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1. 基于主导分量的时间依赖多尺度高对比度问题的两级预条件器
A Two-Level Preconditioner Based on Dominant Components for Time-Dependent Multiscale High-Contrast Problem
作者: Yating Wang, Yibao Li, Wing Tat Leung
类别: math.NA
发布日期: 2026-08-28
arXiv ID: 2608.27958v1
摘要翻译: 本文中，我们为高对比度多尺度介质中的时间依赖问题开发了一种两级重叠预条件器。我们提出了一种基于多尺度方法的粗空间构造，重点放在松弛非局部多连续统（NLMC）方法上。NLMC空间可以分解为代表高渗透率区域和低渗透率背景的分量。我们表明，完整的NLMC空间能有效地预处理异质刚度算子，而仅高渗透率分量就能捕获对比度依赖的全局模态。对于适当的小时间步长，质量矩阵控制低渗透率贡献，在两级预条件器中仅需NLMC空间中的高渗透率分量即可进行全局粗校正。对于一般时间步长，可以通过使用完整的NLMC空间或用标准多尺度空间扩充高渗透率分量来保持性能。所提出的粗空间构造降低了计算成本，同时保持了关于系数对比度和细尺度分辨率的稳健性。为了进一步提高效率，可以通过迭代求解松弛能量最小化公式来构造多尺度基函数。我们通过几个数值实验证明了所提出方法的稳健性、效率和可扩展性。
Comment: 无
链接: http://arxiv.org/abs/2608.27958v1
1. 研究对象和背景
本文研究时间依赖的多尺度高对比扩散问题. 设 $\Omega\subset\mathbb{R}^d$ ($d=2,3$), 渗透率或热导率 $\kappa\in L^\infty(\Omega)$ 满足 $0<\kappa_{\min}\le \kappa(x)\le \kappa_{\max}$, 对比度 $\eta:=\kappa_{\max}/\kappa_{\min}\gg1$. 方程为
$$\partial_t u-\nabla\cdot(\kappa\nabla u)=f\quad\text{in }\Omega\times(0,T],$$配以零 Dirichlet 边界条件 $u=0$ 于 $\partial\Omega\times[0,T]$ 和初始条件 $u|_{t=0}=u_0$. 这类方程描述地下渗流、污染物输运以及复合材料中的热传导. 由于 $\kappa$ 在细小空间尺度上剧烈变化, 并形成高渗透通道、裂缝或长导电通道, 精确数值模拟需要在细网格上离散, 导致每个时间步需要求解大规模且严重病态的代数系统.
采用全隐式 Euler 时间离散后, 问题化为
$$(\Delta t A+M)U^{n+1}=MU^n+\Delta t F,$$其中 $A$ 和 $M$ 分别是细网格刚度矩阵和质量矩阵, $\Delta t$ 为时间步长. 已知 $A$ 的条件数随对比度 $\eta$ 和 $h^{-2}$ 增长, 其中 $h$ 为细网格尺寸. 经典的多项式粗空间无法表示高渗透通道产生的对比度相关近零空间模态, 因此需要基于多尺度信息的粗空间. 已有的 GMsFEM、CEM-GMsFEM、NLMC 等方法的粗空间构造和理论分析大多面向稳态椭圆算子或定常不可压流系统, 针对时间依赖问题全隐式线性系统的系统化粗空间构造较少. 本文试图填补这一空白: 利用质量矩阵的作用, 为 $\Delta t A+M$ 构造只含主导分量的两级重叠预条件子, 降低粗空间维数和离线构造代价, 同时保持对 $\eta$ 和细网格分辨率的鲁棒性.
2. 主要定理或主要结果
记 $\mathcal{T}^H$ 为粗网格, $V_H$ 为松弛 NLMC 空间. 对每个粗单元 $K$, 先定义辅助空间 $V_{\rm aux}^K=V_{\rm aux,1}^K+V_{\rm aux,2}^K$, 其中 $V_{\rm aux,1}^K=\operatorname{span}\{\chi_{K_i^h}:K_i^h\subset K\}$ 对应高渗透连通分量, $V_{\rm aux,2}^K=\operatorname{span}\{\chi_{K^l}\}$ 对应低渗透背景. 设 $\pi_K:H^1(K)\to L^2(K)$ 满足
$$\|(I-\pi_K)u\|_{s(K)}^2\le C_\pi\|u\|_{a(K)}^2,$$则松弛 NLMC 基函数定义为
$$\phi_{j,K}=\arg\min_{\phi\in V_{h,0}}\{a(\phi,\phi)+s(\pi(\phi)-\psi_{j,K},\pi(\phi)-\psi_{j,K})\},$$等价于变分问题 $a(\phi_{j,K},v)+s(\pi\phi_{j,K},\pi v)=s(\psi_{j,K},\pi v)$ 对所有 $v\in V_{h,0}$ 成立. 令 $V_H=V_{H,1}+V_{H,2}$, 其中 $V_{H,1}$ 由高渗透通道或裂缝对应的基函数生成, $V_{H,2}$ 表示低渗透矩阵连续介质.
