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1. WaveHoltz 非均匀多尺度方法

The WaveHoltz Heterogeneous Multiscale Method

作者: Amit Rotem, Olof Runborg, Daniel Appelö

类别: math.NA

发布日期: 2026-07-07

arXiv ID: 2607.05811v1

摘要翻译: 我们考虑在具有快速变化系数的材料中，波方程带有时间谐波源的数值解。对于这类问题，直接离散化的代价过高，转而使用多尺度方法。已有若干直接在频域中离散化的多尺度方法。在本文中，我们从时域出发，将波方程的有限差分非均匀多尺度方法（HMM）与 WaveHoltz 方法相结合。每次 WaveHoltz 迭代都将波方程推进到时间周期的 Helmholtz 解。WaveHoltz 方法相对于传统 Helmholtz 求解器的优势直接传递到了这里考虑的多尺度问题中。此外，由于时域求解器不会在微观问题上人为施加边界条件，因此在均匀化的频域解中不存在由微观问题带来的边界误差。

Comment: 无

链接: http://arxiv.org/abs/2607.05811v1

1. 研究对象和背景

本文研究具有快速变化系数的 Helmholtz 方程

$$ \nabla \cdot (a^\varepsilon(x) \nabla u) + \omega^2 u = b(x), \quad x \in \Omega, $$

并考虑 Dirichlet 或 Neumann 边界条件，其中计算域 $\Omega \subset \mathbb{R}^d$ ($d=1$ 或 $2$) 是有界光滑单连通区域，$b \in L^2(\Omega)$ 不依赖于 $\varepsilon$，频率 $\omega$ 假定非共振。系数 $a^\varepsilon(x)$ 是一致有界且正定的标量函数，其最小尺度 $\varepsilon$ 远小于解的波长 $\propto \lVert \sqrt{a^\varepsilon} \rVert_{L^\infty} / \omega$，即介质具有快速振荡特征。

在此类多尺度问题中，直接对整个区域进行足够精细的离散将导致计算代价过高，因此需要采用均匀化或多尺度数值方法。经典的均匀化理论表明，当 $a^\varepsilon$ 满足尺度分离条件时，真解 $u$ 可按 $\varepsilon$ 展开为

$$ u(x) = \bar u(x) + \varepsilon u_1(x, x/\varepsilon) + O(\varepsilon^2), $$

其中零阶项 $\bar u$ 不依赖于 $\varepsilon$。特别地，若 $a^\varepsilon(x) = a(x, x/\varepsilon)$ 关于快变量 $x/\varepsilon$ 周期，则 $\bar u$ 满足均匀化方程

$$ \nabla \cdot (\bar a(x) \nabla \bar u) + \omega^2 \bar u = b(x), \quad x \in \Omega, $$

其中均匀化系数 $\bar a(x)$ 不再快速变化，从而可在远大于 $\varepsilon$ 的网格上求解。在许多应用中，$\bar u$ 可作为 $u$ 的高精度近似，且计算量大幅降低。

然而，直接对均匀化方程离散需要预先知道 $\bar a$；当 $a^\varepsilon$ 仅依赖于快变量 $a(x/\varepsilon)$ 时，$\bar a$ 可通过求解带有周期边界条件的 cell 问题得到。当系数同时依赖于慢变量和快变量 $a(x, x/\varepsilon)$ 时，cell 问题与慢变量耦合，增加了均匀化的难度。更一般地，数值均匀化方法的目标是在无需显式构造 $\bar a$ 的情况下有效逼近 $u$，并放宽对系数周期性的严格假设。

已有的数值均匀化方法包括多尺度有限元方法 (MsFEM)、广义多尺度有限元方法 (GMsFEM)、局部正交分解 (LOD) 以及异构多尺度方法 (HMM)。这些方法在处理椭圆型方程时已取得显著成功，但对 Helmholtz 方程这类非强制、非正定的问题仍面临额外困难：方程在共振频率处可能不适定，且常用的强制椭圆型先验估计不再成立。在频率域中，MsFEM 通过求解局部边值问题构造多尺度基函数，但必须引入人工边界条件，由此产生的边界误差常需使用过采样技术补偿。LOD 利用插值算子的核构造局部化多尺度空间，误差随过采样区域半径指数衰减，但仍需在整个区域上求解局部问题。HMM 则利用了尺度分离假定，通过局部微观模拟隐式逼近均匀化算子，并通过稀疏的微尺度求解获得宏观通量；但其在 Helmholtz 方程上的应用较少，且局部问题的截断同样引入近似误差。

另一方面，WaveHoltz 方法将时谐 Helmholtz 方程的求解转化为一个时间域波方程上的不动点迭代：

$$ w_{tt} = \nabla \cdot (a(x) \nabla w) + f(t, x), \quad w(x, 0) = v^{(\ell)}(x), \quad w_t(x, 0) = 0, $$

其中 $f(t, x) = -b(x) \cos(\omega t)$，然后通过时间滤波

$$ v^{(\ell+1)} = \Pi v^{(\ell)} = \frac{2}{T} \int_0^T \left( \cos(\omega t) - \frac{1}{4} \right) w(x, t) \, dt, $$

构造迭代序列。当 $\omega$ 非共振时，该迭代收敛到 Helmholtz 方程的解。WaveHoltz 方法的主要优势在于：波方程的有限传播速度避免了人造边界条件的需要；时间推进中使用的矩阵通常是对称正定的，从而可借助共轭梯度法等高效加速收敛。

本文的目标是填补以下空白：将 WaveHoltz 迭代框架与针对波方程的 HMM 相结合，发展一种无需显式知道 $\bar a$、可继承上述优势的 Helmholtz 方程数值均匀化方法。与已有的频率域多尺度方法不同，该方法在时域中进行微尺度模拟，从而彻底消除微尺度问题中的非物理边界条件误差。

2. 主要定理或主要结果

本文针对一维周期系数情形给出了严格的收敛性分析，主要结果如下。

假设 $a^\varepsilon(x) = a(x/\varepsilon)$ 是 $\varepsilon$-周期的，且宏观网格步长 $H$ 是 $\varepsilon$ 的整数倍，微观求解区域半长 $\eta = \tau$。设 $\omega$ 非共振，即至最近共振距离

$$ \delta = \min_{j = 1,\dots, M_x} \bigl\lvert \omega - \sqrt{\bar a}\lambda_j \bigr\rvert > 0, $$

其中 $-\lambda_j^2$ 是离散 Laplacian $D^2$ 的特征值，且 $\omega \ge 1$，$H\omega \le \sqrt{\bar a}$，以及微观-宏观逼近误差满足 $\lvert\bar a - \alpha\rvert / \bar a \le 1/2$，这里 $\alpha$ 是 HMM 局部通量平均给出的近似均匀化系数。

