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1. 中子学反应扩散特征值问题的多尺度有限元方法
A Multiscale Finite Element Method for reaction-diffusion eigenproblems arising from neutronics
作者: Claude Le Bris, Albéric Lefort, Frédéric Legoll
类别: math.NA
发布日期: 2026-09-02
arXiv ID: 2609.02420v1
摘要翻译: 我们考虑具有振荡扩散系数和反应系数的反应扩散特征值问题。反应系数的量级很大：周期均匀化所识别的校正子方程同时涉及扩散算子和反应算子。我们使用多尺度有限元方法（MsFEM）研究该问题的数值逼近。这种经典方法是一种有限元型方法，它在一个特定的、与问题相关的基函数集上对振荡问题进行Galerkin逼近。基函数在离线阶段预先计算。受均匀化理论的启发，并利用一些滤波思想，我们证明了如何定义这些基函数以获得一种高效的方法。我们在周期情形和非周期情形下，对问题的标量值版本（此时为自伴的）和向量值版本（此时一般为非自伴的）进行了全面的数值实验，实验结果验证了该方法的有效性。一些理论论证补充了数值观察结果。
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链接: http://arxiv.org/abs/2609.02420v1
1. 研究对象和背景
我们考虑定义在有界区域 $\Omega\subset\mathbb{R}^d$ 上的反应-扩散特征值问题(1):
$$\Sigma_\varepsilon u_\varepsilon-\varepsilon^2\,\mathrm{div}(A_\varepsilon\nabla u_\varepsilon)=\lambda_\varepsilon\sigma_\varepsilon u_\varepsilon\;\text{in }\Omega,\qquad u_\varepsilon=0\;\text{on }\partial\Omega,$$目标是求最小特征对 $(\lambda_\varepsilon,u_\varepsilon)\in\mathbb{R}\times H^1_0(\Omega)$。系数 $A_\varepsilon,\Sigma_\varepsilon,\sigma_\varepsilon$ 以远小于区域尺寸的典型长度 $\varepsilon$ 振荡；$A_\varepsilon$ 可取值于对称 $d\times d$ 矩阵或四阶张量, $\Sigma_\varepsilon,\sigma_\varepsilon$ 为标量或矩阵, 且在以下意义上一致椭圆和有界: 存在 $\beta\ge\alpha>0$ 使 $\alpha|\xi|^2\le\xi^\mathsf{T}A_\varepsilon(x)\xi$, $|\eta^\mathsf{T}A_\varepsilon(x)\xi|\le\beta|\eta||\xi|$, 且 $\alpha\le\sigma_\varepsilon(x),\Sigma_\varepsilon(x)$ a.e. 因而问题适定, 最小特征值 $\lambda_\varepsilon$ 为实且单重, 特征函数可取正。该问题出现在中子学中: $u_\varepsilon$ 表示中子密度, 振荡系数源于介质非均匀性; 标量版本对应单能中子, 向量版本对应多能中子(多个能量族), 后者一般不再自伴。直接采用经典 $P_1$ 有限元需要网格尺度 $h\ll\varepsilon$ 以分辨系数振荡, 即使只计算最小特征值或特征函数的宏观形态, 计算代价也过高。为此需要多尺度方法。已有的多尺度方法, 如 HMM [1,13], LOD [6,21] 和 MsFEM [16,14,17], 主要用于纯扩散问题或相应特征问题, 尚未被系统适应到这里带大反应系数的反应-扩散特征问题(1); 本文填补这一空白。在周期框架下, 均质化理论给出如下关键事实(定理4, 源自[5]): 设 $(\lambda_\infty,\psi)$ 是胞元特征值问题(4)的最小正解,
$$\Sigma\psi-\mathrm{div}(A\nabla\psi)=\lambda_\infty\sigma\psi\quad\text{in }\mathbb{R}^d,\qquad \psi\ Y\text{-周期},$$则 $v_\varepsilon=u_\varepsilon/\psi(\cdot/\varepsilon)$ 满足纯扩散型广义特征问题(6):
$$-\mathrm{div}\big(\psi^2(\cdot/\varepsilon)A(\cdot/\varepsilon)\nabla v_\varepsilon\big)=\nu_\varepsilon\,\sigma(\cdot/\varepsilon)\psi^2(\cdot/\varepsilon)v_\varepsilon,$$其中 $\nu_\varepsilon=(\lambda_\varepsilon-\lambda_\infty)/\varepsilon^2$。这个变换把原问题中与反应算子的耦合转嫁到权重函数 $\psi$ 上, 是后面构造数值方法的基础。正文随后分别处理标量自伴、向量非自伴以及非周期情形, 提出两种 MsFEM 型离散并附理论估计与数值验证。
2. 主要定理或主要结果
在标量周期情形, 作者先给出两个已知均质化定理作为工具。定理4确认: 对所有 $m$ 有收敛 $\nu_{\varepsilon,m}\to\nu_{\star,m}$ 与 $v_{\varepsilon,m}\rightharpoonup v_{\star,m}$ in $H^1(\Omega)$, 其中 $(\nu_{\star,m},v_{\star,m})$ 是均匀化纯扩散问题(7)
$$-\mathrm{div}(\tilde A_\star\nabla v_\star)=\nu_\star\sigma_\star v_\star\;\text{in }\Omega,\quad v_\star=0\;\text{on }\partial\Omega$$的特征对, 且 $\sigma_\star=\int_Y\sigma\psi^2$, $[\tilde A_\star]_{ij}=\int_Y\psi^2 e_i^\mathsf{T}A(e_j+\nabla\tilde w_j)$, 其中校正子 $\tilde w_j$ 满足 (8):
