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1. 麦克斯韦方程组的结构保持最优控制及其在源隐身中的应用

Structure-Preserving Optimal Control of Maxwell’s Equations with Applications to Source Cloaking

作者: Harbir Antil, Yaw Owusu-Agyemang, Rohit Khandelwal, Jimmie Adriazola, Denis Ridzal

类别: math.OC, math.AP, math.NA, physics.comp-ph

发布日期: 2026-04-30

arXiv ID: 2605.00212v1

摘要翻译: 我们提出一种针对含时麦克斯韦方程最优控制的结构保持求解框架。基于正问题弱形式的适定性理论，我们首先分析了一种将Nédélec元和Raviart-Thomas元与Crank-Nicolson时间步进耦合的正向求解器。该求解器保持de Rham结构，施加离散高斯定律，精确满足每个时间步的能量平衡，并在最优控制设置所给定的问题数据的低正则性假设下收敛到弱解。为了控制麦克斯韦系统，我们在安培定律中加入时空电流密度的旋度作为源。旋度形式无需辅助约束即可保证电荷守恒。我们证明了控制到状态映射的适定性和连续性，推导了伴随系统以及跟踪型目标泛函的梯度表示，并构造了一个继承正向求解器结构保持性质的离散优化格式。我们的离散平稳性条件与其连续对应形式一致，并且随着网格和时间步细化，离散最优控制收敛到连续最优解。我们通过数值求解一系列源隐身模型问题，展示了我们的最优控制公式和理论发展的优点。

Comment: 无

链接: http://arxiv.org/abs/2605.00212v1

1. 研究对象和背景

本文研究时间依赖 Maxwell 方程的最优控制问题, 其状态方程为在有限时间区间 $(0,T)$ 和有界 Lipschitz 区域 $\Omega\subset\mathbb{R}^3$ 上的电磁场演化:

$$ \begin{cases} \varepsilon \partial_t \mathbf{E} - \operatorname{curl}(\mu^{-1}\mathbf{B}) + \sigma \mathbf{E} = \chi_{\Omega_{\text{src}}}\mathbf{I}_{\text{src}} + \chi_{\Omega_{\text{ctrl}}}\operatorname{curl} \mathbf{z}, & \text{in } \Omega\times(0,T),\\ \partial_t \mathbf{B} + \operatorname{curl} \mathbf{E} = 0, & \text{in } \Omega\times(0,T),\\ \mathbf{E}\times\boldsymbol{\nu} = 0, & \text{on } \partial\Omega\times(0,T),\\ \mathbf{E}(0)=\mathbf{E}_0,\; \mathbf{B}(0)=\mathbf{B}_0, & \text{in } \Omega. \end{cases} $$

其中 $\mathbf{E}$ 为电场, $\mathbf{B}$ 为磁通量密度; $\varepsilon,\mu,\sigma$ 分别为电容率、磁导率和电导率张量, 满足一致椭圆性和对称性假设 (Assumption 1); $\mathbf{I}_{\text{src}}\in L^2(0,T;L^2(\Omega_{\text{src}})^3)$ 为给定源电流密度, 支撑在 $\Omega_{\text{src}}\subseteq\Omega$ 上; 控制变量 $\mathbf{z}$ 定义在区域 $\Omega_{\text{ctrl}}\subseteq\Omega$ 上, 并以 $\operatorname{curl}\mathbf{z}$ 形式进入 Ampère 定律, 从而自动保证 $\operatorname{div}(\operatorname{curl}\mathbf{z})=0$, 无需额外约束即实现电荷守恒。目标是通过选择 $\mathbf{z}$ 极小化如下跟踪型泛函:

$$ J(\mathbf{E},\mathbf{B},\mathbf{z}) = \frac12 \int_0^T \Big( \|\varepsilon^{1/2}(\mathbf{E}-\mathbf{E}_d)\|_{L^2(\Omega_{\text{obs}})^3}^2 + \|\mu^{-1/2}(\mathbf{B}-\mathbf{B}_d)\|_{L^2(\Omega_{\text{obs}})^3}^2 \Big) \,dt + \frac12 \|\mathbf{z}\|_Z^2, $$

其中 $\mathbf{E}_d,\mathbf{B}_d$ 为目标场, $\Omega_{\text{obs}}$ 为观测区域, 控制空间范数定义为

$$ \|\mathbf{z}\|_Z^2 = \int_0^T \big( \alpha_1\|\mathbf{z}(t)\|_{L^2(\Omega_{\text{ctrl}})^3}^2 + \alpha_2\|\operatorname{curl}\mathbf{z}(t)\|_{L^2(\Omega_{\text{ctrl}})^3}^2 \big) \,dt, \quad \alpha_1,\alpha_2>0. $$

此类最优控制问题在纳米光学、电磁屏蔽、无损检测、医学成像等工程应用中具有重要意义。从数学角度看, 核心困难在于数据仅有最低可积性 (源 $\mathbf{I}_{\text{src}}\in L^2$, 初始数据 $(\mathbf{E}_0,\mathbf{B}_0)\in L^2$), 且区域仅要求 Lipschitz 正则性, 这使得标准的高正则性假设下的数值分析不再适用。已有的工作 (如 [1]) 已在此低正则框架下建立了 Maxwell 方程弱解的适定性 (定理 1), 并给出了基于 Nédélec 元和 Raviart–Thomas 元的半离散格式的收敛性。然而, 全离散格式 (包括时间步进) 的构造及其在同样低正则条件下的收敛性尚未建立; 将此类结构保持离散化方法系统地融入最优控制框架, 并证明离散最优控制收敛到连续最优控制, 更是未被探索的空白。本文正是为填补这一空白而作, 旨在发展一个从连续问题到离散优化的全链条结构保持理论, 并应用于源隐身 (Ed = 0, Bd = 0) 模型。

2. 主要定理或主要结果

本文的主要结果可概括为以下六个方面, 均在 Assumption 1 (材料张量对称、一致正定, σ 半正定, 源与初值属 L²) 下成立。

全离散前向格式的适定性与稳定性。采用 Crank–Nicolson 时间离散与 Nédélec/RT 空间离散的耦合格式 (6): 给定 $(\mathbf{E}_h^n,\mathbf{B}_h^n)$, 求 $(\mathbf{E}_h^{n+1},\mathbf{B}_h^{n+1})$ 满足
$$(\varepsilon \delta_t \mathbf{E}_h^{n+1}, \boldsymbol{\psi}_h) + (\sigma \mathbf{E}_h^{n+1/2}, \boldsymbol{\psi}_h) - (\mu^{-1}\mathbf{B}_h^{n+1/2}, \operatorname{curl}\boldsymbol{\psi}_h) = (\mathbf{f}^{n+1/2}, \boldsymbol{\psi}_h) \quad \forall \boldsymbol{\psi}_h\in N_h^0,$$
$$(\delta_t \mathbf{B}_h^{n+1}, \boldsymbol{\phi}_h) + (\operatorname{curl}\mathbf{E}_h^{n+1/2}, \boldsymbol{\phi}_h) = 0 \quad \forall \boldsymbol{\phi}_h\in RT_h,$$
其中 $\mathbf{f}^{n+1/2}$ 为时间区间上的平均。该格式存在唯一解 (引理 1, 通过选取特殊测试函数及投影 $P_h$ 证明), 并精确满足每步能量平衡:
$$\frac{\mathcal{E}^{n+1} - \mathcal{E}^n}{\Delta t} + \|\sqrt{\sigma} \mathbf{E}_h^{n+1/2}\|_{L^2(\Omega)}^2 = (\mathbf{f}^{n+1/2}, \mathbf{E}_h^{n+1/2}),$$
其中 $\mathcal{E}^n = \frac12(\|\varepsilon^{1/2}\mathbf{E}_h^n\|_{L^2}^2 + \|\mu^{-1/2}\mathbf{B}_h^n\|_{L^2}^2)$。由此导出稳定性估计 (与 h, Δt 无关):
$$\mathcal{E}^m + \Delta t \sum_{n=0}^{m-1} \|\sqrt{\sigma} \mathbf{E}_h^{n+1/2}\|_{L^2}^2 \le C_T \Big( \mathcal{E}^0 + \sum_{n=0}^{m-1} \Delta t \|\varepsilon^{-1/2}\mathbf{f}^{n+1/2}\|_{L^2}^2 \Big).$$
在 f ≡ 0, σ ≥ 0 时能量耗散, σ ≡ 0 时能量守恒。此外, 格式保持离散 Gauss 定律、遵循 de Rham 结构。全离散解在数据低正则假设下收敛到弱解 (定理陈述未在片段中出现, 但作者在引言中声明)。
控制到状态映射的适定性。对每个控制 $\mathbf{z}\in Z := L^2(0,T;H(\operatorname{curl};\Omega_{\text{ctrl}}))$, Maxwell 系统 (2) 存在唯一弱解 $(\mathbf{E},\mathbf{B})$ 于能量类
$$\mathbf{E} \in H^1(0,T;H_0(\operatorname{curl};\Omega)^*) \cap C([0,T];L^2(\Omega)^3),$$
$$\mathbf{B} \in H^1(0,T;H(\operatorname{curl};\Omega)^*) \cap C([0,T];L^2(\Omega)^3).$$
映射 $\mathcal{S}: \mathbf{z} \mapsto (\mathbf{E},\mathbf{B})$ 为仿射并有界, 因而对凸二次目标泛函和闭凸容许集, 最优控制存在。
伴随系统与梯度表示。约化目标泛函 $\mathcal{J}(\mathbf{z}) := J(\mathcal{S}(\mathbf{z}),\mathbf{z})$ 在 $Z$ 中 Gâteaux 可微, 其梯度由伴随 Maxwell 系统 (在低正则对偶空间中陈述) 的解表达。作者推导了强形式的伴随方程并得到梯度公式 (具体表达式未在片段中给出), 这给出了最优控制的必要 (在凸性下亦为充分) 条件。
离散伴随格式与一致梯度。将前向 CN‑Nédélec/RT 格式应用于伴随方程, 得到结构保持的离散伴随格式; 通过交换投影和 de Rham 复形, 证明了离散梯度的正确性, 即离散最优性条件与 Galerkin 最优性系统一致 (``the discrete stationarity conditions match the Galerkin optimality system’’)。
离散最优控制的收敛性。当空间网格尺寸 h → 0 和时间步长 Δt → 0 时, 离散最优三元组 $(\mathbf{E}_{h,\Delta t}, \mathbf{B}_{h,\Delta t}, \mathbf{z}_{h,\Delta t})$ 存在子序列弱/强收敛到连续最优三元组 $(\mathbf{E},\mathbf{B},\mathbf{z})$; 若目标泛函对 $\mathbf{z}$ 严格凸, 则全序列收敛。
数值验证。运用该格式求解一维、二维和三维的源隐身算例 ($\mathbf{E}_d = \mathbf{B}_d = 0$), 展示了控制效果。图 1 比较了非分离变量控制与因子化变量控制的优劣, 非分离控制能有效消除源辐射。

3. 研究方法、关键技术和核心工具

本文的技术核心在于在低正则数据 (L² 型源与初值) 下建立结构保持离散格式的收敛性, 并系统地将这种保持物理性质的离散化纳入最优控制理论。主要障碍是: 标准能量方法在只有 $L^2$ 正则性时不足以直接导出对测试函数的紧性; 同时, 最优控制场景要求控制-状态映射的稳定性和可微性, 以及离散状态和伴随状态在同一框架下的收敛性。作者的创新与关键技术手段可概括如下。

1. de Rham‑协调有限元与离散正交投影的巧妙运用
空间离散采用 Nédélec 元 ($N_h^0\subset H_0(\operatorname{curl};\Omega)$) 和 Raviart–Thomas 元 ($RT_h\subset H(\operatorname{div};\Omega)$) 配对, 精确保持 de Rham 复形: $\operatorname{curl} N_h^0 \subset RT_h$ 且 $\operatorname{div}(\operatorname{curl} N_h^0)=0$。该性质保证了离散 Gauss 定律的成立, 并且 (1) 能在每一步中通过取恰当的测试函数实现关键项的消去。在证明 CN 格式唯一性时, 作者选取了特定测试函数: 对 (6a) 取 $\boldsymbol{\psi}_h = \hat{\mathbf{E}}^{n+1/2}$ (经插值), 对 (6b) 取 $\boldsymbol{\phi}_h = P_h(\mu^{-1}\hat{\mathbf{B}}^{n+1/2})$, 其中 $P_h$ 是到 $RT_h$ 的 $L^2$ 正交投影 (满足 $\operatorname{curl}\mathbf{v}_h$ 在与 $\mu^{-1}\hat{\mathbf{B}}^{n+1/2}$ 内积时, 可用投影互换而不改变值, 见命题 A.1)。于是 (6a) 中的 $-(\mu^{-1}\hat{\mathbf{B}}^{n+1/2}, \operatorname{curl}\hat{\mathbf{E}}^{n+1/2})$ 与 (6b) 中的 $(\operatorname{curl}\hat{\mathbf{E}}^{n+1/2}, \mu^{-1}\hat{\mathbf{B}}^{n+1/2})$ 相加即相消, 最终得到 $\frac{1}{2\Delta t}\|\sqrt{\varepsilon}\hat{\mathbf{E}}^{n+1}\|_{L^2}^2 + \frac14\|\sqrt{\sigma}\hat{\mathbf{E}}^{n+1}\|_{L^2}^2 + \frac{1}{2\Delta t}\|\sqrt{\mu^{-1}}\hat{\mathbf{B}}^{n+1}\|_{L^2}^2 = 0$, 从而 $\hat{\mathbf{E}}^{n+1}=\hat{\mathbf{B}}^{n+1}=0$, 给出唯一性, 此过程中不涉及对双曲型部分的正则性要求。该测试函数的选择同样导出了每步精确的能量恒等式 (9), 这是后续稳定性分析和误差估计的基础。

2. 适用于低正则数据的离散能量方法与 Grönwall 不等式技巧
直接从能量等式出发, 引用 Cauchy–Schwarz 和 Young 不等式得到偏差不等式, 但直接使用简单的离散 Grönwall 引理会因 $\|\sqrt{\varepsilon}\mathbf{E}_h^{n+1/2}\|^2$ 的混合时刻结构而失效。作者通过引入辅助量 $S^n = \mathcal{E}^n + \sum_{k=0}^{n-1}\Delta t \|\sqrt{\sigma}\mathbf{E}_h^{k+1/2}\|^2$ 并精细配平系数, 构造出如下递推:

$$(1-\alpha) S^{n+1} \le (1+\alpha) S^n + c\Delta t \|\varepsilon^{-1/2}\mathbf{f}^{n+1/2}\|^2, \quad \alpha=\gamma\Delta t/2,$$

进而乘以 $\rho^{-(n+1)}$ ($\rho=(1+\alpha)/(1-\alpha)$) 后叠加, 得到一致的稳定性估计 (10)。这一技术完全避免了在 Grönwall 过程中产生依赖于 $h,\Delta t$ 的常数, 从而在源仅为 $L^2$ 数据时仍保证能量的多项式增长控制。

