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重点关注论文（完整分析）
1. 亥姆霍兹方程的方向自适应平面波间断伽辽金方法
Direction-Adaptive Plane-Wave Discontinuous Galerkin Methods for the Helmholtz Equation
作者: Shelvean Kapita
类别: math.NA
发布日期: 2026-08-18
arXiv ID: 2608.18380v1
摘要翻译: 我们考虑具有自适应局部传播方向的亥姆霍兹方程的平面波间断伽辽金（PWDG）近似。通过最小化网格骨架上的加权残差来选择方向。我们研究两种形式：在部分A中，在固定方向下求解PWDG系统的系数；而在部分B中，相同的残差在系数和方向上联合最小化，其中系数通过变量投影消去。复角度统一了传播和倏逝的Trefftz波。离散系统在Trefftz-DG范数下归一化，并使用局部Cauchy-迹Gram矩阵来移除数值相关的方向。在直边上，残差积分是精确的。对于全Dirichlet问题，残差等于DG误差的平方，并提供局部自适应指示子。我们证明了在固定秩的可识别零残差解附近，约化方向泛函具有局部二次增长。在数值上，一个精确的圆形DtN测试表明，即使小的平面波扇产生较大的场误差，Hankel波的主相位方向也可以被精确恢复。我们还分离了系数基、迹截断和迹谱算术对高$p$ DtN底限的影响。最后，对于平面波的有限和，基于残差的ENRICH–MOVE延拓在$M=1,\ldots,19$时恢复所有方向至舍入误差。在$M=20$时，自动出生步骤进入一个错误的盆地，而附近的出生再次达到舍入误差。结果表明，方向自适应对于低方向复杂性的解最为有效。
Comment: 无
链接: http://arxiv.org/abs/2608.18380v1
1. 研究对象和背景
本文研究对象为截断声波散射问题中的 Helmholtz 方程。设 $\Omega\subset\mathbb R^2$ 为有界 Lipschitz 区域，边界 $\partial\Omega=\Gamma_D\dot{\cup}\Gamma_R$，其中 $\Gamma_D$ 为 Dirichlet 边界，$\Gamma_R$ 为人工外边界。问题为
$$\Delta u+\kappa^2u=0\ \text{in }\Omega,\quad u=g\ \text{on }\Gamma_D,\quad \partial_nu+\mathrm i\kappa u=0\ \text{on }\Gamma_R,$$其中 $\kappa>0$ 为波数，时间因子取 $\mathrm e^{+\mathrm i\omega t}$。对大的 $\kappa$，数值求解非常昂贵；平面波间断 Galerkin (PWDG) 方法使用满足方程本身的 Trefftz 基函数，只需在网格骨架 (mesh skeleton) 上离散，见 Cessenat–Després 的超弱变分格式 [1] 及其后续分析 [4–6]。传统 PWDG 在构造离散空间时事先固定每个单元上的平面波个数与传播方向，常用等间距方向；当解只由少数几个传播方向主导时，这种均匀方向基是浪费的；而且随着局部维数增加，平面波迹函数趋于线性相关，代数问题严重病态 [11]。本文试图填补的空白是：将传播方向本身作为优化变量，由同一个网格骨架上的加权残量自动确定方向，而不是事先固定；同时处理方向优化带来的病态性、数值秩亏损和强非线性/非凸性。文中用复角度参数化同时涵盖传播波 ($\eta=0$) 与倏逝波 ($\eta\neq0$)，从而无需单独添加倏逝波基或临界角判据。
2. 主要定理或主要结果
主要结果按假设和结论表述如下。
复角度 Trefftz 族：对 $z=\theta+\mathrm i\eta\in\mathbb C$，定义 $q_K(z)=\kappa_K(\cos z,\sin z)$，$\phi_{K,z}(x)=\exp(\mathrm i q_K(z)\cdot(x-x_K))$，则 $\Delta\phi_{K,z}+\kappa_K^2\phi_{K,z}=0$；反过来，每个满足 $q\cdot q=\kappa_K^2$ 的复向量都可写成 $q_K(z)$（命题 2.1）。因此局部空间 $V_K(Z_K)=\operatorname{span}\{\phi_{K,z_{K,1}},\ldots,\phi_{K,z_{K,r_K}}\}$ 同时包含传播波与倏逝波。
图–Riesz 归一化：令 $G=(A-A^*)/(2\mathrm i)$ 为 DG/图范数 Gram 矩阵，取 $B^*GB=I$，则 $\widehat A=B^*AB=S+\mathrm i I$ 是正规矩阵且 $\sigma_{\min}(\widehat A)\ge1$；图–Riesz 条件数
$$\kappa_{\mathrm{GR}}=\kappa_2(B^*AB)\le\sqrt{1+\gamma_h^2},\quad \gamma_h=\|S\|_2.$$数值表 1 显示，普通 Euclid 条件数 $\kappa_2(A)$ 随 $p$ 增长极快（$p=27$ 时为 $3.43\times10^{14}$），而 $\kappa_{\mathrm{GR}}$ 增长慢得多（$p=23$ 时为 $8.12\times10^2$），直到局部迹秩亏损。
局部 Cauchy-迹 Gram 压缩：定义
