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1. 各向同性弹性Helmholtz方程MAC格式离散的渐近色散修正
Asymptotic dispersion correction for the isotropic elastic Helmholtz equation discretized with a MAC scheme
作者: Pierre-Henri Cocquet, Antoine Tonnoir, Rachel Yovel
类别: math.NA
发布日期: 2026-08-18
arXiv ID: 2608.17909v1
摘要翻译: 弹性介质中时谐波传播的数值模拟在地球物理学和无损检测等应用中发挥着重要作用。由于数值色散和污染效应，高频下弹性Helmholtz方程的精确离散具有挑战性。在本工作中，我们为各向同性弹性Helmholtz方程的MAC格式离散提出了一种渐近色散修正。我们刻画了该格式的离散色散关系，并确定了二维和三维空间中色散误差的首阶项。基于此分析，我们推导出了一个在网格尺寸趋于零的极限下渐近最优的修正。所提出的方法提高了离散与连续波传播性质之间的一致性，同时保持了底层离散格式的结构。我们还建立了色散关系的因式分解与grad-div算子符号结构之间的联系，为弹性问题的代数结构提供了进一步的洞见。数值实验最终表明相对误差显著减小，并确认了所提出修正的有效性。我们进一步提供了数值证据，表明修正后的离散格式改善了多重网格求解器的收敛行为。
Comment: 26 pages, 7 figures
链接: http://arxiv.org/abs/2608.17909v1
1. 研究对象和背景
本文研究对象是各向同性弹性介质中的时谐波传播, 其强形式为
$$ -\operatorname{div}(\mu(\nabla u+\nabla u^T))-\nabla(\lambda\operatorname{div}u)-\rho(x)\omega^2 u=f\quad\text{in }\Omega, $$其中 $u=(u_x,u_y)$ (二维) 或 $u=(u_x,u_y,u_z)$ (三维) 为位移场, $\lambda,\mu$ 为 Lam'e 系数, $\rho$ 为密度, $\omega$ 为角频率. 在均匀介质中, 该方程等价于
$$ -\mu\Delta u-\nabla((\lambda+\mu)\operatorname{div}u)-\rho\omega^2 u=0. $$对平面波解 $u=X e^{i\xi\cdot x}$ 做色散分析, 得到连续符号
$$ \sigma(\xi)=\mu\|\xi\|^2 I_d+(\lambda+\mu)\xi\xi^T-\rho\omega^2 I_d, $$其行列式可分解为
$$ \det\sigma(\xi)=(\mu\|\xi\|^2-\rho\omega^2)^{d-1}((\lambda+2\mu)\|\xi\|^2-\rho\omega^2), $$从而存在两种波数: 剪切波 $k_S=\omega\sqrt{\rho/\mu}$ 和压力波 $k_P=\omega\sqrt{\rho/(\lambda+2\mu)}$.
数值求解该方程时, 离散色散关系与连续色散关系的失配会导致污染效应: 即使每个波长网格点数固定, 误差仍随频率增长. 弹性波时谐模拟在地球物理和无损检测中需要反复计算多个频率, 频域公式尤其重要, 因此控制高频数值色散是关键. 已有大量工作针对标量 Helmholtz 方程发展高阶格式、边界元、以及通过修改有限差分格式中自由参数来最小化色散误差的方法. 然而, 对矢量弹性 Helmholtz 方程, 尚未有基于"修改物理参数"的渐近最优色散修正. 本文使用 MAC (Marker-And-Cell) 交错网格有限差分格式, 在二维和三维中刻画其离散色散关系, 确定色散误差的首项, 并构造修正的 Lam'e 系数 $\hat\lambda=\lambda+h^2\lambda_2$, $\hat\mu=\mu+h^2\mu_2$, 使得 $h\to0$ 时色散误差在最大范数意义下渐近最优. MAC 格式将 $u_x$ 和 $u_y$ 分别放在两套交错网格上, 保持离散 De Rham 复形结构, 这种结构是本文离散 Helmholtz 分解和符号分解的关键.
2. 主要定理或主要结果
假设介质均匀 ($\lambda,\mu,\rho$ 为正常数), 源项 $f=0$, 网格均匀 $h_x=h_y=h$. 对 MAC 格式, 设 $\sigma_{\Delta_h}(\xi)$ 为离散 Laplacian 的符号, 在二维有
$$ \sigma_{\Delta_h}(\xi)=\frac{2\cos(\xi_1h)+2\cos(\xi_2h)-4}{h^2}, $$三维类似. 离散弹性算子的符号矩阵为
$$ \sigma_h(\xi)=-\mu\sigma_{\Delta_h}(\xi)I_d-(\lambda+\mu)\sigma_{\nabla_h}(\xi)\sigma_{\operatorname{div}_h}(\xi)-\rho\omega^2 I_d. $$作者证明了如下行列式分解:
$$ \det\sigma_h(\xi)=\big(-\mu\sigma_{\Delta_h}(\xi)-\rho\omega^2\big)^{d-1}\big(-(\lambda+2\mu)\sigma_{\Delta_h}(\xi)-\rho\omega^2\big), $$因此离散色散关系为
$$ D_h=\left\{\xi:\ \left(-\sigma_{\Delta_h}(\xi)-k_S^2\right)^{d-1}\left(-\sigma_{\Delta_h}(\xi)-k_P^2\right)=0\right\}, $$即剪切波与压力波分别对应两个标量离散 Helmholtz 色散关系, 且可分解为
$$ D_h=\{\xi:\ -\sigma_{\Delta_h}(\xi)-k_S^2=0\}\cup\{\xi:\ -\sigma_{\Delta_h}(\xi)-k_P^2=0\}. $$令 $\theta\in S^{d-1}$ 表示传播方向, 离散波数 $\tilde{k}$ 满足
$$ \tilde{k}=k+\frac{k^3 h^2}{24}F(\theta)+O(h^4), $$其中 $k$ 取 $k_S$ 或 $k_P$, 而 $F(\theta)=\sum_{j=1}^d \theta_j^4$ (在二维可写为 $F(\theta)=\cos^4\theta+\sin^4\theta$). 未修正时相对色散误差的首项为
$$ \max_{\theta\in S^{d-1}}\frac{|\tilde{k}-k|}{k}=\frac{(kh)^2}{24}\max_{\theta\in S^{d-1}}|F(\theta)|+O(h^4). $$为构造渐近最优修正, 作者引入 $\hat\lambda=\lambda+h^2\lambda_2$, $\hat\mu=\mu+h^2\mu_2$, 并推导出修正后离散波数的展开:
$$ \tilde{\hat{k}}_S=k_S+h^2\left(\frac{k_S^3}{24}F(\theta)-\frac{k_S}{2\mu}\mu_2\right)+O(h^4), $$$$ \tilde{\hat{k}}_P=k_P+h^2\left(\frac{k_P^3}{24}F(\theta)-\frac{k_P(2\mu_2+\lambda_2)}{4\mu+2\lambda}\right)+O(h^4). $$进而将 $\mu_2^{\mathrm{asy}}$ 和 $\lambda_2^{\mathrm{asy}}$ 定义为如下最小最大问题的解:
$$ \mu_2^{\mathrm{asy}}=\arg\min_{\mu_2}\max_{\theta\in S^{d-1}}\left|F(\theta)-\frac{12\mu_2}{\mu k_S^2}\right|, $$$$ \lambda_2^{\mathrm{asy}}=\arg\min_{\lambda_2,\mu_2}\max_{\theta\in S^{d-1}}\left|F(\theta)-\frac{12(2\mu_2+\lambda_2)}{(4\mu+2\lambda)k_P^2}\right|. $$所提供 PDF 摘录中列出了这些最小最大问题并声明下一节定理给出闭式, 但未包含闭式本身; 从推导可以看出, 最优参数使首项色散误差在最大范数意义下达到最小, 而不是完全消除 $O(h^2)$ 项. 数值实验表明, 该修正显著降低了相对误差, 并且修正后的离散格式改善了多重网格求解器的收敛行为.
