生成时间: 2026年06月23日 06:37:24

重点关注论文数量: 1

了解领域论文数量: 4

不相关论文数量: 31

重点关注论文（完整分析）

1. 一阶时间依赖麦克斯韦方程的 Hagstrom-Warburton 无反射边界条件的 Kreiss 稳定性分析

Kreiss stability analysis of Hagstrom-Warburton nonreflecting boundary conditions for the first-order time-dependent Maxwell equations

作者: Kaifang Liu, Matthias Schlottbom, J. J. W. van der Vegt, Yan Xu

类别: math.AP, math.NA

发布日期: 2026-06-22

arXiv ID: 2606.22754v1

摘要翻译: 本文分析了一阶时间依赖麦克斯韦方程的 Hagstrom-Warburton 无反射边界条件（HW-NRBCs）的稳定性。HW-NRBCs 具有非常小的反射系数，且不使用高阶导数或非局部边界算子，这使其非常适合高阶精确数值离散。本文的主要结果是解在 $L^2$ 意义下的先验估计，该估计用半空间中的初始和边界数据以及体积源表示。为得到此结果，我们首先推导了一个映射，该映射将具有 HW-NRBCs 的麦克斯韦方程解的出射分量映射为入射分量，这表明 Kreiss 条件并非一致成立。随后，为证明稳定性，构造了若干对称化子，从而在广义意义上建立了适定性。

Comment: 53 pages, 1 figure

链接: http://arxiv.org/abs/2606.22754v1

1. 研究对象和背景

在许多应用领域，如地震波模拟、声波与电磁波传输，散射问题被建模为无界域上的偏微分方程。实际计算中，无界域常被截断为有限计算域，并在人工边界上施加无反射边界条件（NRBCs）或完美匹配层（PMLs）以最小化非物理反射。Hagstrom 与 Warburton 提出的 HW-NRBCs 具有极低的反射系数，且避免使用高阶导数或非局部算子，非常适合高阶精度离散化。然而，对于一阶时变麦克斯韦方程，HW-NRBCs 的稳定性分析此前尚未建立。本文针对半空间域 $\Omega = \{ x \in \mathbb{R}^3 \mid x_1 < 0 \}$ 上的麦克斯韦方程进行分析，方程形式为（经尺度变换后）

$$ B \frac{\partial u}{\partial t} + \sum_{i=1}^3 A_i \frac{\partial u}{\partial x_i} = f, $$

其中 $u = (\sqrt{\bar{\varepsilon}_r \bar{\mu}_r} E^{\mathsf{T}}, \bar{\mu}_r H^{\mathsf{T}})^{\mathsf{T}} \in \mathbb{R}^6$ 是包含电磁场的未知向量，$A_i$ 为对称矩阵，$B$ 为正定对称矩阵（当 $\varepsilon_r, \mu_r$ 为常数时 $B = I$）。边界 $\Gamma = \{x_1 = 0\}$ 上施加 HW-NRBCs：

$$ \begin{aligned} A_1 \frac{\partial \phi_1}{\partial t} - L_0 \phi_0 &= L_0 g, \\ L_j \phi_j - K_j \phi_{j+1} &= 0, \quad j = 1, \dots, J, \\ B_\Gamma^- \phi_{J+1} &= 0, \\ B_\Gamma^0 L_1 \phi_1 &= 0, \end{aligned} $$

其中 $\phi_0 = u|_\Gamma$，$\phi_j$ 为辅助变量，算子 $L_j, K_j$ 包含时空导数，参数 $a_j \in (0,1)$ 用于优化吸收效果。

稳定性方面，能量估计方法对耗散边界条件有效，但 HW-NRBCs 不是最大耗散的，能量估计失效。Kreiss 理论（均匀 Lopatinskii 条件）为双曲系统提供了强适定性的代数判据，Majda 和 Osher 将其推广到特征边界。本文发现，对应 HW-NRBCs 的均匀 Kreiss 条件不成立，因此不能直接应用 Majda-Osher 的强适定性结果。

以往工作：对于标量波动方程和声波方程，Higdon-NRBCs 与 HW-NRBCs 的能量估计已有分析；对于 Maxwell 方程，[12] 中理想导体边界条件、[13] 中保守边界条件等出现了导数损失现象。本文旨在填补 Maxwell 方程 HW-NRBCs 稳定性分析的空白，证明一个 L2 先验界，并在更一般的材料系数和边界数据下建立广义适定性。

2. 主要定理或主要结果

主要结果是在适当假设下，证明了以下 L2 稳定性估计（定理 2.3）：

设 $\eta > 0$ 充分大，材料参数满足 $\varepsilon_r, \mu_r \in L^\infty(\Omega)$ 且在边界邻域为常数，源项 $f \in L^2(\Omega \times (0,\infty))^6$，初值 $u_0 \in L^2(\Omega)^6$，边界数据 $g_- \in H_\eta^1(\Gamma \times (0,\infty))^6$，并假设解 $u \in H_\eta^1(\Omega \times (0,\infty))^6$ 且迹属于 $H_\eta^1(\Gamma \times (0,\infty))^6$。则存在与 $u, \eta$ 无关的常数 $C > 0$，使得

$$ \|u\|_{0,\Gamma \times [0,\infty),\eta}^2 + \eta \|u\|_{0,\Omega \times [0,\infty),\eta}^2 \le C \left( \frac{1}{\eta} \|f\|_{0,\Omega \times [0,\infty),\eta}^2 + \|u_0\|_{0,\Omega}^2 + \|g_-\|_{1,\Gamma \times [0,\infty),\eta}^2 \right). $$

该估计表明解在加权 L2 范数下由初始数据、源项和边界数据控制。作为副产品，文中还给出了 HW-NRBCs 的显式反射系数，推广了标量波动方程和声波方程的相关结论。

