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1. 基于Young测度线性规划形式的非线性和随机均匀化量子算法
Quantum Algorithm for Nonlinear and Stochastic Homogenization via a Young-Measure based Linear Programming Formulation
作者: Siqi Chen, Shi Jin, Lei Zhang
类别: math.NA
发布日期: 2026-06-04
arXiv ID: 2606.06165v1
摘要翻译: 我们研究基于Young测度的线性规划（LP）形式的非线性和随机均匀化量子算法，该形式通过将微尺度、梯度和可能的随机变量视为独立变量，将非线性问题提升到更高维度的线性问题，从而捕获有效的宏观量，而无需直接解析细尺度振荡。生成的LP规模大但有结构，其高维特性创造了量子LP求解器优于直接经典求解器的情形：在确定性情形下，当适中的均匀化精度足够时，多项式量子加速出现；在随机情形下，将所有随机实现同时编码到单个LP中，产生随随机变量数量增长的随机采样成本的量子平方根缩减。Young测度的正则性或稀疏性可能进一步将这些优势扩展到精细尺度精度。在一维和二维基准上的数值实验证实了Young测度LP形式的正确性。
Comment: 无
链接: http://arxiv.org/abs/2606.06165v1
1. 研究对象和背景
本文研究非线性椭圆偏微分方程在周期性和随机介质中的均匀化问题，并发展基于Young测度的线性规划量子算法。具体考虑两类模型方程。
一、非线性周期性均匀化：设$\Omega \subset \mathbb{R}^d$为有界区域，$Y = (0,1)^d$为周期单元，对$\varepsilon > 0$，考虑
$$-\nabla\cdot a\left(\frac{x}{\varepsilon}, \nabla u_\varepsilon(x)\right) = f(x) \quad \text{in } \Omega, \quad u_\varepsilon = 0 \quad \text{on } \partial\Omega,$$其中$a(y,\xi)$关于$y$是$Y$-周期的，并满足单调$L^p$条件：存在$p>1$及常数$c_0, c_1>0$，使得
$$a(y,\xi)\cdot\xi \ge c_0|\xi|^p - c_1, \quad |a(y,\xi)| \le c_1(1+|\xi|^{p-1}), \quad (a(y,\xi)-a(y,\eta))\cdot(\xi-\eta) \ge 0.$$在这些假设下，解序列$\{u_\varepsilon\}$在$W^{1,p}_0(\Omega)$中一致有界，从而子序列弱收敛到某个$u_0$，该极限满足均匀化方程
$$-\nabla\cdot a^{\hom}(\nabla u_0) = f \quad \text{in } \Omega.$$有效算子$a^{\hom}$通过单元问题定义：对每个$\xi \in \mathbb{R}^d$，求$w_\xi \in W^{1,p}_{\#}(Y)$满足$\int_Y w_\xi = 0$及
$$-\nabla_y\cdot a(y, \xi+\nabla_y w_\xi(y)) = 0 \quad \text{in } Y,$$则$a^{\hom}(\xi) = \int_Y a(y, \xi+\nabla_y w_\xi(y))\,\mathrm{d}y$。
二、随机均匀化：设概率空间$(\Theta, \mathcal{F}, \mathbb{P})$带有保测遍历群作用$\tau_x$，随机单调算子$a(\omega, \xi)$满足平稳性。考虑
$$-\nabla\cdot a\left(\tau_{x/\varepsilon}\omega, \nabla u_\varepsilon(x,\omega)\right) = f(x) \quad \text{in } \Omega, \quad u_\varepsilon = 0 \quad \text{on } \partial\Omega.$$在平稳性、遍历性、一致单调性和多项式增长条件下，对几乎处处的$\omega$，$u_\varepsilon(\cdot,\omega)$弱收敛到确定性极限$u_0$，满足$ -\nabla\cdot a^{\hom}(\nabla u_0) = f $。本文主要处理有限状态随机介质：$\omega_i \in \Theta$，$i=1,\dots,N$，对应概率$p_i$，算子$a_i(\xi) = a(\omega_i, \xi)$。
直接数值求解这些问题需在细尺度$\varepsilon$上离散，经典复杂度为$\tilde O(\varepsilon^{-d})$，在$\varepsilon$很小时计算代价极高。为避开直接解析微观振荡，本文利用Young测度框架：引入概率测度$\mu_{x,y}$（周期情形）或$\mu_x$（随机情形），它们描述微观梯度的局部分布，从而将原始非线性PDE转化为关于测度的线性约束优化问题。约束包括归一化（总质量守恒）、重心一致性（宏观梯度等于微观梯度的平均）和弱力平衡（散度形式的宏观方程）。若本构关系具有位势，则该问题等价于能量最小化，且目标函数关于测度是线性的。离散化后，该问题成为一个大规模但结构化的线性规划（LP）。这种提法将非线性困难转移到高维线性结构上，而量子线性规划求解器有望处理这类高维LP并实现加速。
本文试图填补的空白是：将量子计算应用于非线性和随机均匀化问题，通过Young测度LP重整化，在不直接求解细尺度PDE的情况下提取有效宏观量，并严格分析量子算法相对于经典算法的优势条件。
2. 主要定理或主要结果
本文在将Young测度线性规划离散化后，分析了量子中心路径（QCP）算法的复杂度，并与直接经典求解器（DCS）及经典LP方法比较，得到以下主要定理和结论。
确定性非线性均匀化的量子优势：设空间维度为$d$，微观尺度为$\varepsilon$，目标精度$\delta = \varepsilon^\alpha$（$\alpha \in (0,1)$反映均匀化收敛速率）。采用一阶代数离散，直接经典求解器（DCS）的查询复杂度为$\tilde O(\varepsilon^{-d})$；而QCP求解Young测度LP的复杂度为$\tilde O(R_1 \varepsilon^{-\alpha(3d+4)/2})$，其中$R_1$是中心路径长度。当
$$\alpha < \frac{2d}{3d+4}$$时，QCP相对于DCS具有多项式加速。此结果说明了在中等均匀化精度需求下（即$\alpha$较小时），量子算法有优势。
随机均匀化情形：随机变量个数为$r$，总随机离散点数$N_\omega^r$看作与$\varepsilon$独立的自由参数，目标精度取$\delta = \varepsilon$。DCS查询复杂度为$\tilde O(N_\omega^r \varepsilon^{-d})$；QCP复杂度为$\tilde O(R_1 (N_\omega^r)^{1/2} \varepsilon^{-(3d/2+1)})$。当
$$N_\omega^r \gtrsim \varepsilon^{-(d+2)}$$时，QCP在随机抽样成本上实现平方根加速，且加速因子随$N_\omega^r$增长而增大。
猜想（解析正则性下的精细精度优势）：若Young测度密度在$x$和$y$上解析，则可采用谱离散化替代代数离散。在此情形下，对一切$d \ge 2$，确定性情形下可将量子优势拓展到精细精度$\delta = \varepsilon$，即QCP复杂度为$\tilde O(\varepsilon^{-(3d/2+1)})$，而DCS仍为$\tilde O(\varepsilon^{-d})$，当$d\ge 2$时QCP总是更优；随机情形也有类似改进。这一猜想指出高阶正则性能显著增强量子优势。
数值验证：作者对一维和二维的确定性、随机基准问题进行了数值实验，证实了Young测度LP公式的正确性（即验证了线性规划重构的有效性，而非直接测试量子算法）。具体案例包括：一维线性均匀化（谐振平均）、一维非线性$p$-Laplacian型均匀化、以及二维随机均匀化等。实验表明LP解与理论均匀化解吻合良好。
3. 研究方法、关键技术和核心工具
本文的核心创新点在于将非线性/随机均匀化问题提升为Young测度上的线性规划，并设计量子中心路径算法求解该LP，从而实现相对于经典多尺度直接求解器的加速。关键技术障碍与解决方法分述如下。
- Young测度LP重构： 将原PDE的解$u_\varepsilon$的微观振荡信息编码为概率测度$\mu$。在周期情形，定义$\mu_{x,y} \in \mathcal{P}(\mathbb{R}^d)$为宏观点$x$、微观相$y$处的梯度分布；在随机情形，定义$\mu_x \in \mathcal{P}(\Theta \times \mathbb{R}^d)$。通过引入这些测度作为独立变量，原非线性PDE转化为以下线性约束：
- 归一化：$\int_{\mathbb{R}^d} \mathrm{d}\mu_{x,y}(\xi) = 1$（或随机情形$\int_{\mathbb{R}^d} \mathrm{d}\mu_{x,i}(\xi) = p_i$）；
- 重心一致性：$\nabla u(x) = \int_Y \int_{\mathbb{R}^d} \xi \,\mathrm{d}\mu_{x,y}(\xi)\,\mathrm{d}y$（或随机情形$\nabla u(x) = \sum_{i=1}^{N_\omega} \int_{\mathbb{R}^d} \xi \,\mathrm{d}\mu_{x,i}(\xi)$）；
- 弱力平衡：$-\nabla\cdot \left(\int_Y \int_{\mathbb{R}^d} a(y,\xi) \,\mathrm{d}\mu_{x,y}(\xi)\,\mathrm{d}y \right) = f(x)$（或随机情形类似）；
- 边界条件。 若本构关系有势，目标函数也为$\mu$的线性泛函。因此，离散化后得到标准形式的LP：$\min c^\top z \; \text{s.t.} \; A_{\mathrm{eq}}z = b_{\mathrm{eq}}, \; F \ge 0$，其中向量$z = [u; F]$包含节点值和测度质量分量。这一重构将非线性困难转移到高维线性结构上，使得量子线性规划求解器得以应用。
