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1. 非拟合网格上基于重构间断逼近的椭圆界面问题的预处理间断Galerkin方法

Preconditioned Discontinuous Galerkin Method for Elliptic Interface Problem on Unfitted Mesh with Reconstructed Discontinuous Approximation

作者: Ruo Li, Qicheng Liu, Fanyi Yang, Shuhai Zhao

类别: math.NA

发布日期: 2026-05-27

arXiv ID: 2605.28380v1

摘要翻译: 本文基于重构间断逼近，提出了一种高效的预处理非拟合有限元方法求解椭圆界面问题。该界面问题的逼近方法最初由[Li et al. SIAM J. Sci. Comput. 42(2), 2020]提出，其中通过求解局部最小二乘拟合问题，构造了一个每个单元一个自由度的任意高阶逼近空间。该空间可应用于切割间断Galerkin框架，在该框架中，界面上的跳跃条件通过Nitsche罚方法弱施加。在本工作中，通过引入适当的约束修改了局部最小二乘问题，这使得我们能够利用重构空间自然地保证界面附近的稳定性，并进一步使得我们能够建立高阶空间与最低阶空间之间的范数等价性。这一等价性质促使我们从分片常数空间构造预条件子，并且证明了该预处理方法是最优的，即预处理系统的条件数上界与网格尺寸、系数以及界面相对于非拟合网格的位置无关。我们还给出了作为最低阶系统矩阵之逆的多重网格算法。二维和三维的数值实验在误差度量下证实了最优收敛率，并展示了该预处理方法的有效性和鲁棒性。

Comment: 无

链接: http://arxiv.org/abs/2605.28380v1

1. 研究对象和背景

本文研究椭圆界面问题（elliptic interface problem）的高效数值方法。问题设定为：令 $\Omega\subset\mathbb{R}^d$ ($d=2,3$) 为具有边界 $\partial\Omega$ 的凸多边形（或多面体）区域，$\Omega_0\Subset\Omega$ 为具有 $C^2$ 光滑边界 $\Gamma=\partial\Omega_0$ 的开子域，并记 $\Omega_1=\Omega\setminus\bar\Omega_0$。所求未知函数 $u$ 满足如下方程和跳跃条件：

$$ -\nabla\cdot(\alpha\nabla u) = f \quad \text{在 } \Omega_0\cup\Omega_1,\quad u = g \quad \text{在 } \partial\Omega,\quad [u]=a n_\Gamma,\; [\alpha\nabla u]=b \quad \text{在 } \Gamma, $$

其中扩散系数 $\alpha$ 在 $\Omega_i$ 上为正常数 $\alpha_i$ 但跨界面 $\Gamma$ 可能发生跳跃，$f\in L^2(\Omega)$, $g\in H^{3/2}(\partial\Omega)$, $a\in H^{3/2}(\Gamma)$, $b\in H^{1/2}(\Gamma)$。

此类问题广泛出现于材料科学、多孔介质流、多相流等对含不同介质或材料的物理现象建模中。数值求解的主要挑战在于需要精确捕捉解跨界面的不连续性以及可能的大系数比带来的刚度。有限元方法大致分为贴体网格（body‑fitted）与非贴合网格（unfitted）两类。贴体方法要求网格线对齐界面，这对复杂几何生成网格十分困难；因此非贴合方法因网格生成与几何描述完全解耦而受到重视。典型的非贴合方法包括浸入有限元（IFEM）和切割有限元（CutFEM，又称Nitsche‑XFEM）。后者对每个子域采用独立的有限元空间，并通过Nitsche罚方法弱施加跳跃条件。然而，由于界面可能任意切割单元，会出现很小的切割片，若不加特殊处理将导致条件数无界甚至收敛性恶化。为此常需引入ghost penalty、单元聚合或局部扩展等稳定化技术。

即便经过稳定化，所得离散线性系统的条件数通常仍为 $O(h^{-2})$，当网格加密时仍呈现病态。对于非贴合方法，尤其高阶格式，关于（多重网格类型的）预处理研究非常有限。文献中，[32] 在二维对原始Nitsche有限元方法提出了基于稳定空间分裂的预处理器；[44] 针对稳定化Nitsche有限元设计了专门处理非嵌套层次网格的多重网格预处理器，但缺乏严格谱分析；另有少数工作涉及IFEM的多重网格或区域分解 [16,1,19,4]。

本文基于重构间断近似（reconstructed discontinuous approximation）发展一种高效预处理的非贴合间断Galerkin（DG）方法。该方法源于 [39]，通过在每个单元片（patch）上求解局部最小二乘拟合问题，从分片常数空间重构出每个单元仅需一个自由度的高阶近似空间 $U^m_h$。然而，[39] 仅处理了界面问题，并未涉及预条件。本文的根本推进在于修改局部最小二乘问题：对内部元素和切割元素分别引入适当的约束，使得重构算子自动保证界面附近的稳定性而无需额外稳定化，进而能够建立高阶空间 $U^m_{h,i}$ 与最低阶分片常数空间 $V_{h,i}$ 之间的范数等价性。这一等价性使我们可以从分片常数空间出发构造预条件子，并证明预处理系统的条件数上界与网格尺寸、扩散系数以及界面与网格的相对位置均无关，从而实现最优预处理。同时，在最低阶空间上采用光滑聚合多重网格思想构造了两类多重网格求解器，以高效求得低阶系统矩阵的逆。数值实验在二维和三维均验证了逼近误差的最优收敛率以及预处理方法的效率和鲁棒性。