3. 研究方法、关键技术和核心工具

证明的核心挑战在于 HW-NRBCs 不满足均匀 Kreiss 条件，导致算子对角化或奇异性破坏，无法直接应用 Majda-Osher 的框架。为此，作者发展了一套基于对称子的构造性方法，具体步骤如下：

  1. 半空间常数系数问题的模态分析：通过双侧拉普拉斯-傅里叶变换，将 Maxwell 系统化为关于空间变量 $x_1$ 的常微分方程组（特征模式）。引入变换 $\rho = S^{\mathsf{T}} u$（$S$ 由 $A_1$ 的特征向量构成）后，方程变为

    $$ \frac{\partial \rho}{\partial t} + \Lambda \frac{\partial \rho}{\partial x_1} + \tilde{A}_2 \frac{\partial \rho}{\partial x_2} + \tilde{A}_3 \frac{\partial \rho}{\partial x_3} = S^{\mathsf{T}} f, \quad \Lambda = \operatorname{diag}(1,1,0,0,-1,-1). $$

    在频域 $(\omega_2, \omega_3, s)$ 中求解得到特征解，建立出射模式与入射模式之间的映射关系，并给出反射矩阵 $\hat{R}(\omega, s)$。

  2. 均匀 Kreiss 条件的破坏与分情况处理：由反射矩阵可知，均匀 Kreiss 条件不能一致成立，原因在于某些参数点出现广义特征值使得矩阵 $\hat{M}$（定义于式(3.6)）不可对角化或奇异。为此，依据 $\hat{M}$ 的谱性质将分析分为三种情况：（i）矩阵可对角化且无奇异；（ii）矩阵奇异但可对角化；（iii）矩阵出现 Jordan 块。每种情况都需要分别构造对称子。

  3. 对称子的构造：对称子 $\hat{P}(\omega, s)$ 是一个与 $\hat{M}$ 相关的 Hermite 矩阵，需满足

    • 在每种情况下保持正定性或满足 Garding 不等式条件；
    • 与边界条件配合给出所需的符号控制。 构造的关键是处理虚轴附近的奇性，尤其是当 $\hat{M}$ 的零空间与边界条件的相互作用。作者通过细致的线性代数分析，给出了显式的对称子表达式（见第5.3–5.5 节引理 5.4, 5.10, 5.14）。这是本文最核心的技术贡献，对称子的结构仅依赖于 $\hat{M}$ 和边界条件的形式，因而可推广到其他边界条件。

  4. 先验估计的导出：利用对称子，通过伪微分算子演算和 Garding 不等式，在边界邻域 $\Omega_\epsilon$ 上得到加权 L2 估计。再结合内部区域 $\Omega \setminus \Omega_\epsilon$ 上的能量估计（那里材料系数可非光滑，使用标准能量方法），最终拼接得到全空间上的稳定性界。整个过程依赖于由 Garding 不等式推广得到的半空间版本及其对奇性的处理。

  5. 伪微分算子框架：文中广泛使用了双变量符号 $S_{1,0}^{m_1,m_2}$ 类伪微分算子及其运算法则，以处理拉普拉斯域中出现的符号奇性和权函数。这一套工具使得证明可以系统化，并适用于将来推广到一般光滑域。

4. 与之前工作的比较

本文与现有工作的对比可从以下维度展开：

  1. 对 HW-NRBCs 稳定性分析的推进：

    • 此前，HW-NRBCs 的稳定性结果仅限于标量波动方程 [4] 和声波方程 [30]。对于 Maxwell 方程，Hagstrom-Warburton 在 [30] 中给出了边界条件的形式，但未涉及稳定性分析。本文首次为 Maxwell 方程 HW-NRBCs 建立了 L2 稳定性估计。
    • 与标量情况不同，Maxwell 系统是多重特征（特征边界），且 HW-NRBCs 导致非对角反射矩阵，直接能量估计不能封闭。

  2. 与 Kreiss 理论和 Majda-Osher 工作的关系：

    • 经典 Kreiss 理论 [39] 针对非特征边界，Majda-Osher [42] 推广到特征边界，但均要求均匀 Lopatinskii 条件满足。本文明确证明均匀 Kreiss 条件在此失效，因此不能直接应用 [42] 的定理。这指出了此前理论的局限性。
    • 在均匀条件失效时，通常仅能得到弱适定性（带导数损失）。例如 [14] 中通过构造显式解得到结果，但该方法无法处理变系数和低阶项。本文改用对称子技术，无需显式解，且能容纳边界附近常数系数、内部变系数的一般情形。

  3. 与使用对称子处理非特征边界的工作：

    • [11, Section 2] 对非特征边界构造了对称子，本文将其思想扩展到特征边界并克服了奇性。文中特意说明，所构造的对称子并非针对 HW-NRBCs 特有，对不同边界条件仅需验证引理 5.4、5.10、5.14 的条件，形成了一个灵活的分析框架。

  4. 反射系数的显式表达：

    • 作为副产品，本文给出了 Maxwell 方程 HW-NRBCs 的反射系数，对比了 [19, 29, 30, 31] 中标量/声波的结果，表明 Maxwell 情形具有类似的高阶吸收特性，但推导更复杂。

  5. 适用范围的扩展：

    • 假设方面，本文允许内部材料系数非光滑（仅要求 L∞ 正定性），优于许多要求全局光滑系数的研究。代价是需要假设在边界邻域系数为常数，这是一种常见的工程折衷。
    • 未来工作将扩展到一般光滑域，这也是本文方法较显式解方法的优势所在。

了解领域论文（摘要翻译）

1. 基于投影的重建实现低阶DGSEM模拟的高阶精度

Projection-Based Reconstruction for Achieving High-Order Accuracy from Low-Order DGSEM Simulations