离散化与LP规模： 对空间变量$x$、微观变量$y$（周期情形）、梯度变量$\xi$进行离散，网格点数分别记为$n_x, n_y, n_\xi$。为保证均匀化精度$\delta$，通常取$n_x \sim \delta^{-d}$，$n_y \sim \delta^{-d}$，$n_\xi \sim \delta^{-d}$，故LP变量总数约为$n_x n_y n_\xi = O(\delta^{-3d})$，约束数约为$O(\delta^{-2d})$。然而，LP具有很强的稀疏性和块结构：归一化和重心一致性约束仅涉及局部的测度变量；弱平衡条件涉及宏观节点邻域的少量测度。这为量子算法的加速提供了可能。若测度密度具有解析性，可采用谱离散化，此时规模的对数依赖可进一步降低。
量子中心路径（QCP）算法框架： 采用量子线性规划求解器中的中心路径法。将原始LP嵌入齐次自对偶嵌入（HSDE），然后沿着中心路径迭代更新。每一步需要求解一个牛顿线性系统，其系数矩阵具有分块稀疏结构。作者利用量子线性系统求解器（如QLSA）来高效求解该牛顿系统，每次求解的查询复杂度为$\tilde O(\sqrt{n_{\mathrm{var}}}/\kappa)$的量级（忽略对数因子），其中$\kappa$为条件数。总迭代次数与中心路径长度$R_1$成正比。结合LP的稀疏性和矩阵块结构，最终得到总查询复杂度分别为$\tilde O(R_1 \varepsilon^{-\alpha(3d+4)/2})$（确定性情形的代数离散）和$\tilde O(R_1 (N_\omega^r)^{1/2} \varepsilon^{-(3d/2+1)})$（随机情形的代数离散）。
优势分析的关键不等式： 通过比较QCP复杂度与DCS复杂度$\tilde O(\varepsilon^{-d})$，得出需要满足的条件。确定性情形的优势条件$\alpha < 2d/(3d+4)$源于求解稠密LP的典型量子平方根加速，并结合了离散化带来的额外维度因子。随机情形中，$N_\omega^r$独立于$\varepsilon$，其平方根加速在小$\varepsilon$时占主导，因此条件$N_\omega^r \gtrsim \varepsilon^{-(d+2)}$相对容易满足。
数值验证部分： 作者并未在本文中实现量子算法，而是用经典LP求解器（如Gurobi）求解离散化的Young测度LP，验证了所得均匀化系数或宏观解与理论预测一致。例如一维线性情形得到谐振平均，$p$-Laplacian情形得到与理论一致的$k_{\hom}$。这些实验证实了Young测度LP公式本身的有效性。
4. 与之前工作的比较
本文在均匀化问题的量子算法、Young测度数值方法以及非线性随机均匀化理论之间进行了交叉，现将其与相关工作的比较分述如下。
与经典均匀化数值方法的比较： 直接经典多尺度求解器（DCS）需要在细尺度$\varepsilon$解析微结构，例如用有限元方法在尺度$\varepsilon$的网格上求解原始PDE，其计算代价至少为$\tilde O(\varepsilon^{-d})$。当$\varepsilon$很小时计算量巨大。本文提出的Young测度LP方法避免直接解析$\varepsilon$尺度的振荡，代之以在高维空间（$x, y, \xi$或$x, \omega, \xi$）上求解线性规划。在经典计算机上，这种LP的规模本身也随目标精度$\delta$增长（$O(\delta^{-3d})$），若采用经典内点法，复杂度约为$\tilde O(\delta^{-4.5d})$，甚至比DCS更差。因此经典LP并无优势。本文的核心突破在于引入量子中心路径算法，利用量子线性系统求解器的平方根加速，使得总复杂度降至$\tilde O(\delta^{-(3d+4)/2})$（确定性）和$\tilde O((N_\omega^r)^{1/2} \delta^{-(3d/2+1)})$（随机），从而在一定参数区域超越DCS。与经典LP内点法相比，量子算法实现了维数上的平方根加速，这是本质性的改进。
与已有量子多尺度算法的比较： Hu等人[28]提出了线性多尺度PDE的量子算法，基于直接振荡模型或均匀化方程结合Schrödingerization技术。他们的工作针对线性方程，且利用尺度分离直接构建有效方程，量子加速体现在利用量子线性系统求解器处理均匀化后的大规模线性系统。本文则处理非线性问题，其均匀化不能仅靠系数平均，必须通过Young测度捕捉振荡的分布性质。Balazi等人[7]提出了基于局部正交分解（LOD）的量子增强数值均匀化方法，利用量子局部求解器实现对数尺度的精细分辨率。但他们的方法同样局限于线性问题。本文首次将量子算法拓展到非线性和随机均匀化领域，核心思路是通过Young测度将非线性转化为线性，再用量子LP求解器处理。与上述工作相比，本文不要求底层PDE是线性的，但代价是需要求解一个高维LP而非线性方程组。
与Young测度数值方法的比较： Young测度在非线性PDE数值计算中已被用于松弛、凸化、多尺度模型降阶等[13, 33]。传统做法是采用原子测度近似、自适应离散、或神经网络参数化。这些方法均在经典计算机上执行，难以处理高维（尤其是包含随机参数）情形下的组合爆炸。本文的贡献在于提出用量子计算加速Young测度的高维线性规划，使得测度离散化可以直接在乘积空间上进行，而无需降阶或稀疏假设。但是，量子算法要求LP具有稀疏性和良好的条件数，这依赖于离散方式。本文讨论了一阶代数离散和谱离散（需解析性假设）两种情形。
关于“注意”与“猜想”的说明： 论文明确区分了严格证明的定理结果和基于附加正则性假设的猜想。定理（主要结果1和2）仅依赖于Young测度存在性和有限维LP的构造，且优势区域是精确给出的。猜想1假设Young测度密度解析，从而可采用谱离散化大幅减少变量个数，优势区域可扩展至$\delta = \varepsilon$对所有$d\ge 2$。该猜想基于对谱方法收敛速度的期望，但并未在本文中证明。
局限与代价：
- 量子优势仅在特定参数区域成立：确定性情形要求$\alpha < 2d/(3d+4)$，即中等精度需求；随机情形要求$N_\omega^r$足够大。若$\alpha$接近1（需要非常精细的均匀化解），或$N_\omega^r$较小，量子算法可能不如DCS。
- 量子算法的成功依赖于LP的稀疏性；若测度离散导致稠密约束矩阵，加速可能消失。
- 目前量子计算硬件尚处于早期，本文的分析基于量子查询复杂度的理论估计，实际实现仍有距离。
- 与经典LP相比，QCP要求每次迭代求解量子线性系统，其成功概率和误差管理带来额外对数因子。
- 本文仅讨论了少数量子优势情形，并未证明所有情形下量子算法都优于最优经典算法。
2. 声学逆Born散射的直接重建
Direct reconstruction for acoustic inverse Born scattering
作者: Nuutti Hyvönen, Lisa Schätzle
类别: math.NA
发布日期: 2026-06-04
arXiv ID: 2606.05977v1
摘要翻译: 我们考虑二维亥姆霍兹方程的逆介质散射问题，即从已知的全远场数据或等价的远场算子恢复紧支撑可穿透二维散射体。虽然该问题唯一可解，但它是严重不适定且非线性的。在弱散射情形下，Born近似给出了对比度与远场数据之间的线性化关系，从而克服了第二个困难。这一线性框架允许我们基于最近关于线性化电阻抗层析成像的工作（其中依赖三角Zernike分解），导出一个显式重建公式，将对比度的展开系数用远场数据的展开系数表示出来。通过适当选择展开函数，所得系统矩阵解耦为对比度中空间角频率的独立（无限）三角系统。因此，每个系统均可通过前向代入独立求解。数值实验表明，即使应用于超越Born近似的完全非线性远场数据，这种方法结合适当的正则化方法仍然有效。
Comment: 无
链接: http://arxiv.org/abs/2606.05977v1
1. 研究对象和背景
本文研究二维时间谐波声波逆介质散射问题，声波入射到具有未知折射率分布的紧支集穿透性散射体上时，其总声场 $u(x,d)$ 满足 Helmholtz 方程
$$ \Delta u(\cdot,d)+\kappa^2(1+q)u(\cdot,d)=0 \quad \text{in } \mathbb{R}^2, $$其中 $\kappa>0$ 为波数，入射场为平面波 $u^i(x,d)=e^{i\kappa x\cdot d}$，$d\in S^1$ 为入射方向。实值对比度函数 $q\in L^\infty(\mathbb{R}^2)$ 满足 $q>-1$ a.e. 且支集 $\Omega\subset\mathbb{R}^2$ 紧；散射场 $u^s=u-u^i$ 需满足 Sommerfeld 辐射条件。远场模式 $u^\infty(\hat{x},d)$ 由散射场的渐近展开定义
$$ u^s(x,d)=\frac{e^{i\pi/4}}{\sqrt{8\pi}}\frac{e^{i\kappa r}}{\sqrt{\kappa r}} u^\infty(\hat{x},d)+\mathcal{O}(r^{-3/2}),\quad r=|x|\to\infty, $$并可通过积分表示
$$ u^\infty(\hat{x},d)=\kappa^2\int_\Omega q(y) u(y,d) e^{-i\kappa \hat{x}\cdot y}\,dy,\qquad \hat{x}\in S^1. $$于是可定义远场算子 $F:L^2(S^1)\to L^2(S^1)$ 为 $(Fg)(\hat{x})=\int_{S^1} u^\infty(\hat{x},d) g(d)\,ds(d)$，它是一个 Hilbert–Schmidt 算子。逆散射的目标是由已知的 $F$（或等价地，全远场数据 $u^\infty(\hat{x},d)$ 对所有 $\hat{x},d\in S^1$）恢复对比度 $q$。该问题理论上唯一可解，但高度非线性且严重不适定——多次散射导致非线性依赖，且 $F$ 的特征值迅速衰减使反演对噪声极其敏感。在弱散射情形，即积分算子 $L_q$ 的范数很小时，可仅保留 Neumann 级数前两项得到散射体的线性化模型，称为 Born 近似，此时远场模式简化为对比度的二维 Fourier 变换在圆盘上的取样：
$$ u_B^\infty(\hat{x},d)=\kappa^2\int_\Omega q(y) e^{-i\kappa(\hat{x}-d)\cdot y}\,dy, $$相应的 Born 远场算子 $F_B$ 仍为 Hilbert–Schmidt。尽管 Born 近似消除了非线性，其反演依然不适定（$F_B$ 的特征值衰减速度与原问题相同）。