论文的主要结论如下. (1) 完整 NLMC 空间 $V_H$ 可作为异质刚度算子 $A$ 的有效粗空间; $V_{H,1}$ 单独即可捕捉对比度相关的全局模态. (2) 对两级重叠 Schwarz 预条件子
$$\tilde A_{\rm pre}^{-1}=R_0^T\tilde A_0^{-1}R_0+\sum_{i=1}^N R_i^T\tilde A_i^{-1}R_i,\quad \tilde A=\Delta tA+M,$$当 $\Delta t\lesssim H^2$ 时, 质量矩阵 $M$ 控制低渗透部分对刚度矩阵的贡献, 因此粗空间只需取 $V_{H,1}$ 即可; 对一般时间步长, 应取 $V_H=V_{H,1}+W_H$, 其中 $W_H$ 可以是 $V_{H,2}$ 或其他标准多尺度有限元空间. (3) 基函数可通过迭代求解构造: 令 $A_{S,\rm loc}^{-1}=\sum_i R_i^T(A_i+S_i)^{-1}R_i$, 其中 $S$ 为 $s(\pi\cdot,\pi\cdot)$ 对应的矩阵, 定理 1 证明
$$\operatorname{cond}(A_{S,\rm loc}^{-1}(A+S))=O(1),$$常数与对比度 $\eta$, 细网格尺寸 $h$ 和粗网格尺寸 $H$ 均无关. 因此可用 PCG 迭代求解基函数. 在假设 1 下, 若近似粗空间 $\hat V_H^{(m)}$ 满足
$$|||v_0-T_mv_0|||\le\varepsilon_m\|v_0\|_{\tilde a},\quad \varepsilon_m<\varepsilon_*<1,$$则由指数收敛性, $m=O(\log\eta)$ 个迭代步即可满足要求. 数值实验展示了方法对对比度、细网格分辨率的鲁棒性, 以及并行可扩展性.
3. 研究方法、关键技术和核心工具
核心数学障碍在于高对比度会产生沿高渗透通道传播的全局近零空间模态, 这些模态既不能被多项式粗空间局部化, 又使刚度矩阵条件数随 $\eta$ 增长; 同时, 多尺度粗空间的离线构造通常需要求解大范围 oversampling 区域上的局部问题, 代价很高. 作者真正的创新点不是简单地引入更多谱模态, 而是将 NLMC 空间按物理连续介质分为高渗透分量 $V_{H,1}$ 和低渗透分量 $V_{H,2}$, 并利用时间离散中质量矩阵 $M$ 的局部控制作用: 在 $\Delta t\lesssim H^2$ 时, $M$ 已经对低渗透区域的刚度贡献提供了足够控制, 因而全局粗校正只需保留 $V_{H,1}$. 这体现了多尺度逼近与多尺度预条件之间的本质区别: 直接降阶模型需要为每个连续介质配置基函数, 但预条件子中若离散算子的其余部分已控制相应模态, 则可以省略该基函数.
证明的核心是定理 1 的稳定分解论证, 而非泛泛的能量估计. 定义 $b(u,v)=a(u,v)+s(\pi u,\pi v)$, 则 $\|u\|_b^2=\|u\|_a^2+\|\pi u\|_s^2$. 首先由辅助空间逼近性质 (3.1) 可得
$$\|u\|_s^2\le2\|\pi u\|_s^2+2C_\pi\|u\|_a^2\le C\|u\|_b^2.$$对任意 $u\in V_h$, 取 $u_i=I_h(\chi_i u)\in V_i$, 其中 $\chi_i$ 是满足 $\sum_i\chi_i=1$, $|\nabla\chi_i|\le CH^{-1}$, $\sum_i\chi_i^2\le1$ 的单位分解, $I_h$ 为满足稳定性 $\|I_h(v)\|_a\le C_I\|v\|_a$, $\|I_h(v)\|_s\le C_I\|v\|_s$ 的节点插值. 利用 $\nabla(\chi_i u)=\chi_i\nabla u+u\nabla\chi_i$ 和上述不等式, 得到
$$\sum_i b(u_i,u_i)\le C_{sd}b(u,u).$$另一方面, 由 $u=\sum_i u_i$ 及局部支撑的有限重叠性质, 有
$$b(u,u)\le C_{ov}\sum_i b(u_i,u_i).$$合并两式即得 $\operatorname{cond}(A_{S,\rm loc}^{-1}(A+S))\le C_{sd}C_{ov}$, 且常数与 $h,H,\eta$ 无关. 这保证了用局部子域求解器作为预条件子时, PCG 迭代求解基函数的收敛性不依赖于对比度.
