生成时间: 2026年05月29日 07:29:21
重点关注论文数量: 1
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重点关注论文（完整分析）
1. 富集高阶多尺度方法及其在波动传播中的应用
Enriched higher-order multiscale approaches with applications to wave propagation
作者: Balaje Kalyanaraman, Felix Krumbiegel, Roland Maier, Siyang Wang
类别: math.NA
发布日期: 2026-05-28
arXiv ID: 2605.30118v1
摘要翻译: 我们考虑空间系数高度异质的偏微分方程的数值求解。我们概述椭圆情形下的高阶局部正交分解（LOD）方法及其最新进展，然后将该策略推广至线性双曲多尺度问题。我们针对先前波动方程构造的局限性，该方法在空间上仅能达到二阶收敛，且与所选多项式次数无关。基于近期针对抛物多尺度问题发展的富集校正方法，我们引出并提出波动方程的富集高阶LOD方法。富集校正呈现指数衰减，且可在局部子域上计算。在对系数的最小假设及对数据的标准良态性条件下，我们推导出可实现最优高阶收敛速率的先验误差估计，从而克服了此前观察到的收敛率饱和现象。借助五阶Rosenbrock-Wanner（ROW）时间积分器，我们进行一系列数值算例以验证理论结果。我们给出算例展示方法的最优空间收敛性，包括不同多项式次数下的局部化误差。我们也给出算例展示时间离散的最优收敛率。
Comment: 无
链接: http://arxiv.org/abs/2605.30118v1
1. 研究对象和背景
本文关注具有空间强非均匀系数的声波方程的数值解法. 考虑定义在 Lipschitz 区域 $\Omega\subset\mathbb{R}^d$ 上的线性双曲问题: 求 $u:[0,T]\to H^1_0(\Omega)$ 使得对几乎所有 $t\in[0,T]$,
$$(\ddot u(t), v)_{H^{-1}(\Omega)\times H^1_0(\Omega)} + a(u(t), v) = (f(t), v)_{L^2(\Omega)}\quad\forall v\in H^1_0(\Omega),$$其中 $a(u,v) = (A\nabla u,\nabla v)_{L^2(\Omega)}$, 系数 $A\in L^\infty(\Omega)$ 满足 $0<\alpha\le A(x)\le\beta<\infty$ 几乎处处, 且可能呈现多尺度剧烈变化. 初始条件为 $u(0)=u_0\in H^1_0(\Omega)$ 和 $\dot u(0)=v_0\in L^2(\Omega)$. 当介质内部存在细微非均匀性时, 传统的有限元方法需要网格尺寸足够精细以解析所有细尺度特征, 导致计算代价难以承受. 多尺度方法通过构造问题自适应的粗空间来降低计算量, 其中局部正交分解 (LOD) 方法对粗糙 $L^\infty$-系数的椭圆问题已取得巨大成功.
在椭圆情形, 高阶 LOD 方法基于粗网格 $\mathcal{T}_H$ 上的间断高阶多项式空间 $V_H$ (局部次数 $p$), 利用 $L^2$ 投影 $\Pi_H$ 和约束能量最小化构造多尺度空间 $\mathcal{R} V_H$. 关键性质是 $\mathcal{R} V_H$ 为 $H^1_0$ 协调的, 且 $\mathcal{R}$ 是 $a$-正交投影, 从而误差仅依赖于右端项 $f$ 的正则性: 若 $f\in H^k(\mathcal{T}_H)$ ($k\le p+1$), 则能量误差满足 $\|\nabla(u-\mathcal{R}u)\|_{L^2}\lesssim H^{k+1}\|f\|_{H^k(\mathcal{T}_H)}$. 这一构造不要求系数 $A$ 或解 $u$ 的高正则性, 特别适合多尺度问题. 通过利用基函数的指数衰减性质, 可将全局最小化问题局部化到尺度为 $\ell$ 的补丁上, 仅引入指数小的局部化误差, 从而保持计算效率.
然而, 直接将上述椭圆型构造移植到波方程 (2.21) 时, 数值实验和理论分析均表明空间收敛阶不超过 $2$, 与多项式次数 $p$ 无关. 其原因在于: 标准能量论证将误差转化为 $\|\nabla(u-\mathcal{R}u)\|_{L^2}$, 而正交性给出
$$\|\nabla(u-\mathcal{R}u)\|_{L^2}^2 \lesssim (f,u-\mathcal{R}u) - (\partial_t^2 u, u-\mathcal{R}u).$$第一项可利用 $f$ 的正则性得到 $H^{k+1}$ 阶, 但第二项仅能利用 $\partial_t^2 u\in H^1_0$ 和 $u-\mathcal{R}u\in\ker\Pi_H$ 得到 $H^2$ 阶, 无法提取更高阶, 因为粗糙系数下解的空间正则性不会超过 $H^1$. 这便是收敛率饱和的根源. 最近, 针对抛物问题发展的富集高阶 LOD 方法通过引入额外的、利用时间正则性的校正, 成功恢复了空间高阶收敛. 本文旨在将这一富集策略推广到双曲型波方程, 以突破空间收敛率的饱和, 在系数仅为 $L^\infty$ 的极低正则性假设下获得最优高阶收敛率.