已有的逆散射重构方法可分为两范：定性方法（如线性抽样法 [1,7]、因子分解法 [16,25] 等）仅恢复散射体的形状或支撑集；定量方法则试图重建完整的对比度函数，例如基于迭代优化或对线性化模型直接求逆 [10,21,30]。最近，文献 [30] 提出一种低秩方法，将远场数据在广义长椭球波函数下展开以利用其低秩结构。本文受近期线性化电阻抗成像（EIT）中基于三角 Zernike 分解的直接重建法 [2,14] 的启发，旨在为二维逆 Born 散射发展同样拥有显式重构公式的算法。具体地，假设对比度的支集包含于某个已知球 $B_R(c)$ 内，将远场算子视为 $L^2(S^1)$ 上的 Hilbert–Schmidt 算子，并对 $L^2(B_R(c))$ 和 $HS(L^2(S^1))$ 选择合适的基，使正演映射具有结构化稀疏性。通过将对比度和远场数据展开于适当的角 Fourier 基及径向基，可发现不同的空间角频率在远场数据中被完全解耦：对比度的第 $j$ 个角频率仅影响数据矩阵的第 $-j$ 对角线。为使径向分支的系统矩阵呈现下三角形式——从而可通过简单的前向代入求解对比度系数——本文的关键创新在于径向基的构造：不采用 EIT 中的 Zernike 多项式（对 Helmholtz 核不适用），而是对 Bessel 函数乘积 $J_m(\kappa R\,r)J_{m-|j|}(\kappa R\,r)$ 施行 Gram–Schmidt 正交化。这一选择自然导致各角频率对应一个无限维下三角系统，进而可导出对比度系数的显式递推公式。
本文试图填补的空白即：将 EIT 中成功的三角分解直接重建架构移植到声学逆 Born 散射，并提供完整的理论分析和可稳定实现的数值策略，包括基于 Bessel 函数超指数衰减的低秩截断以及利用数据对称性进行降噪。数值实验进一步表明，即使将本文基于 Born 近似推导的公式直接应用于非线性全波远场数据，配合适当正则化后仍能获得有效重建，这大大扩展了方法的实用范围。
2. 主要定理或主要结果
假设波数 $\kappa>0$ 和包含对比度支集的先验 ROI 球 $B_R(c)$ 已知。定义 $L^2(B_1(0))$ 的规范正交基 $\{\Psi_{j,k}\}_{j\in\mathbb{Z},k\in\mathbb{N}_0}$ 为
$$ \Psi_{j,k}(y)=e_j(\hat{y})\,R_k^{|j|}(|y|),\qquad y\in B_1(0), $$其中 $\{e_j\}_{j\in\mathbb{Z}}$ 是 $L^2(S^1)$ 的标准 Fourier 基，而径向函数 $\{R_k^{|j|}\}_{k\in\mathbb{N}_0}$ 由对序列 $\{P_m^{|j|}(r)=J_m(\kappa R\,r)J_{m-|j|}(\kappa R\,r)\}_{m\ge |j|/2}$ 在加权空间 $L^2_r(0,1)$（内积 $\langle f,g\rangle_{L^2_r(0,1)}=\int_0^1 f(r)\overline{g(r)}\,r\,dr$）中执行 Gram–Schmidt 正交化得到（见定义 3.3）。记 Born 远场算子 $F_B$ 在调制 Fourier 基 $\{e_m^c\}_{m\in\mathbb{Z}}$（$e_m^c(\phi)=e^{-i\kappa c\cdot\phi}e_m(\phi)$）下的矩阵元素为
$$ a_{m,n}=\langle F_B e_n^c, e_m^c\rangle_{L^2(S^1)},\qquad m,n\in\mathbb{Z}, $$则根据 (3.1) 有
$$ a_{m,n}=2\pi\kappa^2 i^{n-m}\int_{B_R(0)} q(y+c) e^{i(n-m)\arg y} J_m(\kappa|y|) J_n(\kappa|y|)\,dy. $$本文主要定理 3.4 表明：对比度平移缩放 $q(R\cdot+c)$ 在 $\{\Psi_{j,k}\}$ 下的展开系数 $c_{j,k}=\langle q(R\cdot+c),\Psi_{j,k}\rangle_{L^2(B_1(0))}$ 可由 $a_{m,n}$ 按如下显式递推公式逐一解出（对每个 $j\in\mathbb{Z}$ 独立计算）：
$$ c_{j,k}= \frac{1}{(2\pi)^{3/2}(\kappa R)^2 (-i)^j \|\widetilde{R}_k^{|j|}\|_{L^2_r(0,1)}} \left( a_{k+\lceil j/2\rceil,\,k-\lfloor j/2\rfloor} - \sum_{i=0}^{k-1} \langle R_i^{|j|}, P_{k+\lceil |j|/2\rceil}^{|j|}\rangle_{L^2_r(0,1)} \,c_{j,i} \right),\quad k\in\mathbb{N}_0, $$其中 $\widetilde{R}_k^{|j|}$ 是 Gram–Schmidt 过程中得到的未归一化正交函数（$R_k^{|j|}=\widetilde{R}_k^{|j|}/\|\widetilde{R}_k^{|j|}\|$），求和项在 $k=0$ 时视作 0。
这一结果源于每个角频率 $j$ 对应的径向重建问题等价于一个下三角的无限线性系统：
$$ F^j c_j = a_j, $$其中向量 $c_j=(c_{j,0},c_{j,1},\ldots)^\top$，$a_j=(a_{\lceil j/2\rceil,\lfloor -j/2\rfloor}, a_{\lceil j/2\rceil+1,\lfloor -j/2\rfloor+1},\ldots)^\top$，下三角矩阵 $F^j$ 的元素为
$$ F^j_{m,k}= \begin{cases} (2\pi)^{3/2}(-i)^j(\kappa R)^2 \langle R_{k-1}^{|j|}, P_{(m-1)+\lceil |j|/2\rceil}^{|j|}\rangle_{L^2_r(0,1)} , & k\le m,\\ 0 , & \text{其他}. \end{cases} $$(见 (3.19))。因此，只要测得了数据矩阵 $(a_{m,n})$，便可先由上述递推计算所有 $c_{j,k}$，再通过 $q(Ry+c)=\sum_{j\in\mathbb{Z}}\sum_{k=0}^\infty c_{j,k}\Psi_{j,k}(y)$ 合成对比度。递推所需的内积与范数均可在线下阶段预先计算并存储。
此外，利用数据固有的对称性 $a_{-m+j,-m}=(-1)^j a_{m,m-j}$，可将递推中的 $a_{k+\lceil j/2\rceil,\,k-\lfloor j/2\rfloor}$ 替换为平均形式 $\tilde{a}_{m,m-j}=(a_{m,m-j}+(-1)^j a_{-m+j,-m})/2$，从而提高信噪比。由于 Bessel 函数 $J_m$ 随阶数 $m$ 超指数衰减，数据矩阵仅有有限个元素显著非零，自然引出截断策略：只需保留前 $N$ 个对角线条目并求解相应大小的三角系统，即可稳定地获得可恢复的分辨率。
3. 研究方法、关键技术和核心工具
本文的技术核心在于通过巧妙构造基函数，将 Born 正演算子的反演转化为一系列解耦下三角系统的直接求解，避免了迭代和大型矩阵求逆。其主要创新与关键工具分为以下三个层次。
1. 基于调幅 Fourier 基的角频率解耦 从 Born 远场模式 (2.6) 出发，对平面波因子应用 Jacobi–Anger 展开
$$ e^{-i\kappa(\hat{x}-d)\cdot y} = 2\pi \sum_{m,n\in\mathbb{Z}} i^{n-m} e^{i(n-m)\arg y} J_m(\kappa|y|) J_n(\kappa|y|) e_m(\hat{x}) e_n(d), $$并引入调制 Fourier 基 $e_m^c(\phi)=e^{-i\kappa c\cdot\phi}e_m(\phi)$，可将 Born 远场算子的矩阵元 $a_{m,n}$ 表为 (3.1) 的形式，其中明显出现了角频率依赖因子 $e_{m-n}(\hat{y})$。于是，当固定 $j=m-n$ 时，$a_{m,m-j}$ 仅由对比度的傅里叶系数 $\widehat{q}_{-j}$ 相关的积分决定，即实现了不同角频率的完全解耦。这一观察虽然简单，却是后续构造下三角结构的前提。注意此解耦不依赖于任何基的特殊选择，而仅由 Helmholtz 核的平移不变性和平面波展开决定，与 [17] 中远场数据稀疏性的分析相容。
2. 径向基的构造与下三角化 要使每个角频率 $j$ 对应的径向反演变为可直接递推求解的下三角系统，关键在径向基函数的选择。定义 $P_m^{|j|}(r)=J_m(\kappa R\,r)J_{m-|j|}(\kappa R\,r)$，这些函数直接出现在 $a_{m,m-j}$ 的表达式中（对比 (3.7)）。命题 3.2 证明了 $\{P_m^{|j|}\}_{m\ge |j|/2}$ 在 $L^2_r(0,1)$ 中是线性无关的，且其张成子空间稠密。线性无关性通过考察 $r\to 0$ 时的渐近行为 $P_m^{|j|}(r)\sim r^{2m-|j|}$ 得到；稠密性则利用反证法——若某个函数 $\rho$ 垂直所有 $P_m^{|j|}$，构造对比度 $q(Ry+c)=e_j(\hat{y})\rho(|y|)$，根据解耦结论它产生零远场数据，从而由逆问题唯一性推出 $\rho=0$。
在此基础上，定义 $\{R_k^{|j|}\}_{k\in\mathbb{N}_0}$ 为对 $\{P_m^{|j|}\}$ 按 $m$ 递增顺序进行 Gram–Schmidt 正交化的结果（定义 3.3）。这一过程确保了当 $k>m-\lceil |j|/2\rceil$ 时 $\langle R_k^{|j|}, P_m^{|j|}\rangle_{L^2_r(0,1)}=0$，即矩阵 $[\,\langle R_k^{|j|}, P_m^{|j|}\rangle\,]_{k,m}$ 为下三角。将此性质代入 (3.10)，即得下三角系统 (3.17)。
