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1. 时间调制 Drude–Lorentz 介质的可杂交间断伽辽金方法
A Hybridizable Discontinuous Galerkin Method for Time-Modulated Drude–Lorentz Media
作者: Jeonghun J. Lee
类别: math.NA
发布日期: 2026-10-15
arXiv ID: 2610.21991v1
摘要翻译: 我们构造并分析了一种用于 Maxwell 方程组与四类时间调制 Drude–Lorentz 模型耦合的可杂交间断伽辽金 (HDG) 方法：调制入耦合、调制极化密度、调制电流密度和调制有效质量。使用电位移和依赖于模型的连续材料状态，为时间界面提供统一处理，同时保持各自不同的物理透射规则。我们推导了能量与稳定性估计，建立了空间误差界以及与事件对齐的时间误差界，并将时间积分误差与切换时刻错置所导致的误差分离。三维及约化模态系统中的数值实验验证了所预测的空间、时间以及切换时序行为，包括极化密度模型特有的场跳跃效应。
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链接: http://arxiv.org/abs/2610.21991v1
1. 研究对象和背景
本文研究有界多面体 $\Omega\subset\mathbb{R}^3$ 上时间域 Maxwell 方程与四类时空调制 Drude–Lorentz 色散材料模型的耦合. 边界取理想电导体 (PEC) 条件 $n\times E=0$, 也可替换为成对周期边界. 背景介电常数与磁导率 $\epsilon_0,\mu_0>0$ 固定, 设 $s=\omega_p^2$ 为等离子体频率平方 (振子强度), $b=\omega_0^2$ 为 Lorentz 共振频率平方, $\gamma\ge0$ 为阻尼. Maxwell 方程组为
$$\partial_t D-\nabla\times H=0,\quad \mu_0\partial_t H+\nabla\times E=0,\quad D=\epsilon_0 E+P,$$无外加源项, 以隔离调制引起的能量转移. 材料是各向同性, 仅有一个电 Lorentz 极点, 无磁色散. 时间调制由全局事件时间 $0=\theta_0<\theta_1<\cdots<\theta_{N_{\rm ev}} 连续状态为 $Y_1=(D,H,P,J)$, 事件处 $[D]=[H]=[P]=[J]=0$. 模型 2 为调制极化密度: 令 $Q=P/s$, $V=\partial_t Q$, 则 连续状态为 $Y_2=(D,H,Q,V)$, 事件处 $[D]=[H]=[Q]=[V]=0$, 但 因而 $P$ 不连续; 若强加 $P$ 连续会与 $P=sQ$ 矛盾. 模型 3 为调制电流密度且带独立记忆: 连续状态为 $Y_3=(D,H,P,Q,V)$, 五个分量在事件处连续, 因此 $E$ 连续而 $\partial_t P=sV$ 可以跳, 一般 $P\ne sQ$. 模型 4 为调制有效质量: 连续状态为 $Y_4=(D,H,P,V)$. 这些模型的重要性在于, 它们覆盖了同一种 Drude–Lorentz 响应在时空调制下四类不同物理传输规则; 对数值方法而言, 时间界面处连续变量不同, 能量泛函不同, 稳定性和误差阶也会不同. 例如模型 2 的物理场跳变, 模型 3 的能量不控制独立极化状态 $P$, 都会直接影响离散格式的初始化和误差度量. 已有数值理论基础包括: 时域 Maxwell 的 HDG 隐式格式 (Christophe–Descombes–Lanteri; Berardocco–Kronbichler–Gravemeier) 和显式格式 (Nehmetallah 等), 时间调和 Maxwell HDG 与超收敛构造 (Chen–Qiu–Shi–Solano), 频域非局部 Drude HDG, 以及色散 Maxwell 系统的 DG 稳定性与空间误差分析 (Wang–Xie–Zhang; Xie–Wang–Zhang; Bokil 等). 对时间依赖系数的二阶双曲问题, Al Jarroudi 与 Grote 近期用时间依赖 Ritz 型投影证明了光滑系数下最优能量范数半离散有限元收敛; Renner 与 Yousept 分析了时间依赖介电常数和磁导率、非线性电磁耦合与非线性 Silver–Müller 边界的 Maxwell 问题, 用边元/分片常数和隐式 Euler 得到有限时间强收敛及附加耗散下的指数衰减. 现有工作的空白在于: 四种时空调制机制具有不同时间传输规则, 需要一种把它们统一处理、同时在事件处保持正确连续状态和物理跳变的 HDG 离散, 并给出半离散能量稳定性、空间误差、事件对齐时间误差, 以及把时间积分误差与切换时间错位导致的误差分离的定量分析. 本文正是针对这一空白, 以电位移 $D$ 与模型依赖的连续材料状态作为未知量, 构造 Maxwell–材料 HDG 格式. 由于所用局部极化方程不引入额外空间导数, 材料未知量不需要额外骨架变量; 这使得方法可作为时空调制波传播和后续逆设计的参考求解器. 作者也指出, 与显式 DG 等求解器相比的计算优势必须在相同响应精度下另行证明, 本文并未声称已经证明. 在 Assumption 2.1 下, 作者首先给出连续能量平衡. 对模型 1,2,4 可写 $E_i(Y)=\frac12\int Y^T M_i(t)Y$, 其中 $M_i(t)$ 为正定对称化矩阵. 定义调制系数 $C_i(t)=\operatorname{ess\ sup}_{x\in\Omega}\max\{0,\lambda_{\max}(M_i^{-1/2}(\partial_t M_i)M_i^{-1/2})\}$ 和事件能量比 $\rho_{i,j}=\operatorname{ess\ sup}_x\lambda_{\max}((M_i^-)^{-1/2}M_i^+(M_i^-)^{-1/2})$, 则 $E_i^+\le\rho_{i,j}E_i^-$, 并定义放大界 $G_i(T)=\exp(\sum_j\int_{I_j}C_i(t)dt)\prod_{j=1}^{N_{\rm ev}}\max\{1,\rho_{i,j}\}$. Proposition 2.2 证明在无事件区间上 $E_i'=-L_i+W_i$, 其中 $L_i\ge0$ 为阻尼损耗, $W_i$ 为调制功. 例如模型 1 的 $L_1=\int_\Omega\frac{\gamma}{\epsilon_0s}|J|^2dx$, $W_1=\frac{1}{2\epsilon_0}\int_\Omega[\partial_t(s^{-1})|J|^2+\partial_t(b/s)|P|^2]dx$; 模型 2 的 $L_2=\int_\Omega\frac{\gamma s}{\epsilon_0}|V|^2dx$, $W_2=\frac{1}{2\epsilon_0}\int_\Omega[(\partial_t s)|V|^2+\partial_t(sb)|Q|^2]dx-\int_\Omega(\partial_t s)E\cdot Qdx$; 模型 3 的 $W_3=\frac{1}{2\epsilon_0}\int_\Omega[(\partial_t s)|V|^2+\partial_t(sb)|Q|^2]dx$, $L_3=\int_\Omega\frac{\gamma s}{\epsilon_0}|V|^2dx$; 模型 4 的 $W_4=\frac{1}{2\epsilon_0}\int_\Omega[(\partial_t s)|V|^2+\partial_t(b/s)|P|^2]dx$, $L_4=\int_\Omega\frac{\gamma s}{\epsilon_0}|V|^2dx$. 在事件处, 用共同连续状态计算单侧能量得到跳跃公式, 如 $[E_1]=\frac{1}{2\epsilon_0}\int([s^{-1}]|J|^2+[b/s]|P|^2)dx$, $[E_3]=\frac{1}{2\epsilon_0}\int([s]|V|^2+[sb]|Q|^2)dx$, $[E_4]=\frac{1}{2\epsilon_0}\int([s]|V|^2+[b/s]|P|^2)dx$; 模型 2 必须包含电磁跳变: $[E_2]=\frac{1}{2\epsilon_0}\int_\Omega(|D-s^+Q|^2-|D-s^-Q|^2+[s]|V|^2+[sb]|Q|^2)dx$. 这些连续估计给出 $E_i(t)\le G_i(T)E_i(0)$. 其次, 作者构造混合 HDG 格式. 令 $W_h^E=\{v:v|_K\in[P_m(K)]^3\}$, $W_h^H=\{w:w|_K\in[P_k(K)]^3\}$, $M_h^0$ 为 facet 上次数 $m$ 的单值切向向量多项式且 PEC facet 上为零. 变体 A: $m=k$, $\tau_K\simeq1$; 变体 B: $m=k+1$, $\tau_K=\theta_K/h_K$, 其中 $\theta_K$ 有与网格和时间无关的正上下界. 所有材料状态和 $D_h$ 在 $W_h^E$ 中, $H_h$ 在 $W_h^H$ 中, 唯一独立 hybrid 变量是 $\hat E_h\in M_h^0$. 半离散方程 (26a-c) 为: $(\partial_t D_h,v)-(H_h,\nabla\times v)-\langle n\times\hat H_h,v\rangle=0$, $(\mu_0\partial_t H_h,w)+(E_h,\nabla\times w)+\langle n\times\hat E_h,w\rangle=0$, $\langle n\times\hat H_h,\hat v\rangle=0$, 对 $v\in W_h^E,w\in W_h^H,\hat v\in M_h^0$, 数值通量 $n\times\hat H_h=n\times H_h-\tau_K(E_{h,T}-\hat E_h)$. 材料方程弱式与所有 $W_h^E$ 测试函数成立. 时间事件处精确复制连续状态: $Y_{i,h}(\theta_j^+)=Y_{i,h}(\theta_j^-)$, 然后用新系数重算 $E_h$ 和代数 trace; trace 不是独立微分状态, 不必复制. 初始用逐元素 $L^2$ 投影, $D$ 和材料状态用次数 $m$, $H$ 用次数 $k$. Theorem 3.1 给出半离散能量平衡: 在每个无事件区间上 $\partial_t E_{i,h}+D_h+L_{i,h}=W_{i,h}$, 其中 $D_h=\sum_K\tau_K\|E_{h,T}-\hat E_h\|_{\partial K}^2$ 为数值耗散, $L_{i,h},W_{i,h}$ 为离散版损耗和调制功; 因此 $\sup_{0\le t\le T}E_{i,h}(t)\le G_i(T)E_{i,h}(0)$. 