only on active interface variables and preserves the original traction–separation law, residual assembly, tangent evaluation, history update, and dissipation checks. Two realizations are developed: IA-NNP-Init for learned initial-guess lifting and IA-NNP-NL for iteration-level nonlinear right preconditioning. Interface graph features encode opening, traction, tangent, damage/history variables, mode mixity, residuals, and neighboring states. The correction is bounded, confidence-gated, and accepted only through the original CZM Newton solve. A root-equivalence property shows that IA-NNP changes the path to convergence but not the discrete CZM solution set. Tests on horizontal, circular, two-interface, and active-front benchmarks show improved difficult-increment convergence, better branch recovery, and fewer failures than standard NR and manual NR modification, while preserving the force–displacement response.

4. 拉普拉斯算子谱的等几何离散：无异常值样条基

Isogeometric discretizations for the spectrum of the Laplace operator: outlier-free spline bases

作者: Damiano Ricci, Carla Manni, Hendrik Speleers

类别: math.NA

发布日期: 2026-06-30

arXiv ID: 2606.31843v1

链接: http://arxiv.org/abs/2606.31843v1

摘要: Optimal spline subspaces are an elegant and efficient tool to remove spurious outliers in isogeometric Galerkin discretizations for the approximation of the spectrum of the Laplace operator. For practical purposes, it is valuable to have a basis construction for such spaces with good computational and spectral properties. We provide a characterization of the bases that enjoy a B-spline-like support structure and whose mass and stiffness matrices are simultaneously diagonalizable. It turns out that these mass and stiffness matrices admit explicitly known closed-form expressions for their eigenvalues, implying that the considered bases are outlier-free. A numerical procedure for constructing such bases is also presented.

5. 互补逼近多项式族及其在任意偶数阶空间微分方程有限元方法中的应用

Complementary families of approximating polynomials with applications to finite element methods applied to differential equations of arbitrary even spatial order

作者: Peter K Moore

类别: math.NA

发布日期: 2026-06-30

arXiv ID: 2606.31829v1

链接: http://arxiv.org/abs/2606.31829v1

摘要: Complementary families of polynomials are introduced to generate $C^m$ finite element basis functions of order $p \geq 2m+2$ for arbitrary $m \ge 0$. One family consists of the Hermite splines that serve as the nodal basis functions by ensuring $C^m$ continuity across element boundaries. Explicit formulas for these splines for any $m \ge 0$ are presented on the canonical interval $[0,1]$. The second family is derived on the interval $[-1,1]$ from derivatives of order $m+1$ of the Legendre polynomials of degree $p-m-1$ multiplied by binomial powers of degree $m+1$ at -1 and 1, respectively. These polynomials, related to the ultraspherical polynomials, serve as the interior or bubble basis functions. A relationship between the two families of polynomials is demonstrated. For a particular $m$ and $p$, an interpolant is constructed using these basis pairs together with the roots of the related ultraspherical polynomial and the interval endpoints. A formula for the interpolation error that extends the results for $m=0$ and $m=1$ is given. To prove the formula extensions of the Lagrange interpolants are introduced. A superconvergence result along with the related asymptotic equivalence of the interpolant and finite element solution is proved in the linear case in $H^{m+1}$. Computational results demonstrate the theory for a model problem.

6. 利用强形式噪声自适应微分的偏微分方程识别（S-IDENT）

PDE Identification Using Noise Adaptive Differentiation in Strong Form (S-IDENT)

作者: Roy Y. He, Sung Ha Kang

类别: math.NA

发布日期: 2026-06-30

arXiv ID: 2606.31776v1

链接: http://arxiv.org/abs/2606.31776v1

摘要: We explore identifying partial differential equations (PDEs) from noisy observations of single time-space trajectories. Recent developments show the benefits of identifying PDEs in their weak forms. We investigate the use of differential Strong-form dictionaries for PDE IDENTification (S-IDENT), which enables finding more general linear and nonlinear PDEs. Building on an extensive exploration of integral-type denoised differentiation approaches, we propose to use Savitzky–Golay (SG) differentiation with an adaptive window length chosen based on Stein’s Unbiased Risk Estimate (SURE). This offers a guaranteed order of accuracy while producing estimators with minimal variance. The identification process is further refined and stabilized through trimming and reduction-in-residual model selection. Numerical evidence shows that S-IDENT can successfully identify nonlinear PDEs at higher levels of noise than existing strong-form methods, while also yielding results comparable to weak-form approaches. We further verify the effectiveness of S-IDENT through comparisons with various strategies to approximate differential features. We provide numerical evidence that general differential-form dictionaries are larger and more ill-conditioned than those used for weak-form identification, yet S-IDENT does not significantly suffer from this combinatorial increase in dictionary size.

