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1. 一种用于 CutFEM 的多重网格方法及其收敛性
A Multigrid Method for CutFEM and its Convergence
作者: Michal Wichrowski
类别: math.NA
发布日期: 2026-09-03
arXiv ID: 2609.04067v1
摘要翻译: 我们为带顶点块光滑子的几何多重网格发展了收敛性理论，该多重网格应用于 Poisson 问题的切割有限元离散。该框架处理非继承的层形式以及物理区域与活动区域之间的不匹配。利用离散延拓性质，我们证明了关于网格尺寸和切割几何一致的两层收敛界，并在两层收敛率的附加小性假设下证明了 W-循环界。数值实验有意采用更强的 V-循环，对于该循环此处不宣称收敛界。
Comment: 无
链接: http://arxiv.org/abs/2609.04067v1
1. 研究对象和背景
本文研究基于笛卡尔背景网格的切割有限元法 (CutFEM) 在高阶多项式离散下的 Poisson 问题的几何多重网格求解。设 $\Omega$ 为有界区域, 具有 $C^2$ 边界 $\Gamma = \partial\Omega$; 将 $\Omega$ 嵌入到均匀笛卡尔网格族 $\{\mathcal{M}_\ell\}$ 中, 网格尺寸递推 $h_\ell = h_{\ell-1}/2$。对每个水平 $\ell$, 活动网格由与 $\Omega$ 相交的单元组成, $\mathcal{M}_{\ell,\Omega}:=\{K\in\mathcal{M}_\ell : K\cap\Omega\neq\emptyset\}$, 其并集为活动域 $\Omega_\ell\supset\Omega$; 在活动域上定义 $Q_p$ 张量积有限元空间 $V_\ell$。连续问题为经典 Poisson 边值问题
$$ -\Delta u = f \quad \text{in } \Omega, \qquad u = g \quad \text{on } \Gamma . $$离散使用 Nitsche 方法配合 ghost penalty 稳定化: 对所有 $u,v\in V_\ell$, 定义
$$ A_\ell(u,v)=a_\ell(u,v)+g_\ell(u,v), $$其中
$$ a_\ell(u,v)=(\nabla u,\nabla v)_\Omega-(\partial_n u,v)_\Gamma-(u,\partial_n v)_\Gamma+\gamma_D h_\ell^{-1}(u,v)_\Gamma, $$以及 ghost penalty
$$ g_\ell(u,v)=\gamma \sum_{F\in\mathcal{F}^G_\ell}\sum_{k=1}^{p}\frac{h_F^{2k-1}}{(k!)^2}\left(\llbracket\partial_n^k u\rrbracket,\llbracket\partial_n^k v\rrbracket\right)_F,\quad \gamma\in(0,1], $$其中 $\mathcal{F}^G_\ell$ 为边界区域中的面集, 即至少有一个相邻单元是被切割单元格的活跃面。难题在于: 当 $\Gamma$ 穿过背景网格时, 被切割单元在物理域内的支撑可能退化, 使未加稳定的能量模不再一致强制。Ghost penalty 通过约束相邻单元间法向导数的跳跃, 让被切单元从相邻完整单元借取稳定性, 在固定多项式次数下恢复对切割位置一致的离散延拓性、强制性和条件数界。本文要建立的多重网格收敛理论面向上述格式, 期望产生网格尺寸和切割位置一致的收缩率。但由于水平间 ghost penalty 面集 $\mathcal{F}^G_\ell$ 不同、活动域 $\Omega_\ell$ 与物理域 $\Omega$ 不一致, 双线性形式 $A_\ell$ 不是由粗层形式 Galerkin 继承得到的, 因此标准 fitted-mesh 多重网格分析失效。特别地, 延长算子不能按 Galerkin 选取, 而是自然注入; 同时顶点 patch smoother 的子空间之间的正交性也会被 ghost penalty 破坏。本文试图弥补的空白是为这种带顶点 patch smoother 的几何多重网格提供严格的收敛性基础, 并显式追踪多项式次数与 ghost 权重 $\gamma$ 的影响。这也是随时间演化或移动边界问题高效求解的重要一步: CutFEM 的优点是避免随边界演化重新网格化, 而有效的多重网格求解器正是其大规模应用所需要的。
2. 主要定理或主要结果
作者证明的主要结果可归结为以下几个定理与引理。 引理1 (延拓稳定性): 取自然注入延拓 $I_\ell^\uparrow: V_{\ell-1}\to V_\ell$, 即函数在 $\Omega_\ell$ 上逐点相等。设 $w=I_\ell^\uparrow v$。在 Assumption 1 的几何条件下, 存在与水平 $\ell$ 和切割位置无关的常数 $C_{\mathrm{pr}}=(2C_A/c_A)^{1/2}$, 使得能量模估计
$$ |||I_\ell^\uparrow v|||_\ell \le C_{\mathrm{pr}}|||v|||_{\ell-1}, \qquad \forall v\in V_{\ell-1}, $$成立。证明中关键事实是细层 ghost penalty 不超过粗层 ghost penalty 的一半, 即 $g_\ell(w,w)\le \frac12 g_{\ell-1}(v,v)$, 该估计不含迹、Markov 或逆不等式, 因而转移稳定性在网格范数意义下与 $p$ 无关。 引理2 (ghost 能与 $H^1$ 半范的控制): 对单一网格立方体 $K$ 的边 $H$, 面 $F\subset\partial K$, $q\in Q_p(K)$, 有
$$ \|\partial_n^k q\|_{0,F}^2 \le C_T (p+1)^2 H^{-1} (C_M^2 p^4 H^{-2})^{k-1} |q|_{1,K}^2,\qquad 1\le k\le p, $$其中 $C_T$ 与 $C_M$ 分别是迹常数和 $L^2$ Markov 常数。由此得到对任何 $u\in V_m$, 全局 ghost 能估计 $g_m(u,u)\le \gamma\,2d\,\Theta'_p |u|_{1,\Omega_m}^2$, 且