作者: Xukun Wang, Suyash Shrestha, Oscar A. Marino, Esteban Ferrer

类别: math.NA, physics.comp-ph

发布日期: 2026-06-22

arXiv ID: 2606.23504v1

链接: http://arxiv.org/abs/2606.23504v1

摘要翻译

中文标题: 基于投影的重建实现低阶DGSEM模拟的高阶精度

摘要翻译: 基于Legendre-Gauss-Lobatto (LGL)节点的高阶间断Galerkin谱元方法(DGSEM)为守恒律提供了精确且高效的离散化。然而，当多项式次数增加时，它们的计算成本、内存占用和时间步长限制迅速增长。本文针对DGSEM-LGL离散提出了一种修正的$\mathbb{P}_n\mathbb{P}_m$ ($c\mathbb{P}_n\mathbb{P}_m$)方法，旨在恢复$m^{th}$阶逼近的精度，同时仅演化$n^{th}$阶表示($n

2. 基于龙格-库塔卷积求积的分数阶变分积分器

On Runge-Kutta convolution quadrature based fractional variational integrators

作者: Khaled Hariz Belgacem, Sina Ober-BlÖbaum, Fernando JimÉnez

类别: math.NA

发布日期: 2026-06-22

arXiv ID: 2606.23302v1

链接: http://arxiv.org/abs/2606.23302v1

摘要翻译

中文标题: 基于龙格-库塔卷积求积的分数阶变分积分器

摘要翻译: 受分数阶阻尼作用的拉格朗日系统可以通过将状态变量加倍并引入分数阶导数而纳入变分框架。基于向后差分卷积求积（BD-FCQ）并结合高阶伽辽金方法的分数阶变分积分器，由于BDFCQ的多步结构未考虑伽辽金离散的内部阶段，其精度只能达到二阶。本文的主要目标是通过将用于逼近分数阶导数的龙格-库塔卷积求积（RKCQ）与高阶伽辽金方法相结合，来发展分数阶变分积分器（FVI）。RKCQ方法天然地与这类基于阶段的离散相兼容，因此更适合于构造高阶格式。我们尤其关注基于Lobatto IIIC的卷积求积。能量衰减等保持性质以及收敛性质通过数值方法进行了研究，并针对二阶格式给出了证明。所提出的格式达到了二阶、四阶和六阶精度。还包括关于中点分数阶积分器的简要讨论。

3. 面向反应扩散系统的矩阵导向分裂方法中的图灵区域保持性

Turing-region preservation in matrix-oriented splitting methods for reaction-diffusion systems

作者: Angela Monti, Fasma Diele, Carmela Marangi

类别: math.NA, math.DS

发布日期: 2026-06-22

arXiv ID: 2606.23211v1

链接: http://arxiv.org/abs/2606.23211v1

摘要翻译

中文标题: 面向反应扩散系统的矩阵导向分裂方法中的图灵区域保持性

摘要翻译: 我们针对二维区域上的两物种反应扩散系统，发展了其一阶分裂积分子的面向矩阵的公式，并以Gierer-Meinhardt系统作为离散图灵不稳定性的基准问题。利用与张量积空间离散相关联的微分矩阵方程，我们得到IE-S和EX-S两族方法，其中扩散流分别通过隐式Euler或精确方式处理，而反应子步则由显式的、辛的、伴随辛的、Poisson的以及显式变体的局部映射近似。从连续扩散驱动的不稳定性阈值出发，该阈值通过模态关系 $J_\mu = J^* - \mu D$ 表达，我们推导出完全离散的模态放大矩阵及其Jury条件。这些条件将连续的图灵机制（由第一Jury条件承载）与离散效应（由第二Jury条件承载）分离开来。将分析具体于Gierer-Meinhardt模型，我们展示出两种相反的异常情形。首先，在连续图灵稳定区，IMEX可能因违反第二Jury条件而生成本不应出现的稳定伪模式。其次，在真实的连续图灵区域中，伴随辛方法族可能是伪稳定的，从而抑制了IMEX本应正确捕捉到的模式。对于IMEX，其第一Jury条件精确再现了连续图灵多项式的符号，我们给出了保证连续图灵区域得以保持的显式时间步长条件，并表明仅在图灵稳定区才需要控制第二Jury条件。这些例子表明，每种积分器都会诱导出其自身的离散图灵区域，在解释数值模式之前，应将其与连续图灵区域进行比较；我们将这一要求归结为保持连续问题的某种定性性质，遵循保结构数值积分的宗旨。

4. 高维嵌入流形上椭圆方程的投影球面行走法

A projected walk on spheres method for elliptic equations on high-dimensional embedded manifolds

作者: Zhiyuan Hui, Changtao Sheng, Bihao Su, Tao Zhou

类别: math.NA

发布日期: 2026-06-20

arXiv ID: 2606.21883v1

链接: http://arxiv.org/abs/2606.21883v1

摘要翻译

中文标题: 高维嵌入流形上椭圆方程的投影球面行走法

摘要翻译: 本文针对嵌入流形上的屏蔽泊松方程，提出了一种投影球面行走法（PWoS）。该方法利用局部延拓与局部欧氏球中的格林表示，并结合最近点投影将边界采样点映射回流形。这一构造在欧氏外围空间中产生一个无网格且高度可并行的随机递归，而不是对流形上的Laplace-Beltrami算子直接离散。为恢复问题的内在几何结构，我们针对最近点延拓的外围Laplacian与流形上的Laplace-Beltrami算子之间的差异引入补偿项，并结合由局部几何和边界信息决定的自适应半径策略。在几何投影和指定的补偿精度下，我们针对所提出的蒙特卡罗方法在边界和闭流形两种情形下建立了均方误差估计。在参数化、隐式、高维（高达1000维）及点云流形上的大量数值算例展示了该方法在不同几何情形下的收敛性和效率。

