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1. Interface Control Domain Decomposition (ICDD) 方法的代数收敛性分析

Algebraic convergence analysis for the Interface Control Domain Decomposition (ICDD) method

作者: Marco Discacciati, Paola Gervasio, Alfio Quarteroni

类别: math.NA

发布日期: 2026-09-15

arXiv ID: 2609.17271v1

摘要翻译: 我们发展了对 Interface Control Domain Decomposition (ICDD) 方法的收敛性分析，该方法是一种基于最优控制框架的重叠区域分解方法，采用 Dirichlet 界面控制函数和界面观测。我们考虑具有可能不连续系数的椭圆问题，并在每个子区域中由 $hp-$FEM 逼近。当离散化在二维区域之间的重叠部分上协调时，我们给出了求解与 ICDD 相关联的非对称界面 Schur 补系统所需的 GMRES 迭代次数的理论估计。我们的结果是通过结合 ICDD 的 Schur 补矩阵的新谱估计与经典 GMRES 收敛理论得到的。我们证明收敛率表现为 $\mathcal{O}(δ^{-1} p^{3/2}\log p)$，其中 $δ$ 表示重叠宽度，$p$ 表示局部多项式次数，同时保持与网格尺寸 $h$ 无关。数值实验验证了理论预测，并表明在计算域中存在大系数跳跃时 ICDD 的有效性。由于在协调情形下，所考虑的 ICDD 格式与 Substructured Restricted Additive Schwarz (SRAS) 方法一致，该分析也为二维情形下的 SRAS 提供了收敛估计，并通过其已知等价性，为 Restricted Additive Schwarz (RAS) 方法提供了收敛估计。

Comment: 无

链接: http://arxiv.org/abs/2609.17271v1

1. 研究对象和背景

本文研究重叠型区域分解方法 ICDD (Interface Control Domain Decomposition) 的代数收敛性. 考虑定义在开有界 Lipschitz 区域 $\Omega\subset\mathbb R^d$ ($d=1,2,3$) 上的二阶线性椭圆边值问题

$$Lu=f \quad \text{in }\Omega,\qquad u=0 \quad \text{on }\partial\Omega,$$

其中 $f\in L^2(\Omega)$，$L$ 是二阶线性椭圆强制算子，典型例子为

$$Lu=-\nabla\cdot(\nu\nabla u)+\gamma u,$$

$\nu,\gamma\in L^\infty(\Omega)$，且存在 $\nu>0$ 使得 $\nu(x)\ge \nu$ a.e.，$\gamma(x)\ge 0$ a.e.。区域 $\Omega$ 被分成两个非重叠子域 $\tilde\Omega_1,\tilde\Omega_2$，系数 $\nu,\gamma$ 可能在公共边界 $\partial\tilde\Omega_1\cap\partial\tilde\Omega_2$ 上发生间断。随后将每个子域向外扩展形成两个重叠子域 $\Omega_1,\Omega_2$，满足 $\Omega=\Omega_1\cup\Omega_2$，并令内部边界（界面）为 $\Gamma_k=\partial\Omega_k\setminus\partial\Omega$，假设

$$\delta:=\operatorname{dist}(\Gamma_1,\Gamma_2)>0.$$

在经典多域问题中，求 $u_k$ 使得

$$ \begin{cases} L_k u_k=f_k & \text{in }\Omega_k,\ k=1,2,\\ u_k=0 & \text{on }\partial\Omega_k\setminus\Gamma_k,\\ u_1=u_2 & \text{on }\Gamma_1\cup\Gamma_2, \end{cases} $$

其中 $L_k u=-\nabla\cdot(\nu_k\nabla u)+\gamma_k u$，$\nu_k=\nu|_{\Omega_k}$，$\gamma_k=\gamma|_{\Omega_k}$，$f_k=f|_{\Omega_k}$。该多域问题与原问题等价。注意局部系数 $\nu_k,\gamma_k$ 可能在 $\Omega_k$ 内仍不连续；作者强调不能为了消除内部间断而将系数连续延拓，否则等价性不成立。

ICDD 方法将上述多域问题重新表述为界面上的最优控制问题。具体地，引入两个未知界面函数 $\lambda_1$ 在 $\Gamma_1$ 上、$\lambda_2$ 在 $\Gamma_2$ 上，称为虚拟控制；对 $k=1,2$ 定义状态问题

$$ \begin{cases} L_k u_k=f_k & \text{in }\Omega_k,\\ u_k=\lambda_k & \text{on }\Gamma_k,\\ u_k=0 & \text{on }\partial\Omega_k\setminus\Gamma_k, \end{cases} $$

其中 $u_k=u_k(\lambda_k)$。成本泛函为界面观测

$$J_\Gamma(\lambda_1,\lambda_2)=\frac12\|u_1(\lambda_1)-u_2(\lambda_2)\|^2_{L^2(\Gamma_1\cup\Gamma_2)},$$

并通过

$$\inf_{\mu_1,\mu_2} J_\Gamma(\mu_1,\mu_2)$$

定义最优控制问题。其最优性系统（不含对偶状态问题）为：对 $k=1,2$，$\ell=3-k$，求 $u_k^0\in V_k^0$ 和 $\lambda_k\in\widehat\Lambda_k$ 使得

$$ \begin{cases} a_k(u_k^0,v_k)+a_k(E_k\lambda_k,v_k)=F_k(v_k) & \forall v_k\in V_k^0,\\ u_k=u_k^0+E_k\lambda_k,\\ \lambda_k=u_\ell & \text{on }\Gamma_k, \end{cases} $$

其中 $a_k(w,v)=\int_{\Omega_k}\nu_k\nabla w\cdot\nabla v+\int_{\Omega_k}\gamma_k wv$，$F_k(v)=\int_{\Omega_k}fv$，$E_k:\widehat\Lambda_k\to V_k$ 是提升算子。连续层面还等价地可写出含对偶状态问题的系统，对应 dual ICDD 方法，但本文主要分析上述无对偶问题的 ICDD 离散格式。

数值实现中，在每个子域采用独立的 $hp$-FEM 离散，允许网格在重叠区域非协调、网格尺寸 $h_1,h_2$ 不同、局部多项式阶 $p_1,p_2$ 不同。离散后消去内部自由度，得到关于界面控制变量的 Schur 补系统

$$\Sigma\lambda=\chi,$$

其中

$$\Sigma=I-T_\Gamma+T_0 A_{II}^{-1}A_{I\Gamma},\qquad \chi=T_0 A_{II}^{-1}f.$$

当两个网格在重叠上协调时，$T_\Gamma=0$，且 $T_0$ 变为限制矩阵 $R_{\Gamma I}$，于是

$$\Sigma=I+R_{\Gamma I}A_{II}^{-1}A_{I\Gamma}.$$

该 Schur 补矩阵非对称，因此用 GMRES 求解。矩阵向量乘只需在子域内解 Dirichlet 问题，从而 ICDD 可以非侵入式实现，复用局部求解器。

本文试图填补的空白是：尽管 ICDD 方法的适定性和应用已有研究，但缺乏对其迭代求解收敛率的理论分析；特别是，需要给出 GMRES 迭代次数关于重叠宽度 $\delta$、网格尺寸 $h$ 和局部多项式阶 $p$ 的显式估计。由于在协调情形下 ICDD 格式与 Substructured Restricted Additive Schwarz (SRAS) 方法一致，而 SRAS 的收敛分析此前尚未建立，本文的分析也推进了 SRAS 以及通过已知等价性推广到 Restricted Additive Schwarz (RAS) 的理论。