$$-\mathrm{div}_y(\psi^2A(e_j+\nabla_y\tilde w_j))=0,\quad \tilde w_j\ Y\text{-周期}.$$若极限特征值简单(如第一个), 则整列收敛。定理5进一步保证, 若 $v_\star\in H^1_0\cap H^2$, 则一阶展开 $v_{\varepsilon,1}=v_\star+\varepsilon\sum_{j=1}^d\tilde w_j(\cdot/\varepsilon)\partial_jv_\star$ 满足 $\|v_\varepsilon-v_{\varepsilon,1}\|_{L^2}\to0$ 与 $\|\nabla v_\varepsilon-\nabla v_{\varepsilon,1}\|_{L^2}\to0$。基于这些结果, 作者在第2.2节提出初步 MsFEM: 取每个粗元 $K$ 上的局部问题(12)
$$-\mathrm{div}(\psi^2(\cdot/\varepsilon)A(\cdot/\varepsilon)\nabla\chi^{\varepsilon,\psi,i})=0\;\text{in }K,\quad \chi^{\varepsilon,\psi,i}=\chi^{P_1,i}\;\text{on }\partial K$$的解组成空间 $V_{\varepsilon,\psi,H}^{\text{interm}}$, 再乘上 $\psi(\cdot/\varepsilon)$ 得到原变量的基函数 $\phi^{\varepsilon,\psi,i}=\psi(\cdot/\varepsilon)\chi^{\varepsilon,\psi,i}$ 和空间 $V_{\varepsilon,\psi,H}$。在 $V_{\varepsilon,\psi,H}$ 上做 Galerkin 离散(16), 所得最小特征对 $(\lambda_H^{\varepsilon,\psi},u_H^{\varepsilon,\psi})$ 满足误差界(见定理14):
$$\frac{\|u_\varepsilon-u_H^{\varepsilon,\psi}\|_{H^1(\Omega)}}{\|u_\varepsilon\|_{H^1(\Omega)}}\le C\bigl(\varepsilon+R(\varepsilon)+\varepsilon\sqrt{\varepsilon/H}+H^2\bigr),$$其中 $R(\varepsilon)$ 由(9)定义, 随 $\varepsilon\to0$ 趋于零。第2.3节把方法改造成不依赖周期假设的实际方法, 思路是用一种滤波技术构造 $\psi(\cdot/\varepsilon)$ 的代理; 具体构造见原文(摘录未展开), 但数值结果显示: 在周期情形下两种方法精度几乎相同; 固定粗网格尺度 $H$ 时精度随 $\varepsilon$ 减小而提高; 当周期与准周期问题在标准 $P_1$ 下精度同样差时, 本方法给出相近良好精度; 缩小 $H$ 会提高精度, 但要求每个粗元仍包含足够多小尺度振荡。第3节将实际方法推广到向量值问题(多能中子), 其在一般情况下非自伴; 数值实验表明所得结论对标量和向量版本均成立。最后, 第4节给出初步方法在标量情形下的收敛性证明(即上述误差界的严格陈述), 第5节在受限设定下给出滤波方法的完整分析, 并在一般设定下辅以数值比较。需要指出, 从提供内容看, 向量情形的理论误差估计和一般非周期情形的完整收敛证明并未显式列为定理, 作者主要依赖 comprehensive numerical experiments。
3. 研究方法、关键技术和核心工具
核心困难在于原问题并不是以 $\mathrm{div}(A_\varepsilon\nabla)$ 为主导的纯扩散问题: 反应项 $\Sigma_\varepsilon u_\varepsilon$ 与已用 $\varepsilon^2$ 缩放的扩散项同阶, 因此经典的 MsFEM 基函数(在粗元内部求解 $\mathrm{div}(A_\varepsilon\nabla\chi)=0$)不能直接用来逼近特征对。作者的第一个关键转化是引入胞元特征函数 $\psi$ 做未知函数替换 $u_\varepsilon=\psi(\cdot/\varepsilon)v_\varepsilon$。由定理4, $v_\varepsilon$ 满足纯扩散广义特征问题(6)。这一步把大反应效应“吸收”到权重 $\psi^2$ 之中, 使问题变成标准的、扩散占优的算子, 从而可以套用 MsFEM 的离线在线框架。在初步方法中, 作者在粗网格上为每个节点 $i$ 求解局部 Dirichlet 问题(12):
$$-\mathrm{div}(\psi^2A\nabla\chi^{\varepsilon,\psi,i})=0\ \text{in }K,\quad \chi^{\varepsilon,\psi,i}=\chi^{P_1,i}\ \text{on }\partial K,$$用这些局部解代替多项式形函数; 然后为了回到原变量, 将每个基函数乘以 $\psi(\cdot/\varepsilon)$ 得到 $\phi^{\varepsilon,\psi,i}$。注意乘 $\psi$ 这一步不是任意的, 它保证了 Galerkin 空间 $V_{\varepsilon,\psi,H}$ 能表示特征函数 $u_\varepsilon$ 的振荡包络; 若直接用 $v_\varepsilon$ 的 MsFEM 空间, 并不能同时逼近 $u_\varepsilon$。这里的创新点在于把均质化理论的已知结果(定理4)用作数值格式的设计原理。
第二个重要创新是处理非周期问题: 实际方法不再需要知道周期 cell 的 $\psi$。他们从 [10,8] 的滤波思想出发, 构造一个代理函数来近似 $\psi(\cdot/\varepsilon)$, 该代理应满足两条性质: (i) 构造过程不依赖周期; (ii) 在周期情形下它可靠逼近 $\psi(\cdot/\varepsilon)$。滤波的作用是从振荡的系数和局部解中提取慢变的调制包络; 摘录没有给出滤波算子的具体公式和算法流程, 但强调了这个代理一旦构造出来, 就可以像初步方法那样生成 $\phi^{\varepsilon,\psi}_{i}$ 型基函数。这样 MsFEM 就摆脱了周期性假设, 可以用于任意微观结构。