2. 主要定理或主要结果
本文在 Assumption 2.2 给出的兼容性及良好准备条件下 (即 $f\in C^m([0,T]; H^k(\mathcal{T}_H))$, 初始值 $u_0,v_0$ 满足一定的相容关系, 且时间导数初值一致有界), 对无局部化的理想情形, 证明了富集高阶 LOD 方法的空间半离散误差估计. 具体地, 令 $\tilde u_H$ 为半离散解 (2.23) 在富集多尺度空间 $\mathcal{R}V_H^j$ 中的 Galerkin 逼近, 则有
$$\|\partial_t(u(t)-\tilde u_H(t))\|_{L^2} + \|\nabla(u(t)-\tilde u_H(t))\|_{L^2} \lesssim_T H^{k+1},$$其中 $k$ 为右端项 $f$ 的空间 Sobolev 正则性指标, $1\le k\le p+1$. 当 $k=p+1$ 时, 收敛阶达到最优 $H^{p+2}$. 该结果完全消除了原 LOD 构造中收敛阶饱和于 $H^2$ 的限制, 且仅需系数 $A\in L^\infty(\Omega)$, 无需任何额外光滑性.
富集空间由基函数 $\mathcal{R}\Lambda_{K,i}$ 及其经校正算子 $D^{[\ell]}_K$ 多次作用得到的函数 $\mathcal{R}\Lambda^\nu_{K,i}$ ($\nu=1,\dots,j$) 张成. 校正算子 $D^{[\ell]}_K$ 在补丁 $N_\ell(K)$ 上求解局部边值问题:
$$a(D^{[\ell]}_K v, w) = -(v,w)_{L^2(N_\ell(K))}\quad\forall w\in H^1_0(N_\ell(K))\cap\ker\Pi_H,$$且满足指数衰减性质: 存在常数 $C>0$ 使得
$$\|\nabla D^{[\ell]}_K v\|_{L^2(\Omega\setminus N_\ell(K))}\lesssim \exp(-C\ell)\|\nabla D^{[\ell]}_K v\|_{L^2(\Omega)}.$$因此, 校正计算可在相同补丁上进行, 不增加额外通信开销. 理论分析表明, 适当地选取局部化参数 $\ell\sim C_p|\log H|$, 局部化后的逼近误差仍保持高阶收敛.
在时间离散方面, 作者采用 $5$ 阶 Rosenbrock-Wanner (Rodas5P) 格式, 数值实验验证了空间误差对于不同多项式次数 $p$ 均达到理论预测的最优阶, 并展现了时间离散的高阶精度.
3. 研究方法、关键技术和核心工具
本文的核心障碍在于: 椭圆型高阶 LOD 空间应用到波方程时, 误差表达式中出现了 $\partial_t^2 u$ 项, 而粗糙系数下解 $u$ 在空间上至多仅有 $H^1$ 正则性, 导致无法利用投影近似性质提取高阶收敛. 作者真正的创新在于引入基于时间正则性的富集校正策略, 通过扩大多尺度空间来吸收 $\partial_t^2 u$ 的低正则性影响, 从而恢复高阶率.
证明框架推进如下:
修正 Galerkin 正交性误差. 令 $\mathcal{R}^{[\ell]}$ 为原局部化 LOD 算子, 其像空间 $\mathcal{R}V_H$ 满足 $a$-正交性 $\mathcal{R}V_H\perp_a \ker\Pi_H$. 但直接用于波方程半离散时, 误差 $e_H = u - \tilde u_H$ 满足能量等式
$$\frac12\frac{d}{dt}\big(\|\dot e_H\|_{L^2}^2 + a(e_H,e_H)\big) = a(\mathcal{R}u - \tilde u_H, \dot e_H) + (\ddot u - \partial_t^2 \mathcal{R}u, \dot e_H).$$第二项是降低收敛阶的根源. 引入富集空间 $\mathcal{R}V_H^j$ 后, 误差被分裂为投影到 $\mathcal{R}V_H^j$ 和其正交补的部分. 通过定义校正算子 $D^{[\ell]}_K$, 构造额外的基函数, 使得新空间包含 $D^{[\ell]}_K\mathcal{R}\Lambda_{K,i}$ 等. 这些函数与 $\ker\Pi_H$ 的适当正交性使得 $(\ddot u - \partial_t^2 \mathcal{R}u)$ 能被新空间捕获, 从而在能量估计中该不良项转化为高阶量.
关键估计. 在定理 3.1 的证明中, 将 $u - \tilde u_H$ 写成 $u - \mathcal{R}^j u$ (投影误差) 和 $\mathcal{R}^j u - \tilde u_H$ (Galerkin 误差) 之和. 对于投影误差, 利用 $\mathcal{R}^j$ 的构造和假设 $f$ 具有空间正则性 $H^k$, 以及时间导数到初始时刻的延拓, 递归地使用校正方程将 $\partial_t^\nu u$ 逐步转换为仅依赖左右端项的高阶项, 得到
$$\|\nabla(u - \mathcal{R}^j u)\|_{L^2} \lesssim H^{k+1} \sum_{\nu=0}^{j} \|\partial_t^{2\nu} f\|_{H^k}.$$对 Galerkin 误差则应用标准能量估计和上面投影误差的界限. 整个分析不依赖解 $u$ 的高空间正则性, 只使用了时间正则性 (通过兼容性条件保证).
局部化与指数衰减. 校正问题 (3.1) 被限制在补丁上求解, 其解具有指数衰减性质 (见 Theorem 2.1 的推广). 证明基于 Caccioppoli 型不等式和迭代论证, 与经典 LOD 分析类似. 因此富集基函数的计算仍然是局部的, 保持了原有 LOD 方法的并行性和计算规模.