不相关论文（基本信息）

1. $k$-局部 Lindbladian 的鲁棒结构学习

Robust Structure Learning of $k$-local Lindbladians

作者: Tim Möbus, Thiago Bergamaschi, Daniel Stilck França, Cambyse Rouzé

类别: quant-ph, cs.IT, math.NA, math.ST

发布日期: 2026-06-22

arXiv ID: 2606.23652v1

链接: http://arxiv.org/abs/2606.23652v1

摘要: We present an efficient protocol for learning an unknown $k$-local Lindblad generator on $n$ qubits using only product-state preparations, short-time evolution, and single-qubit Pauli measurements, without prior knowledge of the interaction structure. For fixed $k$ and bounded weighted interaction strength, the protocol estimates all Hamiltonian and dissipative Pauli–GKSL coefficients to entrywise accuracy $\varepsilon$ with probability at least $1-δ$ using $\widetilde{\mathcal O}_k(\varepsilon^{-2}n^{2k}\log(1/δ))$ samples and polylogarithmically many evolution times. A semidefinite projection converts these estimates into a valid $k$-local Lindblad generator with diamond-norm error at most $\varepsilon$ using $\widetilde{\mathcal O}_k(\varepsilon^{-2}n^{4k}\log(1/δ))$ samples and polynomial-time classical postprocessing. If a suitable set of influential coefficients is supplied and satisfies a stable sparsity condition, the dependence on $n$ can improve from polynomial to logarithmic; in particular, exact supports of bounded intersection degree require only $\widetilde{\mathcal O}_k(\varepsilon^{-2}\log(n/δ))$ samples, with analogous reductions in system-size dependence for sufficiently decaying long-range interactions. We also provide a robust structure-learning procedure, extend the guarantees to model misspecification, and prove complementary sample-complexity lower bounds. To our knowledge, these are the first efficient learning guarantees for general $k$-local dissipative quantum dynamics under such limited experimental control.

2. 时间依赖平均场博弈有限元逼近的后验误差界

A posteriori error bounds for finite element approximations of time-dependent mean field games

作者: Iain Smears, Harry Wells

类别: math.NA

发布日期: 2026-06-22

arXiv ID: 2606.23613v1

链接: http://arxiv.org/abs/2606.23613v1

摘要: We present a posteriori error bounds for a general class of stabilized finite element approximations of time-dependent mean field games. We first show the equivalence between the norm of the error and the dual norm of the residual in the coupled Hamilton-Jacobi-Bellman and Kolmogorov-Fokker-Planck equations. We then derive a reliable and efficient a posteriori error estimator that is based on residual estimators, along with the temporal jump estimator, and an estimator for the stabilization terms in the numerical discretization. Finally, for stabilizations based on mass-lumping in time and affine-preserving spatial stabilizations, we show that the stabilization estimators can be bounded in terms of the residual and temporal jump estimators, thus yielding an improved reliable, locally computable, and locally efficient estimator.

3. 求解超参数，免去搜索：样条回归的柯尔莫哥洛夫最优标度律

Solve for the Hyperparameter, Skip the Search: Kolmogorov-Optimal Scaling Laws for Spline Regression

作者: Yong Yi Bay, Kathleen A. Yearick

类别: cs.LG, cs.AI, math.NA, stat.ML

发布日期: 2026-06-22

arXiv ID: 2606.23575v1

链接: http://arxiv.org/abs/2606.23575v1

摘要: Hyperparameter tuning almost always means search: fit the model at every value on a grid, score each by cross-validation, and keep the winner. For spline regression that search is unnecessary. The optimal resolution can be solved for in closed form, to the accuracy an exhaustive search reaches, at a fraction of the compute. Three ingredients make this possible: classical approximation theory pins the squared bias to a known power of the resolution G, exactly the Kolmogorov n-width of the smoothness class; the basis dimension is an explicit polynomial in G; and leave-one-out error follows from a single fit via the PRESS identity. Balancing the two known curves gives the minimizer analytically. We extend this calculus to many coordinates by replacing ambient input dimension with interaction order, the number of active low-order components in an ANOVA decomposition, yielding a scaling law in which the optimal resolution and error are power functions of the effective density (sample size per active component), with input dimension absent from the exponent. The law becomes an algorithm. KORE (Kolmogorov-optimal Order-aware Resolution Estimation) fits two pilot resolutions, solves a leverage-calibrated 2x2 system for the bias and noise scales, and evaluates the closed-form plug-in resolution with a tiny leave-one-out certificate: about a dozen fits instead of a full grid sweep, with a consistency guarantee as the sample grows. Across additive and sparse pairwise targets up to 80 input dimensions, KORE matches exhaustive 3-fold cross-validation and the full classical ladder (GCV, Mallows’ Cp, AIC, BIC) while fitting roughly 8x fewer models; on 36 real tabular datasets it ranks first among 21 methods in accuracy per unit of compute, ahead of tuned boosters and kernel machines. When complexity lives in low interaction order, solving for the resolution beats searching for it.

4. 扰动的切比雪夫-洛巴托点何时仍为切比雪夫点？

When do perturbed Chebyshev–Lobatto points remain Chebyshev?

作者: Hao-Ning Wu

类别: math.NA

发布日期: 2026-06-22

arXiv ID: 2606.23409v1

链接: http://arxiv.org/abs/2606.23409v1

摘要: Chebyshev points are distinguished in polynomial interpolation by the logarithmic growth of their Lebesgue constants. This paper asks a simple question: how much can Chebyshev points be perturbed before they cease to behave like Chebyshev points? We study perturbed Chebyshev–Lobatto nodes $x_j=\cos(jπ/n+\varepsilon_j)$, with angular perturbations $|\varepsilon_j|\leq σ_n$. The study is motivated by numerical experiments showing a broad stable region when the mesh fraction $nσ_n$ is small and rapid amplification for larger perturbations; the observed transition region is consistent with the curve $nσ_n\asymp(\log n)^{-1}$. The main result is a deterministic worst-case stability estimate: if $nσ_n(\log n+1)$ is bounded by a sufficiently small constant, then the Lebesgue constant remains logarithmic. The proof uses the cosine parametrization and Bernstein’s inequality for trigonometric polynomials, thereby exploiting the angular geometry of the Chebyshev–Lobatto grid rather than a Markov inequality in the physical variable. We also give a worst-case obstruction at the angular mesh scale, showing that perturbations of order $1/n$ cannot be allowed uniformly. Consequences are derived for analytic interpolation in Bernstein ellipses, for the absence of Runge-type divergence in the stable analytic regime, and for pseudospectral differentiation. Numerical experiments illustrate the transition in the Lebesgue constants, the shape of the associated Lebesgue functions, Runge-function interpolants, and finite-precision differentiation errors.