3. 显式递推公式的推导与数值稳定化 下三角系统可通过前向代入求解，最终得到显式递推 (3.21)。该公式中的分母为 $\|\widetilde{R}_k^{|j|}\|_{L^2_r(0,1)}$，恰为 Gram–Schmidt 过程中自然产生的量，从而无需额外计算。所有内积和范数均可离线计算并存储，在线重建仅涉及线性组合，计算量极低——本质上是对应于 QR 分解的 $R$ 矩阵的求逆。
然而 Gram–Schmidt 过程在 $k$ 增大时会出现数值不稳定，因为 $P_m^{|j|}$ 中包含的 Bessel 函数随阶数超指数衰减，导致高阶 $\widetilde{R}_k^{|j|}$ 的范数极小而极易受舍入误差影响。解决策略是采用截断正则化：由于远场数据矩阵的非零结构同样受 Bessel 函数衰减控制（仅有前若干对角线显著），故可自然选取截断参数 $N$，只生成前 $N$ 个基函数，对应的三角系统大小也为 $N\times N$，从而避免了不稳定区间。这一截断建立在文献 [17,18,20] 关于远场数据稀疏性结果之上，并与低秩方法 [30] 的出发点相似，但实现方式不同。
附加的创新点还包括对称性的利用：通过 (3.9) 的平均公式可降低数据噪声，且不改变方程组的解。整体构建的证明框架是：解耦 → 基选择 → 下三角化 → 递推求解 → 截断稳定，逻辑清晰且每个步骤均有严格线性代数与泛函分析支撑。
4. 与之前工作的比较
本文在理论推导、算法框架和适用范围上与既有研究存在多方面关联与推进。
1. 与线性化 EIT 直接重建方法的比较 [2,14] 本文的核心灵感来自 Gardner 等 [2] 和 Hanke 等 [14] 在电阻抗成像（EIT）中发展的三角 Zernike 分解法。EIT 的线性化模型（Calderón 问题）同样可解耦角频率，并使用径向 Zernike 多项式将系统矩阵下三角化，从而得到显式系数递推。但将这一框架搬至逆 Born 散射时，积分核完全不同：EIT 中的核涉及超几何函数乘积，而此处的核为 $J_m(\kappa R\,r)J_{m-|j|}(\kappa R\,r)$。直接套用 Zernike 多项式无法使矩阵三角化。本文的关键推进在于识别出应以 $J_m J_{m-|j|}$ 自身作为“种子”进行 Gram–Schmidt 正交化，从而构造出适用于 Helmholtz 核的径向基，并严格证明了这样得到的基使系统矩阵化为下三角。此外，EIT 相关文献未讨论 Gram–Schmidt 过程中因核函数衰减引发的数值不稳定及截断问题，而本文对此进行了详细分析并给出了稳定性指导。在求解方式上，两种方法均归结为下三角系统的前向代入，但本文的递推公式 (3.21) 利用了 Gram–Schmidt 中间量，形式上更为精简。
2. 与低秩逆散射方法 [30] 的比较 文献 [30] 提出利用广义长椭球波函数展开远场数据，利用其低秩特性进行反演。本文同样依靠低秩性质（源自 Bessel 函数的超指数衰减），但采取了截然不同的基函数与算法：我们使用三角基（Fourier + Gram–Schmidt 基），将反演转化为解小型三角系统，而非特征分解。从计算效率看，本文方法的 QR 分解部分可完全离线，在线阶段仅为递推代入，计算量远小于特征分解或迭代优化。数值实验中作者将本文方法与 [30] 进行了直接比较，初步结果显示本文方法在重构质量和速度上均有竞争力，但具体数值细节在摘要中未展开。值得注意的是，本文重构公式是在 Born 近似下严格推导的；然而实验观察到该方法“即使应用于 Born 区域外的全非线性远场数据”，结合正则化后依然有效。这暗示本文方法或许对中等非线性的散射也有实用价值，可能超越 [30] 仅分析线性模型的适用范围（若 [30] 主要针对 Born 情形）。
3. 与傅里叶重构（如 NUFFT）的比较 公式 (2.6) 表明 Born 远场模式等同于对比度在圆盘 $B_{2\kappa}(0)$ 上的非均匀 Fourier 采样（见图 2.1）。因此，一种自然的反演思路是使用非均匀 FFT（NUFFT）将数据重网格化后再做逆 Fourier 变换。本文在第 5 节数值实验中与 MATLAB 内置 NUFFT 方法对比，后者的局限性在于需要插值且通常伴随着伪影；而本文方法通过构造正交基下的展开，无需处理非均匀采样，直接给出了比直接 NUFFT 更精确的重建，且避免了非凸优化。
4. 与常规迭代反演方法的差异 经典定量逆散射反演多采用高斯–牛顿等迭代法，每次迭代需解大型、不适定线性系统，计算成本极高。本文算法提供了一种直接重建，既可用于 Born 数据一步到位，也可为非线性反演提供优良的初始猜测。其代价是依赖先验信息（已知支集球 $B_R(c)$）和波数 $\kappa$，且仅保证弱散射下的精确性。与近期结合深度学习的物理驱动方法相比，本文方法具有严格的数学推导和可解释性，且不依赖训练数据。
5. 理论视角的推进 本文将逆 Born 散射的“显式反演公式”这一目标从 EIT 成功推广到 Helmholtz 方程，并解决了径向基存在的理论障碍（命题 3.2）。对比关于逆散射唯一性、稳定性等性质的众多文献（如 [24]），本文首次给出了基于特殊基分解的直接代数重建途径，并且论证了该基在无限维意义下确实构成 $L^2$ 的规范正交基，使得重构具有完备的理论框架。不足之处在于，基的构造依赖于波数 $\kappa$ 和 ROI 半径 $R$ 的精确知情，若这些参数估计不准，重建质量会下降；另外 Gram–Schmidt 过程的稳定性仍受 Bessel 函数衰减制约，需要妥善选取截断参数，这在一定程度上限制了可恢复的分辨率。
3. 任意维数下H^k型张量积空间的加权hp一致分解
Weighted hp-Uniform Decompositions for H^k-Type Tensor-Product Spaces in Arbitrary Dimension
作者: Situan Li, Weiying Zheng
类别: math.NA
发布日期: 2026-06-04
arXiv ID: 2606.05615v1
摘要翻译: 我们针对任意空间维数 $d \ge 1$，在拟合界面网格上，为 $H^k$ 型协调与非协调空间的张量积离散建立了加权 $hp$ 一致的顶点块分解，其中 Sobolev 阶 $k \ge 1$ 任意固定。单元为坐标兼容长方体，局部空间为 $Q_{p_K}(K)$，单元次数满足 $p_K \ge 2k-1$，且系数在材料界面处可有任意大的跳跃。在局部系数振荡界和局部高侧连通条件下，协调 $H^k$ 空间和非协调空间 $V_h^{(s)}$（$0 \le s \le k$）均容许稳定分解，其常数可能依赖于固定参数 $d$ 和 $k$，但与网格尺寸、所有多项式次数、相邻次数比以及全局系数反差无关。该论证结合了 $0,\ldots,k-1$ 阶端点射流的 Hermite 端点变换、其张量积延拓、加权破碎块 Poincaré 不等式以及法向导数跳跃的连续校正。针对具有大系数跳跃和强变化局部多项式次数的三维 DG 问题的数值实验支持了预测的鲁棒性。对于 $k=1$，在一致正则映射立方体网格上（其相邻单元映射在每个公共面上一致）可得到相同的结论。
Comment: 20 pages, 2 tables
链接: http://arxiv.org/abs/2606.05615v1
1. 研究对象和背景
在 d 维有界多面体区域 Ω 上, 考虑扩散系数 a(x) 分片连续的拟合界面问题. 具体地, 区域被剖分为 M 个互不相交的开材料子区域 Ω = ∪{m=1}^M Ω_m, 在每个子区域内 a 恒为常数 a_m(或连续有界), 但跨界面可以任意跳跃. 网格 T_h 是由坐标兼容的正交长方体组成的协调网格: 每个单元 K 经仿射映射与参考立方体相连, 相邻单元的面坐标一致, 且横向坐标方向平行于物理法向(Assumption 2.1). 局部多项式空间采用张量积多项式 Q{p_K}(K), 其中 p_K 是单元次数, 满足 p_K ≥ 2k-1, k 为固定的 Sobolev 指标(k ≥ 1). 定义 H^k 型协调空间 V_h^{(k)} = {v ∈ H^k_0(Ω) : v|K ∈ Q{p_K}(K)}, 以及非协调空间 V_h^{(s)}(0 ≤ s ≤ k)为分段 Q_{p_K} 且满足 H^s_0(Ω) 协调的函数空间. 特别地, 低阶协调空间 V_0^{(k)} 的每个单元上多项式次数取为 2k-1.
稳定分解是 hp 有限元方法理论中的基本工具, 它将有限元空间表示为若干子空间(通常索引于网格实体或节点)的直和或非直和, 并保证函数可以分解为这些子空间分量的叠加, 且各分量范数平方和与原函数范数等价. 在 hp 自适应计算中, 稳定分解是构造可扩展区域分解前条件, 证明 hp 版本插值误差估计以及分析非协调方法(如间断 Galerkin 方法)收敛性的核心引理. 分解常数若依赖于网格尺寸 h 或局部多项式次数 p, 则可能导致迭代次数随问题规模增长; 若进一步依赖于系数对比度, 则在实际高对比度问题中前条件的效率会急剧恶化. 因此, 构建对 h, p 和系数同时鲁棒的分解意义重大.
然而, 在本文研究之前, 对于 H^k 协调及非协调张量积空间的 hp 稳定分解存在两个主要困难. 其一, 高阶协调性(k ≥ 2)要求跨单元面匹配直到 k-1 阶的法向导数. 将函数局部化到顶点片时, 必须设计满足这些迹条件的局部子空间; 若处理不当, 则会引起界面跳跃, 破坏协调性或范数等价性. 其二, 对于带有高对比度系数的能量泛函, 标准加权的 Poincaré 不等式常数通常随局部系数的反差增长, 若直接用于分解稳定性, 将导致常数取决于全局系数比. 如何在保证高阶迹结构的同时引入系数鲁棒的估计, 是尚未解决的问题.
本文在前人工作的基础上, 针对拟合界面网格, 在任意空间维数 d ≥ 1 和任意 Sobolev 阶数 k ≥ 1 下, 建立了两个关键假设: 局部系数振荡有界(2.1)和高侧连通性(2.2). 前者要求每个单元上系数变化有限, 后者则要求每个 patch 内高系数区域通过面连接链与一个高侧锚单元相通, 且链上系数拟单调增. 在上述假设下, 作者证明了加权顶点片分解的存在性, 且分解常数只依赖于 d, k 和局部几何/系数参数, 不依赖于全局系数对比度. 这填补了 hp 理论中一项重要空白.
2. 主要定理或主要结果
主要结果(完整定理):
设 d ≥ 1, k ≥ 1, 网格满足坐标兼容性(Assumption 2.1), 系数满足(2.1)与(2.2), 并满足 Assumption 2.2(局部高侧连通性). 则存在仅依赖于 d, k, 形状正则常数 γ, C_{cell}, C_{reg}, N_0 和 C_{qm} 的常数 C, 使得以下成立.