与稳定化局部化的结合. 为避免直接使用 (2.15) 局部化时出现的 $H^{-1}$-污染因子, 本文采用 [HLM26] 中的稳定化局部化策略 $\mathcal{R}^{[\ell]} = I_H - \mathcal{K}^{[\ell]}$, 其中 $I_H$ 为局部拟插值算子, $\mathcal{K}^{[\ell]}$ 通过局部鞍点问题定义. 这样得到的基函数可直接输入富集流程, 且在 $\ell$ 足够大时仍保持指数小的局部化误差. 证明中未显式展开此结合, 但指出该方法框架是统一的.
4. 与之前工作的比较
与高阶椭圆 LOD 方法的比较. 椭圆型高阶 LOD (如 [Mai21], [DHM23], [HLM26]) 的收敛率分析完全依赖右端项 $f$ 的正则性, 因为 $a$-正交性直接给出 $\|\nabla(u-\mathcal{R}u)\|_{L^2}\lesssim H^{k+1}\|f\|_{H^k}$. 在波方程中, 等效的“右端项”变为 $f - \partial_t^2 u$, 而 $\partial_t^2 u$ 的正则性受限于系数粗糙性, 至多 $H^1$, 导致收敛阶饱和在 $2$. 本文引入的富集校正本质上是在渐近展开中逐项补偿 $\partial_t^2 u$ 的低正则部分, 使截断后的残差获得高阶, 从而将收敛率恢复为 $H^{k+1}$. 相比椭圆 LOD, 本文的富集空间维数增加, 但增幅可控, 且构造的局部化性质不变.
与抛物富集 LOD [KKMW25] 的比较. 抛物问题中, 类似饱和现象源自 $\partial_t u$ 的低正则性. [KKMW25] 提出用校正算子处理 $\partial_t u$, 得到空间高阶收敛. 本文将其思想迁移到双曲方程, 需要处理二阶时间导数, 因此富集层数 $j$ 与时间导数阶数的关系有所不同: 抛物情形每消除一阶时间导数需要一层富集, 而波方程需两层富集才能抵消一个 $\partial_t^2$ 带来的低正则效应? 从构造看, 校正算子 $D^{[\ell]}_K$ 每次作用对应提升匹配阶数, 理论上 $j=1$ 即可恢复 $H^{k+1}$, 但文中未详细讨论 $j$ 与所需收敛阶的关系. 此外, 能量估计与抛物情形不同, 本文需处理电感–电容能量范数, 论证更为精细.
与已有波方程多尺度方法的比较. 文献中处理波方程多尺度问题的方法包括 [JEG10] 的 FE-HMM, [AG11] 的变量分离多尺度有限元, [EHR11] 的局部多尺度基, [OZ17] 的广义多尺度有限元, [AH17] 的 LOD 波方程方法等. 其中, 基于 LOD 的 [AH17] 在空间上仅能达到二阶收敛 (若使用标准一阶 LOD, 收敛阶为 $H$; 采用高阶多项式也无法超越 $H^2$). 本文显著推进了 LOD 框架下波方程的空间收敛率上限, 实现了任意高阶. 相比其他方法, 本工作的优势在于: (i) 仅需要系数 $A\in L^\infty$, 无需分离尺度或周期假设; (ii) 误差分析严格, 常数显式依赖于正则性参数; (iii) 方法完全基于局部子问题, 适合大规模并行. 代价是基函数的计算量增加 (多层校正), 但仍在可接受范围内, 且数值实验表明, 对于给定的收敛阶, 所需局部化参数 $\ell$ 随 $p$ 增长, 但实际标度比理论预测的更温和.
局限与未来方向. 从提供的内容中无法确认本文是否分析了长期时间稳定性以及 $j$ 的最优选择. 作者也未显式展开与其他高阶时间离散方案的比较, 仅采用了 ROW 方法. 此外, 对 $t$ 依赖的边界条件和非零初值波源的处理也未提及. 文中提示未来可将富集思想扩展至更多类型的时间依赖问题.