定理 3.1 (离散近似误差). 令 $\bar u_h$ 为均匀化方程 (3.3) $(\bar a D_- D_+ \bar u_h + \omega^2 \bar u_h = b)$ 的解，$v_h$ 为 WaveHoltz HMM (WHMM) 迭代的不动点，即满足

$$ D_- \bar A D_+ v_h + \omega^2 v_h = b, $$

其中 $\bar A = \operatorname{diag}(\alpha_{1/2}, \dots, \alpha_{M_x+1/2})$，且在一维周期设定下有 $\alpha_{j+1/2} \equiv \alpha$。则成立如下误差估计：

$$ \lVert \bar u_h - v_h \rVert_H \le \frac{C_1}{\delta^2} \left( \frac{\varepsilon}{\eta} \right)^q \lVert b \rVert_H, ) ]

$$ \lVert \bar u_h - v_h \rVert_H \le \frac{C_2}{\omega^2 \delta^2} \left( \frac{\varepsilon}{\eta} \right)^q \lVert D^2 b \rVert_H, $$

其中 $\lVert \cdot \rVert_H$ 为离散 $L^2$ 范数 $\lVert f \rVert_H = (H \sum f_j^2)^{1/2}$，$q$ 取决于所用局部平均核 $K \in \mathcal{K}_{p,q}$ 的消失矩阶数，常数 $C_1, C_2$ 与 $\varepsilon, \eta, H, \omega$ 无关。

此外，虽然文中定理 3.3 的内容未全部给出，但据其上下文，它进一步将 $v_h$ 与连续均匀化解 $\bar u$ 的误差归结为两项之和：有限差分截断误差与 HMM 均匀化系数的逼近误差，最终可得到形如 $\lVert v_h - \bar u \rVert_{H^2_\kappa}$ 的显式估计，其中 $H^2_\kappa$ 为按物理波数 $\kappa = \omega/\sqrt{\bar a}$ 加权的 $H^2$ 范数。

方法的关键性质包括：

均匀化系数 $\alpha$ 可通过离线计算得到（命题 2.1），其逼近精度为 $\lvert \alpha - \bar a \rvert \le C_{a,q} [(\varepsilon/\eta)^q + (\varepsilon/\tau)^q]$，且常数独立于 $\varepsilon, \eta, \tau$。
离散 WaveHoltz 算子 $\Pi_h$ 的不动点 $v_h$ 即为离散 Helmholtz 方程的解（命题 2.2），且迭代在稳定时间步长下收敛。
在更高维情况下，若使用交错网格离散，宏观矩阵 $D_- \bar A D_+$ 保持对称正定，从而可使用共轭梯度法加速迭代。

以上结果表明 WHMM 能够以与 $\varepsilon$ 无关的宏观自由度获得高精度均匀化解，逼近阶由核的消失矩阶数 $q$ 控制。

3. 研究方法、关键技术和核心工具

本文的核心技术创新在于将时间域的异构多尺度方法与 WaveHoltz 迭代有机整合。证明中面临的主要障碍有：① 如何将多尺度模拟得到的近似通量无缝嵌入到 WaveHoltz 循环中，并保证迭代的收敛性与精度；② 如何克服 Helmholtz 方程的非强制性带来的稳定性困难，从而获得非共振假设下的最优误差估计。作者的分析框架和关键工具如下。

1. 基于局部平均核的时域 HMM 通量重构 微尺度问题在每个宏观单元界面 $X_{j+1/2}$ 处独立求解，初始条件由宏观解线性插值 $P_{j+1/2}$ 给出。在局部域 $[-\eta, \eta] \times [-\tau, \tau]$ 上求解原始波动方程后，通量 $F_{j+1/2}$ 通过带核的时空平均获得：

$$ F_{j+1/2} = \frac{1}{\eta \tau} \int_{-\tau}^{\tau} \!\!\int_{-\eta}^{\eta} K(x/\eta) K(t/\tau) \, a^\varepsilon(X_{j+1/2} + x) \frac{\partial w}{\partial x} \, dx \, dt. $$

核 $K \in \mathcal{K}_{p,q}$ 具有 $q$ 阶连续导数、紧支于 $[-1, 1]$ 且前 $p$ 阶矩为零。其作用是在微观模拟的有限窗口内投影掉快尺度振荡，使得均匀化系数的逼近达到高阶多项式精度 $O((\varepsilon/\eta)^q + (\varepsilon/\tau)^q)$。值得注意的是，波方程的有限传播速度确保了微观区域按照依赖域 $[-\hat \eta, \hat \eta]$（其中 $\hat \eta = \eta + \tau \lVert \sqrt{a^\varepsilon} \rVert_{L^\infty}$）自然截断，无需施加任何人工边界条件，彻底消除了传统多尺度方法中微尺度边界误差的困扰。

2. 修正的离散 WaveHoltz 迭代与不动点理论 将上述 HMM 近似通量的线性关系 $F_{j+1/2} = \alpha_{j+1/2} P_{j+1/2}$ 代入宏观波动方程的半离散格式，得到线性系统

$$ \frac{d^2 \bar w}{dt^2} = D_- \bar A D_+ \bar w - b \cos(\omega t). $$

时间方向采用改进的蛙跳格式，引入参数 $\theta = \frac{\sin(\omega K/2)}{\omega/2}$ 以保证纯时间调和解的精确再现。离散滤波算子定义为

$$ \Pi_h v^{(\ell)} = \frac{2}{M_t} \sum_{n=0}^{M_t-1} \left( \cos\left(\frac{2\pi n}{M_t}\right) - \rho_0 \right) \bar w^n, $$

其中 $\rho_0 = \frac14 - \frac14 \tan^2\frac{\omega K}{2}$ 为修正后的滤波系数，确保离散迭代收缩。命题 2.2 借助 WaveHoltz 的已有结果 [9, 39, 38] 证明了不动点 $v_h = \Pi_h v_h$ 的存在唯一性，且它恰为离散 Helmholtz 方程 $D_- \bar A D_+ v_h + \omega^2 v_h = b$ 的解。迭代的收敛性依赖于频率 $\omega$ 与离散算子特征值之差的严格正下界 $\delta$。

3. 误差分析的关键估计 为导出定理 3.1，作者首先建立逼近系数 $\alpha$ 与真实 $\bar a$ 的偏差对解的影响。设 $\bar u_h$ 是使用精确 $\bar a$ 的离散均匀化解，将 $v_h$ 看作 $\bar u_h$ 加扰动，通过 Helmholtz 算子的稳定性引理（Lemma 3.5）控制扰动传播。引理的核心是：在非共振条件下，离散 Helmholtz 算子 $\bar a D_- D_+ + \omega^2 I$ 的逆有界，且范数由 $1/\delta$ 决定。再结合离散 Sobolev 不等式和插值误差界，最终将 $\lVert v_h - \bar u_h \rVert_H$ 归结为 $\lvert \alpha - \bar a \rvert$ 的 $q$ 阶小量。

至于 $v_h$ 与连续 $\bar u$ 的误差，文中指出利用标准有限差分离散误差和上述逼近误差的三角分解即可得到定理 3.3 所述估计，具体证明虽未全文给出，但推理路线遵循 HMM 分析的典型范式。