2. 主要定理或主要结果

本文的主要理论结果分两部分：数值格式的误差估计和预处理方法的性质。

  1. 数值格式的误差估计 对于系数分片常数且可能大跳跃的椭圆界面问题(1)，采用修改约束的局部最小二乘重构空间 $U^m_h$ 与对称内罚DG框架结合，定义离散双线性形式 $a_h(\cdot,\cdot)$（式(14)）和相应的能量范数 $\|\cdot\|_{DG}$ 与 $\||\cdot\||_{DG}$。在网格满足几何假设（Assumption 1,2,3）且惩罚参数 $\mu$ 充分大时，有如下结果：
  • 正交性：$a_h(u-u_h, v_h)=0$, $\forall v_h\in U^m_h$。
  • 强制性（Lemma 5）：$a_h(v_h,v_h) \ge C\Lambda_m^{-2} \||v_h\||_{DG}^2$, $\forall v_h\in U^m_h$，其中常数 $\Lambda_m$ 仅依赖于多项式次数 $m$ 和网格的 quasi‑uniform 常数，与 $h$、系数 $\alpha$ 和界面位置无关。
  • 逼近性质（Theorem 1）：对任意 $v\in H^s(\Omega)$ ($s\ge 2$)，重构算子满足 $$ \|v_i - R^{m,i}_K v_i\|_{H^t(K)} \le C\Lambda_m h^{s-t}_K \|v\|_{H^s(D^i_K)},\quad 0\le t\le s-1,\; 2\le s\le m+1. $$
  • 收敛阶（Theorem 2）：设精确解 $u\in H^t(\Omega_0\cup\Omega_1)$ ($t\ge 2$)，数值解 $u_h\in U^m_h$ 满足 $$ \|u-u_h\|_{DG} \le C\Lambda_m^3 h^s \|\alpha^{1/2} u\|_{H^t(\Omega_0\cup\Omega_1)}, $$ $$ \|u-u_h\|_{L^2(\Omega)} \le C\Lambda_m^4 \alpha_{\min}^{-1/2} h^{s+1} \|\alpha^{1/2} u\|_{H^t(\Omega_0\cup\Omega_1)}, $$ 其中 $s=\min(m, t-1)$。这表明在能量范数和 $L^2$ 范数下均获得最优收敛率。
  1. 预处理方法
  • 约束条件保证了重构算子 $R^{m,i}$ 是非退化的，且 $\dim(U^m_{h,i}) = \dim(V_{h,i})$（Lemma 2）。
  • 通过约束可建立高阶空间与最低阶分片常数空间的谱等价性：存在正常数 $c,C$ 使得对任意 $v_h\in V_{h,i}$ 有 $$ c\|v_h\|_{\ell^2}^2 \le \|R^{m,i} v_h\|_{L^2}^2 \le C\|v_h\|_{\ell^2}^2, $$ 或者更一般地，在加权范数下的等价性。基于此，若记 $A_h$ 为高阶格式对应的刚度矩阵，$A_0$ 为最低阶（分片常数）格式的刚度矩阵，则 $A_0$ 和 $A_h$ 是谱等价的，即 $\kappa(A_0^{-1}A_h)$ 关于 $h$、$\alpha$ 和界面位置一致有界。
  • 因此，以 $A_0^{-1}$ 作为 $A_h$ 的预处理器可得到最优的预处理效果。
  • 对于最低阶系统 $A_0$ 的求逆，文中基于非贴合网格的逐次加密序列，设计了光滑聚合多重网格（smoothed aggregation multigrid）算法，并证明了其在分片常数空间上的收敛性。 数值实验验证了 $L^2$ 误差和能量误差的 $h^{m}$ 和 $h^{m+1}$ 收敛阶，且预处理后共轭梯度法的迭代次数几乎不随 $h$、系数比和界面位置变化。

3. 研究方法、关键技术和核心工具

本文的核心技术贡献在于通过对局部最小二乘问题引入适当的约束，同时实现了界面附近的自然稳定性、高低阶空间的范数等价以及最优预处理的构造。关键技术点分述如下。

  1. 带约束的局部最小二乘重构 对每个子区域 $\Omega_{h,i}$ 和每个单元 $K\in\mathcal{T}_{h,i}$，收集一个规模为 $N_m$ 的单元片 $\mathcal{S}^i_K$。约束设计为：若 $K$ 是由切割单元经某种映射关联的内部元素（即 $K\in\mathcal{T}_{h,i}^{\Gamma,\circ}$），则要求重构多项式 $p$ 同时在 $K$ 及其对应的切割单元 $K^{\varsigma_i}$ 的重心处等于给定的分片常数值 $v_{h,i}$；否则只要求 $p$ 在 $K$ 的重心处匹配。这种“点匹配”约束是构造的核心。
  • 适定性：通过要求 $N_m > \dim(P_m)$ 且重心不落在 $m$ 次代数曲线上（Assumption 3），保证最小二乘问题有唯一解（Lemma 1）。
  • 稳定性：对重构局部多项式 $p = R^{m,i}_K v_{h,i}$，通过正交关系 $$ \sum_{K'\in\mathcal{S}^i_K} q(x_{K'})(p(x_{K'})-v_{h,i}(x_{K'})) = 0 $$ （对任意满足约束的测试多项式 $q$）可导出 $L^2$ 稳定性估计（Lemma 3） $$ \|R^{m,i}_K v_{h,i}\|_{L^2(K)} \le C\Lambda_m h^{d/2}_K \max_{K'\in\mathcal{S}^i_K} |v_{h,i}(x_{K'})|, $$ 其中 $\Lambda_m$ 涉及局部矩阵的最小奇异值，当 patch 足够大时与 $h$ 无关。由此可得逼近性质（Theorem 1）。
  • 范数等价性：证明 $\||v_h\||_{DG} \le C\Lambda_m \|v_h\|_{DG}$ 是保证无额外稳定化下格式稳定的关键（Lemma 4）。其主要障碍在于控制切割单元上的梯度，而这些梯度的平方和 $\sum_{K\in\mathcal{T}^\Gamma_h} \alpha_i\|\nabla v_h^{\pi_i}\|_{L^2(K)}^2$ 必须被 $\|v_h\|_{DG}$ 中的界面罚项所控制。 借助约束中内部元素与切割元素的配对以及 patch 内分片常数函数的分解（式(16)‑(18)），可以将邻近内部元素的梯度差与界面跳量联系起来，最终得到该梯度项被 $\Lambda_m^2\|v_h\|_{DG}^2$ 所界。这里的关键细节是：将待估梯度用局部极小值中心化，并利用内部元素逐链到邻近面的跳跃建立 $|w_{\max}-w_{\min}|$ 与 $\|\nabla v_h^{\pi_i}\|$ 及 $\|[v_h^{\pi_i}]\|$ 的关系。

  1. 误差分析 基于由上述范数等价性导出的强制性（Lemma 5），通过标准DG框架的Céa引理型的论证，结合重构算子的逼近性质即可得到能量范数误差；进一步采用对偶论证得到 $L^2$ 范数的最优阶估计。

  2. 预处理构建

  • 由于对任意 $v_{h,i}\in V_{h,i}$，重构函数满足 $R^{m,i} v_{h,i}(x_K) = v_{h,i}(x_K)$，且算子满秩，因此 $V_{h,i}$ 与 $U^m_{h,i}$ 之间存在自然同构。进一步分析表明，两个空间在 $L^2$ 范数（以及带权能量范数）下是谱等价的：即存在不依赖 $h$ 和 $\alpha$ 的常数使得 $$ c\sum_{K\in\mathcal{T}_{h,i}} h^d_K v_{h,i}(x_K)^2 \le \|R^{m,i} v_{h,i}\|_{L^2(\Omega_{h,i})}^2 \le C \sum_{K\in\mathcal{T}_{h,i}} h^d_K v_{h,i}(x_K)^2. $$ 这一等价性源于重构的稳定性及约束条件；其直接推论是刚度矩阵 $A_h$ 与 $A_0$ 谱等价。因此可用 $A_0^{-1}$ 作为 $A_h$ 的预处理器，且条件数 $\kappa(A_0^{-1}A_h)$ 与 $h$、系数跳和界面位置均无关。
  • 为高效求 $A_0^{-1}$，文中针对分片常数空间 $V_{h,i}$ 设计了基于非贴合网格多重光滑聚合的多重网格方法。由于背景网格是结构化（或半结构化）加密的，可构造出一族嵌套（或非嵌套）的网格层次；光滑聚合技术通过构造合适的 prolongator 和 smoother，将原本用于 $P_1$ 连续有限元的算法移植到分片常数非贴合空间上。文中给出了算法的细节及其收敛性证明，表明该多重网格求解器具有最优或近乎最优的复杂度。

4. 与之前工作的比较

本文在重构间断近似框架下，为无贴合网格椭圆界面问题发展了一种具有最优预处理的高阶DG方法。与已有工作的区别和推进可归纳为以下几个方面。

  1. 与作者前序工作 [39] 的比较 文献 [39] 首次将重构间断近似应用于椭圆界面问题，通过局部最小二乘重构从分片常数空间得到每个单元一个自由度的高阶空间。然而，[39] 中并未引入本文的约束条件，因此在证明界面附近的稳定性时可能仍需借助类似 ghost penalty 的附加稳定化，或者其分析未明确建立高阶与低阶空间的范数等价。本文通过修改最小二乘问题，加入与内部‑切割元素配对相关的重心点约束，使得重构空间天然具有“自稳定”性质，从而无需额外稳定化即可确保 $\||\cdot\||_{DG}$ 与 $\|\cdot\|_{DG}$ 的等价性（Lemma 4）。这一改变是根本性的，不仅简化了方法，更是后续构造最优预处理的基石。此外，[39] 未涉及代数系统的预处理。