对于模型 3, 能量不控制 $P_h$ 单独, Remark 3.2 单独估计 $\|P_h(t)\|\le\|P_h(0)\|+s^*\int_0^t\|V_h(r)\|dr\le\|P_h(0)\|+CT\sqrt{G_3(T)E_{3,h}(0)}$, 从而控制 $D_h=\epsilon_0E_h+P_h$. 空间误差分析在 Assumption 4.1 下进行: 每个无事件区间连续状态属于 $W^{1,1}(I_j;L^2(\Omega))$ 并有单侧端点值; 满足精确时间传输规则; 切向迹单值; 变体 B 要求 $H$ 分片 $H^{k+1}$, $E$ 和独立材料状态分片 $H^{k+2}$ 一致有界, 变体 A 要求所有场分片 $H^{k+1}$ 有界. 用与时间无关的 $L^2$ 投影 $\Pi^E,\Pi^H,\Pi^M$, 材料方程投影精确. Lemma 4.2 给出投影电磁误差方程 (33)-(36), 右端出现边界 remainder $R_K=n\times e_H^I-\tau_K(e_{E,T}^I-e_{ET}^I)$, 并定义 $F_h^2(t)=\sum_K\tau_K^{-1}\|R_K\|_{\partial K}^2$. Lemma 4.3 给出投影误差能量恒等式 $E_i(e_i^h)'+D_h(e_i^h)+L_i(e_i^h)=W_i(e_i^h)-\sum_K\langle R_K,e_{E,T}^h-\hat e_{ET}^h\rangle_{\partial K}$, 进而 $E_i(e_i^h)'+\frac12D_h(e_i^h)+L_i(e_i^h)\le C_i(t)E_i(e_i^h)+\frac12F_h^2(t)$. 作者指出变体 B 用块 $L^2$ 投影和平衡两个边界残差, 达到 $h^{k+1}$ 的共同场/状态率; 这是次数 $k$ 磁场的最优率, 不是次数 $k+1$ 电场的额外 $h^{k+2}$ 率. 变体 A 是更便宜的不同变体, 保留 baseline estimate; 但提供文本未给出该 baseline 的具体阶. 摘要还声明建立了空间和 event-aligned 时间误差界, 并把时间积分误差与错误切换时间引起的误差分离; 还报告三维和 reduced modal 数值实验, 确认空间、时间、切换时间行为以及模型 2 特有的场跳变. 然而提供的 PDF 摘录没有包含这些时间误差定理、切换时间位移定理和数值实验的完整陈述, 因此其具体假设、误差阶和常数从提供内容中无法确认. 主要技术障碍有三类. 第一, 时间调制使材料系数 $s,b,\gamma$ 在事件处跳变, 而四个模型的连续变量不同, 不能把所有模型都按同一个守恒或同一个传输条件处理. 例如模型 2 的连续材料坐标是 $Q=P/s$, 但重构 $P=sQ$; 事件处应保持 $Q,V$ 连续, 而不是保持 $P$ 连续, 并且电场满足 $E^+=E^--(s^+-s^-)Q/\epsilon_0$, 因此 $E$ 和 $P$ 都可能跳. 模型 3 的能量只控制 $(E,H,Q,V)$, 不控制独立极化 $P$, 所以即使有能量稳定性, 也必须单独估计 $P$. 第二, 系数跳变意味着不能把步进系数的导数当作分布展开, 能量恒等式和误差方程必须在每个无事件区间上建立, 再通过事件处的状态复制和能量比 $\rho_{i,j}$ 连接. 第三, 电磁场与材料 ODE 耦合, 但材料方程不含额外空间导数; 如何在不引入多余骨架变量的前提下, 使事件处只需复制连续状态, 并把时间积分误差与切换时间错位误差分开, 是关键设计问题.\n\n作者的真正创新点是把电位移 $D$ 和模型依赖的连续材料状态作为主未知量, 而不是直接以 $E$ 和 $P$ 为所有模型的共同状态. 这样四个模型可以共享一个 HDG 框架, 同时保留各自的时间传输规则. 半离散格式中, 唯一独立 hybrid 变量是切向电场 trace $\hat E_h$, 材料状态和 $D_h$ 都在 $W_h^E$ 中, $H_h$ 在 $W_h^H$ 中. 由于局部极化方程没有空间导数, 材料未知量不需要额外 skeleton 变量; 这使元素局部消除成为可能, 隐式时间步最终只需解关于 $\hat E_h$ 的全局系统. 事件处直接复制模型连续状态 $Y_{i,h}(\theta_j^+)=Y_{i,h}(\theta_j^-)$, 然后用新系数重新计算 $E_h$ 和 trace. 作者特别强调, 在模型 2 中 trace 不是独立微分状态, 不必复制, 这一点对正确处理场跳变很重要.\n\n稳定性证明的核心是能量法与对称化. 对模型 1,2,4 定义正定能量矩阵 $M_i(t)$, 把调制功写成 $\frac12\int Y^T(\partial_t M_i)Y$, 并用 $C_i(t)$ 控制其相对能量增长率; 事件处用 $\rho_{i,j}$ 控制能量放大. 定义 $G_i(T)$ 后, 连续和半离散稳定性都归结为逐区间积分和事件乘积. 半离散证明中, 取测试函数 $v=E_h,w=H_h,\hat v=\hat E_h$; 分部积分后边界项为 $\langle n\times H_h-n\times\hat H_h,E_h-\hat E_h\rangle$, 由通量定义和 conservativity 方程化为数值耗散 $D_h$; 材料项消去后得到 Theorem 3.1. 这个框架避免了在系数跳变处做分布求导, 把跳变影响完全放入能量比.\n\n空间误差分析的技术路线是投影误差分解. 作者使用与时间无关的逐元素 $L^2$ 投影 $\Pi^E,\Pi^H$ 和 facet 切向投影 $\Pi^M$. 关键观察是材料方程是电/材料状态的线性组合, 系数在元素上空间常数, 所以投影后材料方程精确成立, 不需要对 $E$ 和 $H$ 做共同投影. 对电磁场, 由于 $\nabla\times W_h^E\subset W_h^H$ 和 $\nabla\times W_h^H\subset W_h^E$ (对 $m=k$ 或 $m=k+1$), 质量投影缺陷在与对应体积测试函数配对时消失. 投影方程与离散方程相减得到投影误差方程, 右端只剩 magnetic flux 的边界 remainder $R_K$. Lemma 4.3 把该 remainder 通过加权 Cauchy–Schwarz 和 Young 不等式吸收为 $\frac12F_h^2(t)$. 变体 B 的 $m=k+1,\tau_K=\theta_K/h_K$ 选择用于平衡边界残差, 使共同场/状态误差达到 $h^{k+1}$; 但这不是对次数 $k+1$ 电场的 $h^{k+2}$ 超收敛结果. 变体 A 的 $m=k,\tau_K\simeq1$ 更便宜, 保 baseline estimate.\n\n时间离散方面, 摘要强调 event-aligned 时间误差界和切换时间错位误差的分离. 从提供的前十页可以看出, 作者把事件对齐时间离散与切换时间位移视为两个不同误差源: 前者是普通时间积分误差, 后者来自数值切换时间与真实事件时间不重合. Section 6 声称量化错位切换导致的状态误差, 说明它何时把 midpoint 收敛限制到一阶, 并识别简单模态稳定边界的位移; 模型 2 的电场不连续重构还要求区分状态收敛和场收敛. 然而提供文本未包含 Section 6 的具体引理、定理和证明细节, 因此无法确认所用时间离散格式的精确形式、稳定性函数、误差范数和常数. 同样, 隐式 Euler 和 midpoint 的存在唯一性由某种能量不等式得到, 但具体不等式未在摘录中展开. 与 Dixon 等 [9] 的关系. Dixon 等展示四种调制机制在相同冻结 Drude–Lorentz 介电常数下可产生不同的时间散射系数, 并把区别归于材料演化和时间界面连续量. 本文的主要推进是把这四种机制写成四个连续状态模型, 逐一给出传输条件、能量泛函和能量跳跃, 并在统一 HDG 框架下分析. 因此本文不是重新发现物理机制, 而是把物理上不同的时间传输规则转化为数值格式必须遵守的连续状态选择. 尤其是模型 2, 本文强调必须保持 $Q=P/s$ 连续而非 $P$ 连续, 并给出 $E$ 的跳变公式和电场不连续重构; 这在既有一般色散 Maxwell 数值分析中通常不会作为核心区分点出现.\n\n与已有 HDG 时域 Maxwell 工作比较. Christophe–Descombes–Lanteri 和 Berardocco–Kronbichler–Gravemeier 给出隐式 HDG 格式, Nehmetallah 等给出显式 HDG 格式; Berardocco 等的两个公式分别基于电磁扩散方程和一阶 Maxwell 方程. 本文离散的是一阶时变 Maxwell–材料耦合系统, 且以 $D$ 和模型依赖材料状态为主变量, 引入唯一 hybrid trace $\hat E_h$. 与这些工作的直接比较中, 作者没有在提供文本中列出计算量或精度上的数值对比; 相反, 作者明确说与显式 DG 或其他求解器的计算优势必须在相同响应精度下另行证明. 因此本文的相对优势主要是适定性、能量稳定性和误差分析框架, 而不是已证明的性能优势.\n\n与时间调和/超收敛 Maxwell HDG 比较. 变体 B 与 Chen–Qiu–Shi–Solano [12] 共享 $k+1/k$ 多项式次数模式和 $h^{-1}$ 尺度稳定化. 但作者指出它是不同方法: 本文离散一阶时变 Maxwell–材料系统, 没有 divergence multiplier, 使用完整切向电场 trace, 分析基于半离散能量恒等式, 而不是定常 curl–curl 对偶论证. 这意味着变体 B 的 $h^{k+1}$ 共同率不能直接看作已有超收敛 HDG 结果的时域推广; 它的最优性是对次数 $k$ 磁场而言, 并非对次数 $k+1$ 电场给出额外一阶. 与频域非局部 Drude HDG 和一般非局部光学响应模型相比, 本文处理的是时间调制而非空间非局部性, 并且材料系数可以在时间上跳变.\n\n与 DG 色散 Maxwell 分析比较. Wang–Xie–Zhang 对冷等离子体、Debye 和 Lorentz 模型建立稳定性与空间误差估计; Xie–Wang–Zhang 研究时空 DG; Bokil 等给出与非线性光学材料 ODE 耦合的能量稳定 DG 格式, 包括 Lorentz 响应. 这些工作为色散 Maxwell 系统提供了广泛数值基础. 本文关注的是不同问题: 四种时间调制机制具有不同的时间传输规则, 需要 event-aligned Maxwell–材料 HDG 方案, 并分析切换时间错位误差. 