对任意 0 ≤ s ≤ k, 令 V_h^{(s)} 为如上定义的空间. 对于任意 v ∈ V_h^{(s)}, 存在分解 v = v_0 + Σ_{z ∈ V_h} v_z, 其中 v_0 ∈ V_0^{(k)}, v_z ∈ V_h^{(s)}, 且 supp v_z ⊆ ω_z.
(i) 若 s = k(即 v 在 H^k_0 协调空间), 则有加权半范稳定性: |v_0|{a,k}^2 + Σ{z ∈ V_h} |v_z|{a,k}^2 ≤ C |v|{a,k}^2. (1) 这里 |v|{a,k}^2 = Σ{K ∈ T_h} \bar{a}K |v|{H^k(K)}^2, 而 \bar{a}_K 是单元代表系数(如单元平均).
(ii) 若 0 ≤ s ≤ k, 则对于由(2.7)定义的非协调加权范数 ||·||{b,s,k,h}, 有 ||v_0||{b,s,k,h}^2 + Σ_{z ∈ V_h} ||v_z||{b,s,k,h}^2 ≤ C ||v||{b,s,k,h}^2, (2) 其中 ||v||{b,s,k,h}^2 = Σ{K ∈ T_h} \bar{a}K |v|{H^k(K)}^2 + Σ_{F ∈ F_h} Σ_{ℓ=s}^{k-1} η_{F,ℓ} J_{F,ℓ}(v)^2, J_{F,ℓ}(v)^2 = Σ_{q=0}^{ℓ} Σ_{|α_τ|=ℓ-q} ||D_τ^{α_τ} J_F^q(v)||{L^2(F)}^2, 面子数 b_F 可取为 b_F^{max} = max{ \bar{a}{K^+}, \bar{a}{K^-} } 或调和平均 b_F^{har} = 2\bar{a}{K^+}\bar{a}{K^-} / (\bar{a}{K^+} + \bar{a}{K^-}); 惩罚权重 η{F,ℓ} 是满足下界条件(2.8)和嵌套关系(2.9)的任意正数. 例如最小选择 η_{F,ℓ}^{min} = b_F h_F^{-2(k-ℓ)+1}. 常数 C 独立于 h, 所有局部多项式度 p_K, 相邻度比 p_K / p_{K’}, 以及全局系数对比度 max \bar{a}_K / min \bar{a}_K.
当 k = 1 时, 上述结论可推广至一致正则的映射长方体网格, 只要相邻单元在公共面上具有相同的映射(参见第 6.2 节).
3. 研究方法、关键技术和核心工具
证明的主要技术障碍在于: 既要将任意函数 v 分解到支集限于顶点片的局部函数, 又要保证它们满足 H^s 协调性所要求的跨面法向迹连续条件, 同时确保加权稳定性常数对系数跳跃完全鲁棒. 作者为此设计了三个相互配合的核心工具.
一维 Hermite 端点变换及其张量积扩展. 参考文献区间 I = (0,1). 选取特定多项式 ψ_k ∈ P_{2k-2}(I) 满足矩条件 ∫0^1 s^r ψ_k(s) ds = 1, r = 0,…,k-1, 且 ψ_k^{(j)}(1) = 0, j = 0,…,k-2. 基于 ψ_k 定义算子 A_0 和 A_1: (A_0 u)(t) = ∫0^1 ψ_k(s) u(st) ds, (A_1 u)(t) = ∫0^1 ψ_k(s) u(1 - s(1-t)) ds. 它们具有端点导数保持性质: (A_0 u)^{(r)}(0) = u^{(r)}(0), (A_1 u)^{(r)}(1) = u^{(r)}(1), r = 0,…,k-1. 利用 Hermite 基函数 ρ{i,r}^{(k)} ∈ P{2k-1} 校正另一端的导数, 得到算子 C_0, C_1, C⋆: C_0 u = A_0 u - Σ_{r=0}^{k-1} λ_{0,r}(u) ρ_{1,r}^{(k)}, C_1 u = A_1 u - Σ_{r=0}^{k-1} λ_{1,r}(u) ρ_{0,r}^{(k)}, C_⋆ u = u - C_0 u - C_1 u, 其中 λ_{i,r} 为适当的泛函. 由构造, 它们满足(Lemma 3.1): Tr_i^k(C_j u) = δ_{ij} Tr_i^k u, i,j ∈ {0,1}, 而 Tr_0^k(C_⋆ u) = Tr_1^k(C_⋆ u) = 0. 这些算子是 H^k(I) 到自身的有界线性的, 且保持 ≥ 2k-1 次多项式不变(Lemma 3.2). 通过张量积, 在参考立方体(0,1)^d 上定义实体算子 B_σ = C_{σ_1} ⊗ … ⊗ C_{σ_d}, σ ∈ {0,1,⋆}^d. 每一个参考实体 ^E_σ 对应一个算子. 将其拉回到物理单元 K 得到 B_K^E. 关键性质(Lemma 3.4): 对于单元 K 的面 F, 若实体 E ⊆ F, 则法向导数 ∂{n_F}^q B_K^E w 在 F 上等于该面实体分量 B_F^E (∂{n_F}^q w); 若 E ⊄ F, 则 ∂_{n_F}^q B_K^E w = 0, q = 0,…,k-1. 这意味着实体分量天然地分离了法向迹: 只有属于该面的实体分量才在该面上有非零法向导数. 这为按实体逐层修正跳跃提供了代数基础.
加权破碎片 Poincaré 不等式与高侧连通性路径. 为了控制系数跳跃, 作者引入了局部拟单调链条件(Assumption 2.2). 在每个实体 S 的 patch ω_S 中, 存在一个高侧锚单元 K_S^+, 使得任意单元 K 均可经过至多 N_0 个面连接到达它, 且链路上系数满足 \bar{a}{K_j} ≤ C{qm} \bar{a}{K{j+1}}. 这一条件保证了沿链面子数 b_{F_j} 不低于 \bar{a}{K_j}(见(2.5)). 对于在该 patch 内满足前 r 阶法向跳跃为零的分片 H^k 函数 w, 定义投影 P{S,r}^a w = Π_{K_S^+} w(Π_K 为单元 K 上的 k-1 次平均泰勒投影). 要估计 w - P_{S,r}^a w 的加权低阶导数. 利用链式分解: 任取单元 K_0, 将 Π_{K_0} w - Π_{K_S^+} w 写作沿链相邻单元投影差之和. 对每一相邻对应用两单元转移引理(Lemma 4.2), 该引理基于有限维面到体估计(Lemma 4.1)和投影误差的迹估计(4.2), 将投影差用源单元和目标单元的 H^k 半范及它们之间面的跳跃 J_{F,ℓ}(w)(ℓ ≥ r)控制. 由于上述链上系数关系, 源单元的系数 \bar{a}{K_0} 被面子数 b{F_j} 一路控制, 从而整个加权和最终由 Σ_{K⊂ω_S} \bar{a}K |w|{H^k(K)}^2 与 Σ_{F∈F_h(S)} Σ_{ℓ=r}^{k-1} b_F h_F^{-2(k-ℓ)+1} J_{F,ℓ}(w)^2 控制, 得到加权破碎片 Poincaré 不等式(4.5). 此估计常数的系数鲁棒性正是依赖于链条件所确保的 \bar{a}_{K} ≲ b_F 关系.
连续修正法向跳跃的分解构造. 具体构造时, 先利用 Hermite 插值在顶点处定义低阶函数 v_0 ∈ V_0^{(k)}, 使余量 w = v - v_0 在顶点处的所有直到 k-1 阶导数均为零. 由此 w 在每个面内部的法向跳跃从低阶开始逐步修正. 作者按实体维数从高到低(例如, 先在内部面, 后在边, 最后在顶点)循环: 在每一步, 针对实体 S, 利用加权 Poincaré 不等式构造一个支集在 ω_S 内, 且法向导数跳跃恰好能抵消当前 w 在 S 上相应跳跃的修正项. 因为实体算子的迹分离性质, 加入此修正项不会破坏已处理的高维实体上的零跳跃状态. 最终将各修正项与 v_0 重新组合为顶点片分量 v_z. 范数稳定性由 Poincaré 不等式和迹估计联合导出, 且由于 Poincaré 估计中系数的处理完全基于局部链, 不引入全局对比度依赖.
这三者的有机结合, 攻克了高对比度系数下任意阶 hp 分解的难题.
4. 与之前工作的比较
与已有文献的详细比较:
系数鲁棒性与高侧连通性假设. 经典 hp 分解研究(如[6]中的 DG 子空间分解,[7]中的内罚项分析)通常假定系数为常数或光滑, 或者即使允许跳跃, 其稳定常数也包含全局对比度因子. 在区域分解或预条件上下文中, 如果能量范数中使用调和平均或类似加权, 常数往往随系数比恶化. 本文首次在张量积设置下, 通过 Assumption 2.2(高侧连通性)实现了完全对比度独立的常数. 代价是网格必须拟合界面, 并且 patch 内的系数链必须拟单调. 这在实际问题中对应于: 在每一种材料内的单元片内部, 系数变化有限; 且如果存在高传导区域, 它必须通过一连串面连接的单元与 patch 内最高系数锚单元相连, 中间不能有"低传导瓶颈"阻断. 作者在 Remark 4.5 中明确指出, 若缺少此连通条件, 则低阶粗空间加一层顶点片的分解无法实现常数独立, 从而说明该假设是必要而非仅是方便.
任意 Sobolev 阶数与维度的推广. 已有工作大多局限于 k = 1(H^1 协调或完全间断)或 k = 2(H^2 协调板问题). 对于更高阶协调空间, 由于法向迹的阶数增加, 实体局部化的困难急剧上升. Karakashian 和 Pascal[3]曾研究矩形网格上 H^k 协调元的基, 但并未建立 hp 稳定分解, 尤其未涉及系数. 本文利用 Hermite 变换的张量积本质, 将构造系统推广到了任意 k≥1 和任意 d≥1, 且常数仅显式依赖于 d 和 k. 对于 k ≥ 2, 要求物理网格为正交长方体, 以保证法向量与张量坐标对齐; 对于 k = 1, 仅需值连续性, 故可推广至映射长方体网格, 但需要相邻单元映射在公共面上相一致(第 6.2 节). 这一延伸保留了张量积坐标的局部结构.
