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1. 基于细分的等几何分析用于轴对称电磁问题
Subdivision-based isogeometric analysis for axisymmetric electromagnetic problems
作者: Devin Balian, Sebastian Schöps, Melina Merkel
类别: cs.CE, math.NA
发布日期: 2026-06-11
arXiv ID: 2606.13308v1
摘要翻译: 本文应用一种基于细分的等几何方法求解轴对称Maxwell特征值问题。化归到$H^1$-形式允许使用Catmull-Clark构造同时进行几何和场离散。该方法给出的电场数值解在除异常顶点外的所有地方是$C^1$连续的。这一点通过对TESLA 9-cell腔的本征模的计算得以展示，结果显示与传统方法相比场更光滑且数值噪声更小。对该方法的收敛速率进行了数值分析，并与文献中观测到的速率一致。
Comment: 无
链接: http://arxiv.org/abs/2606.13308v1
1. 研究对象和背景
本文的研究对象是轴对称时谐 Maxwell 特征值问题。Maxwell 方程组在无源、常材料参数 $\varepsilon,\mu>0$ 下, 时谐场满足
$$\operatorname{curl}\mathbf{E}=-\mathrm{i}\omega\mu\mathbf{H},\qquad \operatorname{curl}\mathbf{H}=\mathrm{i}\omega\varepsilon\mathbf{E},$$消去电场得到磁场波动方程 $\operatorname{curl}\operatorname{curl}\mathbf{H}=\omega^2\mu\varepsilon\mathbf{H}$。通过弱形式, 求特征对 $(\lambda,\mathbf{H})\in\mathbb{R}_{>0}\times H(\operatorname{curl},\Omega)$ 使
$$\forall\mathbf{H}'\in H(\operatorname{curl},\Omega):\quad (\operatorname{curl}\mathbf{H},\operatorname{curl}\mathbf{H}')=\lambda(\mathbf{H},\mathbf{H}'),\qquad \lambda=\omega^2\varepsilon\mu.$$该问题描述了理想导体腔内的谐振模式, 在加速器物理等工程中至关重要。
从数值离散的角度, 需要相容的 $H(\operatorname{curl})$ 有限元空间, 通常采用 Nédélec 棱单元。等几何分析 (IGA) 追求 CAD 与仿真使用相同基函数, 而细分曲面近年来因其可处理任意拓扑四边形网格且几乎处处 $C^2$ 连续而受关注。然而, 经典的 Catmull-Clark 细分产生标量 $H^1(\Omega)$ 极限函数, 无法直接用于 $H(\operatorname{curl})$ 的一致离散。文献中已提出边缘细分方案 (Wang 等, De Goes 等, Piel 和 Bauer) 以构造矢量基函数, 但这些工作并未使用细分极限函数作为 IGA 的基函数, 因而数值解不具备高的单元间正则性; 且其三维修正和精确求值仍未解决。此外, 三维细分体的分析性质远未清楚, 边缘细分方案更无三维推广。因此, 直接将细分 IGA 应用于三维电磁问题存在重大障碍。
本文试图填补的空白是: 利用轴对称性将三维 Maxwell 特征值问题降维到二维标量问题, 从而避开 $H(\operatorname{curl})$ 离散的困难, 使标准 Catmull-Clark 细分基函数可直接应用。设区域 $\Omega$ 为旋转体, 子午面
$$S:=\{(\rho,z)\in\mathbb{R}_{\ge0}\times\mathbb{R}:(\rho,0,z)\in\Omega\},$$并假设磁场仅有 $\varphi$ 分量 $\mathbf{H}=u\,\mathbf{e}_\varphi$ (TM 模式)。推导后, 弱形式化为: 求 $(\lambda,u)\in\mathbb{R}_{>0}\times H^1_\rho(S)$ 满足
$$\forall v\in H^1_\rho(S):\quad (\operatorname{rot} u,\operatorname{rot} v)_\rho=\lambda(u,v)_\rho,$$其中 $\operatorname{rot} u=(\partial_z u,-\partial_\rho u)^\mathsf{T}$ 是旋转梯度, $(\cdot,\cdot)_\rho$ 是带权重 $\rho$ 的 $L^2$ 内积:
$$(u,v)_\rho=\iint_S uv\,\rho\,\mathrm{d}\rho\,\mathrm{d}z.$$电场由安培定律 $\mathbf{E}=-\frac{\mathrm{i}}{\omega\varepsilon}\overline{\operatorname{rot}}u$ 给出, $\overline{\operatorname{rot}}(u_1,u_2)=\partial_\rho u_2-\partial_z u_1$ 为标量旋度。该 $H^1_\rho(S)$ 问题可用标量细分基函数直接离散, 且电场作为导数自动获得单元间 $C^1$ 连续性 (奇异点外)。这一约化为细分 IGA 在电磁仿真中的实际应用开辟了可行途径, 同时保留了细分曲面对拓扑的灵活性。
2. 主要定理或主要结果
本文通过数值实验验证了所提方法的收敛性和场光滑性, 主要结果如下。
圆柱谐振腔 (pillbox) TM$_{010}$ 模式收敛性。几何参数 $R=35$ mm, $d=100$ mm, 采用含奇异点的非结构化网格, Catmull-Clark 基函数离散。正则单元用 $p=4$ 高斯求积, 非正则单元用 $L=10$ 层分级求积。误差-自由度图显示: $H^1_\rho(S)$ 误差收敛速度约为 $O(\sqrt{N_{\mathrm{dof}}}^{-1.5})$, $L^2_\rho(S)$ 误差约为 $O(\sqrt{N_{\mathrm{dof}}}^{-2.5})$, 特征频率相对误差约为 $O(\sqrt{N_{\mathrm{dof}}}^{-3.0})$。这些速率与文献中 Catmull–Clark 离散二维 Poisson 问题观察到的速率一致。
单 cell TESLA 腔收敛性。几何参数取自文献 [25], 包含圆弧和椭圆弧, 不能精确用 Catmull–Clark 曲面表示, 采用三次样条拟插值 $c_i=\frac{1}{6}(-p_{i-1}+8p_i-p_{i+1})$ 拟合边界。对前五个 TM 模式, 频率相对误差均以 $O(\sqrt{N_{\mathrm{dof}}}^{-3.0})$ 趋势下降 (图 7), 证实方法对复杂几何有效。
9-cell TESLA 腔场光滑性。在轴线 $\rho=R_{\mathrm{iris}}/2=17.5$ mm 和 $\rho=R_{\mathrm{eq}}/2=51.65$ mm 上评估 $E_\rho$ 分量, 并与最低阶 $C^0$ 有限元比较。细分方法所得电场光滑, 单元交界处无跳跃或振荡; 最低阶有限元则在峰值处出现明显的非物理噪声。这是因为 Catmull–Clark 基函数跨单元 $C^2$ 连续, 保证了导数 (电场) 的连续性, 从而获得更高质量的数值解。
3. 研究方法、关键技术和核心工具
本文方法的核心在于通过轴对称降维将 $H(\operatorname{curl})$ 问题转化为 $H^1$ 问题, 从而允许使用标量 Catmull–Clark 细分基函数。关键技术点和分析框架分述如下。