  2. 与其他无贴合有限元方法的预处理研究比较 目前针对无贴合方法的预处理研究尚处于起步阶段。

  • [32] 对二维原始 Nitsche‑XFEM（无稳定化）提出了一种预处理器，但其构造依赖于将有限元空间显式分裂为稳定的子空间，分析较为复杂，且只限于二维。
  • [44] 为稳定化 Nitsche‑XFEM 设计了基于非嵌套网格的多重网格预处理器，虽然数值有效，但关于谱质量的严格理论分析缺失；且其 prolongator 需特别设计以处理非嵌套网格。
  • IFEM 的多重网格 [16,1] 针对的是浸入有限元空间，其基函数在切割单元上被修改以满足界面条件，而本文的重构空间是在每个单元片上通过最小二乘全局性构造，自由度极少（每个单元一个），基函数具有紧支撑且更易于进行范数等价性分析。
  • 域分解方法 [19,4] 也为无贴合方法提供了并行求解途径，但通常不追求条件数对 $h$ 的一致有界性。 相较之下，本文的预处理方法具备完整的理论保证：在简单而明确的约束条件下证明了高阶与低阶空间的谱等价性，从而得到预处理器的最优性，且该性质对 $h$、系数跳和界面位置均鲁棒。此外，由于高低阶矩阵都建立在同一网格上，避免了传统低阶预处理策略（如 [14,47,46]）中为匹配自由度而必须借助网格细化的高阶到低阶映射，大大简化了实现。

  1. 在稳定性和收敛性方面的推进 已有稳定化无贴合方法通常需要引入 ghost penalty、单元聚合或局部扩展来达到条件数的一致有界性和最优收敛。这类稳定化往往增加实施复杂度，并且在预处理分析中引入了额外耦合。本文的约束最小二乘重构在保持高精度的同时，自然保证了逼近空间在界面附近的稳定，从而直接推导出 $\||\cdot\||_{DG}$ 范数下的强制性以及最优误差估计，且证明中不依赖任何附加的稳定项。从定理2可见，误差界中的常数 $\Lambda_m$ 只取决于 patch 的大小和多项式次数，与界面如何切割网格无关。

  2. 适用范围与局限 本文方法目前假定界面 $\Gamma$ 为 $C^2$ 光滑，且网格是满足一定 quasi‑uniform 条件的单纯形剖分；这些假设与多数无贴合方法相当。然而，约束的引入虽然带来了理想的理论性质，但在实现上需要识别内部元素和切割元素的配对关系（Assumption 1），在界面拓扑复杂时可能会限制适用性。此外，局部最小二乘问题的求解在每个单元片上都会产生一个小规模线性系统，这增加了装配计算量，不过这部分开销是离线完成的，且由于自由度大幅减少，总体求解效率依然优于标准高阶DG。文中进行了二维和三维数值实验，包括系数大跳跃情形，数值观察确认了误差的最优收敛和预处理迭代次数的稳健性，但在三维极复杂几何下的规模可扩展性未作深入探讨。

2. 基于投影方法的二维准周期亥姆霍兹特征值问题的新计算与分析框架

A Novel Computational and Analytical Framework for 2D Quasiperiodic Helmholtz Eigenvalue Problems via the Projection Method

作者: Teng-Chao Sun, Tiexiang Li, Wen-Wei Lin, Xing-Long Lyu

类别: math.NA

发布日期: 2026-05-27

arXiv ID: 2605.27842v1

摘要翻译: 本文提出一种谱框架，通过投影方法\cite{jiang2014numerical, jiang2024numerical}将一维和二维准周期亥姆霍兹特征值问题嵌入高维（二维和四维）周期空间。为将提升后的高维状态有效映射回原始物理空间，我们基于逐点Rayleigh商的加权期望建立了一种新的验证框架。在全面的误差和谱分析支持下，我们证明了由该期望导出的特征值与原始准周期模型更真实地一致，最终得到更合适且可靠的特征对解。连续介质的数值实验表明，我们的方法为求解准周期亥姆霍兹特征值问题提供了一种精确、稳健且可扩展的工具。

Comment: 无

链接: http://arxiv.org/abs/2605.27842v1

1. 研究对象和背景

拟周期结构因缺乏平移周期性，在物理和数学中诱发临界局域态、康托尔集能谱及霍夫施塔特蝴蝶图案等奇特波动现象。在电磁学与光子学中，光子准晶可调控光传播与自发辐射，通过拟周期介质调制呈现完全光子带隙、慢光效应和鲁棒拓扑边界态，在高 Q 腔、完美吸收体和光学传感器设计中有巨大潜力。精确理解这些现象依赖于高效求解带拟周期系数的偏微分方程特征值问题。

考虑二维横向磁场(TM)模式，设磁导率 $\mu_\perp \equiv 1$，标量场 $u(\mathbf{r}) = E_z(\mathbf{r})$ 满足亥姆霍兹方程

$$ -\Delta u(\mathbf{r}) = \lambda \varepsilon(\mathbf{r}) u(\mathbf{r}), \quad \mathbf{r} \in \mathbb{R}^2, $$

其中 $\varepsilon(\mathbf{r})$ 为拟周期函数，$\lambda$ 为特征值。此问题可退化到一维 $\mathbb{R}^1$。直接使用有限差分(FD)离散得到无限维广义特征值问题 $\mathbf{A}\mathbf{u} = \lambda \mathbf{B}\mathbf{u}$，其中 $\mathbf{A}$ 为离散拉普拉斯算子，$\mathbf{B}$ 为对角系数矩阵。传统晶体近似法(CAM)通过截断区域并施以人工周期边界，利用同时丢番图逼近(SDA)将不可公度频率近似为可公度频率；然而，为达到高保真度需要指数级大的超胞，导致严重的有限尺寸效应和极高计算成本，尤其对多维或高间断问题。

cut-and-project 方法揭示，任何 $d$ 维拟周期结构均可通过将高维 ($n>d$) 周期格点沿无理取向窗口投影得到，从而将非周期结构转化为高维周期结构。投影方法(PM) [10,12] 据此将低维拟周期函数嵌入高维周期空间，彻底绕过丢番图近似误差，恢复快速傅里叶变换和谱方法的适用性。具体地，通过投影矩阵 $\mathbf{P} \in \mathbb{R}^{n \times d}$ 将物理坐标 $\mathbf{r}$ 映射为高维周期坐标 $\mathbf{x} = \mathbf{P} \mathbf{r}$，拟周期系数 $\varepsilon(\mathbf{r})$ 提升为周期函数 $E(\mathbf{x}) = \varepsilon(\mathbf{P}^T\mathbf{r})$，解提升为 $U(\mathbf{x}) = u(\mathbf{P}\mathbf{r})$。原算子经链式法则变为沿投影方向的方向导数 $\nabla_{\mathbf{P}}$，得到高维周期特征值问题

$$ -\nabla_{\mathbf{P}}^2 U(\mathbf{x}) = \lambda E(\mathbf{x}) U(\mathbf{x}), \quad \mathbf{x} \in \mathbb{T}^n. $$