2. 主要定理或主要结果

在 Assumption 1 和 Assumption 2 下，作者对 ICDD 的界面 Schur 补系统进行了谱分析，并结合经典 GMRES 收敛理论，证明了 GMRES 迭代次数的上界估计。Assumption 1 要求两个子域 $\Omega_k\subset\mathbb R^2$ 为相同尺寸 $H_x\times H_y$ 的矩形，界面 $\Gamma_1,\Gamma_2$ 为竖直平行线段，重叠宽度 $\delta=\operatorname{dist}(\Gamma_1,\Gamma_2)>0$ 且 $\delta\ll H_x$。Assumption 2 要求每个子域内的 $hp$-FEM 离散采用仿射正则拟一致剖分；单纯形时使用 Legendre-Gauss-Lobatto (LGL) 节点，内部点（$p_k\ge3$）为 electrostatic 点；四边形时使用张量积 LGL 节点，局部空间为 $Q_{p_k}$ 或 $P_{p_k}$。

核心结论是：当两个子域内的离散在重叠区域上协调时，ICDD 的 GMRES 收敛速率满足

$$\mathcal{O}(\delta^{-1}p^{3/2}\log p),$$

其中 $\delta$ 为重叠宽度，$p$ 为局部多项式阶，且该估计关于网格尺寸 $h$ 一致，即与 $h$ 无关。这里“收敛速率”指保证相对残差低于给定容差所需的最小迭代次数的上界估计。该结果基于作者提出的新 Schur 补矩阵谱估计。需要强调，该结论是在协调离散、二维矩形子域、竖直平行界面的假设下严格证明的；摘要明确说“we prove that the convergence rate behaves as $\mathcal{O}(\delta^{-1}p^{3/2}\log p)$”。

此外，连续层面作者证明了调和延拓的内部迹的 $L^2$ 估计（Theorem 1 和 Corollary 2）。在矩形 $D=(0,H_x)\times(0,H_y)$，$\gamma_t=\{(x,y)\in D:x=t\}$，$\Gamma=\{H_x\}\times(0,H_y)$，对 $\lambda\in H^{1/2}_{00}(\Gamma)$ 及其调和延拓 $u_\lambda$，有

$$\|u_\lambda\|_{L^2(\gamma_t)}\le C_M(t)\|\lambda\|_{L^2(\Gamma)},$$

其中 $C_M(t)$ 的显式表达式为

$$C_M(t)=\left(1-\frac{1}{\frac{\|\nu\|_\infty}{\nu}+C(t)}\right)^{1/2}<1,\qquad C(t)=\frac{c^2\|\nu\|_\infty}{c_1}\frac{\max\{H_t,H_t^{-1}\}}{H_x-t},$$

$H_t=\max\{t,H_y\}$，$c>0$ 仅依赖区域形状，$c_1$ 由 (14) 定义：当 $\gamma=0$ 时 $c_1=\frac{4\nu}{4+2tH_x+(H_x-t)^2}$，当 $\gamma\ne0$ 时 $c_1=\min\{\nu,\gamma\}$。在 Assumption 1 下，对 $k=1,2$，$\ell=3-k$，

$$\|u_{\lambda_k}^k\|_{L^2(\Gamma_\ell)}\le C_{\delta,k}\|\lambda_k\|_{L^2(\Gamma_k)},$$

其中

$$C_{\delta,k}=\left(\frac{\delta\left(\frac{\|\nu_k\|_\infty}{\nu_k}-1\right)+c_{L,k}c_{H,k}}{\delta\frac{\|\nu_k\|_\infty}{\nu_k}+c_{L,k}c_{H,k}}\right)^{1/2},$$

$c_{L,k}$ 仅依赖局部算子物理量，$c_{H,k}$ 仅依赖局部域几何，二者均与 $\delta$ 无关。当 $\delta\to0$ 时 $C_{\delta,k}\to1$，表明界面耦合增强。这些估计是证明 Schur 补谱性质、进而得到 GMRES 迭代次数估计的关键。

由于在协调情形下 ICDD 格式与 SRAS 一致，上述收敛估计也直接给出 SRAS 方法在二维、$hp$-FEM 离散下的收敛率。再结合 [8] 中 SRAS 与 RAS 的等价性，本文结果为 RAS 方法提供了收敛估计，这是此前文献中缺乏的。作者在引言中明确指出：“In those cases where ICDD and SRAS coincide, the theory developed here advances the state of the art by providing a convergence rate for the latter method as well.” 并且“Given the equivalence between SRAS and RAS discussed above, our results also apply to the RAS method.”

作者还报告了数值实验，验证理论预测，并观察到大系数跳跃时 ICDD 的收敛率趋于失去对 $\delta$ 的依赖：跳跃越大，迭代次数越低，对 $\delta$ 的依赖越弱。这一现象在摘要中以数值观察形式给出：“the larger the jump, the lower the number of iterations and the weaker the dependence on $\delta$.” 此外，数值结果还显示收敛率实际上似乎独立于 $p$，尽管理论估计中含 $p^{3/2}\log p$；这是数值观察，严格证明的估计仍为 $\mathcal{O}(\delta^{-1}p^{3/2}\log p)$。需要区分：严格证明的结论是 GMRES 迭代次数上界；数值观察包括对 $p$ 的独立性和对系数跳跃的弱化依赖。从提供内容中无法确认第 4 节中 GMRES 估计的具体定理编号和完整证明细节。

3. 研究方法、关键技术和核心工具

主要技术障碍在于 ICDD 的界面 Schur 补矩阵 $\Sigma$ 非对称，因此条件数不能直接控制 GMRES 的收敛行为；需要估计 GMRES 残差多项式在谱上的表现，或者使用非对称矩阵的字段值 (field of values) 分析。作者采用 [18,30] 的 GMRES 收敛理论，将迭代次数估计转化为对 Schur 补矩阵谱性质或字段值的估计。

第一个关键工具是调和延拓的内部迹的尖锐 $L^2$ 估计。连续层面，界面控制到另一界面的传播由状态解的迹控制。作者证明 Theorem 1，给出 $C_M(t)$ 的显式表达式，其中依赖域尺寸 $H$、系数 $\nu,\gamma$ 和重叠宽度相关变量 $t$。该结果源于 [27] 的更一般引理，但本文的创新在于显式追踪常数对 $H$、$\nu,\gamma$ 和 $t$ 的依赖，特别是当 $t$ 接近 $H_x$（即重叠宽度 $\delta=H_x-t$ 很小时）给出非常尖锐的界。Corollary 2 进一步在 Assumption 1 下将跨界面传播估计写成 $C_{\delta,k}$，其结构显示当 $\delta\to0$ 时 $C_{\delta,k}\to1$，从而界面耦合增强，这最终导致 GMRES 迭代次数随 $\delta^{-1}$ 增长。具体地，$C_{\delta,k}$ 的分母含 $\delta\frac{\|\nu_k\|_\infty}{\nu_k}+c_{L,k}c_{H,k}$，分子含 $\delta(\frac{\|\nu_k\|_\infty}{\nu_k}-1)+c_{L,k}c_{H,k}$，当 $\delta$ 很小时二者接近，因此 $C_{\delta,k}$ 接近 1；但 $1-C_{\delta,k}$ 的行为与 $\delta$ 成正比，这给出了衰减率。