4. 离线均匀化系数计算与算法加速 文中还提出，利用线性特性和时间无关性，可离线求解微观问题一次而得到 $\alpha$，此后所有 WaveHoltz 迭代均直接使用该常数矩阵 $\bar A$，避免重复微观计算。对于周期系数，均匀化后 $\bar a$ 为常数，只需计算一个标量；对于更快或更复杂的系数，可逐点计算并存储 $\alpha_{j+1/2}$。这一离线策略显著降低了整体计算量。

4. 与之前工作的比较

与既有文献相比，WHMM 在以下多个维度上展现出实质性的改进或差异。

1. 与频率域多尺度方法的区别 常见的 Helmholtz 方程多尺度方法（如 MsFEM、GMsFEM、LOD 以及部分 HMM 工作）直接在频率域离散。为构造多尺度基或计算局部通量，必须在局部子域上施加某种人工边界条件（如 Dirichlet、Neumann 或周期），由此引入不可避免的边界层误差。这些方法通常需通过扩大过采样区域或使用吸收层来抑制误差，增加了实现复杂度和计算代价。例如，MsFEM 需仔细设计过采样区域来缓解边界污染；LOD 方法的误差虽随过采样半径指数衰减，但基函数中的局部问题仍依靠人工边界截断。而 WHMM 从时间域波方程出发，利用信息以有限速度传播的物理性质，自然地将微观计算区域限制在依赖域内，域外无需任何边界条件，因而微观问题完全无人工边界误差。这是该方法相对于频率域多尺度方法的一个核心优势。

2. 与传统 HMM 的对比 传统 HMM 已在椭圆型方程和抛物型方程中广泛应用，其思想是假定已知均匀化方程的形式，并通过局部细网格模拟逼近通量。针对波方程，已有的 HMM（例如 [24]）能够在时域中求解宏观波方程，但并未涉及频域 Helmholtz 方程的求解。WHMM 将时域 HMM 与 WaveHoltz 迭代结合，巧妙地将多尺度波方程求解器嵌入到一个求解 Helmholtz 方程的迭代框架中。这样做既保留了 HMM 对微尺度信息的压缩能力（通过核平均获得高阶逼近），又继承了 WaveHoltz 的优势：迭代矩阵对称正定、可结合共轭梯度法大幅加速收敛。相比直接对均匀化 Helmholtz 方程采用 HMM 离散（如 [20, 29, 36]），WHMM 避免了直接在频率域处理非强制问题以及由此带来的稳定性困难，因为时间推进中的算子（即使在半离散层面）通常是强制且易处理的。

3. 与 WaveHoltz 方法本身的比较 原始的 WaveHoltz 方法 [9, 39] 假定式 (2.12) 中的系数 $a(x)$ 为已知且光滑，无法直接处理快速振荡系数。WHMM 通过嵌入 HMM 通量估计，将 WaveHoltz 的适用范围扩展到多尺度 Helmholtz 方程。值得注意的是，在 WHMM 中，WaveHoltz 迭代的每一次传播都是在宏观粗网格上进行的，成本远低于在细网格上运行原始 WaveHoltz；微观模拟仅在迭代开始前离线完成一次，即均匀化系数预先算好后，后续迭代不再涉及时空细尺度。因此，WHMM 的整体计算复杂度与 $\varepsilon$ 无关，仅由宏观网格和所需精度决定。

4. 误差分析与假设条件 WHMM 的一维误差分析在周期性假设下进行，得出 $\lVert v_h - \bar u_h \rVert_H = O((\varepsilon/\eta)^q)$。这一代数收敛阶与传统 HMM 的理论相符，但将其成功移植到了非强制 Helmholtz 背景中。与之相对，MsFEM 或 LOD 方法的误差常由基函数的逼近性质主导，且通常依赖于更精细的局部解和过采样参数，较少出现如此清晰的单参数幂次收敛形式。

5. 实现的简洁性与潜在局限 WHMM 方法的一个显著优点是其实现相对直接：微观求解器可使用任何稳定的波动方程离散，而且因为仅在微小区域上短时模拟，编程和调优负担较轻。离线计算均匀化系数后，宏观迭代仅涉及稀疏矩阵的反复应用，可以有效利用现有 Helmholtz 求解器或 Krylov 子空间方法。但文中也指出，该方法目前是基于尺度分离假设的：它本质上是 HMM，因此主要用于能够合理均匀化的问题，不适用于无尺度分离的“粗糙”系数。这与 MsFEM/LOD 等原则上可处理更一般的粗糙介质形成对比。此外，一维周期性假设下的分析可能无法直接推广至更一般的系数，尽管作者声称在高维下使用交错网格等方法可以保持对称性，且离线计算策略依然适用，但其严格误差分析仍是未来工作。

6. 与具体文献的对比细节 文中明确提及以下前人工作：

文献 [24]：将 HMM 用于时域波动方程，给出了均匀化系数的逼近理论。本文在此基础上将其扩展至 Helmholtz 方程，并通过 WaveHoltz 迭代构建了一种频域解的工具。
文献 [9, 39, 38]：奠定了 WaveHoltz 方法的理论基础，包括连续和离散迭代的收敛性。本文直接借用了这些结论，并将其中的 Helmholtz 算子替换为 HMM 近似算子，并分析了近似带来的额外误差。
文献 [20, 29, 36] 等：为频率域 Helmholtz 问题的 HMM 或其它多尺度方法，这些工作必须在微尺度问题上施加边界条件，WHMM 则从时域完全绕开了这一困难。

中文标题: 演化流体生物膜的统一能量稳定有限元逼近

摘要翻译: 我们提出了一种用于流体生物膜动力学的统一有限元方法。该模型由体内纳维-斯托克斯方程与演化生物膜表面上的表面纳维-斯托克斯方程耦合控制，并带有由 Willmore 能量产生的弯曲力。通过允许体网格速度独立于流体速度，并允许自由的切向表面速度，我们能够推导出耦合体-表面纳维-斯托克斯系统的统一弱形式。为了处理弯曲力，我们考虑了曲率的演化方程，并提出了表面任意拉格朗日-欧拉（ALE）弱形式。使用拟合或非拟合有限元离散化，得到了适定的全离散线性格式，这些格式是无条件能量稳定的。我们展示了一系列数值算例，以证明所提方法的良好性质。

7. 双曲型方程组在物理能量度量下模态相容 Galerkin 封闭的设计原理

Design Principle for Mode-Consistent Galerkin Closure under a Physical Energy Metric for Hyperbolic Systems