了解领域论文（摘要翻译）
1. 一种用于含旋转大气动力学的非协调网格细化隐式-显式间断伽辽金求解器
An IMEX-DG solver with non-conforming mesh refinement for atmospheric dynamics with rotation
作者: Letizia Bottani, Tommaso Benacchio, Giuseppe Orlando, Luca Bonaventura, Allan Peter Engsig-Karup
类别: math.NA, physics.ao-ph
发布日期: 2026-05-28
arXiv ID: 2605.29598v1
链接: http://arxiv.org/abs/2605.29598v1
摘要翻译
中文标题: 一种用于含旋转大气动力学的非协调网格细化隐式-显式间断伽辽金求解器
摘要翻译: 我们提出了一种用于可压缩欧拉方程的高阶隐式-显式间断伽辽金（IMEX-DG）求解器，旨在完全可压缩大气框架下计入旋转效应。时间积分采用二阶加性龙格-库塔格式，对刚性声学模态隐式处理，对平流项显式处理。该求解器基于 deal.II 有限元库构建，结合了无矩阵算子计算、自适应非协调网格能力和分布式内存并行。在求解策略中，我们引入并比较了两种基于非线性不动点迭代的旋转和重力源项处理方案，从精度、鲁棒性和计算效率方面进行了对比。还对旋转算子进行了离散分析，以推导出适合高效无矩阵实现的公式，并避免不一致的朴素离散化。通过旋转惯性重力波基准算例的收敛性研究验证了所提格式，并在均匀和自适应网格上进行了完全三维分层流越过地形的模拟，进一步评估。数值结果表明，旋转 IMEX-DG 框架具有预期的精度和稳定性性质，同时正确捕捉了旋转在大尺度大气流中引起的非对称性和波动结构。
2. 非线性Gross-Pitaevskii方程的高阶时间分裂方法
High-order time splitting methods for the nonlinear Gross-Pitaevskii equation
作者: Roberto Ben, Agustín Besteiro, Diego Rial
类别: math.NA, math.AP
发布日期: 2026-05-28
arXiv ID: 2605.29362v1
链接: http://arxiv.org/abs/2605.29362v1
摘要翻译
中文标题: 非线性Gross-Pitaevskii方程的高阶时间分裂方法
摘要翻译: 我们提出一种高阶数值方法，用于计算具有谐振子势阱的二维Gross-Pitaevskii方程的基态和时间演化。基态通过将归一化梯度流与具有严格正系数的时间分裂格式相结合而获得，这使它们既适用于耗散区域也适用于哈密顿区域，并避免了经典辛方法所需的负时间步长。这些方法的计算代价随阶数呈二次增长，而辛方法的计算代价呈指数增长。数值基准测试评估了在不同初始化下基态的收敛性、不同质量约束下质量和哈密顿能量的长时间保持，以及对于阶数 q = 2, 4, …, 14 的代价-精度权衡。
不相关论文（基本信息）
1. IGA-ODIL：利用等几何分析优化离散鲁棒损失以借助机器学习工具更快求解正反问题
IGA-ODIL: Optimizing DIscretre robust Loss with Isogeometric Analysis to solve forward and inverse problems faster using machine learning tools
作者: Maciej Paszyński, Tomasz Służalec
类别: math.NA
发布日期: 2026-05-28
arXiv ID: 2605.30272v1
链接: http://arxiv.org/abs/2605.30272v1
摘要: Physics-informed neural networks (PINNs) formulate the solution of partial differential equations as residual minimization problems over neural network parameterizations. Although highly flexible, optimization of PINNs using modern variants of Stochastic Gradient Descent algorithms is expensive. On the other hand, iterative computation of PINN parameterization using the Gauss-Newton method suffers from convergence difficulties, dense Jacobian structures, and poor conditioning that limit the effectiveness of second-order optimization methods. In this work, we introduce IGA-ODIL, a spline-based residual minimization framework combining ideas from Optimizing DIscrete Loss (ODIL), robust variational residual minimization, and Isogeometric Analysis (IGA). Instead of neural-network parameterizations of PINNs, the unknown solution is represented by smooth B-spline basis functions, leading to sparse structured Jacobians and efficient Gauss–Newton optimization. We also derive robust residual formulations based on weighted Gram operators, making the loss function related with the true error. The resulting systems inherit locality, sparsity, and approximation-theoretic properties of classical finite element and isogeometric methods while preserving the residual-learning philosophy of scientific machine learning. The proposed methodology is evaluated on several benchmark problems, including Poisson equations, convection-dominated advection–diffusion equations, Helmholtz problems with highly oscillatory solutions, nonlinear Allen–Cahn equations, and inverse Helmholtz parameter identification. Numerical experiments demonstrate orders-of-magnitude speedups compared with PINNs and CRVPINNs while maintaining high accuracy and robustness.
2. 广义矩阵邻近问题 II
Generalized matrix nearness problems II
作者: Rongbiao Thomas Wang, Chi-Kwong Li, Lek-Heng Lim
类别: math.NA, math.OC
发布日期: 2026-05-28
arXiv ID: 2605.30181v1
链接: http://arxiv.org/abs/2605.30181v1
摘要: Given a matrix $A$, a matrix nearness problem seeks an $X$ that most closely approximates $A$ in the sense of minimizing $\lVert A - X\rVert$ under a variety of constraints on $X$. A generalized matrix nearness problem seeks the same but with three given matrices $A,B,C$ and $\lVert A - BXC\rVert$ in place of $\lVert A - X\rVert$. We extend previous studies of the latter problem in three directions: incorporating an affine term, replacing matrix product by Kronecker product in various manners, and generalizing Frobenius norm to any orthogonally invariant norm. We will solve several of these in closed form. For the rest, we develop an iterative algorithm that works for any Schatten norm, proving that it converges to a global minimizer regardless of the initial point. In addition, the algorithm relies purely on numerical linear algebra, and notably does not compute any explicit gradients or subgradients. Along the way, we will also show that there is no Mirsky-type theorem for rank constrained generalized matrix nearness problems.