此外, 本文材料系数在事件处可以跳变, 但背景 $\epsilon_0,\mu_0$ 保持常数; 能量泛函中的权重 $s,b$ 随时间和空间逐元素变化, 通过 $C_i(t)$ 和 $\rho_{i,j}$ 控制增长. 与上述工作相比, 本文没有处理非线性材料 ODE, 也没有处理磁色散或多极点; 它限定为各向同性、一个电 Lorentz 极点、无源项, 以隔离调制效应.\n\n与时间依赖系数二阶双曲和 Maxwell 的有限元结果比较. Al Jarroudi 与 Grote [1] 对系数在空间和时间光滑变化的二阶双曲方程, 用时间依赖 Ritz 型投影证明半离散最优能量范数收敛. 本文的关键差别是材料系数可以在事件处跳变, 且未知量包含色散材料状态; 因此不能直接使用他们的光滑系数投影论证, 而需要逐事件区间分析和事件能量比. Renner 与 Yousept [16] 分析时间依赖介电常数和磁导率、非线性电–磁耦合和非线性 Silver–Müller 边界, 用边元/分片常数和隐式 Euler 得到有限时间强收敛, 并在附加结构和耗散假设下得到指数衰减. 本文则对背景常数、材料系数可跳变、线性 Drude–Lorentz 辅助状态的 HDG 方法给出定量误差估计, 并明确分离事件对齐积分与切换时间位移. 作者认为这些结果与 [16] 互补: [16] 强调非线性边界和强收敛/指数衰减, 本文强调四种时间传输规则、模型 2 的场跳变和切换时间错位导致的误差与稳定性边界移动.\n\n适用范围与代价. 本文的框架限定在 $\Omega\subset\mathbb{R}^3$ 的多面体、共形四面体网格、PEC 或成对周期边界; 初始数据需满足边界相容性和正则性假设; 变体 B 的 $h^{k+1}$ 共同率需要更强分片正则性 ( $H$ 为 $H^{k+1}$, $E$ 和独立材料状态为 $H^{k+2}$ ), 变体 A 的 baseline 未在提供文本中给出具体阶. 作者不声称 HDG 保持完整 Gauss 约束, 只处理连续层面的 $\nabla\cdot D$ 和 $\nabla\cdot(\mu_0H)$ 传播; 离散格式不宣称精确保持. 模型 3 的能量不控制 $P$, 需要额外估计. 时间误差和切换时间错位误差的具体阶、数值实验的详细设置和三维结果未在提供的 PDF 摘录中展开, 因此无法与已有工作做更细的定量比较. 最后, 作者把逆设计作为远期应用, 但没有给出计算成本、内存或条件数方面的比较; 与显式 DG 等方法的性能优势仍待展示. High-order positivity-preserving well-balanced schemes: combining flux globalization with modified Patankar deferred correction methods 作者: Philipp Öffner, Michael Redle, Davide Torlo 类别: math.NA 发布日期: 2026-10-15 arXiv ID: 2610.21857v1 链接: http://arxiv.org/abs/2610.21857v1 中文标题: 高阶保正良平衡格式：结合通量全局化与修正 Patankar 延迟修正方法 摘要翻译: 用于双曲平衡律的保正格式通常基于 Zhang-Shu 限制器和强稳定性保持 Runge-Kutta (SSP-RK) 时间推进构造，代价是限制性的时间步条件。修正 Patankar 延迟修正 (mPDeC) 方法消除了这一限制，具有高阶精度、守恒性和无条件保正性。然而，其依赖于解的 Patankar 权重使时间更新变为非线性，因此，与 RK 方法不同，保持（定常和移动）平衡态，即良平衡 (WB)，不再直接。本文将 mPDeC 与通量全局化方法相结合，构造高阶、良平衡 (WB)、保正的 mPDeC 格式。我们在一维浅水方程上展示该思想，采用良平衡 (WB) A-WENO 空间离散，并引入一种在存在干状态时仍保持稳定的修正局部特征分解。一维基准问题上的数值实验证实该方法满足所需性质。 Greedy multiscale kernel algorithms 作者: Sara Avesani, Armin Iske, Michael Multerer, Tizian Wenzel 类别: math.NA 发布日期: 2026-10-15 arXiv ID: 2610.21615v1 链接: http://arxiv.org/abs/2610.21615v1 中文标题: 贪婪多尺度核算法 摘要翻译: 多尺度核逼近与贪婪核算法是散乱数据逼近的两种成熟工具。我们将它们结合为贪婪多尺度核算法，其中多尺度残差校正的每一层均由贪婪核算法构造。对于多尺度 $f$-贪婪算法，我们证明了以所选中心点数量表示的收敛率，它们与单尺度 $f$-贪婪算法的收敛率一致（在指数上至多有一个任意小的损失）。多尺度构造的收益在于稳定性；我们通过带有重缩放核的核矩阵条件数的双边估计来精确刻画这一点。下界表明，任何单尺度方法，尤其是 $f$-贪婪算法，都为其收敛率付出条件数至少按中心点数量的幂增长的代价。相反，核的多尺度缩放使该下界在每一层保持不变，同时不改变收敛率。上界指出，中心点在核尺度下的分离性是所需控制的量。我们给出了所有考虑方法的运算次数，并在误差、时间和条件性方面，将贪婪多尺度算法与单尺度贪婪方法及经典多尺度方法进行了数值比较。 A priori error estimates for a Hybrid High-Order method on variable-exponent Leray-Lions problems 作者: André Harnist, Romain Mottier 类别: math.NA, math.AP 发布日期: 2026-10-15 arXiv ID: 2610.21611v1 链接: http://arxiv.org/abs/2610.21611v1 中文标题: 变指数 Leray-Lions 问题的 Hybrid High-Order 方法的先验误差估计 摘要翻译: 我们为在一项配套收敛性研究中引入的变指数 Leray–Lions 问题的 Hybrid High-Order (HHO) 离散化推导先验误差估计。HHO 方法属于杂交非协调方法族，基于任意次数的单元和面多项式未知量。在精确解及其通量具有额外分片正则性的条件下，我们建立了一个保留各个单元尺寸的局部化误差估计，以及一个全局估计，其收敛阶区分纯次二次、混合次二次–超二次和纯超二次情形。在次二次范围内，我们还推导了一种依赖于情形的加密策略，该策略比较局部逼近尺度与局部非退化尺度，并在该尺度被网格分辨时恢复最优多项式阶。该分析依赖于多项式投影的变指数逼近性质、离散多项式空间的平均指数工具以及加权单调性估计。该框架还覆盖正则化且轻度非对称的各向异性 Carreau–Yasuda 型通量。在笛卡尔和 Voronoi 网格上的数值实验支持了所预测的收敛阶，并表明关于通量的正则性假设不能省略。 Stochastic Latent Twins: Learning Stochastic Solution Operators 作者: Matthias Chung, Jack Michael Solomon, Deepanshu Verma 类别: cs.CE, math.NA 发布日期: 2026-10-15 arXiv ID: 2610.21084v1 链接: http://arxiv.org/abs/2610.21084v1 中文标题: 随机潜变量孪生：学习随机解算子 摘要翻译: 不确定性下的科学预测需要的不仅仅是单一未来状态：它需要可能结果的分布。我们表明，这种分布值演化可以直接学习为任意源时间和目标时间之间的随机解算子。我们引入随机潜变量孪生（Stochastic Latent Twins），它通过确定性编码器-解码器表示物理状态，同时将随机性完全置于概率值潜演化映射中。这种分离使得随机转移律独立于状态表示，并使得无需递归时间步进即可进行直接概率预测。 我们建立了一种端到端 Wasserstein 近似理论，将表示误差和潜转移误差分离，推导了稳定性保证，并发展了物理一致性的分布理论，具有潜变量到物理的传递界以及由构造保证的一致性。所得框架可以直接从观测到的转移中学习，并在同一表述中支持前向预测、后向条件演化、单次生成以及物理一致的随机动力学。数值实验从线性和非高斯 SDEs 到随机 Lorenz 动力学、生成建模、随机 Hamiltonian 系统和二维随机 Burgers 方程，验证了该理论，揭示了状态表示与转移族表达能力的独特作用，与近期随机方法和一步生成方法相比具有优势，并展示了由构造实现的精确物理守恒。更广泛地，随机潜变量孪生朝着科学代理模型迈出了一步，这些代理模型不仅表示系统的演化，还表示其不确定性和物理结构，为不确定性量化、数据同化、反问题、控制和数字孪生开辟了新机遇。 Adaptive Randomized Neural Networks for PDEs: Structure-Aware Feature Redistribution and Residual-Driven Block Enrichment 作者: You Yang, Fei Wang 类别: math.NA, cs.LG 发布日期: 2026-10-15 arXiv ID: 2610.21077v1 链接: http://arxiv.org/abs/2610.21077v1 中文标题: 面向偏微分方程的自适应随机神经网络：结构感知的特征重分配与残差驱动的块富集 摘要翻译: 随机神经网络保留了输出线性的最小二乘求解，但全局采样的随机特征可能无法捕捉偏微分方程解的空间和尺度结构。我们提出 SFRBE–RaNN，一种两阶段自适应试探空间构造。SFR 根据区域指示量的质量重新分配固定的特征预算，并通过低维的基于残差的优化调整区域权重尺度。RBE 通过筛选多个激活函数族、移除已表示的响应方向，并选择最小的容许前缀来构造非冗余富集。对于解依赖算子，候选通过冻结算子进行筛选，并且仅在非线性残差验证之后才被接受。对于固定线性问题，我们建立了一个精确的离散残差下降恒等式以及一个条件解误差估计。基于该恒等式，我们结合带保护的 SFR 与 RBE 捕获因子，推导出一个涉及基线最佳逼近和经验一致性的有限阶段误差界，以及一个条件随机搜索成功概率界。针对局部化、多尺度和非线性时空问题的数值实验证明了该方法的有效性。 Galerkin finite element method for time fractional diffusion-wave equations with constant time delay 作者: Xuan Li, Kaile Qiu, Chaobao Huang, Weiping Bu 类别: math.NA 发布日期: 2026-10-15 arXiv ID: 2610.21056v1 链接: http://arxiv.org/abs/2610.21056v1 中文标题: 带常数时滞的时间分数阶扩散波方程的 Galerkin 有限元方法 摘要翻译: 本文研究带常数时滞的时间分数阶扩散波方程的数值方法。