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1. 复杂几何区域上波动方程的一种基于能量的间断 Galerkin 方法

An energy-based discontinuous Galerkin method for wave equations on complex geometric domains

作者: Yangxin Fu, Yan Jiang, Siyang Wang, Lu Zhang

类别: math.NA

发布日期: 2026-09-29

arXiv ID: 2609.38378v1

摘要翻译: 本文提出了一种高阶切割型基于能量的间断 Galerkin (CutEDG) 方法，用于求解复杂区域上以及跨越静态材料界面的二阶声波方程。该方法采用非贴体笛卡尔网格，并通过在定义 EDG 能量的局部双线性形式中直接引入 ghost-penalty 稳定化来处理小切割单元问题，同时保持原始空间耦合和数值通量结构不变。我们建立了半离散能量稳定性以及先验误差估计，表明高阶精度得以保持。该格式进一步推广到具有不连续波速的界面问题。数值实验验证了预期的收敛阶，并展示了对任意小切割单元的鲁棒性。特别地，显式时间步长限制由背景网格尺寸决定，而不是由最小物理切割单元决定；并且稳定化能量 Gram 矩阵具有与切割无关的条件数。

Comment: 34 pages

链接: http://arxiv.org/abs/2609.38378v1

1. 研究对象和背景

本文研究复杂几何区域以及静止材料界面上的二阶声波方程的高阶未贴合数值离散. 设物理区域 $\Omega\subset\mathbb{R}^d$ 为有界区域, 边界为 $\Gamma=\partial\Omega$, 未知位移 $u(x,t)$ 满足

$$u_{tt}=\Delta u+f(x,t),\quad x\in\Omega,\ t\in(0,T),$$

$$u=0,\quad x\in\Gamma,\ t\in[0,T],$$

$$u(x,0)=u_0(x),\quad u_t(x,0)=v_0(x),\quad x\in\Omega.$$

引入速度变量 $v=u_t$ 后, 方程等价于时间一阶系统

$$u_t=v,\quad v_t=\Delta u+f.$$

这类声波传播问题出现在声学、地震成像、无损检测和波与结构相互作用等应用中. 在许多场景中, 波要传播很多波长或很长时间, 数值色散和相位误差会累积, 因此高阶离散比低阶方法更有吸引力, 因为达到给定精度所需的每波长自由度更少. 与此同时, 实际计算区域常包含曲边界、嵌入障碍物或复杂材料界面. 如何同时获得高阶精度和几何灵活性, 是计算波传播中的重要问题.

传统高阶有限元和间断 Galerkin 方法通常依赖贴合物理边界或界面的网格. 当合适的网格可以构造时, body-fitted 和曲线网格很有效, 但对复杂几何, 尤其是高阶情形, 网格构造可能昂贵且技术困难. 当需要考察许多几何构型或界面在模拟之间变化时, 这一问题更突出. 未贴合和浸入方法提供了有吸引力的替代: 把物理区域嵌入简单的背景网格, 通常是结构化或 Cartesian 网格, 允许物理边界和界面切割网格而不要求网格贴合. 本文正是在这一背景下, 研究二阶声波方程在复杂区域和静止材料界面上的未贴合高阶离散.

未贴合网格的灵活性伴随着著名的小切胞问题. 当边界或界面任意接近网格顶点或边时, 切割单元中属于物理区域的部分可以任意小. 定义体积分数 $\eta_K=|K_\Omega|/|K|$, 其中 $K_\Omega=K\cap\Omega$. 对内部单元 $\eta_K=1$, 对切割单元 $\eta_K\in(0,1)$, 并可能任意接近 $0$. 当 $\eta_K\ll1$ 时, 物理体积积分对定义在完整背景单元 $K$ 上的多项式自由度只提供很弱的控制, 主要位于虚拟部分 $K\setminus\Omega$ 的模式可能几乎不受约束, 从而导致严重病态和一致稳定性丧失. 对时间依赖波问题, 这个问题尤其重要, 因为任意小的切割单元还可能对显式时间积分施加过于严格的步长限制. 一个有效的未贴合波离散必须对切割构型保持鲁棒, 包括物理部分比例趋于零的极限情形.

Ghost penalty 稳定提供了恢复这种鲁棒性的系统机制, 并且是 Cut Finite Element Methods (CutFEM) 的核心成分. 已有稳定 CutFEM 格式被用于椭圆、界面和波问题. 对波方程, 高阶扩展已在未贴合格网上展示高阶精度和良好的显式稳定性质; 特别地, 前人工作 [26] 数值显示 CFL 限制基本不受浸入边界影响. 相关思想也被扩展到间断有限元空间. 文献 [13] 发展了稳定 CutDG 框架用于椭圆边值问题和界面问题, 给出关于切割构型鲁棒的误差和条件数估计. 随后 ghost-penalty CutDG 也被用于双曲问题. 特别是, 高阶 CutDG 用于双曲守恒律时, 显式时间推进的时间步限制可以独立于最小切割单元的尺寸. 这些发展说明 ghost 稳定是未贴合离散的有效工具, 但它们本身并未确定应如何把稳定引入某个特定波方程格式.

本文在 energy-based discontinuous Galerkin (EDG) 框架内考虑这一问题. DG 方法由于局部结构、高阶精度和与显式时间积分的自然兼容性, 对高阶波传播很有吸引力. 在用于二阶波方程的各种 DG 格式中, EDG 特别适合本文目标, 因为它的构造直接围绕波能量的离散对应物组织. 与网格无关的数值通量可以选择得使半离散格式满足能量守恒或能量耗散关系. 因此能量不仅是分析最终格式的工具, 也是设计离散的自然原则. 这一点在未贴合格网上很重要. 在 EDG 形式中, 离散能量与两个局部双线性形式相关, 一个涉及位移变量的梯度, 另一个涉及速度变量的 $L^2$ 内积. 当单元物理部分任意小时, 这两个形式都失去对该单元上自由度的一致控制. 小切胞问题因此直接作用在定义 EDG 能量的局部形式上. 这提示应在这些能量分量的层面设计切割稳定, 而不是把连续 CutFEM 的质量-刚度稳定直接转移到 EDG 空间算子. 本文试图补的空白正是: 发展高阶 cut energy-based discontinuous Galerkin (CutEDG) 方法, 在复杂区域和静止材料界面上求解二阶声波方程; 用未贴合 Cartesian 网格; 通过把 ghost-penalty 稳定直接引入定义 EDG 能量的局部双线性形式来解决小切胞问题; 保持原始空间耦合和数值通量结构; 建立半离散能量稳定性和先验误差估计; 并把格式扩展到波速不连续的界面问题.

2. 主要定理或主要结果

从摘要和 PDF 摘录中能够确认的主要结果如下. 需要区分作者声称严格证明的结论、摘要层面的声明以及数值观察; 由于提供的 PDF 摘录只到稳定 CutEDG 的抽象形式 (2.20), 未包含后续定理、误差估计细节和数值实验数据, 因此具体误差阶、范数、定理编号和稳定权假设无法确认.