第二个关键工具是对 ICDD Schur 补矩阵的谱估计。利用 $\Sigma=I-T_\Gamma+T_0 A_{II}^{-1}A_{I\Gamma}$（协调时 $\Sigma=I+R_{\Gamma I}A_{II}^{-1}A_{I\Gamma}$），作者将 Schur 补的谱与局部子问题的解算子以及界面迹的传播联系起来。$A_{II}^{-1}A_{I\Gamma}$ 对应于在子域内以单位 Dirichlet 数据求解局部问题并在界面观测其解；再通过 $R_{\Gamma I}$ 限制到另一界面。上述调和延拓迹估计给出了该复合算子的 $L^2$ 范数界。结合有限元逆估计和 $hp$-FEM 的逼近性质，可以得到 Schur 补矩阵的谱聚集区域或字段值包含关系。摘要明确指出“novel spectral estimates for the Schur complement matrix of ICDD”是核心贡献。从提供内容中无法确认完整的谱定理证明细节，但可以推断其思路是将 $\Sigma$ 分解为 $I$ 加上一个由跨界面传播算子构成的扰动，并利用 $C_{\delta,k}$ 和 $p$ 的显式依赖控制该扰动的谱半径或字段值，从而应用 GMRES 收敛理论。

第三个工具是 $hp$-FEM 离散的节点选择与质量矩阵性质。Assumption 2 要求使用 Legendre-Gauss-Lobatto (LGL) 节点（单纯形内部用 electrostatic 点），四边形时张量积 LGL 节点。作者指出这能保证 Lebesgue 插值常数较小、稳定性更好，并且四边形时插值节点与 LGL 求积节点一致，可高效组装刚度矩阵和质量矩阵，降低计算成本。文中提到 Assumption 2 用于建立质量矩阵的性质（如 Lemma 4）。这些性质可能在谱等价和逆估计中用于控制 $L^2$ 范数与节点值向量之间的关系，从而将连续估计转化为矩阵谱估计。例如，质量矩阵的谱等价可给出节点值向量的 $\ell^2$ 范数与有限元函数 $L^2$ 范数之间的双向界，使得连续层面的 $L^2$ 估计能转化为 Schur 补矩阵的范数或谱界。

第四个工具是代数消元与 Schur 补框架。从离散最优性系统 (23) 或协调情形 (31) 中消去内部自由度 $u_k^0$，得到界面控制变量 $\lambda$ 的 Schur 补系统。该系统的矩阵向量乘只需解局部 Dirichlet 问题，因此分析可以聚焦于界面算子。作者还给出 dual ICDD 方法的代数形式 (33)，并指出对偶问题可解释为预处理步骤，能改善极薄重叠时的稳定性；但本文的主要分析针对无对偶问题的 ICDD。dual ICDD 的矩阵结构更复杂，包含额外的对偶未知量 $p_1^0,p_2^0$，作者未在主要收敛率分析中展开。

创新点总结：首次为 ICDD 建立代数收敛率理论；给出显式依赖于 $\delta$ 和 $p$、独立于 $h$ 的 GMRES 迭代次数估计；提出新的 Schur 补谱估计和尖锐的调和延拓迹估计；在协调情形下将结果推广至 SRAS 和 RAS。技术障碍的突破依赖于将连续层面的界面传播估计与矩阵谱分析结合，并利用 GMRES 的非对称收敛理论绕过条件数。需要指出，作者明确区分了严格证明的估计与数值观察：理论估计为 $\mathcal{O}(\delta^{-1}p^{3/2}\log p)$，而数值上观察到对 $p$ 的独立性以及对大系数跳跃时对 $\delta$ 依赖的减弱，这些是启发式/数值现象，而非已证明的定理。

4. 与之前工作的比较

与经典重叠 Schwarz 方法相比，ICDD 是一种基于最优控制框架的重叠区域分解方法，控制变量取为界面上的 Dirichlet 迹，观测也在界面上。经典重叠 Schwarz 方法（如 Additive Schwarz、RAS）通常在体积自由度上操作，利用原变量的连续性作为耦合条件；而 ICDD 只引入界面控制，不需要额外 Lagrange 乘子或对偶变量，局部求解器可作为黑箱，因而非侵入性更强。ICDD 与 SRAS 的关系是本文比较的关键：在离散协调时，本文所考虑的 ICDD 格式与 SRAS 一致；但 SRAS 此前缺乏收敛分析。本文通过 ICDD 的 Schur 补分析间接给出了 SRAS 在二维的收敛估计。再通过 [8] 中 SRAS 与 RAS 的已知等价性，结果也适用于 RAS。作者在引言中明确说：“In those cases where ICDD and SRAS coincide, the theory developed here advances the state of the art by providing a convergence rate for the latter method as well.”

与 RAS 的已有收敛分析相比：RAS 的收敛分析在 [19] 中针对 M 矩阵，用加权最大范数证明了收敛，但没有显式给出对重叠宽度 $\delta$ 的依赖；[31] 在系统矩阵对应于 1D Laplace 方程二阶中心差分离散时，证明了带粗网格的 RAS 收敛率约为 $(1+H/h)$，其中 $H$ 为子域尺寸，$h$ 为网格尺寸。这些分析局限于一维、特定差分格式或 M 矩阵，对不连续系数、更一般的离散和二维/三维区域仍是开放问题。本文在二维、可能不连续系数、$hp$-FEM 离散、协调网格的设定下，给出了 ICDD/SRAS 的 GMRES 迭代次数估计 $\mathcal{O}(\delta^{-1}p^{3/2}\log p)$，显式依赖于 $\delta$ 和 $p$，且独立于 $h$。这一结果推进了适用范围：从 M 矩阵/一维差分到二维椭圆问题的不连续系数和 $hp$-FEM；从无显式 $\delta$ 依赖到显式 $\delta^{-1}$ 依赖；从条件数分析到非对称 GMRES 收敛分析。

与 ICDD 自身已有工作相比：ICDD 方法此前在 [14,16,15,13,12] 等中被提出并应用于各类偏微分方程，其适定性通过最优控制理论在 [14] 中证明。但此前没有迭代求解器的收敛率分析。本文填补了这一空白，首次给出 ICDD 的代数收敛估计。此外，ICDD 最初的最优性系统包含两对 PDE（两个原始问题和两个对偶问题）加两个界面方程；本文分析的是不含对偶问题的 ICDD 格式（系统 (10)），而在极薄重叠时 dual ICDD 可能稳定性更好但更昂贵。作者在引言中说：“When the overlap width is really tiny, solving the optimality system of ICDD with the dual problems performs better than solving the optimality system without dual problems, but it is, in general, more expensive, so we omit them in the formulation considered in this work.” 本文的主要理论结果针对无对偶问题的 ICDD；dual ICDD 的收敛分析未在提供内容中展开，作者也未显式比较两者的收敛率。