3. 研究方法、关键技术和核心工具
证明和推导的核心技术可以归纳为以下几点.
第一, 连续和离散符号的统一处理. 作者先用平面波 $u=Xe^{i\xi\cdot x}$ 得到连续符号 $\sigma(\xi)$, 并使用矩阵行列式引理
$$ \det(A+cd^T)=\det(A)(1+d^T A^{-1}c) $$将 $\det\sigma(\xi)$ 分解为剪切波和压力波两个因子的乘积. 这一引理在离散情形被再次使用: 对 MAC 格式的符号矩阵 $\sigma_h(\xi)$, 取 $A=(-\mu\sigma_{\Delta_h}-\rho\omega^2)I_d$, $c=-(\lambda+\mu)\sigma_{\nabla_h}$, $d^T=\sigma_{\operatorname{div}_h}$, 便得到离散行列式的同样分解. 这个代数步骤揭示了弹性问题色散关系可分解为两个标量 Helmholtz 色散关系的本质.
第二, 离散 Helmholtz 分解与离散 De Rham 复形. 作者利用 [12] 中的离散恒等式
$$ \operatorname{div}_h^\pm \nabla_h^\pm\times=0,\quad \nabla_h^\pm\times\nabla_h^\pm=0,\quad \operatorname{div}_h^\pm\nabla_h^\mp=\Delta_h, $$以及
$$ \Delta_h=\nabla_h^+\operatorname{div}_h^- -\nabla_h^-\times\nabla_h^+\times, $$证明 MAC 离散解 $u_h$ 可写为
$$ u_h=\Phi_h-\frac{1}{k_P^2}\nabla_h^-\psi_h-\frac{1}{\rho\omega^2}f_h, $$其中 $\psi_h=\operatorname{div}_h^+ u_h$ 满足以 $k_P$ 为波数的标量 Helmholtz 方程, 而 $\Phi_h$ 满足 $\operatorname{div}_h^+\Phi_h=0$ 和以 $k_S$ 为波数的矢量 Helmholtz 方程. 这一分解不仅在概念上解释了为什么离散色散关系是两个标量色散关系的并集, 而且为后续将修正分别作用于 P 波和 S 波提供了理论依据.
第三, 渐近展开与最小最大修正. 在均匀网格上, 离散 Laplacian 的符号已知, 作者引用 [11, 12] 中离散波数的渐近展开
$$ \tilde{k}=k+\frac{k^3 h^2}{24}F(\theta)+O(h^4), $$并据此得到未修正色散误差的首项. 作者的真正创新点不是增加 stencil 自由参数, 而是对物理参数 $\lambda,\mu$ 引入一致的 $h^2$ 扰动 $\hat\lambda=\lambda+h^2\lambda_2$, $\hat\mu=\mu+h^2\mu_2$. 这种做法的优点是: (1) 适用于原本没有自由参数的格式; (2) 不需要数值优化; (3) 保持底层 MAC 格式的结构不变. 由于 $F(\theta)$ 随方向变化, 单个标量修正无法对所有方向同时消除误差, 因此作者采用 Chebyshev 意义下的最小最大策略, 即选择 $\lambda_2,\mu_2$ 使色散误差首项的 $L^\infty$ 范数最小. 这正是"渐近最优"的准确含义.
第四, 高维推广与代数结构洞察. 论文将二维推导推广到三维, 并指出离散色散关系因子的重数与 grad-div 算子符号的结构直接相关: 剪切波因子具有重数 $d-1$, 压力波因子重数为 1. 这种联系在摘要中被明确强调, 是从离散 De Rham 复形和符号矩阵分解中获得的额外代数洞察. 此外, 作者还讨论了对 $\lambda,\mu$ 同时移位的必要性: 它允许同时修正剪切波和压力波, 这在可压缩介质中尤其重要, 因为两种波的波数之比与 Poisson 比有关.
4. 与之前工作的比较
与已有工作的比较可以从以下几个层面展开.
第一, 与标量 Helmholtz 方程的渐近色散修正相比. [14, 11] 对标量 Helmholtz 方程提出了通过修改物理参数实现的渐近最优色散修正, [12] 将其推广到 Maxwell 方程. 本文是该思路在矢量弹性 Helmholtz 方程上的首次推广. 弹性问题的新困难在于压力和剪切波以不同速度传播, 存在两个不同波数 $k_P$ 和 $k_S$. 通过对 $\lambda$ 和 $\mu$ 同时做 $h^2$ 扰动, 本文能够同时作用于两个波数, 这是标量情形不需要处理的. 代价是最小最大问题涉及两个参数 $\lambda_2,\mu_2$, 且色散误差无法被完全消除, 只能使首项的最大值最小.
第二, 与已有的弹性波色散最小化格式相比. 文中提到 [25, 21, 22] 给出了弹性波传播的色散最小化离散格式, 但这些工作通常依赖具有自由参数的有限差分模板, 并通过数值优化确定参数. 本文方法不引入模板自由参数, 而是通过物理参数的一致扰动实现修正, 因此可以应用于原本没有自由参数的 MAC 格式, 并且修正参数具有闭式渐近表达式, 不需要求解优化问题. 这是方法层面上的本质区别.
第三, 与 MAC 格式及离散 De Rham 复形文献的关系. 本文使用的 MAC 格式和离散恒等式来自 [12, 32]. 已有文献利用离散 De Rham 复形构造保持结构的格式, 但本文进一步将离散 Helmholtz 分解与色散关系的代数分解联系起来, 指出离散色散关系因式分解正是 grad-div 算子符号结构的体现. 这一观察对此类交错网格格式的代数理解是新的.