第一, 模型与离散格式. 作者考虑二阶声波方程 (2.1a-c) 及其一阶时间系统 (2.2). 在边界拟合网格上回顾 EDG 格式 (2.5a-c), 使用离散能量

$$E_h(t)=\frac12\sum_{K\in\mathcal{T}_h^{fit}}\int_K(|v_h|^2+|\nabla u_h|^2)\,dx.$$

在未贴合 Cartesian 背景网格 $\tilde{\mathcal{T}}_h$ 上定义活跃网格

$$\mathcal{T}_h=\{K\in\tilde{\mathcal{T}}_h: |K\cap\Omega|>0\},\quad \Omega_h=\bigcup_{K\in\mathcal{T}_h}K,$$

以及每个背景单元的物理部分 $K_\Omega=K\cap\Omega$. 未稳定 CutEDG 格式由把拟合 EDG 中的每个 $K$ 替换为 $K_\Omega$ 得到, 见 (2.16a-c). 在内部单元上它退化为拟合 EDG, 只在切割单元附近不同.

第二, 小切胞稳定和能量稳定性. 作者引入体积分数 $\eta_K=|K_\Omega|/|K|$, 并指出当 $\eta_K\ll1$ 时, 物理体积贡献变小, 离散系统可能失去对主要位于虚拟部分 $K\setminus\Omega$ 的模式的一致控制. 为恢复控制, 定义切相关背景面集合

$$F_h^g=\{F=\partial K^+\cap\partial K^-\in F_h^{int}: K^+\in T_h^\Gamma \text{ 或 } K^-\in T_h^\Gamma\},$$

并引入 ghost-penalty 形式

$$J_r(w,z)=\sum_{F\in F_h^g}\sum_{\ell=0}^r\frac{\omega_\ell}{h^{2\ell+1}(2\ell+1)(\ell!)^2}\int_F [[\partial_{n_F}^\ell w]][[\partial_{n_F}^\ell z]]\,dS.$$

该稳定项在完整背景面上积分, 而不是只在物理部分 $F\cap\Omega$ 上积分, 因此把控制从物理域传递到相邻虚拟部分. 由于 DG 空间不连续, $\ell=0$ 项被保留, 惩罚函数值跳变. 稳定化被加入到定义 EDG 能量的两个局部双线性形式中, 而原始空间耦合和数值通量结构保持不变. 作者在摘要和第 4 页声明: 若 $E_h(t)$ 表示物理域上的标准 EDG 能量, $D_h(t)\ge0$ 表示切稳定贡献, 则半离散 CutEDG 方法满足

$$\frac{d}{dt}(E_h(t)+D_h(t))\le0\quad(3.23).$$

这是提供内容中可以确认的核心稳定性估计形式. 它说明总能量 $E_h+D_h$ 不增, 并且额外项 $D_h$ 恢复在任意小物理交叠上失去的控制. 作者还明确称建立了半离散能量稳定性.

第三, 先验误差估计. 摘要声明建立了 a priori error estimates, 显示高阶精度被保留. 然而, 提供内容没有给出误差定理的具体陈述, 例如误差是在哪个范数中度量, 时间半离散误差是否包含时间导数, 网格和多项式阶 $p,q$ 的关系, 正则性假设, 稳定权 $\omega_\ell$ 的下界条件, 以及误差阶是否为预期的高阶收敛率. 因此不能从提供内容中确认 $\|u-u_h\|_{L^2}=O(h^{p+1})$ 或类似具体估计. 只能确认作者声称高阶精度保持.

第四, 界面问题. 摘要称格式进一步扩展到具有不连续波速的界面问题, 界面可以任意切割 Cartesian 背景网格. 但摘录中没有给出界面问题的变分形式、界面条件、稳定项在界面附近的修改、误差估计或稳定性定理. 因此界面扩展的严格理论结论从提供内容中无法确认.

第五, 数值结果和条件数. 摘要声称数值实验确认了预测的收敛率, 并展示对任意小切割单元的鲁棒性. 特别是, 显式时间步限制由背景网格尺寸而不是最小物理切割单元控制, 稳定后的能量 Gram 矩阵表现出与切割无关的条件数. 作者还比较说, 在所考虑的多项式阶下, 相应条件数关于切割位置保持有界, 并且与类似高阶连续 CutFEM 离散同量级, 但略小. 这些是摘要层面的数值观察或引言中的陈述; 是否在正文中被严格证明, 从提供内容中无法确认. 此外, 作者在引言中把主要贡献概括为三点: 发展单区域和静止界面波问题的高阶未贴合 EDG 格式; 构造适合 EDG 局部能量形式的 ghost 稳定, 在不改变原始空间耦合和数值通量结构的情况下恢复小切胞上的一致控制; 建立稳定性和高阶误差估计, 并数值展示切割无关的显式稳定性和有利条件数.

3. 研究方法、关键技术和核心工具

核心障碍是小切胞同时破坏 EDG 能量的两个局部双线性形式. EDG 的能量不是事后分析量, 而是离散设计原则. 离散能量由

$$E_h(t)=\frac12\sum_K\int_K(|v_h|^2+|\nabla u_h|^2)\,dx$$

给出, 其中空间部分和速度部分分别对应两个局部双线性形式. 在未贴合格网上, 物理积分只在 $K_\Omega$ 上, 当 $\eta_K=|K_\Omega|/|K|$ 任意小时, 定义在完整背景单元 $K$ 上的多项式模式中主要位于虚拟部分 $K\setminus\Omega$ 的分量不能被物理积分控制. 这导致稳定性、条件数以及显式时间步限制的恶化. 因此需要稳定虚拟部分, 但不能破坏 EDG 原有的能量守恒或耗散机制和数值通量结构. 直接从连续 CutFEM 移植质量-刚度 ghost penalty 到 EDG 空间算子并不自然, 因为 EDG 的稳定性来自能量形式, 而不是来自单独的空间算子.

作者的真正创新点是把 ghost penalty 直接加入定义 EDG 能量的局部双线性形式, 而不是修改原空间耦合或数值通量. 具体地, 切相关背景面集合取

$$F_h^g=\{F=\partial K^+\cap\partial K^-\in F_h^{int}: K^+\in T_h^\Gamma \text{ 或 } K^-\in T_h^\Gamma\},$$

稳定形式为

$$J_r(w,z)=\sum_{F\in F_h^g}\sum_{\ell=0}^r\frac{\omega_\ell}{h^{2\ell+1}(2\ell+1)(\ell!)^2}\int_F [[\partial_{n_F}^\ell w]][[\partial_{n_F}^\ell z]]\,dS.$$

它在完整背景面 $F$ 上积分, 而不是只在物理部分 $F\cap\Omega$ 上积分. 因此, 稳定项把物理侧的控制传递到相邻虚拟部分. 由于近似空间是间断的, $\ell=0$ 项被保留, 惩罚函数值本身的跳变; $\ell\ge1$ 项惩罚法向导数跳变. 高阶项与多项式阶 $r$ 匹配, 权重 $\omega_\ell>0$ 和网格尺度 $h$ 进入稳定. 在抽象形式中, 稳定贡献如 $\gamma_u M h^{-2}J_p((u_h)_t,\phi_u)$ 被加入到演化方程, 见 (2.20), 而原 EDG 的空间耦合项和数值通量不变. 这样稳定化可解释为把 EDG 能量扩展到切割区域: 若 $D_h$ 是 $J_r$ 对应的非负贡献, 则得到 $d/dt(E_h+D_h)\le0$. 这是与连续 CutFEM 中把 mass-stiffness 稳定加到空间算子的关键区别.