7. 不同聚类方法对亥姆霍兹方程多层快速多极子方法的影响

Effect of different clustering approaches on the multilevel fast multipole method for the Helmholtz equation

作者: Wolfgang Kreuzer, Christian Kasess

类别: math.NA

发布日期: 2026-06-30

arXiv ID: 2606.31771v1

链接: http://arxiv.org/abs/2606.31771v1

摘要: The fast multipole method (FMM) is an important component for the boundary element method (BEM), because with the FMM the efficiency and feasibility of the BEM can be enhanced to a large degree. Part of the FMM is grouping the elements of the boundary element mesh into different clusters. The size of these clusters in terms of number of elements and spatial expansion has a huge impact on the efficiency and stability of the method. However, while the theory behind the multipole expansion has been broadly researched, the clustering process itself and its effect on the FMM has been neglected in comparison. Most of the time, for example, it is implicitly assumed that the elements of the mesh have about the same size, which is often not the case in practical applications, e.g., when calculating the sound field around the human head. In this study we compare different types of clustering approaches with respect to stability and efficiency of the underlying FMM applied to meshes that have uniform as well as non-uniform element sizes. Also, some examples are provided for cases where a wrong clustering can lead to numerical problems and instabilities of the FMM-BEM.

8. 用于流形-仿射交集问题的几何自适应正则化牛顿型方法

A Geometry-Adaptive Regularized Newton-Type Method for Manifold-Affine Intersection Problems

作者: Dengyu Zheng, Shixiang Chen

类别: math.OC, math.NA

发布日期: 2026-06-30

arXiv ID: 2606.31738v1

链接: http://arxiv.org/abs/2606.31738v1

摘要: We propose a regularized algorithm, Regularized Newton-SLRA (RN-SLRA), for local manifold–affine intersection problems under weak intersection conditions, motivated in particular by structured low-rank approximation (SLRA). Newton-SLRA is an efficient method for manifold–affine intersection problems, but its well-definedness relies on the transversality condition between the manifold and the affine subspace, a condition that may fail in practice. RN-SLRA overcomes this difficulty by introducing a regularization term. We prove that, under the intrinsic transversality condition, RN-SLRA converges linearly to the intersection, while under the transversality condition it achieves higher-order convergence, including quadratic convergence for a suitable choice of the regularization parameter. We also study an inexact-projection variant, in which the projection onto the manifold is computed approximately, and show that the same local linear and quadratic convergence properties are preserved under the corresponding assumptions. Numerical experiments on constructed degenerate instances and Hankel-structured examples illustrate improved robustness in settings where Newton-SLRA may fail.

9. 一种结合相干振幅放大的量子配置法求解一维边值问题

A Quantum Collocation Approach to One-Dimensional Boundary Value Problems with Coherent Amplitude Amplification

作者: Daniel Jaroszewski, Bastian Harrach

类别: quant-ph, math.NA

发布日期: 2026-06-30

arXiv ID: 2606.31709v1

链接: http://arxiv.org/abs/2606.31709v1

摘要: We propose a quantum collocation framework for approximating solutions of one-dimensional linear and nonlinear boundary value problems. The method formulates the search for admissible solutions as a residual-based quantum search over a discretized ansatz space, where candidate solutions are evaluated through residual conditions imposed at collocation points. A residual-threshold oracle is constructed that acts jointly on spatial and parameter registers. This joint oracle structure leads to amplification dynamics that decompose into a coherent superposition of spatially conditioned amplitude-amplification processes rather than a single global amplification mechanism. We derive the corresponding amplification geometry and show that the success probability is governed by a weighted combination of spatially dependent amplification angles. Furthermore, we prove that the reversible residual oracle can be implemented with gate complexity polynomial in the logarithm of the number of collocation points, while retaining the quadratic search acceleration associated with amplitude amplification in the parameter space. We analyze how the spatially dependent oracle structure influences the amplification dynamics and corresponding success probabilities. Furthermore, we investigate how discretization, ansatz expressivity, oracle tolerance, and finite-precision effects influence both approximation quality and amplification behavior. Numerical experiments validate the theoretical predictions and illustrate the resulting search dynamics across different discretization and precision regimes.