利用对称分数阶降阶技术并引入辅助变量 $w$，我们将原问题重新表述为一个等价的耦合系统，并首次对 $w$ 进行严格的正则性分析。基于对称分级时间网格，该耦合问题在时间上采用 $L1$ 格式离散，在空间上采用标准有限元方法离散，并借助离散分数阶 Grönwall 不等式建立了所建立全离散有限元格式的稳定性结果。此外，我们推导了一个新的比较原理，并根据对称分级网格的性质，在每个区间 $((k-1)τ,kτ]$，$1\le k\le K$ 上重构了一个新的障碍函数。基于新的障碍函数，我们获得了新的相关性质，这些性质导出了离散分数阶算子的一个新的且关键的稳定性性质。结合所得到的时间局部截断误差，给出了全离散有限元格式的局部误差估计，这表明我们的收敛结果提供了一个统一框架，无缝地同时涵盖均匀网格上的时间逐点误差估计和对称分级网格上的全局时间误差估计。数值实验验证了理论结果。 A Coupled Fluid Naghdi Shell Windkessel Model for Pulmonary Artery Banding: Partitioned Numerical Scheme and Parameter Estimation 作者: Yifan Wang, Josip Tambaca, Suncica Canic 类别: math.NA, math-ph 发布日期: 2026-10-14 arXiv ID: 2610.20966v1 链接: http://arxiv.org/abs/2610.20966v1 中文标题: 用于肺动脉环扎的耦合流体-Naghdi壳-Windkessel模型：分区数值格式与参数估计 摘要翻译: 我们建立了一个三维流固耦合模型，用于模拟血液流经具有局部缩窄的可变形血管，其动机来自先天性心脏病新生儿的肺动脉环扎。该模型将任意拉格朗日-欧拉框架下的不可压缩 Navier-Stokes 方程与用于动脉壁的 Naghdi 壳模型以及用于下游循环的三元件 Windkessel 模型耦合起来。该壳描述具有可变半径的三维圆柱血管壁，并定义在二维参考域上。其低正则性形式保留了膜、弯曲和横向剪切变形，同时仅要求参考中面的第一阶导数，从而可处理具有不连续表面斜率的分片光滑几何。我们推导了耦合变分形式，并发展了一种非迭代分区格式，其中带有包含动脉壁惯性的 Robin 型界面条件。对于线性化的固定区域问题，我们在适当假设下建立了离散能量估计。无迹卡尔曼滤波器根据入口压力观测估计近端和远端阻力以及顺应性。数值实验表明，在空间和时间加密下，压力波传播和壁变形具有一致性。标准 Naghdi 壳与一个修改的、以径向为主的壳模型之间的比较，揭示了结构变形机制如何影响波速和位移。一个理想化的肺动脉环扎示例展示了带参数估计的耦合流体-壳-Windkessel 模型。该框架能够研究局部缩窄的血流动力学与力学效应及其对下游血管性质的依赖性。 Real-Arithmetic Sketched Eigensolvers for Symmetric Matrices 作者: Jingyu Liu, Yuqi Liu 类别: math.NA 发布日期: 2026-10-15 arXiv ID: 2610.22261v1 链接: http://arxiv.org/abs/2610.22261v1 摘要: In this work, we develop sketched variants of subspace iteration and TraceMIN, called sSI and sTraceMIN, for real symmetric eigenvalue problems by combining basis whitening with the sketched Rayleigh-Ritz (sRR) procedure. The nonsymmetric projected matrix in sRR may have nonreal eigenpairs, leading to complex arithmetic and additional computational costs. We overcome this difficulty using a real Schur formulation of sRR that keeps the iterations in real arithmetic. We further develop procedures for characteristic subspace selection and convergence monitoring within this formulation. On the theoretical side, we discuss suitable subspace embedding assumptions for sketched eigensolvers that use a fixed sketching matrix throughout the iterations. Under a mild embedding assumption, we establish local convergence results for sSI and sTraceMIN. Numerical experiments demonstrate runtime savings of up to 39% for sSI and 21% for sTraceMIN relative to their standard counterparts on the test problems. The ReLU Matrix Decomposition: NP-Hardness, Ill-Posedness, and ReLU-Rank 作者: Giovanni Seraghiti, Arnaud Vandaele, Nicolas Gillis 类别: math.NA 发布日期: 2026-10-15 arXiv ID: 2610.22255v1 链接: http://arxiv.org/abs/2610.22255v1 摘要: The ReLU Matrix Decomposition (RMD) approximates a sparse, nonnegative m-by-n matrix X by max(0, WH), where W is an m-by-r matrix, H is an r-by-n matrix, and r is the rank of the decomposition. The most common formulation, referred to as Fro-RMD, minimizes the Frobenius norm of X - max(0, WH) over W and H. We prove that Fro-RMD is NP-hard, even for binary matrices and r = 1; to the best of our knowledge, this is the first complexity result for RMD. We also show that Fro-RMD is ill-posed, in the sense that its infimum may not be attained. We then turn to exact RMD, that is, X = max(0, WH), and define the ReLU-rank of X as the smallest r for which such a decomposition exists. We interpret exact RMDs geometrically as configurations of points and hyperplanes in an r-dimensional space, associated with the rows of W and the columns of H, respectively. Building on this interpretation, we derive upper bounds on the ReLU-rank of diagonal, block-diagonal, and block-sparse matrices and, under an additional structural assumption, improve the best known bound for block-diagonal matrices. Finally, we propose two formulations of exact RMD as feasibility problems: a nonconvex quadratically constrained program, which can be tackled by global solvers such as Gurobi, and a formulation based on the latent model of Saul, (``A nonlinear matrix decomposition for mining the zeros of sparse data’, SIMODS, 2022), which we tackle using alternating projections for which we establish local convergence together with asymptotic convergence rates. We compare these two approaches on some numerical examples. Change the Number, Change the Answer: Assessing Rating Scale Sensitivity in LLM-Based Text Scoring 作者: Wonjeong Jo, Gyuwan Kim, Ming Shan Hee 类别: cs.CL, cs.AI, math.NA 发布日期: 2026-10-15 arXiv ID: 2610.22187v1 链接: http://arxiv.org/abs/2610.22187v1 摘要: Large language models (LLMs) are increasingly used in computational social science to assign scalar scores to social and psychological constructs. However, studies use different rating scales, and