$$m_K(u,v)=\sum_{e\subset\partial K}\left(\kappa_K\int_e uv\,ds+\kappa_K^{-1}\int_e\partial_nu\,\partial_nv\,ds\right),$$对局部 Gram 矩阵 $M_K$ 做特征分解，保留满足 $\lambda_{K,j}>\tau_{\mathrm{rank}}\lambda_{K,1}$ 的特征方向，其中 $\tau_{\mathrm{rank}}=10^{-12}$，并用 $C_K=Q_{K,r}\Lambda_{K,r}^{-1/2}$ 压缩局部基。该压缩不只是坐标缩放，而是确实改变了保留的 Galerkin 空间。
全 Dirichlet 情形残量等于精确 DG 误差（命题 4.1）：当 $\Gamma_R=\emptyset$ 时，对任意 broken Trefftz 场 $v_h(c,Z)$，
$$J(c,Z)=G_h(u-v_h,u-v_h)=\|u-v_h\|_{G,h}^2,$$因此 $J$ 的边贡献 $\eta_e^2$ 可直接作为局部自适应指示子。
混合 Dirichlet–阻抗边界：在 Assumption 4.2（broken Trefftz 稳定性估计）下，
$$\lVert u-v_h\rVert_{L^2(\Omega)}\le C_J(\alpha,\beta)C_{\mathrm{stab}}(\kappa,\Omega)J(v_h)^{1/2},$$其中 $C_J=\sqrt{\max\{\alpha^{-1},\beta^{-1},2\}}$；当 $\alpha=\beta=1/2$ 时 $C_J=\sqrt2$。这是上界而非等价估计，文中未证明逆向效率估计。
直线边上的闭式积分：对边参数化 $x(t)=a+t\ell$，$0\le t\le1$，有
$$\int_e \phi_j\phi_i\,ds=|e|\,\mathrm e^{\mathrm i q_j\cdot(a-x_{K_j})-\mathrm i q_i\cdot(a-x_{K_i})}\Psi(\mathrm i(q_j-q_i)\cdot\ell),\quad \Psi(\zeta)=\frac{\mathrm e^\zeta-1}{\zeta},$$从而残量矩阵可精确组装，方向导数只需一阶矩 $\int_0^1 t\mathrm e^{\zeta t}dt$。
数值结论：对 Hankel 波的圆形 DtN 边界实验，主导相位方向能被准确恢复，但小的平面波扇并未给出相应小的场误差；高 $p$ DtN 地板分别受系数基、迹截断和迹谱算术精度影响；对有限个平面波和 $M=1,\ldots,19$ 的族，基于残量的 ENRICH–MOVE 连续法把全部方向恢复到舍入误差；在 $M=20$ 时自动 birth 步进入假盆地，而附近 birth 又能达到舍入误差。作者据此认为方向自适应仅对低方向复杂度解最有效。
3. 研究方法、关键技术和核心工具
核心障碍有三：(i) 方向优化引入的非线性最小二乘目标高度非凸，特别当解含多个方向时；(ii) 高维 PWDG 基的 Cauchy 迹接近线性相关，导致全局矩阵病态；(iii) 固定方向 Galerkin 约束与方向搜索耦合，系数变量和方向变量的尺度差异大。作者的真正创新点在于将 Trefftz 空间的方向参数化、离散系统的归一化、数值秩压缩和残量精确积分放在同一个框架中处理。
复角度参数化：$z=\theta+\mathrm i\eta$ 使 $q_K(z)=\kappa_K(\cos z,\sin z)$ 自动满足色散关系 $q\cdot q=\kappa_K^2$。这统一了传播波和倏逝波，且 $\operatorname{Im}q_K=\kappa_K\sinh\eta(-\sin\theta,\cos\theta)$ 给出指数变化率的物理解释 $\lVert\operatorname{Im}q_K\rVert_2=\kappa_K|\sinh\eta|$。这一参数化避免了单独的倏逝波基或临界角测试。
图–Riesz 归一化：作者指出 Euclid 条件数依赖于平面波系数的缩放，不能反映离散算子在自然范数下的性质。他们使用标准 coercive Trefftz-DG 范数作为 Riesz 度量，即 $G=(A-A^*)/(2\mathrm i)$，并选择 $B$ 使 $B^*GB=I$，从而得到精确正规形式 $\widehat A=S+\mathrm i I$。这样 $"graph--Riesz"$ 条件数 $\kappa_{\mathrm{GR}}$ 本质上度量的是离散算子 Hermit 部分相对于骨架度量的大小，而不是原系数基的病态程度。
局部 Cauchy-迹 Gram 压缩：作者用局部矩阵 $M_K$ 检测数值上线性相关的迹组合。对每个单元做特征分解后，只保留特征值超过 $\tau_{\mathrm{rank}}\lambda_{K,1}$ 的模态，并用 $\Lambda_{K,r}^{-1/2}$ 使保留的迹基在 $m_K$ 下正交。这一步骤在全局求解之前完成，且明确改变了保留的 Galerkin 空间。高 $p$ 下有用模式可能低于 $10^{-12}$，因此截断阈值必须与工作精度和所需精度一起选取。
全 Dirichlet 情形的精确残量恒等式：命题 4.1 说明当 $\Gamma_R=\emptyset$ 时，残量 $J$ 就是 DG/图范数下的精确误差平方。该恒等式不依赖系数如何得到，因此无论方向如何更新，局部边贡献 $\eta_K$ 都给出局部化误差估计，无需第二估计子。
直线边的精确积分：由于平面波乘积在直边上的积分可写成指数矩 $\Psi(\zeta)$，且关于方向参数的导数只引入一阶矩，因此残量矩阵及其导数可以精确组装，无需对内部边残量进行点采样。数值求积只用于 $g$ 等无闭式表达的数据项。