细分基函数的构造与精确求值。正则单元 (不含奇异点) 上, Catmull–Clark 基函数退化为双三次 B 样条张量积:
$$\widehat{b}_{ij}(\xi,\eta)=N_i(\xi)N_j(\eta),\quad i,j=1,\dots,4,$$$N_i$ 为均匀三次 B 样条基函数。在包含奇异点 (价数 $n\neq4$) 的非正则单元上, 基函数通过无限细分定义。将单位正方形参数域划分为无穷多瓦片 $\Sigma_k^\ell$ (图 2), 在每个瓦片上结合仿射变换 $t_k^\ell$ 和扩展细分矩阵 $A,\bar{A}$, 局部基函数行向量 $\widehat{\mathbf{b}}_\Sigma$ 在 $\Sigma_k^\ell$ 上定义为
$$\widehat{\mathbf{b}}_{\Sigma_k^\ell}=\bigl(\widehat{\mathbf{b}}_k\circ t_k^\ell\bigr)\,\bar{A}A^{k-1},$$其中 $\widehat{\mathbf{b}}_k$ 是按细分顶点排序的双三次 B 样条行向量。这是 Stam 精确求值算法的核心, 使得在任意参数点可计算基函数值, 是 IGA 中必需的。
分层数值积分。非正则单元在奇异点附近不是分片多项式, 标准高斯求积精度不足。本文采用分级求积策略: 对单元进行 $L$ 层细分, 在每个规则子瓦片内应用标准高斯求积, 再累加近似积分。该方法能准确捕获基函数在奇异点附近的行为, 实现中需处理细分拓扑映射和数据结构。
轴对称弱形式推导与离散。从三维 Maxwell 方程出发, 利用 $\partial_\phi=0$ 和轴对称假设, 将矢量旋度和散度转化为标量算子 $\operatorname{rot}$ 和 $\overline{\operatorname{rot}}$, 并引入权重 $\rho$ 以保证内积维数一致。弱形式 (5) 的离散得到广义特征值问题:
$$K\mathbf{u}=\lambda M\mathbf{u},$$其中
$$k_{ij}=(\operatorname{rot} b_i,\operatorname{rot} b_j)_\rho,\quad m_{ij}=(b_i,b_j)_\rho,$$$b_i$ 为物理域上的 Catmull–Clark 极限基函数, 通过等参映射 $b_i=b_i\circ \Phi^{-1}$ 关联到参数域, $\Phi$ 为细分曲面参数化。积分在物理网格上计算, 需处理非正则单元上的奇异映射。
几何拟合。当 CAD 几何无法精确表示为 Catmull–Clark 曲面时 (如圆弧), 采用三次样条拟插值修正边界控制点:
$$c_i = \frac{1}{6}(-p_{i-1}+8p_i-p_{i+1}),$$$p_i$ 为真实边界采样点。这是一种简便的几何逼近, 虽非最优, 但确保了实现可行性。
整体而言, 作者的主要创新在于将物理维度约化与细分基函数特性结合: 利用轴对称性将电场求解转化为磁场标量解的梯度后处理, 避免了构造矢量细分空间的困难, 同时仍能获得高光滑性的场分布。这一思路使得标准细分 IGA 可直接用于电磁问题, 无需边缘细分方案。
4. 与之前工作的比较
本文工作与已有研究在多个维度存在推进和差异, 详细比较如下。
与经典有限元法比较。对于同一轴对称 TESLA 腔问题, 使用最低阶 $C^0$ 有限元在相同网格上求解, 得到的电场 $E_\rho$ 在单元界面处出现不连续跳跃和明显数值噪声 (图 8, 9)。而本文的 Catmull–Clark 细分 IGA 因基函数在正则单元间 $C^2$ 连续 (奇异点外 $C^1$), 电场作为解的导数自动具有跨单元连续性, 消除了虚假振荡, 且不增加自由度。若采用高阶拉格朗日元可部分抑制噪声, 但仍无法根除导数间断, 且会显著增加自由度。本文的细分方法在相同自由度下提供了更光滑可靠的场解。
与细分 IGA 现有研究比较。细分曲面在 IGA 中的早期应用集中于 Kirchhoff–Love 薄壳等需要 $C^1$ 连续性的问题 (Cirak 等)。而在电磁问题上, 主要的障碍是 $H(\operatorname{curl})$ 相容离散需要矢量基函数。Wang 等 (2006) 提出边缘细分方案以构造光滑矢量场, 随后 De Goes 等 (2016) 发展了细分外微积分。但这些工作并未将细分极限函数直接用作有限元基函数, 而是用于构造离散 exterior calculus, 因此数值解不具备单元间的高阶正则性。Piel 和 Bauer (2024) 建立了 k 形细分空间的严格数学框架, 并应用于二维 Maxwell 特征值问题, 但他们没有利用极限函数的 IGA 离散, 而是称其获得的是“细分诱导正则性”而非 Sobolev 正则性。与这些工作相比, 本文的突破在于利用轴对称约化, 首次把 Catmull–Clark 极限函数作为 $H^1$ 标量基函数直接用于电磁特征值问题的 Galerkin 离散, 得到了电场真正 $C^1$ 连续的数值解, 充分发挥了细分的高光滑性优势。
与轴对称电磁求解器比较。传统轴对称 Maxwell 问题通常在子午面上采用棱单元离散 $H(\operatorname{rot},S)$ 等空间 (如 Simona 等)。这类方法虽可在二维求解, 但基函数连续性仅限于 $H(\operatorname{rot})$ 最低要求, 场光滑性受限。本文通过降维到 $H^1$, 允许使用高光滑性基函数, 同时仍能恢复矢量场。这与傅里叶模态展开的通用框架并行, 但本文聚焦 TM 模式, 并首次展示了细分 IGA 在轴对称电磁仿真实例中的可行性。与 Simona 等工作的 B 样条 IGA 相比, 本文进一步处理了非结构化网格 (含奇异点) 和复杂几何逼近, 扩展了 IGA 的适用范围。
局限性比较与代价。本文方法依赖于严格的轴对称假设和 TM 模式, 不能直接处理三维一般问题或 TE 模式。与边缘细分方案相比, 虽避免了复杂矢量构造, 但损失了一般性。在几何表示上, 由于 Catmull–Clark 曲面无法精确表示圆弧等二次曲线, 需进行边界拟合, 引入额外的几何逼近误差; 文中仅用了较简单的拟插值, 更高级的拟合 (如 $L^2$ 投影) 尚未探讨。在收敛性方面, 目前仅有数值验证, 缺乏理论证明, 这与 B 样条 IGA 有完备的误差分析不同。此外, 奇异点处电场连续性可能降至 $C^0$, 文中未定量分析其影响。与 Piel 和 Bauer 的框架相比, 本文未涉及 de Rham 复形稳定性等结构问题。尽管如此, 该工作为细分 IGA 在真实电磁部件仿真中打开了一扇窗, 并提供了清晰的实现策略。
了解领域论文（摘要翻译）
1. 面向稳态Navier-Stokes流的嵌入式Trefftz-DG方法. 第一部分: Oseen线性化
Embedded Trefftz DG method for steady Navier-Stokes flow. Part I: Oseen linearization
作者: Paul Stocker, Igor Voulis, Christoph Lehrenfeld, Philip L. Lederer
类别: math.NA
发布日期: 2026-06-11
arXiv ID: 2606.13229v1
链接: http://arxiv.org/abs/2606.13229v1
摘要翻译
中文标题: 面向稳态Navier-Stokes流的嵌入式Trefftz-DG方法. 第一部分: Oseen线性化
摘要翻译: 我们为Oseen问题开发了一种嵌入式Trefftz-DG方法，并证明了在标准DG范数下的完全稳定性和拟最优性理论。关键要素是构造了Trefftz空间的一个合适的局部补空间，在该补空间上Oseen算子是可稳定求逆的。我们还推导出该方法的降阶形式，所得方程组仅以速度未知量表示，这是分析中尤其对第二部分中的非线性Navier-Stokes问题至关重要的步骤。