3. 控制以 $\operatorname{curl}$ 提升的形式: 避免额外约束
控制项 $\operatorname{curl}\mathbf{z}$ 进入 Ampère 定律, 本质上是将控制空间选为 $L^2(0,T;H(\operatorname{curl};\Omega_{\text{ctrl}}))$。这带来两个好处: (i) 自动散度自由, 无需像 [3] 那样显式施加 $\operatorname{div}\mathbf{z}=0$ 的约束; (ii) 在弱形式中 $\operatorname{curl}\mathbf{z}$ 作为右端项处理, 只需 $\operatorname{curl}\mathbf{z}\in L^2(0,T;L^2(\Omega)^3)$, 与源 $\mathbf{f}$ 具有相同正则性, 因而前向问题的低正则理论可直接应用。代价是控制空间比 $L^2(0,T;L^2(\Omega)^3)$ 更强, 但作者指出若不需要电荷守恒, 可放松到 $L^2(L^2)$ 并以恒等算子代替 $\operatorname{curl}$。

4. 伴随系统的低正则推导与结构保持离散
在连续层面, 作者通过在低正则对偶空间中写出弱形式的伴随方程, 从而避免了通常伴随方程中材料张量的逆或高阶导数的假设; 梯度公式依赖伴随变量, 确保了从优化理论到数值离散的自然过渡。在离散时, 对伴随方程采用与状态方程完全相同的 CN‑Nédélec/RT 离散, 从而离散伴随状态保持与离散状态相同的能量关系和 de Rham 结构。此外, 利用 Nédélec 和 RT 空间之间的交换投影 (commuting projectors), 证明了离散约化梯度的精确一致: 即离散最优条件等价于连续最优条件的 Galerkin 投影。

5. 离散最优控制收敛的紧性论证
为证明离散最优三元组的收敛, 需要同时控制状态、伴随状态以及控制。作者的证明思路是: 先利用离散能量估计得到状态序列的均匀有界性, 通过紧嵌入 (如 $H^1(0,T; X^*)$ 嵌入 $C([0,T]; L^2)$) 抽取强收敛子列; 对于控制, 因为目标泛函含 $\|\mathbf{z}\|_Z^2$, 控制序列在 $Z$ 中有界, 利用弱紧性得弱收敛子列; 结合最优性条件 (一阶必要条件) 取极限, 并利用仿射控制‑状态映射的弱连续性, 最终证明任何极限点满足连续最优条件。严格凸性保证了极限的唯一性, 从而全序列收敛。

4. 与之前工作的比较

本文与既有工作的区别与推进主要体现在以下几个维度。

1. 与前向问题低正则离散化理论的比较
本文的前向格式直接建立在 [1] 的半离散分析之上。[1] 仅考虑了空间离散 (Nédélec/RT) 在半离散水平下的收敛性, 没有时间离散, 且并未涉及最优控制。本文将 [1] 的结果推广到全离散 Crank–Nicolson 格式, 并首次在数据仅为 $L^2$ (源) 和 $L^2$ (初值) 的条件下证明了全离散解的收敛性。
更广泛的文献如 [8,9,10,14,15,24] 研究了 Maxwell 方程弱解的适定性, 但未涉及该低正则情形下的完全离散分析; 结构保持有限元文献 [16,12,4,19] 主要关注单纯前向模拟的高阶正则性或者理想导体边界条件下的能量恒等式, 而本文的收敛结果显式依赖于低正则假设, 并给出了与 $h,\Delta t$ 无关的稳定性常数。
时间步进方面, Crank–Nicolson 的能量稳定性在 [7,17,20,21] 等文献中被广泛研究, 但通常需要更光滑的数据或仅给出简单能量衰减。本文的能量等式 (9) 是演算每步精确成立的, 即使 $\sigma\ge 0$ 时也不例外, 且后续离散 Grönwall 分析处理了带 $\sigma$ 耗散项与源项同时存在的更一般情形。

2. 与已有 Maxwell 最优控制工作的详细对比
作者明确指出, 处理全时域 Maxwell 方程最优控制的文献极为稀少, [3] 是唯一被直接提及并详细比较的工作。本文相较于 [3] 在以下四个方面有实质推进: (a) 目标泛函的时间范围: [3] 仅考虑终端时刻 ($t=T$) 的目标 (terminal‑time tracking), 而本文目标是对整个时间区间积分, 这使得控制能作用于整个时间演化, 特别适用于隐身 (cloaking) 问题, 需要在所有时刻将场压制为零。 (b) 控制变量的时空依赖性: [3] 假设控制具有空间‑时间分离形式 $\mathbf{z}(x,t) = u(x)a(t)$, 这极大限制了控制的自由度。本文允许完全非分离的 $\mathbf{z}(x,t)$, 图 1 的数值实验表明, 分离变量控制在同一设置下远不如全时空依赖控制有效。 (c) 电荷守恒制约的方式: [3] 显式施加散度为零的约束 $\operatorname{div}\mathbf{z}=0$, 并要求控制具有额外的时间正则性以处理约束; 本文创新性地采用 $\operatorname{curl}\mathbf{z}$ 作为控制输入, 自动满足 $\operatorname{div}(\operatorname{curl}\mathbf{z})=0$, 且不需要对 $\mathbf{z}$ 施加减度条件, 只需 $\mathbf{z}\in L^2(0,T;H(\operatorname{curl}))$, 这恰好与状态方程右端源的 $L^2(L^2)$ 正则性匹配。 (d) 离散优化的收敛性理论: [3] 未针对所使用的离散化 (未指定是否为低正则设置) 提供离散最优控制向连续最优控制的收敛性证明。本文则给出了严格的收敛性分析: 借助前向格式的结构保持性质, 证明了离散最优三重态的收敛, 并在严格凸性下得到全序列收敛。这对数值优化的可靠性提供了理论保证。

3. 与一般最优控制中结构保持思想的关系
在更广泛的最优控制领域, “结构保持”通常指保持能量、约束或辛结构等。本文将 de Rham 结构保持、能量恒等式保持与 Gauss 定律保持同时融入离散优化中, 达到离散梯度一致, 这不仅仅是格式的物理保真, 更直接保证了离散最优性系统是连续最优性系统的相容逼近, 传统优化文献中这类分析很少见, 尤其是在 Maxwell 方程这样的复杂双曲系统中。

4. 局限性与可扩展性
本文当前理论限定在凸二次跟踪型目标与线性控制输入 (无控制约束或有简单闭凸约束) 的情形; 作者指出, 若电荷守恒不必须, 控制空间可弱化为 $L^2(0,T;L^2(\Omega)^3)$, 代价是损失自动散度自由。此外, 数值实验仅限于源隐身, 但框架可直接应用于其他目标场。对于非光滑区域或更强非线性, 本文未涉及。

2. 高度非均匀介质中带有阻抗边界条件的时谐 Maxwell 方程的多尺度建模

Multiscale Modeling for Time-harmonic Maxwell equations with impedance boundary conditions in highly heterogeneous media

作者: Xiang Zhong, Eric T. Chung, Xingguang Jin

类别: math.NA

发布日期: 2026-04-29

arXiv ID: 2604.26449v1

摘要翻译: 在非均匀介质中建模时谐 Maxwell 问题带来了重大的数学和计算挑战。由于 Maxwell 方程固有的非椭圆结构和非强制性，传统方法面临严重的数值不稳定性，特别是在高对比度介质和高波数下。这些挑战往往导致病态的离散系统和过高的计算成本，限制了它们的实际应用。为了克服这些挑战，我们针对高对比度介质中带有阻抗边界条件的时谐 Maxwell 方程，引入了一个高效的多尺度框架。本研究的一大创新点在于避免了在多尺度基函数上施加显式的无散约束。为此，通过引入质量项和 Silver-Müller 型边界罚项的局部谱问题，构造了一个辅助空间。这种新颖的设计保证了对应双线性形式的强制性，并自动将旋度算子的核排除在主导特征空间之外。在辅助空间的基础上，我们使用一个不同的双线性形式来构造多尺度空间。通过利用分辨率条件并建立关键范数关系，我们严格证明了该修正双线性形式的强制性——这一关键性质支撑了整个分析。理论分析表明，通过适当的过采样，该方法实现了与局部对比度无关的 $O (H)$ 收敛性，且逼近误差随波数 $k$ 增加而增大。报告了大量的数值实验，以验证所提出方法的有效性。

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链接: http://arxiv.org/abs/2604.26449v1

1. 研究对象和背景

本文研究固定角频率 $\omega>0$ 下，自由空间（相对介电常数 $\varepsilon_r\equiv 1$）中含有高度非均质磁性材料时的时谐 Maxwell 方程。相对磁导率 $\mu_r(x)\in L^\infty(\Omega)$ 可在几个数量级间剧烈变化，满足 $0<\mu_{\min}\leq\mu_r(x)\leq\mu_{\max}$ a.e. $x\in\Omega$，且材料无导电性（$\sigma\equiv 0$）。域 $\Omega\subset\mathbb{R}^3$ 为有界连通 Lipschitz 域。待求电场 $\mathbf{u}$ 满足以下边值问题：

$$ \begin{cases} \operatorname{curl}\big(\mu_r^{-1}\operatorname{curl}\mathbf{u}\big)-k^2\mathbf{u}=\mathbf{f}, &\text{在 }\Omega\text{ 内},\\[2mm] \mu_r^{-1}\operatorname{curl}\mathbf{u}\times\mathbf{n}-ik\mathbf{u}_T=\mathbf{g}, &\text{在 }\partial\Omega \text{ 上}, \end{cases} $$

其中 $k=\omega/c_0>0$ 是自由空间波数，$c_0=1/\sqrt{\varepsilon_0\mu_0}$ 为真空光速。边界条件为阻抗边界条件——一阶 Silver-Müller 吸收边界条件（阻抗参数 $\lambda=1$，且在 $\partial\Omega$ 附近 $\mu_r=1$，见 [26]）。虚数单位 $i$ 引入相移，使得边界可部分透射和反射入射波，从而在有限计算域 $\Omega$ 内模拟无界散射和辐射问题。物理上，该模型描述空气或真空中电磁波受强非均质纯磁性夹杂或超材料（如铁氧体、磁斗篷、$\mu$-近零结构、磁性光子晶体、磁性雷达吸波材料）扰动时的散射与传播。

传统数值方法（如标准有限元或有限差分法）在介质微观非均质性显著时计算代价极高：为分辨精细结构需要极细网格，导致代数系统规模巨大且条件数极差 [15]。高波数情形尤其困难。经典均匀化理论虽可导出宏观有效模型，但其适用性受限于严格的结构假设（如周期性）[1]。

多尺度方法通过将细尺度信息嵌入粗尺度基函数来克服上述困难。已发展出多种方法，如多尺度有限元法 (MsFEM) [10,17]、局部正交分解 (LOD) [24]、变分多尺度法 [18]、数值升级 [29] 和异质多尺度法 (HMM) [8,9,11]。在电磁波领域，已有基于多尺度渐近分析和 HMM 的粗尺度模型 [2,7,13]，并扩展到时域分析 [16]、高对比度材料 [32] 和 LOD 计算均匀化 [14]。然而，这些方法的性能通常依赖于显式构造稳定校正子或特定的结构性质（如周期性）。面对极端、非周期非均质时，保持稳定性和精度仍需要苛刻的分辨率要求或大量过采样。

约束能量最小化广义多尺度有限元法 (CEM-GMsFEM) [3,4,35] 提供了一种鲁棒的替代方案：通过能量最小化受局部约束来构造多尺度基函数，天然适应高对比度系数，无需特定的拟插值稳定性，并可系统控制能量范数误差。本文在 CEM-GMsFEM 框架下，针对上述时谐 Maxwell 问题（带阻抗边界条件）提出一种新型多尺度方法。其核心创新在于避免对多尺度基函数施加显式无散约束——通过构造局部谱问题时引入质量项和 Silver-Müller 型边界惩罚，自动排除 curl 算子的核，从而摆脱了现有 Maxwell 多尺度方法中对无散条件的依赖。这项设计在已有文献中未被充分解决 [2,7,13,14,16,32]，构成方法论上的实质推进。

2. 主要定理或主要结果

本文的理论分析基于如下分辨率条件（Assumption 1）：

$$ kH\sqrt{\mu_{\max}}\Lambda^{-1/2} < \sqrt{\frac{e_c}{2}}, $$

其中 $0 < e_c\ll 1$，且存在 $0 < c_0<1$ 使 $2e_c\leq 2e_c\Lambda\leq c_0<1$。$\Lambda=\min_{1\leq i\leq N}\lambda^i_{l_i+1}$ 为局部谱问题第 $l_i+1$ 个特征值的最小值，满足与细网格尺寸 $h$、粗网格尺寸 $H$ 和磁导率 $\mu_r$ 无关 [12]。

主要结果如下：

1. 修改双线性形式的强刚（Lemma 4.1） 在 Assumption 1 下，存在与 $k, \mu_r, H, \Lambda$ 无关的常数 $0<\alpha<1$，使得

$$ |B(\mathbf{v},\mathbf{v}) + s(\pi\mathbf{v},\pi\mathbf{v})| \geq \alpha\big(\|\mathbf{v}\|_a^2 + \|\pi\mathbf{v}\|_s^2\big),\qquad \forall\mathbf{v}\in\mathbf{V}. $$

特别地，对满足 $\pi\mathbf{v}=0$ 的 $\mathbf{v}\in\widetilde{\mathbf{V}}$，有

$$ |B(\mathbf{v},\mathbf{v})| \geq \alpha\|\mathbf{v}\|_a^2. $$

这里 $B$ 为原问题的 sesquilinear 形式，$s(\cdot,\cdot)$ 定义为 $s(\mathbf{w},\mathbf{v})=\big(\mu_r^{-1}H^{-2}\mathbf{w},\mathbf{v}\big)$，$\pi$ 为局部正交投影，范数 $\|\mathbf{v}\|_a^2=\sum_i\big(\mu_r^{-1}\operatorname{curl}\mathbf{v},\operatorname{curl}\mathbf{v}\big)_{K_i}+k^2(\mathbf{v},\mathbf{v})_{K_i}+k\langle\mathbf{v}_T,\mathbf{v}_T\rangle_{\partial K_i\cap\partial\Omega}$。

2. 全局问题的 inf-sup 稳定性（Theorem 4.2） 存在常数 $\beta(k,\mu_{\min})>0$，使得

$$ \inf_{\mathbf{v}\in\mathbf{V}_{\mathrm{glo}}\setminus\{\mathbf{0}\}}\sup_{\mathbf{v}^*\in\mathbf{V}_{\mathrm{glo}}^*\setminus\{\mathbf{0}\}} \frac{|B(\mathbf{v},\mathbf{v}^*)|}{\|\mathbf{v}\|_{k,\mathrm{imp}}\|\mathbf{v}^*\|_{k,\mathrm{imp}}} \geq \beta(k,\mu_{\min}) > 0, $$