10. Vlasov方程的保正动态低秩方法

Positivity-preserving dynamical low-rank methods for the Vlasov equation

作者: Katharina Kormann, Murtazo Nazarov, Junjie Wen

类别: math.NA

发布日期: 2026-06-30

arXiv ID: 2606.31662v1

链接: http://arxiv.org/abs/2606.31662v1

摘要: In this manuscript, we introduce positivity-preserving correction methods for low-rank approximations of the Vlasov equation. The key idea is to formulate structural properties, including positivity-preservation, as constraints and to seek a minimal correction term that is added to the low-rank solution, by solving a quadratic programming problem. As a result, the corrected solution satisfies the constraints and preserve these properties, while remaining close to the original low-rank solution. Two positivity-preserving schemes are proposed in this work, and one of them also preserves the total mass and momentum of the system. We apply the proposed methods to a Vlasov–Poisson and Vlasov–Poisson-BGK employing a spectral discretization in space and an explicit Runge–Kutta scheme in time. Numerical experiments demonstrate the effectiveness of the proposed methods.

11. 圆盘中V线张量层析成像：理论与数值重建

V-Line Tensor Tomography in a Disk: Theoretical and Numerical Reconstruction

作者: Rahul Bhardwaj, Madhu Gupta

类别: math.NA, math.AP

发布日期: 2026-06-30

arXiv ID: 2606.31632v1

链接: http://arxiv.org/abs/2606.31632v1

摘要: In this article, we investigate V-line transforms for symmetric $m$-tensor fields whose support lies inside a disk of radius $R$ and centered at the origin. We provide an explicit characterization of the kernel of the V-line transforms acting on a symmetric $m$-tensor field and derive a new inversion formula using a decomposition result. In addition, we present a comprehensive numerical verification and validation of the inversion algorithms for these V-line transforms for vector fields and symmetric $2$-tensor fields, which were recently developed in \cite{bhardwaj_2024,bhardwaj2025tensor}. The reconstruction results obtained for various phantoms demonstrate the effectiveness and robustness of the proposed numerical methods, including in the presence of noise.

12. PEERS：一种基于隐式求逆与增强符号分析的并行精确有效电阻求解器

PEERS: A Parallel and Exact Effective Resistance Solver via Implicit Inversion and Augmented Symbolic Analysis

作者: Baiyu Chen, Lin Gan, Guangwen Yang, Wenjian Yu

类别: cs.AR, cs.DC, math.NA

发布日期: 2026-06-30

arXiv ID: 2606.31535v1

链接: http://arxiv.org/abs/2606.31535v1

摘要: High-precision effective resistance computation is a cornerstone of Electronic Design Automation (EDA) sign-off, yet it remains a fundamental bottleneck in large-scale power grid analysis, spectral sparsification, and circuit reliability. Existing approaches face a prohibitive “precision-memory impasse”: approximate methods lack the stringent accuracy required for high-stakes industrial sign-off, while exact methods either suffer from redundant query overheads or trigger $O(n^2)$ memory explosions. To resolve this, we propose PEERS, a Parallel and Exact Effective Resistance Solver powered by an implicit inverse computing model of the Cholesky factor. By integrating a state-inherited augmented depth-first search (DFS) with a dynamic query update mechanism, PEERS eliminates numerical redundancy and evaluates all-edge resistance queries in a single parallel sweep. We provide a rigorous Work-Span analysis, proving that for graphs satisfying an $O(n^α)$ separator theorem, PEERS achieves a theoretically optimal parallel span of $O(n^α)$ while strictly maintaining $O(nnz(L))$ space complexity. Numerical evaluations on industrial benchmarks demonstrate that PEERS achieves an average speedup of 83.3x over state-of-the-art parallel solvers under identical memory constraints. Notably, PEERS processes a 1-million-node industrial graph in just 18.8 seconds and scales to 17 million nodes in under an hour, providing the first computationally feasible path for exact all-edge resistance analysis in multi-million-gate designs.