whether their scores remain comparable across scale designs is understudied. To address this gap, we define and systematically investigate scale sensitivity, the variation in LLM-generated scores caused by rating scale design. We distinguish numerical sensitivity, changes in normalized scores, from semantic sensitivity, changes in score interpretation. To assess both, our experimental design crosses 2 formats (continuous vs. ordinal), 4 ranges (3, 5, 11, and 21 points), and 2 signs (signed vs. unsigned) across 3 tasks (political leaning, sentiment, and toxicity), 2 prompting strategies (zero-shot and rubric-based), and 5 LLMs. We find that changing one design element shifts normalized scores by about 0.10 on average and changes the category of roughly 20% of texts. Sensitivity is greatest for toxicity, the only unipolar construct examined. Also, rubric-based prompting does not reduce sensitivity compared with zero-shot prompting. We provide practical guidance for designing more reliable LLM-based text scoring and highlight the need for robustness checks across alternative scales. Finite Volume Convergence for a Heat Equation with a Measure-Valued Source 作者: Matthias Gobbert, Martin Kružík, Andreas Meister, Michael Muscedere 类别: math.NA, math.AP 发布日期: 2026-10-15 arXiv ID: 2610.22063v1 链接: http://arxiv.org/abs/2610.22063v1 摘要: Past work with Tom Seidman analyzed the spatial convergence order of the numerical finite element method applied to time-dependent diffusion-reaction equations with a measure-valued source term. For smooth source terms, the numerical method has second-order accuracy, but the non-smooth source terms introduce a limitation of the order that depends on the spatial dimension. This work in honor of Tom extends the analysis to the finite volume method in its clas- sical two-point flux form and shows that the regularity of the source term reduces the convergence order to 1 in two and 0.5 in three spatial dimensions which is reduced from the nominal second-order and depends on the space dimension. Computational tests with a point source modeled by a Dirac delta distribution exhibit the behavior predicted by the analytical results. An IMEX multi-level domain-decomposition method for the Boltzmann equation on unstructured meshes 作者: Domenico Caparello, Lorenzo Pareschi, Thomas Rey 类别: math.NA 发布日期: 2026-10-15 arXiv ID: 2610.21740v1 链接: http://arxiv.org/abs/2610.21740v1 摘要: We introduce a dynamically adaptive multi-level method for the multiscale Boltzmann equation on unstructured meshes, coupling the compressible Euler, Navier-Stokes, and full Boltzmann descriptions within a single asymptotic-preserving IMEX framework. Unlike standard domain-decomposition approaches, where independent solvers exchange data only through interface conditions, the three models form a unified multiregime scheme. They share the same globally stiffly accurate IMEX Runge-Kutta discretization and are synchronized at every internal stage through kinetic-to-fluid moment projections and fluid-to-kinetic lifting. This stage-wise coupling maintains temporal consistency across model interfaces, while the asymptotic-preserving structure ensures compatibility with the fluid limit as the Knudsen number becomes small. The active description is selected in each cell through moment-realizability indicators derived from successive Chapman-Enskog approximations. The hierarchy uses Euler in near-equilibrium regions, Navier-Stokes where first-order nonequilibrium corrections are sufficient, and the full Boltzmann equation where a kinetic description is required. Using Navier-Stokes as the intermediate level avoids evolving a velocity distribution in mildly nonequilibrium regions, which is particularly advantageous on unstructured meshes, where kinetic transport dominates the cost. The spatial discretization combines finite-volume WENO reconstruction on triangular meshes with a conservative fast spectral method for the Boltzmann collision operator. Numerical experiments show that the method closely reproduces the full Boltzmann reference while significantly reducing the computational cost. These results suggest that tightly coupled AP model hierarchies are a viable alternative to interface-based domain decomposition for deterministic kinetic simulations in complex geometries. Stability analysis of the steady solution for incompressible lid-driven cavity flow 作者: Dmitri Lomasov, Petr Vabishchevich 类别: math.NA 发布日期: 2026-10-15 arXiv ID: 2610.21721v1 链接: http://arxiv.org/abs/2610.21721v1 摘要: Steady solutions at high Reynolds numbers are computed using an iterative method that linearizes the nonlinear convective terms. This linearization reduces the original nonlinear boundary-value