第四, 就数值性能而言, 本文给出的是数值实验证据而非理论证明. 数值实验显示修正后相对误差大幅降低, 并且多重网格求解器的收敛行为得到改善. 但需要指出, 从提供内容中无法确认作者是否给出了多重网格收敛改善的严格理论分析; 摘要和正文均使用"numerical evidence"表述, 因此该结论应视为经验观察. 此外, 色散分析本身只在均匀介质和零源项下进行, 对于有界区域上的非均匀介质问题, 边界条件会耦合 P 波与 S 波, 作者也明确说明没有利用离散 Helmholtz 分解来设计求解器.
第五, 适用范围与局限. 修正的推导依赖于 $h\to0$ 的渐近展开, 因此要求每个波长有足够多的网格点: $kh$ 足够小. 在近不可压缩极限 $\nu\to0.5$ 时, $k_S/k_P$ 变大, 压力波可能远易于剪切波; 本文方法不依赖 Poisson 比, 但若剪切波每个波长网格点太少, 渐近修正不再可靠. 与仅修正波数的标量方法类似, 本文的常数修正无法消除与方向相关的各向异性色散误差, 只能使最大误差最小化. 尽管如此, 由于修正不改变线性系统的稀疏结构和对称性, 其额外计算代价几乎可以忽略, 这是该方法相对于高阶格式和边界元方法的一个实际优势.
2. 非线性磁场问题中涡流的有效计算
Efficient computation of eddy-currents for nonlinear magnetic field problems
作者: Herbert Egger, Nepomuk Krenn, Andreas Schafelner
类别: math.NA
发布日期: 2026-08-18
arXiv ID: 2608.17860v1
摘要翻译: 估算电气设备中非叠片导电部件的涡流损耗需要进行昂贵的三维模拟。因此，在实际应用中会采用各种近似方法，以降低早期设计阶段的计算成本。我们回顾了一些方法，并讨论了它们的建模假设及由此产生的近似。特别地，我们指出涡流反应场是一个应在全局范围内加以考虑的重要贡献。这些反应场可以通过求解一个线性化时间周期问题，从二维静磁模拟中近似重构。我们进一步讨论了求解这一后处理问题的不同策略。数值结果表明，与标准后处理方法相比，在适度的额外成本下，损耗预测得到了改进。
Comment: 无
链接: http://arxiv.org/abs/2608.17860v1
1. 研究对象和背景
本文研究电气设备中非叠片导电部件（如电机机壳、屏蔽件、永磁体）内的涡流损耗预测问题. 这类部件的电流路径本质上为三维, 精确预测需要求解三维非线性时变涡流问题. 设三维区域 $\Omega_{3D}\subset\mathbb{R}^3$ 及其导电区 $\Sigma_{3D}\subset\Omega_{3D}$, 采用 $A$-$\phi$ 公式:
$$\sigma(\partial_t A+\nabla\phi)+\operatorname{curl}(\partial_B w(\operatorname{curl} A))=J_s\ \text{in}\ \Omega_{3D},\qquad \operatorname{div} A=0\ \text{in}\ \Omega_{3D},$$其中 $A$ 是磁矢势, $\phi$ 是电标势（支撑在 $\Sigma_{3D}$ 内）, $B=\operatorname{curl} A$, 磁场强度 $H=\partial_B w(B)$, $w(B)$ 为磁能密度, 各向同性材料有 $\partial_B w(B)=\nu(|B|)B$（$\nu$ 为磁阻率）, $\sigma$ 为电导率且在 $\Sigma_{3D}$ 外为零, $J_s$ 为无散源电流. 涡流密度与损耗密度为 $J_{ec}=-\sigma(\partial_t A+\nabla\phi)$ 与 $p_{ec}=\sigma|\partial_t A+\nabla\phi|^2$. 由于三维模拟成本过高, 常在轴向对称假设下采用二维模型: 设 $\Omega\subset\mathbb{R}^2$ 为横截面、$\Sigma\subset\Omega$ 为导体截面, 轴向分量 $a$ 与常数 $g$ 满足
$$\sigma(\partial_t a+g)+\operatorname{curl}(\partial_B w(\operatorname{Curl} a))=j_s\ \text{in}\ \Omega,\qquad \int_\Sigma\sigma(\partial_t a+g)\,dx=0,$$其中 $b=\operatorname{Curl} a=(\partial_y a,-\partial_x a)$, $\operatorname{curl}h=\partial_x h_y-\partial_y h_x$. 定理1表明该二维模型是三维模型在柱域 $\Omega_{3D}=\Omega\times(0,L)$、$\Sigma_{3D}=\Sigma\times(0,L)$ 上的精确约化（取 $A=(0,0,a)$、$\phi=gz$、$J_s=(0,0,j_s)$）, 但二维模型的导体轴向端面位于边界上, 仅能保证总轴向电流为零, 无法满足逐点 $n\cdot J_{ec}=0$, 因此隐含着轴向端面短路的近似. 工程标准做法进一步求解序列非线性静磁问题 $\operatorname{curl}(\partial_B w(\operatorname{Curl}\hat a))=j_s$, 再用后处理估计损耗, 如定义 $\hat j_{ec}=-\sigma(\partial_t\hat a+\hat g)$ 且 $\int_\Sigma\sigma(\partial_t\hat a+\hat g)\,dx=0$, 相应的损耗为 $\hat p_{ec}=\sigma|\partial_t\hat a+\hat g|^2$. 这种后处理完全忽略涡流反应场（涡流对主磁场的反馈）, 在高频、高电流工况下造成严重偏差. 已有改进如三维混合方法 [10,11] 和避免三维计算的方法 [12] 缺乏严谨论证. 本文的空白在于: 在二维框架内, 通过仅求解一个线性化时间周期问题来一致恢复全局反应场, 从而以远低于完整非线性时变模型的代价获得精确的损耗预测.