在粗空间分析中, 作者引入 $s$-正交投影 $\pi_1,\pi_2$ 使得 $\pi=\pi_1+\pi_2$, 并定义 $V_{H,1}$ 上的投影 $P_1$ 满足 $s(\pi_1P_1u,\pi_1v)=s(\pi_1u,\pi_1v)$, 由此导出 $a(P_1u,P_1u)+s(\pi P_1u,\pi P_1u)=a(P_1u,u)+s(\pi P_1u,\pi u)$. 这类恒等式用于说明 $V_{H,1}$ 能够捕获主导模态. 基函数离线构造采用松弛 NLMC 的罚函数形式, 用 $s(\pi\phi-\psi,\pi\phi-\psi)$ 代替严格正交约束, 从而允许在较小的 oversampling 区域上计算; 再以 $A+S$ 为系统矩阵、$A_{S,\rm loc}$ 为预条件子做 PCG 迭代. 对于最终两级预条件子的条件数估计, 提供内容中未展示显式定理; 作者通过数值实验验证其鲁棒性.
4. 与之前工作的比较
与既有工作的比较可从以下几个层面展开.
第一, 与 GMsFEM 类两级预条件子相比. 已有工作如文献 [17,14,15,31,32] 利用 GMsFEM 局部特征函数或谱模态构造粗空间, 在稳态椭圆问题中可获得与对比度无关的条件数上界. 但这些构造大多针对 $A$ 本身, 且较少处理时间依赖问题的全隐式矩阵 $\Delta tA+M$. 本文将时间项纳入粗空间设计, 明确指出质量矩阵的存在使得所需粗空间可以显著小于纯刚度矩阵所需粗空间, 这是对稳态情形的重要推进.
第二, 与 CEM-GMsFEM [13] 的比较. CEM-GMsFEM 通过严格正交约束和充分大的 oversampling 区域得到接近 1 的条件数, 但需要在扩大后的区域上求解约束能量极小化问题, 离线计算成本较高. 本文采用松弛 NLMC 罚形式, 以罚项代替硬约束, 可在较小 oversampling 区域上构造基函数; 进一步地, 定理 1 对迭代构造基函数时的局部预条件系统给出 $O(1)$ 条件数, 因此基函数可以通过 PCG 迭代高效求解. 从收敛性角度看, 松弛处理保留了对比度无关的逼近性质, 但代价是基函数不再严格满足正交约束, 需要指数收敛性假设 (假设 1) 保证有限次迭代后的粗空间仍然有效.
第三, 与原始 NLMC 方法作为降阶模型的比较. NLMC 方法通常为每个物理连续介质构造独立的粗变量和基函数, 并在 reduced approximation space 中直接求解粗尺度问题. 本文的视角不同: NLMC 基函数被用作预条件子的粗空间, 而不是逼近空间. 作者强调, 一个基函数对于直接降阶模型可能是必要的, 但当离散算子中的质量矩阵或其他局部求解器已经对低渗透背景提供足够控制时, 它在预条件子中不一定是必要的. 对于 $\Delta t\lesssim H^2$, 低渗透分量 $V_{H,2}$ 完全可以不进入全局粗空间; 对于一般时间步长, 使用完整 $V_H=V_{H,1}+W_H$, 其中 $W_H$ 可取 $V_{H,2}$ 或标准多尺度有限元空间. 这放宽了以往多尺度预条件子必须包含全部连续介质模态的隐含假设.
第四, 代价与局限. 当高渗透区域体积较小时, 只保留 $V_{H,1}$ 能显著降低全局粗问题的维数和计算成本; 若高渗透区域体积较大, $V_{H,1}$ 本身维数也会增大, 降维收益受限. 此外, 迭代构造基函数的效率依赖指数收敛性, 实际迭代次数约为 $O(\log\eta)$, 仍需在实现中选取合适的阈值. 对于最终两级预条件子 $\tilde A_{\rm pre}^{-1}(\Delta tA+M)$ 的条件数是否具有显式理论界, 从提供的 PDF 摘录中无法确认; 作者主要通过数值实验说明其鲁棒性、效率和可扩展性.