problem to a sequence of linear problems, one solved at each iteration, thereby improving the numerical robustness of the algorithm and enabling convergence over a wide range of governing parameters. The stability of steady incompressible fluid flows is studied within the framework of a linear problem for perturbations. The original steady velocity and pressure fields are subjected to small perturbations, after which the equations are linearized with respect to the perturbations. The resulting linear problem allows one to analyze the evolution of small perturbations and to establish whether the steady solutions under consideration are stable with respect to infinitesimal perturbations. In this work, the computational implementation is based on spatial discretization by the finite element method on triangular meshes. The stability of the obtained solutions is analyzed by solving a generalized algebraic eigenvalue problem formed after discretization of the linear problem for perturbations. The presence of eigenvalues with positive real part indicates instability, whereas their absence indicates stability. Calculations were performed on a sequence of refined meshes at various Reynolds numbers for the flow in a square cavity. The $L\log L$-Luxemburg Discrepancy for Nonnegative Weights Suffers from the Curse of Dimensionality 作者: Josef Dick, Friedrich Pillichshammer 类别: math.NA, math.NT 发布日期: 2026-10-15 arXiv ID: 2610.21717v1 链接: http://arxiv.org/abs/2610.21717v1 摘要: We study the normalized discrepancy in the Luxemburg norm generated by $Φ(s)=s\log(1+s)$. For arbitrary nonnegative quadrature weights, we prove that the associated information complexity suffers from the curse of dimensionality: for every $\varepsilon\in(0,1)$, there exist constants $c_\varepsilon>0$ and $d_0=d_0(\varepsilon)\in\mathbb{N}$ such that [N_{Φ,+}(\varepsilon,d)\ge c_{\varepsilon} d^{-1}(1.07735\ldots)^d \qquad \mbox{for all } d\ge d_0. ] The result therefore also applies to equal-weight quasi-Monte Carlo rules. Projection-Based Parareal for Nonlinear Initial-Value Problems 作者: Chenyi Tan, Yuncheng Xu, Yehao Zhang, Yangfeng Su 类别: math.NA 发布日期: 2026-10-15 arXiv ID: 2610.21680v1 链接: http://arxiv.org/abs/2610.21680v1 摘要: Parareal is a parallel-in-time method for initial-value problems that has found extensive applications and given rise to many variants. Krylov-enhanced parareal is a variant of parareal that uses projection techniques to accelerate convergence substantially for linear problems. In this paper, we study a projection-based parareal algorithm for nonlinear initial-value problems with a modified iteration scheme and an arbitrary orthogonal projection. For a general nonlinear problem, we prove a convergence estimate for each iteration step on a fixed time grid. We analyze the errors of the modified scheme and the scheme of Krylov-enhanced parareal relative to the old error as the coarse grid is refined. For a coarse solver of order $p$ and step size $h$, we prove error bounds of $O(h^{p+1})$ for the modified scheme and $O(h)$ for that of Krylov-enhanced parareal at the first potentially inexact time point under mild assumptions. Numerical experiments on two nonlinear problems are consistent with the theoretical analysis. Data-driven Smooth-to-hard Spectral Regularization for General Linear Inverse Problems 作者: Guanzhong Yang, Zi Wang, Guang Yang 类别: math.OC, eess.IV, eess.SP, math.NA 发布日期: 2026-10-15 arXiv ID: 2610.21664v1 链接: http://arxiv.org/abs/2610.21664v1 摘要: The logistic smooth-to-hard shrinker is the proximal map of an explicit saturating spectral penalty for general linear inverse problems. Its exact weak convexity yields a unique scaled update and proximal-gradient stationarity guarantees, while independent validation provides finite-grid parameter-selection control. We use MRI as an illustrative example. On the stability of basis construction in extended Krylov subspace methods 作者: Kathryn Lund, Marcel Schweitzer 类别: math.NA 发布日期: 2026-10-15 arXiv ID: 2610.21603v1 链接: http://arxiv.org/abs/2610.21603v1 摘要: Extended Krylov subspace methods (EKSMs) are a well known approach for approximating the action of a matrix function on a vector $f(A)\mathbf{b}$, and their block versions are often used for solving Lyapunov and Sylvester matrix equations. EKSMs produce Laurent approximations to the underlying operator, which are generally more accurate per iteration than polynomial KSMs, without requiring a pole selection procedure as rational KSMs do. At the same time, due to the expansion of the search space by not only $A$ but also by its inverse, a number of implementation choices must be made that can affect the stability of the algorithm in floating-point arithmetic. We