2. 主要定理或主要结果
主要结果如下. 定理1（二维嵌入三维）: 设 $(a,g)$ 是二维涡流问题 (4)–(5) 的解, 则 $A=(0,0,a)$、$\phi=gz$ 是三维问题 (1)–(2) 在 $\Omega_{3D}=\Omega\times(0,L)$ 与 $\Sigma_{3D}=\Sigma\times(0,L)$ 上的解, 源电流为 $J_s=(0,0,j_s)$. 该定理给出二维模型的严格含义, 但注1指出轴向端面只满足总电流为零, 产生一致性误差. 定理2（线性材料精确性）: 设 $w$ 是二次函数, 若取 $\tilde\nu=\partial_{BB}w(\operatorname{Curl}\hat a)$, 并令 $\tilde a=\hat a+\tilde z$, 其中 $(\tilde z,\tilde g)$ 为线性化问题
$$\sigma(\partial_t\tilde z+\tilde g)+\operatorname{curl}(\tilde\nu\,\operatorname{Curl}\tilde z)=-\sigma\partial_t\hat a\ \text{in}\ \Omega,\qquad \int_\Sigma\sigma(\partial_t\tilde z+\tilde g)\,dx=-\int_\Sigma\sigma\partial_t\hat a\,dx$$的解, 则 $\tilde a=a$、$\tilde g=g$, 即后处理对线性问题精确. 定理3（非线性与牛顿迭代）: 对非线性 $w$, 取同样的 $\tilde\nu$, 则 $(\tilde a,\tilde g)$ 恰为牛顿法求解非线性系统 (4)–(5) 从初值 $(\hat a,0)$ 出发的第一步迭代. 证明基于抽象算子形式 $F(x)=y$, 残差为 $r=(-\sigma\partial_t\hat a,\ -\int_\Sigma\sigma\partial_t\hat a)$, 而雅可比方程 $DF(\hat a,0)v=r$ 正是 (10)–(11). 由此, 该校正方案对线性材料严格精确, 对非线性材料具有明确的迭代解释. 数值实验: 针对永磁同步电机（PMSM）[16], 在 16000 RPM、电流密度 $23.7\,\mathrm{A/mm^2}$ 下, 比较四种模型: M1 完整非线性时间周期模型 (4)–(5), M2 静磁后处理 (7)–(8), M3 局部线性校正 [10,12], M4 本文全局线性校正 (9)–(12). 损耗分别为 926.44、2218.34、803.53、926.92 W, 以 M1 为参考, M2 高估约 139%, M3 低估约 13%, M4 误差低于 0.1%. 计算时间（单核）分别为 250.72、92.30、93.61、118.08 s; 30 核分别为 10.62、4.67、5.89、7.73 s. M4 相比 M2 额外消耗约 30%–60% 时间, 但精度接近 M1; 相比 M1 单核减少约 53% 时间, 30 核减少约 27% 时间.
3. 研究方法、关键技术和核心工具
核心障碍: 标准磁静后处理忽略了涡流反应场, 且局部校正无法捕捉全局返回路径. 本文的创新在于: 从三维模型出发识别反应场的全局性, 提出校正场 $\tilde z$ 满足的线性化涡流问题 (10)–(11), 并将其解释为牛顿法第一步. 证明框架: 将 (4)–(5) 抽象为 $F(x)=y$, 在 $x_0=(\hat a,0)$ 计算残差, 利用静磁方程得出 $r=(-\sigma\partial_t\hat a,\ -\int_\Sigma\sigma\partial_t\hat a)$, 并验证 $DF(\hat a,0)(\tilde z,\tilde g)=r$ 正是 (10)–(11), 从而得到定理2和定理3. 该论证不依赖复杂分析, 纯属代数推导, 但巧妙地将后处理升级为有明确意义的迭代步. 计算工具: 采用经典时间步进加周期循环, 或并行-in-time 的 multigrid-in-time 方法 [14,15] 求解时间周期线性问题. 后者将问题化为一系列可并行求解的广义静磁问题, 在数值测试中表现出良好的并行扩展性. 作者强调校正场无需高精度求解, 只需合理估计即可. 此外, 有限元离散使用最低阶单元, 时间离散用隐式欧拉, 转子运动经谐波 mortar 方法处理.
4. 与之前工作的比较
与标准磁静后处理 [8,9] 相比: M2 仅通过常数 $\hat g$ 强制总电流为零, 未计入反应场, 导致损耗高估超过 100%; 本文 M4 通过全局线性化校正将误差降至 1% 以下, 而额外时间成本仅约 30%–60%. 与局部校正 [10,12] 相比: M3 只在导体区域求解校正, 低估损耗约 13%, 证明反应场是全局的; M4 在全局域求解, 从而捕获完整反应. 与完整非线性时间周期模型 [2,9] 相比: M4 以约一半的单核时间和约 70% 的并行时间达到几乎相同的损耗预测, 显著降低计算成本. 与三维混合方法 [10,11] 相比: 本文在二维情形下给出了线性化校正的牛顿解释, 为这类后处理提供了理论依据, 但三维推广仍开放, 且需额外建模. 本文未进行三维数值比较. 理论局限: 仅保证线性材料精确和对非线性为第一步牛顿迭代, 未给出误差界或收敛性分析; 未讨论 $\tilde\nu$ 的不同选择; 未定量评估二维模型的轴向短路一致误差. 作者明确将三维重建的严格误差估计列为未来工作.
了解领域论文（摘要翻译）
1. 输运主导问题的概率 Gegenbauer 重建的物理信息学习
Physics-Informed Learning of Probabilistic Gegenbauer Reconstruction for Transport-Dominated Problems
作者: Lei Yan, Yan Jiang
类别: math.NA
发布日期: 2026-08-18
arXiv ID: 2608.18001v1
链接: http://arxiv.org/abs/2608.18001v1
摘要翻译
中文标题: 输运主导问题的概率 Gegenbauer 重建的物理信息学习
摘要翻译: 输运主导问题对数据驱动方法仍具挑战性，这些方法由于全局支撑基函数或过于光滑的假设空间，在激波或陡峭梯度附近常表现出严重的数值振荡。Gegenbauer 重建在缓解此类振荡方面已显示出潜力，但其有效性关键取决于重建参数，特别是权重参数 $λ$ 和截断阶数 $m$。对于数据驱动模型，控制方程、训练数据和模型架构的变化使得系统性参数选择尤为困难。为解决这一问题，我们提出了一种物理信息机器学习框架，用于预测候选 Gegenbauer 参数对上的概率分布，从而在考虑参数不确定性的同时实现概率加权重建。采用两阶段策略，先预训练一个通用预测器，再针对目标问题进行微调，以平衡精度和计算成本。该框架分别以 POD-Galerkin 和 DeepONet 为代表的降阶模型和神经算子模型进行评估。在一维和二维输运主导问题上的数值实验表明，该框架能够学习有效的空间自适应参数分布。与传统重建策略相比，它将数值误差降低了一到两个数量级，并实现了比针对特定问题的模型重新训练更优的精度-成本权衡。
2. 保面积参数化：变分原理、梯度流与离散逼近
Area-Preserving Parameterization: Variational Principle, Gradient Flow, and Discrete Approximation
作者: Shu-Yung Liu, Kento Sakai, Mei-Heng Yueh
类别: math.NA
发布日期: 2026-08-18
arXiv ID: 2608.17873v1
链接: http://arxiv.org/abs/2608.17873v1
摘要翻译
中文标题: 保面积参数化：变分原理、梯度流与离散逼近
摘要翻译: 保面积参数化用于必须保持相对表面面积的应用中。我们通过拉伸能量来研究这一问题。对于总面积相等的紧致黎曼二维流形之间的保定向微分同胚，我们证明拉伸能量由面积比的方差刻画，且其临界点是保面积的。这一变分刻画自然导出一个$L^2$-梯度流，我们称之为等面积流。随后，我们基于离散拉伸能量发展其单纯形对应物，并获得针对若干拓扑类型的开曲面和闭曲面的计算方法。为了将离散公式与光滑理论联系起来，我们证明了拉伸能量关于网格细化的相容性达到一阶精度，并在所述几何逼近假设下，建立了离散全局极小化子的一阶$L^2$面积畸变界。在基准网格上的数值实验在所有报告的测试中均产生无折叠映射，并显示出与现有方法相比具有竞争力的面积保持效果。