证明中的关键分析任务包括: 用 ghost penalty 恢复切割单元上虚拟部分的控制, 使稳定后的能量 $E_h+D_h$ 对完整背景单元上的自由度一致有意义; 将物理体积项、物理面数值通量和 ghost penalty 结合, 得到某种一致强制或范数等价; 在 EDG 混合阶空间 $U_h^p\times V_h^q$ 中处理 $u_h$ 和 $v_h$ 的不同多项式阶; 保持原 EDG 数值通量的一致性, 从而使先验误差分析可以采用 Galerkin 型误差分解; 对时间半离散问题建立能量不增估计, 再结合逼近和投影得到误差估计. 从提供内容可确认作者使用了 EDG 能量方法、背景面跳变罚、高阶法向导数罚和稳定权; 但具体引理、迹不等式、逆不等式、谱等价或投影算子在摘录中没有展开, 无法确认其精确形式.

界面扩展的方法是在界面附近施加类似的切稳定, 以处理波速不连续和界面任意切割背景网格. 从摘要看, 稳定构造仍保持原 EDG 空间耦合和数值通量结构, 并扩展能量稳定机制. 但界面处的具体数值通量、稳定项权重和误差分析在提供内容中未给出.

条件数和显式时间步方面的机制可以由稳定 Gram 矩阵的谱行为解释: 稳定后有效质量或刚度矩阵在切割单元上的尺度不再由 $\eta_K$ 决定, 而由背景网格 $h$ 和稳定权决定. 因此显式 CFL 限制由背景网格尺寸 $h$ 控制, 而不是由最小物理切割单元尺寸控制. 稳定能量 Gram 矩阵的条件数关于切位置有界, 并且在所考察的多项式阶下与高阶连续 CutFEM 同量级而略小. 这些性质在摘要中作为数值结果陈述; 是否完全由严格定理保证, 从提供内容中无法确认.

4. 与之前工作的比较

与已有工作的比较可以从几个维度展开. 首先, 与拟合网格 EDG 的关系. 本文以 [2] 的 EDG 格式为参考, 保留其围绕离散能量构造空间离散的思想以及数值通量 (2.7), 但把网格从边界拟合 $T_h^{fit}$ 换成未贴合 Cartesian 活跃网格 $T_h$. 未稳定时在内部单元上格式完全退回拟合 EDG, 只在切割单元处出现小切胞问题. 因此本文推进的是几何灵活性: 不需要构造高阶曲线拟合格网, 同时试图保持 EDG 的能量稳定机制. 代价是引入 ghost penalty 和额外的背景面耦合, 稳定参数 $\omega_\ell$ 需要选择, 代数系统带宽或耦合增加, 并且切割几何积分和边界条件施加带来额外复杂性. 与 body-fitted 或曲线 DG 和传统高阶有限元相比, 那些方法在合适网格可用时有效, 但复杂几何或界面变化时网格生成代价高. 本文用 Cartesian 背景网格避免拟合, 但必须面对小切胞、稳定参数、切割几何积分和边界条件施加等额外复杂性. 显式时间步从可能依赖最小切胞 $\eta_K h$ 改善为依赖背景网格 $h$ 是主要优势之一; 代价是每步增加 ghost penalty 项, 且稳定性或精度依赖稳定权足够大, 过大可能影响精度常数和条件数.

第二, 与连续 CutFEM 的波方程工作 [8,7,25,26,27] 的比较. CutFEM 通常在连续有限元空间中对质量矩阵和刚度矩阵加 ghost penalty, 以恢复切割单元上的一致强制. 本文明确指出, 不把连续 CutFEM 的 mass-stiffness 稳定直接迁移到 EDG 空间算子, 而是在定义 EDG 能量的两个局部双线性形式中加稳定. 这是一个结构性差别: EDG 的能量同时决定离散设计, 因此稳定必须保持能量形式. 与 [26] 的高阶未贴合波离散相比, 两者都追求显式时间步不受浸入边界控制; [26] 数值显示 CFL 限制基本不受浸入边界影响, 本文摘要则声称时间步限制由背景网格尺寸而非最小物理切胞控制. 本文的不同在于针对 EDG 或 DG 空间, 并且强调稳定能量 Gram 矩阵条件数 cut-independent, 在所用多项式阶下与高阶连续 CutFEM 同量级且略小. 但这是数值观察, 严格证明在摘录中未见. 此外, 本文的稳定项放在 EDG 能量双线性形式中, 而连续 CutFEM 通常直接加在质量与刚度矩阵上, 因此两者在实现、能量解释和条件数行为上并不相同.

第三, 与 CutDG 椭圆和界面问题 [13] 的比较. [13] 发展了稳定 CutDG 框架用于椭圆边值问题和界面问题, 并给出关于切割配置鲁棒的误差和条件数估计. 本文处理的是二阶声波时间依赖问题, 而不是椭圆问题; 稳定项被嵌入 EDG 能量而非一般 CutDG 空间算子; 目标是半离散稳定、先验误差和显式时间步. 因此本文可看作把 CutDG 的 ghost 稳定思想专门化到 EDG 波方程框架. 与 [13] 相比, 本文的推进在于二阶波方程的能量设计和时间依赖稳定性, 但摘录未给出界面误差定理和完整数值结果, 因此无法比较界面情形与 [13] 等界面 CutDG 的具体优劣.

第四, 与双曲守恒律的 ghost-penalty CutDG [14,10,9,11] 的比较. 这些工作处理一阶双曲守恒律, 并显示显式时间推进的时间步限制独立于最小切元 [10,11]. 本文处理二阶声波方程, 使用 EDG 的能量形式, 数值通量和耦合结构与守恒律 CutDG 不同. 本文的贡献在于说明在 EDG 能量中直接加 ghost penalty 可以实现类似效果, 同时不改变原 EDG 通量结构. 在适用范围上, 本文包括单区域和静止材料界面; 但摘录未给出界面误差定理和完整数值结果, 因此无法比较界面情形与 [13] 等界面 CutDG 的具体优劣. 与双曲 CutDG 相比, 本文的未知量是二阶位移及其速度, 能量由梯度项和速度 $L^2$ 项组成, 而双曲守恒律 CutDG 的稳定和误差分析针对一阶守恒量, 因此不能直接互推.