了解领域论文（摘要翻译）
1. QGPINNs：量子图上非局部微分方程的物理信息神经网络框架
QGPINNs: A Physics-Informed Neural Network Framework for Nonlocal Differential Equations on Quantum Graphs
作者: Vaibhav Mehandiratta, Saket Ramchandra
类别: cs.LG, math.NA
发布日期: 2026-08-28
arXiv ID: 2608.28589v1
链接: http://arxiv.org/abs/2608.28589v1
摘要翻译
中文标题: QGPINNs：量子图上非局部微分方程的物理信息神经网络框架
摘要翻译: 我们提出QGPINNs，一个在PyTorch中开发的物理信息神经网络框架，用于数值求解量子图上的非局部微分方程。该框架被设计为一个通用的计算实现，其中图上的每条边上的解由神经网络逼近，而一个统一的基于图的损失函数同时强制执行控制方程以及初始条件、边界条件和顶点传输条件。特别地，该公式将标准连续性条件、Kirchhoff-Neumann顶点条件和Dirichlet边界条件纳入学习过程，从而将局部的逐边神经逼近耦合为图上的全局解。该框架针对两类具有代表性的非线性模型开发：量子图上的多阶分数阶椭圆问题和时间分数阶演化方程。为了提高精度和训练稳定性，QGPINNs集成了多种图自适应学习策略，包括软约束和硬约束强制执行、动态损失平衡、Fourier特征嵌入，以及一个可学习的奇异性捕获特征，用于处理所考虑问题中出现的弱奇异解。该框架还自然地扩展到反问题，包括从含噪声的观测数据中识别分数阶算子的阶数和物理参数。我们通过在基准图结构和真实网络（包括IEEE 14节点系统和一个明渠农业排水网络）上的数值实验，验证了所提出框架的精度、计算效率和物理一致性。
2. 神经算子的保形不确定性量化保证
Conformal Uncertainty Quantification Guarantees for Neural Operators
作者: Tom Stent, Nicolas Boullé
类别: math.NA, cs.AI, math.PR
发布日期: 2026-08-28
arXiv ID: 2608.28515v1
链接: http://arxiv.org/abs/2608.28515v1
摘要翻译
中文标题: 神经算子的保形不确定性量化保证
摘要翻译: 神经算子为逼近函数空间之间的算子提供了快速替代模型，但其预测往往缺乏不确定性量化。我们开发了一个分裂保形框架，以保证在至少 $1-γ$ 的评估域内，神经算子输出周围的校准逐点带包含真实解，且对测试和校准输入的概率至少为 $1-α$，其中 $α,γ\in(0,1)$。我们的方法将归一化残差场简化为其空间上的 $(1-γ)$ 分位数，并使用留出的校准数据集计算缩放因子。我们证明了定义在任意概率空间上的可测残差场的边际覆盖保证，涵盖了连续域和固定离散化。在对数据分布的温和假设下，我们表明以校准集为条件的覆盖率服从 Beta 分布，并通过 Darcy 流和 Navier–Stokes 方程的数值实验验证了这一点，其中我们的校准产生的带始终比现有校正更紧，同时保持目标覆盖率。
3. 修正的Camassa–Holm方程的一种局部间断Galerkin方法
A Local Discontinuous Galerkin method for the modified Camassa–Holm equation
作者: Xiang-Ke Chang, Yong Liu, Qi Tao
类别: math.NA
发布日期: 2026-08-28
arXiv ID: 2608.28077v1
链接: http://arxiv.org/abs/2608.28077v1
摘要翻译
中文标题: 修正的Camassa–Holm方程的一种局部间断Galerkin方法
摘要翻译: 在本文中，我们针对修正的Camassa-Holm（mCH）方程提出了一种局部间断Galerkin（LDG）方法，该方程包含三次非线性和高阶导数项。利用守恒通量和耗散通量分别得到了能量守恒性和稳定性。还建立了半离散格式的误差估计。数值算例验证了我们的理论结果是精确的，并且所提出的LDG方法能够捕获mCH方程的peakon解。
4. 奇异Zienkiewicz四面体：定义与积分
The singular Zienkiewicz tetrahedron: Definition and Integration
作者: Johannes Storn
类别: math.NA
发布日期: 2026-08-27
arXiv ID: 2608.27741v1
链接: http://arxiv.org/abs/2608.27741v1
摘要翻译
中文标题: 奇异Zienkiewicz四面体：定义与积分
摘要翻译: 我们将二维奇异Zienkiewicz单元推广到三维，从而在四面体网格上得到一种基于有理形函数的新型$H^2$-协调有限元。除有限元构造及其协调性分析外，我们还为相应的有理函数类发展了一种精确的迭代积分方法。所得公式能够精确积分基函数、其导数以及有限元组装中出现的乘积。
不相关论文（基本信息）
1. 利用黎曼优化再次求解最近正规矩阵
Closest Normal Matrix Found Again Using Riemannian Optimization
作者: Vanni Noferini, Matvei Zhukov
类别: math.NA
发布日期: 2026-08-28
arXiv ID: 2608.28545v1
链接: http://arxiv.org/abs/2608.28545v1
摘要: We propose an approach based on Riemannian optimization to compute a nearest normal matrix to a given one. The problem can be formulated as the minimization of a smooth function either on the manifold $U(n)$ of unitary matrices of size n or on the flag manifold $U (n)/U (1)^n$. The flag manifold is particularly suitable for theoretical analysis; we characterize the global maximum of the objective function and prove that, for generic inputs, its local minimizers are finitely many and isolated; in turn, this implies the original nearest normal matrix problem generically has finitely many local minimizers, all with distinct eigenvalues. We also develop a Riemannian trust-region method that improves substantially on classical algorithms and can handle considerably larger matrices, as well as a variant for computing the nearest real normal matrix. The paper is complemented by extensive numerical experiments.