多项式次数要求与相邻度比独立性. 分解要求 p_K ≥ 2k-1, 这与标准 Hermite 插值的最低次数要求一致, 此前 hp 分析中也常采用(如内罚 DG 的惩罚项[2,4]). 本文进一步证明了稳定常数不依赖于相邻单元的次数比 p_K / p_{K’}, 这对于自适应计算中单元间次数大幅度变化的情形尤为重要. 之前的许多研究成果仅对全局一致次数成立, 或在常数中包含次数比. 本文的估计通过局部投影和迹不等式的仔细标度, 消去了这种依赖.
面权重与 DG 的联系. 在非协调空间(s < k)情形, 范数包含面跳跃惩罚项, 本文给出了两种面子数选择 b_F^{max} 和 b_F^{har}, 并讨论了它们与直接 IPDG 方法中常用权重的对应(第 6.1 节). 选择 b_F^{har} 时, 结合链条件仍满足 \bar{a}_K ≲ b_F, 这确保了 Poincaré 常数的一致性. 相比以前 DG 分析中往往需要专门设计惩罚参数来吸收系数比, 本文的结果表明, 在满足高侧连通性的网格上, 标准的面权形式自然具有系数鲁棒的分解稳定性.
局限性. 本文的结果适用于张量积多项式空间(Q_{p_K}), 不能直接移植到单纯形或一般混合单元. 对于 k ≥ 2, 网格必须由正交长方体组成且满足坐标兼容, 这限制了区域几何. 尽管可以通过区域分解在多边形区域上使用, 但全局坐标兼容难以保证. 对于一般光滑区域, 通常需要等参映射, 但高阶法向导数在映射下的变换会复杂化. 文中的 k=1 推广是第一步, 更高阶的映射情形仍是开放问题. 另外, 高侧连通性假设在某些复杂材料分布(如高度曲折的界面或孤岛)下可能过于严格, 需要更复杂的粗空间设计才能绕过.
总之, 本文在 hp 稳定分解的理论中迈出了重要一步, 将适用范围拓宽到任意阶数, 任意维度和高对比度系数, 所引入的 Hermite 变换和高侧链技术为后续研究提供了可借鉴的范例.
不相关论文（基本信息）
1. 机会约束规划的q-Tsallis安全逼近
A q-Tsallis Safe Approximation for Chance-Constrained Programs
作者: Sergio Assunção Monteiro, Fabricio Alves Barbosa da Silva
类别: math.NA, math.OC
发布日期: 2026-06-04
arXiv ID: 2606.06401v1
链接: http://arxiv.org/abs/2606.06401v1
摘要: Classical chance-constrained programs are solved by safe approximations based on the empirical CVaR, which uses a uniform measure over scenarios and systematically underweights tail events under heavy-tailed distributions. We introduce \emph{q-CCP}, a non-extensive safe approximation grounded in the Riemannian geometry of the Tsallis statistical manifold: the rank-based q-CVaR escort weights are the $g^{(q)}$-geodesic projection onto the tail simplex face, and the q-CCP feasible set is a Tsallis-divergence ball (Proposition12). This geometric foundation yields three results. First, q-CCP is a provable strict tightening of CVaR-CCP for all $q > 1$ (Theorem7). Second, the empirical violation ratio satisfies $ρ(q) = [1-(1-\varepsilon)^{q+1}]/\varepsilon$, independent of the tail index $ν$ (Proposition10). Third, the feasible-region volume cost is monotone increasing in $q$ and $ν$ (Proposition11), providing a data-adaptive safety knob. The formulation inherits convexity and coherence from the q-CVaR functional and admits an iterative LP reformulation converging in 2–3 iterations. Experiments on 15 Ibovespa equities confirm the theory (violation ratio $0.241$, $q^* = 1.50$); an M5 inventory newsvendor experiment generalises the method to supply chain ($q^* = 1.88$, cost premium $1.155\times$, zero OOS stockout violations).
2. 面向高维偏微分方程的DAS-PINNs：将深度自适应采样扩展到时空域
DAS-PINNs for high-dimensional partial differential equations: extending deep adaptive sampling to spacetime domains
作者: Anshima Singh, David J. Silvester
类别: math.NA, cs.LG, stat.ML
发布日期: 2026-06-04
arXiv ID: 2606.06314v1
链接: http://arxiv.org/abs/2606.06314v1
摘要: Time-dependent high-dimensional partial differential equations (PDEs) with spatially localised and dynamically evolving solutions pose a fundamental challenge for physics-informed neural networks (PINNs), as uniform collocation sampling becomes increasingly ineffective in high-dimensional spatiotemporal domains. In this work, a deep adaptive sampling framework for PINNs is extended to the time-dependent setting by treating space and time as a unified domain without any explicit time marching. A normalising flow neural network model effectively learns the distribution induced by the PDE residual and generates new collocation points concentrated in regions where the solution is most difficult to learn. Unlike conventional adaptive strategies that require explicit time stepping or moving meshes, high-residual regions are automatically identified and tracked across both space and time, driven purely by the PDE residual distribution. The effectiveness of the proposed strategy is assessed on a range of benchmark problems, from sharp and moving features in two spatial dimensions to localised structures in up to eight spatial dimensions.
3. 带非齐次边界条件的二维不可压缩Navier-Stokes方程的Tr-PINNs算法误差分析
Error Analysis of Tr-PINNs Algorithm for 2D Incompressible Navier-Stokes Equations with Non-Homogeneous Boundary Conditions
作者: Dongjie Liu, Xuebo Li, Rong Yang
类别: math.NA
发布日期: 2026-06-04
arXiv ID: 2606.06268v1
链接: http://arxiv.org/abs/2606.06268v1
摘要: Physics-informed neural networks (PINNs) have been widely applied to solve Navier-Stokes equations by enforcing outputs and gradients of deep models to satisfy target equations. However, conventional PINNs only constrain the boundary terms by means of the $L^2$-norm when addressing the equations with non-homogeneous boundary conditions. This single constraint strategy may cause inaccurate boundary simulation, further resulting in the decline of prediction accuracy. To resolve this critical issue, this paper proposes an improved physics-informed neural network by correcting the error of the boundary value, which is called Tr-PINNs. Based on the results of nonhomogeneous Stokes problem, the algorithm error analysis of Tr-PINNs is established. The efficacy of the Tr-PINNs algorithm is demonstrated via numerical experiments, which further demonstrate that the Tr-PINNs algorithm achieves a remarkable improvement in computational accuracy.
4. 有效维度作为物理信息神经网络中保物理约束自适应的算子不变量
Effective Dimensionality as an Operator Invariant for Physics-Preserving Constraint Adaptation in Physics-Informed Neural Networks
作者: Cornelius Otchere, Michael Shields
类别: stat.ML, cs.LG, math.NA, physics.comp-ph
发布日期: 2026-06-04
arXiv ID: 2606.06171v1
链接: http://arxiv.org/abs/2606.06171v1
摘要: Physics-Informed Neural Networks inherently suffer from task interference because they rely on a shared parameter space to satisfy both governing differential equations and boundary conditions. We analyze this structural conflict using the Fisher Information Matrix to quantify the effective degrees of freedom ($d_{eff}$) in a physics-constrained model. Unlike the classical $d_{eff}$ which measures how many parameter directions are informed by data against a statistical prior, our $d_{eff}$ measures the dimension of the parameter directions unconstrained by the differential operator. For operators with finite-dimensional kernel, we show that $d_{eff}$ converges to the kernel dimension exactly, independent of network width, depth, or activation function, recasting it from a fit diagnostic into a structural invariant of the underlying continuous operator. For operators with infinite-dimensional kernel, $d_{eff}$ instead measures the network’s finite-dimensional representational bandwidth for that kernel rather than recovering an integer invariant. Importantly, $d_{eff}$ also serves as an a priori structural diagnostic. Driving $d_{eff}$ of a well-posed problem to zero certifies that the physics and boundary constraints have absorbed the network’s free directions. Building on this characterization, we introduce subspace projection strategies for boundary adaptation. Rather than retraining from scratch, we project parameter updates into the null space of the pre-trained physics operator so that new boundary conditions are satisfied without disturbing the learned physics. Gradient-based fine-tuning can match or exceed this but needs more wall-clock time and tuning, whereas subspace projection delivers near-equivalent quality in seconds to minutes. We validate on linear and nonlinear operators, demonstrating accurate adaptation to initial and boundary shifts and unencountered constraint types.
5. 基于JR分解的电路模型结构保持算子分裂
Structure-Preserving Operator Splitting via JR-Decomposition for Circuit Models
作者: Andreas Bartel, Malak Diab
类别: math.NA
发布日期: 2026-06-04
arXiv ID: 2606.06153v1
链接: http://arxiv.org/abs/2606.06153v1
摘要: We investigate circuit models, namely, modified nodal analysis (MNA) in the port-Hamiltonian framework. Based on this, the JR-decomposition for the numerical treatment would offer an energy conform splitting. However, for circuit models, the application of the standard JR-decomposition is restricted. To enable a JR-decomposition for MNA, we need to relax the decomposition. To this end, we introduce the enhanced JR-decomposition, which is particularly tailored to the application to circuits. We conclude with a numerical example that illustrates the applicability of the proposed approach as well as its convergence and structure-preserving properties.
6. 用于黎曼流形上遍历动力学长时间采样的后处理冻结流方法
Post-processed frozen-flow methods for the long time sampling of ergodic dynamics on Riemannian manifolds
作者: Adrien Busnot Laurent, Sébastien Macé
类别: math.NA, math.CO, math.DG, math.PR
发布日期: 2026-06-04
arXiv ID: 2606.06150v1
链接: http://arxiv.org/abs/2606.06150v1
摘要: In this work, we propose a novel intrinsic approach to the approximation of ergodic SDEs on Riemannian manifolds, which include Riemannian Langevin dynamics. In opposition to the standard extrinsic approaches such as penalization methods and projection methods, our methodology does not use embeddings or coordinates and only relies on natural geometric operations: geodesics, parallel transport,… We give a criterion for high order of accuracy for the invariant measure, develop new intrinsic numerical methods designed solely for sampling the invariant measure, and derive high order conditions using a new algebraic operation on exotic Lie-Butcher series. In the spirit of the Leimkuhler-Matthews method, our approach prioritizes long time sampling efficiency over finite time accuracy, and outperforms the previous extrinsic and intrinsic approaches in terms of cost for a given accuracy, which we illustrate with several numerical experiments.