作者: Hirofumi Tomita

类别: math.NA, physics.comp-ph

发布日期: 2026-07-07

arXiv ID: 2607.05781v1

链接: http://arxiv.org/abs/2607.05781v1

摘要翻译

中文标题: 双曲型方程组在物理能量度量下模态相容 Galerkin 封闭的设计原理

摘要翻译: 本文遵循 Arakawa 保持结构的理念，推导了守恒能量双曲型方程组 Galerkin 逼近的一个设计原理。其目标是在一个有限个模态的解析空间中重现连续系统的模态能量交换结构，从而使得总能量守恒作为一个推论得以保持。我们引入一个表示物理能量密度的状态依赖度量 $H(U)$，并推导相应的能量相容恒等式。在精确积分无穷多模态参考模型中，$H$-正交化使体积算子成为反对称的，因而模态能量平衡被表示为模态之间的两两交换。边界和界面上的贡献同样表示为与相邻单元模态的交换，其中内部交换两两抵消。为了在半离散有限模态系统中重现这一结构，我们结合两种构造：一个与物理能量度量耦合的 Galerkin 投影，它保证了 $H$-度量的分部求和恒等式；以及一个能量相容封闭，它通过修正 $H$-度量质量矩阵的演化来消除相容残差。所得的有限模态系统恢复了模态能量交换结构。对于间断的单元边界迹，界面贡献通过一个满足相同两两平衡的共享数值能量通量来封闭。我们还将实际的算子构造与精确积分有限模态参考模型进行了比较。反对称模态能量交换算子中的缺陷被分解为固定求积和投影求积两份贡献，从而得到一个 $O(h^{p+1})$ 相容的估计。最后，变换回原 Galerkin 基底给出一个等价的固定基底系数方程，该方程可直接实施。

不相关论文（基本信息）

1. 高维泊松方程有限元离散化的量子启发方法

Quantum-inspired methods for finite-element discretizations of the high-dimensional Poisson equation

作者: Xue Wang, James H. Adler, Xiaozhe Hu

类别: math.NA

发布日期: 2026-07-07

arXiv ID: 2607.06533v1

链接: http://arxiv.org/abs/2607.06533v1

摘要: In recent years, quantum linear system algorithms have been applied to partial differential equations (PDEs), particularly in high-dimensional settings, demonstrating an exponential speedup in dimension. Concurrently, randomized and quantum-inspired classical linear solvers have emerged, showing computational complexity comparable to their quantum counterparts in many application areas. In this paper, we investigate the applicability of these quantum-inspired classical algorithms to PDEs. We provide both upper and lower bounds on their computational complexity, proving that these methods cannot achieve exponential speedup in dimension for discretizations of high-dimensional Poisson problems. Our theoretical findings definitively demonstrate that quantum-inspired classical algorithms are not competitive with quantum algorithms for solving PDEs, confirming that quantum methods retain a significant advantage for high-dimensional problems.

2. 关于线性最近邻系统的低秩张量列可逼近性

On low-rank tensor train approximability for linear nearest neighbor systems

作者: Patrick Gelß, Sebastian Matera, Reinhold Schneider, André Uschmajew

类别: math.NA, math-ph

发布日期: 2026-07-07

arXiv ID: 2607.06453v1

链接: http://arxiv.org/abs/2607.06453v1

摘要: Low-rank tensor methods are an important tool in the numerical treatment of equations with a high-dimensional state space. Nearest neighbor interaction systems like the Ising model or more general Markov jump processes, as well as 1D finite-state quantum systems are examples of such problems. While low-rank tensor train/matrix product state models have been shown to be highly efficient for the simulation of such systems, providing theoretical justification for this remains a challenging task. One approach for obtaining estimates on required ranks for certain accuracies is to investigate the rank increase in Krylov subspace methods for solving the problem at hand. In the context of area laws for ground states of 1D spin systems, nontrivial results on rank-increasing properties of nearest neighbor operator polynomials have been obtained in work of Arad et al. [arXiv:1301.1162] by studying the partial commutativity of local operators. In the present work, this technique is applied to polynomial methods for definite linear equations and dissipative linear ODEs with nearest neighbor structure. This allows to derive corresponding low-rank approximability statements for solutions of such problems which are independent of the system size. Numerical simulations of high-dimensional nearest neighbor systems illustrate the theoretical findings.

3. 稀疏时空谱方法可通过剥离-传递实现时间步进

Sparse space-time spectral methods can time-step by peel and pass

作者: Timon S. Gutleb

类别: math.NA

发布日期: 2026-07-07

arXiv ID: 2607.06449v1

链接: http://arxiv.org/abs/2607.06449v1

摘要: Global space-time spectral methods give spectral accuracy in time but typically require the whole space-time history to be resolved and stored on a single tensor-product domain $T \times Ω$. We record that in an endpoint-benign Legendre or Chebyshev-$T$ time basis, whose polynomials all equal one at the right endpoint, the final time slice of a space-time block is recovered exactly by summing the stored coefficients along the time index. This peel-and-pass step is a special case of a Jacobi endpoint identity, which also gives derivative formulae for higher-order equations. Writing such higher-order equations as first-order systems preserves the benign value-passing structure. The result is a sparse space-time spectral element method that advances block by block, stores only one block, and needs far fewer time coefficients per solve for long-time problems. We prove the identities, give resident-memory, solve-cost and error-propagation models, and demonstrate the method on $(1{+}1)$D heat, wave and Klein–Gordon equations, and on $(2{+}1)$D fractional heat on the disk with weighted Zernike polynomials in space.

4. 嵌套体积-表面积分方程的声学应用

Nested Volume-Surface Integral Equations for Acoustics

作者: Danilo Aballay, Elwin van ’t Wout

类别: math.NA, physics.comp-ph

发布日期: 2026-07-07

arXiv ID: 2607.06429v1

链接: http://arxiv.org/abs/2607.06429v1

摘要: The simulation of high-frequency acoustic wave propagation in unbounded domains with local heterogeneous materials and high-contrast interfaces poses significant challenges to numerical methods. The volume-surface integral equation (VSIE) method is an attractive approach as it automatically satisfies the radiation condition at infinity via Green’s functions, handles heterogeneous materials via Newton potentials, and models scattering at high-contrast interfaces via surface integral operators. However, its effectiveness in practical simulations has been limited by high computational costs, sensitivity to sharp interfaces, and insufficient computational verification. This study extends the applicability of VSIE by deriving integral formulations for nested heterogeneous materials with parameter jumps at interfaces. We also develop extensive benchmarks against coupled finite-element and boundary-element methods to verify the VSIE’s accuracy and mesh convergence. The various benchmarks using open-source software demonstrate the effectiveness of VSIE for large-scale acoustic simulations.

5. 周期性CMV矩阵特征值的快速计算

Fast computation of eigenvalues of periodic CMV matrices

作者: Thomas Mach, Michele Rinelli, Raf Vandebril, David Watkins

类别: math.NA

发布日期: 2026-07-07

arXiv ID: 2607.06400v1

链接: http://arxiv.org/abs/2607.06400v1

摘要: Periodic CMV matrices are unitary matrices that can be specified by $O(n)$ data. Their eigenvalues can be computed by standard methods, storing them as conventional matrices (using $O(n^{2})$ data) in $O(n^{3})$ time. Here a fast method that computes the eigenvalues in $O(n^{2})$ time (using $O(n)$ data) is presented.