2. 稳态Navier-Stokes流的嵌入式Trefftz间断伽辽金方法：第二部分 非线性问题
Embedded Trefftz DG method for steady Navier-Stokes flow. Part II: Nonlinear problem
作者: Paul Stocker, Igor Voulis, Christoph Lehrenfeld, Philip L. Lederer
类别: math.NA
发布日期: 2026-06-11
arXiv ID: 2606.13219v1
链接: http://arxiv.org/abs/2606.13219v1
摘要翻译
中文标题: 稳态Navier-Stokes流的嵌入式Trefftz间断伽辽金方法：第二部分 非线性问题
摘要翻译: 我们针对稳态不可压缩Navier-Stokes方程，基于第一部分中的简化Oseen离散，提出并分析了一种嵌入式Trefftz间断伽辽金方法。主要困难在于简化Trefftz空间依赖于对流场，因此相继的Picard迭代处于不同的离散空间。我们通过构造依赖于对流的Trefftz空间之间的投影，并利用它们来控制简化Oseen解映射，解决了这一问题。在合适的网格分辨率和小数据假设下，我们证明了离散解的存在性、唯一性以及Picard迭代的收敛性。我们还通过将该方法与底层的DG离散相联系，推导了先验误差分析，从而从相容的DG Navier-Stokes分析中继承了收敛性质。在标准不可压缩流基准测试上的数值实验验证了该理论。
3. 弱阻尼半线性波动方程 SIPG 方法的 Lyapunov 稳定性与最优误差估计
Lyapunov Stability and Optimal Error Estimates for an SIPG Method for Weakly Damped Semilinear Wave Equations
作者: Ajeet Singh, Abhinav Jha
类别: math.NA
发布日期: 2026-06-11
arXiv ID: 2606.13185v1
链接: http://arxiv.org/abs/2606.13185v1
摘要翻译
中文标题: 弱阻尼半线性波动方程 SIPG 方法的 Lyapunov 稳定性与最优误差估计
摘要翻译: 本文针对弱阻尼半线性波动方程，提出并分析了一种全离散格式，该格式将对称内部惩罚间断 Galerkin（SIPG）空间离散与混合的 Crank–Nicolson/二阶向后微分公式（CN–BDF2）时间积分器相结合。对非线性反应项采用弦斜率线性化，既保持了精确的离散梯度结构，又关键地不需要对非线性项施加任何全局 Lipschitz 连续假设。通过构造离散 Lyapunov 泛函，基于 Lyapunov 稳定性分析（而非谱论证）建立了全离散解的稳定性，从而得到数值解的存在性、唯一性与一致有界性。在标准正则性假设下，证明了 DG 能量范数下 $\mathcal{O}(h^{k}+τ^{2})$ 与 $L^{2}$ 范数下 $\mathcal{O}(h^{k+1}+τ^{2})$ 的最优先验误差估计，其中 $h$ 为网格尺寸，$τ$ 为时间步长，$k$ 为多项式次数。针对具有线性、三次及三角非线性的二维问题进行的数值实验，验证了理论收敛速率，并展示了由 Lyapunov 结构所保证的长时间能量耗散性质。
4. 基于亥姆霍兹分解的鲁棒实压缩层方法用于时谐弹性波散射
A Robust Helmholtz-Decomposition-Based Real Compressed Layer Method for Time-Harmonic Elastic Wave Scattering
作者: Li-Lian Wang, Lu Zhang
类别: math.NA
发布日期: 2026-06-11
arXiv ID: 2606.12974v1
链接: http://arxiv.org/abs/2606.12974v1
摘要翻译
中文标题: 基于亥姆霍兹分解的鲁棒实压缩层方法用于时谐弹性波散射
摘要翻译: 时谐弹性波散射同时涉及压缩波（P波）和剪切波（S波），它们以不同的波数和偏振特性传播。基于复坐标拉伸的完美匹配层（PML）类方法的朴素构造可能缺乏鲁棒性，甚至失败，特别是当波数对比度很大时。最近为时谐亥姆霍兹问题发展的实压缩层（RCL）技术建立在实压缩变换和显式提取所产生的振荡模式的基础上，可能无法工作，因为这些振荡无法通过单一的变量替换显式提取出。本文旨在通过为具有紧支撑非均匀性的无界区域中的二维时谐弹性波散射发展一种鲁棒的RCL方法来弥补这一不足。一个关键的观察是，通过亥姆霍兹分解，外部均匀区域中的位移场解耦为P波和S波，并且每一部分在极坐标下都有其振荡模式和衰减行为的独特分离。然后我们对每一分量在径向应用实压缩坐标变换。我们进一步提出了一种耦合位移-势的RCL公式，它将亥姆霍兹分解后的波分量与内部位移场无缝集成。我们证明，在此框架下，层中的本质振荡可以被有效移除。我们证明了所得耦合问题的适定性，并建立了在所关心的截断区域内RCL解到原始散射解的指数收敛性。我们使用高阶谱元法离散RCL系统，并通过丰富的数值结果展示了所提方法的有效性和鲁棒性。
5. 三维一般曲面上的Neumann Green函数：奇异渐近与边界积分方法
The three dimensional Neumann Green’s function for general surfaces: singular asymptotics and boundary integral methods
作者: Alan E. Lindsay, Andrew J. Bernoff, Tristan Goodwill, Jeremy G. Hoskins
类别: math.NA, math.AP, physics.app-ph, physics.comp-ph
发布日期: 2026-06-10
arXiv ID: 2606.12714v1
链接: http://arxiv.org/abs/2606.12714v1
摘要翻译
中文标题: 三维一般曲面上的Neumann Green函数：奇异渐近与边界积分方法
摘要翻译: 我们提出了一般几何中三维Neumann Green函数的渐近分析和高阶边界积分方法。Neumann Green函数是众多科学和工程领域中涌现的一个基本量。在将奇异摄动方法应用于强局部化反应和扩散输运时，该Green函数在介导全局动力学中起着关键作用。然而，这一重要量仅在有限的几何形状下可得到封闭形式。Laplace算子的Green函数是一个具有Dirac强迫项的椭圆问题。精确求解该解需要将其仔细分解为奇异部分和正则部分。体情形是指源点放置在曲面之外，奇性由自由空间函数给出。在表面情形中，源点放置在边界上的弯曲点处，我们利用渐近分析确定一个三项奇异结构。利用对这些奇性的显式知识，我们开发了一种高阶边界积分方法来确定剩余的正则部分。为处理奇异边界数据，我们的积分方法在源点附近使用了带有Duffy片的定制离散化。我们通过几个测试用例验证了我们的方法，在这些用例中可得到封闭形式解，包括球体、长球体和构造区域。我们展示了我们的方法在解决狭窄捕获理论中一些公开问题上的适用性。
6. 牵引驱动指状形态发生的 Cahn-Hilliard 模型的保结构离散与指状化动力学
Structure-preserving discretization and fingering dynamics of a Cahn-Hilliard model for traction-driven digit morphogenesis
作者: Marvin Fritz
类别: math.NA
发布日期: 2026-06-10
arXiv ID: 2606.12574v1
链接: http://arxiv.org/abs/2606.12574v1
摘要翻译
中文标题: 牵引驱动指状形态发生的 Cahn-Hilliard 模型的保结构离散与指状化动力学