5. 具有递归效用的连续时间异质性代理模型的有限差分方法

Finite difference methods for a continuous-time heterogeneous agent model with recursive utility

作者: Yves Achdou, Qing Tang

类别: math.OC, math.NA

发布日期: 2026-06-22

arXiv ID: 2606.23408v1

链接: http://arxiv.org/abs/2606.23408v1

摘要: We propose, analyze and test computational methods for solving a continuous-time heterogenous agent model with Epstein-Zin utility. Such recursive utilities allow the model to disentangle between risk aversion and intertemporal substitution. Having discretized the Hamilton-Jacobi-Bellman (HJB) equation arising in the model, we propose a Howard-Newton algorithm for the late resolution preference case, and a Howard-Tarski-Kantorovich algorithm for the early resolution preference case. We prove the convergence of the iterative algorithms. We obtain as a consequence the existence of solutions to the discretized HJB equations. In the late resolution case, we supply a priori estimates between the unique solutions of the continuous and discretized HJB equations.

6. 通过图-泊松凸限制的高阶不变域保持连续有限元

High-Order Invariant-Domain Preserving Continuous Finite Elements via Graph-Poisson Convex Limiting

作者: Murtazo Nazarov

类别: math.NA

发布日期: 2026-06-22

arXiv ID: 2606.23397v1

链接: http://arxiv.org/abs/2606.23397v1

摘要: We develop a high-order invariant-domain preserving continuous finite element method for nonlinear scalar conservation laws. The method combines a residual-viscosity high-order discretization with a low-order invariant-domain scheme constructed on a fine $\mathbb P_1$ submesh induced by the high-order nodal points. This separation avoids the restrictions caused by nonpositive lumped masses and overly wide high-order graph stencils. The high- and low-order updates are connected by a graph-Poisson flux reconstruction, which represents their difference as conservative antisymmetric graph fluxes. These fluxes are limited using convex limiting coefficients, followed by a capacity-based mass redistribution step that restores conservation while preserving the prescribed bounds whenever sufficient admissible capacity is available. Numerical experiments for smooth and nonsmooth scalar conservation laws demonstrate high-order accuracy in smooth regimes, robustness near discontinuities, and convergence to the entropy solution for challenging benchmarks.

7. 具有混合兼容性、高隐式求解效率和增强刚性衰减的紧凑两阶段四阶双导数IMEX方法

A Compact Two-Stage Fourth-Order Two-Derivative IMEX Method with Mixed Compatibility, High Implicit-Solve Efficiency,and Enhanced Stiff Decay

作者: Zhixin Huo

类别: math.NA

发布日期: 2026-06-22

arXiv ID: 2606.23319v1

链接: http://arxiv.org/abs/2606.23319v1

摘要: For additively split stiff evolution problems, classical fourth-order IMEX Runge-Kutta methods usually require several stages and complicated coupling order conditions. This paper proposes a compact fourth-order IMEX-type method based on a two-derivative formulation and analyzes its accuracy and stability properties. The proposed scheme incorporates the mixed explicit-implicit interaction directly through temporal derivatives evaluated along the full vector field, which ensures mixed compatibility for non-commuting split systems. With only one intermediate stage and two implicit solves per time step, the method achieves fourth-order accuracy while improving the accuracy obtained per implicit solve compared with classical multi-stage fourth-order IMEX-RK schemes. In addition, the method exhibits stronger damping of stiff modes in the purely implicit scalar limit and in the strong implicit-stiffness limit with a fixed explicit component. In the purely implicit scalar limit, its stability factor decays quadratically as the stiffness increases, whereas a representative classical fourth-order IMEX-RK reference method shows only linear decay. Numerical experiments on non-commuting split systems, scalar stiff-mode damping, increasing stiffness tests, and one- and two-dimensional advection-diffusion high-mode problems confirm the mixed consistency, the predicted stiff decay, and the smaller errors obtained under equal implicit-solve budgets in strongly stiff regimes.

8. EML树是万有逼近器

EML Trees Are Universal Approximators

作者: Joe Germany, Elie Abdo, Joseph Bakarji

类别: cs.LG, cs.NE, cs.SC, math.NA

发布日期: 2026-06-22

arXiv ID: 2606.23179v1

链接: http://arxiv.org/abs/2606.23179v1

摘要: The recently introduced EML (Exp-Minus-Log) function acts as continuous analogue of NAND gates, providing a compositional building block capable of representing elementary functions. In this work, we study the expressive power of tree-structured compositions of EML functions. We show that such trees enjoy a universal approximation property for functions in $W^{k, \infty}$ for $k \in \mathbb N$, drawing on classical neural network approximation arguments while exploiting the ability to explicitly construct EML trees that mimic polynomial representations. We further propose a learning algorithm for EML-type trees equipped with fitting parameters, and demonstrate its feasibility in practical optimization problems. Our results establish EML trees as a theoretically grounded framework for function approximation.