Part A / Part B 两种非线性形式：Part A 保留 Galerkin 方程 $A(Z)c^{\mathrm{PW}}(Z)=f(Z)$，对约化目标 $\hat J^{\mathrm{PW}}(Z)=J(c^{\mathrm{PW}}(Z),Z)$ 做方向优化。Part B 去掉 Galerkin 约束，对 broken Trefftz 场本身最小化骨架残量；固定方向时是 Monk–Wang 型线性最小二乘，再用 variable projection 消去线性系数。Part B 避免每个非线性迭代都求解 PWDG 系统，但目标函数的 Jacobian 需要方向导数。
连续化 ENRICH–MOVE：对高维非凸问题，作者用前一问题的方向初始化更难的后续问题，逐步增加方向数。数值实验显示自动 birth 步在 $M=20$ 时进入假盆地，而邻近 birth 成功，说明初始化与连续化策略对避免局部极小至关重要。
证明框架方面，摘要与第 4 节预告了“在可辨识的零残量解的固定秩邻域内，约化方向泛函局部二次增长”的定理。这表明在理想情形下目标函数在解附近有正定的曲率，为梯度类优化提供理论依据。但具体定理的假设、固定秩的定义和证明细节在提供的 PDF 内容中未完整出现。
4. 与之前工作的比较
与既有 PWDG 文献相比，本文的推进主要有以下几个层面。
方向参数化的推广：Agrawal–Hoppe [13] 在保持 PWDG 变分方程作为约束的前提下优化实方向；Amara 等 [14] 在最小二乘方法中旋转局部平面波基。本文将方向参数推广到复角度 $z=\theta+\mathrm i\eta$，从而统一处理传播波和倏逝波。这是对 [13,14] 的重要扩展，因为倏逝波在多层或近临界角问题中不可忽略，而传统方法需要额外引入倏逝波基或临界角判据。作者明确指出“无需单独 evanescent basis 或 critical-angle test”。
约束形式的比较：Part A 与 Agrawal–Hoppe [13] 类似，保留 PWDG 方程作为约束；Part B 则去掉该约束，直接在 broken Trefftz 场上最小化残量，并用 variable projection [3] 消去系数变量。这更接近 Monk–Wang [2] 的固定方向线性最小二乘问题。Part B 的好处是避免每步求解非线性约束问题，但目标函数的几何不同，局部二次增长定理是在约化方向泛函上证明的。作者没有显式比较 Part A 与 Part B 的收敛速度或计算代价。
归一化与条件数：已有 PWDG 分析 [4–6,11] 常关注 Euclid 条件数，[11] 指出高维平面波迹近似相关导致病态。本文引入 graph–Riesz 条件数 $\kappa_{\mathrm{GR}}$，并证明其有上界 $\sqrt{1+\gamma_h^2}$。数值表 1 显示在 $p$ 增大时 $\kappa_{\mathrm{GR}}$ 远小于 $\kappa_2(A)$，直到局部迹真正失去数值秩。这是对 [11,12,18–20] 中条件数和冗余性讨论的具体算法化：先用局部 Cauchy-迹 Gram 矩阵压缩，再做全局归一化。
残量–误差关系：全 Dirichlet 情形下，命题 4.1 给出精确等式 $J=\|u-v_h\|_{G,h}^2$。这比一般残量估计强得多：通常残量只是误差的上界或等价估计，而这里是等式。混合 Dirichlet–阻抗情形只能得到上界 (34)，且需要 Assumption 4.2 的 broken Trefftz 稳定性常数。作者未证明逆向效率估计，因此混合问题中 $J$ 的局部贡献不能严格解释为局部误差。
数值现象的剖析：本文的实验设计专门展示方向自适应的适用边界。对精确圆形 DtN 边界上的 Hankel 波，学习到的方向能很好地跟踪局部相位，但小平面波扇的场误差并不相应减小；高 $p$ DtN 地板分别受系数基、迹截断和迹谱算术精度影响。这些现象在以往 PWDG 方向优化文献 [13,14] 中未见系统讨论。作者通过 $M=1,\ldots,19$ 的精确恢复实验说明方向自适应在解由少量可辨识方向主导时非常有效，而在 $M=20$ 时因非凸性出现假盆地，说明方向自适应的适用性是“低方向复杂度”，而非一般高频散射问题。
代价与局限：方向优化需要计算残量矩阵及其方向导数，并要求每个单元做局部 Gram 特征分解和全局图–Riesz 归一化；这些都比固定均匀方向的 PWDG 昂贵。作者在文中未给出与均匀方向方法的计算时间对比，但实验显示对方向复杂度高的解，方向自适应带来的场误差改善有限。从提供内容无法确认作者是否对 $M=20$ 的假盆地做了全局优化策略的系统比较；他们仅报告“nearby birth again reaches roundoff”，表明初始化细节对结果影响显著。
了解领域论文（摘要翻译）
1. 多阶段神经算子学习及其在卷积中的应用
Multi-stage neural operator learning with application for convolutions
作者: Zhiping Mao, Zhenye Wen, Yong Zhang, Xiaofei Zhao
类别: cs.LG, math.NA
发布日期: 2026-08-19
arXiv ID: 2608.18851v1
链接: http://arxiv.org/abs/2608.18851v1
摘要翻译
中文标题: 多阶段神经算子学习及其在卷积中的应用