在误差分析中, 作者用到了几个关键工具。一是 $v_\varepsilon$ 满足的问题(6)在标量情况下是自伴的纯扩散型问题, 因而可借助 min-max 原理对特征值 $\nu_\varepsilon$ 建立先验界, 并用能量估计控制 $v_\varepsilon$ 的 $H^1$ 模; 这些估计在 [5] 中已经得到, 本文直接使用。二是运用周期均质化的一阶校正子展开式 $v_{\varepsilon,1}=v_\star+\varepsilon\sum\tilde w_j(\cdot/\varepsilon)\partial_jv_\star$ (定理5), 该展开式不仅给出精确特征函数的近似, 还是构造局部 Dirichlet 问题边界条件与误差分解的出发点。初步方法误差界中出现的 $\varepsilon\sqrt{\varepsilon/H}$ 项是 MsFEM 局部谱误差与均质化误差相互作用的结果, 而 $H^2$ 项反映粗有限元空间对光滑极限解的逼近, $\varepsilon+R(\varepsilon)$ 项来自均质化及校正子展开误差。定理14的证明细节不在摘录中; 从文章结构看, 它应是在标准能量估计框架中逐项控制上述误差分量。对于滤波方法, 第5节的分析被刻意限制在一个简化设定中, 即可以严格控制代理与真 $\psi(\cdot/\varepsilon)$ 之差的情形; 在更一般的周期、准周期和非周期情形, 作者用数值实验说明鲁棒性。此外, 向量值情形本质非自伴, 特征问题和谱隙的结构更难处理, 作者没有给出其完整收敛定理, 只是通过数值实验展示方法可用。从文本可见, 作者真正的方法论贡献不是提出新的有限元空间或后处理技巧, 而是将均质化中的“除以最小特征函数”技巧与滤波近似相结合, 为反应-扩散特征问题设计出可离线预处理、对微结构不敏感的多尺度基函数。
4. 与之前工作的比较
与经典多尺度有限元文献相比: 本文所研究的问题不是常见的二阶椭圆算子特征问题(例如 $-\mathrm{div}(A_\varepsilon\nabla u)=\lambda u$ 一类), 而是带大反应项 $\Sigma_\varepsilon$ 的方程(1)。在均质化极限中, 该方程的最小特征值 $\lambda_\varepsilon$ 不以 $\varepsilon^2$ 量级趋近某个平凡极限, 而是逼近胞元特征值 $\lambda_\infty$; 同时特征函数 $u_\varepsilon$ 含有周期胞元特征函数 $\psi(\cdot/\varepsilon)$ 作为强振荡包络。正因为如此, 如果只是沿用标准 MsFEM 中求解 $\mathrm{div}(A_\varepsilon\nabla\chi)=0$ 的基函数, 离散空间无法再现 $\psi$ 因子; 也不像 HMM/LOD 方法那样可以将微分算子直接视为标准二阶形式。作者特意指出, 在他们知识范围内, 没有现成的 MsFEM 推广覆盖该问题, 这一空白正是本文贡献所在。
与均质化理论 [2,5,20] 的关系: 定理4来自 [5], 定理5来自 [2], 这些是纯理论结果。本文的推进在于把它们从理论刻画变成可计算的数值算法: 先用它们定义基函数(初步方法), 再引入滤波代理消除对胞元解的依赖(实际方法)。在周期情形, 初步方法数值结果与滤波方法几乎一致, 这反过来验证了滤波代理的正确性。但在一般非周期或向量值情形, 极限问题本身尚未有像周期情形那样完备的显式均匀化公式, 因此作者没有提出新的均匀化理论, 而是依托数值实验。
与只做滤波或均匀化的工作比较: 滤波技术最初由 [10] 提出, 后来 [8] 在周期均匀化背景下使用。本文将该思想用于 MsFEM 离线基函数构造, 其作用不是简单平滑短波, 而是提取局部问题解的慢变调制以逼近未知的 $\psi(\cdot/\varepsilon)$, 从而使格式可应用于非周期几何。
误差尺度与代价: 初步方法的误差界 $C(\varepsilon+R(\varepsilon)+\varepsilon\sqrt{\varepsilon/H}+H^2)$ 可视为 MsFEM 针对该问题的自然推广。若固定 $\varepsilon$, 标准 $P_1$ 方法需要网格尺度 $h\ll\varepsilon$, 而本方法允许粗网格尺度 $H$ 大于或等于 $\varepsilon$, 只需 offline 阶段在每个粗元上用更细网格计算局部基函数。数值结果显示, 当 $H$ 固定时误差随 $\varepsilon$ 减小而减小, 且减小 $H$ 会提高精度, 但要求每个粗元含足够多的振荡周期。这些是多尺度方法的常见性质, 本文在反应-扩散特征问题中给出了定量验证。
局限与未竟之处: 从提供内容无法确认滤波方法在一般设定下的完整收敛定理; 作者明确说明第5节分析限于 restricted setting。向量值问题由于非自伴, 文中主要提供数值证据; 理论证明是否完全覆盖这一情形未有明确断言。此外, 误差界中的常数 $C$ 是否依赖于 $\varepsilon/H$ 之比, 以及如何依赖 $\alpha,\beta$ 和谱隙, 摘录没有展开。与直接使用极小网格的标准有限元相比, MsFEM 的优势是多查询场景: 计算多个特征对、时间依赖演化或系数空间重组时, 离线基函数可重复使用(参 [19, Chapter 4-5]), 这正是中子学应用(如 [4] 所述)中需要反复求解问题(1)的典型情况。该计算收益是文章开篇强调的动机, 但本文没有给出实际 wall-clock 时间对比。
了解领域论文（摘要翻译）
1. 神经算子逼近强连续凸单调半群
Neural operators approximate strongly continuous convex monotone semigroups
作者: Jonas Blessing, Philipp Schmocker, Alessandro Sgarabottolo
类别: math.NA, cs.LG, math.AP, math.PR, stat.ML
发布日期: 2026-09-02
arXiv ID: 2609.02727v1
链接: http://arxiv.org/abs/2609.02727v1
摘要翻译
中文标题: 神经算子逼近强连续凸单调半群
摘要翻译: 我们通过用神经算子学习其Chernoff型一步算子来逼近强连续凸单调半群。首先，我们引入所谓Chernoff-神经算子的一般类，并在一个万能逼近定理中证明它们可以任意好地逼近Chernoff一步算子。通过利用加权Hölder空间之间的稳定性估计，一步逼近误差可以通过迭代传播，从而得到相应半群的万能逼近。其次，我们引入更专门的包络-神经算子类用于包络半群，这使得我们能够推导出定量逼近率。最后，我们在来自非线性偏微分方程、随机最优控制和模型不确定性下的随机过程的几个数值例子中展示了这些神经算子的有效性。
2. 面向自组织表面几何的耦合体-表面问题的无网格求解器
A meshfree solver for coupled bulk-surface problems with self-organizing surface geometry
作者: Lennart J. Schulze, Alejandra Foggia, Ivo F. Sbalzarini