9. 一种基于WKB的固定网格方法用于捕捉扩散演化模型中的性状浓度

A WKB-based fixed-grid method for capturing trait concentration in a dispersal evolution model

作者: Weijie Huang, Xinran Ruan

类别: math.NA

发布日期: 2026-06-22

arXiv ID: 2606.23093v1

链接: http://arxiv.org/abs/2606.23093v1

摘要: The evolution of dispersal traits is a fundamental topic in evolutionary ecology, where natural selection may drive the trait distribution toward concentration in the rare-mutation regime. This singular behavior poses a serious numerical difficulty, since direct discretizations of the population density require very fine trait grids to identify the fittest trait and to resolve the concentrated profile accurately. In this paper, we develop a WKB-based numerical framework for a dispersal evolution model. By separating the exponentially concentrated trait dependence from a smoother amplitude variable and combining this WKB representation with dual trait-grid implementation and other specially designed techniques, the method recovers the selected trait and the associated concentration structure accurately and efficiently on fixed trait grids. We establish a semi-discrete stationary fixed-grid asymptotic-preserving structure for the rare-mutation limit of the steady-state problem. Numerical experiments compare the proposed method with direct density discretizations and confirm its advantage in the small-mutation regime.

10. 基于双参数幂尺度变分最小化的稳定图像重建

Stable Image Reconstruction via Two-Parameter Power-Scale Variation Minimization

作者: Ziwei Li, Wengu Chen, Huanmin Ge, Limei Huo, Dachun Yang

类别: cs.IT, math.FA, math.NA

发布日期: 2026-06-22

arXiv ID: 2606.23083v1

链接: http://arxiv.org/abs/2606.23083v1

摘要: In this article, we introduce a power-scale variation (PSV$_{a,p}$) with two tunable parameters: the sparsity-inducing exponent $p\in(0,1]$ and the scaling factor $a\in(0,\infty)$. By minimizing the PSV$_{a,p}$, we establish stable reconstructions in both the gradient and the image domains under the restricted isometry property (RIP) framework. Furthermore, we design an iteratively re-weighted least squares algorithm IRLSPSV to solve the unconstrained PSV$_{a,p}$ minimization. Numerical experiments demonstrate its superior performance and broad applicability. The main novelties are: (i) the PSV$_{a,p}$ minimization enjoys great flexibility and wide applicability due to its two tunable parameters $a$ and $p$, (ii) as $a\to\infty$, the PSV$_{a,p}$ minimization reduces to the $p$-th power total variation (TV$_p$) minimization and, even in this limiting case, the established RIP condition for image reconstruction is also new, (iii) the derived RIP upper bound $\overlineδ$ is proved to be asymptotically optimal in $a$ for gradient recovery, (iv) sensitivity analysis confirms the distinct roles of $a$ and $p$, thereby motivating a practical parameter tuning scheme for the proposed model.

11. Bojanov–Naidenov 关于四次多项式与二阶导数的不等式

The Bojanov–Naidenov inequality for quartics and second derivatives

作者: Gentian Zavalani

类别: math.NA

发布日期: 2026-06-22

arXiv ID: 2606.23020v1

链接: http://arxiv.org/abs/2606.23020v1

摘要: We settle the case $n=4$, $k=2$ of the Bojanov–Naidenov problem for algebraic polynomials. Let $P$ be a real polynomial of degree at most four with $\left\lVert{P}\right\rVert_{C[-1,1]}\leq 1$, and let $T_4(x)=8x^4-8x^2+1$. We prove that, for every $t\geq0$, [ \int_{-1}^{1} \bigl(|P’’(x)|-t\bigr)+,dx \leq \int{-1}^{1} \bigl(|T_4’’(x)|-t\bigr)+,dx . ] This tail estimate implies [ \int{-1}^{1}\varphi(|P’’(x)|),dx \leq \int_{-1}^{1}\varphi(|T_4’’(x)|),dx ] for every nondecreasing convex function $\varphi:[0,\infty)\to\mathbb{R}$. If $\varphi$ is strictly increasing and convex, equality can occur only for $P=\pm T_4$. The proof is elementary and finite. We interpolate at the five extremal points of $T_4$; convexity then reduces the problem to the $32$ sign choices at these nodes. At each vertex the second derivative is a quadratic polynomial, so the remaining work is an explicit comparison of level sets.

12. 通过NFFT计算连续可积函数的傅里叶变换

Computation of the Fourier transform for a continuous integrable function via NFFT

作者: Daniel Potts, Manfred Tasche

类别: math.NA

发布日期: 2026-06-22

arXiv ID: 2606.22970v1

链接: http://arxiv.org/abs/2606.22970v1

摘要: We investigate the approximation of continuous Fourier transforms by trigonometric sampling polynomials and their efficient evaluation by the nonequispaced fast Fourier transform (NFFT). While the NFFT is traditionally used for the evaluation of trigonometric polynomials, we show that it can also serve as an effective computational tool for the approximation of Fourier transform values. Building on ideas of M.~Ehler, K.~Gröchenig, and A.~Klotz \cite{EhGrKl24}, we derive explicit $\ell_\infty$ error bounds between the Fourier transform and suitable sampling polynomials. The resulting estimates quantify the influence of the sampling width and truncation parameter and provide rigorous accuracy guarantees on entire frequency intervals. In contrast to previous analyses focusing mainly on discrete or $L_2$-type errors, our results yield uniform approximation bounds that are directly relevant for practical computations. The derived theory leads to a simple algorithmic framework: first approximate the Fourier transform by a trigonometric sampling polynomial and then evaluate this polynomial efficiently by the NFFT. Numerical experiments confirm the theoretical convergence rates and demonstrate that accurate approximations of continuous Fourier transforms can be obtained with moderate computational effort.

13. 局部守恒重分配限制及其在守恒方程逼近中的应用，第二部分

Locally conservative redistribution limiting and applications to the approximation of conservation equations, Part II

作者: Jean-Luc Guermond

类别: math.NA

发布日期: 2026-06-22

arXiv ID: 2606.22839v1

链接: http://arxiv.org/abs/2606.22839v1

摘要: A non-intrusive, discretization and PDE agnostic limiting technique is introduced. The method is conservative and can be applied to high-order approximations including spectral methods. It can be applied to time-dependent and time-independent problems. It can be used for elliptic, parabolic and hyperbolic problems.