13. 离散基尔霍夫元的误差分析

An error analysis of discrete Kirchhoff elements

作者: Dietmar Gallistl, Ngoc Tien Tran

类别: math.NA

发布日期: 2026-06-30

arXiv ID: 2606.31463v1

链接: http://arxiv.org/abs/2606.31463v1

摘要: The Discrete Kirchhoff Triangle (DKT) method for the biharmonic equation is analyzed in the discrete energy norm. The error is bounded by the best approximation of the Hessian by piecewise constants and the oscillation of the right-hand side, without additional regularity assumptions on the exact solution. This result implies first-order convergence of the classical DKT element and the analysis yields a canonical extension to three space dimensions with the same approximation properties. Residual-based a posteriori error estimates are derived. The analysis is formulated within a general framework for low-order nonconforming methods, which also applies to various classical elements and yields best-approximation results by constants. It is furthermore shown how known stable pairs for the planar Stokes system have discrete stream functions in discrete Kirchhoff spaces. This yields variants of the known schemes with positive definite formulations and pressure-robust error bounds.

14. 无界区域上偏微分方程的区域分解随机神经网络

Domain-Decomposed Randomized Neural Networks for Partial Differential Equations in Unbounded Domains

作者: Haixin Wang, Haoning Dang, Fei Wang, Shimin Guo

类别: math.NA, cs.LG

发布日期: 2026-06-30

arXiv ID: 2606.31342v1

链接: http://arxiv.org/abs/2606.31342v1

摘要: Partial differential equations on unbounded domains are challenging because the exterior region must be represented without excessive truncation error. Truncation-based methods often require problem-dependent artificial boundary conditions, while global spectral bases may be inefficient for localized structures, irregular geometries, or solutions with different near-field and far-field behaviors. We propose a domain-decomposed randomized neural network framework for such problems. Different randomized subnetworks are assigned to different spatial regimes: a near-field subnetwork captures local and geometric features, whereas a far-field subnetwork represents exterior decay. The subnetworks are coupled by boundary and interface conditions, and only the output-layer coefficients are solved from linear least-squares systems arising from Petrov–Galerkin or collocation formulations. We develop a Petrov–Galerkin method for semi-unbounded elliptic problems and a collocation method for fully unbounded, perforated, and time-dependent problems. A conditional bounded-parameter approximation result is proved in a broken Sobolev norm, together with an error decomposition covering approximation, empirical-consistency/quadrature, and least-squares optimization errors. Numerical experiments for Poisson and time-dependent Schrödinger equations demonstrate the accuracy and flexibility of the proposed method.

15. 高阶指数Runge-Kutta Galerkin有限元法用于带非光滑数据的半线性抛物问题

Higher-order exponential Runge-Kutta Galerkin finite element method for semilinear parabolic problems with nonsmooth data

作者: Shuo Yang, Runjie Zhang, Zhe Yu, Jinwei Fang

类别: math.NA

发布日期: 2026-06-30

arXiv ID: 2606.31231v1

链接: http://arxiv.org/abs/2606.31231v1

摘要: We develop a rigorous numerical analysis framework for a class of semilinear parabolic problems with nonsmooth initial data. We employ a linear Galerkin finite element method for spatial discretization coupled with a high-order explicit exponential Runge-Kutta (EERK) temporal integration scheme. In contrast to conventional smooth error analysis, the nonsmooth case lacks a priori estimates for the higher-order derivatives of both the nonlinear term and the exact solution. By combining analytic semigroup techniques with fractional power space theory, we establish rigorous bounds for these derivatives. Finally, our analysis proves that the $p$th-order EERK method achieves a convergence rate of $\min(1 + γ/2 + ρ_1(γ)/2,\:p)$, where $γ$ characterizes the initial data regularity and $ρ_1(γ)$ quantifies the boundedness of the nonlinearity’s first Fréchet derivative. Numerical experiments confirm the sharpness of these estimates.