2. 带积分约束的线性抛物问题源项的适定性与数值重构
Well-posedness and numerical reconstruction of a source term for linear parabolic problems with an integral constraint
作者: Jason R. Morris, Sedar Ngoma
类别: math.AP, math.NA
发布日期: 2026-08-28
arXiv ID: 2608.28171v1
链接: http://arxiv.org/abs/2608.28171v1
摘要: This work investigates a time-dependent source identification problem for linear parabolic equations subject to an integral constraint and Neumann boundary conditions in a domain of $\mathbb{R}^d$, $d\ge 1$. We establish well-posedness and higher regularity of the solution pair in parabolic Hölder spaces. A numerical algorithm based on a finite element discretization in space and an implicit time-stepping scheme is then developed for the reconstruction of the unknown source. The resulting discrete inverse problem involves a well-conditioned operator. We show that identity Tikhonov regularization provides only uniform, nonselective shrinkage in this setting, whereas Tikhonov regularization with derivative-based penalties, analyzed through the generalized singular value decomposition, provides an effective denoising strategy. The regularization parameter is selected using the Morozov discrepancy principle. Numerical errors are evaluated using full parabolic Hölder norms, which provide a more comprehensive assessment of the reconstruction by incorporating errors in the solution, its derivatives, and the associated Hölder seminorms. Numerical experiments for smooth and piecewise constant sources demonstrate accurate and robust reconstructions under increasing levels of noise.
3. 正迹类算子的随机迹估计
Stochastic trace estimation for positive trace-class operators
作者: Zvonimir Bujanović, Luka Grubišić, Daniel Kressner, Hrvoje Olić
类别: math.NA
发布日期: 2026-08-28
arXiv ID: 2608.28135v1
链接: http://arxiv.org/abs/2608.28135v1
摘要: Implicit trace estimation aims to approximate the trace of a matrix or linear operator accessible only through matrix-vector or operator-vector products. In the matrix setting, the Girard-Hutchinson estimator typically requires $\mathcal{O}(\varepsilon^{-2})$ products to achieve accuracy $\varepsilon$, while the variance-reduced Hutch++ algorithm reduces this sample complexity to $\mathcal{O}(\varepsilon^{-1})$ for positive semidefinite matrices. We develop infinite-dimensional analogues of these estimators for positive trace-class operators on separable Hilbert spaces. The idealized estimators use Gaussian random elements whose covariance is determined by the target operator, leading to unbiased operator versions of Girard-Hutchinson and Hutch++. We prove high-probability error bounds analogous to the finite-dimensional matrix results; in particular, idealized infHutch++ achieves $\mathcal{O}(\varepsilon^{-1})$ sample complexity. For practical computation, we introduce truncated implementations that restrict the random samples to finite-dimensional subspaces; for fixed sample budget, we show that truncated infHutch++ converges in distribution to its idealized counterpart as the truncation dimension tends to infinity. Numerical experiments with integral operators, density-of-states approximations, and spectral filtering for a radial Dirac operator show that these truncated estimators can achieve accuracy comparable to the ContHutch++ algorithm by Zvonek, Horning & Townsend while using lower-degree function representations and smaller internal discretizations in chebfun.
4. 一种用于模拟完全非线性水波的高阶多项式修正移动边界方法
A high-order polynomial-corrected shifted boundary method for simulating fully nonlinear water waves
作者: Jens Visbech, Allan P. Engsig-Karup, Harry B. Bingham, Mario Ricchiuto
类别: physics.flu-dyn, math.NA
发布日期: 2026-08-28
arXiv ID: 2608.28123v1
链接: http://arxiv.org/abs/2608.28123v1
摘要: We present a novel unfitted computational framework for simulating fully nonlinear potential flow-based water waves. Focusing on wave propagation, we describe the core methodology, which involves a high-order polynomial-corrected shifted-boundary approximation on unfitted spectral elements. This approach allows for the simulation of a curved, highly time-dependent (moving and deforming) free surface affected by bathymetric changes. All that on a simple Cartesian background mesh without re-meshing or further approximations of non-affine geometric features. In addition, we highlight the importance of proper gradient recovery using a polynomial-preserving technique to accurately capture the vertical free-surface velocity. Ultimately, the goal is to develop a high-order convergent numerical scheme capable of simulating highly nonlinear waves over long periods. This is achieved through an arbitrary-order finite-difference approximation of the free surface combined with added hyperviscosity for numerical stability. We present verification and validation test cases for wave propagation in both periodic and finite domains. Emphasis is placed on convergence studies, the justification for using high-order approximations, the importance of optimal gradient recovery, long-time simulation of highly nonlinear waves, and nonlinear wave interactions with both affine and curved bathymetry changes, as well as with vertical walls, for highly nonlinear stream function waves and high-amplitude solitons. The novel computational model offers a comprehensive, all-in-one framework for simulating ocean waves and their interactions with offshore structures. It provides significant geometric flexibility, enabling boundaries such as the free surface to move and deform over time without the need to re-mesh a boundary-fitted mesh.