中文标题: 一种用于模拟具有时间与空间依赖性质的石墨烯上等离激元的近似方法

摘要翻译: 本文提出了近似计算方法，用于模拟具有时间与空间依赖材料性质的石墨烯片上的表面等离激元传播。从 Maxwell 方程组出发，在空间和时间调制的 Drude 权重下，电流密度的演化被表述为一个偏积分微分方程 (PIDE)。假设 Drude 权重的小振幅扰动，建立了正则摄动展开。在适当的 Sobolev 空间中，建立了所得逐阶积分方程的可解性与 Fourier 可逆性。介绍了一种将 Volterra 积分方程表述与快速 Fourier 变换 (FFT) 相结合的数值求解器，并给出了逐点误差估计和计算复杂度分析。此外，针对行波 Drude 权重调制，推导了利用围道积分和 Laplace 反演的基于变换的解析解。数值模拟展示了多种波动现象，包括行波、驻波和指数增长的等离激元电流密度。

不相关论文（基本信息）

1. 圆盘上泊松方程的一种快速非均匀求解器

A Fast Nonuniform Solver for the Poisson Equation over a Disk

作者: Charlie Pyle, Prabir Daripa

类别: math.NA

发布日期: 2026-09-30

arXiv ID: 2609.40344v1

链接: http://arxiv.org/abs/2609.40344v1

摘要: We study fast numerical methods for the Poisson equation on a disk within the FFTRR (Fast Fourier Transform Radial Recurrence) framework, which is built on Green’s function representations. Classical FFTRR schemes apply FFTs in the azimuthal variable and evaluate mode-by-mode radial recurrences, achieving a complexity of (O(MN\log N)) on an (N\times M) uniform grid. However, they require a uniformly spaced azimuthal mesh of (N) points. In this work we develop a Nonuniform (NUFFTRR) solver that admits nonuniform grids in both the radial and azimuthal directions while retaining the favorable structure of the original FFTRR formulation. The azimuthal analysis-synthesis step is implemented using either a dense NUDFT least-squares solver or one of two NUFFT-based iterative schemes: a Toeplitz solver using circulant-preconditioned conjugate gradients (PCG), and a preconditioned conjugate gradient for least squares (PCGLS) solver with Pipe–Menon density compensation. These yield azimuthal complexities (O(N^3 + MN^2)) for the NUDFT variant and (O(K_{\mathrm{iter}} MN\log N)) for the NUFFT-based variants on an (N\times M) grid and Krylov iteration count (K_{\mathrm{iter}}). Numerical experiments on nonuniform meshes demonstrate that the proposed method is robust, spectrally accurate in the azimuthal variable, and competitive in runtime with existing fast Poisson solvers. We make use of vectorization and batched BLAS/GPU-accelerated linear algebra operations to eliminate explicit loops while also formulating azimuthal and radial steps entirely in terms of dense array operations, FFTs, and NUFFTs, allowing for straightforward GPU acceleration. The implementation is released as an open-source Python package at https://github.com/CharliePyle4/NUFFTRR_Poisson, and its methodology can be extended directly to related elliptic problems such as the Helmholtz equation.

2. adapol: 面向量子多体物理的自适应极点拟合

adapol: Adaptive pole-fitting for quantum many-body physics

作者: Zhen Huang, Chia-Nan Yeh, Lin Lin, Nils Wentzell, Jason Kaye, Hugo U. R. Strand

类别: cond-mat.str-el, math.NA

发布日期: 2026-09-30

arXiv ID: 2609.40150v1

链接: http://arxiv.org/abs/2609.40150v1

摘要: We describe adapol, a Python package for fitting Matsubara Green’s functions, self-energies, and hybridization functions by a compact sum of simple poles with real pole locations and matrix-valued residues. This pole-fitting step arises in many contexts, including hybridization fitting for quantum impurity solvers based on exact diagonalization, perturbation theory, and tensor networks, the compression of existing pole expansions such as the discrete Lehmann representation, diagrammatic methods based on discrete pole representations, and the analytic continuation of Matsubara Green’s functions. adapol uses a modified version of the AAA rational approximation algorithm to estimate pole locations, and non-convex optimization to refine them. This procedure yields accurate and compact fits in a black-box and noise-robust manner. adapol provides a simple, self-contained interface with extensive documentation and examples, as well as an interface to the TRIQS package.

3. 有限测量下测地射线变换的反演：稳定性与重建

Inverting the geodesic ray transform with finite measurements: stability and reconstruction

作者: Yernat M. Assylbekov

类别: math.AP, math.DG, math.NA

发布日期: 2026-09-30

arXiv ID: 2609.40015v1

链接: http://arxiv.org/abs/2609.40015v1

摘要: We study the inversion of the geodesic ray transform from finitely many local averages of its data. Under suitable geometric assumptions, we prove Lipschitz stability on any fixed finite-dimensional reconstruction space when the measurement partition is sufficiently fine. We develop convergent reconstruction algorithms based on Steepest Gradient Descent and Conjugate Gradients and implement them using piecewise constant finite element spaces. Numerical experiments in 2D and 3D illustrate the influence of geometry and measurement discretization on reconstruction quality.

4. 计算群体感应反应网络中的灭绝屏障

Computing Extinction Barriers in a Quorum-Sensing Reaction Network

作者: Mario Ayala, Johannes Zimmer

类别: math.PR, cond-mat.stat-mech, math.NA, q-bio.PE

发布日期: 2026-09-30

arXiv ID: 2609.39946v1

链接: http://arxiv.org/abs/2609.39946v1

摘要: We introduce a simple reaction-network model of a quorum-sensing population that couples the cell density $x$ to the signal density $w$. In the four-channel cell-signal network studied here, scaling signal production and removal by the same factor $r$ leaves all deterministic equilibria, and their stability types, unchanged. Nevertheless, we show that $r$ shifts the quasipotential barrier $ΔV(r)$ for rare transitions towards extinction, and thus, under metastable exit assumptions, the mean time to reach a fixed neighbourhood of the extinction state on the exponential scale $e^{NΔV(r)}$. We compute the barrier by minimization of the path action with the signal retained as a fluctuating coordinate, and we compare it with exact stochastic simulation of population-threshold crossing times regressed in $N$. Over a range of $r$, the minimum-action barriers satisfy $ΔV(r)=ΔV_\infty+O(1/r)$, where $ΔV_\infty$ is obtained by eliminating the signal first. At $r=1$, the barrier is 55% larger than $ΔV_\infty$. The saddle barrier $ΔV(r)$ is also the least action needed to enter the basin of extinction, but the density threshold can be crossed more cheaply: at $r=0.5$ the cheapest crossing costs 7.7% less action, keeps the signal high, and is usually followed by recovery. Simulated arrival times in this neighbourhood, which include failed attempts, grow with slopes within two fitted standard errors of the saddle barrier at every tested rate, and within $0.004$ of it if the logarithmic prefactor term is omitted. Under either regression model they exclude $ΔV_\infty$ at $r\le2$ by at least $4.4$ fitted standard errors, without using the action solver.

5. 双隐层多层感知机对分段多项式有限元函数的精确表示

Exact Representation of Piecewise Polynomial Finite Element Functions by Two-Hidden-Layer Multilayer Perceptrons

作者: Jun Hu, Pengzhan Jin, Shengyang Xu

类别: math.NA

发布日期: 2026-09-30

arXiv ID: 2609.39856v1

链接: http://arxiv.org/abs/2609.39856v1

摘要: We construct exact representations of finite element functions of arbitrary polynomial degree on simplicial meshes in any spatial dimension by two-hidden-layer multilayer perceptrons (MLPs). The first hidden layer uses $\mathrm{ReLU}^0+\mathrm{ReLU}$, and the second uses the degree-dependent activation $\mathrm{ReLU}^k$. The neural networks can realize both zero-skeleton and prescribed-skeleton representatives of discontinuous finite element functions, where the former vanish on the mesh skeleton and the latter admit independently prescribed polynomial data on each relatively open lower-dimensional mesh subsimplex; in particular, continuous finite element functions can be represented pointwise on the closed domain. In the continuous case, a half-open decomposition reduces the second-hidden-layer width by assigning each subsimplex to a single incident element. The number of nonzero parameters grows linearly with the number of elements for fixed spatial dimension, polynomial degree, and output dimension, and it is of the same order as the number of global degrees of freedom of the finite element space in the continuous case, provided that the mesh is shape-regular. All parameters in the neural network can be explicitly computed via the information of the mesh and finite element function without training.