16. 用于具有不匹配反投影器的 X 射线计算机断层成像的 GKB 方法

GKB Methods for X-Ray Computed Tomography with an Unmatched Back Projector

作者: Abdulmajeed Alsubhi

类别: math.NA

发布日期: 2026-06-30

arXiv ID: 2606.31153v1

链接: http://arxiv.org/abs/2606.31153v1

摘要: In large scale X ray Computed Tomography (CT) inverse problems, the forward and back projectors are often generated using different discretizations. This discrepancy leads to unmatched pairs of projections, resulting in inconsistent normal equations. Consequently, employing the Conjugate Gradient method does not produce a useful solution. For matched operator pairs, the Golub Kahan bidiagonalization (GKB) method provides an efficient solution strategy. It works by projecting the original large-scale problem onto a lower-dimensional subspace, enabling the solution to be computed via a singular value decomposition of a sparse lower bidiagonal matrix. To address unmatched-pair problems in CT, we propose the AB and BA GKB algorithms as preconditioned forms of the GKB. These methods are straightforward to implement and allow for parameter tuning. We provide a discussion on the theoretical computational costs of our proposed algorithms in terms of floating point operations and compare with existing methods. While many Krylov methods tend to amplify noise in solutions, leading to semiconvergence, our proposed algorithms demonstrate greater resilience against this effect. We validate the effectiveness of our approach through numerical examples across various CT problems, showcasing its ability to deliver more stable solutions.

17. 用于扰动Boussinesq系统的相容分裂广义标量辅助变量格式

A consistent-splitting generalized scalar auxiliary variable scheme for the perturbed Boussinesq system

作者: M Nader Alhomsi, Jiahong Wu, Xiaoming Zheng

类别: math.NA

发布日期: 2026-06-30

arXiv ID: 2606.31152v1

链接: http://arxiv.org/abs/2606.31152v1

摘要: We propose and analyze a second-order consistent-splitting scheme, based on the generalized scalar auxiliary variable (GSAV) approach, for the two-dimensional perturbed Boussinesq system. The system is obtained by subtracting a stable, linearly stratified hydrostatic equilibrium from the standard Boussinesq system. The time discretization extends the consistent-splitting generalized BDF2 framework of Huang and Shen [17] for the Navier-Stokes equations, treating the nonlinear convection and advection together with the linear buoyancy and stratification couplings explicitly, so that each time step reduces to a small number of decoupled linear systems. We prove an unconditional weak stability theorem for the GSAV scheme and derive optimal second-order error estimates for the velocity, pressure, and temperature. A careful tracing reveals that the error constant depends on the inverse viscosity and inverse thermal diffusivity through a quadruply-nested exponential, so the scheme is not robust as either tends to zero. Numerical experiments confirm the second-order convergence and reproduce the expected internal-wave dynamics and exponential relaxation toward hydrostatic balance in a long-time stratified-flow simulation.

18. 通过BDSDE表示对SPDE进行量子衍生品定价

Quantum Derivative Pricing for SPDEs via BDSDE Representation

作者: Xinmiao Li, Yanqiao Wang, Rundi Lu, Zhengwei Liu, Jin-Peng Liu

类别: quant-ph, math.NA

发布日期: 2026-06-30

arXiv ID: 2606.31076v1

链接: http://arxiv.org/abs/2606.31076v1

摘要: We study quantum speedups of derivative pricing for stochastic partial differential equation (SPDE) models through their backward doubly stochastic differential equation (BDSDE) representations. We develop conditional and nested quantum-accelerated multilevel Monte Carlo (QA-MLMC) methods for estimating the resulting conditional and nested expectations, improving the sampling complexity of classical Monte Carlo methods from $\widetilde{O}(ε^{-2})$ to $\widetilde{O}(ε^{-1})$ within additive error $ε$. We apply the framework to derivative pricing and sensitivity analysis, providing quantum-accelerated estimators for prices as well as first-order and second-order Greeks, likelihood-ratio and Malliavin-weight representations for Greeks, and Heston-type stochastic-volatility models. To enable efficient multilevel coupling, we construct a family of Forward–Backward Taylor discretization schemes for the stochastic integrals arising in the BDSDE representations and establish global strong-error order one convergence for pricing and Greek estimators. Numerical experiments showcase our schemes for first-order and second-order Greeks can reach the required orders for the full quadratic quantum speedups.