当 $d=1$ 时嵌入至2维环面 $\mathbb{T}^2$，$d=2$ 时嵌入至4维环面 $\mathbb{T}^4$。高维周期问题可用傅里叶伪谱法高效离散：将 $U$ 和 $E$ 展为截断傅里叶级数，利用无理方向导数在傅里叶空间可解析计算的优势，避免有限差分在无理斜率上的阶梯误差。离散后得到高维广义特征值问题 $\mathbf{K}\mathbf{U} = \tilde{\lambda} \mathbf{M}\mathbf{U}$，其中刚度矩阵 $\mathbf{K}$ 为对角阵，质量矩阵 $\mathbf{M}$ 为块托普利兹矩阵。

然而，从截断高维空间映射回原始低维系统的误差分析尚不完整[15]。高维计算得到的特征值和特征向量投影回物理空间后，其能否真实代表原拟周期系统的特征值，缺乏严格的数值一致性验证。本文试图填补这一空白：基于投影方法，建立了从高维周期空间到低维物理空间的点态瑞利商加权期望验证框架，并提供系统的误差与谱分析，确保所得特征对的可靠性与准确性。

2. 主要定理或主要结果

本文的主要理论结果包括：

  1. 加权平均关系 (定理4.1) ：对于无限维离散向量 $\mathbf{u}$，若 $0 < \varepsilon_{\min} \le \varepsilon_i \le \varepsilon_{\max}$ 且 $\mathbf{u}^T\mathbf{B}\mathbf{u} < \infty$，$\varepsilon_i u_i \neq 0$，则整体瑞利商

    $$ \rho(\mathbf{u}) = \frac{\mathbf{u}^T\mathbf{A}\mathbf{u}}{\mathbf{u}^T\mathbf{B}\mathbf{u}} $$

    可表示为点态瑞利商 $r_i = \frac{(\mathbf{A}\mathbf{u})_i}{(\mathbf{B}\mathbf{u})_i}$ 的加权平均：

    $$ \rho(\mathbf{u}) = \sum_{i=-\infty}^{\infty} p_i r_i = \mathbb{E}_p[\{r_i\}], \quad p_i = \frac{\varepsilon_i u_i^2}{\mathbf{u}^T\mathbf{B}\mathbf{u}}. $$

  2. 误差稳定性 (定理4.2) ：设近似特征对 $(\lambda, \mathbf{u})$ 满足扰动方程 $\mathbf{A}\mathbf{u} = \lambda \mathbf{B}\mathbf{u} + \mathbf{e}$，其中残差 $\mathbf{e} = \{e_i\}$ 满足 $\max_i |e_i| = e_{\max} < \infty$。若存在常数 $C_0 > 0$、$\alpha > 1$ 和 $N_0 \in \mathbb{N}$，使得 $|u_i| \le C_0 |i|^{-\alpha}$ 对 $|i| \ge N_0$ 成立，则存在常数 $C > 0$ 使得

    $$ \mathbb{E}_p\left[ \left| \frac{e_i}{\varepsilon_i u_i} \right| \right] \le C e_{\max}. $$

    进而，加权期望特征值估计满足

    $$ |\mathbb{E}_p[r_i] - \lambda| \le C e_{\max}. $$

    该定理表明，尽管局部点态商在 $|u_i|$ 较小时可能对残差敏感，但权重 $\propto \varepsilon_i u_i^2$ 自动抑制这些点的贡献，从而加权期望提供稳定的特征值估计。

此外，数值实验验证了所提框架对一维和二维连续介质拟周期亥姆霍兹特征值问题的有效性、精度和可扩展性，计算特征值与参考解吻合良好，且点态瑞利商估计的计算开销微小。

3. 研究方法、关键技术和核心工具

本文的方法围绕两个核心创新：将拟周期特征值问题通过投影方法升维为周期问题，并利用傅里叶伪谱法高效离散；提出基于点态瑞利商加权期望的后验验证框架，实现对映射误差的控制。

1. 高维嵌入与傅里叶伪谱离散

投影方法的关键在于利用链式法则将低维微分算子转化为高维周期区域上的方向导数。对于一维情况，投影方向由矩阵 $\mathbf{P} = (p_1, p_2)^T$ 给出，其中 $p_1, p_2$ 有理无关。原算子 $-d^2/dr^2$ 变为 $\nabla_{\mathbf{P}}^2 = (p_1 \partial_{x_1} + p_2 \partial_{x_2})^2$。二维情况推广至 $\mathbf{P} \in \mathbb{R}^{4 \times 2}$，算子变为 $\nabla_{\mathbf{P}}^2 = (\nabla_{\mathbf{x}}^T \mathbf{P}) (\mathbf{P}^T \nabla_{\mathbf{x}})$。传统有限差分法在处理无理斜率方向导数时会产生严重的阶梯误差，而傅里叶伪谱法在频率空间精确处理方向导数，因为 $\nabla_{\mathbf{P}}$ 作用于傅里叶模 $e^{\mathrm{i} (\mathbf{k}+\boldsymbol{\xi})\cdot\mathbf{x}}$ 仅产生因子 $\mathrm{i} \mathbf{P}^T (\mathbf{k}+\boldsymbol{\xi})$ 或 $\mathbf{P}^T(\mathbf{k}+\boldsymbol{\xi})$ 的平方项。因此，离散后得到的广义特征值问题为 $\mathbf{K}\mathbf{U} = \tilde{\lambda} \mathbf{M}\mathbf{U}$，其中 $\mathbf{K}$ 是对角矩阵，元素为 $(\mathbf{P}^T(\mathbf{k}+\boldsymbol{\xi}))^2$ (1D) 或 $(\mathbf{k}+\boldsymbol{\xi})^T\mathbf{P}\mathbf{P}^T(\mathbf{k}+\boldsymbol{\xi})$ (2D)；$\mathbf{M}$ 为块托普利兹卷积矩阵，可通过快速傅里叶变换 (FFT) 实现 $\mathcal{O}(N_{\text{F}} \log N_{\text{F}})$ 的矩阵-向量乘法。求解该特征值问题采用逆形式 $\mathbf{M}\mathbf{U} = \mu \mathbf{K}\mathbf{U}$，用迭代特征求解器提取最大特征值 $\mu = \tilde{\lambda}^{-1}$，充分利用了矩阵结构和稀疏性。

2. 点态瑞利商验证框架

投影回物理空间的解 $\hat{\mathbf{u}}$ 作为原问题近似特征向量的质量需要量化。本文创新地引入原始无限维有限差分离散算子 $\mathbf{A}, \mathbf{B}$ 在采样网格上的点态瑞利商

$$ r_i = \frac{(\mathbf{A}\hat{\mathbf{u}})_i}{(\mathbf{B}\hat{\mathbf{u}})_i} = \frac{(\mathbf{A}\hat{\mathbf{u}})_i}{\varepsilon_i \hat{u}_i}. $$