摘要翻译: 卷积积分在应用中广泛存在，为了实现快速且精确的计算，本文引入了两种通用的多阶段神经算子学习框架。第一种是深度配置神经算子（DCNO），这是一种监督方法，通过从输入-输出数据对中学习残差来迭代改进算子逼近。第二种是深度伽辽金神经算子（DGNO），这是一种无监督框架，适用于目标算子可由PDE表示的情形，利用PDE残差的弱形式进行训练。两种方法均通过多个训练阶段逐步构建基算子以丰富逼近空间，从而相较于标准的一次性算子学习显著提高了精度。我们对其逼近能力进行了理论分析，并将其应用于学习卷积。大量数值实验表明，DCNO和DGNO均能达到高精度，在卷积问题中接近单精度下的机器精度，并且在大量查询或参数变化情况下，与传统求解器相比可提供显著的效率提升。我们还将这些框架扩展到涉及卷积密度和核均变化的多输入算子学习场景。
2. 基于逆Born级数的神经算子
Inverse Born series based neural operators
作者: John C Schotland, Aseel Titi, Jenn-Nan Wang
类别: math.NA
发布日期: 2026-08-18
arXiv ID: 2608.18262v1
链接: http://arxiv.org/abs/2608.18262v1
摘要翻译
中文标题: 基于逆Born级数的神经算子
摘要翻译: 逆Born级数为表示无限维函数空间之间的非线性逆映射提供了一个通用的微扰框架，并在由偏微分方程和积分方程支配的逆问题中得到了广泛应用。受其算子理论结构的启发，我们开发了一个系统框架，用于构造逼近逆Born级数中出现的算子展开的神经算子。我们的方法将逆Born级数的解析表示与神经算子的表达能力相结合，在保持底层算子结构的同时，生成非线性逆映射的数据驱动逼近。所提出的框架适用于广泛类别的逆问题，并在一般的泛函分析环境中呈现。为了展示其实际性能，我们考虑了三个代表性问题：逆散射和Calderón（电阻抗断层成像）问题。数值实验表明，所构造的神经算子能够准确逼近逆Born展开，并在各种测试案例中产生高质量的重建结果。这些结果表明，所提出的方法为学习非线性逆算子提供了一种有效且计算高效的途径，并为将经典算子展开与现代神经算子架构相结合指明了一个有前景的方向。
不相关论文（基本信息）
1. 分段光滑逆源重建的局部傅里叶延拓方法
A Localized Fourier Extension Method for Piecewise-Smooth Inverse Source Reconstruction
作者: Zhihong Dou, Zhenyu Zhao
类别: math.NA
发布日期: 2026-08-19
arXiv ID: 2608.19193v1
链接: http://arxiv.org/abs/2608.19193v1
摘要: We reconstruct a piecewise-smooth source in a Poisson equation on a semi-infinite strip from noisy solution values measured along an interior line. Applying the one-dimensional Dirichlet Laplacian to the observation reduces the inverse problem to regularized second-order differentiation followed by a boundedly invertible correction. The differentiated trace and the source differ by an analytic smoothing term and therefore have the same interior singular support and jump data. This structure motivates a localized Fourier extension method that combines two staggered detection partitions, GTSVD-regularized local derivative coefficients, mollified conjugate Fourier sums, and structure-aligned numerical differentiation. The source is then recovered by an exponentially decaying spectral correction. For an exact partition, the reconstruction inherits the piecewise differentiation rate $\mathcal O(δ^{(\bar s-2)/\bar s})$; local peak and partition perturbation estimates describe the additional effect of breakpoint errors. Comparisons with full-grid total-variation regularization and truncated Fourier inversion show competitive high-noise performance and a pronounced low-noise advantage of the localized method. Repeated Gaussian-noise tests confirm robustness in the moderate- and low-noise regimes.