3. 分段光滑非均匀介质中Lippmann–Schwinger方程的快速高阶求解器
Fast high-order solvers for the Lippmann–Schwinger equation in piecewise-smooth heterogeneous media
作者: Thomas G. Anderson, Juan Burbano-Gallegos, Luiz M. Faria, Carlos Pérez-Arancibia
类别: math.NA, physics.comp-ph
发布日期: 2026-08-18
arXiv ID: 2608.17615v1
链接: http://arxiv.org/abs/2608.17615v1
摘要翻译
中文标题: 分段光滑非均匀介质中Lippmann–Schwinger方程的快速高阶求解器
摘要翻译: 本文提出了一种快速、高阶的Nyström求解器，用于求解由可穿透、分段光滑非均匀介质中的时谐散射产生的二维Lippmann–Schwinger方程。该方法依赖于在不连续界面上自适应非结构化网格上的牛顿势高阶求积，结合快速多极子等现有快速算法，实现了高阶精度。作为一种迭代方法，该求解器与一类结构化网格求解器——提供拟线性时间和内存复杂度以及几乎恒定迭代次数的快速求解方法，但精度长期受限——所构成的预处理策略相结合时，表现出快速收敛。该预处理策略通过一对传递算子，将Nyström离散（由符合空间变化对比度支撑域的弯曲非结构化网格上的高阶求积节点给出）与底层快速预处理器所采用的均匀笛卡尔网格联系起来。由此得到的预处理器继承了笛卡尔对应物的频率鲁棒性，同时不牺牲非结构化离散的几何灵活性和高阶精度。我们证明了在网格尺寸和笛卡尔预处理器满足显式条件时，所提出的预处理器是可逆的。数值实验表明，预处理系统的GMRES迭代次数远少于未预处理系统，迭代次数几乎与网格尺寸和波数无关，并展示了该方法对具有分段光滑折射率和跨界面跳跃间断的非均匀介质的鲁棒性。
4. 狄拉克源的混合有限元方法：散度形式分裂与 L^p 误差分析
Mixed Finite Element Methods for a Dirac Source: Divergence-Form Splitting and L^p Error Analysis
作者: Yueyao Wu, Shun Zhang
类别: math.NA
发布日期: 2026-08-18
arXiv ID: 2608.17575v1
链接: http://arxiv.org/abs/2608.17575v1
摘要翻译
中文标题: 狄拉克源的混合有限元方法：散度形式分裂与 L^p 误差分析
摘要翻译: 对于混合有限元方法，狄拉克源首先是对偶性的失败，而非正则性的失败：守恒方程以 Lebesgue 空间为测试空间，而狄拉克测度不属于任何一个 Lebesgue 空间的对偶。因此，我们通过散度形式分裂将测度从守恒律中移除。一个其散度等于狄拉克测度的显式场被从物理通量中减去，修正后的通量被取为混合未知量，从而守恒方程中仅保留载荷的正则部分。等价地，并且独立于任何离散化，狄拉克问题被改写为数据呈散度形式的椭圆方程，该数据由某个 L^p 场生成。被减去的场仅依赖于源的位置，而不依赖于系数。不需要依赖于系数的奇异解，也不需要离散狄拉克，仅需改变 RT_0-P_0 系统的载荷向量，且源可以位于相对于网格的任何位置：在顶点上、在系数界面上或在单元内部。除非分裂在源处与算子匹配，否则修正后的通量对每个 p<2 都属于 L^p，但不在 L^2 中，因此通量误差分析必须离开 Hilbert 尺度。我们证明了通量的拟最优逼近界，并由此证明在拟一致网格族上，通量误差恰好是 h^(2/p-1) 阶：匹配的下界已经来自承载源的单个单元。在该处对网格进行加密可恢复一阶复杂度，即关于单元数 N 的 N^(-1/2) 阶，自适应计算达到了这一阶。标量变量仅受解的逐片常数近似的限制，并且达到了该近似。我们还证明了 Lebesgue 尺度下的残差范数等价性，得到一个可计算的 L^p 估计子，对混合通量以及恢复势可靠且局部有效。
不相关论文（基本信息）
1. 有序扩散核
Ordered Diffusion Kernels
作者: Jack H. Soulsby, Andreas C. S. Jørgensen, Atiyo Ghosh, Vahid Shahrezaei
类别: math.NA, math.DS
发布日期: 2026-08-18
arXiv ID: 2608.18019v1
链接: http://arxiv.org/abs/2608.18019v1
摘要: We introduce Ordered Diffusion Kernels (ODKs), a novel class of local kernels that can approximate the infinitesimal generator of an arbitrary Itô Stochastic Differential Equation (SDE). ODKs are designed to be applied to data sampled from dynamical systems where little dynamical information is available a priori. The Laplacian of classical diffusion kernels approximates the Laplace-Beltrami operator on the underlying manifold; adjusting the normalisation introduces an advection term that depends on the sampling density; recently, TMDmap generalised this normalisation to target an arbitrary measure, but at the cost of coupling advection to diffusion. More general local kernels can learn arbitrary second-order elliptic operators but are formulated in terms of known velocity fields — making the first step of any analysis a potentially ill-posed inference problem. To formulate ODK, we first relax the problem of potential estimation to the more tractable task of inferring an ordering of the data, which we represent through an ordering function. We prove ODK’s Laplacian converges to the infinitesimal generator of a gradient-flow SDE with state-dependent isotropic diffusion, without coupling advection and diffusion. We provide various extensions of ODK to: arbitrary drifts via local ordering functions; anisotropic diffusions via a Strang splitting scheme; multiple ordering functions; and self-tuning bandwidths. In addition, we introduce two loss functions which exploit the structure of ODKs to solve a non-parametric inference problem. We validate this framework on synthetic data from deterministic and stochastic systems, demonstrating accurate recovery of operators, velocity fields, extrinsic curvature, and spatially dependent drift and diffusion coefficients.