19. 带增量正交化的流式张量分解在线 TT-ALS

Online TT-ALS for Streaming Tensor Decomposition with Incremental Orthogonalization

作者: Hiroki Takeda, Yuto Miyatake, Daisuke Furihata

类别: math.NA, cs.CV, cs.LG

发布日期: 2026-06-30

arXiv ID: 2606.31061v1

链接: http://arxiv.org/abs/2606.31061v1

摘要: Tensor Train (TT) decomposition is a powerful technique for analyzing high-dimensional data. Existing algorithms for computing TT decompositions can be categorized into two main types: conventional batch-based approaches and recursive online methods. In the context of streaming data, batch methods typically achieve higher reconstruction accuracy but often suffer from memory exhaustion, while online methods provide greater computational efficiency. In this work, we introduce Online TT-ALS (Alternating Least Squares), an algorithm that sequentially enforces orthogonality constraints. This approach allows for efficient and exact updates of the core tensor while maintaining high reconstruction accuracy. Theoretically, we prove that enforcing these orthogonal gauge constraints guarantees monotonic decrease of the local objective function and temporal smoothness. Computationally, our deterministic single-sweep update reduces the rank dependence from quadratic to linear, achieving an overall complexity of $\mathcal{O}(I^{n-1} r)$. Experimental results demonstrate that the proposed method outperforms existing online techniques not only in terms of mathematical approximation accuracy but also in human perception-based video quality metrics. Furthermore, compared to recent deep learning-based paradigms, our algebraic approach achieves speedups of several orders of magnitude. Consequently, our method exhibits high computational efficiency and is suitable for low-latency real-time processing applications.

20. 超越二分裂：一种用于蠕虫状胶束溶液的广义三种类级联断裂模型

Beyond binary scission: a generalized three-species cascade breakage model for wormlike micellar solutions

作者: Rongxin Lu, Jiwei Jia, Young Ju Lee

类别: cond-mat.soft, math.NA

发布日期: 2026-06-30

arXiv ID: 2606.31059v1

链接: http://arxiv.org/abs/2606.31059v1

摘要: Wormlike micellar fluids exhibit complex rheological behavior driven by the continuous breakage and recombination of self-assembled micellar networks. Existing two-species models provide a coarse binary representation of the micellar population, limiting their ability to resolve intermediate structural states and broad relaxation spectra. To address this limitation, we develop a three-species cascade breakage model consisting of gel-network, long chains, and short chains. By introducing an intermediate micellar state, the model links the rapid relaxation of short fragments to the slow recovery of the gel-network within a unified kinetic framework. This additional structural pathway gives rise to a three-mode viscoelastic response, improves the high-frequency description of the dynamic moduli, and produces a non-monotone constitutive curve that evolves into a stress plateau with coexisting shear bands in Couette flow. This cascade mechanism also governs the transient response, including stress overshoot, hysteresis, and multistep relaxation after shear cessation. Overall, the proposed three-species model provides a physically interpretable framework for worm-like micellar shear banding, capturing the connection between cascade microstructural evolution, broad relaxation dynamics, and macroscopic flow localization.

21. 基于任意阶酉三角分解的数值代数几何的Ricci记号张量框架

Ricci-Notation Tensor Framework for Numerical Algebraic Geometry via Any-Degree Unitary-Triangular Factorization