类别: cs.CE, math.NA, physics.comp-ph
发布日期: 2026-09-02
arXiv ID: 2609.02489v1
链接: http://arxiv.org/abs/2609.02489v1
摘要翻译
中文标题: 面向自组织表面几何的耦合体-表面问题的无网格求解器
摘要翻译: 在许多系统中，可变形表面或界面与周围体相流体之间的相互作用，与运动表面内部固有的时空动力学相耦合。实例包括肿瘤生长、生物组织形态发生、心脏力学、多相表面活性剂化学、增材制造、衣物磨损以及反应性燃烧流。求解此类问题既需要用于跟踪和分辨表面的几何计算算法，也需要用于求解表面及周围体相中耦合控制方程的数值方法。在此，我们提出一种用于此类具有可变形界面的耦合体-表面问题的全无网格数值求解器。该求解器隐式地跟踪表面，基于与周围流体相耦合的应力平衡来求解动态表面几何。我们展示了在增长球体上质量守恒情形的收敛性，并求解了不可压缩Navier-Stokes流体与表面内非线性反应-扩散动力学相耦合的体-表面问题。最后，我们展示了一个生物形态发生模型，通过双向耦合同时求解动态表面形状和弯曲表面上的场。
3. 表面Stokes-Helfrich模型的流函数公式——表面有限元离散与验证
A Stream-Function Formulation for the Surface Stokes-Helfrich Model - Surface Finite Element Discretization and Validation
作者: Enno Igel, Maik Porrmann, Axel Voigt
类别: math.NA
发布日期: 2026-09-02
arXiv ID: 2609.02347v1
链接: http://arxiv.org/abs/2609.02347v1
摘要翻译
中文标题: 表面Stokes-Helfrich模型的流函数公式——表面有限元离散与验证
摘要翻译: 我们引入了表面Stokes-Helfrich模型的一种流函数公式，并研究了这些方程在空间上的任意拉格朗日-欧拉（ALE）表面有限元（SFEM）离散和时间上的半隐式离散。与静止或给定演化表面上的表面Stokes模型的特殊情形相比，表面压力仍然作为未知量，并且需要在每个时间步重构。如同在给定演化表面上的表面Stokes模型一样，该公式还需要一个额外的梯度势。针对单连通表面的代表性问题设定，我们进行了详细的数值比较：将所提流函数公式的ALE-SFEM空间离散和类似半隐式时间离散与相应的速度-压力公式的ALE-SFEM空间离散和类似半隐式时间离散进行比较，证明了该方法的有效性。
不相关论文（基本信息）
1. GRADSOLVE：GPU上ODE集成的快速精确梯度
GRADSOLVE: fast exact gradients for ODE ensembles on GPUs
作者: Alessio Spurio Mancini
类别: cs.MS, cs.DC, cs.LG, math.NA
发布日期: 2026-09-02
arXiv ID: 2609.02876v1
链接: http://arxiv.org/abs/2609.02876v1
摘要: Ordinary differential equations (ODEs) underlie models in science and engineering, and many applications need derivatives of their solutions with respect to parameters. Ensembles of independent trajectories suit graphics processing units (GPUs), but current GPU software forces a trade-off: the fastest ensemble solvers cannot be differentiated in reverse mode at the speed they solve, and the solvers built for differentiation solve more slowly. No single tool has yet offered a reverse-mode gradient at the speed of a fused-kernel solve. We present GRADSOLVE, an open-source JAX library for solving and reverse-mode differentiating low-dimensional ODE ensembles on NVIDIA GPUs. It records the steps an adaptive solver accepts and differentiates a fixed-step replay of them; the returned gradient is the exact discrete adjoint of those steps, the same derivative Diffrax returns by default, obtained more cheaply from a fixed-length chain than from an adaptive loop. It targets ensembles differentiated many times against one recorded mesh, keeps Diffrax as a fallback, and supports explicit and Rosenbrock integrators. Used as a solver, GRADSOLVE’s forward-only kernel ran 2.8x faster than DiffEqGPU.jl; used for gradients, once a record exists, it computed them 5.6-14.1x faster than Diffrax’s checkpointed adjoint at matched forward-state accuracy across three GPU generations, the advantage narrowing on large ensembles and, on stiff systems, down to parity at tight accuracy. GRADSOLVE is released at https://github.com/ECLIPSE-AI4Science/gradsolve.