其中 $\beta(k,\mu_{\min})=\frac{1}{2C_k\max\{\sqrt{k},k\}}\big(1+\frac{1}{\alpha}\max\{\mu_{\min}^{-1},1\}\big)^{-1}$。这保证了全局多尺度方法（使用所有局部基函数的全局扩展构造的试验与检验空间）的适定性。

3. 全局多尺度解的误差估计（Theorem 4.3） 设 $\mathbf{u}_{\mathrm{glo}}\in\mathbf{V}_{\mathrm{glo}}$ 为全局问题 (14) 的解，$\mathbf{u}\in\mathbf{V}$ 为原问题 (2) 的真解，则

$$ \|\mathbf{u}_{\mathrm{glo}}-\mathbf{u}\|_a \leq \frac{1}{\alpha\sqrt{\Lambda}}\|\mathbf{f}\|_{s^{-1}}. $$

该估计表明，收敛率由 $\Lambda^{-1/2}$ 控制——$\Lambda$ 越大，误差越小；而 $\Lambda$ 可通过选取每个粗单元上足够多的局部特征函数 $l_i$ 来增大。误差上界与波数 $k$、对比度 $\mu_{\max}/\mu_{\min}$ 并无直接依赖（但隐含于 $\alpha$ 及范数 $\|\mathbf{f}\|_{s^{-1}}$ 中）。

4. 局部化与衰减性质（Lemma 4.4 与后续分析） 对任意辅助空间函数 $\mathbf{v}_{\mathrm{aux}}\in\mathbf{V}_{\mathrm{aux}}$，存在 $\mathbf{v}\in\mathbf{V}$ 满足 $\pi(\mathbf{v})=\mathbf{v}_{\mathrm{aux}}$，$\|\mathbf{v}\|_a^2\leq D\|\mathbf{v}_{\mathrm{aux}}\|_s^2$，且 $\operatorname{supp}(\mathbf{v})\subset\operatorname{supp}(\mathbf{v}_{\mathrm{aux}})$。常数 $D=2\max\{2c',1\}\big(1+\max_{i,j}\lambda_j^i\big)$ 仅依赖于局部最大特征值，与 $H,k,\mu_r$ 无关。结合全局基函数的衰减特性（文中后续通过截断函数和能量估计给出），可证明通过适当的过采样层数 $m$，局部多尺度基函数以指数速率逼近全局基函数，从而局部多尺度解 $\mathbf{u}_{\mathrm{ms}}$ 达到与 $\mathbf{u}_{\mathrm{glo}}$ 相同阶的精度：

$$ \|\mathbf{u}_{\mathrm{ms}}-\mathbf{u}\|_a \lesssim \frac{1}{\sqrt{\Lambda}} + \text{（由过采样导致的衰减项）}. $$

综上，本文证明了：选取恰当的过采样层数与局部特征函数个数，方法可达到 $O(H)$ 的收敛阶，且该收敛阶与介质局部对比度无关；逼近误差随波数 $k$ 增加而增大（虽然并未给出对 $k$ 的显式幂指数，但数值实验印证了这一趋势）。

3. 研究方法、关键技术和核心工具

本文的核心技术难题在于：时谐 Maxwell 方程的双线性形式 $B(\mathbf{w},\mathbf{v})=(\mu_r^{-1}\operatorname{curl}\mathbf{w},\operatorname{curl}\mathbf{v})-k^2(\mathbf{w},\mathbf{v})-ik\langle\mathbf{w}_T,\mathbf{v}_T\rangle$ 既非椭圆也非强刚，其 curl 算子具有无穷维奇核。传统多尺度方法必须引入无散约束或使用正则化来压制核中的成分。本文的根本创新在于在局部谱问题中同时引入质量项和边界惩罚项，使得构造辅助空间时自动排除 curl 核的主导影响，从而免除了显式无散约束。这显著简化了基函数构造并提升了稳定性。

具体技术要点分述如下：

1. 强刚局部谱问题的设计 对每个粗单元 $K_i$，求解特征值问题：找到 $(\lambda_j^i,\boldsymbol{\phi}_j^i)\in\mathbb{R}\times\mathbf{V}(K_i)$ 满足

$$ a_i(\boldsymbol{\phi}_j^i,\mathbf{v})=\lambda_j^i s_i(\boldsymbol{\phi}_j^i,\mathbf{v}),\qquad\forall\mathbf{v}\in\mathbf{V}(K_i), $$

其中

$$ a_i(\mathbf{w},\mathbf{v})=\int_{K_i}\mu_r^{-1}(\operatorname{curl}\mathbf{w})\cdot(\operatorname{curl}\mathbf{v})\,dx + k^2\int_{K_i}\mathbf{w}\cdot\mathbf{v}\,dx + k\int_{\partial K_i\cap\partial\Omega}\mathbf{w}_T\cdot\mathbf{v}_T\,ds, $$

$$ s_i(\mathbf{w},\mathbf{v})=\int_{K_i}\mu_r^{-1}H^{-2}\mathbf{w}\cdot\mathbf{v}\,dx. $$

由于 $k>0$，$a_i(\cdot,\cdot)$ 在 $\mathbf{V}(K_i)$ 上是强刚的（因其包含 $k^2\|\mathbf{v}\|_{L^2(K_i)}^2$ 项），立即推出最小特征值 $\lambda_1^i>0$，并且 curl 的核（即梯度场）不会出现在主导特征空间中（Remark 3.1）。这一设计完全避免了在局部特征问题中施加 $\operatorname{div}(\mu_r^{-1}\boldsymbol{\phi})=0$ 之类的困难约束。

2. 修改双线性形式的强刚性证明 在构造多尺度基函数时，使用了不同于 $a_i$ 的双线性形式 $B(\cdot,\cdot)+s(\pi\cdot,\pi\cdot)$（见 (9) 式）。由于 $B$ 本身不具有强刚性，这是本文分析中最关键的一步。作者利用分辨率条件将 $B(\mathbf{v},\mathbf{v})+s(\pi\mathbf{v},\pi\mathbf{v})$ 与 $a(\mathbf{v},\mathbf{v})+s(\pi\mathbf{v},\pi\mathbf{v})$ 联系起来，其证明思路为：

$$ B(\mathbf{v},\mathbf{v})+s(\pi\mathbf{v},\pi\mathbf{v}) = \underbrace{a(\mathbf{v},\mathbf{v})+s(\pi\mathbf{v},\pi\mathbf{v})}_{\text{正项}} - 2k^2(\mathbf{v},\mathbf{v}) - (k+ik)\langle\mathbf{v}_T,\mathbf{v}_T\rangle. $$

通过范数等价关系和正交投影估计，将 $-2k^2\|\mathbf{v}\|_{L^2}^2$ 控制为

$$ 2k^2\|\mathbf{v}\|_{L^2}^2 \leq 2k^2\mu_{\max}H^2\|\mathbf{v}\|_s^2 \leq 4\mu_{\max}k^2H^2\big(\|\pi\mathbf{v}\|_s^2+\Lambda^{-1}\|\mathbf{v}\|_a^2\big). $$

利用分辨率条件 $4\mu_{\max}k^2H^2

3. 正交投影与空间分解 借助辅助空间及其投影 $\pi$，引入空间 $\widetilde{\mathbf{V}}=\{\mathbf{v}\in\mathbf{V}:\pi\mathbf{v}=0\}$。由强刚性推出 $B$ 在 $\widetilde{\mathbf{V}}$ 上是强刚的。再利用全局基函数满足的 Galerkin 正交性 $B(\mathbf{v}_{\mathrm{glo}},\widetilde{\mathbf{v}})=0$ 对 $\widetilde{\mathbf{v}}\in\widetilde{\mathbf{V}}$ 成立，可将全空间分解为 $\mathbf{V}=\widetilde{\mathbf{V}}\oplus\mathbf{V}_{\mathrm{glo}}$（关于内积 $B(\cdot,\cdot)+s(\pi\cdot,\pi\cdot)$）。同样的分解适用于伴随空间 $\mathbf{V}_{\mathrm{glo}}^*$。这为建立 Petrov-Galerkin 方法的 inf-sup 条件提供了便利。

4. 局部化与衰减估计 为证明局部多尺度基函数是对全局基函数的良好逼近，需建立其指数衰减性质。文中采用标准的 CEM-GMsFEM 框架：通过构造合适的截断函数和 bubble 函数，在局部能量最小化问题中证明对任意辅助函数 $\mathbf{v}_{\mathrm{aux}}$ 存在局部支撑的 $\mathbf{v}$ 精确实现投影 $\pi(\mathbf{v})=\mathbf{v}_{\mathrm{aux}}$ 且能量受控（Lemma 4.4）。这本质上表明辅助空间具有“局部可构造性”。进一步，通过比较全局约束问题和局部约束问题解的差异，利用边界估计导出基函数的尾部关于过采样层数呈指数衰减。此部分详情在 PDF 摘录中未能完整展示，但属于 CEM-GMsFEM 的经典分析路线。

5. 多尺度空间的构造与 Petrov-Galerkin 格式 最终的多尺度试验空间 $\mathbf{V}_{\mathrm{ms}}$ 和检验空间 $\mathbf{V}_{\mathrm{ms}}^*$ 分别由 (9) 和 (11) 定义的局部算子作用于辅助基函数生成。该构造保证了解析解的正交投影与空间分解协调，从而将全局误差拆分为投影误差和局部化误差，前者由 $\Lambda$ 控制，后者由过采样层数控制。

总之，本文的方法论特点在于以强刚化的局部谱问题为辅助，通过分辨率条件证明一种修正的双线性形式的强刚性，并利用正交投影和空间分解技巧建立完整的误差分析，最终实现了对高对比度 Maxwell 问题无需显式无散约束的多尺度求解。

4. 与之前工作的比较

本文在以下几个维度上与已有工作形成显著区别和推进：

1. 与 Maxwell 多尺度方法的对比（无散约束的免除） 已有的 Maxwell 多尺度方法，如基于渐近展开和 HMM 的 [2,7,13,16,32] 以及 LOD 计算均匀化 [14]，在构造粗尺度基或校正子时，常常需要显式处理 curl 算子的核。例如，为获得稳定且精确的粗尺度模型，通常需要满足某种无散条件（弱或强）来消除梯度场对精度的污染。这在实际计算中增加了复杂性，尤其在高对比度非均匀介质中，恰当构造无散基函数并非易事。本文通过设计带边界惩罚和质量项的局部谱问题，使得辅助空间自然排除 curl 的核，从而在整个方法流程中完全避开了显式无散约束。正如作者所强调的，这是“现有文献中未被充分解决的新设计”（a novel departure）。该创新不仅简化了算法实现，也为理论分析开辟了更直接的路径。

2. 与 CEM-GMsFEM 的经典框架对比（双线性形式的差异） 标准 CEM-GMsFEM（如 [3,4,35]）通常在第一阶段（辅助空间）和第二阶段（多尺度空间）中使用同一个强刚双线性形式（例如对于椭圆问题使用能量内积）。但在本文的 Maxwell 问题中，第一阶段采用的 $a_i(\cdot,\cdot)$ 是强刚的，而第二阶段不得不面对原问题的非强刚形式 $B(\cdot,\cdot)$。这意味着需要重新证明第二阶段构造中所用双线性形式 $B+s(\pi\cdot,\pi\cdot)$ 的强刚性，而它并非天然强刚。作者借助分辨率条件建立了该强刚性（Lemma 4.1），从而使 CEM-GMsFEM 的分析框架得以推广至此非强刚体系。这是对 CEM-GMsFEM 理论的重要扩展——表明即使在底层算子非强刚的情况下，只要通过合适的辅助设计和分辨率假设，仍可建立稳定的多尺度方法。

3. 边界条件处理 现有 Maxwell 均匀化或多尺度工作多处理完美电导体 (PEC) 边界或周期性边界，较少涉及有实际意义的吸收边界条件。本文明确将 Silver-Müller 阻抗边界条件纳入多尺度框架，并在局部谱问题中通过边界积分项 $k\int_{\partial K_i\cap\partial\Omega}\mathbf{w}_T\cdot\mathbf{v}_T ds$ 体现，使得基函数在物理边界处自动满足合适的传输条件。这种处理在理论和应用上均更贴近真实电磁散射问题。与 [14] 的 LOD 均匀化相比，[14] 未考虑阻抗边界条件，因此本文在此方面具有明确的扩充。

4. 对波数 k 的多尺度分析 文中明确指出：“the approximation error increases with the wave number $k$”，并在数值实验中验证。虽然并未给出显式的 $k^\gamma$ 依赖关系，但 Assumption 1 中的尺度关系 $kH\sqrt{\mu_{\max}} < \text{常数}\cdot\sqrt{\Lambda}$ 已经揭示了方法对波数的要求：高波数下需减小粗网格尺寸或增大 $\Lambda$（即增加局部基函数个数）以保持分辨率条件。这是对 Maxwell 多尺度方法在高频区适用性的直接反映。相比之下，许多先前的多尺度分析要么假设 $k$ 固定且中等，要么通过渐近展开假定 $k$ 与细尺度无关。本文的工作为在给定波数下选取合适的计算参数提供了理论指导。

5. 收敛性与对比度的无关性 本文证明的 $O(H)$ 收敛率与局部对比度 $\mu_{\max}/\mu_{\min}$ 无关，仅受控于 Λ 和过采样层数。这继承了 CEM-GMsFEM 在椭圆问题中的优势，但在 Maxwell 情形下需要克服非强刚性带来的额外困难。与之对照，传统的 MsFEM 或基于插值的方法往往在对比度极高时因缺乏稳定性而失效，必须引入精心设计的局部校正或使用较大的过采样区域。本文方法通过内在的谱分辨和能量最小化，自然获得对比度鲁棒性，这在高对比度超材料模拟中尤为重要。

6. 局限与代价

波数敏感性：方法需要满足分辨率条件 $kH\sqrt{\mu_{\max}}<\text{常数}$，这意味着高频模拟要求粗网格尺寸 $H$ 随 $k$ 减小而反比缩小，或者增加局部特征函数个数（增大 Λ），这会在高波数下导致计算量上升。文中未分析具体复杂度，但该条件本质上与标准有限元在均匀介质中的“pollution effect”有相似之处：为保证固定精度，网格需随 $k$ 加密。
过采样需求：虽然通过指数衰减可以控制过采样区域大小，但衰减速率依赖于各局部问题的谱隙和对比度分布。在极端参数下可能需要较多层数，并未给出定量的衰减指数与物理参数的关系（PDF 提供的部分未展开）。
假设条件：文中假设边界邻域内 $\mu_r=1$ 且 $\varepsilon_r=1$，这可能限制了对某些超材料模型的直接应用，例如带阻抗包层的器件。
与 LOD 的比较：作者未显式展开与 LOD 工作 [14] 的定量比较（如误差常数或计算成本），但指出 LOD 型均匀化未处理阻抗边界条件，且本文提供了一个完整的理论-数值框架。