14. 通过谱优化的分解稀疏近似逆预处理

Factored Sparse Approximate Inverse Preconditioning via Spectral Optimization

作者: Francesco Brarda, Tianshi Xu, Vassilis Kalantzis, Rui Peng Li, Yuanzhe Xi

类别: math.NA

发布日期: 2026-06-22

arXiv ID: 2606.22742v1

链接: http://arxiv.org/abs/2606.22742v1

摘要: In this paper, we study value selection for fixed-pattern factorized sparse approximate inverse preconditioners. Given a prescribed sparsity pattern for a factor $G,$ we choose its admissible entries by optimizing spectral objectives of the congruent preconditioned operator $P(G)=GAG^T.$ This differs from classical sparse approximate inverse and FSAI constructions, which choose entries through algebraic Frobenius-residual criteria. For symmetric positive definite systems, the spectral target is a cluster near $+1.$ For symmetric indefinite systems, where congruence preserves inertia, we introduce a bimodal loss that drives positive and negative eigenvalues toward separated clusters near $+1$ and $-1,$ while penalizing eigenvalues near zero. To make these objectives practical for large sparse matrices, we derive projected Krylov support-gradients. Lanczos runs provide both a stochastic trace estimate of the spectral objective and a Ritz approximation to the exact gradient. We implement the resulting gradient through a detached Rayleigh surrogate: the Lanczos data are computed without gradient tracking and held fixed, while the backward pass differentiates only recomputed Rayleigh quotients with respect to the admissible entries of $G.$ This avoids differentiating through the Lanczos recurrence while returning a matrix-free gradient on the prescribed support. We also discuss a projected Kernel Polynomial Method rule as a finite polynomial comparison. Experiments on finite-element test problems show that spectral value selection improves fixed-support preconditioners, especially for symmetric indefinite saddle-point systems. We further demonstrate a graph neural network model for predicting admissible factor entries across related matrices.

15. 插值：一种线性代数方法

Interpolations: a linear algebra approach

作者: Andronick Arutyunov, Nikolay Korgin

类别: math.NA, math.OC

发布日期: 2026-06-21

arXiv ID: 2606.22671v1

链接: http://arxiv.org/abs/2606.22671v1

摘要: This paper presents a unified linear-algebraic framework for interpolation problems, viewing different standart problems (Lagrange polynomials, Hermite splines, and trigonometric interpolation) with the help of dual spaces. By representing interpolation conditions as linear functionals in the dual of a finite-dimensional function space, we establish a general existence and uniqueness theorem. We derive explicit formulas for basis functions dual to given interpolation schemes, enabling efficient construction of interpolants as linear combinations of these bases. The approach is extended to trigonometric interpolation, where we provide closed-form expressions for dual bases and analyze degeneracy conditions. We also address practical scenarios involving heterogeneous sensors, where function values and derivatives are specified at distinct points, and propose a corresponding interpolation scheme. The method is systematic, dimension-aware, and decouples the analysis of interpolation conditions from the choice of basis functions. We cosider examples from kinematics and periodic function interpolation demonstrate the versatility of the approach, suggesting applications in trajectory planning, signal processing, and beyond.

16. 基于不完全矩阵-向量乘积的随机化迹估计进行图三角形计数

Counting Triangles of Graphs via Randomized Trace Estimation with Incomplete Matrix-Vector Products

作者: Soumyadip Ghosh, Lior Horesh, Vasileios Kalantzis, Yingdong Lu, Tomasz Nowicki, Shashanka Ubaru

类别: math.NA

发布日期: 2026-06-21

arXiv ID: 2606.22629v1

链接: http://arxiv.org/abs/2606.22629v1

摘要: Counting triangles in graphs is a fundamental operation in network analysis, underpinning metrics such as clustering coefficients and serving as a signal for community detection, link prediction, and anomaly detection. The standard approach computes the trace of the cube of the adjacency matrix, but explicitly forming $\mathbf{A}^3$ is infeasible for large graphs. Hutchinson randomized trace estimator offers an efficient alternative by approximating the trace through stochastic averaging of quadratic forms, requiring only matrix vector products with $\mathbf{A}$. However, in distributed and heterogeneous computing environments, observing all entries of these products can be costly due to communication overhead and straggler effects. To address this, we propose a new variant of Hutchinson estimator that operates under partial observation constraints, where both the number and identities of observed entries are random. We provide theoretical guarantees on unbiasedness, variance bounds, and sample complexity, and demonstrate through experiments on synthetic and real world graphs that our method achieves accurate triangle count estimates while reducing synchronization costs. This work highlights the adaptability of randomized algorithms to modern computational architectures and opens avenues for efficient motif counting in large scale network analytics.

17. 非线性Burgers方程的鲁棒单元中心节点积分法（RCCNIM）：精确表述、高效实现与验证

A Robust Cell-Centered Nodal Integral Method (RCCNIM) for Nonlinear Burgers’ Equation: Accurate Formulation, Efficient Implementation, and Validation

作者: Nadeem Ahmed, Ram Prakash Bharti

类别: math.NA, physics.flu-dyn

发布日期: 2026-06-21

arXiv ID: 2606.22549v1

链接: http://arxiv.org/abs/2606.22549v1

摘要: An improved formulation of the recently developed cell-centered nodal integral method (MCCNIM) is proposed for the numerical solution of the nonlinear Burgers’ equation. The improved scheme, referred to as RCCNIM, reformulates the nonlinear convection term prior to discretization by evaluating the convective velocity using cell-averaged values from the previous time level, rather than the present-time approximation used in the original MCCNIM formulation. This reformulation leads to a fully algebraic system whose coefficients depend only on known quantities and are therefore evaluated once per time step. As a result, the proposed method significantly reduces computational cost while retaining the accuracy of the original MCCNIM. The performance of RCCNIM is assessed through systematic numerical comparisons with MCCNIM for the nonlinear Burgers’ equation. The results demonstrate that RCCNIM achieves comparable accuracy with improved computational efficiency, indicating its potential for extension to more complex nonlinear fluid flow problems.