3. 部分相干下的全息X射线相衬成像：唯一性及基于强度关联的重建
Holographic X-ray Phase Contrast Imaging with Partial Coherence: Uniqueness and Reconstructions from Intensity Correlations
作者: Thorsten Hohage, Milad Karimi, Björn Müller
类别: math.AP, math.NA
发布日期: 2026-05-28
arXiv ID: 2605.30146v1
链接: http://arxiv.org/abs/2605.30146v1
摘要: Holographic coherent X-ray imaging enables nanoscale imaging of biological cells and tissues, rendering both phase and absorption contrast, i.e. real and imaginary parts of the refractive index. Unlike the standard model, which assumes a perfectly coherent incident beam, we consider partial coherence characterized by a known covariance operator. In addition, we assume time-resolved intensity measurements, granting access not only to expected intensities but also to their correlations. We investigate the information content of these correlations and analytically demonstrate that, under a symmetry-breaking condition on the sample and the illumination area, both phase and absorption contrast can be uniquely recovered in both the full and the linearized models. A key challenge in numerical reconstruction is the substantial increase in data dimensionality caused by computing intensity correlations during preprocessing. We propose a novel approach that leverages a low-rank assumption on the incident beam covariance operator, bypassing explicit correlation computation while still exploiting its full information. Numerical experiments demonstrate its feasibility, yielding accurate simultaneous reconstructions of phase and absorption contrast.
4. 一种新颖的基于球多项式的混合谱方法求解单位球上的双调和方程
A novel mixed spectral method with ball polynomials for the Biharmonic equation on a unit ball
作者: Mengxue Gao, Bing Su, Jianwei Zhou
类别: math.NA
发布日期: 2026-05-28
arXiv ID: 2605.30037v1
链接: http://arxiv.org/abs/2605.30037v1
摘要: A novel mixed spectral-Galerkin method based on generalized ball polynomials is proposed for solving the biharmonic equation on a unit ball. By introducing an auxiliary variable to decouple the biharmonic equation into a system of second-order equations, the corresponding discrete scheme yields a strictly diagonal stiffness matrix, which significantly enhances the computational efficiency. Rigorous a-priori error estimates are established to demonstrate the exponential convergence rates in both the $L^2$- and $H^1$-norms. Extensive numerical experiments are conducted to verify the theoretical analysis and confirm the high efficiency and accuracy of the proposed scheme.
5. 多孔介质中两相流的全解耦、保界且能量耗散格式的统一框架
A unified framework of fully decoupled, bound-preserving and energy-dissipative schemes for two-phase flow in porous media
作者: Xiaoli Li, Cheng Wang, Yujing Yan, Nan Zheng
类别: math.NA
发布日期: 2026-05-28
arXiv ID: 2605.30004v1
链接: http://arxiv.org/abs/2605.30004v1
摘要: Developing high-order numerical schemes for two-phase flow in porous media that preserve key physical properties remains a significant challenge in numerical analysis. In this article, we propose a general framework to construct fully discrete first- and second-order numerical schemes for thermodynamically consistent model of incompressible and immiscible two-phase flow in porous media. The proposed schemes are rigorously proved to ensure five fundamental properties: (i) unique solvability; (ii) full decoupling; (iii) bound preservation for both phases; (iv) original energy dissipation; (v) local mass conservation for both phases. The key to ensure the unique solvability lies in guaranteeing the strict convexity of the discrete energy functionals associated with the constructed schemes. Departing from the coupled solution approach for the pressure and saturation variables, the proposed approach breaks traditional paradigm by subtracting the two-phase mass conservation equations to derive a fully decoupled system. In addition, the bound-preserving property for both phases is established by leveraging the singular nature of the logarithmic term around the limit values of $0$ and $1$. A rigorous error estimate for the first-order scheme, in the $\ell^{\infty}(0,T; H_h^{-1} (Ω)) \cap \ell^{2}(0,T; \ell^2(Ω))$ norm for the saturations of two phases, is established. Finally, various numerical examples are presented to verify the theoretical results and demonstrate the efficiency of the proposed schemes.
6. 含泥沙卷吸和沉积的一阶矩浅水Exner矩模型的静水平衡稳定性分析：扩展技术报告
Water-at-Rest Equilibrium Stability Analysis of a first-moment Shallow Water Exner Moment Model with Sediment Entrainment and Deposition: Extended Technical Report
作者: Afroja Parvin, Giovanni Samaey, Julian Koellermeier
类别: math.AP, math.NA
发布日期: 2026-05-28
arXiv ID: 2605.29554v1
链接: http://arxiv.org/abs/2605.29554v1
摘要: We derive the first-moment Shallow Water Exner Moment model with sediment entrainment and deposition (SWEMED1) and show that the full source term has a fully-settled water-at-rest equilibrium manifold. We prove that the model is only weakly hyperbolic at this equilibrium, which prevents the use of Yong’s structural stability framework. However, a linear spectral analysis and numerical results do not indicate instability. Based on numerical results, we introduce a fast-slow scaling of the source term, and for the fast limit, we derive a new suspended water-at-rest equilibrium manifold, which has a different structure but is still only weakly hyperbolic. Our results show that the remaining obstruction is linked to the transport closure of the SWEMED1, and we give a constructive direction for the derivation of new closures leading to models with more desirable analytical properties.