6. 解析函数的精确重心有理插值及其在贝塞尔函数中的应用

Exact barycentric rational interpolation of analytic functions with an application to the Bessel functions

作者: Brett R. Green

类别: math.NA, math.CA

发布日期: 2026-09-30

arXiv ID: 2609.39805v1

链接: http://arxiv.org/abs/2609.39805v1

摘要: We derive a method of analytic function expansion using barycentric rational interpolation, in which an arbitrary analytic weight function defines the interpolation and nodes. Truncating these expansions yields successive approximations, and the derivation of the expansions gives the approximation error as a byproduct. We demonstrate the method with cosine and the Bessel functions of the first kind for several choices of weight functions, and we examine the accuracy and convergence of the approximate interpolations as a function of the weight and the number of nodes used in the interpolation.

7. 关于原子尺度位错模型分析中的线性化力平衡条件

On the linearised force balance condition in the analysis of atomistic dislocation models

作者: Julian Braun, Thomas Hudson

类别: math.NA

发布日期: 2026-09-30

arXiv ID: 2609.39642v1

链接: http://arxiv.org/abs/2609.39642v1

摘要: A technical condition arising in the analysis of atomistic models for dislocations is studied. Key estimates for the far-field strain behaviour proved in Ehrlacher, Ortner, Shapeev (2016) rely on a summation-by-parts argument for linearised forces, under conditions of sufficient decay and of vanishing net force in an infinite system. In particular, the latter condition is required for an application of this theory to the standard far-field dislocation predictor, but the vanishing of the associated force sum has not been fully justified. Here, the missing verification of this condition is provided and its role in determining the far-field decay of the corrector is clarified. It is moreover shown that for more general physically compatible predictors, the linearised force sum need not vanish, leading to slower decay and potentially divergent finite-domain approximations.

8. 深度截断FBSDE方法：一种用于高维非线性偏微分方程与完全耦合FBSDE的鲁棒求解器

Deep Truncated FBSDE Method: A Robust Solver for High-Dimensional Nonlinear PDEs and Fully Coupled FBSDEs

作者: Xinyu Cheng, Yunzhang Li, Weiran Xiong

类别: math.NA, math.PR

发布日期: 2026-09-30

arXiv ID: 2609.39616v1

链接: http://arxiv.org/abs/2609.39616v1

摘要: In this paper, we introduce a deep truncated forward-backward stochastic differential equation (FBSDE) method for high-dimensional partial differential equations (PDEs). Compared with existing deep-learning solvers for fully coupled FBSDEs, where strong coupling may lead to numerical instability, our approach exhibits improved stability. The proposed method combines gradient-truncated iterative decoupling with fictitious-play averaging to separate the forward and backward processes in a coupled framework. This preserves the coupled dynamics while reducing the unstable feedback induced by parameter-dependent forward paths during optimization. Furthermore, we incorporate a pathwise consistency term to create explicit local gradient shortcuts, thereby providing a structural mechanism that may mitigate gradient vanishing. We also derive a residual-based error estimate and establish conditional convergence of the fully discrete numerical approximations under suitable conditions, in which the pathwise consistency loss is not required. Our approach is particularly effective for convection-dominated equations, where the coupled formulation provides a stable representation of nonlinear transport without introducing singular terms into the BSDE. Numerical experiments demonstrate improved accuracy and stability in both low- and high-dimensional problems and robust performance for strongly coupled problems.

9. LampAttention：面向专用加速器的前瞻混合精度 FlashAttention

LampAttention: Look-Ahead Mixed-Precision FlashAttention for Dedicated Accelerators

作者: Stanislav Budzinskiy, Marian Gloser, Tolunay Yilmaz, Ying Hong Tham, Yuanyi Lin, Wenyi Fang, Fan Wu, Philipp Petersen

类别: cs.LG, math.NA

发布日期: 2026-09-30

arXiv ID: 2609.39361v1

链接: http://arxiv.org/abs/2609.39361v1

摘要: While most attention logits can be computed in low precision without degrading numerical stability, current attention kernels fail to exploit this phenomenon. We introduce a novel hardware-algorithm co-design in the form of mixed-precision FlashAttention. Our method accumulates key-query products and evaluates their exponentials in 8-bit formats, then adaptively identifies sensitive sub-blocks and recomputes them in 16-bit formats. We propose the specifications for a dedicated accelerator capable of executing this pipeline efficiently. Simulated experiments with Qwen3 and Gemma 3 show that rerouting a selective minority of sub-blocks to high precision is sufficient to recover the baseline model performance.

10. 关于 Lyapunov 方程共轭梯度法中迭代量秩增长的界

On bounds for rank growth of iterands in conjugate gradients for Lyapunov equation

作者: Jana Lungová, Martin Plešinger

类别: math.NA

发布日期: 2026-09-30

arXiv ID: 2609.39308v1

链接: http://arxiv.org/abs/2609.39308v1

摘要: We focus on solving the Lyapunov equation $AX + XA^T = F$, where $A$, $X$ and $F$ are square matrices, $A$ is symmetric positive definite (SPD) and sparse, and $F$ is symmetric and of a low-rank. The solution $X$ is then also symmetric and in general dense, but it can be approximated by a low-rank matrix. If $n$ is large, $X$ cannot be computed directly, but it is accessible by using the so-called low-rank arithmetics (LRA). The equation is solved by matrix reformulation of the method of conjugate gradients (MCG). We are interested in the behavior of ranks of the solution approximations $X_\ell$, residuals $R_\ell$, and direction vectors $P_\ell$ during iterations.

11. 基于移动最小二乘对分布不良数据的快速稳定调和逼近

Fast and Stable Harmonic Approximation from Ill-distributed Data using Moving Least Squares

作者: Ralf Hielscher, Tim Pöschl, Erik Wünsche

类别: math.NA

发布日期: 2026-09-30

arXiv ID: 2609.39200v1

链接: http://arxiv.org/abs/2609.39200v1

摘要: We consider harmonic approximation on the torus $\mathbb{T}^d$, the sphere $\mathcal{S}^d$, and the rotation group $\mathrm{SO}(3)$. While several global approaches are available for computing a harmonic expansion from scattered data, their stability and, especially, their runtime depend strongly on the geometry of the nodes, in particular on a sufficiently small fill distance. Even a single large hole in the data may cause instability and long runtimes. We propose harmonic approximation via moving least squares (HAMLS), a global harmonic approximation scheme that avoids the ill-conditioned global solve and works well in both the overdetermined and the underdetermined settings. The main idea is an intermediate transition from the scattered data to a quadrature grid, which is realized via moving least squares (MLS). This replaces one large global problem by many small local ones that can be regularized individually. The global harmonic approximation is then obtained via quadrature using a fast Fourier transform. This makes the global stage fast, stable and non-iterative. For sufficiently smooth functions sampled at well-distributed nodes with small fill distance $h$, we bound the resulting $\mathrm{L}^2$-error by the error of the harmonic approximation obtained from exact values on the quadrature grid, plus a term of order $h^{K+1}$, where $K$ is the polynomial degree employed in the MLS step. Numerical experiments on $\mathcal{S}^2$ with ill-distributed nodes show that HAMLS achieves errors comparable to those of global least-squares approximation, while being significantly faster, especially for larger bandwidths.