7. 集合卡尔曼反演作为一种惯性相互作用粒子系统
Ensemble Kalman Inversion as an Inertial Interacting Particle System
作者: Michael Herty, Pierpaolo Porretta, Giuseppe Visconti
类别: math.NA, math.DS, math.OC
发布日期: 2026-06-04
arXiv ID: 2606.06121v1
链接: http://arxiv.org/abs/2606.06121v1
摘要: Ensemble Kalman Inversion (EKI) is a derivative-free, ensemble-based method for inverse and optimization problems. Its continuous-time formulation can be interpreted as an interacting particle system driven by a Kalman-type preconditioned descent direction. A well-known limitation of this dynamics is the possible premature collapse of the covariance of the ensemble, which makes the method sensitive to the initial ensemble. We introduce a second-order particle system in which the particles evolve according to an inertial dynamics. The model combines a Kalman-type relaxation force with damping, attraction towards the ensemble mean, and a short-range repulsive interaction designed to counteract ensemble collapse. The resulting dynamics can be interpreted as a heavy-ball reformulation of continuous-time EKI enriched by competing attractive and repulsive mechanisms. For linear inverse problems, we analyze the induced mean and fluctuation dynamics and identify a parameter regime in which fully collapsed configurations are linearly unstable. We further characterize asymptotic equilibria through a constrained optimality condition on the subspace retained by the limiting ensemble covariance and derive an exponential decay estimate. Numerical experiments illustrate the effect of inertia and repulsion on the ensemble dynamics and compare the proposed second-order method with first-order EKI-type
8. 一种用于无界区域上基于残差的波动方程求解器的微局部开放边界方法
A Microlocal Open-Boundary Method for Residual-Based Wave Solvers on Unbounded Domains
作者: Avy Soffer, Nguyen Gia Hien, Minh-Binh Tran
类别: math.NA
发布日期: 2026-06-04
arXiv ID: 2606.06115v1
链接: http://arxiv.org/abs/2606.06115v1
摘要: We introduce a microlocal phase-space-filtered physics-informed neural network (PINN–TDPSF or Microlocal PINNFilter) framework for wave propagation on unbounded domains. The method combines a slabwise neural residual approximation of the interior evolution with a time-dependent phase-space filter applied in a buffer surrounding the physical computational domain. The central idea is to replace local artificial-boundary penalties by a phase-space radiation mechanism: a component is removed only when it is localized near the artificial boundary and its group velocity points outward. The proposed method is not intended to replace FFT, spectral, or split-step solvers for known-coefficient forward problems where such methods are available and highly accurate. Instead, it embeds the time-dependent phase-space filter into a residual-based neural framework. This coupling is useful when open-domain wave propagation must be combined with nonlinear residuals, sparse or off-grid observations, unknown coefficients, variable interior media, or other non-FFT-diagonalizable physics. Numerical experiments for linear Schrödinger propagation, potential scattering, anisotropic Schrödinger dynamics, nonlinear Schrödinger wave packets, soliton stress tests, linearized Euler waves, and sparse-data recovery of a localized acoustic defect show that the method reduces artificial reflection and wraparound, uses group velocity correctly in anisotropic media, preserves physically incoming branch components, and provides diagnostics when the assumptions behind outgoing-packet filtering are violated.
9. 张量火车多维逆拉普拉斯变换
A tensor-train multidimensional inverse Laplace transform
作者: Martin Mikkelsen, Michael Kastoryano
类别: math.NA, physics.comp-ph
发布日期: 2026-06-04
arXiv ID: 2606.06093v1
链接: http://arxiv.org/abs/2606.06093v1
摘要: Laplace transforms and their numerical inverses arise throughout applied mathematics, physics, finance, and probability theory. Numerical inversion, however, quickly becomes intractable in high dimensions because the number of quadrature evaluations grows exponentially with dimension. We develop a tensor train (TT) formulation of the multidimensional inverse Laplace transform. The method constructs a TT approximation of the transformed function on the complex quadrature grid and then performs the inversion through a sequence of tensor contractions. Under suitable low-rank assumptions, this reduces the computational cost from exponential to polynomial in the dimension, provided that the relevant bond dimensions remain bounded. The method has only a small number of tunable parameters and admits error estimations. We demonstrate its performance in numerical experiments, benchmarked against Monte Carlo estimates and exact references, for multivariate normal-inverse Gaussian, Wishart, and correlated Gamma-type distributions.
10. 含时间依赖奇异矩阵的随机微分代数方程的保结构随机theta方法的弱一阶收敛
Weak order one convergence of structure-preserving stochastic theta methods for stochastic differential algebraic equations with time-dependent singular matrices
作者: Caiyuan Zhu, Ziheng Chen, Lin Chen, Yiwei Zhou
类别: math.NA
发布日期: 2026-06-04
arXiv ID: 2606.06071v1
链接: http://arxiv.org/abs/2606.06071v1
摘要: This paper studies the weak convergence order of structure-preserving stochastic theta methods for a class of index-$1$ stochastic differential algebraic equations with time-dependent singular matrices. The singular matrix is allowed to vary in time but preserves a fixed differential-algebraic splitting, thereby extending the constant singular-matrix setting while retaining the projector structure required for constraint preservation. By exploiting the index-$1$ algebraic-differential decomposition of the exact solution, we establish an abstract weak convergence theorem for constraint-preserving one-step approximations and apply it to the stochastic theta method with $θ\in (0,1]$. Under global Lipschitz, linear growth, and suitable smoothness assumptions, the considered method is proved to be well posed, to preserve the algebraic constraints at all time levels, and to converge with weak order one. Numerical experiments are finally presented to confirm the structure-preserving property and the theoretical convergence order.
11. 基于残差的Kaczmarz方法求解t-乘积张量线性方程组
Residual-based Kaczmarz methods for tensor linear equations with t-product
作者: Li-Lin Ji, Juanjuan Sun, Jun-Feng Yin
类别: math.NA
发布日期: 2026-06-04
arXiv ID: 2606.06057v1
链接: http://arxiv.org/abs/2606.06057v1
摘要: Tensor linear systems widely arise from high-dimensional data mining and computing, for instance, natural language processing and machine learning. A class of residual-based tensor Kaczmarz method is proposed for tensor linear equations with t-product. Theoretical analyses prove the convergence and give an upper bound of the convergence rate of the proposed method. Furthermore, an accelerated residual-based Kaczmarz method with heavy ball momentum is developed. Numerical experiments verify the efficiency of the proposed methods and demonstrate that they are faster than the existing tensor Kaczmarz methods.
12. 用稀疏逼近方法学习偏微分方程解算子
Learning solution operators of PDEs with sparse approximation methods
作者: Sebastian Neumayer, Daniel Potts, Fabian Taubert
类别: math.NA, cs.LG
发布日期: 2026-06-04
arXiv ID: 2606.06046v1
链接: http://arxiv.org/abs/2606.06046v1
摘要: We investigate the approximation of solution operators for partial differential equations (PDEs) using sparse high-dimensional techniques. Building on a dimension-incremental framework, we combine product basis expansions with sparse recovery methods, specifically orthogonal matching pursuit (OMP), to substantially reduce the required sample size compared with a previously considered cubature-based approach. We evaluate the resulting method numerically on several examples, comparing it against both cubature-based sparse approximation and Fourier neural operators in terms of accuracy, runtime, and sample size. The experiments show that our approach considerably reduces the number of required PDE solves relative to its predecessor while maintaining competitive accuracy, particularly when the solution admits a sparse representation in the chosen basis. Furthermore, the recovered sparse index sets yield interpretable insights into the relevant variables and parameter interactions.
13. 广义牛顿模型的辅助梯度流求解器
Auxiliary Gradient-Flow Solvers for Generalized Newtonian Models
作者: Stefano Zampini, Daniele Boffi, Gurt Dovletov, Peter Markowich
类别: math.NA, math.FA
发布日期: 2026-06-04
arXiv ID: 2606.05926v1
链接: http://arxiv.org/abs/2606.05926v1
摘要: We introduce an auxiliary gradient-flow framework for variational problems with generalized Newtonian structure governed by an N-function. The key idea is to replace the nonlinear constitutive dependence on the gradient, or symmetric gradient, by an auxiliary scalar variable representing its squared magnitude. This shifts the nonlinearity from the state equation to the auxiliary variable, yielding a sequence of uniformly elliptic weighted linear problems. At the continuous level, we construct an auxiliary energy on a metric space adapted to the growth of the underlying N-function. In this topology, we prove lower semicontinuity, geodesic $λ$-convexity, and exponential convergence of the associated minimizing-movement scheme. At the finite element level, we derive a metric gradient flow through an explicit Riesz map, prove global well-posedness of the resulting semi-discrete ODE, and establish convergence to the finite element solution of the Euler–Lagrange equations of the generalized Newtonian energy. For the $p$-Laplacian and $p$-Stokes models, this gives a rigorous convergence result for $4/3\le p\le 4$, $p\ne2$, with asymptotic rate estimates beyond this range. We also propose practical time discretizations, including an operator-splitting scheme that gives the \kac iteration as a special case, and an adaptive pseudo-transient method that can be implemented using scalable linear solvers. Numerical experiments for power-law, Carreau–Yasuda, regularized Bingham, and optimal-design models demonstrate robustness, mesh-independent iteration counts in the tested regimes, and performance that matches or outperforms Newton’s method.
14. 离散时间逼近对退化抛物-双曲系统的收敛性
Convergence of a discrete-in-time Approximation to a Degenerate Parabolic-Hyperbolic System
作者: Julia Hauser, Hideki Murakawa, Markus Schmidtchen
类别: math.AP, math.NA
发布日期: 2026-06-04
arXiv ID: 2606.05879v1
链接: http://arxiv.org/abs/2606.05879v1
摘要: In this paper we consider an implicit semi-discrete approximation of a degenerate reaction-cross-diffusion system. Due to the symmetry in the parabolic part, this system is known to preserve segregation of densities – initially non-overlapping densities belonging to different species remain segregated for all times, which leads to internal layers between different species. We show that time-discrete approximations exist and converge to a weak solution, as the timestep goes to zero.