局限与代价：本文的收敛估计在 Assumption 1 下要求两个子域为相同尺寸的矩形、界面为竖直平行线段、$\delta\ll H_x$；在 Assumption 2 下要求局部网格仿射正则拟一致、节点为 LGL/electrostatic 点；最关键的是要求重叠区域上的离散协调（网格和多项式阶一致），此时 $T_\Gamma=0$，Schur 补具有简化形式。对非协调网格、一般区域、三维情形、以及不满足 Assumption 1 的几何，理论分析尚未覆盖。此外，GMRES 迭代次数估计是最坏情况上界，数值实验显示实际收敛率可能独立于 $p$，且大系数跳跃时对 $\delta$ 的依赖减弱，这说明理论估计在系数跳跃很大时可能偏保守。作者未显式展开与 dual ICDD 的收敛比较，也未给出非协调情况的严格结果。从提供内容中无法确认该等价性在非协调或更一般情形是否仍成立。

中文标题: 求解偏微分方程的物理信息随机特征神经网络

摘要翻译: 基于机器学习的偏微分方程 (PDEs) 求解器近年来引起了广泛关注。该领域的大多数进展由深度神经网络推动，例如物理信息神经网络 (PINNs) 和核方法（如物理信息高斯过程）。我们引入一种物理信息随机特征方法，用于对抗基于 PINN 的求解器在某类偏微分方程中面临的部分谱偏置。随机特征方法最初被提出用于逼近大规模核机器，并可视为一种专门的随机化神经网络。与其他需要大量配点的最先进的基于 PINN 的求解器相比，我们提出的方法降低了计算复杂度。本文中，我们建立了严格的逼近误差分析，并推导了 $H^1$ 范数上的高概率误差界。我们提供了广泛的数值实验，以验证我们关于误差衰减率的理论保证，以及若干比较实验，以展示我们所声称的在这些基于深度学习的方法中对抗谱偏置的能力。

不相关论文（基本信息）

1. 随机 Volterra 方程的马尔可夫逼近在 Hölder 范数下的收敛性

Convergence in Hölder norms for Markovian approximations of stochastic Volterra equations

作者: Noé Corneille, Kristin Kirchner, Pietro Pezzoli Frigerio

类别: math.PR, math.NA

发布日期: 2026-09-15

arXiv ID: 2609.17472v1

链接: http://arxiv.org/abs/2609.17472v1

摘要: We bound the difference between two stochastic Volterra processes with identical Lipschitz coefficients but different kernels. For non-convolution kernels, we establish estimates in $C^0([0,T];L^p(Ω))$, $p\geq 2$, and for convolution kernels in $L^p(Ω;L^q(0,T))$, $q \in [1,p]$, and $C^β([0,T];L^p(Ω))$, $L^p(Ω;C^β([0,T]))$, where the range of the Hölder exponent $β\in (0,1]$ is the maximal permitted by the regularity of the processes. For the fractional kernel, we then construct Markovian approximations whose error we show to decay as $e^{-a\sqrt{N}}$ in the aforementioned norms, using an $N$-node quadrature based on sinc methods. Numerical experiments for the fractional Brownian motion verify our findings.

2. 面向大规模反问题的一种可证明收敛的 MM-GKS 变体

A provably convergent MM-GKS variant for large-scale inverse problems

作者: Mirjeta Pasha, Eric de Sturler, Misha Kilmer

类别: math.NA

发布日期: 2026-09-15

arXiv ID: 2609.17229v1

链接: http://arxiv.org/abs/2609.17229v1

摘要: For high-quality images with sharp edges, a popular choice for edge-preserving regularization is using a general(ized) $\ell_q$-norm of the gradient of the image. This can be implemented efficiently using the $\ell_2$-norm and a sequence of weighted gradients, with weights derived from the current solution estimate. We can solve the resulting sequence of regularized least squares problems using hybrid Krylov subspace methods, which efficiently compute the regularization parameter using the problem projected on the Krylov subspace. However, each update of the regularization operator requires a new Krylov subspace. The majorization-minimization generalized Krylov subspace method (MM-GKS) addresses this problem by using a single, generalized, Krylov subspace (GKS). Unfortunately, for large-scale problems, if convergence is not fast, MM-GKS has overwhelming memory requirements and computational costs. We propose a variant of MM-GKS that alternately compresses and expands the search space while maintaining strict monotonic convergence. We show that our method provably converges to the minimum of the selected functional, even if the search space dimension is kept very small. This substantially improves on previous theoretical results for MM-GKS, where the convergence proof relies on the basis for the solution space (eventually) spanning the full space. We show that our method can solve large-scale problems efficiently both in terms of memory requirements and computational complexity. We further generalize our proposed method to handle streaming problems, where the data is either not all available simultaneously or needs to be treated as such because of the extreme memory requirements. We use numerical examples from image deblurring, dynamic photoacoustic tomography, and streaming X-ray computed tomography (CT) to illustrate the effectiveness of our proposed methods.

3. 迈向 Pólya 猜想：通过能量正交性改进单个 Li-Yau 界

Toward Pólya’s Conjecture: Improving the Individual Li-Yau Bound via Energy Orthogonality

作者: Yifan Wang, Hehu Xie

类别: math.NA, math-ph

发布日期: 2026-09-15

arXiv ID: 2609.17219v1

链接: http://arxiv.org/abs/2609.17219v1

摘要: We establish two complementary lower-bound mechanisms for individual eigenvalues of the Dirichlet Laplacian. First, energy orthogonality yields a frequency-dependent cap on the Fourier density of a finite spectral projection. Combining this cap with the standard $L^2$ Bessel estimate and a radial-capacity bathtub principle gives, on every open set of finite positive measure in $\mathbb R^n$ with $n\geq2$, [ λ_k\geq c_n(2π)^2ω_n^{-2/n}|Ω|^{-2/n}k^{2/n}, \qquad \frac{n}{n+2}<c_n<1. ] The constant $c_n$ is characterized by a scalar equation, with $c_2=0.5383068077\ldots$. This estimate preserves Weyl scaling and strictly improves the individual consequence of the Li-Yau sum inequality, although it does not improve the sharp leading coefficient in that sum inequality. Second, we retain part of the spectral deficit discarded when an eigenvalue sum is bounded by its largest term. A lower bound for the counting function, integrated through the exact first Riesz-mean identity, leads to a strictly monotone scalar equation. Its unique positive root is no weaker than the volume-only bound, and we give necessary and sufficient criteria for strict improvement over both that baseline and any independent lower bound. Quantitative estimates of Jiang-Lin provide an explicit implementation on bounded Lipschitz domains. The final comparisons and numerical example distinguish improvements within this framework from stronger estimates available under additional geometric or spectral assumptions.

4. 时间因果带通小波的框架界与带宽铺砌

Frame bounds and bandwidth tiling for time-causal bandpass wavelets

作者: Jens E. Pedersen, Tony Lindeberg, Peter Gerstoft

类别: eess.SP, math.NA

发布日期: 2026-09-15

arXiv ID: 2609.17201v1

链接: http://arxiv.org/abs/2609.17201v1

摘要: Wavelets and frames that provide strong guarantees for signal representations are a cornerstone in signal processing theory, but have long been restricted to non-causal settings. Recent work extended wavelet analysis to time-causal systems that operate on streaming signals, with exact reconstruction and time-recursive implementations. However, the frame guarantees that make such representations reliable under noise and quantization are uncharacterized. We derive closed-form frame bounds and conditioning on band-limited domains, peak frequencies, constant-Q bandwidth tiling, and spectral decay rates for time-causal bandpass wavelets. Our work provides design rules for the scale ratio and channel count, which we validate numerically against discrete implementations of the causal wavelets in terms of recursive filters.