$$ \Theta'_p:=2C_T(p+1)^2\sum_{k=1}^{p}\frac{(C_M^2 p^4)^{k-1}}{(k!)^2}. $$该常数满足 $\log\Theta'_p=2p\log p+O(p)$, 呈超指数增长, 是后文稳定分解常数 $C_{\mathrm{sd}}(p)$ 中唯一的 $p$ 依赖来源。 定理1 (稳定分解): 在 Assumption 1 与离散延拓性质 (9) 下, 顶点 patch 子空间族 $\{V_{\ell,j}\}$ 构成 $V_\ell$ 的稳定加性 Schwarz 分解: 存在常数 $C_{\mathrm{sd}}(p)$, 与 $\ell$ 和切割配置无关, 使得每个 $u\in V_\ell$ 有分解 $u=\sum_j u_j$, $u_j\in V_{\ell,j}$, 且
$$ \sum_j |||u_j|||_\ell^2 \le C_{\mathrm{sd}}(p)\, |||u|||_\ell^2. $$定理2 (两水平收敛): 基于定理1的稳定分解与引理1的转移稳定性, 两水平方法的收缩率存在一个与网格尺寸 $h_\ell$ 和边界切割几何无关的上界 $\rho_{\mathrm{two}}<1$。即对任意水平 $\ell$, 由一次预平滑、一次粗空间修正和一次后平滑组成的迭代收缩因子满足 $\rho \le \rho_{\mathrm{two}}$, 其中 $\rho_{\mathrm{two}}$ 只依赖 $p$、Nitsche 参数和 Assumption 2 中的常数。 推论2 (W 循环): 若两水平收缩率满足小性条件 $q\le 1/4$, 则 W-循环的收缩率一致有界, 且该界与切割几何无关。该条件是证明多层收敛的归纳假设, 文中并未对不满足该条件的参数范围给出无条件 W-循环结论。 并行性与半乘法 smoother: 标准 $2^d$ 着色在 ghost penalty 下不能使同色 patch 精确正交。对同色索引 $i\neq j$, 体积项与 Nitsche 项贡献消失, 但残留耦合 $A_\ell(u,v)=g_\ell^{ij}(u,v)$ 可能非零。为保留廉价 $2^d$ 着色, 作者提出半乘法 smoother: 仅对边界条带的跨色耦合做加性处理, 并对条带更新引入切割无关的阻尼。推论4证明该半乘法更新对应的收敛界与精确乘法更新的估计只差一个与切割无关的常数因子。数值实验采用无阻尼变体, 不属于该推论假设。若改用 stride-3 着色, 则同色 patch 的 ghost 面集互不重叠, 从而恢复精确乘性正交性, 此为推论3。 命题3 ($p$ 依赖的定位): 设 $\sigma_\ell(p)$ 为边界条带上精确 cut sweep 的收缩因子。存在与 $\ell$ 和切割无关的 $\sigma(p)<1$, 使得 $\sigma_\ell(p)\le \sigma(p)<1$ 对所有水平成立。然而 $\sigma(p)$ 关于 $p$ 的精确依赖仍是开放问题。由此导出一个平衡条件 (文中式 (51)), 给出所需的 cut-sweep 重复次数 $n_c^*$, 使条带收缩因子不超过内部/粗网格收缩的参考值; 该诊断与实测迭代次数进行了对比。此外 Remark 11 指出, 在导数跳跃型 ghost penalty 权重族内, 无法通过调整权重使文中建立的组装界成为 $p$ 的多项式。 关于 V-cycle: 文中所有多层数值实验均采用 V-cycle, 但 V-cycle 的收敛界不在证明范围内。作者指出, 对非继承形式建立 V-cycle 界需要 BPX 型工具, 这是一个未解决的方向。
3. 研究方法、关键技术和核心工具
证明的主要障碍来自三层: (i) 多层形式 $A_\ell$ 的非继承性, 使得粗层函数在细层上的能量不能由粗层能量直接控制; (ii) 活动域 $\Omega_\ell$ 与物理域 $\Omega$ 不重合, 被切割单元在物理域外延部分需要由 ghost penalty 和离散延拓性质桥接; (iii) 并行着色下 ghost penalty 使同色 patch 之间出现非零耦合, 破坏乘性 Schwarz 的精确性。 针对障碍 (i), 作者没有使用复杂的拟插值延拓, 而是证明自然注入本身在稳定能量模下是稳定的。证明的关键在于对细层 ghost 面按粗网格分类: 设细面 $F$ 的两个细单元共享同一粗父单元, 则 $F$ 落在粗单元内部, 粗函数在该单元上是整体多项式, 所有跳跃消失; 若细面跨越粗面, 则细面上的法向导数跳跃拼起来恰为粗面上的跳跃, 且权重 $h_\ell^{2k-1}\le \frac12 h_{\ell-1}^{2k-1}$, 从而 $g_\ell(w,w)\le \frac12 g_{\ell-1}(v,v)$。这一估计不依赖任何多项式逆不等式, 是完全组合和几何层面的, 因此转移在网格范数意义下保持 $p$-无关。 针对障碍 (ii), 论文将 ghost 能的局部控制归结为引理2。这里利用笛卡尔网格的张量积结构与 $Q_p$ 空间的特殊性质: 任一坐标方向导数仍是 $Q_p$ 函数, 沿该方向的割线是一维多项式。将一元端点迹估计 (对应 Gauss-Lobatto 节点 Christoffel 数) 张量化, 再对余法向低阶项反复应用一维 $L^2$ Markov 不等式, 得到对第 $k$ 阶法向导数在面 $F$ 上的界。该界中的放大因子 $(C_M^2 p^4)^{k-1}$ 正是超指数增长 $\Theta'_p$ 的来源。作者指出此上界是不紧的, 因为完整 Bessel 级数的主项 $k\approx C_M p^2$ 落在求和截断 $k\le p$ 之外; 但它在现有分析框架中无法避免。 在得到单层 ghost 能与 $H^1$ 半范的控制后, 作者结合 Assumption 2 的离散延拓性质 (9), 利用标准 Schwarz 分解论证证明定理1的稳定分解。具体而言, 对任意 $u\in V_\ell$, 先在物理域内部用内部自由度构造一个在物理域上与 $u$ 相同且跨过边界条带光滑的中间函数, 再把由 ghost penalty 控制的边界条带部分分配到各个 patch。