6. 快速旋转Thomas-Fermi标度下Gross-Pitaevskii方程的数值涡旋分辨率

Numerical vortex resolution for the Gross-Pitaevskii equation in the rapid rotation Thomas-Fermi scaling

作者: Patrick Henning, Anna Persson, Christos Pilichos

类别: math.NA

发布日期: 2026-07-07

arXiv ID: 2607.06362v1

链接: http://arxiv.org/abs/2607.06362v1

摘要: In this paper we analyze finite element approximations of ground states of the Gross-Pitaevskii equation in the rapid rotation Thomas-Fermi scaling. In this regime, the healing length and vortex core size are of order $\eps \ll 1$, while the effective confinement potential may degenerate as the angular velocity approaches a critical value. In this setting, we analyze the $\eps$-dependence of the ground states and show that the local flatness of the energy landscape plays a decisive role for numerical resolution. More precisely, we establish mesh size conditions that guarantee the existence of discrete ground states in finite element spaces which are quasi-best approximations of an exact ground state. In particular, we prove that the absolute $H^1$-error behaves asymptotically like $h/\eps^2$. However, to enter this asymptotic regime, the mesh size must satisfy a significantly stronger resolution condition than the natural requirement $h \lesssim \eps$. The additional restriction is governed by the first spectral gap of the Riemannian Hessian of the energy functional at the ground state, which measures the local flatness of the energy surface. With this, our results provide an explanation of the mesh resolution required to capture vortex structures in rapidly rotating Bose-Einstein condensates and highlight the interplay between vortex core size, spectral stability, and discretization accuracy.

7. 多项式系统奇异解的计算：走向无收缩的超线性收敛

Computing singular solutions of polynomial systems: towards superlinear convergence without deflation

作者: Mikhail Karapetyants, Vladimir Kolmogorov, Jeferson Zapata

类别: math.NA

发布日期: 2026-07-07

arXiv ID: 2607.06329v1

链接: http://arxiv.org/abs/2607.06329v1

摘要: In Numerical Algebraic Geometry (NAG) isolated solutions of polynomial systems are usually computed by tracking a solution curve defined by a homotopy equation. The tracking problem becomes especially challenging close to a singular root (the endgame'' regime). Existing approaches include power series endgames, Cauchy endgames, and various methods that regularize the system via dual-space-based {\em deflation}. We make the following contributions. (1) For corank-1 systems we introduce a new Arclength Endgame’’ which combines the idea of the classical {\em pseudo-arclength continuation method} with the estimation of the Puiseux series of the curve. We formally prove that it has a superlinear rate of convergence in some neighborhood of the root. The method uses only evaluations of the system and its Jacobian, whereas previous techniques with proven superlinear convergence (such as deflation) require computing additional derivatives of the system. (2) For systems with a larger corank we propose a heuristic ``Lifted Arclength Endgame’’, which shows promising experimental results. (3) A key step in our approach (as well as in the standard power series endgame) is estimating the Puiseux series of the curve, which is characterized by fractional exponents $k_i/c$ for $i\ge 1$ together with associated coefficients. Previous work addressed only estimating the ratio $k_1/c$. We present a new method for that which empirically appears to be more stable than previous methods, and also show how to estimate $k_i/c$ for $i\ge 2$.

8. 微分方程的高斯过程逼近的统一视角

A unified perspective of Gaussian process approximation for differential equations

作者: Mengwu Guo

类别: math.NA, cs.CE, stat.ML

发布日期: 2026-07-07

arXiv ID: 2607.06292v1

链接: http://arxiv.org/abs/2607.06292v1

摘要: The use of Gaussian processes for approximating differential equations has expanded rapidly, leading to a growing, diverse, and fragmented body of numerical methods. We present a unified Bayesian perspective that places these techniques within a common probabilistic framework, based on a derivative matching interpretation for incorporating differential equation constraints into likelihood. This unified perspective supports both parameter estimation and solution approximation, and shows how a range of existing methods can be understood within it. This work aims to consolidate current developments and provide a foundation for future research.

9. 关于非线性偏微分方程的对称核配置法

On Symmetric Kernel Collocation for Nonlinear PDEs

作者: Milan Bacchetta, Tobias Ehring, Bernard Haasdonk

类别: math.NA

发布日期: 2026-07-07

arXiv ID: 2607.06276v1

链接: http://arxiv.org/abs/2607.06276v1

摘要: This paper considers kernel-based approximation methods for nonlinear partial differential equations. To this end, the problem is formulated as an optimal-recovery generalized interpolation problem, that is, as an optimization problem in an RKHS with nonlinear functional constraints. This formulation provides the basis for a convergence analysis carried out directly in the RKHS and extends existing results by relaxing the uniqueness assumption on the PDE solution. In the nonunique case, the limiting object is characterized as a minimum-norm solution. Furthermore, a residual-greedy strategy for adaptive collocation point selection is proposed, and convergence of the resulting sequence of generalized interpolants is established. Numerical experiments for a stationary nonlinear heat equation illustrate the method and indicate that residual-greedy point selection can lead to markedly smaller PDE residuals than point sets selected according to fill-distance criteria.

10. 用强Lyapunov函数驯服离散粗糙路径

Taming discrete rough paths via strong Lyapunov functions

作者: Luu Hoang Duc

类别: math.NA, math.DS, math.FA

发布日期: 2026-07-07

arXiv ID: 2607.06266v1

链接: http://arxiv.org/abs/2607.06266v1

摘要: Based on the newly introduced concept of {\it strong Lyapunov functions} for rough differential equations \cite{ducjost25}, we study a tamed numerical scheme to approximate the solutions of the continuous system. We derive explicit estimates of solution norms of the tamed system which look similar to those of the continuous system. As a result, we prove the convergence of the tamed scheme in the $L^1$ sense. For systems with the negative gradient condition, we prove the existence of a numerical pullback attractor for the generated random dynamical system from the tamed numerical scheme which is integrable and upper semi-continuous w.r.t. the scheme step size.

11. PhyRes-MDNF：用于多层离散神经场反演的物理耦合残差图神经网络校正

PhyRes-MDNF: Physics-Coupled Residual GNN Correction for Multilevel Discrete Neural Field Inversion

作者: Zheng Lu, Jiwei Jia, Young Ju Lee

类别: math.NA

发布日期: 2026-07-07

arXiv ID: 2607.06237v1

链接: http://arxiv.org/abs/2607.06237v1

摘要: Coefficient inversion on fine grids under PDE constraints is ill conditioned: sparse observations weakly constrain fine-scale parameters, and direct single-resolution optimization must recover state and coefficient fields across all scales simultaneously. This causes slow, initialization-sensitive convergence; learned transfer models require offline data and can introduce approximation error into the numerical physics. We propose PhyRes-MDNF, a fixed-physics multilevel discrete neural field framework. On each level, a single-level DNF represents the inverse unknowns directly as trainable fields and optimizes the discrete objective. In the full-space Darcy realization, state fields (U) and their shared coefficient field (K) are optimized jointly in one fixed-physics inverse process. Between levels, one zero-initialized PhyRes-GNN jointly performs fixed-stencil prolongation and bounded residual correction to construct an incoming target representation, which a fixed initialization map converts to the next DNF variables. It is fitted anew from the observations and unchanged numerical model, without offline pretraining or fine-grid truth. Coarse levels therefore resolve large-scale structure before refined degrees of freedom are introduced, shortening the fine-grid optimization path while retaining the original discrete operator. Under the same final-grid update budget, the multilevel Darcy realization reduces coefficient and state errors by approximately (85%) and (90%), respectively, demonstrating improved accuracy and final-grid iteration efficiency. On measured KTC2023 EIT data, the full-(W) pipeline improves mean Otsu mIoU by approximately (3.4%) over the official linearized CEM reconstruction and (16.9%) over direct single-level DNF.