12. 半球 $t$-设计的 Marcinkiewicz-Zygmund 不等式

Marcinkiewicz-Zygmund inequalities for hemispherical $t$-designs

作者: Chao Li, Xiaojun Chen

类别: math.NA

发布日期: 2026-09-30

arXiv ID: 2609.39171v1

链接: http://arxiv.org/abs/2609.39171v1

摘要: In this paper, we introduce a new set of $\mathbb{L}_2$-orthogonal polynomials on the hemisphere $\mathbb{S}^d_+:=\{\mathbf{x}\in\mathbb{R}^{d+1}:\|\mathbf{x}\|=1, \mathbf{x}\cdot\mathbf{e}_{d+1}\geq0\}$ and establish $\mathbb{L}_1$ Marcinkiewicz-Zygmund inequalities on the hemisphere for even and odd spherical polynomials. Hemispherical $t$-designs provide equal weight cubature rules on the hemisphere which are exact for polynomials up to degree $t$. We propose variational characterizations of hemispherical $t$-designs by the $\mathbb{L}_2$-orthogonal polynomials. Moreover, based on the Marcinkiewicz-Zygmund inequalities and the $\mathbb{L}_2$-orthogonal polynomials, we prove that for each $n\geq C_d t^d$, there exists a hemispherical $t$-design on the hemisphere with $n$ points, where $C_d$ is a constant depending only on $d$.

13. 带 LCHS 的量子 Frozen–Oseen 同伦分析方法求解非线性偏微分方程

Quantum Frozen–Oseen homotopy analysis method with LCHS for solving nonlinear partial differential equations

作者: Jinhao Yuan, Leyu Chen, Tiegang Liu, Kun Wang

类别: math.NA, quant-ph

发布日期: 2026-09-30

arXiv ID: 2609.39170v1

链接: http://arxiv.org/abs/2609.39170v1

摘要: Nonlinear partial differential equations (PDEs) underpin computational fluid dynamics, yet resolving nonlinear transport on fine grids remains computationally demanding. Quantum linear-evolution algorithms offer a possible route to large-scale simulation but cannot directly propagate nonlinear coupling. Here we develop a Frozen–Oseen quantum homotopy method (FOQHAM) framework that links transport-aware auxiliary-operator selection to the size of the resulting linear representation. We freeze the full Fréchet derivative at a prescribed flow profile, retaining transport and profile-gradient coupling and close each prescribed finite-order homotopy hierarchy as a fixed affine linear system by product lifting. We formulate its propagation through Linear Combinations of Hamiltonian Simulations (LCHS), without outer homotopy iterations or profile updates. For spatially semidiscrete equations, we establish local approximation bounds and exact closure of the finite-order hierarchy. Classical tests on Burgers, Korteweg–de Vries, Zakharov–Kuznetsov, and one- and two-dimensional isothermal compressible Navier–Stokes equations provide numerical evidence of accurate non-iterative approximations in the tested regimes. The framework thus connects nonlinear PDE approximation to quantum linear evolution and offers a conditional path toward quantum acceleration.

14. 通过深度 ReLU 神经网络学习谱 Barron 函数的极小极大速率

Minimax rates for learning spectral Barron functions by deep ReLU neural networks

作者: Songqiu Ma, Yunfei Yang

类别: stat.ML, cs.LG, math.NA

发布日期: 2026-09-30

arXiv ID: 2609.39020v1

链接: http://arxiv.org/abs/2609.39020v1

摘要: We study how well deep neural networks approximate and learn spectral Barron functions. Recent studies have shown that these function classes can be efficiently approximated by shallow neural networks without suffering from the curse of dimensionality. We complement these results by providing new approximation bounds for deep networks with ReLU activation and establishing the minimax rates for learning these function classes. Specifically, we show that $d$-dimensional spectral Barron functions with smoothness index $s>0$ can be approximated by deep ReLU neural networks with approximation rate $\widetilde{\mathcal{O}} (S^{-\frac{1}{2}-\frac{s}{d}})$, where $S$ denotes the number of nonzero parameters in the network. Using this approximation result, we further show that deep ReLU neural networks can learn spectral Barron functions in a fast rate $n^{-\frac{d+2s}{2d+2s}}$ with $n$ training samples. Finally, we prove that this convergence rate is minimax optimal up to logarithmic factors.

15. 一种用于具有高阶相互作用的玻色–爱因斯坦凝聚体基态的保正守恒 Fourier 优化方法

A positive-conservative Fourier optimization method for ground states of Bose–Einstein condensates with higher-order interactions

作者: Bo Lin, Xinran Ruan

类别: math.NA

发布日期: 2026-09-30

arXiv ID: 2609.38965v1

链接: http://arxiv.org/abs/2609.38965v1

摘要: We develop a positive-conservative Fourier optimization (PCFO) method for computing ground states of Bose–Einstein condensates with higher-order interactions. The ground-state problem admits a convex density formulation, but the singular behavior near zero density poses difficulties for high-accuracy computation. We introduce convex regularizations of the density energy and establish their $Γ$-convergence on the computational domain as the regularization parameters vanish. The regularized problem is discretized by a Fourier pseudospectral method with mass conservation and nodal positivity. For the resulting constrained optimization problem, we use an accelerated first-order method for the main energy reduction, followed by a Newton refinement to reduce the remaining optimality residual. Numerical experiments show spectral-type convergence for regularized problems, quantify the effects of the regularization parameters on accuracy and spatial resolution, and demonstrate the effectiveness of the two-stage optimization method.

16. 用于双曲守恒律稳态的、采用 WENO-JS 局部求解器的完全收敛定点快速扫描法

A fully convergent fixed-point fast sweeping method with the WENO-JS local solver for steady state of hyperbolic conservation laws

作者: Liang Li, Jun Zhu, Yong-Tao Zhang

类别: math.NA

发布日期: 2026-09-30

arXiv ID: 2609.38841v1

链接: http://arxiv.org/abs/2609.38841v1

摘要: The fixed-point fast sweeping methods with weighted essentially non-oscillatory (WENO) local solvers are a class of efficient and high-order accuracy numerical methods for solving steady-state solutions of hyperbolic conservation laws. However, with the classical WENO-JS local solver, the iteration residue of high-order fixed-point fast sweeping scheme often has difficulty to settle down to round-off errors. To achieve the full convergence in a fast sweeping method, the fixed-point fast sweeping methods with non-traditional WENO local solvers based on unequal-sized substencils were designed. However, the WENO schemes based on unequal-sized stencils are more complex and in general more expensive in computational costs than the classical WENO-JS schemes. In this paper, we go back to the classical WENO-JS local solver and develop a new fully convergent fifth-order fixed-point fast sweeping method for solving steady-state problems of hyperbolic conservation laws. Based on recent studies on the nonlinear weighting process around discontinuities of solution, we apply the technique of frozen weights for avoiding unnecessary adjustment of nonlinear weights in the WENO-JS local solver, which freezes the nonlinear weights once the residual sequence in the fast sweeping iterations has stabilized. Different from the existing work on frozen weights, we design a simple and robust approach to judge stabilization of iteration residues and determine the iteration step when the nonlinear weights are frozen in the fast sweeping method. Extensive numerical experiments on a wide range of challenging two-dimensional steady-state problems demonstrate that, unlike the previous fast sweeping method with the fifth-order WENO-JS local solver, the proposed new scheme consistently drives the iteration residues to round-off errors and achieves the full convergence.