离散延拓性质保证了分解后 patch 函数的 ghost 能可以被 $|u|_{1,\Omega_\ell}$ 控制, 从而得到与切割无关的常数。这一步提取了稳定分解常数 $C_{\mathrm{sd}}(p)$。 针对障碍 (iii), 作者在分析中把 smoother 对 patch 的访问方式区分为乘性序贯和按色组并行。由于 ghost penalty 的作用, 同色 patch 的能量内积可能非零, 经典的色组加性近似会引入额外误差。论文引入半乘法 smoother, 将色组内部仍作乘性, 但只在边界条带的耦合面上作加性处理, 并利用条带耦合的稀疏性给出收敛估计。对阻尼参数 $\theta$ 的选择, 推论4证明了其界与切割无关。为了提供对照, 又提出 stride-3 着色, 使同色 patch 在背景网格上至少相距三个单元, 此时 ghost 跳跃面完全分离, 能量正交性恢复。作者还比较了两种局部空间的覆盖能力: 经典 Neumann 型 patch 空间 $V^D_{\ell,j}$ (仅取 patch 内部节点自由度) 与全残差空间 $V^{\mathrm{FR}}_{\ell,j}$ (额外包含所有支撑完全落在 patch 内、但节点位于活动域边界上的自由度), 后者迭代次数降低约 1.6–1.8 倍。 数值实验中, 作者故意使用比已证明范围更强的 V-cycle 和半乘法无阻尼变体, 这不是证明对象, 而是用于反映实际行为。实验还采用张量积高阶 ghost penalty 求值, 研究了全局权重 $\gamma$ 的缩放、与 $p$ 有关的半定阈值等。这些实验展示了理论界中出现的 $p$ 依赖在实际迭代中确实出现, 并支持了"将 $p$ 依赖视为边界条带现象"的解释。
4. 与之前工作的比较
与已有工作的差异可从多个维度看。 第一, 对 CutFEM 上多重网格收敛理论本身。此前 [14] 将顶点 patch smoother 推广到 CutFEM 并给出数值实验, 但没有任何收敛性证明; 本文填补了这一空白, 给出了两水平与条件 W-cycle 的严格界。相比 fitted 网格上的顶点 patch 理论 [26,24,35,4,34,12,13], 本文显式量化了因切割几何产生的 $p$ 退化: 在 fitted 情形下该 smoother 是 $p$-鲁棒的, 但 cut 情形下 convergence 常数随 $p$ 恶化, 作者将原因局部化为边界条带上的 ghost-penalty 耦合。这一发现与 [14] 观察到的迭代次数随 $p$ 增长一致, 并解释为 ghost penalty 跨层与跨 patch 作用所致。 第二, 与界面非贴体多重网格工作 [22,21] 相比, 那些方法为 unfitted 界面问题设计了特殊延拓算子和界面校正 smoother, 其界面更新需求解全局界面未知量; 本文保留标准自然注入延拓和顶点 patch 局部问题, 没有引入额外全局界面求解器, 因此更易并行和矩阵-free 实现。代价是需要 Assumption 1 的几何条件 (边界只简单穿过网个, 正 reach, 水平嵌套) 和离散延拓性质 (9), 而 [22,21] 可能允许更复杂的界面拓扑。 第三, 与 [18] 相比, 后者构造了 CutFEM 空间的一个两子空间加性 Schwarz 预处理器, 将空间分解为背景空间与 cut 基函数张成空间, 对界面和 fictitious domain 问题得到最优条件数, 但那是两子空间方法且不含粗空间? 实际上其最优性依赖于界面子空间的性质。本文则使用多水平粗空间校正, 因此条件数不随水平数增长; 同时本文的 smoother 是局部顶点 patch 而非全局界面求解。 第四, 与 ghost penalty 稳定性文献 [5,7,23] 相比, 这些文献证明了离散延拓性质 (9) 并在固定 $p$ 下给出 cut-uniform 的强制性与条件数, 但没有将其嵌入到多网格框架。本文直接调用该性质, 并进一步追踪其常数对 $p$ 的依赖。与 [8] 中通过扩展有限元空间将延拓直接内建到离散空间 (从而避免 ghost penalty) 相比, 本文保留 ghost penalty 是为了不改变每个背景网格上的有限元空间, 从而保持层次间空间的包含关系 $V_{\ell-1}|_{\Omega_\ell} \subset V_\ell$。这个选择使 multigrid 转移简单, 但代价是引入了需要显式分析的 ghost 项。Remark 11 进一步指出, 在导数跳跃型 penalty 族内无法通过权重选择把组装界的 $p$ 依赖变为多项式, 因此若想恢复 $p$-鲁棒, 可能需要像 [8] 那样修改空间而不是只改形式。 第五, 关于 V-cycle 与 W-cycle。经典几何多重网格中 V-cycle 往往与 W-cycle 一样可证; 但本文由于形式非继承, V-cycle 界需要 BPX 型工具 [3], 文中只证明了 W-cycle 在 $q\le1/4$ 小性条件下收敛。数值实验使用 V-cycle 且效果良好, 说明实际 V-cycle 很可能也收敛, 但理论没有覆盖。这个局限是当前方法的明显短处, 也是与 fitted-mesh 多网格理论的重要差异。 第六, 对高阶 ghost penalty 的设计。作者没有采用通常固定 $\gamma=O(1)$ 的做法, 而是允许 $\gamma\in(0,1]$ 并可随 $p$ 改变。实验发现降低 $\gamma$ 能改善迭代次数。摘要中的 aligned two-cell 模型显现半定阈值约 $p^{-2}$: 当 $\gamma$ 低于该量级时, 由两个对齐单元构成的简化模型出现半定失效; 而 $p$ 次 cut 模态的可见尺度则随 $p$ 指数衰减。这提示: 要避免高阶模态被过度惩罚, $\gamma$ 可能需与 $p$ 反向缩放; 但作者说明这些实验并未证明组装修理后的全局算子具有相同的阈值, 只是对设计的启发。这种将 ghost-penalty 权重作为设计变量并与 $p$ 的退化相联系的分析, 在先前 ghost penalty 文献中未见系统展开。
了解领域论文（摘要翻译）
1. 自适应可细化的极样条离散微分形式：层次构造、正合性与应用
Adaptively-refinable polar-spline discrete differential forms: hierarchical construction, exactness, and applications