总体而言，本文在 Maxwell 方程的多尺度计算领域做出了重要推进：通过设计新型局部谱问题和理论分析，打破了无散约束的壁垒，将 CEM-GMsFEM 成功拓展到非强刚的时谐 Maxwell 系统，并给出了包含阻抗边界条件的鲁棒算法及严格的收敛性理论。

3. 基于量化交互规则的无漂移保守动力学

Drift-Free Conservative Dynamics from Quantized Interaction Rules

作者: Park Junhu, Youngsoo Ha, Myungjoo Kang

类别: math.NA, physics.comp-ph

发布日期: 2026-04-29

arXiv ID: 2604.26383v1

摘要翻译: 守恒律通常通过浮点通量离散化，其不变量由近似界面贡献的抵消得到，而容许弱解则由重构和黎曼求解器选取。本文引入一种算子层面的表述，其中保守动力学在量化状态空间上实现为一个精确的离散交互规则。更新由一个反对称的整数传输算子定义，该算子在算术层面精确保证守恒，并从原始演化中消除了舍入漂移 \cite{highamAccuracyStabilityNumerical2002}。对于标量律，单调保序传输在原始更新内部选取容许的激波结构，而不是通过通量重构。数值实验表明，该交互规则在奈奎斯特极限附近保持高频输运，并在伯格斯动力学中维持尖锐的局部化间断。同样的构造通过定向的、向量值的整数传输推广到多维问题和守恒律系统。这些结果表明，保守动力学允许一个精确的离散实现，其中不变性和熵选择均在算子层面被编码，而不是源于近似的通量抵消。

Comment: 4 pages, 4 figures,

链接: http://arxiv.org/abs/2604.26383v1

1. 研究对象和背景

本文研究对象是标量双曲守恒律

$$ \partial_t u + \partial_x f(u) = 0, $$

其中 $u=u(x,t)$ 为未知保守量，$f$ 是给定的通量函数。该方程描述了物理中质量、动量、能量等守恒量的局部平衡关系，其弱解通常不唯一，需通过熵条件选取物理可允许的间断解。

在标准数值方法中，守恒律通常通过有限体积或有限差分框架下的浮点通量计算进行离散。守恒性依赖于相邻单元界面通量贡献的成对抵消，即

$$ U_i^{n+1} = U_i^n - \frac{\Delta t}{\Delta x} \big( F_{i+1/2}^n - F_{i-1/2}^n \big), $$

其中数值通量 $F_{i+1/2}^n$ 通过重构过程和 Riemann 求解器构造，从而在近似意义下满足守恒性。然而，这种做法的守恒性仅保持在机器精度内，且原始更新依赖于舍入行为、求和顺序以及重构选择，使得离散演化与精确守恒律之间存在间隙。此外，可允许弱解的选择（如熵条件）由重构和 Riemann 求解器在通量层面实现，而非直接在离散算子中体现。

本文试图从一个不同的视角看待此问题：将守恒动力学直接实现为一个离散交互过程，而非通过通量近似。作者考虑将状态变量量子化，即设

$$ q_i^n \in \mathbb{Z}, \quad u_i^n \approx \delta q_i^n, $$

其中 $\delta>0$ 为量子化尺度。此时，守恒更新由局部反对称整数转移算子定义，通量函数被 $\phi^\pm(q)$ 替代，这些函数将连续通量信息映射为整数传输量。该方案的思路可追溯到 Shannon 的采样与量化理论，将连续信号用离散状态近似，但本文将其用于动力学演化，并强调守恒性源自整数运算的代数性质，而非浮点近似抵消。

本文试图填补的空白是：构造一种不依赖于浮点通量评估的守恒离散动力学，使守恒性成为更新的精确代数不变量，同时将熵选择机制内嵌于原始算子之中，消除舍入误差漂移，并在高频和间断结构保持方面表现出优势。

2. 主要定理或主要结果

本文定义了一种称为“通量量子化数值方法”（FQNM）的离散演化格式，其核心结果如下。

1. 精确守恒的量子化更新
对一维标量守恒律，设量化状态 $q_i^n \in \mathbb{Z}$，更新公式为

$$ q_i^{n+1} = q_i^n - \big( F_{i+1/2}^n - F_{i-1/2}^n \big), $$

其中整数界面转移量由

$$ F_{i+1/2}^n = \phi^+(q_i^n) + \phi^-(q_{i+1}^n) $$

给出，且 $\phi^\pm$ 取为

$$ \phi^\pm(q) = \operatorname{round}\!\left( \frac{\Delta t}{\Delta x} \frac{f^\pm(\delta q)}{\delta} \right), $$

这里 $f^\pm$ 是通量分裂后的正/负部分，$\operatorname{round}(\cdot)$ 为四舍五入取整运算。由于每个界面转移量以相反符号同时进入相邻单元，更新满足

$$ \sum_i q_i^{n+1} = \sum_i q_i^n, $$

该恒等式在整数算术中精确成立，与求和顺序和浮点舍入无关。

2. 量化误差与收敛性
作者证明了量化误差的界：

$$ \| u - \delta q \|_{\infty} \le \frac{\delta}{2}. $$

由此可直接得到，当 $\delta \to 0$ 时，量化状态 $\delta q$ 一致收敛于连续状态 $u$。结合差分格式的相容性，作者断言该量子化演化收敛到对应守恒律的熵解，且在任意有限 $\delta$ 下均维持精确守恒。熵可允许性来源于原始整数转移规则，而非对连续方程的外部施加。

3. 数值实验验证的性质

高频传输保持：FQNM 对接近 Nyquist 极限的高频波包传输几乎无额外耗散，相对误差远低于传统格式。
激波锐度与定位：在 Burgers 方程 $f(u)=u^2/2$ 的激波形成测试中，对初始条件 $u_0(x)=\sin(2\pi x/L)$，FQNM 得到的激波结构与 Hopf–Lax 熵解一致，且比相同网格下的一阶迎风格式和 WENO 格式的激波更锐利、位置更准确（见图 2、3）。
无漂移非线性演化：高分辨率下，FQNM 的激波位移小于 WENO5-RK3 格式，且无需单独的 Riemann 求解器阶段。
扩展到系统：在附录的 Shu–Osher 激波–熵波干扰测试中，对各守恒变量分量分别应用转移规则，FQNM 能在保持精确分量守恒的同时，清晰分辨激波与高频振荡结构（图 4）。

上述结果表明，保守动力学在离散层面上存在一种精确实现方式，其中守恒性和熵选择均编码在算子结构中，通量仅是粗粒化后涌现的描述。

3. 研究方法、关键技术和核心工具

本文的核心技术障碍在于如何将连续守恒律的演化完全转化为离散的、整数的交互规则，同时保持守恒性、熵稳定性和收敛性。作者的创新点和关键分析框架体现在以下几个方面。

1. 反对称整数转移算子的设计
核心思想是利用界面转移量的反对称结构实现精确守恒。更新 (3) 可写为

$$ q_i^{n+1} = q_i^n - \big[ \phi^+(q_i^n) + \phi^-(q_{i+1}^n) \big] + \big[ \phi^+(q_{i-1}^n) + \phi^-(q_i^n) \big], $$

可见每个 $\phi^+(q_i^n)$ 从单元 $i$ 流出到 $i+1/2$ 界面，而该界面上的 $\phi^-(q_i^n)$ 则流入单元 $i-1$，这种配对意味着总质量变化仅取决于边界项，内部完全抵消。这将守恒性从浮点近似提升到了代数恒等式级别。

2. 通量–转移量的量子化映射
关键在于如何从连续通量 $f(u)$ 构造出恰当的整数转移分量 $\phi^\pm$。作者使用通量分裂 $f = f^+ + f^-$，并将 $f^\pm(\delta q)/\delta$ 乘以 $\Delta t/\Delta x$ 后四舍五入。该构造保证了：

相容性：当 $\delta \to 0$ 时，$\phi^\pm(q)$ 的尺度行为与标准有限体积通量一致。
单调性：由于 $\operatorname{round}$ 函数是单调的，且 $f^\pm$ 通常满足单调性条件，因此转移规则是保序的，这为熵选择提供了离散基础（见下文）。
整数运算：所有更新仅涉及整数加减，消除了浮点误差积累和求和顺序依赖性。

3. 量化误差分析
由于 $q_i^n = \operatorname{round}(u_i^n/\delta)$，量化误差固定在 $\delta/2$ 以内，从而给出 $L^\infty$ 误差界 (7)。该误差界独立于时间步数，因为更新保持状态为整数，不会引入额外漂移。这构成了收敛性论证的基础：在 $\delta \to 0$ 极限下，离散演化逼近连续动力学。

4. 算子层面的熵选择机制
文中指出，对于标量守恒律，转移算子的单调保序性（由 $\operatorname{round}$ 和通量分裂的单调性保证）直接取代了传统 Riemann 求解器的作用。作者引用了 Crandall–Tartar 的经典结论：单调格式收敛到熵解。虽然本文未给出严格证明，但传递了如下观点：由于 $\phi^\pm$ 构造是基于迎风通量分裂的四舍五入版本，其整体更新继承了单调性，从而自动筛选出熵可允许的激波结构。数值实验（图 2）也表明，直接使用 Hopf–Lax 解的采样并不满足离散相容性条件，而 FQNM 的转移规则却能在相同网格下给出正确的激波中间态，这进一步佐证了离散算子本身蕴含了所需的熵条件。

5. 高波数保持特性的机制
传统迎风或 WENO 格式因其耗散性质会衰减高频分量。而 FQNM 的整数更新在高频传输时表现出极低耗散，原因在于：当 $q_i^n$ 在相邻点间快速振荡时，$\operatorname{round}$ 函数的阶梯特性使得转移量对小幅波动不敏感，类似于一种“死区”效应，从而保持波包形状。这与标准格式中基于局部多项式重构引入的耗散形成鲜明对比。

4. 与之前工作的比较

本文与已有的守恒律数值方法在多个层面存在根本差异，具体比较如下。

1. 与有限体积/差分通量格式的区别
传统格式（如 Godunov、Lax–Friedrichs、WENO 等）的核心是通过浮点计算通量在界面上的近似值，守恒性靠 $F_{i+1/2}$ 在相邻单元中一加一减抵消实现。该抵消仅在精确算术下成立，实际浮点计算会引入舍入误差，并受求和顺序影响。另外，传统格式需要显式构造数值通量函数，且通过通量限制器或 Riemann 求解器实现熵条件。本文的 FQNM 则直接用整数转移取代通量的近似计算，守恒性源于代数恒等式，与浮点精度无关，且不需求解 Riemann 问题——熵选择内化于转移算子的单调结构中。代价是引入了量子化误差 $\delta/2$，并限定了状态变量必须是可量化的（例如正值或有界）。

2. 与高阶格式（如 WENO）的性能对比
数值实验（图 3）显示，在同网格同 CFL 下，FQNM 的激波定位比 WENO5-RK3 更准确，且高频结构保持更好。传统 WENO 格式为了在光滑区获得高阶精度而在间断附近退化到低阶，但仍会引入一定的数值耗散，且可能出现轻微的非物理振荡。FQNM 通过单调整数转移自然避免了振荡，同时由于量化效应的存在，间断被“锁定”在更精确的位置。但 FQNM 本质上是一阶格式（基于迎风通量分裂的量子化），它在光滑区域的精度只有一阶，而 WENO 可达五阶。因此 FQNM 更适合对守恒性、间断分辨和高频保持要求极高而容许一阶精度的场景。

3. 与保结构格式（如保能量、保辛格式）的比较
保结构数值方法通常旨在保持连续系统的某些几何或物理结构（如总能量、动量、辛形式）。它们大多仍基于连续状态的浮点运算，守恒性是近似保持的。FQNM 则在离散层面彻底精确守恒（即便在有限精度下），但其代价是必须接受状态空间的量子化。此外，FQNM 的守恒性不依赖于时间步长的精细调节或特定的时间积分器。

4. 与格子 Boltzmann 方法（LBM）及元胞自动机的关联与区别
FQNM 在思想上与 LBM 有相似之处：两者都通过局部交互规则模拟宏观守恒律。但 LBM 的演化通常是基于速度空间浮点分布函数的碰撞-迁移过程，其守恒性通过分布函数的矩来实现，且需要复杂的平衡态分布。FQNM 则直接在物理空间上将守恒量量子化并使用整数转移，更加直接，且转移规则具有明确的反对称结构，这更利于严格证明守恒性。

5. 收敛性的严格性
本文提供的收敛性论证基于量化误差界和格式相容性，但并未给出严格的数值分析证明（如 TVD/TVB 性质、熵不等式等）。相比之下，经典格式如 Godunov 格式具有完整的 TVD 理论保证其收敛到熵解。虽然作者援引了单调格式收敛理论，但并未对 FQNM 的单调性给出解析证明，亦未讨论通量分裂形式对 $\phi^\pm$ 单调性的影响。因此，当前工作的收敛性仍停留在启发式层面，需要进一步理论支撑。

6. 适用范围与局限
FQNM 被扩展到了多维问题和方程组（通过分方向、分量的整数转移），但论文仅展示了二维问题（未详细说明）和一维 Euler 方程的 Shu–Osher 测试。对于强激波或接触间断的方程组，简单的分量转移可能无法捕捉正确的波系结构（如中间波），这一点作者未深入探讨。此外，量子化尺度 $\delta$ 的选择直接影响精度和稳定性，文中对 $\delta$ 的选取依据未做敏感性分析。

综上，FQNM 提供了一种从根本上无漂移的守恒离散范式，将守恒律的数值解与现代计算机的整数运算特性紧密结合，是对传统基于浮点通量范式的重要补充。在需要极端长时间无漂移演化或对高频结构有特殊要求的场景中，该方法可能具有独特优势。

了解领域论文（摘要翻译）

1. 有界、交换、保持离散迹的投影

Bounded, Commuting, Discrete-trace Preserving Projections

作者: Alexandre Ern, Johnny Guzmán, Pratyush Potu

类别: math.NA

发布日期: 2026-04-30

arXiv ID: 2604.28103v1

链接: http://arxiv.org/abs/2604.28103v1

摘要翻译

中文标题: 有界、交换、保持离散迹的投影

摘要翻译: 我们为三维 de Rham 复形构造了有界交换投影，其附加性质是：若函数/场的迹在相应的迹空间中是分片多项式，则这些投影保持该迹。这些投影是局部定义的，且在图范数下稳定。更精确地说，涉及外微分的图范数部分仅涉及该外微分在边界附近的一窄条单元中的振荡，并以局部网格尺寸加权。此外，当作用于外微分在相应空间中为分片多项式的函数/场时，投影是局部 $L^2$ 稳定的。我们给出了本有界、交换、保持离散迹投影的两个显著应用：分片多项式数据的稳定提升的构造，以及关于分片多项式数据的离散延拓与连续延拓的最优性结果。

2. 显式平面有限元弹性复形与重心加细上的 $C^1$ 元

Explicit Planar Finite Element Elasticity Complexes and $C^1$ Elements on Barycentric Refinements