12. 一种基于分解的框架用于具有物理相互作用的互联系统的联合优化与空间包装

A Decomposition-Based Framework for Joint Optimization and Spatial Packaging of Interconnected Systems with Physical Interactions

作者: Julien Bückmann, Jorn van Kampen, Theo Hofman

类别: cs.CE, cs.CG, eess.SY, math.NA

发布日期: 2026-07-07

arXiv ID: 2607.06087v1

链接: http://arxiv.org/abs/2607.06087v1

摘要: This paper presents an approach and application of optimization of spatial packaging of interconnected systems with physical interactions (SPI2) in three-dimensional component placement problems. To enable its application for an automotive use case, SPI2 must support both initial design generation, including component alignment, and robust system-level coordination, requiring improved solution reliability and tractable computational cost. To address these requirements, the proposed methodology improves convergence rate and solution quality by enhancing numerical robustness in gradient-based optimization while reducing computational load. Existing SPI2 approaches are extended through the addition of alignment capabilities, enabling the representation of port-to-port alignments between components. Furthermore, the applicability of SPI2 is expanded by treating component placement locations as design variables, allowing for penalty-based coordination to ensure design feasibility and enabling integration within system-level optimization. The approach is validated using a multi-objective optimization framework based on Nondominated Sorting Genetic Algorithm II (NSGA-II), applied to a combined powertrain optimization and battery chassis integration problem. This demonstrates the effectiveness of the SPI2 in a system-level design context. The results show a twofold application of SPI2 in an automotive use case: first, as a tool for initial design generation, and second, as part of a system-level design coordinator that outperforms a discretized exhaustive search while requiring lower computational cost.

13. 基于等几何分析和二阶导数的电机高效帕累托前沿生成

Efficient Pareto-Front Generation for Electric Machines using IGA and Second Order Derivatives

作者: Theodor Komann, Michael Wiesheu, Stefan Ulbrich, Sebastian Schöps, Peter Gangl

类别: math.NA

发布日期: 2026-07-07

arXiv ID: 2607.06085v1

链接: http://arxiv.org/abs/2607.06085v1

摘要: The multiobjective optimization of electric machines always involves a trade-off caused by various competing objectives such as performance and cost. A suitable design is usually determined by comparing variants from the Pareto front, which has been generated by a large number of simulation runs. This paper addresses the efficient generation of the Pareto front using a continuation method based on a homotopy method that exploits second-order derivative information to achieve superlinear convergence, enabling the fast generation of new Pareto-optimal points within only a few iterations. A key contribution is the derivation of formulas to compute the Hessian with respect to geometry parameters and shape, thus enabling direct modifications of the motor geometry in the context of Isogeometric Analysis. We apply our method to nonlinear 2D magnetostatic simulations of a permanent magnet synchronous motor and demonstrate its effectiveness by optimizing the cost, mean torque and torque ripple of the motor. Compared to a first-order optimization method, this approach reduces the number of iterations and function evaluations needed, making the pareto optimization fast and efficient.

14. 用于Legendre-Gauss-Lobatto间断伽辽金谱元法的保不变域限制与自适应网格细化

Invariant-domain-preserving limiting with Adaptive Mesh Refinement for Legendre-Gauss-Lobatto Discontinuous Galerkin Spectral Element Methods

作者: Benjamin Bolm, Andrés M. Rueda-Ramírez, Dmitri Kuzmin, Gregor J. Gassner

类别: math.NA

发布日期: 2026-07-07

arXiv ID: 2607.06045v1

链接: http://arxiv.org/abs/2607.06045v1

摘要: We present an invariant-domain-preserving (IDP) treatment of nonconforming interfaces for Legendre–Gauss–Lobatto Discontinuous Galerkin Spectral Element Methods (LGL-DGSEM) with adaptive mesh refinement (AMR) on Cartesian meshes. The proposed methodology extends recently developed convex limiting and graph-viscosity frameworks for DGSEM to meshes containing hanging nodes. Starting from a conservative mortar formulation, we derive low-order interface fluxes that satisfy the requirements of invariant-domain-preserving discretizations. To avoid the excessive diffusion associated with fully connected mortar couplings, a sparsification strategy based on LGL subcell characteristic functions is introduced, yielding compact interface stencils. The resulting mortar fluxes remain conservative, reduce to the standard conforming formulation on matching interfaces, and naturally fit into graph-viscosity-based low-order schemes used for convex limiting. The proposed construction provides the missing ingredient required to combine high-order DGSEM discretizations, invariant-domain-preserving limiting, and adaptive mesh refinement within a unified framework for nonlinear hyperbolic conservation laws. We provide numerical verifications of the properties of the proposed scheme and run challenging simulations that require positivity limiting and shock-capturing.

15. SplineNet：一种面向复杂壳体的等几何深度学习方法

SplineNet: An Isogeometric Deep Learning Method for Complex Shells

作者: Shizhou Luo, Xiaodong Wei

类别: cs.LG, math.NA

发布日期: 2026-07-07

arXiv ID: 2607.06026v1

链接: http://arxiv.org/abs/2607.06026v1

摘要: We present a novel isogeometric deep learning method, termed SplineNet, for the seamless design and analysis of shell structures with complex geometries. The proposed approach is built upon watertight spline representations, e.g., analysis-suitable unstructured T-splines, and features exact geometric descriptions of Computer-Aided Design (CAD) models in neural networks. Bézier extraction is used to build the network architecture, where Bernstein polynomials serve as the nonlinear activation functions. SplineNet can be applied in a data-free or data-driven way. In the data-free case, energy-based formulations can be naturally incorporated as loss terms, which fulfill the need of Computer-Aided Engineering (CAE) and can be accurately calculated. In particular, the Kirchhoff–Love (KL) model is adopted to solve for the mechanical behaviors of shell structures. This way, CAD and CAE can be tightly integrated in a deep neural network without the time-consuming model/data exchange process. In the data-driven case, SplineNet can be used as the trunk net of Deep Operator Networks (DeepONet) to provide interpretability. Given such a trained network and unseen input data, results can be immediately obtained without retraining the network or repeatedly performing the traditional workflow for analysis. In the end, a variety of numerical examples are studied to demonstrate the effectiveness of the proposed method, especially when real-world complex geometries are involved.