5. Hamilton–Jacobi–Bellman 方程的边界停止正求积格式的误差界

Error Bounds for Boundary-Stopped Positive Cubature Schemes for Hamilton–Jacobi–Bellman Equations

作者: Haoran Xu, Xingye Yue

类别: math.NA

发布日期: 2026-09-15

arXiv ID: 2609.17186v1

链接: http://arxiv.org/abs/2609.17186v1

摘要: We derive error bounds for a boundary-stopped positive cubature scheme for degenerate parabolic Hamilton–Jacobi–Bellman equations with Dirichlet data on bounded domains. The scheme balances antipodal branches at their first boundary contacts and uses a control-dependent effective time. Its stopped-barrier remainder is (O(τΔt^{γ/2})), and the total mass of interpolated branches is at most (2τ/Δt), where (τ) is the effective time. After division by the effective time, these estimates give a consistency bound that is uniform as (τ\to0). Under a common strict boundary barrier and the stated mesh conditions, interior consistency, coefficient shaking, switching, and localized comparison yield [ |(u-u_h)^+|\infty\le C(Δt^{1/4}+hΔt^{-1/2}), \qquad |(u_h-u)^+|\infty\le C(Δt^{1/10}+h^{1/2}Δt^{-1/4}). ] Thus (Δt\asymp h^{10/7}) gives an (O(h^{1/7})) maximum-norm bound. The analysis applies to any fixed centrally symmetric positive degree-two cubature and permits rank-deficient diffusion. Numerical examples examine parallel heat loss to a cold wall and control-dependent, rank-deficient diffusion.

6. 基于随机动态模态分解和深度学习的高保真数字孪生数据模型及其在流体力学中的应用

High-Fidelity Digital Twin Data Models by Randomized Dynamic Mode Decomposition and Deep Learning with Applications in Fluid Dynamics

作者: Diana A. Bistrian

类别: cs.LG, math.NA

发布日期: 2026-09-15

arXiv ID: 2609.17101v1

链接: http://arxiv.org/abs/2609.17101v1

摘要: The purpose of this paper is the identification of high-fidelity digital twin data models from numerical code outputs by non-intrusive techniques (i.e., not requiring Galerkin projection of the governing equations onto the reduced modes basis). In this paper the author defines the concept of the digital twin data model (DTM) as a model of reduced complexity that has the main feature of mirroring the original process behavior. The significant advantage of a DTM is to reproduce the dynamics with high accuracy and reduced costs in CPU time and hardware for settings difficult to explore because of the complexity of the dynamics over time. This paper introduces a new framework for creating efficient digital twin data models by combining two state-of-the-art tools: randomized dynamic mode decomposition and deep learning artificial intelligence. It is shown that the outputs are consistent with the original source data with the advantage of reduced complexity. The DTMs are investigated in the numerical simulation of three shock wave phenomena with increasing complexity. The author performs a thorough assessment of the performance of the new digital twin data models in terms of numerical accuracy and computational efficiency.

7. 在HLL型多状态近似Riemann解中加入慢磁声模式

Adding slow magnetoacoustic mode to the HLL-type multi-state approximate Riemann solution

作者: Fan Zhang, Stefaan Poedts, Andrea Lani

类别: physics.comp-ph, astro-ph.IM, math.NA

发布日期: 2026-09-15

arXiv ID: 2609.17049v1

链接: http://arxiv.org/abs/2609.17049v1

摘要: Multi-state HLL-type approximate Riemann solutions of ideal magnetohydrodynamics (MHD) typically assume that the medium within the Riemann fan is incompressible, and thus the slow magnetoacoustic mode cannot be included in their space-time wave configurations for the approximated states. We propose a new strategy to design multi-state HLL-type approximate solutions, allowing for the medium within the Riemann fan to be compressible. In particular, for the complete seven-wave configuration of the MHD Riemann problem, we first estimate the slow magnetoacoustic speeds and a longitudinal flow speed between the slow modes, and then follow the conservation laws and Rankine-Hugoniot jump relations across the fast, Alfvén, and slow waves, to calculate all the intermediate states within the Riemann fan. Moreover, we discuss the solutions when certain wave modes degenerate, ensuring well-posedness and smooth transitions between complete and degenerate wave configurations. Numerical simulations using a Finite Volume (FV) solver show that the proposed approximate Riemann solution is less diffusive than the classic HLLD scheme, particularly for slow mode waves. For example, in a 1D test case with a strong longitudinal magnetic field, the new scheme needs one order of magnitude fewer grid cells compared to the classic HLLD scheme to resolve all wave modes.

8. 近乎最优的非凸矩阵补全

Near-Optimal Nonconvex Matrix Completion

作者: Jian-Feng Cai, Xiliang Lu, Juntao You

类别: math.NA, cs.LG

发布日期: 2026-09-15

arXiv ID: 2609.17048v1

链接: http://arxiv.org/abs/2609.17048v1

摘要: We study nonconvex methods for matrix completion, the problem of recovering a low-rank matrix from a subset of its entries. Convex methods achieve sample complexity linear in the matrix dimension and the rank, up to logarithmic factors, whereas global guarantees for commonly used nonconvex methods require a higher polynomial dependence on the rank. We close this gap by analyzing Riemannian gradient descent (RGD) and Riemannian Gauss–Newton (RGN) methods. For an $n\times n$ matrix of rank $r$ with incoherence parameter $μ$ and condition number $κ$, the two methods achieve exact recovery with high probability from $O(μnr\log n\log(nκ))$ and $O(μnr\log n\log(2μrκ))$ observations, respectively. The methods use a multiscale residual initialization, while the analysis simultaneously controls the spectral error and incoherence. The resulting RGD iterates converge linearly, whereas RGN eventually converges Q-quadratically.

9. 从基于共识的优化到粒子群优化：漂移-扩散耦合下的收敛保证

From Consensus-Based Optimization to Particle Swarm Optimization: Convergence Guarantees under Drift-Diffusion Coupling

作者: Franca Hoffmann, Dohyeon Kim, Ritvik Teegavarapu

类别: math.OC, math.NA

发布日期: 2026-09-15

arXiv ID: 2609.17002v1

链接: http://arxiv.org/abs/2609.17002v1

摘要: Particle swarm optimization (PSO) is a widely used algorithm featured in many state-of-the-art optimization tool-kits. However, rigorous performance guarantees are still lacking. The standard PSO dynamics do not admit a natural mean-field description, which would provide an avenue for theoretical analysis. By modifying the PSO formulation, one can recover the consensus-based optimization (CBO) algorithm with memory, which admits a mean-field limit and facilitates rigorous convergence analysis. These theoretical guarantees rely heavily on the fact that for CBO, the drift and noise strengths can be chosen independently, whereas they are coupled for PSO. We analyze how the PSO parameter coupling affects existing convergence guarantees for CBO and its variant with memory effect. We show, by an explicit construction, that the coupling still leaves a non-empty set of admissible parameters for these convergence guarantees to hold. However, the admissible parameter ranges shrink in the limits used to recover PSO. The resulting convergence guarantees from CBO therefore do not directly extend to the classical PSO model. We provide numerical simulations illustrating the parameter tradeoffs shown in the theoretical analysis.