17. 退化预条件下具有最优非遍历收敛的 Halpern 加速主化 ADMM

Halpern-Accelerated Majorized ADMM with Optimal Non-Ergodic Convergence under Degenerate Preconditioning

作者: Ning Zhang, Benqi Liu, Liwei Zhang

类别: math.OC, math.NA

发布日期: 2026-09-30

arXiv ID: 2609.38808v1

链接: http://arxiv.org/abs/2609.38808v1

摘要: We develop a Halpern-accelerated majorized alternating direction method of multipliers (ADMM) with possibly indefinite proximal terms for linearly constrained convex composite optimization. Majorization simplifies the subproblems but introduces a forward perturbation to the standard degenerate proximal point representation. We reformulate majorized ADMM as a perturbed degenerate proximal point method and characterize the fixed points of the induced mapping as Karush–Kuhn–Tucker (KKT) solutions. Under range compatibility and metric cocoercivity conditions, the mapping is nonexpansive in the seminorm induced by a possibly singular preconditioner. In finite dimensions, a verifiable block-matrix condition permits indefinite proximal regularization while retaining the required metric properties. A relaxed Halpern iteration then yields an $\mathcal O(1/k)$ fixed-point residual bound, sharp in the general nonexpansive setting, and an $\mathcal O(1/k)$ nonergodic KKT residual bound at the intermediate iterates. Numerical experiments illustrate the predicted residual decay and the benefits of majorization and indefinite proximal terms.

18. 具有近似少量 ANOVA 项的张量的最坏情况补全

Worst-Case Completion of Tensors with Approximately Few ANOVA Terms

作者: Simon Foucart, Jingchun Shao

类别: math.NA

发布日期: 2026-09-30

arXiv ID: 2609.38749v1

链接: http://arxiv.org/abs/2609.38749v1

摘要: In this article, the problem of completing a tensor from some incomplete knowledge of its entries is treated by adopting a worst-case perspective, given the realistic assumption that the tensor’s low-order ANOVA terms are dominant. We survey and leverage some recent all-purpose results from the field of Optimal Recovery to provide solutions on a theoretical level. But the accompanying constructions of optimal completion procedures, which often feature semidefinite programs, are not directly applicable in the tensor case due to the huge dimensions involved. To resolve the issue, we put forward a storage-friendly way to produce low-order ANOVA projections based on the fast Fourier transform (FFT), while exploiting the specificities of the completion problem to efficiently compute regularizers and extremal eigenvalues. Numerical experiments on synthetic tensors and real-world datasets demonstrate the accuracy and scalability of our FFT-based method.

19. ABrA-GD：面向相对光滑与（强）凸优化的自适应 Bregman 加速梯度下降

ABrA-GD: Adaptive Bregman Accelerated Gradient Descent for Relatively Smooth and (Strongly-)Convex Optimization

作者: Damien Scieur

类别: math.OC, math.NA

发布日期: 2026-09-30

arXiv ID: 2609.38741v1

链接: http://arxiv.org/abs/2609.38741v1

摘要: We propose ABrA-GD (Adaptive Bregman Accelerated Gradient Descent), an adaptive algorithm for relatively smooth convex optimization. The algorithm is derived from a computable primal–dual certificate that guarantees progress by comparing the objective value with a lower bound on the minimum of a regularized objective. This certificate enables adaptation to both smoothness and geometry, using only the relative strong convexity constant as a problem-dependent input. We introduce the dual Bregman Length Distortion Factor (BLDF), which measures how dual Bregman lengths change under an anchor shift or rescaling. Under bounded local dual BLDF, ABrA-GD achieves an accelerated $O(1/k^2)$ rate for convex objectives and an accelerated linear rate for relatively strongly convex objectives.

20. Picard HMC 的高精度模拟，第一部分：高斯云校正

High-accuracy simulation of Picard HMC, part I: Gaussian cloud correction

作者: Fan Chen, Sinho Chewi, Jianfeng Lu, Matthew S. Zhang

类别: math.ST, cs.DS, math.NA

发布日期: 2026-09-30

arXiv ID: 2609.38710v1

链接: http://arxiv.org/abs/2609.38710v1

摘要: We study the problem of sampling from a continuous density $π\propto \exp(-V)$ on $\mathbb R^d$, where $V\in C^2(\mathbb R^d)$ has a $β$-Lipschitz gradient and $π$ satisfies a logarithmic Sobolev inequality with constant $α^{-1}$, and write $κ= β/α$. We introduce the Gaussian cloud sampler, which achieves total variation accuracy $\varepsilon$ using $\widetilde O(κd^{1/5}\,\text{polylog}(1/\varepsilon))$ gradient queries in expectation. The algorithm uses first-order rejection sampling (FORS) to correct the law of smoothed Picard HMC trajectories. To do so, we represent the iterates of the ideal Picard iteration via Gaussian clouds, whose centers are never evaluated, and we develop a suite of likelihood correction gadgets which only use samples from this indirect cloud representation.

21. Null-A 模式：广义逆的组合式计算

The Mode of Null-A: Compositional Computation of a Generalized Inverse

作者: Barak A. Pearlmutter, Jeffrey Mark Siskind

类别: math.NA, cs.PL

发布日期: 2026-09-29

arXiv ID: 2609.38497v1

链接: http://arxiv.org/abs/2609.38497v1

摘要: We present a novel algorithm for calculating the preimage of an affine space through a product ${J}={J}_{T-1}\cdots{J}_0$ of matrices ${J}_t$ of special form: finding the largest input space $\mathbf{X}$ such that $\mathbf{x}\in\mathbf{X}$ implies ${J}\mathbf{x}\in \mathbf{Y}$, where $\mathbf{Y}$ is a given output affine space. These special matrices arise in AD, where the Jacobians $J$ describing the linearized computation have precisely this structure: the product of a series of linearized primitive numeric operations. This allows us to use the new algorithm to formulate Null-A mode preimage AD, which finds the affine preimage through the Jacobian or Jacobian transpose of a numeric computation. This is a generalization of the inverse AD problem of solving ${J}\acute{\mathbf{x}}^{\ast}=\acute{\mathbf{y}}^{\ast}$ or ${J}^{T}\grave{\mathbf{y}}^{\ast}=\grave{\mathbf{x}}^{\ast}$. The key is to represent affine spaces in a fashion which lends itself to efficient preimage calculation, in a compositional and quasi-local fashion, through a succession of matrices ${J}_t$. Unlike previous methods, Null-A preimage mode AD allows the ${J}_t$ matrices to be non-square, corresponding to a computer program whose number of active variables swells and shrinks during the computation. When ${J}$ is square and the initial affine space is a single point, this finds the conventional inverse. But in the more general case, having the entire affine space provides freedom which can be leveraged in a problem-specific manner. We apply the method to small problems on-CPU where the $J_t$ are linearized scalar unary or binary numeric functions; and to larger problems on-GPU where the $J_t$ are linearized aggregate array operations like convolution and attention.