摘要翻译: 我们研究具有各向异性牵引通量的 Cahn-Hilliard 方程，该方程源自形成指状结构的类器官中力学偏置细胞相互作用的约化连续体模型。对于具有严格正有界迁移率的正则化问题，我们引入基于化学势隐式-显式处理的混合有限元离散。我们证明了离散解的存在性，建立了精确质量守恒和离散能量不等式，并证明了全离散逼近到正则化问题弱解的收敛性。数值实验展示了所得动力学，并显示了从经典粗化向牵引诱导的指状化和突刺生长的转变。计算研究辅以质量和能量诊断、能量平衡残差、指状化起始与突刺计数诊断，以及人为解收敛性研究。
不相关论文（基本信息）
1. 三维体积分算子的通用全局正则化方法
A general-purpose global regularization method for 3D volume integral operators
作者: Thomas G. Anderson, Marc Bonnet, Luiz M. Faria, Carlos Pérez-Arancibia
类别: math.NA
发布日期: 2026-06-11
arXiv ID: 2606.13549v1
链接: http://arxiv.org/abs/2606.13549v1
摘要: Singular volume integral operators associated with constant-coefficient partial differential operators extend the applicability of potential theory to inhomogeneous problems, for example arising from nonlinearities or variable coefficients. Typically the PDE kernels in these operators give rise to singularities at all $\mathcal{O}(1/h^3)$ volume discretization/evaluation points in a mesh of characteristic size $h$, while the slowly-decaying nature of such kernels give rise to long-range interactions that require coupling to fast summation algorithms. The presented method uses Green’s identities to regularize a wide variety of both scalar-valued and vector-valued volume integral operators by use of a certain regularizing volume density interpolant. The analysis shows how the regularizing effect of the interpolant is global in the sense that the interpolation quality increases in an exactly compensatory fashion as the distance to the Green’s function singularity decreases. High-order convergence estimates with tabulated simplex quadratures are established, including with exact representation of curved domains.
2. 用于求解电场积分方程的稳定化多层B样条快速积分方法
A Stabilized Multilevel B-Spline-Based Fast Integral Method for the Solution of the Electric Field Integral Equation
作者: Danijel Jukić, Bernd Hofmann, Thomas F. Eibert, Simon B. Adrian
类别: math.NA
发布日期: 2026-06-11
arXiv ID: 2606.13482v1
链接: http://arxiv.org/abs/2606.13482v1
摘要: We present a multilevel B-spline-based fast integral method for the solution of the electric field integral equation (EFIE), combining fast Fourier transformation (FFT)-compatible kernel interpolation with robust high-order interpolation. Existing FFT-accelerated global Lagrange-based approaches rely on equidistant interpolation points and can, therefore, suffer from Runge-type instabilities at high interpolation orders, limiting robust high-accuracy compression. In contrast, B-splines on equidistant knot vectors overcome these instabilities and enable robust high-order interpolation for accurate matrix compression. Replacing Lagrange interpolation by B-spline interpolation is, however, non-trivial: B-spline coefficients do not coincide with function values at the interpolation points, and the associated sampling matrices can become ill-conditioned. To address these challenges, we introduce a knot-removal stabilization strategy, combined with exact interlevel transfers based on knot insertion, yielding accurate, well-conditioned multilevel interpolation. Moreover, we propose a factorization strategy that preserves the null space of the scalar potential operator up to machine precision and is compatible with low-frequency preconditioning techniques. Numerical results for both canonical and realistic geometries demonstrate robust high-order interpolation without the breakdown observed for Lagrange-based approaches and confirm $\mathcal{O}(N)$ complexity.