2. 高斯消去法中的元素增长
Entry growth in Gaussian elimination
作者: Rikhav Shah, John Urschel
类别: math.NA
发布日期: 2026-08-19
arXiv ID: 2608.19189v1
链接: http://arxiv.org/abs/2608.19189v1
摘要: Gaussian elimination is one of the oldest algorithms in mathematics, and the most popular method for solving an unstructured linear system. Its stability in finite precision is controlled by its growth factor, which measures how large the entries produced during elimination can become. Understanding the worst-case behavior of this quantity has been a central problem in numerical analysis since the 1940s. Here we make a significant leap in that understanding, settling several open problems. In particular, we determine the asymptotic behavior of the maximum growth factor under complete and rook pivoting, proving that both are quasi-polynomial in dimension. We also show that the exponential growth under partial pivoting persists for sparse matrices and that randomized partial pivoting suffers the same instability. By contrast, we show that every matrix has a row permutation with polynomial growth, though finding the optimal row permutation is NP-hard.
3. 一种带块卷积的精确矩局部勒让德框架方法用于Caputo分数阶微分
An Exact-Moment Local Legendre Frame Method with Block Convolution for Caputo Fractional Differentiation
作者: Zhenyu Zhao, Benxue Gong, Tinggang Zhao, Xianzheng Jia
类别: math.NA
发布日期: 2026-08-19
arXiv ID: 2608.19157v1
链接: http://arxiv.org/abs/2608.19157v1
摘要: We propose a local Legendre frame method for the accurate computation of Caputo fractional derivatives of order (0<α<1). On each local subinterval, the function is represented by a restricted Legendre frame obtained from scaled Legendre polynomials on an extended interval. The local coefficients are computed from equispaced samples by an exponentially weighted GTSVD regularization. The Caputo derivative is then evaluated by applying the weakly singular fractional integral to the derivatives of the local frame basis functions. Since these derivatives are polynomials, the corresponding Caputo weights can be written in terms of finite weighted moments, so that the singular kernel is treated analytically rather than by a low-order quadrature rule. For uniform partitions, the history weights have a block-dependent structure and can be reused efficiently. The error analysis separates the local frame reconstruction from the Caputo integration. In particular, the Caputo error is bounded by the derivative reconstruction error, while the latter is obtained from the (L^2) reconstruction error and a weighted smoothness bound of the GTSVD approximation through an interpolation argument. For analytic local functions with exponential coefficient decay, this leads to exponential-type convergence of the derivative and hence of the Caputo approximation. Numerical experiments confirm the accuracy of the exact moment weights, the effectiveness of the local weighted reconstruction, and the efficiency of the block implementation.
4. 拟线性系统的局部零能控性及相关数值实验
Local null controllability of a quasi-linear system and related numerical experiments
作者: Enrique Fernández-Cara, Juan Límaco, Yuri Thamsten, Denilson Menezes
类别: math.OC, math.AP, math.NA
发布日期: 2026-08-19
arXiv ID: 2608.19023v1
链接: http://arxiv.org/abs/2608.19023v1
摘要: This paper concerns the null control of quasi-linear parabolic systems where the diffusion coefficient depends on the gradient of the state variable. In our main theoretical result, with some assumptions on the regularity and growth of the diffusion coefficient and regular initial data, we prove that local null controllability holds. To this purpose, we consider the null controllability problem for the linearized system, we deduce new estimates on the control and the state and, then, we apply a Local Inversion Theorem. We also formulate an iterative algorithm of the quasi-Newton kind for the computation of a null control and an associated state. We apply this method to some numerical approximations of the problem and illustrate the results with several experiments.
5. 自适应分辨率框架：Hilbert空间中的多水平框架
Adaptive resolution frames: A multilevel framework in Hilbert spaces
作者: Jahangir Cheshmavar
类别: math.FA, math.NA
发布日期: 2026-08-19
arXiv ID: 2608.18879v1
链接: http://arxiv.org/abs/2608.18879v1
摘要: We introduce adaptive resolution frames (AR-frames), a class of weighted frames equipped with a partitioned index structure that models multiple resolution levels in Hilbert spaces. We establish a block-triangular representation of the associated operators and prove the convergence of an iterative reconstruction method with explicit error estimates. We further derive optimal block-diagonal preconditioners that minimize the iteration error and obtain stability results showing that AR-frames and their canonical reconstruction operators remain stable under sufficiently small weighted perturbations. These results provide a unified framework for efficient reconstruction and perturbation analysis of AR-frames.