2. 从复步进微分到一般重构框架
From complex-step differentiation to a general reconstruction framework
作者: Rafael Abreu, Chahana Nagesh
类别: math.NA, physics.geo-ph
发布日期: 2026-08-18
arXiv ID: 2608.17924v1
链接: http://arxiv.org/abs/2608.17924v1
摘要: The complex-step method is traditionally derived from the Taylor expansion of an analytic function and it is widely used as a numerical technique for derivative approximation. We present an alternative formulation based on the Cauchy–Riemann equations. We show that the classical complex-step approximation arises naturally from the harmonic structure of holomorphic functions and formulate the corresponding boundary reconstruction problem in the upper half-plane. This construction naturally leads to the Poisson, Hilbert, and Cauchy kernels as the elementary reconstruction operators for harmonic and holomorphic functions. Extending the boundary reconstruction from ordinary functions to finite measures on obtains the classical Stieltjes transform and its inversion formula. We further show that the same reconstruction principle naturally extends to spectral theory, where scalar matrix elements of the resolvent are Stieltjes transforms of the associated spectral measures. This provides a direct connection between the complex-step method, Stieltjes inversion, resolvent methods, and semiclassical analysis, where the same complex perturbation underlies the recovery of spectral information. Finally, we discuss an FFT-based implementation for the numerical evaluation of the required analytic continuation.
3. 基于Brézis-Ekeland-Nayroles原理的次梯度流的基于对偶的 $\textit{A Posteriori}$ 误差恒等式
Duality-Based $\textit{A Posteriori}$ Error Identities for Subgradient Flows Based on the Brézis-Ekeland-Nayroles Principle
作者: Harbir Antil, Alex Kaltenbach, Keegan L. A. Kirk
类别: math.NA, math.AP, math.FA, math.OC
发布日期: 2026-08-18
arXiv ID: 2608.17845v1
链接: http://arxiv.org/abs/2608.17845v1
摘要: We derive duality-based $\textit{a posteriori}$ error identities for a broad class of subgradient flows induced by time-dependent convex integral functionals. Starting from the Brézis-Ekeland-Nayroles principle, we identify an unsteady primal energy functional and derive its Fenchel dual formulation, including strong duality and the corresponding optimality system under general normal-integrand assumptions. This Fenchel duality framework is used to derive $\textit{a posteriori}$ error identities for subgradient flows. In doing so, we depart from the usual duality-based $\textit{a posteriori}$ error control framework in the unsteady setting, since the Brézis-Ekeland-Nayroles formulation reveals the following unsteady feature: the minimal primal value and the maximal dual value are both prescribed by the initial datum. This allows us to pass from a combined primal-dual gap identity to separate primal and dual gap identities. These identities quantify the primal and dual errors independently and admit representations in terms of generalized Bregman divergences and, under a spatial convex conjugation formula, as non-negative time-space integral quantities suitable for localization. The abstract framework is applied to a number of variational problems of physical interest, including the unsteady heat equation, the unsteady Stokes equations, the unsteady Navier-Lamé equations, the unsteady Bingham flow through a pipe, the unsteady obstacle problem, and the unsteady elasto-plastic torsion problem.
4. 大有限阵列天线中有源阻抗渐近性质的研究
Investigation of the Asymptotic Properties of Active Impedance in Large Finite Array Antennas
作者: Harald Hultin, Lucas Åkerstedt, B. L. G. Jonsson
类别: physics.comp-ph, math.NA
发布日期: 2026-08-18
arXiv ID: 2608.17844v1
链接: http://arxiv.org/abs/2608.17844v1
摘要: This paper presents an improved full-wave solver for finite array antennas, and uses this solver to examine asymptotic properties of active impedance for large regular arrays. The improved solver is based on a preconditioning scheme that has been adapted for use on more general finite geometries, and an improved data structure. These two improvements result in a fast and stable solver with a lower memory footprint. By investigating the active impedance of finite arrays it is found that, even for arrays with 1000 elements, asymptotic behavior may differ from infinite arrays. Tied to these properties, different predictors on how the active impedance of the center element in the array behaves are presented. The two best predictors work very well for the wideband arrays investigated, and may be used to decide when array approximations, such as unit cell methods, are appropriate, rather than general statements on array size.
5. 关于鲁棒α阻尼粘性格式
On Robust Alpha-Damping Viscous Scheme
作者: Hiroaki Nishikawa
类别: math.NA, physics.comp-ph
发布日期: 2026-08-18
arXiv ID: 2608.17664v1
链接: http://arxiv.org/abs/2608.17664v1
摘要: In this paper, we investigate the convergence of an implicit defect-correction solver for a viscous discretization based on the alpha-damping scheme for unstructured grids. We show that significantly more robust iterative convergence is achieved by evaluating the damping term at the midpoint between two adjacent cell centers (edge midpoint) rather than at the face centroid. A one-dimensional Fourier analysis reveals that the implicit solver tends to be stable when the damping term in the residual is smaller than that used to construct the Jacobian. This observation suggests that the solver can be stabilized by effectively reducing the magnitude of the damping term in the residual - an effect achieved by the edge-midpoint evaluation. Robust convergence is demonstrated numerically for two-dimensional viscous-flow problems on highly irregular mixed-element and triangular grids.
6. 慢变化解析含时哈密顿量的量子模拟
Quantum simulation of slow analytic time-dependent Hamiltonians
作者: Chenhao Zhao, Yinan Li, Dong An
类别: quant-ph, math.NA
发布日期: 2026-08-18
arXiv ID: 2608.17653v1
链接: http://arxiv.org/abs/2608.17653v1
摘要: We develop a quantum algorithm for slow analytic Hamiltonians $\widetilde H(t)=H(t/T)$ with $\|H(s)\|\leqα$ that achieves nearly additive query complexity and low gate overhead. Our main technical contribution is a periodic Gevrey extension of $H(s)$, together with Fourier component decay and truncation bounds that enable an efficient finite-dimensional simulation. Combined with Floquet embedding and optimal time-independent Hamiltonian simulation technique, this gives query complexity $\widetilde{\mathcal O}\!\left(αT+\log(1/\varepsilon)\right)$ and additional gate complexity $\widetilde{\mathcal O}\!\left((αT+\log(1/\varepsilon))^2\log(1/\varepsilon)\right)$, assuming coherent access to $H'(s)$ and endpoint derivatives. For slow analytic control Hamiltonians, only block encodings of the time-independent control operators are required, with the same query complexity and lower gate overhead. Our method also extends to Gevrey Hamiltonians and improves the precision dependence for simulating slow analytic semi-dissipative linear differential equations.