3. 在量子计算机上求解非线性微分方程的降阶基算法
Reduced basis algorithm for solving nonlinear differential equations on quantum computers
作者: Monica Lăcătuş, Matthias Möller, Sauro Succi
类别: math.NA, quant-ph
发布日期: 2026-06-11
arXiv ID: 2606.13457v1
链接: http://arxiv.org/abs/2606.13457v1
摘要: As quantum computing moves toward scientific computing applications, nonlinear differential equations remain a central challenge since quantum evolution is intrinsically linear. In this work, we introduce a reduced basis algorithm (RBA) for polynomial nonlinear ordinary differential equations (ODEs) and spatially discretized partial differential equations (PDEs). After time discretization, the method composes the resulting polynomial update map over $m$ timesteps, identifies the reduced monomial basis appearing in this composed map, and constructs a linear RBA operator whose action recovers the exact $m$-timestep nonlinear dynamics. Thus, at the level of the chosen discrete update rule, the method introduces no additional approximation error beyond the time discretization error. The qubit number requirement is governed by the size of the reduced monomial basis. For an $n$-dimensional polynomial ODE system of degree $p>1$, the lifted register requires at most $q_m^{\mathrm{ODE}} = O(nm\log p)$ qubits in the full basis scenario. For PDEs discretized on $N^D$ grid points, a locality-based construction requires at most $q_m^{\mathrm{PDE}} = O(D\log N + n m^{D+1}\log p)$ qubits. Hence, the dependence on the grid size remains logarithmic, while the nonlinear overhead is controlled by local reduced basis size. The main computational burden is moved from the quantum computer to a classical preprocessing step, where the reduced monomial basis and RBA operator are constructed for the chosen timestep window. Through numerical tests on the Lorenz system and the one-dimensional Burgers equation, we verify that the RBA reproduces the corresponding discrete time nonlinear dynamics exactly, while exposing the trade-off between timestep composition, reduced basis growth, and locality.
4. 双无公度三层石墨烯电子结构的动量空间算法
Momentum Space Algorithm for Electronic Structure of Double-Incommensurate Trilayer Graphene
作者: Ken Beard, Daniel Massatt
类别: math.NA, cond-mat.mtrl-sci
发布日期: 2026-06-11
arXiv ID: 2606.13434v1
链接: http://arxiv.org/abs/2606.13434v1
摘要: Numerical algorithms for computing electronic structure of incommensurate 2D materials using ab initio models is critical for predicting material properties and guiding experiment. For bilayers, momentum space and continuum models have been introduced to approximate observables of ab initio tight-binding models using a momenta description despite the lack of periodicity in the tight-binding model required for Bloch theory. A similar structure has been introduced for double-incommensurate trilayers using a continuum model, where the three lattices are all mutually incommensurate. However, this description leads to a four-dimensional lattice space, and numerical convergence of the density of states was observed to have poor convergence. In this work, we introduce a momentum space framework for double incommensurate trilayer graphene, and introduce an efficient truncation scheme of the four-dimensional lattice to drastically improve convergence of the density of states and momentum local density of states (a parallel object to classical band structure). We implement this algorithm on an ab initio model of twisted trilayer graphene and validate convergence estimates. We further verify numerically that the momentum space algorithm, inherently higher order than the continuum model as it is an exact transformation of the tight-binding model, captures altered band behavior near the flat bands at magic angles.
5. 一种用于时间依赖的伪应力斯托克斯问题的可扩展缩减共轭梯度求解器
A Scalable Deflated Conjugate Gradient Solver for the Time-Dependent Pseudo-Stress Stokes Problem
作者: Alessandra Cancrini, Gabriele Ciaramella, Paola F. Antonietti
类别: math.NA
发布日期: 2026-06-11
arXiv ID: 2606.13429v1
链接: http://arxiv.org/abs/2606.13429v1
摘要: We propose a novel iterative solution framework for the unsteady Stokes equations in the pseudo-stress formulation. When solving this class of problems by using implicit time-integration schemes, standard solvers suffer from deteriorating convergence properties for small time steps, independently of the chosen space discretisation method. This is due to the singular modes of the dev-dev operator. For this reason, we introduce a computational framework obtained by combining a deflated Conjugate Gradient method with a W-cycle multigrid scheme that employs a Restricted Additive Schwarz smoother. The key point is to choose the deflation subspace so that the inner system to be solved within a deflated Conjugate Gradient scheme corresponds to a Laplace problem defined on the singular modes of the original dev-dev operator. This results to be independent of the spatial discretisation method and allows one to use efficient multigrid iterative solvers. Numerical experiments show that the proposed strategy significantly accelerates the Conjugate Gradient convergence and provides stable performance with respect to the time step, confirming its robustness for solving linear systems in the pseudo-stress framework.
6. 基于围道积分的迭代特征求解器求解非线性特征值问题的线性收敛性
Linear convergence of iterative contour integral-based eigensolvers for nonlinear eigenvalue problems
作者: Daniel Kressner, Yuqi Liu, Jose E. Roman, Meiyue Shao, Nian Shao
类别: math.NA
发布日期: 2026-06-11
arXiv ID: 2606.13357v1
链接: http://arxiv.org/abs/2606.13357v1
摘要: Solving nonlinear eigenvalue problems is an important and challenging task in scientific computing. Contour integral-based approaches are attractive for such eigenvalue problems because they reliably target all eigenvalues in a prescribed domain. However, unlike in the linear case, many traditional methods of this type, such as Beyn’s method, lack an inherent iterative refinement mechanism. Consequently, achieving high accuracy requires high-quality quadrature rules for approximating the contour integral, which often leads to prohibitive computational costs. A notable exception is the so-called NLFEAST algorithm, which combines contour integral techniques with a nonlinear Rayleigh–Ritz extraction step. In this work, we propose a general framework of iterative contour integral-based methods for nonlinear eigenvalue problems that includes NLFEAST. This allows us to prove linear convergence of NLFEAST under mild assumptions and also explains why certain nonlinear eigensolvers do not combine well with iterative methods. Numerical experiments confirm our theoretical findings; in particular that NLFEAST can achieve high accuracy even with a limited number of quadrature nodes, significantly outperforming Beyn’s method on challenging problems.
7. 关于最坏情形下线性张量积问题的代数$(s, t)$-弱可处理性的一个注记
A Note About Algebraic $(s, t)$-Weak Tractability Of Linear Tensor Product Problems In The Worst-Case Setting
作者: Zirong Liu, Heping Wang
类别: math.NA
发布日期: 2026-06-11
arXiv ID: 2606.13339v1
链接: http://arxiv.org/abs/2606.13339v1
摘要: This paper is devoted to discussing the linear tensor product problems in the worst case setting. We consider algorithms that use finitely many evaluations of arbitrary continuous linear functionals. We investigate algebraic $(s, t)$-weak tractability (ALG-$(s, t)$-WT) under the absolute error criterion in the case $λ_1 > 1$, where $λ_1$ is the square of the univariate maximal singular value. We solve the problem by giving the necessary and sufficient conditions for ALG-$(s, t)$-WT on univariate singular values and fill the gap left open.