7. 基于薛定谔化的多尺度常微分与偏微分方程量子隐式-显式格式
Quantum Implicit-Explicit Schemes for Multiscale Ordinary and Partial Differential Equations via Schrödingerization
作者: Qitong Hu, Xiaoyang He, Shi Jin, Xiao-Dong Zhang
类别: math.NA, math-ph, quant-ph
发布日期: 2026-05-28
arXiv ID: 2605.29423v1
链接: http://arxiv.org/abs/2605.29423v1
摘要: In this paper, we present a quantum implicit-explicit (IMEX) scheme for multiscale ordinary and partial differential equations whose discretization parameters are independent of the scaling parameter $\varepsilon$. A key ingredient of our approach is a continuous-time formulation of classical IMEX schemes, which decouples the evolution time of the quantum algorithm from the physical time of the differential equation and is therefore particularly useful in multiscale settings. Building on this idea, we employ the Schrödingerization framework [Phys. Rev. Lett. 133 (2024), 230602] to implement IMEX schemes on quantum computers. Compared to previous HHL type quantum AP scheme [J. Comput. Phys. 471 (2022), 111641], this new method requires narrower – an extra logarithmic factor – auxiliary register numerical examples on linear heat and multiscale telegraph equations demonstrate the independence in $\varepsilon$ of the method.
8. 反问题中贝叶斯推断的深度自适应降维
Deep Adaptive Dimension Reduction for Bayesian Inference in Inverse Problems
作者: Yueyang Wang, Xili Wang, Kejun Tang, Xiaoliang Wan, Tao Zhou, Chao Yang
类别: cs.LG, math.NA
发布日期: 2026-05-28
arXiv ID: 2605.29373v1
链接: http://arxiv.org/abs/2605.29373v1
摘要: Solving high-dimensional PDE-governed inverse problems is often challenging due to complex non-Gaussian posterior distributions, expensive forward model evaluations, and misspecified prior information. To address these issues, we propose a deep adaptive dimension-reduction Bayesian inference framework based on the Variational Flow (VF) model. Since standard normalizing flows are restricted by bijective mappings and cannot directly reduce dimensions, VF overcomes this limitation by integrating VAE-based nonlinear dimension reduction with dual normalizing flows for the latent prior and encoder. This design provides a strictly higher evidence lower bound than VAE and allows more flexible approximation of complex posterior distributions. We further introduce an iterative prior updating strategy that gradually moves the prior mean toward high-probability posterior regions, avoiding manual prior tuning. These components form a closed adaptive loop together with an adaptively fine-tuned Fourier Neural Operator (FNO) surrogate: VF generates posterior-concentrated samples to refine the surrogate, while the updated surrogate further improves posterior inference. Numerical experiments on a 100-dimensional Rosenbrock problem and three standard PDE-governed inverse problems show that our method delivers competitive or superior accuracy compared with MCMC, UKI, and SVGD baselines across all tested configurations, with the most pronounced advantages emerging in challenging scenarios such as high-noise observations and high-dimensional parameter spaces.
9. 基于核的势能平均场博弈与无偏随机傅里叶 $U$-统计量
Kernel-based potential mean-field games with unbiased random Fourier $U$-statistics
作者: Yumiharu Nakano
类别: math.OC, cs.LG, math.NA, stat.ML
发布日期: 2026-05-28
arXiv ID: 2605.29371v1
链接: http://arxiv.org/abs/2605.29371v1
摘要: We study the subclass of potential mean-field games in which the running interaction cost and the terminal target cost are both expressed through reproducing-kernel maximum mean discrepancy (MMD) penalties, and develop a computational framework that exploits this kernel structure. Both costs are estimated from finite-sample empirical distributions using a random Fourier U-statistic representation that is unbiased and has linear cost in the batch size. The drift of the controlled diffusion is parametrized by a neural network and trained via stochastic gradient descent. For this subclass we prove a sample-level almost-sure convergence theorem and an explicit almost-sure rate of convergence, under coupled rate conditions on the penalty parameter, the random-feature count, the sample size, and the optimization tolerance. The framework includes the kernel-MMD-penalty Schrödinger bridge problem as the special case of a vanishing interaction cost. Numerical experiments illustrate the method on the Schrödinger bridge problem in dimensions up to one hundred, and on an electric vehicle charging coordination problem with per-vehicle physical heterogeneity, where an aggregate-demand congestion cost represents price-feedback competition at the population level and the terminal MMD penalty shapes the state-of-charge distribution at the deadline.
10. 非结构样条上的加权求积
Weighted Quadrature on Unstructured Splines
作者: Ji Sheng, Xiaodong Wei, Falai Chen
类别: math.NA
发布日期: 2026-05-28
arXiv ID: 2605.29334v1
链接: http://arxiv.org/abs/2605.29334v1
摘要: This work presents a weighted quadrature (WQ) method to fast assemble Galerkin matrices based on unstructured spline surfaces. The method is developed upon a particular variant of unstructured splines, namely the bicubic analysis-suitable unstructured T-splines (ASUTS). While existing WQ approaches have significant speedup for structured splines (e.g., B-splines), their extension to unstructured splines faces several challenges: (1) lack of a global parametric domain for defining quadrature points, (2) a varying number of basis functions across elements that complicates the determination of the optimal number of quadrature points, and (3) ill-conditioned underdetermined linear systems that must be solved to find the quadrature weights. To solve these issues, we first define the WQ rule directly in the physical domain. Second, we specify the number of quadrature points function-wise (rather than element-wise), which naturally satisfies the well-posedness condition, namely the number of unknown weights no less than that of exactness constraints. Third, we employ the truncated Singular Value Decomposition to improve the conditioning of the underdetermined systems by discarding extremely small singular values, which are caused by the splines around extraordinary points. Several different model problems are studied, such as Poisson’s problem, the biharmonic problem, and the nonlinear heat transfer problem. In the end, a variety of numerical tests are performed to demonstrate the accuracy and efficiency of the proposed method.