作者: Dileepan Joseph

类别: math.NA, cs.MS

发布日期: 2026-06-30

arXiv ID: 2606.31003v1

链接: http://arxiv.org/abs/2606.31003v1

摘要: The unitary-triangular (QR) factorization of linear algebra may be used to robustly and efficiently solve a linear system. Toward a comparable numerical method to solve a polynomial system of higher degree, this paper proposes an any-degree unitary-triangular (Qr) factorization, which for a degree-one system reduces to the QR factorization. The work develops a tensor framework, i.e., codesigned algebra and software, where polynomial system coefficients are represented by a vector-shaped sparse tensor, a multidimensional array whose number of Ricci-notation indices, called the tensor degree, equals the highest monomial degree of the system. With the proposed Qr factorization, the coefficient tensor decomposes into a product of unitary and triangular factors that, in general, also have Ricci-notation indices and sparse entries. The unitary factor defines a unitary transform, a generalization of the related linear algebra concept to tensor algebra, that can triangularize a polynomial system while preserving its solution set, whether zero- or positive-dimensional. The work extends the author’s Ricci-notation tensor framework, providing new algebra and new software to model, construct, and factorize polynomial systems in this manner. After applying the approach to numerically triangularize two zero-dimensional systems, chosen for educational value, results are compared to the Gröbner-basis (GB) method for triangularizing polynomial systems symbolically. One problem is of degree three, with three equations and unknowns, and the other of degree two, with four equations and unknowns. Although it resembles GB triangularization, the proposed Qr factorization has a completely different pedigree associated with numerical methods.

22. 非紧区域上参数化偏微分方程的局部Gevrey正则性与拟蒙特卡罗求积

Local Gevrey regularity and Quasi-Monte Carlo quadrature for PDEs parameterized on non-compact domains

作者: Dinh Dũng, Tùng Lê, Alexey Chernov

类别: math.NA

发布日期: 2026-06-29

arXiv ID: 2606.30972v1

链接: http://arxiv.org/abs/2606.30972v1

摘要: We establish local Gevrey regularity for the weak solution to parametric divergence-form diffusion elliptic PDEs, assuming the diffusion coefficient itself possesses local Gevrey parametric regularity over a non-compact domain. Here “local Gevrey regularity” means that the regularity is determined in a neighborhood of each parametric point and depends on that point. Explicit bounds on parametric derivatives of the weak solution are proved in the $H^1$-norm. Building on this local Gevrey framework, we develop a novel theoretical treatment of the dimension truncation error for infinite-dimensional integration. We prove convergence rates of quasi-Monte Carlo quadrature for the finite-dimensional Gaussian-weighted integration of integrands having local Gevrey regularity. Together with the finite element discretization error, the errors arising from dimension truncation and quasi-Monte Carlo quadrature yield a complete error analysis and convergence rates for the fully discrete approximation of a bounded linear functional of the weak solution. Numerical experiments confirming the theoretical convergence rates are presented.

23. 矩阵李群中的向量对齐

Vector alignment in matrix Lie groups

作者: Congzhou M Sha

类别: math.NA, hep-lat, math-ph

发布日期: 2026-06-29

arXiv ID: 2606.30868v1

链接: http://arxiv.org/abs/2606.30868v1

摘要: The difference in gauge between two observers of the same physical system can be thought of as a group element acting on their common vector representations. Recovering that group element from a finite, noisy list of paired observations may be of use in both theory and experiment. The Kabsch and Horn algorithms efficiently align point clouds in $\mathbb R^3$, reconciling rotated frames of reference in Galilean relativity (i.e. $SO(3)$). In a previous work, we proposed an alternative Lie algebra method which extends to the Lorentz group $SO(3,1)_+$, and putatively to all Lie groups. In this work, we report the explicit formulae for applying the Lie algebra method to the classical matrix Lie groups (general linear $GL(n)$, special linear $SL(n)$, special orthogonal $SO(n)$, unitary $U(n)$, indefinite special orthogonal $SO(p,q)$, symplectic $Sp(n)$, spin $Spin(n)$, special Euclidean $SE(n)$) over both the real and complex fields. The four steps (pseudoinverse, matrix logarithm, projection onto the Lie algebra, matrix exponential) are exact in the noiseless case. The only group-dependent step is the projection, which we show produces the unique least squares-optimal element of the Lie algebra whenever its image lies in $\mathfrak g$ and its residual is orthogonal to $\mathfrak g$. Additionally, the Lie algebra method is optimal only to leading order for noisy data, so we refine it with a Newton-style correction. This correction matches the Lie algebra method in the noiseless case and direct least squares optimization in the noisy case, with performance between that of the Lie algebra method without correction and naive least squares optimization. The projections, their optimality, and the identity underlying the correction are formally proven in Lean~4.31.0 (with Mathlib 4.31.0), and numerical experiments are benchmarked in Julia.