8. 基于嵌入方法的线性求解器性能预测
Embedding-based Methods for Linear Solver Performance Prediction
作者: Hayden Liu Weng, Hans-Joachim Bungartz, Felix Dietrich
类别: cs.CE, math.NA
发布日期: 2026-06-11
arXiv ID: 2606.13255v1
链接: http://arxiv.org/abs/2606.13255v1
摘要: The solution of large, sparse linear systems often dominates the computational effort of scientific applications and is a frequent optimization target. Modern libraries provide numerous solver and preconditioner configurations, but their performance varies significantly across problem instances. Previous works have addressed the selection of an optimal solver, but are typically limited by the problem set addressed (e.g., only symmetric positive definite matrices), the use of expensive matrix features, or the complexity of the approach. This work proposes a modular, low-cost embedding-based framework for solver selection that decouples performance modeling from feature representation and downstream prediction. Solver-problem relationships are learned directly from observed performance data, while inexpensive numerical features are used to project unseen problems into the learned embedding space. The framework focuses on multilabel prediction and evaluation using user-centric metrics, such as MAPE and nDCG, which better reflect relative performance. Experiments on 621 matrices from the SuiteSparse matrix collection across 101 PETSc solver configurations demonstrate a 17% increase in top-prediction accuracy over classical feature-based models when expensive numerical features are included, along with reductions of 37% in mean average percentage error (MAPE) and 46% in top-prediction error (1-error). When restricted to a reduced feature set, the embedding approach remains competitive, while still consistently achieving ca. 24% lower MAPE and 1-error across a broad range of problems.
9. SCI层级中的有限查询坍缩与模态精确基
Finite-Query Collapse and Modal Exact Bases in the SCI Hierarchy
作者: Christopher Sorg
类别: math.LO, math.NA, math.SP
发布日期: 2026-06-11
arXiv ID: 2606.13250v1
链接: http://arxiv.org/abs/2606.13250v1
摘要: We study the exact-basis problem for Solvability Complexity Index (SCI) computational problem families through finite-query transports. A raw finite-query reduction permits arbitrary encodings and finite transcript reconstructions, with only a continuous output decoder. For the Colbrook-Hansen (CH23) singleton-window spectral/pseudospectral block, this raw preorder collapses the expected two-source structure: the diagonal exact spectral and fixed-$\varepsilon$ pseudospectral sources are raw- and continuous-finite-query equivalent, and, for computable $\varepsilon$ under the evaluation-name representations, TTE-finite-query equivalent, so the six-problem ambient is raw-principal. We then introduce modal finite-query preorders, whose admissibility conditions may restrict encodings, decoders, reconstructions, uniformity, and geometric naturality. We also characterize TTE finite-query transport as computable point transport with a uniform finite interface trace; after forgetting the trace this gives strong Weihrauch reducibility, and the implication is strict. Under a CH23 geometric modality generated by representation inclusions, unitary and graph relabelings, and neutral stabilizations, the same ambient has exactly two minimal exact sources. This gives a calibrated reformulation of the exact-basis problem: natural SCI families should be classified by modality-indexed exact bases and refinement maps, not by one raw preorder alone.
10. 用于混合分布抽样的切换哈密顿蒙特卡洛方法
Switching Hamiltonian Monte Carlo for sampling from mixture distributions
作者: A. Sharma
类别: stat.CO, math.NA, math.ST
发布日期: 2026-06-11
arXiv ID: 2606.13234v1
链接: http://arxiv.org/abs/2606.13234v1
摘要: We introduce a switching Hamiltonian Monte Carlo method for sampling from finite mixture Boltzmann-Gibbs distributions. We propose symmetric numerical integrators to approximate switching Hamiltonian dynamics interlaced with Poisson jumps, where the regime-switching chain is simulated using the uniformization technique or the stochastic simulation algorithm. We prove geometric ergodicity of the resulting Markov chain. We develop an approach based on the discrete Poisson equation associated with numerical schemes to estimate the error in computing ergodic averages. Using this approach we prove that the proposed numerical integrators have second-order bias. This approach is simple and can be generalized to other settings, for example, kinetic Langevin equations. Finally, we verify the convergence result via numerical experiment.
11. 带状加半可分矩阵的 QR 分解可用线性复杂度计算
The QR factorization of a Banded-Plus-Semiseparable Matrix is Computable with Linear Complexity
作者: Tao Chen, Sheehan Olver
类别: math.NA
发布日期: 2026-06-11
arXiv ID: 2606.13225v1
链接: http://arxiv.org/abs/2606.13225v1
摘要: We show that the QR factorization of a banded-plus-semiseparable (BPS) matrix is computable in optimal linear complexity with respect to the discretization size by showing that the intermediate stages of a QR factorization as computed using Householder reflection maintain a specific structure which has optimal storage. This theoretical result enables the design of stable, linear-complexity algorithms for solving the associated linear systems. For symmetric BPS matrices, we further show that the $RQ$ product – central to eigenvalue computations via the QR algorithm – also preserves the BPS structure, leading to a linear-complexity algorithm for each iteration. Numerical experiments validate the optimal linear complexity, confirm high numerical accuracy, and demonstrate substantial speedups compared with existing hierarchical approaches. The algorithms have been implemented in an open-source Julia package, providing an efficient and accessible platform for practical use.
12. 在实验室规模搅拌槽反应器中追踪计算机模拟的拉格朗日传感器
Tracking in-silico Lagrangian sensors in a lab-scale stirred tank reactor
作者: Vamika Rathi, Fatima Sehar, Finn Sommer, Sebastian Goetschel, Eike Steuwe, Alexandra von Kameke, Daniel Ruprecht
类别: math.NA
发布日期: 2026-06-11
arXiv ID: 2606.13099v1
链接: http://arxiv.org/abs/2606.13099v1
摘要: Lagrangian sensors have shown promise to improve operator awareness of conditions inside a chemical reactor but three-dimensional tracking remains a mostly unsolved challenge. We explore a setup where in-silico sensors, based on a recently proposed real-world design, are tracked using data from an accelerometer and magnetometer available from a built-in inertial measurement unit. Filtering algorithms, using a bespoke dynamical model, are used to process these readings into position estimates. We compare tracking performance of an extended Kalman filter, a particle filter and the unscented Kalman filter implemented in the pykalman library. Our numerical experiments track in-silico particles moving in an analytically given three dimensional vortex as well as in the experimentally measured flow-field of a lab-scale stirred tank reactor. Using the Maxey-Riley-Gatignol equations for the movement of inertial particles as ground-truth, we demonstrate that trajectories can be reconstructed from noisy synthetic data with errors below 10%.