11. 关于随机迭代方法的子空间约束预处理
On subspace-constrained preconditioning for randomized iterative methods
作者: Yonghan Sun, Hou-Duo Qi, Deren Han, Jiaxin Xie
类别: math.NA
发布日期: 2026-05-28
arXiv ID: 2605.29304v1
链接: http://arxiv.org/abs/2605.29304v1
摘要: In this paper, we further investigate and refine the subspace-constrained preconditioning technique to enhance the theoretical and numerical convergence properties of randomized iterative methods for solving linear systems. In particular, we design a QR-like factorization that transforms the original linear system into an equivalent block-orthogonal form, thus avoiding the full-rank assumptions adopted in existing work. Moreover, this reformulation reduces the problem to solving a smaller linear system with a favorable singular value distribution, provided an appropriate initial point is employed. The proposed framework can be implemented implicitly within the iteration and does not require explicitly constructing either a preconditioner matrix or a preconditioned linear system, which eliminates the prohibitive cost of forming a fully preconditioned system. Furthermore, we construct orthogonalized search directions from stochastic gradients and develop accelerated variants of the framework. We prove that the proposed algorithmic framework converges linearly in expectation. Numerical experiments demonstrate the benefits of the proposed preconditioning strategy.
12. 一类 Gross-Pitaevskii 型特征值问题的弱 Galerkin 方法
The weak Galerkin method for a class of Gross-Pitaevskii type eigenvalue problems
作者: Wei Lu, Qilong Zhai
类别: math.NA
发布日期: 2026-05-28
arXiv ID: 2605.29264v1
链接: http://arxiv.org/abs/2605.29264v1
摘要: This paper aims to employ the weak Galerkin method to solve a class of nonlinear eigenvalue problems. We proved the weak Galerkin scheme produces lower bound for the energy. Moreover, by the post-processing technique, we obtain lower bound for the ground state eigenvalue. Finally, numerical experiments are provided to validate the theoretical analysis.
13. 多保真度本征正交分解
Multifidelity Proper Orthogonal Decomposition
作者: Nicole Aretz, Karen Willcox
类别: math.NA, cs.CE, math-ph
发布日期: 2026-05-28
arXiv ID: 2605.29213v1
链接: http://arxiv.org/abs/2605.29213v1
摘要: This paper introduces a multifidelity formulation that reduces the computational cost of the proper orthogonal decomposition (POD) of a high-fidelity model by leveraging data from cheaper, lower-fidelity models. POD is a prevalent technique for extracting a low-dimensional basis from training data to achieve subsequent dimension reduction or reduced-order modeling. In scientific and engineering applications, the training data are typically numerical snapshot solutions of a high-fidelity model, and computation of a sufficiently rich snapshot set can be prohibitively expensive, especially when sampling over a high-dimensional parameter space. Insufficient snapshot training data risks overfitting and poor generalizability of the POD basis to outside the training regime. Our multifidelity POD (MFPOD) formulation reallocates computational budget to cheaper, low-fidelity models that can be sampled more extensively. MFPOD then weights high- and low-fidelity snapshot data via a control-variate formulation to guarantee an unbiased estimate of the expected high-fidelity least-squares projection error. The MFPOD subspace is chosen to minimize the estimate of this projection error, and converges in probability to the same subspace as single-fidelity POD in the limit of an arbitrarily large budget. For restrictive computational budgets, the MFPOD cost function has (under some assumptions) lower variance than the POD cost function, which makes the MFPOD subspace more robust against variations in the training data and thus less prone to overfitting. For a numerical example modeling the velocity of the Pine Island glacier, MFPOD achieves the same accuracy as single-fidelity POD with an order of magnitude reduction in the offline computational cost of snapshot generation.
14. 用于生成随机物理系统轨迹的随机提升方法
Stochastic Lifting for Generating Trajectories of Stochastic Physical Systems
作者: Jules Berman, Tobias Blickhan, Benjamin Peherstorfer
类别: cs.LG, cs.AI, math.NA
发布日期: 2026-05-28
arXiv ID: 2605.29194v1
链接: http://arxiv.org/abs/2605.29194v1
摘要: Many stochastic physical systems evolve smoothly over time in the sense that the distribution of states changes regularly across time steps. The transition from current state to the next state can often be modeled as the combination of a smooth map and an explicit source of randomness. Stochastic Lifting exploits this structure by attaching an independent, high-dimensional random label to each state transition in the training data and fitting a transition map from the current state and label to the next state using a standard regression loss. The labels act as auxiliary coordinates that let the model represent multiple plausible next states from similar current states, avoiding collapse to a mean prediction in the finite-sample size regime. At inference, fresh labels are sampled at each time step and the learned map is rolled forward autoregressively, generating diverse trajectories with a single network evaluation per time step.