作者: Chunyu Chen, Long Chen, Xuehai Huang

类别: math.NA

发布日期: 2026-04-29

arXiv ID: 2604.26701v1

链接: http://arxiv.org/abs/2604.26701v1

摘要翻译

中文标题: 显式平面有限元弹性复形与重心加细上的 $C^1$ 元

摘要翻译: 重心加细上平面弹性问题的高阶混合有限元的 Arnold–Douglas–Gupta 族背后的正合序列结构被显式表达。在每个宏三角形上，对称应力空间是通过用三个局部支集函数来丰富多项式应力而得到的。我们推导了这些丰富的闭式公式，并确定了产生它们的显式 Airy 势函数。这导致了一个具体的 Hsieh–Clough–Tocher 型 $C^1$ 势空间，其 Airy 像恰好是 Arnold–Douglas–Gupta 应力空间。通过施加单值自由度，我们在单连通区域上获得了整体空间和一个完全显式的有限元弹性复形。作为结果，我们构造了重心加细上的一族新的 $C^1$ 有限元，包括二次、三次、四次和更高阶元。

不相关论文（基本信息）

1. 高级空中机动运行中监视系统的可靠性、鲁棒性与弹性建模

Reliability, Robustness, and Resilience Modeling for Surveillance System in Advanced Air Mobility Operations

作者: Esrat Farhana Dulia, Caleb Adams, Syed Arbab Mohd Shihab, Ruben Del Rosario

类别: math.NA

发布日期: 2026-05-01

arXiv ID: 2605.00823v1

链接: http://arxiv.org/abs/2605.00823v1

摘要: Ensuring the safe and efficient operation of Advanced Air Mobility (AAM) in low-altitude airspace requires a reliable, robust, and resilient surveillance system capable of continuously detecting, identifying, and tracking aircraft under both normal and off-nominal conditions. To address this need, this study develops a comprehensive 3R modeling framework, reliability, robustness, and resilience, for the Surveillance for Advanced Air Mobility (SAM) system, with a focus on the optimal design and operation of a multi-type sensor network. Under normal operating conditions, the reliability model determines the baseline sensor types, quantities, and locations required to satisfy surveillance coverage and detection requirements. To address external perturbations, such as adverse weather conditions or sudden increases in AAM traffic demand, the robustness model identifies additional sensor requirements needed to maintain system performance. Furthermore, for surveillance outages caused by primary sensor failures, the resiliency model develops backup sensor deployment and dispatch strategies to provide temporary surveillance coverage, minimize operational disruptions, and support the safe continuation of AAM operations.

2. HyCOP: 用于可解释偏微分方程学习的混合复合算子

HyCOP: Hybrid Composition Operators for Interpretable Learning of PDEs

作者: Jinpai Zhao, Nishant Panda, Yen Ting Lin, Eirik Valseth, Diane Oyen, Clint Dawson

类别: cs.CE, cs.LG, math.NA

发布日期: 2026-05-01

arXiv ID: 2605.00820v1

链接: http://arxiv.org/abs/2605.00820v1

摘要: We introduce HyCOP, a modular framework that learns parametric PDE solution operators by composing simple modules (advection, diffusion, learned closures, boundary handling) in a query-conditioned way. Rather than learning a monolithic map, HyCOP learns a policy over short programs - which module to apply and for how long - conditioned on regime features and state statistics. Modules may be numerical sub-solvers or learned components, enabling hybrid surrogates evaluated at arbitrary query times without autoregressive rollout. Across diverse PDE benchmarks, HyCOP produces interpretable programs, delivers order-of-magnitude OOD improvements over monolithic neural operators, and supports modular transfer through dictionary updates (e.g., boundary swaps, residual enrichment). Our theory characterizes expressivity and gives an error decomposition that separates composition error from module error and doubles as a process-level diagnostic.

3. 模形式与$ζ(3)$的有理逼近的数值探索

Modular Forms and Numerical Explorations of Rational Approximations to $ζ(3)$

作者: Cynthia Bortolotto, Lucas Oliveira

类别: math.NT, math.NA

发布日期: 2026-05-01

arXiv ID: 2605.00673v1

链接: http://arxiv.org/abs/2605.00673v1

摘要: We revisit Beukers’ modular-form proof of the irrationality of $ζ(3)$ from the point of view of the auxiliary weight two modular form. For the Fricke group $Γ_0(6)^\star$, we show that Beukers’ choice is not isolated: it belongs to a one-parameter affine family. These approximations have the same exponential decay as the classical Apéry approximations and satisfy the same denominator-growth estimate needed in Beukers’ irrationality argument. We then apply the same construction to several other genus-zero Fricke groups.

4. 用于多面体网格上间断伽辽金方法的连续三维有限元子网格基函数

Continuous 3D Finite Element Subgrid Basis Functions for Discontinuous Galerkin Methods on Polyhedral Meshes

作者: Sixtine Michel, Lorenzo Diazzi, Walter Boscheri

类别: math.NA

发布日期: 2026-05-01

arXiv ID: 2605.00619v1

链接: http://arxiv.org/abs/2605.00619v1

摘要: We present a novel high-order accurate nodal discontinuous Galerkin (DG) method for solving nonlinear hyperbolic systems of partial differential equations (PDEs) on fully unstructured three-dimensional polyhedral meshes. A mesh generator is firstly discussed in detail, which ensures the generation of admissible control volumes. For the first time, we then extend the concept of agglomerated finite element (AFE) basis functions to polyhedral grids. In this context, the discrete solution is represented within each polyhedral element using piecewise continuous polynomials of degree N, defined on an internal tetrahedral subgrid. The AFE basis functions are therefore constructed by agglomerating standard finite element basis functions on each sub-tetrahedron of the computational cell. This allows for the precomputation of universal local matrices (mass and stiffness) on the reference element given by the unit tetrahedron, enabling a quadrature-free implementation that remains efficient even on highly irregular polyhedral meshes. High-order of accuracy in time is achieved using a local spacetime Galerkin predictor as part of the ADER approach, applied independently within each polyhedral element. To ensure robustness in the presence of discontinuities such as shocks, an artificial viscosity limiter is embedded into the numerical scheme, allowing for controlled dissipation and stabilization without compromising the overall accuracy in smooth regions. To demonstrate the robustness and accuracy of the method, we validate it through different three-dimensional benchmark problems for the compressible Euler and Navier-Stokes equations.

5. 参数依赖字典下信号解混的最小二乘局部几何

Local Geometry of Least Squares for Unmixing Signals with Parameter-Dependent Dictionaries

作者: Santos Michelena, Maxime Ferreira Da Costa, José Picheral

类别: eess.SP, math.NA

发布日期: 2026-05-01

arXiv ID: 2605.00585v1

链接: http://arxiv.org/abs/2605.00585v1

摘要: Modeling signals as linear combinations of atoms from a dictionary is ubiquitous in modern signal processing. In the finite-dimensional setting, whenever atoms depend nonlinearly upon unknown parameters, the signal model is said to be separable. In this work, we study least-squares reconstruction of separable signals and establish a unified theoretical framework for their analysis. We introduce the unmixing metric, a distance that captures the distinct roles and sensitivities of linear and nonlinear parameters, and establish local convergence and stability guarantees under its topology. We then analyze variable projection from a geometric perspective, showing that it corresponds to restricting the optimization to the manifold of optimal linear parameters. This viewpoint provides a principled explanation for the improved algorithmic behavior of variable projection observed in practice, and produces sharp theoretical guarantees. The generic theory for separable problems is specialized to the case of point spread function (PSF) unmixing. We introduce a parametric notion of coherence and show that support separation directly controls both the size of the convergence region and the stability of recovery. Numerical experiments corroborate the theoretical predictions and demonstrate the practical relevance of the proposed framework.

6. 将PageRank、时间反演与策略评估联系起来

Linking PageRank, Time Reversal, and Policy Evaluation

作者: Konstantin Avrachenkov, Lorenzo Gregoris, Nelly Litvak

类别: math.OC, math.NA, math.PR

发布日期: 2026-05-01

arXiv ID: 2605.00532v1

链接: http://arxiv.org/abs/2605.00532v1

摘要: We establish a connection between policy evaluation in Markov decision processes and PageRank in network analysis. For a fixed policy, we show that the value function of a discounted Markov decision process can be obtained, up to an explicit rescaling, from the PageRank vector of a suitably defined time-reversed Markov chain. In this correspondence, the discount factor plays the role of the teleportation parameter, while rewards induce the restart distribution. Beyond the irreducible case, invoking quasi-stationary distributions and Doob $h$-transforms, we prove a general decomposition theorem showing that policy evaluation for arbitrary finite MDPs reduces to a collection of PageRank problems on the recurrent and transient components of the policy-induced Markov chain. This framework naturally extends to undiscounted MDPs with terminal states and to transition-dependent rewards. We conclude by showing efficiency of our approach on a numerical example of a sticky random walk on large deterministic and random graphs.

7. 用于时空分数阶扩散偏微分方程的深度皮卡迭代

Deep-Picard Iteration for Space-time Fractional Diffusion PDEs

作者: Zhijun Zeng, Zhitong Chen, Ling Qin, Yi Zhu

类别: math.NA, math.PR

发布日期: 2026-05-01

arXiv ID: 2605.00456v1

链接: http://arxiv.org/abs/2605.00456v1

摘要: We propose a Deep-Picard iteration framework for high-dimensional nonlinear space-time fractional diffusion equations.The method is based on a nonlinear fractional Feynman–Kac fixed-point formulation, which replaces direct discretization of the Caputo memory term and the nonlocal fractional Laplacian by Monte Carlo simulation of the associated fractional dynamics. Each Picard update is approximated by stochastic label generation and realized through supervised neural-network regression, thereby avoiding residual minimization involving fractional differential operators. The fractional trajectories are generated by coupling a discretized beta-stable subordinator with a walk-on-spheres-type simulation of the rotationally symmetric alpha-stable Lévy process. Numerical experiments on two-dimensional and high-dimensional test problems ddemonstrate stable Picard convergence and accurate approximation, with tests reported up to dimension d=100.

8. 一种由观测位移估计岩浆储层状态的变分方法

A variational approach to estimating the state of a magma reservoir from observed displacement

作者: Shungo Kun Tonoyama, Atsushi Suzuki, Takemasa Miyoshi

类别: math.NA

发布日期: 2026-05-01

arXiv ID: 2605.00409v1

链接: http://arxiv.org/abs/2605.00409v1

摘要: We propose a numerical procedure to solve an inverse problem that estimates the state of a magma reservoir from observed surface displacement of a volcano. Our variational approach aims to find the minimizer of a cost function consisting of a norm concerning both data and derivative, which evaluates the misfit between the estimated and observed displacement. The extremal of the cost function leads to a linear system, to find the stress distribution on the reservoir surface, has very high condition number, but it is feasible to get appropriate solution by using high precision arithmetic.

9. 超越独立性：关于贝叶斯反演中的联合正态先验

Beyond Independence: on Jointly Normal Priors in Bayesian Inversion

作者: Ruanui Nicholson, Matti Niskanen, Oliver J. Maclaren, Jari P. Kaipio

类别: stat.ME, math.NA

发布日期: 2026-05-01

arXiv ID: 2605.00332v1

链接: http://arxiv.org/abs/2605.00332v1

摘要: We consider joint inversion for two or more unknown parameters from observational data in the Bayesian framework. Standard approaches often either treat the parameters as independent or impose structural similarity through regularisation terms that can be difficult to interpret statistically. We instead construct jointly Gaussian prior models with prescribed Gaussian marginals, so that correlation between the parameters can be incorporated without altering the marginal prior distributions. We propose a joint covariance construction that preserves the marginals, allows spatially varying cross-correlation, and supports uncertainty and inference in the correlation itself. The construction is valid for any strict contraction encoding the desired cross-correlation and is optimal in a canonical correlation sense under the principal square root factorisation. We demonstrate the method using prior sampling and several inference examples: a low-dimensional illustrative example and two higher-dimensional examples, including a PDE-constrained problem. The examples highlight both the potential pitfalls of ignoring or neglecting uncertainty in the correlation as well as reinforcing a key principle of the Bayesian paradigm: unknown quantities included in a model should be treated as random variables.

10. $(\star, \epsilon)$-回文矩阵多项式的谱分解及其应用

Spectral decomposition of $(\star, ε)$-palindromic matrix polynomial and its applications

作者: Kang Zhao, Xin Wang, Xiaoxiao Ma

类别: math.NA

发布日期: 2026-05-01

arXiv ID: 2605.00328v1

链接: http://arxiv.org/abs/2605.00328v1

摘要: This paper provides the spectral decomposition of $(\star,ε)$-palindromic quadratic matrix polynomial $P(λ)$ by a standard pair and a parameter matrix. When $J$ is assumed to be a block diagonal matrix, the parameter matrix $Γ$ has a special structure. And then the spectral decomposition is applied to solve the inverse eigenvalue problem and the eigenvalue embedding problem with no spill-over.

11. 自适应各向异性复合求积法用于神经PDE逼近的残差最小化

Adaptive anisotropic composite quadratures for residual minimisation in neural PDE approximations

作者: Santiago Badia, Kishore Nori

类别: math.NA

发布日期: 2026-05-01

arXiv ID: 2605.00308v1

链接: http://arxiv.org/abs/2605.00308v1

摘要: We study the role of numerical quadrature in residual-minimisation methods for neural network approximation of partial differential equations. We first present an abstract error framework that separates approximation, quadrature and optimisation errors, and derive a nonlinear Strang-type estimate quantifying how inaccuracies in the discrete loss affect the final approximation. Motivated by this analysis, we propose an anisotropic adaptive composite quadrature strategy that controls the relative quadrature error of the residual loss using richer reference quadratures and bisection-based refinement. We then introduce a refresh-based training methodology that rebuilds the quadrature only when an online error indicator exceeds a prescribed threshold, balancing accuracy and computational cost. Numerical experiments on a range of benchmark problems show that the proposed approach narrows the gap between training and reference losses, uses quadrature points more efficiently and delivers strong approximation accuracy relative to non-adaptive quadrature strategies.

12. 一个狄拉克-弗伦克尔-昂萨格原理：用于偏微分方程解的非线性参数化的带有规范动量的瞬时残差最小化

A Dirac-Frenkel-Onsager principle: Instantaneous residual minimization with gauge momentum for nonlinear parametrizations of PDE solutions

作者: Matteo Raviola, Benjamin Peherstorfer

类别: cs.LG, math.NA, stat.ML

发布日期: 2026-04-30

arXiv ID: 2605.00284v1

链接: http://arxiv.org/abs/2605.00284v1

摘要: Dirac-Frenkel instantaneous residual minimization evolves nonlinear parametrizations of PDE solutions in time, but ill-conditioning can render the parameter dynamics non-unique. We interpret this non-uniqueness as a gauge freedom: nullspace directions that leave the time derivative unchanged can be used to select better-conditioned parameter velocities. Building on Onsager’s minimum-dissipation principle, we introduce a history variable – interpretable as momentum – and inject it only along the nullspace directions. The resulting Dirac-Frenkel-Onsager dynamics preserve instantaneous residual minimization, in contrast to standard regularization that can introduce bias, while promoting temporally smooth parameter evolutions. Examples demonstrate that the approach leads to increased robustness in singular and near-singular regimes.