6. 色散激波之前无色散极限区域中gKdV方程的一些一致误差分析
Some uniform error analysis for gKdV equations in dispersionless limit regime before dispersive shock
作者: Bing Li, Tingfeng Wang, Xiaofei Zhao
类别: math.NA
发布日期: 2026-08-19
arXiv ID: 2608.18809v1
链接: http://arxiv.org/abs/2608.18809v1
摘要: This work establishes uniform error estimates for classical numerical schemes applied to the generalized Korteweg-de Vries (gKdV) equation in the dispersionless limit regime, specifically before the development of dispersive shock wave (DSW). We analyze the Crank-Nicolson (CN) and a Lawson-type Runge-Kutta (Lawson-RK) methods, when discretized in space via the Fourier pseudo-spectral method. We prove that both fully discrete schemes achieve optimal second-order temporal accuracy and spectral spatial accuracy, with error constants being uniform in the vanishing dispersion parameter $\varepsilon$. The analysis also addresses the unique solvability of the CN scheme in the dispersionless limit regime. Theoretical findings are supported by numerical experiments, demonstrating the $\varepsilon$-uniform accuracy and the necessity of step size restriction for CN. The study in general validates the classical methods for solving the dispersionless gKdV equation before DSW.
7. 一种用于拉伸主导脆性断裂相场建模的平移能垒方法
A shifted energy barrier approach for phase-field modeling of tensile-dominated brittle fracture
作者: Yaode Yina, Luigi Greco, Hongjun Yua, Simone Morganti
类别: math.NA, cond-mat.mtrl-sci
发布日期: 2026-08-19
arXiv ID: 2608.18766v1
链接: http://arxiv.org/abs/2608.18766v1
摘要: The classical AT1 phase-field model contains an intrinsic energy barrier for crack nucle ation, which makes the predicted strength depend on the fracture toughness and the regularization length. For tensile-dominated brittle fracture, this barrier is shifted by mapping the Rankine criterion, evaluated on the effective stress, onto a state-dependent active-energy threshold. The prescribed tensile strength then controls crack nucleation, while the AT1 crack-density functional, stiffness degradation, and degraded stress response remain unchanged. Since the threshold depends on the current stress state, the field equations are derived from a restricted variational principle. A microforce formulation identifies the barrier shift as a dissipative resistance and provides the corresponding lower bound on the regularization length. In one-dimensional tension, closed-form solutions recover the prescribed peak strength and give a cosine-type localization profile that ap proaches the classical AT1 profile as the shift vanishes. Numerical examples show that, within the admissible range, the nucleation load is nearly insensitive to the regularization length and the predicted multiaxial nucleation states follow the Rankine envelope. Under overall compression, crack nucleation remains associated with local tensile stress concen trations. The formulation also captures the transition from strength-controlled failure for small flaws to the LEFM limit for large cracks.
8. 有界域上二阶线性椭圆Dirichlet边值问题的解对称性与区域约简有限元方法
Symmetry of Solutions and Domain-Reduction Finite Element Method for Second-Order Linear Elliptic Dirichlet Boundary Value Problems on Bounded Domains
作者: Xianlong Pan, Wei Jiang
类别: math.NA, math.AP
发布日期: 2026-08-19
arXiv ID: 2608.18549v1
链接: http://arxiv.org/abs/2608.18549v1
摘要: Combining classical group theory and partial differential equation theory, this paper investigates the symmetry group $\operatorname{Sym}(u)$ of the unique solution $u$ to the second-order linear elliptic boundary value problem on an $n$-dimensional bounded domain $Ω$ $-\sum_{i,j=1}^{n} a_{ij}(x)u_{x_ix_j} + \sum_{i=1}^{n} b_i(x)u_{x_i} + c(x)u = f(x), x\in Ω$, $u(x) = h(x), x\in \partial Ω$, The following symmetry groups are defined and characterized respectively: the symmetry group $\operatorname{Sym}(A)$ of the second-order coefficient matrix function $A(x)=(a_{ij}(x))_{n\times n}$; the symmetry group $\operatorname{Sym}(b)$ of the first-order coefficient column vector function $b(x)=(b_{1}(x),b_{2}(x),\cdots,b_{n}(x))^{T}$; the symmetry group $\operatorname{Sym}(c)$ of the zero-order coefficient function $c(x)$; the symmetry group $\operatorname{Sym}(f)$ of the internal source function $f(x)$; and the symmetry group $\operatorname{Sym}(h)$ of the boundary source function $h(x)$. This paper rigorously proves that the common symmetry group $\operatorname{Sym}(A)\cap\operatorname{Sym}(b) \cap\operatorname{Sym}(c) \cap\operatorname{Sym}(f) \cap\operatorname{Sym}(h)$ is a subgroup of $\operatorname{Sym}(u)$. In addition, if the common symmetry group contains several mirror symmetry elements, the original second-order linear elliptic boundary value problem on the entire domain $Ω$ can be reduced to the corresponding boundary value problem on a certain subdomain. It is strictly proven in this paper that the new boundary condition imposed on the boundary of the subdomain is the homogeneous generalized Neumann boundary condition. The linear finite element method is used to numerically solve the second-order linear elliptic boundary value problem on the subdomain, thereby achieving domain reduction and significantly reducing the computational cost.