作者: Diogo C. Cabanas, Deepesh Toshniwal, Rafael Vazquez
类别: math.NA
发布日期: 2026-09-03
arXiv ID: 2609.03461v1
链接: http://arxiv.org/abs/2609.03461v1
摘要翻译
中文标题: 自适应可细化的极样条离散微分形式：层次构造、正合性与应用
摘要翻译: 在等几何分析中，表示圆盘状和球状几何的一种常用方法是在底层参数化中使用塌缩边奇点。在这种奇异参数化上所得的张量积 B 样条空间缺乏等几何分析所需的足够光滑性。极样条（Toshniwal 等，CMAME 2017）通过提取一个更光滑的子空间解决了这一问题，并已成功用于离散高阶偏微分方程，以及对 de Rham 复形进行保结构离散。后者对于电磁学和流体动力学等应用中出现的混合形式尤为重要。 本文的主要贡献有两点。我们将极样条构造与层次样条相结合，得到自适应可细化的极样条。此外，我们是在保结构方法的框架内进行这一工作的。即，我们展示了如何构造层次极样条 $k$-形式空间，证明了相应的基函数是线性无关的，并证明它们构成一个上同调正确的离散 de Rham 复形。 这些贡献为极几何上的自适应保结构模拟提供了坚实的数学基础。我们还提供了数值实验来验证理论，并通过必须保持结构的问题展示该构造的实际行为。这些实验包括：一项关于伪谐波的研究，该研究突出了在极坐标与张量积设定下，破坏上同调的细化方式有何不同；一个在局部细化的极网格上 Maxwell 特征值问题的离散化；以及一个 Hodge–Laplace 问题的自适应收敛性研究。
2. 用于模拟微波加热流的间断Petrov-Galerkin有限元框架
A discontinuous Petrov-Galerkin finite-element framework for the simulation of microwave-heated flows
作者: Oreste Marquis, Matthias Maier, Bruno Blais
类别: math.NA, physics.comp-ph
发布日期: 2026-09-02
arXiv ID: 2609.03155v1
链接: http://arxiv.org/abs/2609.03155v1
摘要翻译
中文标题: 用于模拟微波加热流的间断Petrov-Galerkin有限元框架
摘要翻译: 我们提出了一种用于模拟微波加热流的高阶多物理场求解器。该求解器将用于时谐Maxwell方程的间断Petrov-Galerkin (DPG) 有限元方法与用于热方程和不可压缩Navier-Stokes方程的连续Galerkin有限元方法耦合。我们针对多个基准问题验证了电磁求解器：矩形波导中的波传播、具有奇异解的腔体问题以及微波加热障碍物问题，并将我们的结果与文献中的数值和实验数据进行了比较。结果证实了实现的正确性，并展示了其利用DPG方法内置误差估计器进行自适应网格细化的能力。研究的最后部分通过模拟具有奇异几何特征的障碍物周围的微波加热流，展示了多物理场框架的能力。这些结果突出了所提出的框架在微波辅助化学过程的模拟和优化方面的潜力。最后，所开发的高阶多物理场求解器具有较低的内存占用，因为电磁求解器依赖于共轭梯度 (CG) 迭代求解器，而流体求解器以无矩阵方式实现，使得整体方法具有可扩展性，非常适合大规模并行模拟。
不相关论文（基本信息）
1. 贝叶斯概率数值方法中的Bernstein–von Mises定理
Bernstein–von Mises theorems for Bayesian probabilistic numerics
作者: Sascha Gaudlitz, Sven Wang
类别: math.ST, math.NA
发布日期: 2026-09-03
arXiv ID: 2609.04124v1
链接: http://arxiv.org/abs/2609.04124v1
摘要: We study probabilistic numerical methods for solving nonlinear PDEs from a Bayesian nonparametric perspective. Given noisy evaluations at random collocation points, we place a truncated Gaussian series prior on the unknown solution and establish contraction at the minimax nonparametric rate, up to a logarithmic factor. Our main results give Gaussian approximations of the posterior in positive-order Sobolev spaces and, under suitable conditions, in the uniform topology. This contrasts with classical ill-posed inverse problems, where Bernstein–von Mises theorems typically require substantially weaker topologies. Here, the observation operator is differential rather than smoothing, and inversion of its linearisation gains regularity, making these strong-topology results possible. The posterior may be centred at either the posterior mean or the posterior mode. We further prove that the Gaussian Laplace approximation is asymptotically equivalent to the true posterior at a $\sqrt{N}$-scale.