7. COS-TT-CHF：用于多资产期权定价的张量列特征函数COS方法
COS-TT-CHF: A Tensor-Train Characteristic-Function COS Method for Multi-Asset Option Pricing
作者: Lucas Arenstein, Michael Kastoryano
类别: q-fin.CP, math.NA, q-fin.PR
发布日期: 2026-08-18
arXiv ID: 2608.17636v1
链接: http://arxiv.org/abs/2608.17636v1
摘要: This paper considers European multi-asset option pricing under Lévy and affine characteristic-function models. The main obstruction is the curse of dimensionality: direct multidimensional COS pricing forms tensor-product coefficient arrays whose size grows exponentially with the number of assets. We study and extend COS-TT-CHF, a low-rank construction that uses TT-cross to compress sampled characteristic-function tensors into tensor-train COS coefficients for arithmetic basket and min/max option pricing. Once built, the compressed representation gives fast post-setup strike-grid and selected component Delta/Vega calculations. The numerical study compares with adaptive-quadrature Fourier benchmarks, direct COS, a tensor-Fourier min-option benchmark, and quasi-Monte Carlo (QMC) references based on randomized Sobol points. The reported timings show a low-dimensional crossover against direct COS as the benchmark moves from $d=2$ to $d=4$, favorable timings against the tensor-Fourier min-option benchmark from $d=3$ onward, and favorable timings against the QMC common-Heston reference already at $d=2$. The reported tests reach $d=30$ for GBM and $d=20$ for VG, NIG, and common-Heston benchmark families, with accuracy, rank, runtime, control-sensitivity, and component Delta/Vega diagnostics reported throughout.
8. 基于相位引导的Bregman TV正则化的联合近场全息层析重建
Joint Near-Field Holotomography Reconstruction with a Phase-Guided Bregman TV Regularization
作者: Jin Liu, Johannes Hagemann, Martin Burger
类别: math.NA
发布日期: 2026-08-18
arXiv ID: 2608.17621v1
链接: http://arxiv.org/abs/2608.17621v1
摘要: Near-field holotomography combines coherent diffraction imaging with tomographic acquisition to recover the three-dimensional complex refractive index of a specimen. Since the measured diffraction intensities are generated by a nonlinear object transmission and wave propagation process, the resulting inverse problem is intrinsically nonlinear and ill-posed. We study a direct variational reconstruction framework based on a fully nonlinear wave propagation model that avoids both intermediate phase retrieval and linearization under the weak-object approximation. We analyze the forward operator in appropriate Banach spaces, establish its Fréchet differentiability, and derive explicit gradient expressions for variational reconstruction. To improve quantitative reconstruction of weak absorption features, we further develop a phase-guided Bregman TV regularization framework that exploits structural correlations between phase and absorption components. This enables multi-material reconstruction without imposing a globally fixed ratio between the two components. We perform numerical studies on synthetic phantoms and experimental data. The results demonstrate stable three-dimensional reconstructions and improved recovery of the absorption contrast compared to state-of-the-art methods.
9. 含动态毛细压力的两相流模型有限体积格式的能量耗散与稳定性
Energy dissipation and stability of a finite-volume scheme for two-phase flow models with dynamic capillary pressure
作者: Ansgar Jüngel, Josipa-Pina Milišić, Sara Xhahysa
类别: math.NA
发布日期: 2026-08-18
arXiv ID: 2608.17537v1
链接: http://arxiv.org/abs/2608.17537v1
摘要: An implicit Euler finite-volume scheme for degenerate pseudo-parabolic cross-diffusion equations is proposed and analyzed. The system describes the dynamics of an unsaturated two-phase flow mixture with dynamic capillary pressure in a porous medium. The numerical scheme is based on a two-point flux approximation that preserves the energy structure, ensures the conservation of total mass, and guarantees strict positivity and boundedness of the water saturation. These properties rely on carefully selected mean functions for the nonlinear components. The existence and uniqueness of a discrete solution and additional mesh-uniform bounds are proved. Numerical experiments in two space dimensions illustrate the effects of the dynamic capillary pressure.
10. 非自反Banach空间上统计反问题的正则化
Regularization of Statistical Inverse Problems on Non-Reflexive Banach Spaces
作者: Darrel K Joseph, M P Rajan
类别: math.NA, math.FA, math.ST, stat.ML
发布日期: 2026-08-18
arXiv ID: 2608.17533v1
链接: http://arxiv.org/abs/2608.17533v1
摘要: Inverse learning within a statistical framework has a wide range of applications. It has garnered significant attention in machine learning, artificial intelligence, and related fields, where the goal is to infer unknown parameters from indirect and noisy observations. This work investigates the stable approximation of $u^{\dagger}$ which solves the equation $Au=g$, with $A$ being a linear operator between appropriate vector spaces. We will consider the domain to be a non-reflexive Banach Space and the co-domain to be a space of real-valued functions on a metric space $X$. The function $g$ is characterized by a finite number of independently and identically distributed data points, which are assumed to follow some unknown probability measure $ρ$. We employ Tikhonov regularization with an arbitrary convex functional to obtain the regularized solution corresponding to the given data point. The convergence analysis is carried out with respect to the Bregman distance, and an upper bound for the error is derived in probability terms. The theoretical findings are then supported by numerical experiments.
11. 多层移动最小二乘逼近的改进收敛性
Improved Convergence of Multilevel Moving Least-Squares Approximation
作者: Robert Durst, Holger Wendland
类别: math.NA
发布日期: 2026-08-18
arXiv ID: 2608.17441v1
链接: http://arxiv.org/abs/2608.17441v1
摘要: Moving least-squares approximation is a popular method for approximating multivariate functions from given discrete data. For higher accuracy higher degree polynomials have to be used, resulting also in higher computational cost and numerical instabilities. Recently, the combination of low-order moving least squares with a multilevel scheme showed superior numerical behavior. In this paper we will prove, amongst other things, that such a combination of moving least-squares with a multilevel scheme indeed leads to improved convergence results, at least if the data sites form a regular grid.