16. 基于机器学习框架的亚线性低秩泊松方程求解器用于GPU加速

A Sub-linear Low-Rank Solver for Poisson’s Equation using Machine Learning Frameworks for GPU Acceleration

作者: Måns I. Andersson, Daniel Appelö

类别: cs.PF, math.NA

发布日期: 2026-07-07

arXiv ID: 2607.06021v1

链接: http://arxiv.org/abs/2607.06021v1

摘要: In this paper we explore a fast Poisson solver for problems with a solution that is known to be low-rank. We use an adaptive and warm started cross approximation called Cross-DEIM that iterates between index selection and and cross approximation to generate a low-rank solution. This paper focuses on leveraging a modern machine learning framework, PyTorch, as a general purpose array language to implement low-rank solvers based on Cross-DEIM. PyTorch enables native access to GPUs and accelerators but with a user-friendly high-level interface. We investigate statistical leverage scores for the index selection for the cross approximation due to the cost associated with the pivoted algorithms used with the discrete empirical interpolation methods (DEIM and QDEIM) which are historically preferred. The cross approximation is naturally paired with a Discrete Sine Transform (DST) Poisson solver. This allows the Fast Fourier Transform (FFT) to be evaluated in batches along dimensions independently without any global transpose even in higher dimensions. We present performance results running on a A100 GPU and AMD EPYC CPU demonstrating the usefulness of the approach that enables problems sizes that previously were not feasible.

17. 基于元图的区域分解伽辽金降阶模型

Metagraph-Based Domain-Decomposed Galerkin Reduced-Order Model

作者: Kyohei Shintate, Naoki Morita, Shigeki Kaneko, Nozomi Magome, Naoto Mitsume

类别: math.NA

发布日期: 2026-07-07

arXiv ID: 2607.06011v1

链接: http://arxiv.org/abs/2607.06011v1

摘要: This study proposes a metagraph-based domain-decomposed Galerkin reduced-order model (MBDD-G-ROM) for distributed-memory parallel reduced-order analysis of large-scale problems. The method represents domain-decomposed Galerkin reduced-order models over arbitrary domain decompositions using two graph levels: calculation-point graphs for interactions among discretization points and metagraphs for connectivity among local approximation-space subdomains. In the proper orthogonal decomposition (POD)-based implementation, POD computation subdomains are represented as metanodes, while metaedges encode the block-sparsity induced by overlaps between local POD basis supports. Partitioning the metagraph enables the POD computation subdomains to be decoupled from the parallel computation subdomains, allowing distributed-memory parallelization of both offline and online phases, including reduced-system assembly and iterative linear solution, without requiring the two decompositions to coincide. The metagraph also supports static load balancing through metanode weights that approximate computational costs. The method is evaluated for an unsteady diffusion equation and incompressible Navier-Stokes flow around a three-dimensional cylinder. The results show that MBDD-G-ROM preserves reduced-order solution accuracy while achieving high online parallel efficiency. A load-balancing test further demonstrates that cost-based metanode weights can improve computational efficiency.

18. 矩阵分解证明形式化的统一框架

A Unified Framework for Formalizing Matrix Decomposition Proofs

作者: Wanli Ma, Zichen Wang, Zaiwen Wen

类别: math.NA

发布日期: 2026-07-07

arXiv ID: 2607.05874v1

链接: http://arxiv.org/abs/2607.05874v1

摘要: Existence proofs for many matrix decompositions share a recursive routine: a local transformation prepares the matrix, a slice is selected, a recursive solution is obtained, and the result is lifted and transported back. Formalizing this routine uniformly in dependent type theory is difficult because recursive subproblems may change index types, and reconstruction must preserve structural predicates across block embeddings and reindexings. We develop a Lean~4 framework that separates decomposition schemas, transformations, reduction strategies, measures, lifting, transport, and subtype induction. The framework uses general index types, packages square and rectangular matrices in universe types, and provides a decomposition driver that assembles strategy data into subtype-induction instances. It has been instantiated across PLU, LU, LDL/Cholesky, QR variants, Gauss rank normal form, Hessenberg reductions, Schur variants, normal spectral decomposition, SVD, bidiagonalization, tridiagonalization, UTV, Smith normal form, rational canonical form, and Jordan-type forms at varying levels of statement strength. Across these instances, repeated decomposition proofs are best treated not as separate tasks but as instances of a more general inductive statement whose interface records a certified proof path compatible with the chosen decomposition statement.

19. 关于带有双侧几何包络保护机制的L-BFGS逆海森近似矩阵的条件数上界

On the Condition Number Upper Bound of the L-BFGS Inverse Hessian Approximation Matrix with a Two-Sided Geometric Envelope Safeguarding Mechanism

作者: Don Li

类别: math.OC, cs.LG, math.NA

发布日期: 2026-07-07

arXiv ID: 2607.05836v1

链接: http://arxiv.org/abs/2607.05836v1

摘要: The limited-memory BFGS (L-BFGS) algorithm is a cornerstone of large-scale optimization due to its linear memory and computational costs. However, in ill-conditioned or non-convex landscapes, the implicit inverse Hessian approximation can suffer from an exploding condition number, leading to numerical instability and degraded convergence. To address this, we propose Two-Sided L-BFGS, a safeguarded variant that dynamically constrains the condition number of the inverse Hessian operator via a two-sided geometric envelope. Moreover, we show that Two-Sided L-BFGS preserves accumulated curvature information and maintains standard $O(mn)$ memory and per-iteration time complexities. We prove that this geometric envelope yields a uniform bound on the condition number of every inverse Hessian approximation generated by the algorithm. By tracking the algebraic evolution of the extreme eigenvalues through $m$ consecutive quasi-Newton updates starting from a scaled identity matrix, the resulting bound is expressed explicitly as a function of the memory depth, problem dimension, and envelope hyperparameters. Moreover, we show that Two-Sided L-BFGS preserves asymptotic global convergence in non-convex regimes under standard smoothness and strong Wolfe line-search assumptions, matching the theoretical guarantees of L-BFGS variants utilizing the Li-Fukushima cautious update rule. Numerical experiments on high-dimensional optimization problems demonstrate that the proposed method maintains well-conditioned inverse Hessian approximations and improves robustness and convergence behavior on ill-conditioned benchmarks.