9. 一般区域上具有解析激活的线性化浅层网络的最优Sobolev逼近
Sharp Sobolev Approximation on General Domains by Linearized Shallow Networks with Analytic Activations
作者: Jia Li, Tong Mao, Jinchao Xu
类别: math.NA
发布日期: 2026-08-19
arXiv ID: 2608.18520v1
链接: http://arxiv.org/abs/2608.18520v1
摘要: We study Sobolev approximation on bounded domains by linearized shallow neural networks whose inner parameters are prescribed independently of the target function. Our main step is a one-dimensional construction for analytic activations. We prove that quasi-Chebyshev parameter sets with univariate resolution $m$ generate fixed feature spaces attaining the sharp $H^r$-to-$H^s$ approximation order $m^{-(r-s)}$ for a class of analytic activations satisfying a quantitative non-cancellation condition on their Taylor coefficients. Combining this result with the ridge-function lifting theorem in [SIAM J. Math. Anal. 30 (1998), pp. 155-189] and its extension to arbitrary quasi-uniform direction sets established in this work, we construct tensor-product-type parameter sets that attain the sharp rate
$$\|f-f_n\|_{L^2(Ω)}\lesssim n^{-\frac rd}\|f\|_{H^r(Ω)},\quad f\in H^r(Ω)$$for all $r>0$. In contrast to the finite-difference construction in [Neural Comput. 8 (1996), pp. 164-177], whose explicit admissibility condition may require an extremely small parameter scale, the proposed parameter sets remain distributed over fixed intervals and are therefore more amenable to practical computation.
10. 一种面向参数化Navier-Stokes流的高效时间二阶惩罚-投影系综涡粘性方法
An Efficient Second-Order-in-Time Penalty-Projection Ensemble Eddy Viscosity Method for Parameterized Navier-Stokes Flows
作者: Md Mahmudul Islam, Muhammad Mohebujjaman, Jahrul Alam
类别: math.NA
发布日期: 2026-08-19
arXiv ID: 2608.18487v1
链接: http://arxiv.org/abs/2608.18487v1
摘要: We propose a novel, robust, and second-order-accurate parameterized penalty-projection ensemble algorithm for incompressible Navier–Stokes flow problems. The resulting linearized algorithm, based on the second-order Backward Differentiation Formula (BDF-2), is computationally efficient because it shares the same coefficient matrix across all realizations for each subproblem at every time step. To enhance robustness in convection-dominated flows, the scheme incorporates Ensemble Eddy Viscosity (EEV) regularization. In addition, it is equipped with grad-div stabilization parameter $γ$, which controls the splitting error; under the assumptions of the analysis, the splitting error decreases and vanishes asymptotically as $γ\to\infty$. We establish the stability of the proposed scheme and rigorously prove its optimal convergence by demonstrating that, as $γ\to\infty$, the scheme converges to an equivalent coupled formulation. We further validate the method through a series of numerical experiments designed to verify the theoretically predicted convergence rates and evaluate its performance on benchmark convection-dominated problems. The numerical results are in excellent agreement with the theoretical analysis and confirm the effectiveness of the proposed scheme.
11. 不确定性下提高熔融沉积成型中聚合物丝材粘结质量的过程优化
Process Optimization Under Uncertainty for Improving the Bond Quality of Polymer Filaments in Fused Filament Fabrication
作者: Berkcan Kapusuzoglu, Matthew Sato, Sankaran Mahadevan, Paul Witherell
类别: cs.CE, cs.LG, math.NA, math.PR
发布日期: 2026-08-19
arXiv ID: 2608.18431v1
链接: http://arxiv.org/abs/2608.18431v1
摘要: This paper develops a computational framework to optimize the process parameters such that the bond quality between extruded polymer filaments is maximized in fused filament fabrication (FFF). A transient heat transfer analysis providing an estimate of the temperature profile of the filaments is coupled with a sintering neck growth model to assess the bond quality that occurs at the interfaces between adjacent filaments. Predicting the variability in the FFF process is essential for achieving proactive quality control of the manufactured part; however, the models used to predict the variability are affected by assumptions and approximations. This paper systematically quantifies the uncertainty in the bond quality model prediction due to various sources of uncertainty, both aleatory and epistemic, and includes the uncertainty and the model discrepancy in the process parameter optimization. Variance-based sensitivity analysis based on Sobol indices is used to quantify the relative contributions of the different uncertainty sources to the uncertainty in the bond quality. A Gaussian process (GP) surrogate model is constructed to compute and include the model discrepancy within the optimization. Physical experiments are conducted for calibration and validation of the physics model and also for validation of the optimum solution. The results show that the proposed formulation for process parameter optimization under uncertainty results in high bond quality between adjoining filaments of the FFF product.