2. 具有未知粘性的定常Navier-Stokes方程中的连续数据同化：稳健且高效的求解器与快速参数恢复
Continuous data assimilation in steady Navier-Stokes equations with unknown viscosity: robust and efficient solvers and fast parameter recovery
作者: L. Rebholz, J. Reyes, J. Whitehead
类别: math.NA, math.AP
发布日期: 2026-09-02
arXiv ID: 2609.02862v1
链接: http://arxiv.org/abs/2609.02862v1
摘要: Recent advances in equation discovery methods such as SINDy have highlighted the growing interest in identifying governing parameters and models directly from data. In this work, we take a complementary approach grounded in analysis and numerical PDE methods: we recover an unknown viscosity in steady Navier-Stokes equations (NSE) from partial incompressible flow observations using continuous data assimilation (CDA). We propose a simple and efficient parameter recovery algorithm and also a nonlinear solver for CDA-NSE. Together, this creates a highly efficient technique for recovering an unknown viscosity from partial solution data. Our analysis establishes the well-posedness of steady CDA-NSE, quadratic convergence of the parameter recovery algorithm, and quadratic convergence of a CDA-Picard + CDA-Newton nonlinear solver. Numerical experiments illustrate that the methods are very effective in restoring parameters quickly, even with poor initial guesses.
3. 分段线性边界区域上Vlasov输运的稳定秩自适应步进-截断有限体积法
A stable rank-adaptive step-and-truncate finite volume method for Vlasov transport on domains with piecewise linear boundaries
作者: André Uschmajew, Andreas Zeiser
类别: math.NA
发布日期: 2026-09-02
arXiv ID: 2609.02721v1
链接: http://arxiv.org/abs/2609.02721v1
摘要: We consider the numerical solution of the linear Vlasov transport equation on bounded spatial domains with inflow boundary conditions based on low-rank approximation. We combine a finite volume discretization with a rank-adaptive step-and-truncate scheme for the resulting matrix ODE. The spatial and velocity meshes may be unstructured, while suitable numerical fluxes retain a separated space-velocity representation. For homogeneous inflow, we show that the low-rank scheme inherits the $L_2$ stability and CFL restriction of the underlying full finite volume forward Euler method. In addition, the low-rank approximation error is bounded explicitly in terms of the truncation tolerances, avoiding the modeling error associated with tangent-space projections in dynamical low-rank approximation. Numerical experiments in 1d1v and 2d2v confirm the predicted error behavior. In 2d2v, the method handles an unstructured spatial mesh with nonzero inflow and a full tensor-product discretization of approximately $5.8\cdot10^{10}$ phase-space cells while the numerical rank is at most twelve.
4. 球面上带限谱再现中的正性损失
Positivity loss in bandlimited spectral reproduction on spheres
作者: Hao-Ning Wu
类别: math.NA, math.CA
发布日期: 2026-09-02
arXiv ID: 2609.02695v1
链接: http://arxiv.org/abs/2609.02695v1
摘要: How small can the positivity loss be for an $N$-bandlimited spectral operator that exactly reproduces all modes up to degree $L$? For spherical polynomial approximation on $\mathbb S^d$, we prove that the smallest possible excess of the uniform operator norm above 1, equivalently the least positivity loss, is of sharp order $\left({L}/{(N+1)}\right)^2$ when $1\le L< N$. The lower bound follows from a Fejér peak test and a concentration estimate for bandlimited kernels, while a matching upper bound is obtained by correcting a positive Jackson operator with a smooth filter. We illustrate the result in three settings. On the circle, taking $N=sL-1$, this determines the sharp order of the generalized-projection constant above 1 and identifies the gap between the $s^{-1}$ excess of delayed de la Vallée–Poussin means and the optimal $s^{-2}$ order. For filtered hyperinterpolation, whose operator norm has long been known to be uniformly bounded, we give a quantitative lower bound on its separation from the positivity threshold 1. Finally, we identify an operator-level obstruction to maximum principles.
5. 参数化逼近的本征正交分解方法的计算研究
Computational study of Proper Orthogonal Decomposition methods for parametric approximations
作者: Bosco García-Archilla, Alicia García-Mascaraque, Julia Novo
类别: math.NA
发布日期: 2026-09-02
arXiv ID: 2609.02583v1
链接: http://arxiv.org/abs/2609.02583v1
摘要: This paper studies the computational implementation of proper orthogonal decomposition reduced-order models (POD-ROMs) for evolutionary parametric time-dependent partial differential equations (PDEs). For a one-parameter model, many papers in the literature build the correlation matrix by projecting onto $L^2(Ω)$ even though optimal pointwise error estimates are proved when projecting onto $H^1_0(Ω)$. We compare both scenarios and observe the similar performance in practice. Additionally, to get the POD approximation it is necessary to compute the nonlinear term in the reduced equations. This requires a high computational cost that increases when the problem gets more complex. Numerical results in this paper show different approaches to address this issue. Discrete Empirical Interpolation Method (DEIM) is the most efficient approach. It reduces computational time by approximately half compared to computing the whole FEM formulation by a tensor construction while maintaining the same accuracy. For a multiparameter model, we analyze the new and standard method proposed in \cite{newmethod} using a two-dimensional Brusselator model to complement the results and support the error analysis with a more complex system.