定理 4.1 证明，当将 $\hat{\mathbf{u}}$ 视为无限维向量时，其整体瑞利商恰为这些点态商的加权平均，权重 $p_i = \varepsilon_i \hat{u}_i^2 / \hat{\mathbf{u}}^T\mathbf{B}\hat{\mathbf{u}}$。这一关系为局部信息提供了全局解释。进一步，定理 4.2 在假设特征向量具有多项式衰减 $|\hat{u}_i| \le C_0 |i|^{-\alpha}$ ($\alpha > 1$) 的条件下，证明了加权平均误差被残差上界 $e_{\max}$ 控制，从而保证了估计量的鲁棒性。该衰减条件在实际拟周期系统的局域或扩展态中通常成立。证明的关键是利用 $\sum |\hat{u}_i| < \infty$ 来压制 $1/|\hat{u}_i|$ 的奇异性，同时权重中的 $\hat{u}_i^2$ 因子自动抑制小幅值区域的不良影响。

3. 局部稳定性分析与辛对

为进一步考察点态商的数值稳定性，作者推导了局部比值 $y_i = \hat{u}_{i+1}/\hat{u}_i$ 满足的递推关系

$$ y_i = (2 - h^2 \rho \varepsilon_i) - \frac{1}{y_{i-1}}, $$

并将其改写为转移矩阵形式 $y_i = L_{i}^{-1} M_i y_{i-1}$，其中 $M_i, L_i$ 构成一个辛对。通过引理 4.3 的辛对交换与合并技术，可分析递推的长期行为，揭示特征向量局部振荡特性对点态商波动的贡献。该分析为理解数值误差在空间上的传播提供了代数工具。

4. 算法实现

算法 1 总结了整体流程：生成投影关系、构建傅里叶指标集、组装并求解高维广义特征值问题、反投影得到物理空间解、在低维有限差分网格上计算点态瑞利商并加权平均。点态瑞利商计算仅涉及低维算子作用及分量积商，成本远低于高维特征求解，因此是一种轻量的后验改进方案。

4. 与之前工作的比较

本文与已有工作的比较主要体现在以下几个方面：

1. 与晶体近似法 (CAM) 的对比

CAM 通过截断并施加人工周期边界，依赖同时丢番图逼近将拟周期系数近似为周期函数。为逼近无理频率，需要极长的超胞和极高的计算资源，且有限尺寸效应造成特征值偏移，尤其对高阶能带或奇异介质更为严重。投影方法 [10,12] 从根本上避免 SDA 误差，将拟周期问题精确映射为高维周期问题。本文在此框架上进一步引入点态瑞利商验证，弥补了从高维解到低维特征值的数值映射空白，比 CAM 更准确且避免了 SDA 导致的近似误差。

2. 与先前投影方法工作的对比

前期投影方法工作 [10,12] 主要针对边值问题建立数值框架；[11] 发展了谱方法；[14] 针对薛定谔特征值问题，在高维投影谱方法中嵌入无理窗口滤波器作为预处理；[15] 在连续层面分析了投影特征值问题的谱收敛性，但未给出从截断高维离散空间到低维物理空间的映射误差的定量分析。本文明确指出“a practical numerical analysis detailing the exact mapping error from the truncated high-dimensional space back down to the original low-dimensional system is still not complete”，并填补了这一理论空缺：通过建立点态瑞利商加权期望框架，证明了在一定条件下 (定理 4.2) 从高维解提取的特征值估计具有与残差同阶的精度，从而提供了完整的映射误差控制。这一分析是投影方法在特征值问题中的关键进展。

3. 验证策略与计算成本

与 [14] 的高维预处理不同，本文的验证步骤置于后处理阶段，利用低维有限差分算子计算点态瑞利商，避免了在高维求解过程中引入额外复杂性。计算点态瑞利商的开销仅涉及少量向量操作，相较于高维特征求解可忽略不计，因此是一种高效且易于实现的验证策略。数值实验表明，该方法能校正高维计算中因截断造成的微小偏差，得到更贴近原拟周期系统的特征值。

4. 适用性与局限性

本文处理的系数为连续拟周期函数，且假设特征向量具有多项式衰减 $|u_i| \le C_0 |i|^{-\alpha}$，$\alpha>1$，这一条件在多数物理问题中自然成立。与 CAM 相比，本文方法可处理高维嵌入的普适性更强，但对于更高维的原始问题 (如三维嵌入到六维)，高维离散的维数灾难依然存在，但在 $d \le 2$ 时可行。文中未与有限元方法进行直接比较，亦未讨论系数有间断或奇性的情况，这些可能作为未来工作。

5. 与 [15] 的理论联系

[15] 建立了投影特征值问题的连续谱收敛性，本文则从离散角度补充了映射回物理空间时的数值误差分析，并将连续投影理论与离散验证相结合，形成了完整的求解与认证框架。定理 4.1 和 4.2 可视为对 [15] 中收敛性结果的离散实现与实用化延伸。

综上所述，本文在投影方法的基础上，通过提出点态瑞利商加权期望和相应的误差分析，显著增强了对拟周期亥姆霍兹特征值问题数值解的可靠性，与既有工作形成互补和推进。

了解领域论文（摘要翻译）

1. 求解$p$-Laplace问题的对偶变分神经网络

Dual Variational Neural Network for the $p$-Laplace Problem

作者: Tianhao Hu, Guanglian Li, Fengru Wang, Yifeng Xu, Zhi Zhou

类别: math.NA

发布日期: 2026-05-27

arXiv ID: 2605.28291v1

链接: http://arxiv.org/abs/2605.28291v1

摘要翻译

中文标题: 求解$p$-Laplace问题的对偶变分神经网络

摘要翻译: 在极端情形$p \to 1^{+}$和$p \gg 1$下，$p$-Laplacian 的可靠且精确的数值逼近尤其具有挑战性；此时算子变得高度奇异或强退化，常常导致标准数值方法的严重不稳定。为克服这些困难，我们针对$p$-Laplace问题提出了一种新颖的基于深度学习的框架，称为对偶变分神经网络。该方法基于混合公式和基于$L^q$的亥姆霍兹分解，将原问题解耦为两个凸子问题：一个关于无旋分量的线性 Poisson 问题，以及一个关于无散场的无约束极小化问题。进行分解后，我们使用两个神经网络，借助梯度–旋度表示来近似通量，并进一步建立了神经逼近的误差分析。分析依赖于基本的向量不等式以及统计学习理论的工具。数值实验表明，所提方法在具有挑战性的情形下具有稳健收敛性，包括极端情形$p \to 1^{+}$、$p \gg 1$以及$p(x)$-Laplace方程。

2. 用于移动物体声波散射的结构保持PML–域嵌入方法

A Structure-Preserving PML-Domain-Embedding Method for Acoustic Wave Scattering by Moving Objects

作者: Xuelong Gu, Qi Wang

类别: math.NA

发布日期: 2026-05-26

arXiv ID: 2605.27680v1

链接: http://arxiv.org/abs/2605.27680v1

摘要翻译

中文标题: 用于移动物体声波散射的结构保持PML–域嵌入方法

摘要翻译: 我们建立了一个用于移动物体声波散射的结构保持计算框架，该框架包含一个新的PML–域嵌入模型和一个相容的数值近似。该模型将用于截断声波方程的完美匹配层（PML）与在固定计算域上表示移动物体的域嵌入公式耦合起来。由此得到的PML–域嵌入（PML-DE）系统使得移动边界散射问题无需重新划分网格即可求解。通过匹配渐近展开，我们证明当界面厚度趋于零时，扩散界面公式收敛到相应的尖锐界面系统。随后，我们为PML-DE系统构造了一个保能量耗散率的有限差分格式。为提高计算效率，该格式与依据移动物体位置、固定PML区域以及演化中的波动力学信息的层次化局部细化相结合，一切均在固定计算域内完成。数值实验展示了所求散射解的精度、吸收层与物体嵌入策略的有效性以及自适应算法的高效率。所提出的框架为涉及复杂声波散射问题的工程应用提供了一种实用且稳健的计算方法。