10. 准地转方程的非线性滤波稳定化

Nonlinear filtering stabilizations for the quasi-geostrophic equations

作者: Lander Besabe, Sachin Kumar, Annalisa Quaini

类别: math.NA, physics.comp-ph

发布日期: 2026-09-15

arXiv ID: 2609.16983v1

链接: http://arxiv.org/abs/2609.16983v1

摘要: Numerical simulations of ocean flows typically require fine computational meshes to resolve the Munk scale, leading to high computational costs. Filtering-based large eddy simulation (LES) provides a way to relax the mesh size requirement by modeling the effects of the unresolved scales. For the implementation of this strategy, we propose a three-step algorithm called Evolve-Filter-Relax (EFR) that requires (i) the solution of a QGE problem, (ii) a nonlinear Helmholtz filter for the potential vorticity field leveraging an indicator function, and (iii) a final relaxation step. We show that the EFR algorithm can be interpreted as a splitting scheme for a perturbed QGE problem with additional dissipation and provide a practical choice for the relaxation parameter. For comparison, we also investigate a nonlinear Bardina regularization of the QGE. Numerical results on a classical benchmark show that both the EFR approach and the nonlinear Bardina regularization significantly improve the accuracy and stability of coarse mesh simulations with no LES model. Additionally, the EFR method with a deconvolution-based indicator function delivered the best balance between accuracy, stability, and computational efficiency in a test case involving a more realistic geometry (Mediterranean Sea).

11. 超线性随机反应扩散方程的适定性、正则性与强逼近

Well-posedness, Regularity, and Strong Approximations of Superlinear Stochastic Reaction-Diffusion Equation

作者: Zhihui Liu

类别: math.NA, math.PR

发布日期: 2026-09-15

arXiv ID: 2609.16979v1

链接: http://arxiv.org/abs/2609.16979v1

摘要: This paper develops a general framework for the well-posedness, regularity, and strong approximation of the stochastic reaction–diffusion equation (SRDE) with superlinear drift and diffusion coefficients. We first extend the well-posedness results in \emph{W. Liu and M. Röckner, J. Funct. Anal., 2902–2922, 2010} and \emph{W. Liu, J. Differential Equations, 572–592, 2013} to the case of superlinear diffusion in the Gelfand triple (V \hookrightarrow H \hookrightarrow V^*), with (V) equipped with the norm (|\cdot|_V), and derive a moment estimate by establishing a new Itô formula for (|X|_V^p) with general (p \ge 2). We then apply this abstract result to the SRDE, establish higher spatial regularity (\dot H^{1+γ}) for any (γ\in [0,1]) whenever the initial datum lies in the same Sobolev space, and obtain temporal Hölder regularity. Finally, we construct a family of tamed finite element methods (tamed-FEMs) for the SRDE under general assumptions on the tamed functions, derive their long-time unconditional stability, and establish optimal strong convergence rates. To our knowledge, this is the first strong approximation result for SPDEs with superlinear diffusion coefficients.

12. 基于SAV方法的能量稳定且保精度的有限体积格式及其在壁面距离计算中的应用

An energy stable and accuracy-preserving finite volume scheme based on the SAV method with application to wall-distance computation

作者: Xiaorui Xu, Qian Wang

类别: physics.comp-ph, math.NA

发布日期: 2026-09-15

arXiv ID: 2609.16885v1

链接: http://arxiv.org/abs/2609.16885v1

摘要: A novel semi-implicit second-order finite volume scheme integrating the scalar auxiliary variable (SAV) approach is proposed for solving the pseudo-time Eikonal equation in wall-distance computation. Unconditional energy stability under zero boundary conditions is rigorously proved, eliminating the dependence of the time step on the grid scale and enabling large-time-step computation. The scheme is a priori accuracy-preserving, and its discretization matrix forms an M-matrix, thereby guaranteeing strict non-negativity of the numerical solution inherently. The framework extends readily to any non-conservative scalar equation and, being independent of the specific finite-volume reconstruction, is compatible with schemes of arbitrary order of accuracy. A vanishing artificial viscosity is introduced to smooth the solution without compromising formal accuracy, and upwinding is incorporated through directional weighting in the weighted least-squares (WLS) reconstruction. Numerical experiments confirm that the scheme achieves the designed accuracy and permits stable computations with uniformly large time steps. For complex configurations such as a three-element airfoil and the three-dimensional ONERA M6 wing, accurate results are obtained on high-aspect-ratio grids, with relative errors in the computed wall distance below 3% relative to the search-based reference, except near geometric singularities.

13. 通过流形插值与子空间更新的在线自适应非侵入式模型降阶：在流固耦合收敛加速中的应用

Online adaptive non-intrusive model reduction via manifold interpolation and subspace updates: application to FSI convergence acceleration

作者: Azzeddine Tiba, Florian de Vuyst, Iraj Mortazavi

类别: cs.CE, math.NA

发布日期: 2026-09-15

arXiv ID: 2609.16876v1

链接: http://arxiv.org/abs/2609.16876v1

摘要: We introduce a novel online adaptive non-intrusive reduced-order modeling strategy for parameterized dynamical systems involving parameter and time-dependent reduced bases. The proposed framework is based on a unified Grassmann manifold formulation combining three key components: interpolation of local reduced subspaces for unseen parameters, geodesic online subspace updates driven by incoming high-fidelity snapshots, and a latent-space regression strategy relying on Grassmann-distance weighting and Procrustes alignment to consistently aggregate predictions from multiple local models. The adaptive reduced-order model is embedded in a partitioned fluid-structure interaction framework, where it predicts fluid interface forces to provide accurate initial guesses for the nonlinear coupling iterations, thus achieving computational speedups with no loss of accuracy. The reduced basis and the regression operators are adapted independently during the simulation and without requiring the storage of high-dimensional streaming data, preserving computational efficiency while substantially improving predictive capabilities. Numerical results on reference FSI test cases demonstrate superior accuracy with respect to static and global reduced-order models, leading to a significant reduction in the number of fixed-point iterations required for convergence. The proposed framework offers a flexible and fully non-intrusive approach for the efficient simulation of nonlinear parameter-dependent multiphysics problems.

14. 用于热方程全时程 Runge–Kutta 离散的混合精度求解器

Mixed precision solvers for the all-at-once Runge–Kutta discretization of the heat equation

作者: Santolo Leveque, Luca Bergamaschi, Ángeles Martínez, Erin Carson

类别: math.NA

发布日期: 2026-09-15

arXiv ID: 2609.16863v1

链接: http://arxiv.org/abs/2609.16863v1

摘要: We study the effect of mixed precision on the numerical integration of the heat equation discretized with a Runge–Kutta method in time. A full space-time discretization is applied, which results in a very large and sparse linear system to be solved for the numerical approximations and the Runge–Kutta stages of all time steps. The linear system is solved by applying a suitable preconditioned iterative method that can be run in parallel. In order to speed up the solution process, the preconditioner is applied using a mixed precision framework. Sequential results show the robustness of the preconditioner, even when applied in mixed precision. Finally, we present numerical evidence of the improved performance of the mixed precision strategy when applied in a parallel environment, achieving up to a 50% reduction in CPU time.