6. 参数化Boltzmann方程的降阶模型及其在反问题中的应用
Reduced order model for parametric Boltzmann equation and its application to inverse problems
作者: Shanyin Tong, Jingwei Hu, Fengyan Li, Zhaiming Shen, Yunan Yang, Jiajia Yu
类别: math.NA, math.OC
发布日期: 2026-09-02
arXiv ID: 2609.02578v1
链接: http://arxiv.org/abs/2609.02578v1
摘要: The Boltzmann equation plays an important role in modeling mesoscopic behavior in a wide range of scientific and engineering applications. However, its numerical solution is computationally expensive due to the high dimensionality of the model and the nonlinear nonlocal collision operator, especially for steady-state problems that require iterative solvers. This cost becomes prohibitive for inverse problems, where the induced optimization problem requires repeated forward solves. In this work, we propose a reduced-order model (ROM) for the parametric Boltzmann equation to address this computational challenge. The ROM constructs a low-dimensional approximation space for the parameter-induced solution manifold through a residual-based greedy strategy, and the reduced solution is then obtained via residual minimization over the reduced space, subject to mass conservation. The overall efficiency of the ROM is achieved by exploiting the quadratic structure of the collision operator and a precomputed separable approximation of the collision kernel. The resulting ROM is further applied to a thermally-driven inverse problem for reconstructing collision parameters from the observed macroscopic temperature data. This is accomplished either by directly replacing the PDE constraint with the ROM, leading to a bilevel optimization formulation, or by reformulating the task as a single-level optimization problem through the Karush–Kuhn–Tucker (KKT) conditions. Numerical experiments in both collision-dominated and transport-dominated cases are performed to demonstrate the efficiency and accuracy of the proposed ROM and its effectiveness in inverse problems. In particular, the resulting inverse problem is computationally much more tractable, achieving speedups of several orders of magnitude over that based on the full-order model while maintaining comparable accuracy.
7. Smagorinsky 湍流模型的虚拟元逼近的数值分析
Numerical Analysis of the Virtual Element Approximation for the Smagorinsky Turbulence Model
作者: Karol L. Cascavita, Francesca Marcon, Maria Strazzullo
类别: math.NA
发布日期: 2026-09-02
arXiv ID: 2609.02433v1
链接: http://arxiv.org/abs/2609.02433v1
摘要: In this paper, we consider the Smagorinsky model for the Navier-Stokes equations within a virtual element framework. Under the standard assumption of small data, we prove the existence and uniqueness of a solution. Assuming more regularity to the solutions, we derive the known convergence rates $h$ for the a priori error estimates of the Smagorinsky model in two dimensional domains. We additionally prove that divergence-free virtual discretizations provide improved convergence orders, with weaker regularity assumptions than in the finite element literature. We conclude the paper with numerical results that corroborate the theory.
8. $C^1$三次样条空间中几何依赖的秩亏
Geometry-dependent rank defect in $C^1$ cubic spline space
作者: Xinyu Wu, Jiansong Deng
类别: math.NA, math.CO
发布日期: 2026-09-02
arXiv ID: 2609.02424v1
链接: http://arxiv.org/abs/2609.02424v1
摘要: Determining the dimension of the $C^1$ cubic spline space $S_3^1(\mathcal{T})$ on an arbitrary nondegenerate planar triangulation has remained unresolved since the 1970s. Schumaker’s lower bound includes a local correction $σ$ for singular interior four-stars, and it was conjectured that this bound is always attained. We disprove this conjecture by constructing a one-parameter family of nondegenerate realizations of a fixed 18-triangle complex, with only the central vertex moving as $v_6(t)=(t,0)$ on the admissible interval $I=(-3/4,24/55)$. The family exhibits three distinct cases. For $t\in I\setminus\{1/5,3/83\}$, the lower bound is attained and $\dim S_3^1(\mathcal{T}(t))=33$. At $t=3/83$, the central four-star is singular, $σ=1$, and the resulting dimension 34 is exactly accounted for by the classical local correction. At $t=1/5$, however, all interior vertices are nonsingular and $σ=0$, yet $\dim S_3^1(\mathcal{T}(1/5))=34>P_{\mathcal{T}(1/5)}(1,3)=33$. The smoothing-cofactor calculation shows that the dependence at $t=3/83$ is confined to the central vertex block, whereas the dependence at $t=1/5$ couples all seven interior vertex cycles even though every individual block has full row rank. A complementary Bernstein–Bézier calculation gives the same dimension profile. Thus the singular-four-star correction does not capture every geometry-dependent contribution to $\dim S_3^1(\mathcal{T})$; genuinely global compatibility must also be taken into account.