不相关论文（基本信息）

1. 基于增量奇异值分解的历史感知自适应降阶模型

History-aware adaptive reduced-order models via incremental singular value decomposition

作者: Amirpasha Hedayat, Ali Mohaghegh, Laura Balzano, Cheng Huang, Karthik Duraisamy

类别: cs.LG, cs.CE, math.NA, physics.comp-ph

发布日期: 2026-05-27

arXiv ID: 2605.28684v1

链接: http://arxiv.org/abs/2605.28684v1

摘要: Reduced-order models (ROMs) can accelerate high-dimensional dynamical simulations, but their accuracy often deteriorates when online dynamics leave the regime represented by offline training data. We develop a projection-based adaptive ROM framework based on incremental singular value decomposition (iSVD), in which occasional full-order operator evaluations provide correction snapshots for online basis updates. The intrusive ROMs considered here are fully parameterized by the basis, so each update naturally propagates to reduced operators and hyper-reduction machinery. Through its evolving singular structure, iSVD retains an encoded history of the observed dynamics and is history-aware in this sense. We study the method on three nonlinear problems of increasing complexity: the one-dimensional viscous Burgers equation, the Sod shock tube, and a stiff one-dimensional ten-species rotating detonation engine (RDE). The Burgers problem is used to analyze the method and compare iSVD with alternative basis adaptation rules, showing that history-aware updates outperform instantaneous updates and that iSVD gives the strongest overall performance. The Sod and RDE cases demonstrate that these advantages persist in more challenging compressible-flow settings. For the RDE problem, the iSVD adaptive ROM improves upon the current state-of-the-art Direct adaptive ROM baseline in both predictive accuracy and computational efficiency. A cost analysis shows that the dominant online cost comes from interacting with the full-order model to obtain correction snapshots, while the iSVD update itself is negligible. These results identify iSVD as an effective mechanism for online learning of reduced subspaces and suggest a path toward ROMs that remain predictive over horizons several orders of magnitude longer than their initial training window.

2. 面向AI赋能建模的聚变装置瞬态分析中积分电磁公式的高效精确模型降阶

Efficient and Accurate Model Order Reduction for Integral Electromagnetic Formulations in Fusion Device Transient Analysis Toward AI-Enabled Modeling

作者: Salvatore Ventre

类别: math.NA

发布日期: 2026-05-27

arXiv ID: 2605.28648v1

链接: http://arxiv.org/abs/2605.28648v1

摘要: The numerical simulation of electromagnetic transients in fusion devices is essential for analyzing plasma stability and disruptive events. However, it remains computationally demanding due to the large-scale dense systems arising from integral formulations. This work proposes a model order reduction (MOR) strategy for transient electromagnetic problems based on integral formulations. Unlike operator-based compression techniques (such as $\mathcal{H}$-matrix approaches), the reduced space is constructed directly from the transient excitation. In contrast to classical snapshot- and transfer-function-based MOR approaches, the proposed formulation avoids repeated explicit inversions or factorizations of the dense integral operator during the MOR basis-construction stage. By combining wavelet-based temporal compression with source-driven Krylov projections, the method generates reduced models tailored to the dynamically reachable responses of the prescribed excitation families. Numerical validations on various plasma events and fusion-relevant scenarios demonstrate the robustness of the strategy, achieving substantial computational speedups while accurately preserving the transient electromagnetic response. Finally, the method is successfully applied to the null-field problem to efficiently generate training data for neural-network surrogates, contributing toward physics-consistent AI-enabled fusion modelling.

3. 谱Boltzmann碰撞算子的Wigner-Eckart因子分解

Wigner-Eckart Factorization of the Spectral Boltzmann Collision Operator

作者: René R. Hiemstra, Torsten Keßler, Michael R. A. Abdelmalik

类别: math.NA, physics.comp-ph, physics.flu-dyn

发布日期: 2026-05-27

arXiv ID: 2605.28475v1

链接: http://arxiv.org/abs/2605.28475v1

摘要: We reduce the eight-dimensional weak form of the bilinear Boltzmann collision operator to a five-dimensional kinematic core by rigidly rotating the laboratory frame to align with the colliding pair and integrating over the $\mathrm{SO}(3)$ rotation group. This reduction yields an exact Wigner–Eckart factorization within a spectral Galerkin framework of associated Laguerre polynomials and spherical harmonics. The decomposition decouples the angular geometry from the scattering physics. The former, represented by Clebsch–Gordan coefficients, is evaluated exactly, while the latter is evaluated to machine precision by a spectrally convergent singular quadrature strategy. By explicitly zeroing specific entries, the macroscopic collision invariants are embedded without approximation. Cache-optimized contractions deliver up to a 37-fold single-core speedup and a 1000-fold memory reduction over standard dense Cartesian formulations. The approach is validated against analytical solutions for Maxwell molecules and infinite-order Chapman–Enskog viscosity coefficients for hard spheres.

4. 航天器数字孪生中基于物理约束的图热网络辨识

Physics-constrained identification of graph-based thermal networks for spacecraft digital twins

作者: Luca Sosta, Carlo Ciancarelli, Leonardo Marini, Stefano Pagani, Francesco Regazzoni, Nicola Parolini

类别: math.NA

发布日期: 2026-05-27

arXiv ID: 2605.28452v1

链接: http://arxiv.org/abs/2605.28452v1

摘要: Reconstructing a thermal model capable of efficiently simulating the behavior of a spacecraft from sparse and localized temperature measurements remains a challenging task. To address this, we introduce a physically-constrained calibration framework for Lumped Parameter Thermal Models (LPTMs), formulated as a trajectory-based inverse problem for graph dynamical systems. The model reconstructs thermal dynamics directly from temperature measurements and known inputs, without relying on a priori parameter values derived from material properties or geometric assumptions. Physical admissibility is enforced at the parameterization level: positivity of nodal coefficients and symmetry of conductive interactions are imposed by construction. This guarantees stable dynamics and restricts the identification problem to a physically meaningful parameter space, improving conditioning without the need of additional regularization. The identification problem is addressed through trajectory matching, ensuring stable rollout over extended time horizons. The methodology is validated on synthetic datasets generated from high-fidelity finite element simulations under progressively complex forcing conditions. The calibrated LPTMs accurately reproduce long-term temperature evolution and exhibit robustness to measurement noise. The proposed framework provides a systematic approach to the calibration of reduced-order thermal models by combining physical structure with data-driven identification. The numerical results show a favorable balance between accuracy and computational efficiency, making the models suitable for integration in spacecraft thermal Digital Twin applications.