15. 用于$\ell_{1-2}$正则化稀疏恢复的迭代阈值追踪连续化方法
Iterative Thresholding Pursuit with Continuation for $\ell_{1-2}$-Regularized Sparse Recovery
作者: Junxi Wu, Zeyu Dong, Jun-Feng Yin
类别: math.NA, cs.IT, math.OC
发布日期: 2026-06-04
arXiv ID: 2606.05657v1
链接: http://arxiv.org/abs/2606.05657v1
摘要: Sparse recovery aims to reconstruct sparse signals from underdetermined and possibly noisy linear measurements. Existing $\ell_{1-2}$ iterative thresholding schemes are first-order methods. We propose an iterative thresholding pursuit method with continuation (ITP-C) for $\ell_{1-2}$-regularized sparse recovery. The method goes beyond first-order thresholding by combining the active-set identification capability of the $\ell_{1-2}$ proximal step with a restricted least-squares pursuit step that provides a second-order update on the identified support. The support is generated adaptively by the thresholding update, and no prior knowledge of the true sparsity level is required. To control the possible instability of the pursuit step while preserving the descent structure of the continuation scheme, we impose a strict descent check with respect to the dynamic objective. We establish convergence of the generated sequence under the Kurdyka-Lojasiewicz framework and prove a local oracle-type property after correct support identification. Numerical experiments on synthetic sparse recovery and image reconstruction illustrate the descent preservation of the proposed safeguard and demonstrate the improved recovery performance of ITP-C over the state-of-the-art baselines.
16. 一种求解大规模 T-Sylvester 方程的低秩插值投影算法
A Low-rank Interpolatory Projection Algorithm for Solving Large-scale T-Sylvester Equations
作者: Umair Zulfiqar
类别: math.NA
发布日期: 2026-06-04
arXiv ID: 2606.05640v1
链接: http://arxiv.org/abs/2606.05640v1
摘要: This paper considers large-scale T-Sylvester equations of the form $AX - X^\top E^\top + B_1B_2^\top = 0$, which admit a low-rank solution. It is shown that when the unique solution of the T-Sylvester equation is low-rank, the problem naturally reduces to a tangential interpolation problem via oblique projection. The specific interpolation points and tangential directions needed to obtain the low-rank solution are not known a priori, thus requiring an iterative approach. An iterative interpolatory projection algorithm is proposed based on these interpolation conditions, which iteratively refines the interpolation data as the projection matrices expand in the number of columns. Numerical examples demonstrate that the proposed algorithm converges with projection matrices having significantly fewer columns compared to existing Krylov-subspace-based projection methods, confirming the superiority of the proposed algorithm over existing approaches.
17. 关于基于范数最小化的凸向量优化的并行与批量切割策略
On Parallel and Batch-Cutting Strategies for Norm-Minimization-Based Convex Vector Optimization
作者: Mohammed Alshahrani
类别: math.OC, math.NA
发布日期: 2026-06-04
arXiv ID: 2606.05617v1
链接: http://arxiv.org/abs/2606.05617v1
摘要: We develop parallel and batch-cutting variants of the norm-minimization-based outer approximation algorithm for convex vector optimization. The standard algorithm solves $N_k$ independent subproblems at each iteration~$k$ to evaluate all vertices of the current polyhedral approximation, but processes only the single best cut. We propose two improvements. First, we parallelize the \revise{subproblem evaluations} across $\nw$ workers, reducing per-iteration wall-clock time. Second, we introduce a batch-cutting strategy that adds up to $K$ supporting halfspaces per iteration, using information from all solved subproblems rather than discarding it. We prove that the batch-cutting variant inherits the convergence rate $O(k^{2/(1-q)})$ of the standard algorithm, where $k$ is the number of outer iterations and $q$ is the number of objectives. Computational experiments on eight test problems with $q \in \{2,3,4,5\}$ show that parallelism on 8 cores \revise{increases the speed by a factor of 1.1 to 4.2}, and batch cutting consistently reduces the iteration count by 62–80%. However, the wall-clock benefit of batch cutting is problem-dependent: the additional cuts per iteration accelerate vertex count growth, so batch cutting is most effective when per-vertex subproblem cost dominates.
18. 具有最优离散度、周期 $L_2$ 偏差及更多性质的无穷序列
Infinite sequences with optimal diaphony, periodic $L_2$-discrepancy, and beyond
作者: Peter Kritzer, Nicolas Nagel, Friedrich Pillichshammer
类别: math.NA
发布日期: 2026-06-03
arXiv ID: 2606.05482v1
链接: http://arxiv.org/abs/2606.05482v1
摘要: We investigate the periodic $L_2$-discrepancy of infinite sequences $§_d$ in $[0,1)^d$ and its analytic counterpart, the diaphony. We prove that infinite order-2 digital sequences over $\mathbb{F}_2$ attain the optimal order $L_{2,N}^{\rm per}(§_d) \le C_d (\log N)^{d/2}/N$ for all $N \in \mathbb{N}\setminus \{1\}$, matching known lower bounds for infinitely many $N \in \mathbb{N}$. This confirms the conjectured optimality of order-2 constructions. By this result, we improve upon previously known constructions using order-5 digital sequences, and reduce the underlying dimension for the interlacing construction from $5d$ to $2d$, significantly improving practicality. We establish our bounds within a broader framework of quasi-Monte Carlo integration for periodic Besov spaces $S_{p,q}^rB(\mathbb{T}^d)$ with dominating mixed smoothness $r \in (1/p,2)$, where $p,q\in [1,\infty]$. Rules based on infinite order-2 digital sequences yield worst-case errors of order $(\log N)^{(d-1)(1-1/q)} / N^{ \min(r,1)}$ for $r \not=1$, and $(\log N)^{d(1-1/q)}/N$ for $r=1$, for all $N \in \mathbb{N}\setminus\{1\}$, while preserving extensibility in $N$.
19. 迭代相关动力学中用于早期轨迹表征的双通道F变换表示
A Two-Channel F-Transform Representation for Early Trajectory Characterization in Iterated Correlation Dynamics
作者: Ishrak Alhajj Hassan
类别: math.DS, math.NA, stat.CO, stat.ML
发布日期: 2026-06-03
arXiv ID: 2606.05462v1
链接: http://arxiv.org/abs/2606.05462v1
摘要: Many nonlinear iterative procedures generate high-dimensional trajectories whose early behavior is informative but difficult to compare directly. This paper studies a soft-computing representation problem: how to convert a short early trajectory segment into compact, interpretable, fixed-dimensional fuzzy coordinates that preserve information about subsequent convergence and trajectory geometry. The problem is investigated for iterated Pearson correlation matrices, a nonlinear matrix iteration historically connected with CONCOR-type blockmodeling and repeated-correlation methods. The proposed descriptor uses two logarithmic signals from the early post-transient regime: a step-size signal, measuring contraction magnitude, and a contraction-ratio signal, measuring local contraction evolution. Each signal is projected onto a three-node triangular fuzzy partition using zero-degree F-transform coefficients and one centered first-degree coefficient. This yields an eight-dimensional two-channel representation separating local level from local trend and contraction magnitude from contraction evolution. Across 22 matrix dimensions with 1000 trajectories per dimension, the descriptor is compared with raw trajectory samples, statistical summaries, and PCA-compressed raw features using Random Forest regression for convergence-length approximation. It achieves mean R^2 = 0.6480, close to raw trajectories (0.6518) and statistical summaries (0.6528), while improving over the step-size-only F-transform descriptor (0.5001). Repeated random-split and shifted-window experiments confirm stability. PCA and clustering further show reproducible low-dimensional organization, with the first two principal components explaining 84.26% of variance and k = 3 favored by the mean silhouette criterion.
20. Mamba辅助的非马尔可夫封闭用于降阶建模
Mamba-Assisted Non-Markovian Closure for Reduced-Order Modeling
作者: Zhi-Feng Wei, Saad Qadeer, Panos Stinis
类别: cs.LG, math.NA, stat.ML
发布日期: 2026-06-03
arXiv ID: 2606.05371v1
链接: http://arxiv.org/abs/2606.05371v1
摘要: Reduced-order modeling of high-dimensional dynamical systems is often hindered by the non-Markovian closure term that represents the effect of unresolved variables on the resolved dynamics. Inspired by the Mori–Zwanzig formalism, in which the closure takes the form of a memory functional of the resolved trajectory, we recast closure modeling as a sequence modeling problem and propose the Mamba-Assisted Closure (MAC) framework: a Mamba-based sequence model, trained to predict the closure from the resolved trajectory, is coupled with the reduced-order governing equations through a numerical integrator to advance the resolved variables in time. A key feature of the framework is its exploitation of the dual representation of state-space models – the model is trained in a sequence-to-sequence fashion via the convolutional form, and deployed for step-by-step autoregressive rollout via the recurrent form, yielding both efficient long-trajectory training and constant per-step inference cost. On the viscous Burgers’ equation and the chaotic two-scale Lorenz ‘96 system, the MAC model substantially outperforms the Markovian reduced-order model, the GRU-based sequence model, and the Wilks method in predictive accuracy and long-time rollout stability.
21. 优化未调整朗之万算法的不可逆扰动
Optimizing Irreversible Perturbations of the Unadjusted Langevin Algorithm
作者: Qianyu Julie Zhu, Youssef Marzouk, Konstantinos Spiliopoulos, Benjamin Zhang
类别: math.NA, math.PR, stat.AP, stat.CO, stat.ME
发布日期: 2026-06-03
arXiv ID: 2606.05324v1
链接: http://arxiv.org/abs/2606.05324v1
摘要: Irreversible perturbations accelerate the convergence of Langevin dynamics, breaking detailed balance while preserving the invariant measure. The design of optimal irreversible perturbations has been studied in the continuous-time Gaussian setting, but extensions to non-Gaussian target distributions, and the impact of time discretization on the design of optimal perturbations, have not been well understood. Numerical discretizations of Langevin dynamics introduce bias, which is typically exacerbated by irreversible perturbations; handling this interaction demands a joint treatment of acceleration and accuracy. This paper develops a systematic framework for optimizing position-independent irreversible perturbations of the unadjusted Langevin algorithm (ULA). We formulate a constrained optimization problem that simultaneously accounts for mixing efficiency and discretization bias, where the former is characterized by a spectral gap analogue and the latter is quantified via a weighted expected squared jump distance. Within this framework, we derive an explicit characterization of the optimal position-independent irreversible perturbation. Extensive numerical experiments demonstrate that our design yields faster convergence with controlled bias, and improves mean squared estimation errors compared to other choices of irreversible perturbation.