concentrate on two variants that often appear in the literature and in code, based on whether new basis vectors are built via what we denote as A Hybrid High-Order method for variable-exponent Leray-Lions problems with convergence under minimal regularity 作者: André Harnist, Romain Mottier 类别: math.NA, math.AP 发布日期: 2026-10-15 arXiv ID: 2610.21598v1 链接: http://arxiv.org/abs/2610.21598v1 摘要: We consider a homogeneous Dirichlet problem for nonlinear elliptic equations in divergence form with Leray–Lions fluxes of variable growth. The class includes the variable-exponent $p(\cdot)$-Laplacian and regularized Carreau–Yasuda-type laws. We propose and analyze a Hybrid High-Order (HHO) discretization on general polytopal meshes, a hybrid nonconforming method based on cell and face polynomial unknowns of arbitrary degree. We develop a variable-exponent polynomial and discrete functional toolkit centered on a mean-exponent norm equivalence, together with inverse, trace, Poincaré–Wirtinger, Sobolev–Poincaré, discrete Lebesgue embedding, interpolation, and compactness estimates adapted to HHO unknowns. We then establish well-posedness of the nonlinear discrete problem and convergence without assuming additional regularity of the exact solution beyond its natural energy space. The discrete potential converges strongly in every uniformly subcritical variable-exponent Lebesgue space, the reconstructed gradient converges strongly in the natural variable-exponent space, and the stabilization energy vanishes. Numerical experiments in two, three, and four space dimensions confirm convergence for singular solutions and for exponent fields ranging from close to $1$ up to $6$. Quantitative a priori error estimates for the same method are derived in a companion paper. Reconstructing constant point heat sources from few temperature sensors: sparse selection and spatial resolution 作者: Xiaomei Yang, Zhiliang Deng 类别: stat.ME, math.NA 发布日期: 2026-10-15 arXiv ID: 2610.21301v1 链接: http://arxiv.org/abs/2610.21301v1 摘要: Thermal diffusion and measurement noise can obscure the number and individual parameters of localized heat sources. We study the reconstruction of constant positive point sources from temperature histories at a few sensors, without prescribing the source count. Orthogonal matching pursuit or least angle regression selects candidate locations from a unit-response dictionary. These candidates initialize continuous constrained fits, and a noise-based discrepancy rule selects the smallest accepted model among the attempted fits. The response analysis distinguishes rejection of a merged one-source model from accurate recovery of the individual sources. It gives a projected quadratic residual for a close pair, a weak direction in the two-source parameters, and a conditional bound for recovering the total strength and strength-weighted center of an unresolved cluster. A local Gaussian approximation relates one-source rejection to noise and separation. Paired numerical tests compare sensor and time-sampling budgets, noise levels, and source separations. Finite-plate data generated independently of the inverse response and a three-dimensional example assess the same fitting procedure. In the tested configurations, source-count selection and individual parameter recovery differ substantially, while unresolved pairs can retain accurate aggregate estimates. The framework combines reconstruction with a local interpretation of which heat-source information the observations support. A geometric criterion for the dimension of Morgan-Scott spline spaces $S_{2r}^r(Δ_{\mathrm{MS}})$ 作者: Xinyu Wu, Yi Zheng, Jiansong Deng 类别: math.NA 发布日期: 2026-10-15 arXiv ID: 2610.21192v1 链接: http://arxiv.org/abs/2610.21192v1 摘要: We establish a geometric dimension criterion for spline spaces on nondegenerate standard Morgan-Scott triangulations with four distinct edge directions at each interior vertex. For every integer $r\ge1$, concurrence of the three auxiliary lines joining corresponding interior and boundary vertices implies that $\dim S_{2r}^r(Δ_{\mathrm{MS}})$ exceeds the Schumaker lower bound by one. The proof constructs a quadratic form through the six dual spoke points and uses its $r$th power to define a nonzero functional annihilating the image of the smoothing-cofactor map. Diener’s upper bound then gives the dimension formula. We also give a matrix representation of this functional and an explicit example in the quadratic case. Combined with the known necessary conditions and the negative-branch exclusion of Deng, Feng, and Kozak, the result yields a necessary and sufficient concurrence criterion for $r>2$. Superfast Spectral Compressed Sensing with Quadratic Convergence 作者: Xunmeng