5. 加速异步不动点迭代在灵活计算基础设施上的容错性

Fault Tolerance of Accelerated Asynchronous Fixed-Point Iterations on Flexible Computing Infrastructure

作者: Evan Coleman, Masha Sosonkina

类别: cs.DC, math.NA

发布日期: 2026-05-27

arXiv ID: 2605.28426v1

链接: http://arxiv.org/abs/2605.28426v1

摘要: Asynchronous iterative methods tolerate straggling processors by allowing workers to proceed with stale data, but at a cost: the iterates become inconsistent, potentially degrading convergence. We investigate whether convergence accelerators such as Anderson acceleration compensate for this degradation. We experimentally study three fixed-point iterations: the Jacobi method for sparse linear systems, value iteration for the Bellman equation, and the Hartree–Fock self-consistent field (SCF) iteration. The experiments are conducted using a high-performance execution framework Ray, which abstracts the complexity of distributed systems and enables code parallelization and fault injection with minimal changes. We establish two main results. First, straggler tolerance is universal: asynchronous execution provides wall-clock speedups of $2.9\times$ (Jacobi), $7.7\times$ (VI), and $16.9\times$ (SCF) over synchronous execution with a 100,ms-delayed worker, independent of whether acceleration is used. Second, Anderson acceleration’s effectiveness under asynchrony depends on where staleness enters the computation. We identify two staleness mechanisms: iterate-level corruption, where stale worker returns directly overwrite portions of the accelerated iterate (as in block Jacobi), and evaluation-level perturbation, where staleness acts as a bounded perturbation to the fixed-point map evaluation (as in VI and SCF). Anderson acceleration fails categorically under the first mechanism but retains its benefits under the second, consistent with the perturbation analysis of Toth et al.\ (2017). This distinction, rather than the contraction norm or smoothness of the map, is the primary determinant of whether acceleration survives asynchronous execution.

6. 一维波动方程逆边界问题重建方法的统计比较

Statistical comparison of reconstruction methods for the inverse boundary problem of the one-dimensional wave equation

作者: Samuel Agenorwoth, Emilia Blåsten

类别: math.NA, math.AP

发布日期: 2026-05-27

arXiv ID: 2605.28415v1

链接: http://arxiv.org/abs/2605.28415v1

摘要: Several numerical reconstruction algorithms for the inverse boundary value problem of the 1-dimensional wave equation exist. In this paper we revisit two of them, the Sondhi-Gopinath (SG) method from 1971 and the Korpela-Lassas-Oksanen (KLO) method from 2016. The former is stable enough that it was used in practical applications. The latter has a regularisation scheme with a theoretical proof, and is an evolution of the boundary control method. Both are based on the idea of constructing solutions that are characteristic functions of a set at a given time. This similarity has been pointed out before, but no systematic comparison has been published. We compare the performance of the two algorithms with noisy simulated data. The application in our minds is reconstructing the internal cross-sectional area of a pressurised fluid pipe which corresponds to the first order $\partial_x$ term of the wave equation. Instead of just observing the performance on a few test cases, we generate $n=1000$ random area profiles of various smoothness levels and measurement noise up to $10\%$ of the signal energy and perform statistical tests. SG and KLO have a difference of one time-derivative in their standard boundary data, which complicates the analysis. Our results show that SG performs better in the low noise regime, and KLO with high noise. SG is easier to implement and runs faster.

7. 混合神经世界模型

Hybrid Neural World Models

作者: Pranav Lakshmanan, Paras Chopra

类别: cs.LG, cs.AI, math.NA, physics.comp-ph

发布日期: 2026-05-27

arXiv ID: 2605.28317v1

链接: http://arxiv.org/abs/2605.28317v1

摘要: Neural surrogates promise large speedups over classical solvers for physical dynamics but fail silently at sharp dynamical events such as shocks, fronts, and contact. We present hybrid neural world models for physical dynamics: a recipe for training and deploying multi-horizon surrogates in physical state space, where a single network with continuous horizon conditioning is trained with direct supervision against textbook reference solvers to predict any future state at horizon T in one forward pass. Although no part of the training data, loss function, or architecture supervises discontinuity location, the trained surrogate encodes it implicitly, recoverable from its forward passes alone as a per-trajectory error map that concentrates on shocks, fronts, and contacts, and stays small elsewhere. The map is competitive with or better than standard label-free baselines including deep ensembles, learned error heads, gradient-magnitude indicators, and locally-adaptive conformal prediction, while using only a single trained network and requiring no calibration set or governing-equation knowledge. The recipe supports two operating points. Mode 1 runs the surrogate alone for maximum throughput, with same-hardware CPU speedups of 26x to 72x against textbook solvers on the PDE environments. Mode 2 uses the error map to gate a reference-solver fallback, deferring uncertain trajectories and roughly halving the surrogate’s residual error at the default operating point. The recipe applies without modification across reaction-diffusion, compressible Euler, and rigid-body collision dynamics.

8. 求解大规模Sylvester方程的因子化Krylov子空间方法

Factorized Krylov subspace methods for solving large Sylvester equations

作者: Yuki Satake, Takeshi Fukaya, Tomohiro Sogabe, Shao-Liang Zhang

类别: math.NA

发布日期: 2026-05-27

arXiv ID: 2605.28274v1

链接: http://arxiv.org/abs/2605.28274v1

摘要: Krylov subspace methods, such as the Conjugate Gradient (CG) and BiCGSTAB methods, are widely used in scientific computing for solving linear systems. In this study, we propose a new framework for solving large Sylvester equations in a low-rank format by reconstructing matrix-oriented Krylov subspace methods. The framework realizes efficient algorithms that are mathematically equivalent to the matrix-oriented Krylov subspace methods by exploiting the mathematical properties of the Sylvester operator and the low-rank structure of the right-hand side. Specifically, by leveraging these properties, approximate solutions can be expressed in a low-rank factorized form, enabling efficient computation and reduced memory requirements. The effectiveness of our algorithms is demonstrated through numerical experiments.

9. 条件均值障碍：从确定性回归到条件分布学习

The conditional-mean barrier: From deterministic regression to conditional distribution learning

作者: Junfeng Chen

类别: stat.ML, math.NA, nlin.CD, physics.data-an

发布日期: 2026-05-27

arXiv ID: 2605.28076v1

链接: http://arxiv.org/abs/2605.28076v1

摘要: Many problems in computational science and engineering become one-to-many after coarse graining, partial observation, or inverse reconstruction: a resolved state may not determine a unique subgrid forcing, a structural descriptor may not determine a unique effective response, and a low-resolution observation may correspond to many plausible high-resolution fields. In such settings, deterministic surrogates may learn a well-defined mathematical object while still missing application-relevant uncertainty. This tutorial develops a self-contained module centered on the conditional-mean barrier: the point at which a squared-loss predictor has reached the conditional mean and the remaining error is irreducible aleatoric variance. We give two diagnostics for locating this barrier, residual-feature orthogonality and the coefficient of determination against its explained-variance ceiling, and prove that adding latent randomness to a squared-loss predictor collapses it back to the conditional mean. Crossing the barrier therefore requires a loss that scores distributions rather than point predictions. We briefly organize common distributional objectives, including negative log-likelihood, moment and observable matching, variational objectives, adversarial divergences, and score matching, by the feature of the conditional law each targets. The emphasis is the boundary itself and a finite-data procedure for recognizing it, rather than a survey of methods beyond it. CPU-based demonstrations on a two-branch law and a two-scale Lorenz-96 closure problem show how the diagnostics distinguish deterministic underfitting from residual distributional variability.