12. 熔融沉积制造中零件质量不确定性下的多目标优化
Multi-Objective Optimization Under Uncertainty of Part Quality in Fused Filament Fabrication
作者: Berkcan Kapusuzoglu, Paromita Nath, Matthew Sato, Sankaran Mahadevan, Paul Witherell
类别: cs.CE, cs.LG, math.NA, math.OC
发布日期: 2026-08-19
arXiv ID: 2608.18429v1
链接: http://arxiv.org/abs/2608.18429v1
摘要: This work presents a data-driven methodology for multi-objective optimization under uncertainty of process parameters in the fused filament fabrication (FFF) process. The proposed approach optimizes the process parameters with the objectives of minimizing the geometric inaccuracy and maximizing the filament bond quality of the manufactured part. First, experiments are conducted to collect data pertaining to the part quality. Then, Bayesian neural network (BNN) models are constructed to predict the geometric inaccuracy and bond quality as functions of the process parameters. The BNN model captures the model uncertainty caused by the lack of knowledge about model parameters (neuron weights) and the input variability due to the intrinsic randomness in the input parameters. Using the stochastic predictions from these models, different robustness-based design optimization formulations are investigated, wherein process parameters such as nozzle temperature, nozzle speed, and layer thickness are optimized under uncertainty for different multi-objective scenarios. Epistemic uncertainty in the prediction model and the aleatory uncertainty in the input is considered in the optimization. Finally, Pareto surfaces are constructed to estimate the tradeoffs between the objectives. Both the BNN models and the effectiveness of the proposed optimization methodology are validated using the actual manufacturing of the parts.
13. Navier-Stokes方程的grad-div稳定化Crank-Nicolson人工压缩方法全离散参数鲁棒误差分析
Fully discrete parameter-robust error analysis of a grad-div stabilized Crank-Nicolson artificial compressibility method for the Navier-Stokes equations
作者: Feiyu Chen, Lili Ju, Rihui Lan, Shusen Xie
类别: math.NA
发布日期: 2026-08-18
arXiv ID: 2608.18377v1
链接: http://arxiv.org/abs/2608.18377v1
摘要: The artificial compressibility method (ACM) relaxes the incompressibility constraint in the Navier-Stokes equations by introducing a perturbation term proportional to the time derivative of the pressure, scaled by a small positive parameter $\varepsilon$. Consequently, the ACM system inherently involves two small parameters: the fluid viscosity $ν$ and the artificial compressibility parameter $\varepsilon$. While existing temporal analyses of ACM provide insights into its behavior, rigorous fully discrete error estimates that are robust with respect to both parameters remain a significant gap in the literature. In this paper, we propose a second-order Crank-Nicolson fully discrete ACM scheme and establish its parameter-robust optimal error estimates. To enhance stability and ensure robustness, we incorporate two distinct grad-div stabilization strategies: one facilitates the decoupling of velocity and pressure computations, while the other guarantees robustness with respect to both $ν$ and $\varepsilon$. For spatial discretization, we employ the Scott-Vogelius finite element pair, which is crucial for the decoupling and error analysis. The resulting parameter-uniform bounds are crucial for ensuring the long-time accuracy of the scheme, circumventing the exponential dependence on the Reynolds number typically introduced by Grönwall’s lemma. Numerical experiments are provided to validate the theoretical findings and demonstrate the efficiency of the proposed methods.
14. 无强制收敛的后验收敛：粗糙贝叶斯反问题的分辨率稳定采样
Posterior Convergence without Force Convergence: Resolution-Stable Sampling for Rough Bayesian Inverse Problems
作者: Zhiliang Deng, Xiaomei Yang
类别: math.NA, stat.ME
发布日期: 2026-08-18
arXiv ID: 2608.18365v1
链接: http://arxiv.org/abs/2608.18365v1
摘要: Bayesian posteriors can converge under model refinement even when the exact sensitivities used by gradient-based samplers do not. We study this mismatch for discretely scale-invariant rough potentials and its consequences for Metropolized Hamiltonian proposals. For Weierstrass truncations, adjacent classical-force increments grow geometrically in the rough regime, whereas fixed additive quotients converge uniformly and fixed multiplicative quotients converge locally away from the origin. The matched Jackson quotient is distinguished not by convergence alone but by dilation covariance and exact finite closure at the intrinsic scale. Uniform negative-log-likelihood approximation is shown to imply explicit total-variation, Hellinger, and bounded quantity-of-interest convergence, with a Gaussian forward-map criterion for Bayesian inverse problems. At the algorithmic level, measurable kick–drift–kick maps are triangular shears, so exact Metropolis correction does not require differentiability of the proposal field. Local uniform field convergence yields convergence of fixed-length proposals, acceptance functions, and bounded-Lipschitz kernels on common compact sets, while the first classical HMC half-kick has no fixed-step refinement limit. Numerical experiments verify the predicted force-refinement rates, show that posterior stabilization can coexist with severe resolution-dependent retuning of exact-gradient HMC, and demonstrate forward convergence without sensitivity convergence in two-dimensional Darcy flow. The computations are used to assess refinement consistency rather than to claim a universal efficiency ordering among proposal fields.