Wu, Jian-Feng Cai, Zongben Xu, Zai Yang 类别: cs.IT, eess.SP, math.NA 发布日期: 2026-10-15 arXiv ID: 2610.21068v1 链接: http://arxiv.org/abs/2610.21068v1 摘要: In this paper, we study the spectral compressed sensing problem of recovering a spectrally sparse signal from linear compressive measurements. Exploiting the low-rank Hankel structure associated with the signal, we formulate a nonconvex fixed-rank optimization problem with a Hankel regularization. We propose a Superfast Iterative Hard Thresholding (SIHT) algorithm, where each iteration consists of a tangent-space least-squares update followed by a fixed-rank hard thresholding projection. SIHT is the first nonconvex algorithm with provable quadratic convergence for spectral compressed sensing. For random Gaussian measurements, it achieves exact recovery in the noiseless case and stable recovery in the noisy case with near-optimal sample complexity. To efficiently implement SIHT, we develop a parameterized conjugate gradient method for solving the least-squares subproblem, endowing SIHT with state-of-the-art computational and memory efficiency. Numerical experiments are provided to validate our results. A nonstandard fractional Jacobi pseudospectral method for weakly regular right-sided Caputo terminal problems 作者: Ibrahem G. Ameen, Magda Hammad, Mohsen Razzaghi, Mahmoud A. Zaky 类别: math.NA 发布日期: 2026-10-14 arXiv ID: 2610.20829v1 链接: http://arxiv.org/abs/2610.20829v1 摘要: Right-sided fractional differential equations arising in terminal value problems present substantial analytical and numerical challenges due to their intrinsic nonlocal structure and the presence of weak algebraic-type singularities near the terminal point. These singular behaviors significantly degrade the performance of classical finite-difference, finite-element, and standard spectral methods, which typically rely on smoothness assumptions that break down in the fractional setting. In this work, we develop a comprehensive fractional pseudospectral framework specifically designed for nonlinear right-sided Caputo terminal value systems. The proposed method employs a versatile family of nonstandard fractional Jacobi basis functions whose endpoint-weighted structure naturally reflects the asymptotic behavior of right-sided fractional operators. This construction enables the accurate approximation of solutions with limited regularity near the terminal point, without requiring smoothing transformations or mesh grading. We establish the existence and uniqueness of the discrete numerical solution and derive rigorous convergence estimates that faithfully capture the inherent singularity structure of the problem. A series of numerical experiments validates the theoretical developments and demonstrates that the proposed approach delivers high-order accuracy and superior resolution of terminal-layer behavior compared with traditional spectral techniques. The resulting framework provides an efficient and reliable computational tool for challenging nonlinear fractional terminal value systems characterized by weak solution regularity.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. 高阶保正良平衡格式：结合通量全局化与修正 Patankar 延迟修正方法
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2. 贪婪多尺度核算法
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3. 变指数 Leray-Lions 问题的 Hybrid High-Order 方法的先验误差估计
摘要翻译
4. 随机潜变量孪生：学习随机解算子
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5. 面向偏微分方程的自适应随机神经网络：结构感知的特征重分配与残差驱动的块富集
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6. 带常数时滞的时间分数阶扩散波方程的 Galerkin 有限元方法
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7. 用于肺动脉环扎的耦合流体-Naghdi壳-Windkessel模型：分区数值格式与参数估计
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不相关论文（基本信息）
1. 面向对称矩阵的实算术草图特征求解器
2. ReLU 矩阵分解：NP-困难性、不适定性与 ReLU-秩
3. 改变数字，改变答案：评估基于LLM的文本评分中的评分量表敏感性
4. 具有测度值源项的热方程的有限体积收敛性
5. 非结构网格上 Boltzmann 方程的 IMEX 多层区域分解方法
6. 不可压缩顶盖驱动方腔流稳态解的稳定性分析
7. 非负权重下的 $L\log L$-Luxemburg 偏差受维数灾难困扰
8. 面向非线性初值问题的基于投影的 Parareal 方法
9. 面向一般线性反问题的数据驱动从光滑到硬谱正则化
10. 关于扩展 Krylov 子空间方法中基构造的稳定性
mixed channels" or split channels." We characterize their loss of orthogonality with an easily computable metric and prove that when $A$ is symmetric positive definite, mixed channels have better orthogonality than split channels; however, numerical experiments demonstrate that straightforward predictions are not possible for indefinite or nonsymmetric matrices. We also provide a modular MATLAB toolbox, \texttt{ektoolbox}, that contains new versions of EKSMs with low-synchronization Gram-Schmidt backbones.11. 变指数 Leray-Lions 问题的混合高阶方法及其在最小正则性下的收敛性
12. 从少量温度传感器重构常值点热源：稀疏选择与空间分辨率
13. Morgan-Scott 样条空间 $S_{2r}^r(Δ_{\mathrm{MS}})$ 维数的一个几何判别准则
14. 具有二次收敛的超快速谱压缩感知
15. 一种用于弱正则右侧 Caputo 终值问题的非标准分数阶 Jacobi 拟谱方法