13. 一种二次降阶–高斯过程常微分方程框架用于大型连续动力系统的推断
A Quadratic Order Reduction – Gaussian Process Ordinary Differential Equation framework for the inference of Large Continuous Dynamical Systems
作者: Guglielmo Padula, Michele Girfoglio, Gianluigi Rozza
类别: math.NA, stat.ML
发布日期: 2026-06-11
arXiv ID: 2606.13063v1
链接: http://arxiv.org/abs/2606.13063v1
摘要: Forecasting the evolution of complex dynamical systems remains a fundamentally challenging task, primarily due to pronounced nonlinear interactions, high-dimensional state spaces, and the concomitant requirement for rigorous and reliable uncertainty quantification. Contemporary reduced-order modelling (ROM) frameworks frequently exhibit inherent trade-offs among predictive accuracy, numerical stability, and interpretability, and thus often fail to achieve an optimal balance among these competing objectives. To address these limitations, we propose a framework for forecasting complex dynamical systems via a kernel autonomous ordinary differential equation approach based on Gaussian Processes and Quadratic Order Model Reduction. Our base method, the Gaussian Process Ordinary Differential Equations model, allows accurate short-term forecasting with uncertainty quantification, and it provably converges to the real autonomous equation in the smooth case. We integrate it with quadratic order reduced-order modelling and sphere projection for learning the latent dynamics efficiently while preserving stability. Numerical experiments demonstrate that our full model outperforms ROM forecasting methods such as Extended Dynamic Mode Decomposition, Bagging Optimised Dynamic Mode Decomposition and Linear and Nonlinear Disambiguation Optimisation in terms of accuracy or computational costs. These results demonstrate the potential of the framework as a robust and stable tool for forecasting complex dynamical systems with rigorous uncertainty quantification.
14. 正交多项式具有任意度步长的三项递推关系
Three-term Recurrence Relation with Arbitrary Degree Step for Orthogonal Polynomials
作者: Bo Yang
类别: math.NA
发布日期: 2026-06-11
arXiv ID: 2606.13056v1
链接: http://arxiv.org/abs/2606.13056v1
摘要: An approach to generate three-term recurrence relations with arbitrary degree step is proposed. Specifically, given any class of orthogonal polynomials $\{Q_{p}(x)\}_{p=0}^{\infty}$ defined by Favard’s theorem, we employ the adjacent members $Q_{p}(x)$ and $Q_{p-1}(x)$ to compute the one of high degree $Q_{p+s}(x)$ and that of low degree $Q_{p-t}(x)$, where $(s,t)\in \mathbb{N^{+}}$. Therefore, it is able to derive a three-term recurrence relation with respect to $Q_{p+s}(x)$, $Q_{p}(x)$ and $Q_{p-t}(x)$ by taking $Q_{p-1}(x)$ as the bridge. Moreover, as the extensions of standard recursive formula which is characterized with degree increase, the ones for degree decrease and end-to-middle directions are formulated as well. The explicit recurrence relations with two-degree step are offered for Hermite, Gegenbauer and Legendre polynomials. Precision comparison is also performed between the standard three-term recurrence relation and the proposed ones.
15. 用于SIDER插值局部精度的四元数–BCH框架
A Quaternion–BCH Framework for the Local Accuracy of SIDER Interpolation
作者: Shingyu Leung
类别: math.NA
发布日期: 2026-06-11
arXiv ID: 2606.12833v1
链接: http://arxiv.org/abs/2606.12833v1
摘要: Spherical Interpolation of orDER $n$ (SIDER-$n$) is a recursive high-order interpolation method for data on the unit sphere $\mathbb{S}^2$, built from repeated spherical linear interpolation (SLERP). This paper develops a quaternion–Lie algebra framework for proving the local consistency of SIDER for smooth spherical curves sampled at equally spaced parameter values. Points on $\mathbb{S}$ are represented as pure unit quaternions, and interpolation errors are measured in fixed-base quaternion logarithmic coordinates. In this setting, each SLERP operation admits an exact Baker–Campbell–Hausdorff (BCH) representation, which converts the geometric interpolation problem into an algebraic problem involving filtered Lie-polynomial expansions. The BCH expansion shows that SLERP is affine to leading order, has no quadratic correction, and has a first nonlinear correction that is cubic and commutator-valued. Using this structure, we prove that SIDER2 has a third-order divided-error form with the same leading nodal factor as ordinary quadratic interpolation. We then show that the recursive SIDER step raises the order by one: the affine part gives the Neville-type finite-difference cancellation, while the nonlinear BCH remainder preserves the sharp filtered degree structure after the nodal factor is removed. Consequently, for every fixed $n\geq2$, $d_{\mathbb{S}^2}\bigl(γ(θh),P_i^{[n]}(θ;h)\bigr) = O(h^{n+1}) $under the stated smoothness and small-stencil assumptions. The proof also identifies the shift-invariance of the leading divided-error coefficient as the algebraic compatibility condition underlying the SIDER recurrence.
16. 球面插值的局部一致性与高阶结构
Local Consistency and Higher-Order Structure of Spherical Interpolation
作者: Shingyu Leung
类别: math.NA
发布日期: 2026-06-11
arXiv ID: 2606.12802v1
链接: http://arxiv.org/abs/2606.12802v1
摘要: Spherical Interpolation of orDER $n$ (SIDER-$n$) is a recursive high-order interpolation construction for data on the unit sphere $\mathbb{S}^2$, built from repeated spherical linear interpolation (SLERP). This paper gives a local consistency analysis of SIDER for smooth spherical curves sampled at equally spaced parameter values. The analysis is carried out in geodesic normal coordinates, which allows the SIDER recursion to be compared with classical Neville interpolation while retaining the curvature-dependent corrections introduced by SLERP. We first derive local expansions of SLERP and show that SIDER2 has third-order accuracy; its leading error has the same shifted nodal structure as Euclidean quadratic interpolation. We then prove that the adjacent SIDER2 errors entering SIDER3 have a common leading coefficient, so that the SIDER3 recurrence cancels the cubic term and yields fourth-order accuracy. Carrying the expansion one order further gives the corresponding coefficient compatibility for SIDER3 and proves fifth-order accuracy of SIDER4. Finally, we introduce a degree-filtered formal expansion framework for the general SIDER recursion. This framework proves that, for each fixed $n$, SIDER-$n$ preserves the required polynomial degree structure in the normalized stencil variable. Together with the interpolation conditions at the $n+1$ nodes, this yields the local consistency estimate $d_{\mathbb{S}^2}\bigl(γ(θh),P_i^{[n]}(θ;h)\bigr)=O(h^{n+1})$ under the stated smoothness and small-stencil assumptions.
17. 大规模并行流路由与流域面积确定
Massively parallel flow routing and drainage area determination
作者: Wolfgang Bangerth
类别: math.NA
发布日期: 2026-06-11
arXiv ID: 2606.12800v1
链接: http://arxiv.org/abs/2606.12800v1
摘要: Digital elevation models (DEMs) have reached resolutions and sizes that only parallel computaters can efficiently process. One important application of DEMs is predicting how much water flows where, the so-called ``flow routing problem’’ (a variation of which is the problem of determining the drainage area upstream of a point in a DEM). The traditional algorithm for flow routing is sequential, and attempts to parallelize this method have so far only been moderately successful. Herein, we build on earlier work in Richardson et al. (2014) and propose an algorithm and several variations that can efficiently solve the flow routing problem on very large models with very large numbers of parallel processes. For the largest model we use, with 1.88 billion points, the best algorithm herein can route water in 4.0 seconds on 12,288 processes of a computer cluster.