9. 紧支撑多节点Shepard算子在环面上的散乱数据插值
Scattered data interpolation on the torus by compactly supported multinode Shepard operators
作者: F. Dell’Accio, F. Di Tommaso, R. Lammirato, F. Larosa
类别: math.NA
发布日期: 2026-09-02
arXiv ID: 2609.02259v1
链接: http://arxiv.org/abs/2609.02259v1
摘要: We introduce a compactly supported multinode Shepard operator for the interpolation of scattered data on the torus embedded in $\mathbb{R}^3$. The method combines local polynomial interpolation of total degree $d\in \mathbb{N}$ with compactly supported Shepard-type weights, so that the approximation at each evaluation point depends only on neighbouring stencils of nodes. The torus is treated as an algebraic surface defined by a quartic polynomial, and Gröbner bases are used to construct reduced polynomial spaces on the surface by removing the redundancy induced by the defining equation. This yields local Vandermonde systems adapted to the toroidal geometry. We discuss the metric structure of the torus, show the local equivalence between the periodic parameter distance and the Euclidean distance inherited from $\mathbb{R}^3$, and use this equivalence to motivate the compact support construction. We establish a uniform error estimate in terms of the maximal support radius and the local Lebesgue constants. Under uniform locality and stability assumptions, the method converges with order $d+1$ with respect to the fill distance. Numerical experiments on analytical test functions confirm polynomial reproduction and exhibit an error decay consistent with the theoretical analysis. The approach is also tested on Computational Fluid Dynamics data mapped onto the torus, including the interpolation of the velocity components, the reconstruction of the velocity magnitude from the interpolated components, and the reconstruction of a tangent velocity field through an orthonormal lifting of the interpolated components.
10. 基于多尺度解分解的次扩散方程非均匀BDF3-L2方法的误差估计
Error estimate of the nonuniform BDF3-L2 method for subdiffusion equations via multiscale solution decomposition
作者: Wenlin Qiu, Kexin Li, Yiqun Li, Hao Zhang
类别: math.NA
发布日期: 2026-09-02
arXiv ID: 2609.02178v1
链接: http://arxiv.org/abs/2609.02178v1
摘要: Numerical experiments reported by Quan and Wu [SIAM J Numer Anal 61 (2023) 2106-2132] show that the observed temporal convergence rates of nonuniform L2 methods for subdiffusion models are not consistent with the theoretically predicted order $3-α$. This discrepancy suggests that a more refined analysis is needed and motivates the development of a nonuniform BDF3-L2 method for the subdiffusion equation. To account for the initial solution singularity, we employ the multiscale solution decomposition to decompose the original solution and approximate a smoother unknown variable that satisfies the subdiffusion model with a smoother source term. The resulting formulation, however, involves restrictive high-order boundary conditions on the source term and initial data. To overcome this difficulty, we introduce a spectral truncation technique that requires only slightly stronger regularity of the data and a controllable truncation error. We establish high-order regularity estimates of the solution to the truncated problem and develop a nonuniform BDF3-L2 method for its numerical approximation, based on which we derive a rigorous error estimate of temporal convergence order $2+α$. Numerical experiments are carried out to substantiate the theoretical findings.
11. 随机投影保持几何的精确极限：高斯模型中的距离恢复、最近邻排序与协方差形状
Exact Limits of Random Projections for Preserving Geometry: Distance Recovery, Nearest-Neighbor Rankings, and Covariance Shape in Gaussian Models
作者: Piyush Sao
类别: cs.LG, cs.IT, math.NA, math.PR, math.ST
发布日期: 2026-09-02
arXiv ID: 2609.02155v1
链接: http://arxiv.org/abs/2609.02155v1
摘要: The Johnson-Lindenstrauss (JL) lemma guarantees that a random projection of $n$ points to $m=O(\varepsilon^{-2}\log n)$ dimensions preserves pairwise squared distances within relative error $\varepsilon$ with high probability, and this dimension order is asymptotically optimal. In high dimensions, however, distances concentrate around a baseline while key geometric information lies in much smaller fluctuations. We show that the JL bound can therefore be uninformative about retained geometry: an independent Gaussian replacement map can satisfy it even though the replacement cloud is independent of the original data. We then ask how well any decoder can recover a feature $f(D)$ of a squared distance $D$ from a linear sketch. Under squared-error loss, the optimal decoder is conditional expectation, so recovery defines a linear operator whose singular values quantify feature recovery. For isotropic Gaussian data ($Σ=σ^2 I_d$), we diagonalize this operator in closed form. For fixed $k$ with $m,d-m\to\infty$, its $k$th singular value satisfies $\ell_k\approx(m/ d)^{k/2}$. This yields three sharp consequences. A rank-$m$ sketch retains at most an $m/d$ fraction of the variance of any feature of one squared distance. If $m\to\infty$ and $m/d\to0$, the expected Kendall correlation is $\frac{2}π\sqrt{m/d}(1+o(1))$; for fixed $q$, nearest- neighbor agreement tends to $1/q$. Yet one projection can satisfy the JL bound while mean Kendall correlation vanishes when $\log n\ll m\ll d$. After removing scale, Haar-averaged retained covariance-shape information is $(m/d)^2$. Thus JL distance preservation does not quantify the geometry available for comparison or inference.
12. Fokker-Planck优化的两个伴随视角：微观-宏观对应
Two Adjoint Perspectives on Fokker-Planck Optimization: A Microscopic-Macroscopic Correspondence
作者: Kathrin Hellmuth, Qin Li, Yunan Yang
类别: math.NA, math.AP, math.OC, math.PR
发布日期: 2026-09-02
arXiv ID: 2609.02072v1
链接: http://arxiv.org/abs/2609.02072v1
摘要: The Fokker-Planck equation admits both a macroscopic Eulerian description through probability densities and a microscopic Lagrangian description through stochastic trajectories. Consequently, optimization problems constrained by the Fokker-Planck equation can be formulated from either perspective. Surprisingly, the corresponding adjoint equations appear to be fundamentally different: the macroscopic adjoint is governed by the backward Kolmogorov equation, whereas the microscopic adjoint evolves pathwise along stochastic trajectories. In this note, we reconcile these two formulations by establishing their correspondence in the continuum setting. We further show that, although their discrete gradients no longer coincide after discretization, both provide consistent numerical approximations of the continuum gradient. Explicit convergence rates are established for both discretization strategies.