5. 带奇异逆平方势的时间分数阶扩散波动方程的反源问题
Inverse Source Problem for a Time-Fractional Diffusion-Wave Equation with a Singular Inverse-Square Potential
作者: Zewen Wang, Bin Wu, Yikan Liu, Liangwei Jin, Shufang Qiu
类别: math.NA, math.AP
发布日期: 2026-08-28
arXiv ID: 2608.28103v1
链接: http://arxiv.org/abs/2608.28103v1
摘要: This paper investigates an inverse source problem for a time-fractional diffusion-wave equation with a singular inverse-square potential. The source term is assumed to consist of a known temporal factor and an unknown spatial component, which is to be recovered from terminal-state measurements. The well-posedness and regularity of the forward problem are established within an appropriate energy framework by exploiting Hardy-type inequalities and the spectral properties of the associated singular elliptic operator. The terminal observation operator is then shown to be compact, and uniqueness of the spatial source is established under a suitable nondegeneracy condition on the temporal factor. To stabilize the resulting ill-posed inverse problem, a Tikhonov regularization approach is introduced. The gradient of the regularized functional is derived through an adjoint problem involving a right-sided fractional derivative, leading to an adjoint-based conjugate gradient method with an exact line search for the numerical reconstruction of the unknown source. Numerical experiments are conducted on both one-and two-dimensional spatial domains, using both exact and noisy terminal data, to demonstrate the effectiveness and stability of the proposed source reconstruction method.
6. 支持向量扩展的新兴RISC-V硬件上快速傅里叶变换的性能评估
Performance Evaluation of Fast Fourier Transforms on Emerging RISC-V Hardware with Vector Extension Support
作者: Daniel Seibel, Kaveh Haghighi Mood, Jayesh Badwaik, Prateek Chawla, Stepan Nassyr, Andreas Herten
类别: cs.MS, cs.DC, math.NA
发布日期: 2026-08-28
arXiv ID: 2608.28076v1
链接: http://arxiv.org/abs/2608.28076v1
摘要: This manuscript presents a performance evaluation of Fast Fourier Transform (FFT) implementations on emerging processors supporting the RISC-V Vector Extension (RVV 1.0). By introducing juFFTe, a light-weight high-performance library for discrete Fourier transforms, it is demonstrated how effective vectorization of performance-critical FFT kernels can be achieved on RVV-enabled hardware. Comprehensive benchmarks on three RVV 1.0-ready processors, the SiFive X280, the X100 core of the SpacemiT K3 and the C920v2 core of the Sophon SG2044, reveal substantial performance improvements of juFFTe (https://github.com/FZJ-JSC/juFFTe) over the widely used FFTW3 library. Although RVV-enabled platforms show promising results at this stage of development, a comparison with AMD’s Zen 5 architecture indicates that RISC-V needs further maturing to reach the performance of established micro-architectures.
7. 具有闭式逆的二阶 Stiefel 回缩的条件性与插值误差界
Conditioning and interpolation error bounds for second-order Stiefel retractions with closed-form inverses
作者: Rasmus Jensen, Ralf Zimmermann
类别: math.NA, math.DG
发布日期: 2026-08-28
arXiv ID: 2608.28073v1
链接: http://arxiv.org/abs/2608.28073v1
摘要: Retractions provide a computationally efficient alternative to the Riemannian exponential and logarithm maps for practical data-processing tasks on manifolds. In particular, second-order retractions with closed-form inverse are well-suited for interpolation problems on manifolds. On the Stiefel manifold of orthogonal frames, there are only two retractions of this type: the Cayley retraction, which is second-order accurate under the canonical metric, and the recently proposed polar-light retraction, which is second-order accurate under the Euclidean metric. In this paper, we study the properties of these maps in the context of interpolation on the Stiefel manifold. To obtain explicit interpolation error bounds, we examine the conditioning of the retraction maps and their inverses. We show that the retractions are well-conditioned, and we derive interpolation error bounds similar to those of classical Euclidean interpolation. The inverse retractions are not well-conditioned in general, and we discuss how data can be mapped via an isometric group action to ensure stable computations. As with all retractions on compact manifolds, the inverse canonical Cayley retraction and the inverse polar-light retraction exist only locally, and we construct normal neighborhoods around any point in which either the inverse Cayley retraction or the invese polar-light retraction are guaranteed to be computable. As an application of the retraction maps, we consider Hermite interpolation, where the objective is to reproduce both sampled function values and derivative information. A numerical example demonstrates that retraction-based interpolation is competitive with classical methods based on Riemannian normal coordinates.
8. 多面体间断伽辽金离散的扩散综合加速的输运匹配惩罚
Transport-Matched Penalties for Diffusion Synthetic Acceleration of Polytopic Discontinuous Galerkin Discretisations
作者: Ansar Calloo, Matthew Evans, Francois Madiot, Tristan Pryer
类别: math.NA
发布日期: 2026-08-28
arXiv ID: 2608.28022v1
链接: http://arxiv.org/abs/2608.28022v1
摘要: Diffusion synthetic acceleration is most effective when its diffusion correction reflects the transport discretisation that generates the iteration error. We develop this principle for high-order upwind discontinuous Galerkin discretisations of discrete-ordinates transport on polytopic meshes. From the discrete transport sweep, we derive the exact scalar correction that removes the source-iteration scalar error in one step. We prove that the associated scalar response is positive and self-adjoint, obtain an exact expression for the source-iteration convergence factor, and quantify the additional damping produced by vacuum leakage. Using the exact correction as a reference, we construct a transport-matched modified interior penalty correction whose boundary terms are inherited directly from homogeneous vacuum inflow. In the optically thick regime, the resulting MIP form approximates the exact correction with relative error proportional to the effective cell Knudsen number. This gives a strict acceleration of source iteration, with bounds uniform in mesh size, polynomial degree, and element face count for admissible polytopic meshes. Numerical experiments on Cartesian and centroidal Voronoi meshes confirm the predicted convergence and correction-operator scaling.