15. 状态约束策略迭代的罚参数一致局部化

Penalty-Uniform Localization for State-Constrained Policy Iteration

作者: Yeongjong Kim, Jiwoong Jang, Yeoneung Kim

类别: math.OC, math.NA

发布日期: 2026-09-15

arXiv ID: 2609.16829v1

链接: http://arxiv.org/abs/2609.16829v1

摘要: Penalizing a state constraint creates a singular localization problem: whole-space values may grow like the inverse penalty parameter, while numerical diffusion allows even an inward feedback to cross the boundary. For deterministic discounted optimal control, we show how the penalty itself supplies confinement that offsets this growth. With mesh size $h$ and penalty parameter $\varepsilon$, an inward barrier bounds the additional cost of numerical leakage by $O(h/\varepsilon)$ for a monotone centered-difference scheme with vanishing viscosity. Under $h\le\varepsilon$, an occupation estimate then yields $O(h)$ localization error with a sufficient box margin logarithmic in $1/h$ and independent of $\varepsilon$. We extend the result to bounded squared-distance penalties and combine it with residual-based evaluation bounds and discounted policy-error propagation. Under coupled refinement with vanishing penalization and discretization errors, explicit conditions on evaluation errors and greedy gaps ensure convergence of the neural value approximations to the constrained value, allowing measurable, nonunique greedy selectors. Reference calculations isolate leakage and localization across mesh and penalty scales, and distinguish evaluation, iteration, and approximation errors. An explicit cylindrical state-constraint solution provides a benchmark in arbitrary dimension, tested up to dimension twenty. Paired obstacle-navigation experiments illustrate why policy-value accuracy must be assessed alongside sampled residuals when comparing raw-residual and finite-grid-assisted neural evaluation.

16. 一种适应学习到的多变量结构的加权核逼近方法

A Weighted Kernel Method for Approximation that Adapts to Learned Multivariable Structure

作者: John E. Darges, Laura Weidensager

类别: cs.LG, math.NA

发布日期: 2026-09-15

arXiv ID: 2609.16606v1

链接: http://arxiv.org/abs/2609.16606v1

摘要: Approximating the input-output behavior of a multivariable black-box function from limited data is challenging when blind to the importance of its inputs and their interactions. We introduce total sensitivity kernels (TSKs), a method based on families of weighted ANOVA kernels that learn and adapt to this multivariable structure. TSKs parameterize the weights on each multivariable component of the target function by factors for each input. We propose learning these factors directly from function evaluations by selecting the reproducing kernel Hilbert space (RKHS) in which the target function has minimum norm. Under suitable conditions, we show that this norm-minimization problem admits a unique solution, and we establish consistency of a finite-data formulation based on minimum-norm interpolation. The learned TSK factors characterize the participation of individual inputs across interactions and main effects, providing a kernel-dependent notion of input sensitivity related to total Sobol indices. Numerical experiments demonstrate that adapting the kernel to learned multivariable structure can substantially improve approximation accuracy over a standard product kernel.

17. 结构驱动反演：求解反问题的一种新范式

Structure-Driven Inversion: A New Paradigm for Solving Inverse Problems

作者: Shengchang Chen

类别: math.NA, physics.geo-ph

发布日期: 2026-09-15

arXiv ID: 2609.16536v1

链接: http://arxiv.org/abs/2609.16536v1

摘要: Inverse problems are predominantly solved within the optimization-driven paradigm, which formulates the problem as objective-function minimization and approaches the solution by iterative search. Though universal, it suffers from limitations in efficiency, interpretability, and multi-parameter decoupling. This paper proposes a new paradigm—Structure-Driven Inversion (SDI). Instead of iterative search, SDI identifies and exploits the intrinsic structure of the problem to construct a solution method. It has two types of structure—mathematical structure and physical structure—and two corresponding driving inversion types: Mathematical Structure-Driven Inversion (MSDI) and Physical Structure-Driven Inversion (PSDI). The current representative method of MSDI is Mathematical Structure-Driven Pseudo-Inverse Inversion (MSDPII), which constructs a pseudo-inverse in the spectral domain via unitary diagonalization. The current representative methods of PSDI are Physical Structure-Driven Waveform Inversion Imaging (PSDWII), which performs inversion through virtual-source projection, and Physical Structure-Driven Back-Propagation (PSDBP), which implements three structural projections for deep neural networks via physical-system analogy. SDI is not a rejection but a complement and extension of Optimization-Driven Inversion (ODI). To the best of the author’s knowledge, no existing study has systematically presented structure-driven inversion as an independent paradigm; this paper aims to fill that gap.

18. 基于动理学缺陷测度的结构信息引导数据驱动标量双曲守恒律降阶建模

Structure-Informed Data-Driven Reduced-Order Modeling of Scalar Hyperbolic Conservation Laws via Kinetic Defect Measure

作者: Marissa Llamas, Jan Fuhg, Hannah Lu

类别: math.NA, physics.comp-ph

发布日期: 2026-09-15

arXiv ID: 2609.16525v1

链接: http://arxiv.org/abs/2609.16525v1

摘要: Reduced-order modeling of transport-dominated systems remains challenging because moving fronts and shocks are poorly represented by low-dimensional linear subspaces. We develop a structure-informed data-driven reduced-order model(ROM) for scalar hyperbolic conservation laws based on the kinetic defect formulation. This formulation separates the nonlinear dynamics into known characteristic transport and a kinetic entropy defect localized on the shock manifold. We exploit this structure by first removing the known transport from the solution snapshots. We then extract and register the remaining defect-driven dynamics in a shock-attached coordinate system. Separate ROMs are used to evolve the shock geometry and the registered defect-driven source. During prediction, the predicted shock geometry is used to inverse-register the learned defect-driven source, which advances the kinetic state and recovers the physical solution. Numerical examples in one and two spatial dimensions demonstrate accurate reconstruction and prediction of nonlinear transport with shocks, including evolution beyond the training interval, while accurately capturing the mass and entropy-dissipation behavior of the reference solution.

19. 超越相位约化：耦合振子最优控制中的振幅塌缩

Beyond Phase Reduction: Amplitude Collapse in Optimal Control of Coupled Oscillators

作者: Faranak Rajabi, Frédéric Gibou, Jeff Moehlis

类别: math.OC, eess.SY, math.NA

发布日期: 2026-09-15

arXiv ID: 2609.16449v1

链接: http://arxiv.org/abs/2609.16449v1

摘要: We solve the four-dimensional Hamilton-Jacobi-Bellman (HJB) equation for two diffusively coupled Stuart-Landau-like oscillators to obtain full-state optimal feedback control. A sweep over coupling strength reveals a sharp change in the numerically optimal strategy: below a threshold coupling value, the controller steers the phase difference toward anti-phase while keeping both oscillators near the limit cycle, as reduced-order models would suggest. Above this threshold, the HJB solution changes qualitatively; the controller transiently collapses one oscillator’s amplitude to near zero, thereby enabling large phase repositioning near the origin before rebuilding its amplitude. Direct gradient-based and stochastic optimization do not recover this lower-cost collapse trajectory from the initializations considered, suggesting that it occupies a region of the control landscape that is difficult to access by direct search. A joint sweep over nonisochronicity and coupling shows that collapse can occur even for an isochronous oscillator: phase repositioning near the origin can favor an off-cycle strategy. Nonisochronicity provides an additional energetic benefit through a phase-velocity surplus at small amplitude, quantitatively accounting for the observed reduction in control cost. Comparisons with uncoupled and coupled phase-reduced baselines show that phase models become increasingly inaccurate and cost significantly more energy for strong coupling. Results for coupled Van der Pol oscillators further demonstrate that exploitation of off-cycle dynamics is not specific to the Stuart-Landau-like oscillators.