15. 基于模态矫正的定向边扩散方法用于笛卡尔对流-扩散问题
Modal-Rectification-Based Directional Edge Diffusion for Cartesian Convection–Diffusion Problems
作者: Gossrin Jean-Marc Bomisso, Ali Ouattara Kouma
类别: math.NA
发布日期: 2026-05-27
arXiv ID: 2605.29186v1
链接: http://arxiv.org/abs/2605.29186v1
摘要: Centered finite-difference discretizations of convection–diffusion equations may oscillate when convection dominates at the mesh scale. For homogeneous Dirichlet problems with constant coefficients on uniform Cartesian grids, we derive ADSC (Adaptive Directional Sparse Correction), a local directional edge-diffusion correction guided by modal rectification of the centered-stencil Fourier symbol. The ideal modal reference damps modes independently, but its exact nodal action is nonlocal; ADSC replaces it by a nearest-neighbor positive semidefinite correction. For a regularized operator with activation fixed by an auxiliary sequence, we prove consistency, fixed-epsilon energy stability, and conditional discrete H^1-seminorm convergence. The implemented iteration instead uses activation generated by the computed solution. For that fully coupled nonlinear problem we prove existence and qualitative L^2 compactness/convergence only; uniqueness, convergence of activation updates, and energy-norm rates remain open. Numerical tests show selective extrema control, reduced modal-dominance indicators, and a low-cost few-shot variant. Comparisons with upwinding, SUPG, and AFC-inspired strategies are diagnostic rather than claims of uniform superiority.
16. 一种用于超弹性材料非线性有限元分析的变分量子算法
A Variational Quantum Algorithm for Nonlinear Finite Element Analysis of Hyperelastic Materials
作者: Uditnarayan Kouskiya, Caglar Oskay
类别: quant-ph, math.NA, physics.comp-ph
发布日期: 2026-05-27
arXiv ID: 2605.29181v1
链接: http://arxiv.org/abs/2605.29181v1
摘要: This manuscript explores a variational quantum formulation for nonlinear elasticity problems arising from hyperelastic material models, targeting near term noisy intermediate scale quantum (NISQ) devices. The approach leverages the potential energy structure of hyperelasticity and employs a hybrid quantum classical framework in which the energy functional is evaluated using parameterized quantum circuits and optimized through classical routines. To enable implementation on current quantum hardware, polynomial approximations of the nonlinear strain energy density are introduced, yielding a representation compatible with variational quantum algorithms. The methodology is demonstrated on a one dimensional NeoHookean material model using finite element discretizations with first and second order shape functions and nonhomogeneous boundary conditions. Numerical experiments investigate the influence of the polynomial approximation order on the accuracy and efficiency of the proposed approach, illustrating its feasibility for near term quantum devices.
17. 用于真实数据能源消耗预测校正的集成得分滤波
Ensemble Score Filtering for Real-Data Energy Consumption Forecast Correction
作者: Ruoyu Hu, Dahai Yu, Feng Bao, Guang Wang, Guannan Zhang
类别: cs.LG, math.NA
发布日期: 2026-05-27
arXiv ID: 2605.29072v1
链接: http://arxiv.org/abs/2605.29072v1
摘要: Accurate estimation and forecasting of energy consumption are important for power-system operation, planning, and demand-side management. In practice, however, complete and timely measurements may not always be available, and the observed data can be partial, noisy, or delayed. This motivates the use of learned forecasting models for predicting the evolving consumption state, together with data assimilation methods for sequential forecast correction. In this work, we study a high-dimensional data assimilation problem for real energy-consumption data. \modeltext{The forward prediction is supplied by a pretrained black-box spatio-temporal forecasting model, which is treated as the state propagator in the filtering procedure.} We employ the Ensemble Score Filter (EnSF) to assimilate partial and noisy observations and to correct the forecast trajectory over time. The EnSF uses score-based diffusion models to approximate filtering distributions and avoids retraining neural-network score models during assimilation by using a closed-form score representation and Monte Carlo approximation. Numerical experiments demonstrate that open-loop propagation of the learned forecasting model can become unreliable over long horizons, while EnSF-based correction substantially improves state estimation. Comparisons with the Ensemble Kalman Filter (EnKF) further show that EnSF provides stronger correction under the nonlinear observation setting considered in this work.
18. 基于流形的阿达玛分解算法
Manifold-based Algorithms for the Hadamard Decomposition
作者: Nicolas Gillis, Subhayan Saha, Stefano Sicilia, Arnaud Vandaele
类别: math.OC, cs.LG, eess.SP, math.NA
发布日期: 2026-05-27
arXiv ID: 2605.28980v1
链接: http://arxiv.org/abs/2605.28980v1
摘要: Given a matrix $X$, and two ranks $r_1$ and $r_2$, the Hadamard decomposition (HD) looks for two low-rank matrices, $X_1$ of rank $r_1$ and $X_2$ of rank $r_2$, both of the same size as $X$, such that $X\approx X_1\circ X_2$, where $\circ$ is the Hadamard (element-wise) product. In most cases, HD is more expressive than standard low-rank approximations such as the truncated singular value decomposition (TSVD), as it can represent higher-rank matrices with the same number of parameters; this is because the rank of $X_1 \circ X_2$ is generically equal to $r_1 r_2$. In this paper, we first present some theoretical insights for HD, in particular a useful reformulation $X\approx WH^\top$ where $W$ and $H$ have $r_1 r_2$ columns and belong to certain manifolds. These allow us to develop three new algorithms for computing HD. The first one uses the representation $X\approx X_1\circ X_2$ and relies on the Manopt toolbox. The other two rely on the reformulation $X\approx WH^\top$: one is a block projected gradient method, and the other is a manifold-based gradient descent algorithm that does not require projection onto the feasible set. The last two algorithms are particularly effective for handling large sparse data. We also propose new initializations that allow us to improve the accuracy of the HD. We compare our algorithms and initialization strategies with the TSVD and with the state of the art. Numerical results show that the new methods are efficient and competitive on both synthetic and real data.