20. 一种用于双重退化抛物问题的自适应时空离散线性迭代格式

An adaptive, space-time discretized linear iterative scheme for doubly-degenerate parabolic problems

作者: Ayesha Javed, Koondanibha Mitra, Iuliu Sorin Pop

类别: math.NA

发布日期: 2026-07-07

arXiv ID: 2607.05714v1

链接: http://arxiv.org/abs/2607.05714v1

摘要: Degenerate diffusion problems, where the governing parabolic equation can change type to either an ordinary differential equation or an elliptic equation, model many real life applications. Due to the presence of free-boundaries, accurate numerical simulation of such problems require extremely small mesh and time step sizes locally. To remediate this issue, in this work, we consider a space-time formulation of the problem based on an efficient splitting of the nonlinearities. First, an iterative linearization scheme is proposed to resolve the nonlinearities that effectively reduces to solving a sequence of heat equations. Unconditional convergence of the scheme is proven even for double degenerate cases with linear convergence achieved if the problem is non-degenerate. Next, the dual norm of the nonlinear residual is decomposed into a linearization error component and a discretization error component corresponding to the heat equation. This leads to reliable and fully computable a posteriori estimates for the problem that are robust with respect to the nonlinearities/degeneracies. These estimates are used then in a fully adaptive (discretization + linearization) space-time solver. Numerical experiments for multiple test cases (one and two dimensions in space) demonstrate that this solver efficiently allocates the computational resources in the space-time domain, resulting in a rapid decay of error in terms of total degrees of freedom spent.

21. 关于Bregman近端梯度方法的线性收敛性及其在Kullback–Leibler回归中的应用

On The Linear Convergence of Bregman Proximal Gradient Methods with Applications to Kullback–Leibler regression

作者: Jonathan Chirinos-Rodríguez, Christian Daniele, Cédric Févotte, Emmanuel Soubies

类别: math.OC, math.NA

发布日期: 2026-07-06

arXiv ID: 2607.05539v1

链接: http://arxiv.org/abs/2607.05539v1

摘要: Bregman Proximal Gradient methods (BPGM) exploit the underlying geometry of the objective function through a carefully chosen mirror map. In this work, we introduce a novel notion of strong convexity, termed Restricted Relative Strong Convexity, and establish linear convergence rates for BPGM under this condition. We then exploit the proposed theoretical framework to provide an in-depth analysis of the convergence of BPGM for (regularized) Kullback–Leibler regression problems, covering scenarios with both unique and non-unique minimizers, as well as regularized and unregularized formulations. Specifically, we demonstrate that using the popular Burg’s entropy as a distance-generating function may only yield linear convergence for certain KL regression problems. In contrast, we show that employing a smoothed version of the Burg’s entropy induces the suitable geometry required to guarantee linear convergence. We conclude with numerical experiments that nicely align with our theoretical findings.

重点关注论文（完整分析）#

1. WaveHoltz 非均匀多尺度方法#

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. 周期性分层结构散射的 T 矩阵的 Trefftz DG 近似#

摘要翻译#

2. 偏微分方程解族的物理信息神经嵌入#

摘要翻译#

3. 基于核的算子学习：误差分析、预算分配及物理信息扩展#

摘要翻译#

4. 使用可迁移神经网络模型学习用于线性和非线性重叠型区域分解方法的自适应粗空间#

摘要翻译#

5. 时间不变神经算子及其在求解时间相关偏微分方程中的应用#

摘要翻译#

6. 演化流体生物膜的统一能量稳定有限元逼近#

摘要翻译#

7. 双曲型方程组在物理能量度量下模态相容 Galerkin 封闭的设计原理#

摘要翻译#

不相关论文（基本信息）#

1. 高维泊松方程有限元离散化的量子启发方法#

2. 关于线性最近邻系统的低秩张量列可逼近性#

3. 稀疏时空谱方法可通过剥离-传递实现时间步进#

4. 嵌套体积-表面积分方程的声学应用#

5. 周期性CMV矩阵特征值的快速计算#

6. 快速旋转Thomas-Fermi标度下Gross-Pitaevskii方程的数值涡旋分辨率#

7. 多项式系统奇异解的计算：走向无收缩的超线性收敛#

8. 微分方程的高斯过程逼近的统一视角#

9. 关于非线性偏微分方程的对称核配置法#

10. 用强Lyapunov函数驯服离散粗糙路径#

11. PhyRes-MDNF：用于多层离散神经场反演的物理耦合残差图神经网络校正#

12. 一种基于分解的框架用于具有物理相互作用的互联系统的联合优化与空间包装#

13. 基于等几何分析和二阶导数的电机高效帕累托前沿生成#

14. 用于Legendre-Gauss-Lobatto间断伽辽金谱元法的保不变域限制与自适应网格细化#

15. SplineNet：一种面向复杂壳体的等几何深度学习方法#

16. 基于机器学习框架的亚线性低秩泊松方程求解器用于GPU加速#

17. 基于元图的区域分解伽辽金降阶模型#

18. 矩阵分解证明形式化的统一框架#

19. 关于带有双侧几何包络保护机制的L-BFGS逆海森近似矩阵的条件数上界#

20. 一种用于双重退化抛物问题的自适应时空离散线性迭代格式#

21. 关于Bregman近端梯度方法的线性收敛性及其在Kullback–Leibler回归中的应用#

重点关注论文（完整分析）

1. WaveHoltz 非均匀多尺度方法

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. 周期性分层结构散射的 T 矩阵的 Trefftz DG 近似

摘要翻译

2. 偏微分方程解族的物理信息神经嵌入

摘要翻译

3. 基于核的算子学习：误差分析、预算分配及物理信息扩展

摘要翻译

4. 使用可迁移神经网络模型学习用于线性和非线性重叠型区域分解方法的自适应粗空间

摘要翻译

5. 时间不变神经算子及其在求解时间相关偏微分方程中的应用

摘要翻译

6. 演化流体生物膜的统一能量稳定有限元逼近

摘要翻译

7. 双曲型方程组在物理能量度量下模态相容 Galerkin 封闭的设计原理

摘要翻译

不相关论文（基本信息）

1. 高维泊松方程有限元离散化的量子启发方法

2. 关于线性最近邻系统的低秩张量列可逼近性

3. 稀疏时空谱方法可通过剥离-传递实现时间步进

4. 嵌套体积-表面积分方程的声学应用

5. 周期性CMV矩阵特征值的快速计算

6. 快速旋转Thomas-Fermi标度下Gross-Pitaevskii方程的数值涡旋分辨率

7. 多项式系统奇异解的计算：走向无收缩的超线性收敛

8. 微分方程的高斯过程逼近的统一视角

9. 关于非线性偏微分方程的对称核配置法

10. 用强Lyapunov函数驯服离散粗糙路径

11. PhyRes-MDNF：用于多层离散神经场反演的物理耦合残差图神经网络校正

12. 一种基于分解的框架用于具有物理相互作用的互联系统的联合优化与空间包装

13. 基于等几何分析和二阶导数的电机高效帕累托前沿生成

14. 用于Legendre-Gauss-Lobatto间断伽辽金谱元法的保不变域限制与自适应网格细化

15. SplineNet：一种面向复杂壳体的等几何深度学习方法

16. 基于机器学习框架的亚线性低秩泊松方程求解器用于GPU加速

17. 基于元图的区域分解伽辽金降阶模型

18. 矩阵分解证明形式化的统一框架

19. 关于带有双侧几何包络保护机制的L-BFGS逆海森近似矩阵的条件数上界

20. 一种用于双重退化抛物问题的自适应时空离散线性迭代格式

21. 关于Bregman近端梯度方法的线性收敛性及其在Kullback–Leibler回归中的应用