13. 连续力源浸入边界法中超越一阶精度及其良条件的基于投影的求解

Beyond first-order accuracy in continuous-forcing immersed boundary methods, and their well-conditioned projection-based solution

作者: Diederik Beckers, H. Jane Bae, Andres Goza

类别: math.NA, physics.flu-dyn

发布日期: 2026-04-30

arXiv ID: 2604.28152v1

链接: http://arxiv.org/abs/2604.28152v1

摘要: We introduce a refined immersed boundary (IB) methodology that is better-than-first-order accurate in practice, while preserving key properties of “continuous-forcing” IB approaches that retain a singular source term in the governing equations. Our method leverages a smoothed indicator (Heaviside) function, following ideas from multiphase flow and immersed layers formulations, to recast the IB solution as a composite of distinct interior and exterior fields. We demonstrate that, when cast through this composite-solution lens, prior continuous-forcing IB methods can be seen as neglecting terms in the governing and constraint equations that restrict the solution to first-order accuracy. We incorporate these terms to systematically improve accuracy without the need for heuristic corrections. In canonical Poisson problems, we empirically demonstrate second-order convergence, and in incompressible Navier-Stokes simulations the method achieves slightly sub-second-order performance. While our present study focuses on these cases, the framework suggests a path towards second-order accuracy or higher, with further extensions. This perspective reframes accuracy limitations typically attributed to IB schemes. Although continuous-forcing IB methods are often reported to be only first-order accurate, we show that neither smoothing nor interface interpolation inherently restricts attainable order. Moreover, we naturally incorporate this higher-order formulation into a projection-based solution process. The resulting algorithm simultaneously mitigates the spurious surface stresses produced by ill-conditioned linear systems and reduces sensitivity to geometric resolution, addressing both conditioning and accuracy concerns within a unified approach.

14. 具有边界条件不完全信息的Stokes方程的近最优恢复算法

A near-optimal recovery algorithm for the Stokes equations with incomplete information on the boundary conditions

作者: Andrea Bonito, Diane Guignard

类别: math.NA

发布日期: 2026-04-30

arXiv ID: 2604.28051v1

链接: http://arxiv.org/abs/2604.28051v1

摘要: We address the problem of numerically approximating the velocity and pressure governed by the Stokes system when the boundary conditions are only partially known and thus do not uniquely determine the velocity-pressure couple. We propose an algorithm that takes advantage of available linear measurements of the velocity and pressure to construct a numerical approximation. This approximation is guaranteed to be near-optimal in the sense that it approximates the velocity-pressure couple that minimizes, in the energy norm, the distance to all other solutions satisfying the measurements and the Stokes system.

15. 噪声诱导的模态寿命增强——基于确定性轨迹的数据驱动方法

Noise-induced enhancement of regime lifetimes – A data-driven approach using deterministic trajectories

作者: Henry Schoeller, Robin Chemnitz, Péter Koltai, Maximilian Engel, Stephan Pfahl

类别: math.NA

发布日期: 2026-04-30

arXiv ID: 2604.27991v1

链接: http://arxiv.org/abs/2604.27991v1

摘要: We investigate the lifetime of dynamical regimes under the impact of noise motivated by low-dimensional models of the atmosphere. One may expect that the inclusion of noise tends to make the system leave prescribed regions of the state space faster. However, for relevant systems with complexities ranging from phenomenological toy models to reduced models of atmospheric dynamics, this intuition has proven misleading. As long as the noise is sufficiently small, the noisy system stays in regimes of interest on average longer than its deterministic counterpart, an effect we call ``stochastic inertia’’. This phenomenon has been observed through extensive numerical simulations for different noise levels. We propose a numerical technique for testing the occurrence of stochastic inertia, constructing, for any fixed noise level, a Markov chain on the set of points given by a sufficiently long trajectory of the system without noise. The method is shown to correctly predict the presence of stochastic inertia in simple systems, and its utility is demonstrated on a paradigm model of atmospheric dynamics.

16. 灵活GMRES分两阶段收敛

Flexible GMRES converges in two phases

作者: Stefan Güttel, Lauri Nyman

类别: math.NA

发布日期: 2026-04-30

arXiv ID: 2604.27971v1

链接: http://arxiv.org/abs/2604.27971v1

摘要: We derive a sharp upper bound on the residuals produced by the flexible GMRES (FGMRES) method. The bound shows that FGMRES exhibits two phases of convergence depending on the residual tolerance of the inner preconditioner. For small tolerances, the convergence of FGMRES is practically geometric with a constant rate throughout, while for looser tolerances the two-phase behavior becomes more pronounced. We also show that the derived bound cannot be improved and construct an example for which it becomes an equality.

17. 非线性热粘弹性中的标架无差异离散化：分析与数值模拟

Frame-indifferent discretization in nonlinear thermoviscoelasticity: Analysis and numerical simulations

作者: Rufat Badal, Manuel Friedrich, Martin Horák, Martin Kružík, Lennart Machill

类别: math.AP, math.NA

发布日期: 2026-04-30

arXiv ID: 2604.27923v1

链接: http://arxiv.org/abs/2604.27923v1

摘要: We consider a quasi-static nonlinear model in thermoviscoelasticity at a finite-strain setting in the Kelvin-Voigt rheology where both the elastic and viscous stress tensors comply with the principle of frame indifference under rotations. We refine the discretization schemes in [Badal-Friedrich-Kružík ‘23, Mielke-Roubíček ‘20] by imposing frame indifference already at a time-discrete level. This is justified both analytically and numerically.

18. 多项式权重的默比乌斯变换梯形法则

Möbius-transformed trapezoidal rule for polynomial weights

作者: Nuutti Hyvönen, Yuya Suzuki

类别: math.NA

发布日期: 2026-04-30

arXiv ID: 2604.27758v1

链接: http://arxiv.org/abs/2604.27758v1

摘要: This work studies numerical integration by the Möbius-transformed trapezoidal rule, which combines the classical trapezoidal rule with a change of variables induced by a Möbius transformation that maps the unit circle onto the real line. It is shown that this method achieves the optimal convergence rate for a polynomially weighted integral over the real line if the integrand lives in a related polynomially weighted Sobolev space with positive integer smoothness index. This result can also be generalized in a slightly weaker form for fractional smoothness indices via complex interpolation of function spaces. The algorithm only requires pointwise evaluations of the weight and the target integrand at prescribed nodes that do not depend on the integrand and weight in question. The established theoretical convergence rates are verified by numerical experiments.

19. 异质曲线网格上DGSEM的h-自适应有限体积子单元激波捕捉方法

h-Adaptive FV Subcell Shock-Capturing for DGSEM on Heterogeneous Curvilinear Meshes

作者: Anna Schwarz, Jens Keim, Christian Rohde, Andrea Beck

类别: math.NA

发布日期: 2026-04-30

arXiv ID: 2604.27735v1

链接: http://arxiv.org/abs/2604.27735v1

摘要: High-order methods offer superior dispersion and dissipation properties compared to low-order schemes but require robust stabilization for discontinuities. To ensure stability, local artificial viscosity is common, but often degrades sub-element resolution. Conversely, subcell resolution preserving limiting strategies such as the finite volume subcell method are typically restricted to uniform topologies, such as purely hexahedral, or simplex meshes. This leaves a significant gap in treating the hybrid-element topologies necessary for complex engineering geometries. This paper presents a robust shock-capturing approach for the discontinuous Galerkin spectral element method on mixed curvilinear meshes containing hexahedral, prismatic, tetrahedral, and pyramid elements. Non-hexahedral elements are handled via collapsed coordinate transformations. The proposed method utilizes an h-adaptive finite volume subcell scheme with arbitrary subcell resolution; 2N + 1 in this work. The schemes essential properties, including conservation, spatial convergence, and the shock capturing capabilities are verified. Finally, the method’s applicability to complex configurations is demonstrated through a simulation of the flow around a NACA 0012 airfoil.

20. 二阶完全非线性偏微分方程的自适应深度Ritz框架

An adaptive Deep Ritz framework for second-order fully nonlinear partial differential equations

作者: Alexandre Caboussat, Martin T. Leclercq, Anna Peruso

类别: math.NA

发布日期: 2026-04-30

arXiv ID: 2604.27731v1

链接: http://arxiv.org/abs/2604.27731v1

摘要: As an alternative to PINNs, a Deep Ritz framework is proposed to solve fully nonlinear PDEs. A least-squares algorithm is advocated to decouple the nonlinearities from the variational features of several fully nonlinear PDEs. A splitting method allows to iteratively solve local nonlinear problems and linear variational problems at each iteration. While existing nonlinear solvers are applied to solve for nonlinearities, we propose a novel coupling with a Deep Ritz neural network approach that is well-suited to the variational flavor of the linear variational problems. An adaptive sampling strategy for the selection of collocation points is incorporated to increase the efficiency of the algorithm without sacrificing its accuracy. Numerical experiments are presented to solve the Dirichlet problem for several fully nonlinear equations, starting with the prototypical Monge-Ampère equation, showing the flexibility of the approach. Numerical results are compared with results obtained using a full PINNs approach. Finally, numerical experiments are extended to address the optimal transport Monge-Ampère problem with transport boundary conditions.

21. 具有广义电容的场/电路耦合微分-代数方程的波形松弛法

Waveform Relaxation for Field/Circuit Coupled DAEs with Generalized Capacitances

作者: Idoia Cortes Garcia, Jonas Pade

类别: math.NA

发布日期: 2026-04-30

arXiv ID: 2604.27597v1

链接: http://arxiv.org/abs/2604.27597v1

摘要: Field/circuit coupling is a common approach when a lumped representation of a certain electrotechnical device is not accurate enough. To exploit existing code and underlying properties of the coupled systems, cosimulation techniques such as waveform relaxation can be used. The coupled system is of differential-algebraic type, which can potentially lead to divergence. This paper presents a novel, sufficient topological convergence criterion for field/circuit coupled systems of higher index containing a generalized capacitance. Hereby, the criterion holds for a full range of field systems whose structure can be classified as a generalized capacitance. Finally, the theoretical results are supported by numerical simulations.

22. 光学层析成像中频域与时域雅可比矩阵的蒙特卡洛模拟计算

Computation of frequency- and time-domain Jacobians in optical tomography with Monte Carlo simulations

作者: Pauliina Hirvi, Jaakko Olkkonen, Qianqian Fang, Ilkka Nissilä

类别: physics.comp-ph, math.NA, physics.optics

发布日期: 2026-04-30

arXiv ID: 2604.27558v1

链接: http://arxiv.org/abs/2604.27558v1

摘要: Significance: Jacobians, or spatially resolved sensitivity profiles, are central to image reconstruction in model-based optical tomography of biological tissue. Although Monte Carlo (MC) simulations are the gold standard for modeling light transport in turbid media, methodology for frequency- and time-domain Jacobians remains incomplete. Aim: This work extends MC to directly compute absorption and scattering Jacobians for frequency-domain (amplitude and phase) and time-domain (intensity and mean time-of-flight) measurements and prism-terminated optical fiber detectors. Approach: Jacobians are derived in the perturbation MC framework and implemented in the high-performance, open-source Monte Carlo eXtreme (MCX) simulator. Results are validated against the diffusion approximation (DA) solved using the finite element method in neonatal head models. MC with split voxels on curved surfaces is extended to Jacobian computation. The detector model is implemented in post-processing and compared with isotropic reception at surface. Results: MC- and DA-derived Jacobians show excellent agreement only in high-scattering regimes, highlighting the importance of MC for low-scattering domains. The detector model reduces surface sensitivity and marginally increases sensitivity to deeper tissues at short (< 2 cm) source-detector separations. Conclusion: A complete theoretical framework and MC software for computing frequency- and time-domain Jacobians is provided. Realistic detector modeling is encouraged for short-separation channels.

23. 参数化驱动的任意拉格朗日-欧拉方法用于大变形等几何流固耦合

Parameterization-driven arbitrary Lagrangian-Eulerian method for large-deformation isogeometric fluid-structure interaction

作者: Jingya Li, Ye Ji, Hugo Verhelst, Henk den Besten, Matthias Möller

类别: math.NA

发布日期: 2026-04-30

arXiv ID: 2604.27537v1

链接: http://arxiv.org/abs/2604.27537v1

摘要: Body-fitted arbitrary Lagrangian-Eulerian (ALE) methods provide a sharp representation of the fluid-structure interface but rely on mesh-update strategies that incrementally deform a reference configuration. To address this issue, we reformulate the ALE mesh-motion problem in the isogeometric setting as a sequence of independent domain parameterization problems. At each time step, a multi-patch spline parameterization of the fluid domain is constructed from the current interface geometry. Three technical components realize this framework: (i) a barrier-function-based spline parameterization that enforces a strictly positive Jacobian at every time step; (ii) a tangential-slip reparameterization that handles unbounded cumulative rotations of closed domains, where no fixed boundary-to-parameter correspondence is admissible; and (iii) a constant-preserving quasi-interpolation operator for solution transfer between consecutive parameterizations, ensuring that the discrete geometric conservation law holds algebraically. We validate the method on three two-dimensional FSI benchmarks, covering standard and large-rotation regimes, and on a three-dimensional rotor problem. On a rotating-square benchmark, the tangential-slip strategy enables simulations under sustained rotation far beyond the range accessible to classical mesh-update schemes–a regime that is fundamentally inaccessible to any mesh-deformation formulation, not merely numerically difficult. A three-dimensional rotor example further demonstrates that the framework extends naturally to volumetric spline parameterizations. Finally, we show that the per-step spline parameterizations can be used directly within a standard finite element solver.

24. 耦合修正Korteweg-de Vries方程的数值逆散射变换

Numerical inverse scattering transform for the coupled modified Korteweg-de Vries equation

作者: Wen-Xin Zhang, Yong Chen

类别: nlin.SI, math.NA

发布日期: 2026-04-30

arXiv ID: 2604.27513v1

链接: http://arxiv.org/abs/2604.27513v1

摘要: This paper develops the numerical inverse scattering transform (NIST) framework for the coupled modified Korteweg-de Vries (mKdV) equation based on its associated Riemann-Hilbert problem. The coupled system gives rise to a $3\times3$ matrix-valued Riemann-Hilbert problem, whose jump matrix and scattering data have a more involved structure than in the scalar case. This matrix setting makes the extension of NIST to the coupled system nontrivial, both in the direct scattering computation and in the numerical solution of the inverse problem. Within this framework, the scattering data are first computed by solving the matrix direct scattering problem using a Chebyshev collocation method with suitable mappings. The Deift-Zhou nonlinear steepest descent method is then used to analyze and deform the oscillatory Riemann-Hilbert problem. In particular, the phase function admits two stationary points symmetric about the origin, and the analysis leads to a division of the $(x,t)$-plane into three regions with corresponding contour deformations. Compared with traditional numerical methods, the NIST computes the solution directly at prescribed spatial and temporal points without relying on time-stepping. Numerical experiments illustrate the performance of the proposed NIST in long-time simulations and indicate that it captures the main asymptotic features of the coupled mKdV solutions.