2. 由微分方程定义的边界约束的横截性条件
Transversality Conditions for Boundary Constraints Defined by Differential Equations
作者: Isaac M Ross
类别: math.OC, eess.SY, math.DG, math.NA
发布日期: 2026-09-03
arXiv ID: 2609.04084v1
链接: http://arxiv.org/abs/2609.04084v1
摘要: What are the transversality conditions for an optimal control problem when the boundary conditions are defined by differential equations? This seemingly bizarre question is motivated by trajectory optimization problems in the $N$-body system. The question, however, is more fundamental and goes beyond problems in astrodynamics to nonintegrable dynamical systems in general. The main contribution of this paper is the development of generic initial- and final-time transversality conditions for optimal control problems whose boundary conditions are defined in terms of differential equations with side conditions. The mathematical definition of differential boundary conditions are part of the foundations developed in this paper. To support the new fundamentals, the concept of coordinated/uncoordinated clock times and weak adjoint covectors are introduced. In the case of uncoordinated clock times, the new transversality conditions reveal that there exists a special situation where a weak adjoint covector is orthogonal to the vector field of the boundary differential equation. This condition is sharply different from the classical statement of orthogonality with respect to the endpoint manifold. The theorems developed in this paper are generic. An application of the theorems to several cases in the three-body problem are described in separate papers.
3. Sylvester方程解的条件性
Conditioning of solutions to the Sylvester equation
作者: Massimiliano Fasi, Behnam Hashemi
类别: math.NA
发布日期: 2026-09-03
arXiv ID: 2609.04050v1
链接: http://arxiv.org/abs/2609.04050v1
摘要: We partially answer an open problem, posed by Nick Higham, concerning the conditioning of solutions to Sylvester and Lyapunov equations. The question arises in the backward stability analysis of numerical algorithms for these equations. We first show that the solution to the Sylvester equation $AX-XB = C$ can be arbitrarily ill-conditioned even if $A, B, C$ and the Kronecker sum $I \otimes A - B^T \otimes I$ are all perfectly conditioned. We then derive general a priori bounds on the condition number of the solution, as well as bounds for the Sylvester equation when $A$ and $B$ are diagonalizable. We also provide lower bounds involving matrix exponentials and Zolotarev numbers. For the Lyapunov equation $AX+XA^T = -C$, we obtain upper bounds in two settings: (i) when $A$ is symmetric positive definite while $C$ is symmetric negative definite, and (ii) when $A$ is strictly dissipative and $C$ is symmetric positive definite.
4. 挤压丝材形状对3D混凝土打印可建造性的影响：一种几何信息驱动的深度学习-有限元方法
Influence of Extruded Filament Shape on Buildability in 3D Concrete Printing: A Geometry-Informed Deep Learning-FEM Approach
作者: Giacomo Rizzieri, Saif-Ur-Rehman, Jörg F. Unger, Annika Robens-Radermacher
类别: cs.CE, cs.AI, cs.LG, math.NA
发布日期: 2026-09-03
arXiv ID: 2609.04028v1
链接: http://arxiv.org/abs/2609.04028v1
摘要: The geometric morphology of deposited filaments can significantly influence the structural performance and stability of 3D concrete-printed (3DCP) structures. However, most finite element (FEM)-based approaches for buildability assessment represent printed layers as simplified rectangles, potentially limiting predictive accuracy. This study proposes a geometry-informed modelling framework that integrates the deep-learning-based filament shape prediction tool ShapeGen3DCP with a layer-activation FEM approach to investigate the effect of realistic filament geometries on buildability. The framework generates geometry-aware numerical models directly from material and process parameters, eliminating the need for experimental filament characterization or computationally intensive fluid-flow simulations. Validation against experimental data and a parametric study of rectilinear walls demonstrate that extrusion parameters and the resulting filament geometry can significantly influence buildability predictions. Realistic filament representations are particularly important for free-flow deposition, whereas layer-pressing strategies are less sensitive to geometric simplifications. Among the investigated representations, an elliptical approximation provides an effective balance between geometric fidelity and modelling simplicity. When rectangular representations are preferred to enable regular computational meshes for faster simulations, defining their dimensions based on volume conservation improves prediction reliability compared with calibrating them using either the maximum filament width or the interlayer contact width. Overall, the proposed methodology demonstrates the importance of incorporating filament geometry into 3DCP simulations and provides practical guidance for selecting efficient and accurate geometric representations for buildability assessment.
5. 可微混合建模用于从实验数据学习和优化化学输运过程
Differentiable Hybrid Modelling for Learning and Optimising Chemical Transport Processes from Experimental Data
作者: Arthur Jessop, Mohammed Alsubeihi, Ben Moseley, Ashwin Kumar Rajagopalan
类别: cs.CE, cs.LG, math.NA
发布日期: 2026-09-03
arXiv ID: 2609.04011v1
链接: http://arxiv.org/abs/2609.04011v1
摘要: Reliable transport models are essential when modelling and optimising many chemical engineering processes, yet, most models assume hand-picked constitutive laws which may not reflect reality, and often assume initial conditions are known exactly. Both restrictions can significantly bias model predictions and lead to systematic error when used in predictive and control settings. Black-box neural surrogate alternatives for modelling can better match real example data, but are confined to the task they were trained on and cannot be interrogated for physical consistency. Here we introduce a general-purpose differentiable hybrid modelling framework for transport processes, specifically for the case of population balance equations. Our framework integrates a JAX finite volume population balance solver with learnable neural network components which are trained to both discover constitutive laws and fit initial conditions from real experimental data, allowing us to better model real experimental transport systems. Furthermore, we use our framework for process optimisation, using its differentiability to allow us to direct optimising experimental settings for quantities of interest. This work highlights the huge potential of such differentiable hybrid modelling frameworks for learning and optimising any given chemical separation which involves mass, energy, and/or momentum transport.