13. FlashKAN：基于截断幂形式的B样条KAN
FlashKAN: B-Spline KANs via Truncated Power Form
作者: Naveen Mysore
类别: cs.LG, math.NA
发布日期: 2026-09-02
arXiv ID: 2609.01956v1
链接: http://arxiv.org/abs/2609.01956v1
摘要: Kolmogorov-Arnold Networks (KANs) place learnable B-spline activations on network edges rather than fixed activations on nodes. The standard Cox-de Boor recursion evaluates these activations through k sequential passes for degree-k splines, consuming over 90% of forward-pass time. FlashKAN replaces this recursion with the truncated power form, a classical result from approximation theory that expresses each uniform cubic B-spline as five (x)_+^3 terms at shifted knot positions. This paper makes three contributions: (1) a torch.compile-fused implementation that collapses these operations into a single GPU kernel, eliminating all recursion, span lookup, and scatter-gather operations; (2) a bounded-coordinate stabilization that clamps the normalized input to [0, k+1], preventing the catastrophic cancellation that historically motivated the Cox-de Boor recursion; and (3) a production-ready, open-source package (pip install flashkan) that serves as a drop-in replacement for existing KAN layers.
14. 时间依赖偏微分方程的界约束有限元逼近
Bounds-Constrained Finite Element Approximation of Time-Dependent Partial Differential Equations
作者: Robert C. Kirby, John D. Stephens
类别: math.NA
发布日期: 2026-09-01
arXiv ID: 2609.01915v1
链接: http://arxiv.org/abs/2609.01915v1
摘要: Finite elements provide accurate and efficient methods for the numerical solution of partial differential equations by means of restricting variational problems to finite-dimensional approximating spaces. However, they do not, in general, guarantee enforcement of bounds constraints inherent in the original problem. We propose two approaches to enforcing bounds constraints for time-dependent problems. First, we propose general projective methods which result from a systematic modification of any abstract time-stepping scheme. Second, we present a monolithic technique for which we take a modified formulation of implicit single-stage Runge-Kutta methods and general implicit multistep methods as prototypical examples. By solving a constrained optimization problem, we are able to ensure that the bounds constraints are enforced by the approximate solution at the discrete time levels, obtaining (formally) high order methods both in space and time. Numerical examples for the linear heat and advection equations and nonlinear Allen-Cahn equation are given.
15. 从分子动力学中学习一类一般的可容许多物种碰撞算子
Learning a general class of admissible multi-species collision operators from molecular dynamics
作者: Yue Zhao, Andrew Christlieb, Huan Lei
类别: physics.comp-ph, math.NA, physics.plasm-ph
发布日期: 2026-09-01
arXiv ID: 2609.01845v1
链接: http://arxiv.org/abs/2609.01845v1
摘要: We develop a structure-preserving, data-driven collision operator for spatially homogeneous multi-species kinetic systems from molecular dynamics (MD). The operator consists of diagonal self-collision blocks and ordered off-diagonal cross-species blocks to describe intra- and inter-species momentum and energy exchange. Within a local and point-wise identifiable kernel class, we develop the necessary and sufficient condition for the admissible kernel class satisfying the conservation laws, the H-theorem, and the frame indifference. Unlike the classical Landau operator, the off-diagonal kernels are not restricted to be symmetric under permutation of the two velocity variables. This unique structural freedom captures the distinct responses of different species to unresolved correlations and many-body effects arising from micro-scale particle interactions. The equivalent parameterizable kernel formalization enables us to learn a generalized data-driven collision operator directly from MD, where the low-rank tensor representations and random sampling are used to achieve efficient kernel training and numerical simulation. Numerical experiments show that the learned operator accurately predicts transport coefficients and the non-equilibrium relaxation, while retaining discrete conservation and entropy production. In particular, it captures plasma kinetics in the moderately coupled regime, where the predictions of both the Landau and the data-driven model restricted to velocity-permutation symmetry show significant discrepancies.
16. 抛物型偏微分方程的双重自适应求解器
A Double-Adaptivity Solver for Parabolic PDEs
作者: Gregor Gantner, Robin Smeets, Rob Stevenson
类别: math.NA
发布日期: 2026-09-01
arXiv ID: 2609.01748v1
链接: http://arxiv.org/abs/2609.01748v1
摘要: We study minimal residual space-time finite element discretizations of linear parabolic initial value problems in canonical space-time variational form. To deal with the arising dual norm, we introduce the Riesz lift of the residual as an additional variable. Quasi-optimality of the primal variable of the mixed system follows from a uniform inf-sup condition. This condition is known to be satisfied for finite element spaces w.r.t. prismatic partitions of the space-time cylinder that allow for a decomposition into time-slabs. We prove that this condition cannot be expected to hold otherwise. To recover stability for general partitions and the data at hand, we derive an a posteriori condition on the error between the exact Riesz lift of the residual and its Galerkin approximation – being the secondary variable of our system – under which the primal variable is quasi-optimal. We derive a posteriori error estimators for both variables, and use them in a double-adaptive loop that alternates test-space with trial-space enrichment. We illustrate our findings with numerical experiments in $1+1$ and $2+1$ dimensions.