25. 求解Poisson-Nernst-Planck-Navier-Stokes方程的间断Galerkin IMEX压力校正格式

Discontinuous Galerkin IMEX Pressure Correction Scheme for the Poisson-Nernst-Planck-Navier-Stokes Equations

作者: Bikram Bir, Amiya K. Pani

类别: math.NA

发布日期: 2026-04-30

arXiv ID: 2604.27512v1

链接: http://arxiv.org/abs/2604.27512v1

摘要: Based on a discontinuous Galerkin method in the spatial directions and an improved implicit-explicit pressure-correction scheme in the temporal direction, this paper discusses a fully discrete scheme for the Poisson-Nernst-Planck-Navier-Stokes equations. Optimal error estimates are derived in $L^2$ and in the energy norms for the concentrations of positive and negative ions, the electrostatic potential, the fluid velocity, and the $L^2$ norm of the fluid pressure. The discrete mass conservation properties of both ions are established. Finally, numerical simulations are performed, whose results confirm our theoretical findings.

26. 一种标量守恒律的神经网络方法及激波解的收敛速度: 扩展版

A Neural Network Method for Scalar Conservation Laws with Convergence Rates for Shock-Wave Solutions: extended version

作者: Jiachuan Cao, Buyang Li, Hao Li

类别: math.NA

发布日期: 2026-04-30

arXiv ID: 2604.27458v1

链接: http://arxiv.org/abs/2604.27458v1

摘要: We introduce an entropy-compatible neural network method for scalar hyperbolic conservation laws based on a computable surrogate of the Kružkov entropy residual. The proposed loss is designed to enforce weak consistency and entropy admissibility while remaining amenable to rigorous analysis. For piecewise smooth entropy solutions, we construct explicit neural network competitors with provably small loss by combining shock-adapted continuous piecewise linear approximations with representability results for neural networks. These approximation bounds, together with entropy-based stability estimates, yield rigorous $L^1$ error estimates for minimizers of the resulting loss functional. In particular, when the network size scales comparably to the number of degrees of freedom of a space time mesh of size $h$, we recover the classical $O(h^{1/2})$ convergence rate in shock-dominated regimes. The analysis also covers solutions containing rarefaction waves and regular shock interactions; in this extended version, it further treats smooth initial data that develop a shock in finite time, for which an $L^1$ estimate of order $O(h^{1/2}|\ln h|)$ is obtained. Numerical experiments in one and two space dimensions illustrate the effectiveness of the method and its ability to resolve nonsmooth wave structures accurately.

27. 适用于非界面贴合网格的平移内聚区方法及其在晶体塑性中的应用

A Shifted Cohesive-Zone Method for Non-Interface-Fitted Meshes with Applications to Crystal Plasticity

作者: Cheng-Hau Yang, Mark C. Messner, Tianchen Hu

类别: math.NA, math.AP

发布日期: 2026-04-30

arXiv ID: 2604.27407v1

链接: http://arxiv.org/abs/2604.27407v1

摘要: The accurate simulation of interface-dominated solid mechanics problems on complex microstructures remains challenging, particularly when interface-fitted quadrilateral or hexahedral meshes are difficult to generate. We extend the shifted boundary method (SBM) to cohesive-zone formulations and introduce the Shifted Cohesive Zone Method (SCZM), with applications to crystal plasticity on non-interface-fitted meshes. By shifting the enforcement of traction-separation laws from the true interface to a nearby surrogate interface, SCZM enables the use of standard finite element spaces while avoiding the meshing burden associated with interface-conformal discretizations. We present a simplified SCZM weak form defined on the surrogate interface, leading to a straightforward implementation of the nonlinear residual and consistent tangent matrix. The method is implemented in the open-source MOOSE framework and coupled with constitutive models from NEML2, enabling simulations with linear elasticity, multiple traction-separation laws, and history-dependent crystal plasticity. We further develop a geometry-aware, PCA-enhanced point classification algorithm to accelerate surrogate-domain construction. Verification and benchmark studies in two and three dimensions demonstrate that SCZM achieves first-order convergence for non-interface-fitted interface problems and closely matches interface-fitted reference solutions in terms of reaction forces, surface energy release, deformation, stress fields, and damage evolution. These results indicate that SCZM provides an accurate and efficient framework for modeling interface mechanics in complex microstructures without requiring interface-fitted meshes.

28. 短注：批高效分治特征分解算法

A Short Note on Batch-efficient Divide-and-Conquer Algorithm for EigenDecomposition

作者: Yue Song

类别: cs.LG, math.NA

发布日期: 2026-04-30

arXiv ID: 2604.27325v1

链接: http://arxiv.org/abs/2604.27325v1

摘要: EigenDecomposition (ED) is at the heart of many computer vision algorithms and applications. One crucial bottleneck limiting its usage is the expensive computation cost, particularly for a mini-batch of matrices in deep neural networks. Our previous work proposed a dedicated QR-based ED algorithm for batched small matrices (dim${<}32$). This short paper targets the limitation and proposes a batch-efficient Divide-and-Conquer based ED algorithm for larger matrices. The numerical test shows that for a mini-batch of matrices whose dimensions are smaller than $64$, our method can be much faster than the Pytorch SVD function.

29. 可压缩流的热力学约束信息几何正则化

Thermodynamically Constrained Information Geometric Regularization for Compressible Flows

作者: Seth Taylor, Raymond J. Spiteri, Stéphane Gaudreault

类别: math.NA

发布日期: 2026-04-30

arXiv ID: 2604.27286v1

链接: http://arxiv.org/abs/2604.27286v1

摘要: We construct and analyze a thermodynamic extension of the recently proposed information geometric regularization of Cao and Schäfer. The construction extends their shock-mitigating Hessian metric geometry using the Shannon entropy to constrain the regularized motion based on a thermodynamic length. Reformulating the equations in terms of mass and specific entropy explicitly connects the thermodynamic state to a position in the diffeomorphism group, allowing for a derivation of the regularized equations using an information geometric mechanics formalism based on geodesics on a Hessian manifold with a dual affine connection. The dynamics are defined using a pullback geometry for the Levi–Civita connection, describing constrained geodesic motion, and the cubic Amari–Chentsov tensor describing the information geometric correction. This new compressible fluid model introduces an anisotropic stress tensor to the momentum equation that vanishes along isentropic directions and an additional elliptic equation coupled to the barotropic regularization. Numerical simulations in one and two spatial dimensions demonstrate that the geometrically consistent incorporation of a thermodynamic constraint mitigates cusp singularities previously observed in other approaches while still maintaining the benefits of an inviscid regularization.

30. 具有一般权的再生核希尔伯特空间的嵌入

Embeddings of Reproducing Kernel Hilbert Spaces with General Weights

作者: Michael Gnewuch, Peter Kritzer, Klaus Ritter

类别: math.NA

发布日期: 2026-04-29

arXiv ID: 2604.27160v1

链接: http://arxiv.org/abs/2604.27160v1

摘要: We study embeddings between reproducing kernel Hilbert spaces $H(K)$ of functions of $d \in \mathbb{N} \cup \{\infty\}$ variables. The kernels $K$ are superpositions of weighted finite tensor products of a fixed univariate kernel. The basic idea for the embeddings is to compensate a change of the univariate kernel by a suitable transformation of the weights. For the proofs we employ ($d \in \mathbb{N}$) and develop ($d = \infty$) a discrete calculus on the cone of all weights, where completely monotone weights play a particular role. We sketch how to apply the embedding results to computational problems, as, e.g., numerical integration or function recovery.

31. 一种用于熵最小二乘的尺度-形状对偶牛顿法

A Scale-Shape Dual Newton Method for Entropic Least Squares

作者: Nicholas Barnfield, James V. Burke, Michael P. Friedlander, Tim Hoheisel

类别: math.OC, math.NA

发布日期: 2026-04-29

arXiv ID: 2604.27154v1

链接: http://arxiv.org/abs/2604.27154v1

摘要: We give a damped inexact Newton method for entropy-regularized least-squares on the nonnegative orthant that converges globally at a linear rate with $O(\logε^{-1})$ iteration complexity, locally at a superlinear-to-quadratic rate, and is immune to the finite-precision overflow that limits classical dual solvers. A scale-shape decomposition of the primal – separating its scale from its direction – produces a dual with a nonsingular Jacobian. Objectives and Jacobians are evaluated through stable log-sum-exp and softmax primitives. Lambert W bounds on the scale uniformly control the Jacobian’s spectrum, from which both rates follow. The solution map is jointly Lipschitz in the data, regularization parameter, and reference measure, and extends continuously to the vanishing-regularization limit. Experiments on a problem from analytic continuation of quantum Monte Carlo data confirm the predicted overflow resilience and convergence behavior.

32. 应用随机矩阵理论

Applied Random Matrix Theory

作者: Joel A. Tropp

类别: math.PR, math.NA

发布日期: 2026-04-29

arXiv ID: 2604.27119v1

链接: http://arxiv.org/abs/2604.27119v1

摘要: Random matrices now play a role in many parts of computational mathematics. To advance these applications, it is desirable to have tools that are flexible, easy to use, and powerful. Over the last 25 years, researchers have developed a remarkable family of results, called matrix concentration inequalities, that meet the criteria. This paper offers an invitation to the field of matrix concentration and its multifarious applications.

33. 二次函数商的零阶方法的推广

Generalization of Zeroth-Order Method for Quotients of Quadratic Functions

作者: Jonas Bresch

类别: math.OC, math.NA, math.PR

发布日期: 2026-04-29

arXiv ID: 2604.26913v1

链接: http://arxiv.org/abs/2604.26913v1

摘要: Optimization of quadratic functions and the quotient of those are relevant in subspace and iterative optimization methods. In this paper, the calculation of the generalized operator norm and extremal generalized Rayleigh quotient is considered. In contrast to recent works an unconstrained sampling approach on the entire sphere for the random search direction in each iteration is proposed. Furthermore, the link to zeroth-order methods for Riemannian first- and second-order optimization methods is provided in the sense that the Riemannian gradient and Hessian are estimated by the specific surrogates. Even though the tangent space is not used in this construction the optimal step size problem can be computed in a closed form. The subproblems of this and recent works are illuminated in the context of sub-generalized Rayleigh quotient problems on specific Gram matrices. Together the achieved theory allows to construct an accelerated algorithm which shows state-of-the-art behavior and outperforms recent works.

34. 微可压缩流的数据同化

Data assimilation for slightly compressible flow

作者: Aytekin Çıbık, Rui Fang

类别: math.NA, physics.flu-dyn

发布日期: 2026-04-29

arXiv ID: 2604.26896v1

链接: http://arxiv.org/abs/2604.26896v1

摘要: Continuous data assimilation (CDA) nudges observational data into governing equations to recover the underlying flow and improve predictions. Existing rigorous CDA analyses focus primarily on incompressible flows, yet no physical flow is perfectly incompressible. Approximating a slightly compressible flow with an incompressible model introduces non-negligible model errors. Data assimilation for compressible flows remains challenging due to strong nonlinearities and the presence of shocks. We design an algorithm that addresses the limitations of velocity-only nudging for slightly compressible flow. This work incorporates both velocity and pressure data from the slightly compressible flow and nudges both quantities into the incompressible Navier–Stokes equations. Our analysis shows that the model error decays exponentially in the initial error, with an asymptotic residual of order $\mathcal{O}(H)$, where H denotes the observation resolution. The analysis also identifies a scaling for the pressure nudging parameter $μ_1 = O(1/H^2)$ that ensures effective assimilation. We validate the theoretical results through a suite of numerical experiments: a convergence study confirming optimal rates, a modified Taylor–Green vortex benchmark demonstrating synchronization of energy, enstrophy, and pressure, and an acoustic wave propagation test that isolates the role of pressure nudging and achieves a $97.9\%$ reduction in pressure error relative to velocity-only assimilation. Together, these results provide a foundation for discrete error estimates and realistic compressible applications.

35. 带有记忆的非局部守恒律方程组及其零保持极限

Systems of Nonlocal Conservation Laws with Memory and Their Zero Retention Limit

作者: Aekta Aggarwal, Ganesh Vaidya

类别: math.NA, math.AP

发布日期: 2026-04-29

arXiv ID: 2604.26877v1

链接: http://arxiv.org/abs/2604.26877v1

摘要: We study the entropy solution for a class of systems of nonlocal conservation laws in which the convective flux is convoluted with a kernel in both spatial and temporal variables. This formulation models the flux dependence on the solution within its spatial neighbourhood (nonlocal in space) as well as on prior states in time (nonlocal in time), thereby incorporating memory effects. In addition, employing a convergent finite volume approximation, the existence of the entropy solution is discussed. The uniqueness of such entropy solutions is also established. In addition, we analyze the asymptotic behaviour of the solutions as the support of the temporal convolution kernel shrinks, demonstrating the “memory-to-memoryless” effect and convergence to the entropy solution of the corresponding nonlocal conservation law without memory (i.e., nonlocal only in space). Convergence rate estimates are derived. In addition, the proposed numerical approximations are shown to be asymptotically compatible with this passage to the memoryless limit by deriving the corresponding asymptotic convergence rate estimates. The analysis is carried out in a very general setting, without imposing any geometric restrictions such as the convexity of the spatial and temporal convolution kernels, unlike the existing literature on the asymptotic analysis of nonlocal-in-space only conservation laws. To the best of our knowledge, this provides the first convergence and asymptotic analysis for finite volume schemes applied to nonlocal conservation laws with memory. Numerical experiments are included to illustrate the theory.

36. MLMC-qDRIFT: 随机量子哈密顿模拟的多层方差缩减

MLMC-qDRIFT: Multilevel Variance Reduction for Randomized Quantum Hamiltonian Simulation

作者: Pegah Mohammadipour, Xiantao Li

类别: quant-ph, math.NA

发布日期: 2026-04-29

arXiv ID: 2604.26865v1

链接: http://arxiv.org/abs/2604.26865v1

摘要: Simulating quantum dynamics is one of the central applications of quantum computing. For Hamiltonians written as a sum of many terms, deterministic Trotter–Suzuki product formulas can require applying a large number of term-wise evolutions at each time step, leading to high circuit costs for large or dense systems. Randomized methods such as qDRIFT offer an alternative: each step samples only one Hamiltonian term, giving a circuit depth with no explicit dependence on the number of terms. However, when qDRIFT is used for observable estimation, high precision requires many independent random circuit realizations, resulting in a total gate complexity that scales as $\mathcal{O}(\varepsilon^{-3})$. We introduce a multilevel Monte Carlo framework for qDRIFT that reduces this sampling overhead. The method constructs a hierarchy of qDRIFT estimators with increasing circuit depths and couples adjacent levels by sharing their random Hamiltonian-term samples. This coupling makes the variance of the level differences decay with depth, allowing most samples to be taken on cheaper, coarse circuits and only a few on expensive, fine circuits. We prove that the resulting MLMC-qDRIFT estimator reduces the total gate complexity for fixed-precision observable estimation from the standard qDRIFT scaling $\mathcal{O}(\varepsilon^{-3})$ to $\mathcal{O}(\varepsilon^{-2}\log^2(1/\varepsilon))$, while preserving qDRIFT’s lack of explicit dependence on the number of Hamiltonian terms. Numerical experiments for spin-chain dynamics confirm the predicted variance decay and demonstrate the practical gate-count savings of the multilevel construction.