9. 带马萨特噪声的流式线性系统的分位数随机卡奇马兹方法
Quantile Randomized Kaczmarz for Streaming Linear Systems with Massart Noise
作者: Emeric Battaglia, Jian-Feng Cai, Junren Chen, Anna Ma, Deanna Needell, Tong Wu
类别: math.NA
发布日期: 2026-08-28
arXiv ID: 2608.27968v1
链接: http://arxiv.org/abs/2608.27968v1
摘要: Quantile randomized Kaczmarz (QRK) has proven to be an efficient solver for corrupted linear systems and has received much attention. It was recently shown by Cai et al. (SIAM J. Matrix Anal. Appl. 47(2):802-823, 2026) that using $O(\log T/\log(1/β))$ samples for computing the quantile is necessary and sufficient for QRK to converge linearly over $T$ iterations when solving linear systems with a $β$-fraction of arbitrary corruptions, as long as $β$ is small enough. However, it remains unclear how large the corruption level $β$ can be, and how to compute the required subsample size $D$ explicitly, without hidden constants. This paper studies streaming linear systems with Massart noise via QRK using an order-optimal batch size $D=O(\log T)$ in each update. The independence of samples from previous iterations in the streaming setting enables a sharper analysis, yielding explicit, computable bounds on both the tolerable corruption level and the required subsample size. In particular, we establish linear convergence for corruption levels of up to approximately 7%. We also discuss how the constants improve under oblivious noise.
10. Stokes方程的半步反弹格子Boltzmann方法的最优误差估计
Optimal error estimates for the half-way bounce-back lattice Boltzmann method for the Stokes equations
作者: Kai Koike
类别: math.NA, math.AP
发布日期: 2026-08-28
arXiv ID: 2608.27926v1
链接: http://arxiv.org/abs/2608.27926v1
摘要: We give a mathematical proof of the optimal convergence rates for the D2Q9 BGK lattice Boltzmann method with the half-way bounce-back rule for the incompressible Stokes equations in a flat channel. The convergence rates are second-order for the velocity and first-order for the pressure as the lattice spacing $h$ tends to zero, in agreement with formal analyses and numerical experiments, whereas the available rigorous convergence theorems only yield an $O(h^{1/2})$ bound for the velocity error. A key step in the proof is a decomposition of the leading boundary consistency error into macroscopic and kinetic components. These components are absorbed by suitably constructed Stokes and discrete Knudsen layer correctors, respectively. Incorporating these correctors into the prediction function used in previous rigorous analyses, we obtain a refined prediction function with consistency errors of sufficiently high order. Combined with the known weighted $L^2$-stability estimate, this gives the optimal convergence rates.
11. 面向GPU上可压缩流动直接数值模拟的大规模并行杂交化间断伽辽金求解器
A Massively Parallel Hybridizable Discontinuous Galerkin Solver for Direct Numerical Simulation of Compressible Flows on GPUs
作者: Andrew Welter, Thea Collin, Ngoc Cuong Nguyen, Jaime Peraire
类别: math.NA, physics.flu-dyn
发布日期: 2026-08-28
arXiv ID: 2608.27863v1
链接: http://arxiv.org/abs/2608.27863v1
摘要: Direct numerical simulation (DNS) of compressible transitional and turbulent flows requires numerical methods that combine high-order accuracy, robustness, and computational efficiency to resolve a broad range of spatial and temporal scales. This paper presents a massively parallel hybridizable discontinuous Galerkin (HDG) solver for DNS of the compressible Navier-Stokes equations on GPU-accelerated high-performance computing systems. The proposed solver combines high-order HDG discretization with robust shock capturing, diagonally implicit Runge-Kutta (DIRK) time integration, and an efficient Newton-GMRES solution strategy accelerated by additive Schwarz preconditioning and reduced-basis approximation. A distributed implementation of these methods based on GPU-aware MPI, Kokkos, and CUDA/HIP libraries enables scalable execution on heterogeneous computing platforms. The solver is demonstrated on three canonical benchmark problems covering a wide range of Mach-number flow regimes: subsonic transitional flow over the Eppler 387 airfoil, the supersonic Taylor-Green vortex, and hypersonic boundary-layer transition. Numerical results are compared with available experimental measurements and published DNS data, showing good agreement across distinct flow regimes. The results demonstrate the ability of the proposed solver to resolve laminar-turbulent transition, strong compressibility effects, shock-associated flow structures, and fully three-dimensional turbulent dynamics.