18. 时间分数阶Allen-Cahn方程的变时间步长二阶保MBP线性稳定化格式
A variable time-step, second-order, and MBP-preserving linear stabilized scheme for the time-fractional Allen-Cahn equation
作者: Bingyin Zhang, Ao Zhang, Hongfei Fu
类别: math.NA
发布日期: 2026-06-11
arXiv ID: 2606.12799v1
链接: http://arxiv.org/abs/2606.12799v1
摘要: In this paper, we present a second-order linear scheme based on the variable-step Alikhanov formula and central difference discretization for the time-fractional Allen-Cahn equation. The nonlinear potential is treated explicitly via a second-order extrapolation with preprocessing, which enables the discrete maximum-bound principle (MBP) to be preserved through an appropriate stabilization technique. Moreover, by developing a discrete fractional Grönwall inequality together with the uniform boundedness of numerical solutions guaranteed by the MBP, we establish an $α$-robust and optimal second-order maximum-norm error estimate under initial weak singularity assumption. In addition, energy stability is proved in the sense that the discrete original energy is uniformly bounded by the initial energy plus a high-order spatiotemporal correction term. Finally, extensive numerical experiments are presented to demonstrate the effectiveness of the proposed scheme.
19. 论广义层次样条空间的完备性
On the completeness of generalized hierarchical spline spaces
作者: Ahmed Oufqir, Carla Manni, Hendrik Speleers
类别: math.NA
发布日期: 2026-06-10
arXiv ID: 2606.12656v1
链接: http://arxiv.org/abs/2606.12656v1
摘要: We introduce a general theoretical approach to hierarchical spline spaces that replaces the classical constructive definition - based on basis selection - with a descriptive formulation in terms of regularity constraints. Specifically, we define generalized hierarchical spline spaces on multi-level domains as collections of piecewise functions satisfying hierarchical contact conditions across interfaces between refinement levels. The proposed framework applies to a broad class of local function spaces and relies on a minimal abstract requirement, the extension assumption, rather than on specific polynomial properties. Within this framework, we identify rules under which the hierarchical selection mechanism yields a complete basis, in the sense that it spans exactly the space characterized by the contact conditions. As an application, we consider Tchebycheffian spline spaces. We show that spaces generated by extended complete Tchebycheff (ECT) systems fit in this framework, thereby establishing the completeness of hierarchical Tchebycheffian splines. This demonstrates that the proposed theory naturally extends beyond the polynomial setting and provides a unified foundation for hierarchical constructions in more general spline spaces.
20. 基于多项式空间上结合截断乘积的守恒律保双曲性随机伽辽金方法
Hyperbolicity-Preserving Stochastic Galerkin Methods for Conservation Laws Based on Associative Truncated Products on Polynomial Spaces
作者: Haroun Meghaichi, Yulong Xing
类别: math.NA
发布日期: 2026-06-10
arXiv ID: 2606.12632v1
链接: http://arxiv.org/abs/2606.12632v1
摘要: Stochastic Galerkin discretizations of nonlinear hyperbolic conservation laws may lose hyperbolicity because the standard pseudospectral product is generally nonassociative, leading to non-commuting blocks in the flux Jacobian matrix. We develop a novel framework for constructing hyperbolicity-preserving stochastic Galerkin systems based on associative truncated products on polynomial spaces. In one stochastic dimension, we characterize associative truncated products through a single polynomial datum and identify examples with useful symmetry, positivity, and spectral properties, including collocation products and an associative symmetric product based on Gaussian quadrature nodes. We prove a consistency result showing that, under suitable projection-error assumptions, these products converge to the classical product as the polynomial degree grows. For systems with rational fluxes, we derive sufficient conditions under which the resulting stochastic Galerkin flux remains hyperbolic on the corresponding admissible set. Applications to the one-dimensional isothermal and compressible Euler equations show accurate statistical approximation and robust hyperbolicity preservation of the computed stochastic Galerkin states.
21. 多重网格的级联光滑子
Cascading Smoothers for Multigrid
作者: Robert I. Saye
类别: math.NA
发布日期: 2026-06-10
arXiv ID: 2606.12577v1
链接: http://arxiv.org/abs/2606.12577v1
摘要: Multigrid methods are among the most effective frameworks for solving large-scale sparse systems. However, achieving their hallmark linear scaling and rapid convergence crucially depends on an effective smoother algorithm, whose design is often highly problem-dependent. This paper develops a new approach, referred to as \textit{cascading smoothers} due to their operation as an ordered sequence of single-step block-diagonal smoothers. Each level in the cascade is optimised to maximally damp the output of prior steps via a Frobenius norm minimisation of the corresponding error propagators. In particular, we develop an additive (resp., multiplicative) formulation analogous to Jacobi (resp., Gauss-Seidel). Applied within a standard multigrid V-cycle, we show they are remarkably effective across a wide array of problems, including finite difference, finite element, and discontinuous Galerkin discretisations applied to Poisson, elliptic interface, and Stokes systems as well as multiphase variants. In every case, cascading smoothers closely match or significantly outperform their optimally-damped classical counterparts, yet require no parameter tuning apart from a few discrete solver choices. Additionally, the approach is highly parallelisable and robust to geometric and operator complexities such as unstructured meshes and high-contrast coefficients.
22. 用于含激波流动数据同化的特征保持潜在集合卡尔曼滤波
Feature-preserving Latent-EnKF for Data Assimilation of Flows with Shocks
作者: Hemanth Chandravamsi, Hangchuan Hu, Ponkrshnan Thiagarajan, Tamer A. Zaki
类别: physics.comp-ph, cs.LG, math.NA, physics.flu-dyn
发布日期: 2026-06-10
arXiv ID: 2606.12559v1
链接: http://arxiv.org/abs/2606.12559v1
摘要: The ensemble Kalman filter (EnKF) is widely adopted for sequential data assimilation, but fails for solutions with discontinuities, such as shocks in compressible flows. Uncertainty in shock location induces multimodal ensemble statistics that violate the Gaussian assumptions underlying the EnKF, producing large-scale spurious oscillations in the analysis state. We introduce a feature-preserving latent-EnKF that performs the ensemble update in a learned low-dimensional latent space, where shock and flow features admit a smooth manifold representation, thereby preserving sharp features during EnKF analysis. The updated latent state is mapped back to physical state through a shared decoder for all ensemble members. The algorithm eliminates the member-specific ordered training and positivity flooring used in prior approaches. Numerical experiments on a Sod shock tube and Mach 2 shock interaction with a 2D cylinder, using sparse and noisy observations, show accurate feature recovery of shocks and contact discontinuities without spurious oscillations.