6. 三角形网格最长边二分的闭包复杂度
Closure complexity of longest-edge bisection for triangular meshes
作者: Yuwen Li, Zhiyuan Yang
类别: math.NA
发布日期: 2026-09-03
arXiv ID: 2609.04003v1
链接: http://arxiv.org/abs/2609.04003v1
摘要: On triangular meshes, we analyze local mesh refinement based on longest-edge bisection equipped with the serial longest-edge propagation-path closure. Ties are resolved by terminal priority: if the incoming shared edge is a longest edge of the neighboring triangle, the pair is declared terminal and that edge is bisected. For every adaptive mesh sequence $\mathcal{T}_0, \mathcal{T}_1, \ldots, \mathcal{T}_L$ with marked subset sequence $\mathcal{M}_0, \mathcal{M}_1, \ldots, \mathcal{M}_{L-1}$, we prove the cumulative closure estimate $\#\mathcal{T}_L - \#\mathcal{T}_0 \lesssim \sum_{\ell=0}^{L-1}\#\mathcal{M}_\ell$. The proof has two ingredients. First, the finite-similarity-class theorem for planar longest-edge bisection turns the set of possible diameters into a finite union of $\sqrt{2}$-geometric lattices. Hence diameters grow by a uniform strict factor at every nonterminal step. This yields generation and spatial locality for all triangles created by a single mark. Second, a Binev–Dahmen–DeVore type charging argument converts this single-mark locality into the cumulative estimate.
7. 满足离散最大值原理的时间分数阶对流-扩散-反应方程的一种稳定化格式
A stabilized scheme satisfying the discrete maximum principle for a time fractional convection-diffusion-reaction equation
作者: Christos Pervolianakis
类别: math.NA
发布日期: 2026-09-03
arXiv ID: 2609.03808v1
链接: http://arxiv.org/abs/2609.03808v1
摘要: We study a time fractional convection-diffusion-reaction equation in a bounded domain $Ω\subset\mathbb{R}^2$. A stabilized numerical scheme satisfying a discrete maximum principle is constructed by combining the conforming linear finite element method with the algebraic flux correction method. The resulting semi-discrete scheme is nonlinear, and its well-posedness is established. Assuming nonsmooth initial data, we derive error estimates for the semi-discrete scheme using energy arguments. For the temporal discretization, we employ the L1 method, obtaining a fully discrete scheme for which we prove well-posedness and the discrete maximum principle. We also present numerical experiments that validate the order of convergence as well as we test our schemes to solutions that possess layers.
8. 近不可压缩线弹性问题的MINI混合有限元方法
Mini mixed finite element method for nearly incompressible linear elasticity problems
作者: Zhijin Guan, Yue Feng, Hehu Xie, Chenguang Zhou
类别: math.NA
发布日期: 2026-09-03
arXiv ID: 2609.03780v1
链接: http://arxiv.org/abs/2609.03780v1
摘要: This paper addresses the numerical solution of nearly incompressible linear elasticity boundary value problems and their associated eigenvalue problems. A mixed finite element formulation based on Mini element is proposed to circumvent the locking phenomenon that plagues standard low-order elements in the nearly incompressible limit. For the boundary value problem, we establish the well-posedness of mixed variational formulation and derive a priori error estimates that are uniform with respect to the Lamé constant $\underlineλ$, thereby proving the method’s locking-free property. For the eigenvalue problem, we develop an efficient non-nested augmented subspace algorithm designed within the mixed finite element framework. A comprehensive convergence analysis is provided for the discrete eigenvalue approximation and the proposed iterative solver, demonstrating that the convergence rates remain independent of $\underlineλ$. Numerical experiments on both problems confirm the theoretical conclusions, showing optimal convergence rates and robustness as $\underlineλ \to \infty$. The results validate the effectiveness of Mini element and proposed augmented subspace algorithm for reliable and efficient computation in the nearly incompressible regime.
9. 一个基于Darbo型定理的泛函积分方程的存在性结果，以及一个求解它的迭代算法
An existence result of a functional integral equation via Darbo type theorem and an iterative algorithm to solve it
作者: Bhuban Chandra Deuri, Anupam Das
类别: math.FA, math.NA
发布日期: 2026-09-03
arXiv ID: 2609.03733v1
链接: http://arxiv.org/abs/2609.03733v1
摘要: In our study, Darbo’s fixed point theorem and the coupled fixed point theorem have been extended. The two theorems were generalized using $C-class$ mapping. Using the Darbo type theorem, we gave an existence result for a functional integral equation along with an appropriate illustration. In addition, we explored the numerical solution of a given functional integral problem using modified homotopy perturbation.
10. 精确数据下离散化椭圆系数反问题的完全全局化求解器
A fully globalized solver for discretized inverse elliptic coefficient problems with exact data
作者: Bastian Harrach
类别: math.NA, math.OC
发布日期: 2026-09-03
arXiv ID: 2609.03631v1
链接: http://arxiv.org/abs/2609.03631v1
摘要: We consider finite-dimensional nonlinear inverse problems arising from finite element discretizations of elliptic inverse coefficient problems such as the Calderón problem with finitely many measurements and unknowns. Such inverse coefficient problems are notorious for their nonlinearity and ill-posedness, and numerical solvers tend to depend strongly on good initial values. In this work, we develop a new locally convergent algorithm with an explicit residual criterion that ensures convergence to the inverse problem solution, and a globalized variant that is guaranteed to automatically switch to the faster locally convergent algorithm after finitely many global search steps.