12. 用于具有构型相关动能的耗散系统的投影半显式积分器：接触-赫尔格洛茨表述与基准测试
A Projected Semiexplicit Integrator for Dissipative Systems with Configuration-Dependent Kinetic Energy: Contact-Herglotz Formulation and Benchmarks
作者: Lorena Loera-Galeana, Santiago Mejía, Espartaco Alvarado, Héctor Medel-Cobaxin
类别: math-ph, math.DS, math.NA
发布日期: 2026-08-17
arXiv ID: 2608.17198v1
链接: http://arxiv.org/abs/2608.17198v1
摘要: Contact Hamiltonian dynamics gives dissipative mechanics an intrinsic action variable, but explicit contact splittings reach only kinetic energies whose terms are exactly integrable: frozen-coordinate diagonal metrics (the spherical pendulum, a torus particle) are included, while dense metrics with momentum cross terms, with the double pendulum as flagship, are not. We introduce a projected Pihajoki-contact integrator for this non-separable setting, combining phase-space duplication, symmetric projection onto the physical diagonal, and constant-friction damping half-steps, with the action factor carried by an exact Herglotz update. As in the projected extended-phase-space framework it builds on, the construction needs no binding parameter, returns the copies to the diagonal at every step, and confines the nonlinear solve to the $2n$ projection variables. For constant friction the step rescales $ω=dη$ by the exact factor $e^{-γτ}$ when the projection is solved exactly (a classical conformally symplectic identity, realized here for this class), while time-symmetry, consistency, and smoothness yield an $O(τ^3)$ one-step contact-form residual, a bound not specific to the contact form. On the damped double pendulum, spherical pendulum, and torus particle the method is second-order accurate, reproduces the contact decay law, and controls long-time energy and contact drift in coarse or stiff regimes where the Tao baseline and the unprojected average lose the solution. A head-to-head with exact-contactomorphism splittings delimits the niche: where a frozen-coordinate splitting exists it preserves the contact form exactly and wins at matched cost; for the dense double-pendulum metric the realizable alternative is first-order with a prohibitive constant and the projected method prevails. The contact-form estimate is local, one-step, and constant-friction.
13. 稳健投影分裂龙格-库塔积分器（二阶和三阶）
Robust Projector-Splitting Runge-Kutta Integrators of Orders Two and Three
作者: Shiheng Zhang
类别: math.NA
发布日期: 2026-08-17
arXiv ID: 2608.17157v1
链接: http://arxiv.org/abs/2608.17157v1
摘要: Dynamical low-rank approximation requires time integrators that remain accurate in the presence of small singular values. We construct robust projector-splitting Runge–Kutta methods of orders two and three. Their central feature is a common-base stage construction: every internal stage and the endpoint are obtained by applying the practical projector-splitting algorithm of Lubich and Oseledets to a Runge–Kutta increment, always from the factors and row space at the beginning of the time step. Under uniform boundedness, Lipschitz, smoothness, and normal-component assumptions, every computation in which the projector-splitting factorizations have rank $r$ has local errors $C(h^3+h\varepsilon_r)$ and $C(h^4+h\varepsilon_r)$, and global errors $C(δ+\varepsilon_r+h^2)$ and $C(δ+\varepsilon_r+h^3)$, for the midpoint and third-order methods, respectively. Here $\varepsilon_r$ bounds the normal component of the vector field and $δ$ is the initial error. The constants are independent of small singular values. Every stage and output has rank $r$, with the original basis width retained throughout the calculation.
14. 随机化拟蒙特卡洛积分
Randomized quasi-Monte Carlo integration
作者: Art B. Owen
类别: stat.CO, math.NA
发布日期: 2026-08-17
arXiv ID: 2608.17143v1
链接: http://arxiv.org/abs/2608.17143v1
摘要: Quasi-Monte Carlo sampling is a numerical integration method that uses points with a space-filling property in $[0,1]^s$ designed to give better estimates than plain Monte Carlo methods do. For integrands of bounded variation in the sense of Hardy and Krause, errors of $O(n^{-1+ε})$ for any $ε>0$ are obtained from $n$ sample points. Randomized quasi-Monte Carlo (RQMC) points are individually uniformly distributed but collectively space-filling and then independent replications provide variance estimates. For smooth enough integrands the randomization can give a root mean squared error of $O(n^{-3/2+ε})$. This article explains RQMC for a statistical readership recounting some history and presenting some current directions.
15. 基于小波的 $\ell_1$ 正则化图像去模糊多层框架
Wavelet-based multilevel framework for $\ell_1$-regularized image deblurring
作者: Danyh Tolah, Malena I. Español, Misha E. Kilmer
类别: math.NA
发布日期: 2026-08-17
arXiv ID: 2608.17123v1
链接: http://arxiv.org/abs/2608.17123v1
摘要: Solving large-scale $\ell_1$-regularized image deblurring problems efficiently while preserving sharp edges remains a significant computational challenge. We propose a wavelet-based multilevel framework that embeds three iterative solvers, Iteratively Reweighted Least Squares (IRLS), Split Bregman (SB), and Majorization-Minimization (MM), within a multilevel V-cycle. Discrete wavelet transforms define the interlevel transfer operators, and regularization parameters are selected automatically by Generalized Cross Validation. Two information transfer strategies are introduced and compared: one transfers only the coarse solution to the fine level, while the other transfers solver-specific auxiliary quantities. Numerical experiments demonstrate substantial computational savings for IRLS, with speedups exceeding an order of magnitude, while MM and SB exhibit more modest computational differences. The experiments generally show that transferring auxiliary iterates performs best with Haar wavelets, whereas transferring only the solution performs best with Daubechies wavelets.
16. 改进的Arrow-Hurwicz方法用于稳态无感应磁流体动力学系统
Improved Arrow-Hurwicz method for Stationary Inductionless Magnetohydrodynamics System
作者: Duygu Uludag, Fatma G. Eroglu, Aziz Takhirov, Songul Kaya
类别: math.NA
发布日期: 2026-08-17
arXiv ID: 2608.17035v1
链接: http://arxiv.org/abs/2608.17035v1
摘要: In this work, we propose a new Arrow-Hurwicz iterative scheme designed to solve the steady inductionless magnetohydrodynamics system. The main feature of the proposed scheme is the introduction of new penalty terms in the current density equation. These terms play a central role in effectively controlling the unfavorable mixed terms that commonly arise in such formulations. The proposed method is shown to achieve geometric convergence. Numerical tests affirm the efficiency of the new scheme without compromising accuracy.
17. 从有限DFT样本中恢复整数信号：格方法与稳定性分析
Recovery of Integer Signals from Limited DFT Samples: Lattice Methods and Stability Analysis
作者: Howard Levinson, Isaac Viviano
类别: math.NA
发布日期: 2026-08-17
arXiv ID: 2608.17024v1
链接: http://arxiv.org/abs/2608.17024v1
摘要: We analyze lattice-based algorithms for recovering integer-valued signals from partial discrete Fourier transform (DFT) measurements. These algorithms formulate signal recovery as the problem of finding short vectors in an appropriately constructed lattice. We derive parameter estimates that guarantee successful recovery and quantify how these estimates depend on the signal length, the error of an initial guess, and the number of sampled DFT coefficients. The analysis characterizes the stability of the inversion algorithms, as the lattice parameters are closely related to the required measurement precision. Numerical experiments demonstrate close agreement between the theoretical predictions and observed recovery thresholds over a broad range of problem parameters.