20. 带 BGK 碰撞的 Vlasov-Poisson 方程的特征映射方法

Characteristic Mapping Method for Vlasov-Poisson with BGK-collisions

作者: Xi-Yuan Yin, Philipp Krah, Zetao Lin, Jean-Christophe Nave, Kai Schneider

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

发布日期: 2026-09-14

arXiv ID: 2609.16328v1

链接: http://arxiv.org/abs/2609.16328v1

摘要: This work presents the first steps for simulating kinetic plasmas with collisions using the characteristic mapping method (CMM). The CMM is a semi-Lagrangian method that explores a semi-group structure to store diffeomorphic flow maps efficiently. Using the semi-group structure, individual submaps can be composed to relate the flow backward in time to its initial food point. The novelty of the presented work is handling the source term by storing sub-integrals that correspond to the individual sub-maps and allow efficient integration. Furthermore, we use the Lagrangian structure to avoid implicit time integration in the hydrodynamic regime, which is known to be stiff. We benchmark our method on the Boltzmann-BGK and Vlasov-Poisson-BGK equations and consider different test cases, the Sod shock tube problem and nonlinear Landau damping for different Knudsen numbers. We show third-order spatial and temporal convergence and illustrate the fine-scale zoom property of CMM for the bump-on-tail instability.

重点关注论文（完整分析）#

1. Interface Control Domain Decomposition (ICDD) 方法的代数收敛性分析#

1. 研究对象和背景#

2. 主要定理或主要结果#

3. 研究方法、关键技术和核心工具#

4. 与之前工作的比较#

了解领域论文（摘要翻译）#

1. 关于超临界流体压力平衡保持（Pressure-Equilibrium-Preserving）数值通量的存在性#

摘要翻译#

2. 用于捕捉不可压缩湍流宏观行为的无导数框架#

摘要翻译#

3. 高维输运方程的分片常数稀疏网格间断 Galerkin 方法的尖锐 CFL 条件#

摘要翻译#

4. 用于高阶全局多迹积分方程的准 Helmholtz Calderón 乘性预条件#

摘要翻译#

5. 基于有限测试与神经代理模型的边界数据材料系数潜空间反演#

摘要翻译#

6. 用于多边形网格上 Reissner–Mindlin 板问题的一个基于 BGG 的任意阶离散格式#

摘要翻译#

7. 一种 Reynolds 半鲁棒、全局无散度的 E-HDG/IMEX-SAV 方法，用于 Navier-Stokes 方程的变分初始状态数据同化#

摘要翻译#

8. 保持双线性的代数通量修正#

摘要翻译#

9. 变系数双调和问题的无矩阵增广高阶紧致求解器#

摘要翻译#

10. 求解偏微分方程的物理信息随机特征神经网络#

摘要翻译#

不相关论文（基本信息）#

1. 随机 Volterra 方程的马尔可夫逼近在 Hölder 范数下的收敛性#

2. 面向大规模反问题的一种可证明收敛的 MM-GKS 变体#

3. 迈向 Pólya 猜想：通过能量正交性改进单个 Li-Yau 界#

4. 时间因果带通小波的框架界与带宽铺砌#

5. Hamilton–Jacobi–Bellman 方程的边界停止正求积格式的误差界#

6. 基于随机动态模态分解和深度学习的高保真数字孪生数据模型及其在流体力学中的应用#

7. 在HLL型多状态近似Riemann解中加入慢磁声模式#

8. 近乎最优的非凸矩阵补全#

9. 从基于共识的优化到粒子群优化：漂移-扩散耦合下的收敛保证#

10. 准地转方程的非线性滤波稳定化#

11. 超线性随机反应扩散方程的适定性、正则性与强逼近#

12. 基于SAV方法的能量稳定且保精度的有限体积格式及其在壁面距离计算中的应用#

13. 通过流形插值与子空间更新的在线自适应非侵入式模型降阶：在流固耦合收敛加速中的应用#

14. 用于热方程全时程 Runge–Kutta 离散的混合精度求解器#

15. 状态约束策略迭代的罚参数一致局部化#

16. 一种适应学习到的多变量结构的加权核逼近方法#

17. 结构驱动反演：求解反问题的一种新范式#

18. 基于动理学缺陷测度的结构信息引导数据驱动标量双曲守恒律降阶建模#

19. 超越相位约化：耦合振子最优控制中的振幅塌缩#

20. 带 BGK 碰撞的 Vlasov-Poisson 方程的特征映射方法#

重点关注论文（完整分析）

1. Interface Control Domain Decomposition (ICDD) 方法的代数收敛性分析

1. 研究对象和背景

2. 主要定理或主要结果

3. 研究方法、关键技术和核心工具

4. 与之前工作的比较

了解领域论文（摘要翻译）

1. 关于超临界流体压力平衡保持（Pressure-Equilibrium-Preserving）数值通量的存在性

摘要翻译

2. 用于捕捉不可压缩湍流宏观行为的无导数框架

摘要翻译

3. 高维输运方程的分片常数稀疏网格间断 Galerkin 方法的尖锐 CFL 条件

摘要翻译

4. 用于高阶全局多迹积分方程的准 Helmholtz Calderón 乘性预条件

摘要翻译

5. 基于有限测试与神经代理模型的边界数据材料系数潜空间反演

摘要翻译

6. 用于多边形网格上 Reissner–Mindlin 板问题的一个基于 BGG 的任意阶离散格式

摘要翻译

7. 一种 Reynolds 半鲁棒、全局无散度的 E-HDG/IMEX-SAV 方法，用于 Navier-Stokes 方程的变分初始状态数据同化

摘要翻译

8. 保持双线性的代数通量修正

摘要翻译

9. 变系数双调和问题的无矩阵增广高阶紧致求解器

摘要翻译

10. 求解偏微分方程的物理信息随机特征神经网络

摘要翻译

不相关论文（基本信息）

1. 随机 Volterra 方程的马尔可夫逼近在 Hölder 范数下的收敛性

2. 面向大规模反问题的一种可证明收敛的 MM-GKS 变体

3. 迈向 Pólya 猜想：通过能量正交性改进单个 Li-Yau 界

4. 时间因果带通小波的框架界与带宽铺砌

5. Hamilton–Jacobi–Bellman 方程的边界停止正求积格式的误差界

6. 基于随机动态模态分解和深度学习的高保真数字孪生数据模型及其在流体力学中的应用

7. 在HLL型多状态近似Riemann解中加入慢磁声模式

8. 近乎最优的非凸矩阵补全

9. 从基于共识的优化到粒子群优化：漂移-扩散耦合下的收敛保证

10. 准地转方程的非线性滤波稳定化

11. 超线性随机反应扩散方程的适定性、正则性与强逼近

12. 基于SAV方法的能量稳定且保精度的有限体积格式及其在壁面距离计算中的应用

13. 通过流形插值与子空间更新的在线自适应非侵入式模型降阶：在流固耦合收敛加速中的应用

14. 用于热方程全时程 Runge–Kutta 离散的混合精度求解器

15. 状态约束策略迭代的罚参数一致局部化

16. 一种适应学习到的多变量结构的加权核逼近方法

17. 结构驱动反演：求解反问题的一种新范式

18. 基于动理学缺陷测度的结构信息引导数据驱动标量双曲守恒律降阶建模

19. 超越相位约化：耦合振子最优控制中的振幅塌缩

20. 带 BGK 碰撞的 Vlasov-Poisson 方程的特征映射方法