11. 残差神经网络克服半线性热方程的维数灾难
Residual neural networks overcome the curse of dimensionality for semilinear heat equations
作者: Ilkhom Mukhammadiev, Diyora Salimova
类别: math.NA, cs.LG, math.AP, math.PR
发布日期: 2026-09-03
arXiv ID: 2609.03626v1
链接: http://arxiv.org/abs/2609.03626v1
摘要: Rigorous results show that feedforward neural networks can overcome the curse of dimensionality in the numerical approximation of high-dimensional partial differential equations (PDEs), but comparatively little is known about residual neural networks (ResNets) in the nonlinear PDE setting. We prove that ResNets overcome the curse of dimensionality in the numerical approximation of solutions of semilinear heat equations with globally Lipschitz continuous, gradient-independent nonlinearities: under polynomial growth and network approximability hypotheses on the PDE data, there exist $η\in(0,\infty)$ and ResNets $Ψ_{d,\varepsilon}$, $d\in\mathbb{N}$, $\varepsilon\in(0,1]$, with at most $ηd^η\varepsilon^{-η}$ parameters whose realizations approximate the solution in dimension $d$ with an $L^2$-error of at most $\varepsilon$. The proof represents one deterministic realization of a multilevel Picard estimator by a ResNet whose shortcut connections transmit the spatial variable and a scalar accumulator, while the residual branches successively add the summands of the estimator. For ridge-sum initial conditions, admissible sigmoidal activations, and globally Lipschitz truncations of the nonlinearity, we obtain, for every $ξ>0$, the explicit bound $C_ξd^{4+ξ}\varepsilon^{-(3+ξ)}$ on the number of parameters.
12. 半离散二次Wasserstein能量与状态依赖的Langevin探索
Semi-discrete quadratic Wasserstein energy and state-dependent Langevin exploration
作者: Ran Gu, Gaoyue Guo, Kelvin Shuangjian Zhang
类别: math.NA, math.OC, math.PR
发布日期: 2026-09-03
arXiv ID: 2609.03405v1
链接: http://arxiv.org/abs/2609.03405v1
摘要: We study the semi-discrete quadratic Wasserstein energy. The energy is nonsmooth at collisions of sites. We prove local Lipschitz continuity on the full configuration space, together with global semiconcavity, coercivity, and dissipativity; show that every global minimizer is interior and collision free; and establish $C^2$ regularity on the collision-free configuration space. The gradient is expressed through the barycenters of the balanced Laguerre cells, while the Hessian is given by an explicit facet formula and satisfies a global one-sided bound. We also solve the one-dimensional problem explicitly in each ordering chamber and give a two-site example on the unit square with non-minimizing Lloyd fixed points. For $d\ge 2$, we then formulate an entropy-regularized relaxed control of the Langevin temperature. The controlled dynamics is strongly well posed, nonexplosive, and collision free. Its value function is a classical interior solution of the exploratory Hamilton-Jacobi-Bellman equation; the Laplacian of the value function is locally $C^1$, which yields a locally Lipschitz optimal temperature feedback. Independently of this optimal-control result, for every fixed Borel temperature rule bounded away from zero, and every sufficiently small step size, the associated Gaussian Euler chain is geometrically ergodic with a full-support invariant law. The raw iterates do not converge, whereas the best-so-far energy converges almost surely to the global minimum and the running record approaches the set of global minimizers.
13. 多维情形下随机特征方法的谱收敛
Spectral Convergence of Random Feature Method in Multiple Dimensions
作者: Pingbing Ming, Hao Yu
类别: math.NA, cs.AI, cs.LG, math.ST
发布日期: 2026-09-03
arXiv ID: 2609.03401v1
链接: http://arxiv.org/abs/2609.03401v1
摘要: We first prove spectral convergence of the random feature method (RFM) for multidimensional targets in Sobolev, Gevrey, ultra-analytic, and bandlimited classes. The analysis establishes general high-probability approximation estimates in the interpolation scale generated by a kernel integral operator. On a single event determined only by the sampled features, one random space approximates every target in a prescribed source ball; moreover, for each target, a single coefficient vector defines an approximant that attains spectral accuracy simultaneously in all admissible error norms. For both regularity-adapted frequency distributions and uniform distributions on growing frequency windows, the resulting rates range from super-exponential to algebraic, depending on the regularity of the target. Second, we establish abstract error estimates for strong- and weak-form RFM discretizations, thereby converting the preceding approximation bounds into convergence estimates for multidimensional second-order elliptic boundary value and eigenvalue problems. Finally, for random feature matrices (RFMtxs), we prove super-exponential singular-value decay with Fourier features and exponential decay with $\tanh$ features, together with corresponding condition-number lower bounds. The analysis identifies a common mechanism: the same spectral approximation that yields high accuracy also drives severe ill-conditioning.
14. 面向贝叶斯反问题的基于无限维连续归一化流的信息先验学习
Learning Informative Prior with Infinite-Dimensional Continuous Normalizing Flow for Bayesian Inverse Problem
作者: Yang Zhao, Junxiong Jia, Tao Zhou
类别: math.NA, cs.LG
发布日期: 2026-09-03
arXiv ID: 2609.03343v1
链接: http://arxiv.org/abs/2609.03343v1
摘要: This paper addresses infinite-dimensional Bayesian inference for inverse problem of partial differential equations with model parameters in infinite-dimensional Hilbert space. To effectively incorporate prior information, we propose a novel continuous normalizing flows based infinite-dimensional model. Specifically, by introducing a well-defined neural ordinary differential equation in infinite-dimensional space, a simple reference measure can be transformed into a more complex measure which encodes the prior information. A corresponding theoretical framework is established to ensure the well-posedness of our proposed Bayesian prior in infinite-dimensional space. We also provide training methods of the prior for two distinct data settings, along with two sampling algorithms for the resulting Bayesian posterior. The proposed framework is applied to three representative inverse problems: the simple smooth inverse problem, inverse scattering problem, and the inverse heat conduction problem. Numerical experiments support the theoretical analysis and demonstrate the efficiency of the proposed algorithms.