22. 具有滞留和吸收性动脉壁的微极环形流中导管调控的瞬态溶质弥散

Catheter-modulated transient solute dispersion in micropolar annular flow with a retentive and absorptive arterial wall

作者: Sohel Ahmed, Nanda Poddar, Kajal Kumar Mondal, Subham Dhar

类别: physics.flu-dyn, math-ph, math.NA, physics.bio-ph, q-bio.TO

发布日期: 2026-09-29

arXiv ID: 2609.38423v1

链接: http://arxiv.org/abs/2609.38423v1

摘要: We study transient solute dispersion in pressure-driven micropolar flow through a concentric annulus representing a catheterized artery. The catheter is impermeable to solute, whereas the arterial wall combines irreversible removal with reversible surface retention. The exact steady micropolar velocity field is coupled to the Gill-Sankarasubramanian generalized-dispersion framework; the resulting bulk-surface hierarchy is advanced with Rannacher-damped Crank-Nicolson time stepping to determine $K_0(t)$, $K_1(t)$ and $K_2(t)$. The exchange coefficient is independent of the micropolar parameters and satisfies $-K_0(0^+)=2(β+θDa)/(1-λ^2)$, whereas $K_2-Pe^{-2}\simσ_v^2t$ initially, so the leading short-time dispersion is hydrodynamic and independent of wall kinetics. In the weak-reaction regime, micropolarity modifies convection approximately linearly and shear dispersion quadratically through the velocity-amplitude factor. A narrow-gap analysis with $\varepsilon=1-λ$ gives $\bar v\sim(2-N_c)\varepsilon^2/6$ and $K_2-Pe^{-2}\sim(2-N_c)^2\varepsilon^6/7560$ for a fixed pressure gradient, revealing a sixth-power suppression of shear dispersion as the catheter approaches the arterial wall. Increasing $λ$ from $0.01$ to $0.30$ reduces effective convection by about a factor of $2.45$, while chemically passive shear-induced dispersion falls by a factor of $24$. The coupled formulation also yields the exact partition $Φ_m+Φ_s+Φ_a=1$, separating mobile, reversibly retained and irreversibly absorbed solute. The reconstructed field quantifies reaction-modulated transverse non-uniformity, while the results distinguish hydrodynamic effects of confinement and microrotation from kinetic wall effects.

重点关注论文（完整分析）#

1. 复杂几何区域上波动方程的一种基于能量的间断 Galerkin 方法#

1. 研究对象和背景#

2. 主要定理或主要结果#

3. 研究方法、关键技术和核心工具#

4. 与之前工作的比较#

了解领域论文（摘要翻译）#

1. 一种用于虚拟区域界面问题的约化耦合空间多重网格 Schur 预条件子#

摘要翻译#

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3. 有限元外微积分中的旋转基：顶点置换下的闭式基变换#

摘要翻译#

4. 通过优化的 Kolmogorov–Arnold 神经网络对双调和方程进行无网格数值逼近#

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5. 求解多维椭圆型问题的良态 Birkhoff 配点法#

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8. 一种用于模拟具有时间与空间依赖性质的石墨烯上等离激元的近似方法#

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3. 有限测量下测地射线变换的反演：稳定性与重建#

4. 计算群体感应反应网络中的灭绝屏障#

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7. 关于原子尺度位错模型分析中的线性化力平衡条件#

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9. LampAttention：面向专用加速器的前瞻混合精度 FlashAttention#

10. 关于 Lyapunov 方程共轭梯度法中迭代量秩增长的界#

11. 基于移动最小二乘对分布不良数据的快速稳定调和逼近#

12. 半球 $t$-设计的 Marcinkiewicz-Zygmund 不等式#

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14. 通过深度 ReLU 神经网络学习谱 Barron 函数的极小极大速率#

15. 一种用于具有高阶相互作用的玻色–爱因斯坦凝聚体基态的保正守恒 Fourier 优化方法#

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18. 具有近似少量 ANOVA 项的张量的最坏情况补全#

19. ABrA-GD：面向相对光滑与（强）凸优化的自适应 Bregman 加速梯度下降#

20. Picard HMC 的高精度模拟，第一部分：高斯云校正#

21. Null-A 模式：广义逆的组合式计算#

22. 具有滞留和吸收性动脉壁的微极环形流中导管调控的瞬态溶质弥散#

重点关注论文（完整分析）

1. 复杂几何区域上波动方程的一种基于能量的间断 Galerkin 方法

1. 研究对象和背景

2. 主要定理或主要结果

3. 研究方法、关键技术和核心工具

4. 与之前工作的比较

了解领域论文（摘要翻译）

1. 一种用于虚拟区域界面问题的约化耦合空间多重网格 Schur 预条件子

摘要翻译

2. 曲面 Stokes 问题的无压力虚拟元方法

摘要翻译

3. 有限元外微积分中的旋转基：顶点置换下的闭式基变换

摘要翻译

4. 通过优化的 Kolmogorov–Arnold 神经网络对双调和方程进行无网格数值逼近

摘要翻译

5. 求解多维椭圆型问题的良态 Birkhoff 配点法

摘要翻译

6. 粗时间监督下长时间 Cahn-Hilliard 动力学的保结构傅里叶神经算子

摘要翻译

7. 面向单元中心近不可压弹性问题的核感知两重网格预条件子

摘要翻译

8. 一种用于模拟具有时间与空间依赖性质的石墨烯上等离激元的近似方法

摘要翻译

不相关论文（基本信息）

1. 圆盘上泊松方程的一种快速非均匀求解器

2. adapol: 面向量子多体物理的自适应极点拟合

3. 有限测量下测地射线变换的反演：稳定性与重建

4. 计算群体感应反应网络中的灭绝屏障

5. 双隐层多层感知机对分段多项式有限元函数的精确表示

6. 解析函数的精确重心有理插值及其在贝塞尔函数中的应用

7. 关于原子尺度位错模型分析中的线性化力平衡条件

8. 深度截断FBSDE方法：一种用于高维非线性偏微分方程与完全耦合FBSDE的鲁棒求解器

9. LampAttention：面向专用加速器的前瞻混合精度 FlashAttention

10. 关于 Lyapunov 方程共轭梯度法中迭代量秩增长的界

11. 基于移动最小二乘对分布不良数据的快速稳定调和逼近

12. 半球 $t$-设计的 Marcinkiewicz-Zygmund 不等式

13. 带 LCHS 的量子 Frozen–Oseen 同伦分析方法求解非线性偏微分方程

14. 通过深度 ReLU 神经网络学习谱 Barron 函数的极小极大速率

15. 一种用于具有高阶相互作用的玻色–爱因斯坦凝聚体基态的保正守恒 Fourier 优化方法

16. 用于双曲守恒律稳态的、采用 WENO-JS 局部求解器的完全收敛定点快速扫描法

17. 退化预条件下具有最优非遍历收敛的 Halpern 加速主化 ADMM

18. 具有近似少量 ANOVA 项的张量的最坏情况补全

19. ABrA-GD：面向相对光滑与（强）凸优化的自适应 Bregman 加速梯度下降

20. Picard HMC 的高精度模拟，第一部分：高斯云校正

21. Null-A 模式：广义逆的组合式计算

22. 具有滞留和吸收性动脉壁的微极环形流中导管调控的瞬态溶质弥散