12. 带Dirac测度的分数阶扩散最优控制
Optimal control of fractional diffusion with Dirac measures
作者: Enrique Otarola, Abner J. Salgado
类别: math.OC, math.NA
发布日期: 2026-08-27
arXiv ID: 2608.27776v1
链接: http://arxiv.org/abs/2608.27776v1
摘要: We study a PDE-constrained optimization problem for an elliptic equation with the spectral fractional Laplacian and a linear combination of Dirac measures as the forcing term; the controls are the amplitudes of these singular sources. We prove existence and uniqueness of an optimal solution and derive first-order optimality conditions. We then propose a discretization based on finite elements. Since the set of admissible controls is finite dimensional, the control variable itself does not require discretization. We conclude by deriving a priori error bounds
13. 高阶鲁棒基更新与Galerkin积分器用于动力低秩近似
High-order robust basis-update & Galerkin integrators for dynamical low-rank approximation
作者: Cory D. Hauck, Jonas Kusch, Steffen Schotthöfer
类别: math.NA
发布日期: 2026-08-27
arXiv ID: 2608.27749v1
链接: http://arxiv.org/abs/2608.27749v1
摘要: Dynamical low-rank approximation has become a widely used numerical method in diverse disciplines. Its main idea is to represent the matrix or tensor-valued solution to a time-dependent differential equation as a low-rank factorization. The evolution of the factorization leads to highly stiff dynamics and requires the derivation of novel time integration methods that are not prone to this stiffness. A promising family of integrators are basis-update & Galerkin (BUG) integrators as they enable implicit time integration and structure–preservation. However, current BUG integrators are limited to second–order accuracy while general-order BUG integrators are designed as projections of explicit time integration methods, thus severely limiting their use. In this work, we propose general-order BUG integrators that do not rely on an explicit time integration scheme while requiring a smaller number of basis functions to achieve high-order accuracy. We prove a general order error bound for the proposed augmented and parallel BUG integrators and demonstrate their behaviour for a series of stiff and non-stiff numerical benchmarks in which they significantly outperform previous BUG versions.
14. 带连续源注入的变系数输运的高效量子模拟
Efficient Quantum Simulation of Variable-Coefficient Transport with Continuous Source Injection
作者: Mohammad Mehedi Hasan Akash, Turag Dev, Nhat-Quang Nguyen, Yanzhu Chen, Huixuan Wu, Kourosh Shoele
类别: quant-ph, math.NA
发布日期: 2026-08-27
arXiv ID: 2608.27712v1
链接: http://arxiv.org/abs/2608.27712v1
摘要: Quantum time-marching algorithms for transport PDEs often represent variable coefficients and forcing through register-expanding dilations, block-encoding oracles, or repeated postselection. We present an alternative algorithm for a forced variable-coefficient advection-diffusion equation in flow-inspired skew-symmetric form that incorporates spatially varying velocity, viscous dissipation, and persistent source injection with a peak logical requirement of $n_q+1$ qubits. A centered skew-symmetric discretization makes the advection operator strictly skew-Hermitian for arbitrary velocity profiles, enabling an ancilla-free unitary realization using a Gray-code Trotter sequence of controlled-$R_y$ rotations. Diffusion is applied in the Fourier basis through a uniformly controlled rotation on one postselected ancilla, which is measured, reset, and reused between the two diffusion half-steps, while the source is incorporated classically through second-order Strang splitting. Statevector simulations for $N=16$ and $32$ recover second-order temporal convergence against high-accuracy classical solutions, while Richardson extrapolation gives fourth-order accuracy and reduces kernel calls by factors of four to fourteen. Independent tests through $N=256$ confirm second-order spatial consistency. We further show that the per-step ancilla failure probability is proportional to the instantaneous viscous dissipation rate, making postselection cost self-regulating over a fifty-fold viscosity range. Stable evolution is demonstrated for $5\times10^4$ time steps without observable secular error growth, while Gray-code advection accounts for $71$–$95\%$ of transpiled controlled-NOT gates. The fixed-width kernel provides a qubit-efficient building block for near-term hardware studies, although classical readout and state re-preparation remain the main obstacles to coherent multistep evolution.
15. 微可压缩流数据同化的数值分析
Numerical analysis of data assimilation for slightly compressible flow
作者: Aytekin Çıbık, Rui Fang
类别: math.NA
发布日期: 2026-08-27
arXiv ID: 2608.27647v1
链接: http://arxiv.org/abs/2608.27647v1
摘要: Continuous data assimilation improves flow predictions by continually nudging a model toward available observational data. For slightly compressible flow, a recent model addresses the limitations of velocity-only nudging by assimilating both velocity and pressure data and nudging both quantities into the incompressible Navier-Stokes equations [5]; continuous-in-time error estimates and preliminary experiments show that this joint nudging is effective and substantially reduces the model error relative to velocity-only nudging. Motivated by these results, we carry out the numerical analysis of the model and its finite element discretizations. We establish stability and error estimates for the semi-discrete scheme and for the fully discrete, linearized backward Euler scheme. The analysis shows an infinite predictability horizon: the effect of the initial error decays exponentially in time, and the model error is first order in the observation resolution H and of order $μ_1^{-1/2}$ in the pressure nudging parameter $μ_1$. Balancing these two error terms, we choose $μ_1=\mathcal{O}(H^{-2})$, which yields the optimal convergence rate. Numerical experiments confirm the predicted rates.