15. 仿射不变集成Langevin及其离散时间变体的几何遍历性
Geometric Ergodicity of Affine Invariant Ensemble Langevin and its Discrete Time Variants
作者: Hong Ye Tan, Yifan Chen
类别: math.ST, math.NA, math.OC
发布日期: 2026-09-03
arXiv ID: 2609.03326v1
链接: http://arxiv.org/abs/2609.03326v1
摘要: Affine-invariant ensemble samplers are widely used in Bayesian applications. However, their quantitative convergence theory, in particular geometric ergodicity, remains a basic open question. We study the affine invariant ensemble Langevin dynamics, an interacting particle system that uses the empirical covariance of the whole ensemble as a preconditioner. While effective in practice, theoretical understanding of this method is not available beyond plain qualitative convergence in total variation; a central difficulty is that the empirical covariance can approach singularity. This paper addresses this challenge. For potentials with bounded Hessian that are strongly convex outside a ball, we prove geometric ergodicity using a novel Lyapunov function that combines an inverse-covariance barrier with a coercive exponential energy. We then show that directly applying the Euler–Maruyama scheme can diverge with positive probability, even for a one-dimensional Gaussian target. This motivates a covariance-trace time regularization. We prove geometric ergodicity of the regularized diffusion and, for sufficiently small step size, of its unadjusted Euler–Maruyama discretization. We also show that the invariant distributions of the discretization converge weakly to the product target distribution as the step size tends to zero.
16. 一种求解某些自由边界值问题的新型并行方法
A novel parallel approach for solving some free boundary value problems
作者: Peeyush Singh, Amboru Yalmanda
类别: math.NA
发布日期: 2026-09-02
arXiv ID: 2609.03214v1
链接: http://arxiv.org/abs/2609.03214v1
摘要: This paper introduces a novel, efficient class of parallel direct and indirect iterative schemes to solve general obstacle and free boundary value problems. The uniqueness of the solution for the direct parallel method is established under the assumption that the model problem yields an $M$-matrix. The convergence analysis of the indirect approach is predicated on minimizing the discrete energy functional at each iterative stage of the linear approximation process. Under the aforementioned setting, we establish theoretical convergence results for both smooth and nonsmooth energy functionals defined over a convex set, in the sense of Ferris and Mangasarian \cite{ferris1994parallel}. For the numerical computation of the one-dimensional obstacle problem, we adopt a direct generalized parallel approach based on the SPIKE algorithm \cite{polizzi2006parallel}. To accelerate convergence and reduce computational complexity, a fast recursive version of the scheme is generalized to the nonsmooth case. For two- and three-dimensional obstacle problems, we employ a directional splitting method that treats each directional subproblem as a one-dimensional obstacle minimization problem. By framing the energy functional minimization as a parabolic time-dependent problem and utilizing an Armijo time-stepping rule during the solution update process, we successfully obtain results for higher-dimensional obstacles. Additionally, this study explores the possibility of extending the algorithm into a constrained quadratic programming optimization solver. We also evaluate the framework on image deblurring phenomena under the aforementioned setting, successfully recovering the original images. Finally, numerical illustrations are provided to validate the theoretical results.
17. 耦合张量-张量补全方法及其在药物重定位中的应用
Coupled Tensor-Tensor Completion Method with Applications in Drug Repurposing
作者: Maryam Bagherian, Albert Hung, Ivo Dinov, Joshua Welch
类别: math.NA, cs.LG
发布日期: 2026-09-02
arXiv ID: 2609.03190v1
链接: http://arxiv.org/abs/2609.03190v1
摘要: Many biomedical challenges can be posed as tensor completion problems where the observed entries of a multidimensional array (a tensor) are used to impute the missing values. In such settings, incorporating side information about the modes of the tensor, such as gene-gene similarity, can significantly enhance the solutions of the completion problem. Most existing tensor completion methods can only incorporate side information in the form of matrices. In this study, we introduce a novel framework to incorporate side information in the form of tensors. Our new approach, called Coupled Tensor-Tensor Completion (CTTC), leverages the hidden connections among multimodal tensors to improve tensor completion performance. In addition to practical utility, CTTC has theoretical foundations in distance metric learning and group theory. We derive an alternating algorithm to solve the CTTC optimization problem and establish its convergence to a stationary point. Finally, we show that CTTC outperforms state-of-the-art tensor completion methods at predicting drug effects. Results: Compared with other tensor completion methods, including HaLRTC, CTRC, Cell, and NTDDR, CTTC demonstrates superior run-time and RSE tensor completion accuracy on two benchmark datasets, DTD and LINCS.
18. 非线性有限元模型的维度超缩减：基于流形自适应权重的经验求积
Dimensional hyperreduction of nonlinear finite element models via empirical cubature with manifold-adaptive weights
作者: Joaquín A. Hernández, S. Ares de Parga, Riccardo Rossi
类别: math.NA, cs.CE
发布日期: 2026-09-02
arXiv ID: 2609.03068v1
链接: http://arxiv.org/abs/2609.03068v1
摘要: Nonlinear-manifold reduced-order models for parametrized finite element problems can achieve substantial compression both in the number of generalized (latent) coordinates and, through sampling-and-weighting hyperreduction, in the number of sampled elements/integration points. Yet current sampling-and-weighting approaches employ weights that remain fixed over the solution manifold. We contend that this restriction leaves hyperreduction potential untapped: allowing the weights to vary continuously and nonlinearly with the latent coordinates can further decrease the number of sampled spatial entities. To exploit this possibility, we propose the Manifold-Adaptive-Weight Empirical Cubature Method (MAW-ECM). Starting from a feasible fixed-weight ECM rule, a greedy pruning strategy removes sampled entities through convex quadratic weight-redistribution problems enforcing local conditions and positivity. The method is assessed on two nonlinear benchmarks: homogenization of a metamaterial unit cell exhibiting negative incremental stiffness, and a history-dependent continuum-damage problem. In both cases, the nonlinear manifold is constructed from an initial linear compression followed by an input-informed identification of the latent coordinates as general linear combinations of the retained modal amplitudes, incorporating graph information when relevant to seek the intrinsic dimensionality of the solution manifold. We show that combining the nonlinear-manifold representation with MAW-ECM reduces the number of sampled integration points by more than two orders of magnitude relative to the corresponding standard linear reduced model. Relative to the fixed-weight manifold models alone, the adaptive weights eliminate approximately 80% of the remaining points in the homogenization benchmark and more than 97% in the damage benchmark, while essentially preserving their accuracy.