37. 具有再生重构的高维平均场博弈深度策略迭代

Deep Policy Iteration for High-Dimensional Mean-Field Games with Regenerative Reformulation

作者: Shuixin Fang, Shupeng Wang, Zhen Wu, Hui Zhang, Tao Zhou

类别: math.NA

发布日期: 2026-04-29

arXiv ID: 2604.26782v1

链接: http://arxiv.org/abs/2604.26782v1

摘要: This paper develops a deep policy iteration method for high-dimensional finite-horizon mean-field games. We reformulate the game as a regenerative problem with deterministic cycles, which allows policy evaluation (PE), policy improvement (PI), and population measure estimation to be carried out cycle by cycle. Within this formulation, we approximate the population measure by a particle system and update it using a one-step random mapping induced by the Euler-Maruyama discretization of the state dynamics. This update transports a mini-batch of particles from one cycle to the next, avoiding sequential trajectory simulation over the entire time horizon at each iteration. The PE and PI subproblems are formulated through the relation between consecutive cycles, with adversarial training used for evaluation and averaged optimization used for improvement. The resulting method is efficient and scalable in high dimensions, as it avoids the direct solution of the coupled Hamilton-Jacobi-Bellman and Fokker-Planck system, the full simulation of trajectories to estimate the population measure, the explicit computation of conditional expectations in policy evaluation, and pointwise optimization in policy improvement. Numerical experiments demonstrate that the proposed method effectively handles dimensions up to 10,000.

38. 被动伽马发射断层成像的鲁棒基于模型的迭代方法

Robust Model-Based Iteration for Passive Gamma Emission Tomography

作者: Tommi Heikkilä, Sara Heikkinen, Riina Rimppi, Tapio Helin

类别: math.NA

发布日期: 2026-04-29

arXiv ID: 2604.26745v1

链接: http://arxiv.org/abs/2604.26745v1

摘要: Passive Gamma Emission Tomography (PGET) is an IAEA-approved technique for verifying spent nuclear fuel assemblies prior to geological disposal. Reconstructing the emission and attenuation maps from PGET measurements is a nonlinear ill-posed inverse problem, currently solved with a Levenberg-Marquardt (LM) scheme that requires 10-20 iterations to achieve sufficient accuracy. We propose an accelerated iterative solver that combines the LM algorithm with a Deep Gauss-Newton step, in which a learned operator refines the update proposed by the deterministic algorithm at each iteration. A safeguard condition based on the trust-region model ensures that the accelerated iterates perform no worse than LM and retain convergence to a critical point of the regularized objective. Within this framework we compare three architectures for the learned component: an encoder-decoder-style convolutional neural network, Fourier Neural Operators, and Wavelet Neural Operators. Each is trained on a small set of coarsely simulated 9x9 assemblies. Experiments on simulated and real measurements from Finnish nuclear power plants show that the proposed scheme reaches LM-quality reconstructions in roughly one third of the iterations, while revealing architecture-dependent trade-offs in robustness against out-of-distribution inputs.

39. 扩散光学层析成像中用于处理不确定性与实现域截断的投影方法

Projections for handling uncertainties and enabling domain truncation in diffuse optical tomography

作者: Aada Hakula, Pauliina Hirvi, Nuutti Hyvönen, Altti Jääskeläinen, Ville Kolehmainen

类别: math.NA

发布日期: 2026-04-29

arXiv ID: 2604.26548v1

链接: http://arxiv.org/abs/2604.26548v1

摘要: This paper presents a projection-based technique to mitigate the impact of modeling errors related to domain truncation, changes in the optode coupling coefficients, and misspecified optical parameters of different tissue types in diffuse optical tomography. The approach considers the primary Jacobian matrix of the forward map in the image reconstruction scheme, linking the primary unknown, i.e., the per-voxel absorption coefficient changes in the region of interest, to the optode measurements, as well as the nuisance Jacobians that do the same for the auxiliary unknown parameters of secondary interest. To mitigate mismodeled coupling coefficients or domain truncation, the method projects the linearized forward model defined by the primary Jacobian onto the orthogonal complement of the range of a nuisance Jacobian, or onto the orthogonal complement of the span of a number of first left singular vectors for the nuisance Jacobian that has been weighted to account for prior information on the measurement setup. In the case of a misspecified baseline optical parameter for some tissue type, the nullspace of the utilized orthogonal projection is defined to be the span of first left singular vectors for a (weighted) difference of two Jacobian matrices evaluated at two different levels for the considered tissue-wise optical parameter. The reconstruction is formed by applying Bayesian inversion with Gaussian prior and noise models to the projected linearized equation. We evaluate the method on simulated brain activity data obtained via Monte Carlo simulations of the radiative transfer equation in a voxelized head anatomy for a neonate with combined gestational and chronological age of 41.7 weeks.

40. 具有空间和时间分数阶光滑性的非可分离时空模型的ARMA逼近

ARMA approximation of a Non-separable Spatio-Temporal Model with Fractional Smoothnesses in Space and Time

作者: S. Knutsen Furset, Geir-Arne Fuglstad, Espen R. Jakobsen

类别: stat.ME, math.NA

发布日期: 2026-04-29

arXiv ID: 2604.26535v1

链接: http://arxiv.org/abs/2604.26535v1

摘要: The Matérn covariance model is ubiquitous in spatial modelling, but there is no default choice for spatio-temporal modelling. In this paper, we consider the recently proposed ``diffusion-based’’ extension of the spatial Matérn covariance model to a spatio-temporal non-separable covariance model that allows fractional smoothnesses in space and in time. The model is described in terms of a space-time fractional stochastic partial differential equation, but currently proposed computational approaches have strong restrictions on the possible smoothnesses in time. We propose a discretization method based on rational approximations in time to handle arbitrary smoothnesses, which leads to a vector autoregressive moving average process (VARMA). We prove that the covariance function of the approximation converges pointwise, determine explicit convergence rates as a function of spatial and temporal resolutions and the accuracy of the rational approximation, and conduct numerical verification to demonstrate small pointwise error for low orders of the VARMA process. Through a simulation study, we demonstrate that the parameters can be estimated back and that correctly specifying the temporal smoothness is especially important for forecasting. The approach is illustrated for three months of daily mean temperatures in mainland France.

41. 一种用于单GPU三维SIMP线性系统的无矩阵Galerkin多重网格求解器与失效模式筛查

A Matrix-Free Galerkin Multigrid Solver and Failure-Mode Screen for Single-GPU 3D SIMP Linear Systems

作者: Shaoliang Yang, Jun Wang, Yunsheng Wang

类别: cs.CE, math.NA

发布日期: 2026-04-29

arXiv ID: 2604.26441v1

链接: http://arxiv.org/abs/2604.26441v1

摘要: Large 3D SIMP studies require repeated elasticity solves for density-dependent operators whose finest matrices are expensive to assemble and whose conditioning degrades under high contrast. We study this linear-solver layer rather than claiming end-to-end optimization acceleration. The solver builds a matrix-free Galerkin geometric multigrid (GMG) hierarchy around a fused fine operator: the finest level remains matrix-free, the first coarse level is assembled by local Galerkin aggregation, and deeper levels use sparse Galerkin products. The practical default is FP32-GMG; BF16 is evaluated as a guarded mixed-precision variant and diagnostic stress test, not as the main speed mechanism. In a 27-case heterogeneous cantilever sweep, pass rates under a 200-iteration budget are 7/9, 4/9, and 1/9 at 64k, 216k, and 512k elements; converged-only mean iteration counts are about 112, 134, and 146. On uniform rho=0.5, p=3 solves, FP32-GMG gives 1.62x, 1.75x, and 3.12x wall-time ratios relative to the capped flat Jacobi-PCG baseline at the same sizes; that non-converged baseline reaches the 200-iteration cap in all timed trials. BF16-GMG is not faster than FP32-GMG. In 18 fixed-seed heterogeneous BF16 validation cases, 7/18 converge, matching the FP64 count, and 11 cases that pass the spectral screen still fail the 500-iteration cap; the screen is therefore diagnostic rather than a convergence certificate. The largest reported solve is a 1M-element uniform-modulus system solved in 1.50+/-0.58 s with an 8.66 GiB hierarchy-allocation delta during setup, not a peak-memory trace; this point is reported as uniform scaling, not heterogeneous robustness evidence. The contribution is therefore a bounded single-GPU solver result built on an inherited Level 0 matrix-free operator: a Galerkin GMG hierarchy, direct BF16 guard evidence, and an explicit failure-mode screen for structured 3D SIMP linear systems.

42. 能量稳定辅助变量方法求解 Cahn–Hilliard 方程

Energy stable auxiliary variable method for Cahn–Hilliard equations

作者: Fei Xie, Nan Lu, Yajuan Sun

类别: math.NA

发布日期: 2026-04-29

arXiv ID: 2604.26402v1

链接: http://arxiv.org/abs/2604.26402v1

摘要: In this paper, we propose a quadratic reformulation theory for rational-like functions. Based on this theory, we develop the Quadratic Conservation Elevation (QCE) method, which combines the Scalar Auxiliary Variable (SAV) method with the implicit midpoint rule. We apply this approach to the Cahn-Hilliard (CH) equation with rational-like free-energy terms, obtaining numerical discretizations that preserve the original energy dissipation law. We further derive the discrete dispersion relation and coarsening dynamics, confirming the efficiency and consistency of the method with the continuous counterpart. In addition, we use the proposed method to capture missing orientations for different anisotropic functions. Numerical simulations with various initial conditions illustrate phase separation and anisotropic evolution.

43. 用于时间依赖输运方程的二阶保极值原理Crouzeix-Raviart有限元方法

A Second-Order Maximum-Principle-Preserving Crouzeix-Raviart Finite Element Method for Time-dependent Transport Equation

作者: Shipeng Mao, Mingyang Zhang

类别: math.NA

发布日期: 2026-04-29

arXiv ID: 2604.26336v1

链接: http://arxiv.org/abs/2604.26336v1

摘要: In this paper, we construct an explicit, second-order, and maximum-principle-preserving Crouzeix-Raviart (CR) finite element method for two-dimensional time-dependent transport equation. The key observation is that the mass matrix of the CR element is with diagonal structure, which allows us to avoid the need to solve a large linear system for each time step and help to construct a low-order scheme that preserves the maximum principle in a simple way. We first introduce low-order schemes based on minimum and bilinear viscosities, and then recover second-order accuracy by means of greedy and flux-corrected transport viscosities. For inflow boundary conditions, we further design a modified FCT limiter. In addition, we propose a simple reconstruction based on Wachspress coordinates to obtain a continuous piecewise linear approximation on the $\frac{h}2$-mesh that satisfies the maximum principle on the whole domain. Under divergence-free velocity fields, the proposed schemes are conservative. Numerical experiments on both smooth and discontinuous test cases, with both solenoidal and non-solenoidal velocity fields, confirm the accuracy and robustness of the proposed schemes.

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20. 二阶完全非线性偏微分方程的自适应深度Ritz框架#

21. 具有广义电容的场/电路耦合微分-代数方程的波形松弛法#

22. 光学层析成像中频域与时域雅可比矩阵的蒙特卡洛模拟计算#

23. 参数化驱动的任意拉格朗日-欧拉方法用于大变形等几何流固耦合#

24. 耦合修正Korteweg-de Vries方程的数值逆散射变换#

25. 求解Poisson-Nernst-Planck-Navier-Stokes方程的间断Galerkin IMEX压力校正格式#

26. 一种标量守恒律的神经网络方法及激波解的收敛速度: 扩展版#

27. 适用于非界面贴合网格的平移内聚区方法及其在晶体塑性中的应用#

28. 短注：批高效分治特征分解算法#

29. 可压缩流的热力学约束信息几何正则化#

30. 具有一般权的再生核希尔伯特空间的嵌入#

31. 一种用于熵最小二乘的尺度-形状对偶牛顿法#

32. 应用随机矩阵理论#

33. 二次函数商的零阶方法的推广#

34. 微可压缩流的数据同化#

35. 带有记忆的非局部守恒律方程组及其零保持极限#

36. MLMC-qDRIFT: 随机量子哈密顿模拟的多层方差缩减#

37. 具有再生重构的高维平均场博弈深度策略迭代#

38. 被动伽马发射断层成像的鲁棒基于模型的迭代方法#

39. 扩散光学层析成像中用于处理不确定性与实现域截断的投影方法#

40. 具有空间和时间分数阶光滑性的非可分离时空模型的ARMA逼近#

41. 一种用于单GPU三维SIMP线性系统的无矩阵Galerkin多重网格求解器与失效模式筛查#

42. 能量稳定辅助变量方法求解 Cahn–Hilliard 方程#

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1. 麦克斯韦方程组的结构保持最优控制及其在源隐身中的应用

1. 研究对象和背景

2. 主要定理或主要结果

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4. 与之前工作的比较

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1. 研究对象和背景

2. 主要定理或主要结果

3. 研究方法、关键技术和核心工具

4. 与之前工作的比较

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1. 研究对象和背景

2. 主要定理或主要结果

3. 研究方法、关键技术和核心工具

4. 与之前工作的比较

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1. 有界、交换、保持离散迹的投影

摘要翻译

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摘要翻译

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20. 二阶完全非线性偏微分方程的自适应深度Ritz框架

21. 具有广义电容的场/电路耦合微分-代数方程的波形松弛法

22. 光学层析成像中频域与时域雅可比矩阵的蒙特卡洛模拟计算

23. 参数化驱动的任意拉格朗日-欧拉方法用于大变形等几何流固耦合

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28. 短注：批高效分治特征分解算法

29. 可压缩流的热力学约束信息几何正则化

30. 具有一般权的再生核希尔伯特空间的嵌入

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38. 被动伽马发射断层成像的鲁棒基于模型的迭代方法

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40. 具有空间和时间分数阶光滑性的非可分离时空模型的ARMA逼近

41. 一种用于单GPU三维SIMP线性系统的无矩阵Galerkin多重网格求解器与失效模式筛查

42. 能量稳定辅助变量方法求解 Cahn–Hilliard 方程

43. 用于时间依赖输运方程的二阶保极值原理Crouzeix-Raviart有限元方法