10. 带有多步细传播子的Parareal算法的收敛性分析

Convergence analysis of a parareal algorithm with multistep fine propagator

作者: Georgios Akrivis, Qingle Lin, Zhi Zhou

类别: math.NA

发布日期: 2026-05-27

arXiv ID: 2605.28059v1

链接: http://arxiv.org/abs/2605.28059v1

摘要: The parareal algorithm is a powerful parallel-in-time integration method that accelerates the numerical solution of evolution equations by iteratively combining a fine propagator and a coarse propagator. Although the convergence of the parareal algorithm has been extensively studied, most existing analyses assume that the fine propagator is either an exact solver or a single-step method. In this paper, we construct and analyze a parareal algorithm for solving parabolic equations, where the fine propagator is based on the two-step backward differentiation formula (BDF2), while the coarse propagator remains a single-step method. We propose a novel approach to design an effective correction for the initialization steps and establish linear convergence of the iteration. Numerical results fully support the theoretical findings, show clear improvements over existing multistep parareal strategies, and indicate that the proposed approach extends effectively to higher-order BDF methods and to nonlinear problems.

11. 协调/非协调虚拟元素及其在弯曲三维区域弹性问题中的应用

Conforming/Non-conforming Virtual Elements and application to elasticity problems in curved three-dimensional domains

作者: L. Beirão da Veiga, F. Dassi, A. Russo, M. Trezzi

类别: math.NA

发布日期: 2026-05-27

arXiv ID: 2605.28041v1

链接: http://arxiv.org/abs/2605.28041v1

摘要: The Virtual Element Method (VEM) is a well-established framework for solving partial differential equations on polygonal and polyhedral meshes. In this paper, we introduce a novel hybrid VEM that integrates both conforming and nonconforming virtual spaces. We apply this formulation to a three-dimensional linear elasticity problem, providing rigorous theoretical analysis to demonstrate optimal convergence rates. Furthermore, we explore the extension of this approach to domains with curved boundaries.

12. 一种用于一般区域四边形网格生成的高效且稳定的扩散生成方法

An efficient and stable diffusion generated method for quadrilateral mesh generation in general domains

作者: Jingwen Dai, Zhonghua Qiao, Dong Wang

类别: math.NA

发布日期: 2026-05-27

arXiv ID: 2605.27854v1

链接: http://arxiv.org/abs/2605.27854v1

摘要: This paper introduces a novel, robust, and computationally efficient framework for high-quality quadrilateral mesh generation on general two-dimensional domains. The core of the proposed approach is a novel method for computing cross fields by minimizing a modified and relaxed Ginzburg–Landau-type energy functional. A key innovation is the extension of the problem from the original, potentially complex domain to a larger regular computational domain. This extension transforms the central computational procedure into an iterative scheme that requires only two straightforward and efficient operations: linear diffusion solved globally via the Fast Fourier Transform (FFT) and point-wise normalization. Notably, our method eliminates the conventional need for generating an intermediate triangular mesh or solving complex nonlinear optimization problems on the irregular domain. We provide a rigorous theoretical analysis, proving that the proposed iterative algorithm guarantees unconditional monotonic decay of the objective functional. Comprehensive numerical experiments demonstrate the method’s robustness across a wide range of complex geometries, its significant computational efficiency afforded by the FFT-based diffusion, and its consistent generation of high-quality quadrilateral meshes. This work presents a reliable and theoretically sound alternative to existing mesh generation techniques, with strong potential for practical applications in scientific computing.

13. 神经梯度流中的全局收敛性与误差传播：一个黎曼优化框架

Global Convergence and Error Propagation in Neural Gradient Flows: A Riemannian Optimization Framework

作者: Shixin Zheng, Yiwei Wang, Haizhao Yang

类别: math.OC, math.NA

发布日期: 2026-05-26

arXiv ID: 2605.27779v1

链接: http://arxiv.org/abs/2605.27779v1

摘要: We develop a geometric convergence theory for neural-network optimization within the minimizing movement scheme (MMS) framework. Reformulating each neural MMS step as a minimization over the set of increments in a Hilbert space, we show that under a $C^2$ network with locally non-degenerate Jacobian this increment set is a boundaryless smooth embedded submanifold, on which a natural preconditioned (Gauss–Newton-type) gradient flow in parameter space induces exactly the Riemannian gradient flow. Under a strict interior-localization condition and an explicit data condition, the reached sublevel set is geodesically convex and the subproblem objective is geodesically strongly convex on it; both the continuous Riemannian gradient flow and its discrete companion via the exponential map converge linearly to the unique subproblem minimizer. Propagating finite-time inner-solver inexactness and neural-approximation error through the MMS iterations yields a uniform function-space tracking bound and an explicit trajectory budget, so the inexact neural iterates converge to an $O(δ)$-neighborhood of the global minimum. Numerical experiments on nonlinear regression and a small-scale latent-diffusion testbed indicate that the Gauss–Newton-type inner solver achieves smaller trajectory errors with substantially fewer inner iterations than first-order baselines.

14. 基于神经网络的稀疏POD模态选择与流形降维

Sparse POD Mode Selection and Manifold Dimensionality Reduction with Neural Networks

作者: Tomoki Koike, Prakash Mohan, Marc T. Henry de Frahan, Elizabeth Qian, Julie Bessac

类别: physics.flu-dyn, cs.LG, math.DS, math.NA

发布日期: 2026-05-26

arXiv ID: 2605.27756v1

链接: http://arxiv.org/abs/2605.27756v1

摘要: High-performance computing enables simulation of high-dimensional physical systems, but downstream analyses such as inverse problems and control remain computationally expensive, motivating model order reduction (MOR) to construct efficient low-dimensional surrogates. Proper Orthogonal Decomposition (POD), a widely adopted data-driven MOR method, projects dynamics onto linear subspaces spanned by the most energetic modes. However, POD struggles for problems with slowly decaying Kolmogorov (n)-widths, such as advection-dominated and turbulent flows, requiring many modes for accurate reconstruction. Moreover, energy-based selection can discard crucial low-energy modes needed to capture small-scale features. Recent nonlinear manifold methods using polynomial mappings with alternating or greedy mode selection achieve better reconstruction with fewer modes. However, these methods fix the nonlinear mapping form a priori, limiting expressivity. Conversely, neural network (NN) manifolds offer greater expressivity but employ energy-based selection. We present SparseModesNet, a dimensionality reduction framework that employs linear encoding via POD modes and nonlinear NN decoding. The decoder leverages LassoNet, a method enforcing hierarchical sparsity through residual connections with linear skip layers, to simultaneously select informative POD modes and learn a nonlinear mapping that minimizes reconstruction error. On benchmark advection-dominated and chaotic flows, SparseModesNet matches or exceeds state-of-the-art performance. For turbulent channel flow at friction Reynolds number (Re_τ=5200), we reduce reconstruction error by 51–78% compared to existing polynomial manifold methods while maintaining interpretability through physically meaningful mode selection.

15. 乘法矩阵复合算子的求逆

Inversion of the Multiplicative Matrix Compound Operator

作者: Debojyoti Dey, Ron Ofir, Christian Grussler

类别: math.RA, eess.SY, math.AG, math.NA

发布日期: 2026-05-26

arXiv ID: 2605.27682v1

链接: http://arxiv.org/abs/2605.27682v1

摘要: We study the problem of determining a matrix whose $k$th multiplicative compound is a prescribed matrix~$M$. The cardinality of the set of matrices whose $k$th multiplicative compound equals~$M$ is characterized in terms of $\rank(M)$. On the one hand, if $\rank(M)\le 1$, it is shown that there exist infinitely many such matrices for which a complete characterization is determined. On the other hand, if $\rank(M)>1$, then there exists a unique matrix – up to an overall sign – whose compound is~$M$. An algorithm for finding a matrix whose compound equals~$M$ is detailed, and its time complexity is analyzed.