18. PI-DOSnet: 一种用于演化偏微分方程的物理信息深度算子分裂网络

PI-DOSnet: A Physics-Informed Deep Operator-Splitting Network for Evolution Partial Differential Equations

作者: Jizu Huang, Yue Qian, Tao Zhou

类别: math.NA

发布日期: 2026-06-21

arXiv ID: 2606.22514v1

链接: http://arxiv.org/abs/2606.22514v1

摘要: Evolution partial differential equations (PDEs) describe time-dependent physical systems governed by differential laws and arise widely across science and engineering. In recent years, operator learning has emerged as a powerful and efficient paradigm for solving evolution PDEs by learning mappings between infinite-dimensional function spaces, enabling solution prediction without explicit time-step integration. In this work, we propose PI-DOSnet, a physics-informed operator learning framework built upon DOSnet and operator splitting. Unlike purely data-driven operator learning methods, PI-DOSnet incorporates physical constraints during training, allowing it to operate even in the absence of paired input-output data. Once trained, PI-DOSnet performs long-time inference of PDE solutions through an iterative strategy. We analyze the linear stability and approximation error of PI-DOSnet and demonstrate its accuracy, efficiency, and robustness through multiple numerical experiments. Moreover, for the Allen–Cahn equation, PI-DOSnet achieves energy stable solutions even with a large time-step size.

19. Poisson方程的杂交高阶离散化的最优$hp$误差估计与$p$-多重网格收敛

Optimal $hp$-error estimates and $p$-multigrid convergence for Hybrid High-Order discretizations of the Poisson equation

作者: Daniele A. Di Pietro, Emil Hössjer

类别: math.NA

发布日期: 2026-06-21

arXiv ID: 2606.22492v1

链接: http://arxiv.org/abs/2606.22492v1

摘要: This paper presents two new theoretical results for Hybrid High-Order (HHO) methods applied to elliptic problems. First, we establish $hp$-error estimates for the HHO discretization of the Poisson problem that achieve optimal approximation rates with respect to both the mesh size $h$ and the polynomial degree $k$. These results improve upon previous analyses of hybrid methods, whose convergence estimates were suboptimal in $k$. Second, building on these estimates, we develop and analyze a non-inherited $p$-multigrid solver for the statically condensed HHO system. We prove results that improve upon the corresponding theory available for other non-conforming methods and constitute, to the best of our knowledge, the first rigorous convergence analysis of a $p$-multigrid algorithm for HHO discretizations.

20. 通过锥射线见证的精确非负矩阵分解：钝角排序、饱和曲线与一种增广交替LP突破

Exact Nonnegative Matrix Factorization via Cone-Ray Witnesses: Obtuseness Ranking, Saturation Curves, and an Augmented Alt-LP Breakthrough

作者: Mithil Ramteke

类别: math.NA, cs.CG

发布日期: 2026-06-21

arXiv ID: 2606.22451v1

链接: http://arxiv.org/abs/2606.22451v1

摘要: We study exact nonnegative matrix factorization (NMF) of small exact-rank-r matrices via a cone-ray pipeline combining the truncated SVD, the polyhedral cone of nonnegative preimages, the Double Description Method (DDM, via Fukuda’s cddlib), and an alternating linear program (alt-LP) for slack minimisation. Under a uniform-support restriction the factorisation constraint Q P^T = I_r reduces to entrywise nonnegativity of an r x r witness matrix M_T = R_T^{-1} (R_K^T)^{-1} for an r-subset pair (T, K) of cone rays; this closed-form witness recovers an exact NMF in microseconds when feasible. We characterise feasibility by ranking r-subsets via geometric near-orthogonality (“obtuseness”) and walking the top of each list. A 100-trial Monte Carlo at m=n=10 exposes a clean saturation curve: success 44/32/8, 79/85/58, and 79/87/59 of 100 at top-5/200/400 for r=4,5,6 – beyond top-200 the failures are structural, not budget-limited. Enlarging m,n at fixed r hurts: at m=n=15 success collapses to 37/7/0/0/0 for r=4..8. On Olivetti faces (400x4096) the DDM step itself times out. Our main contribution is a hybrid that breaks this ceiling: at each pair we first try the closed-form M_T, and when it is infeasible we augment both supports by k=2 maximally angularly-separated rays and solve for mu,nu>=0 by a short slack-LP alternation. On the same m=n=10 Monte Carlo this lifts success from 79/85/58 to 99/95/75 at r=4,5,6, with cone reconstruction error at or near machine precision. We close with the four scaling walls the pipeline faces and concrete next steps.

21. 使用图神经网络的多重网格训练用于分子生成

Multigrid Training for Molecular Generation using Graph Neural Networks

作者: Zixuan Ling, Paula Mercurio, Di Liu

类别: cs.LG, math.NA

发布日期: 2026-06-21

arXiv ID: 2606.22377v1

链接: http://arxiv.org/abs/2606.22377v1

摘要: Deep learning has demonstrated significant success for modeling biochemical molecular systems, where inputs are commonly represented as graphs or 3D grids. A major challenge is that computational cost scales with resolution, making full graph/grid computation of molecular densities expensive and often unstable. We introduce a multigrid training strategy that leverages low-resolution optimization to accelerate learning at higher resolution through parameter transfer across discretizations. For graph molecular representations, we progressively transfer parameters learned from a coarse graph to a sequence of increasingly finer graphs via biased random walk upsampling. For 3D molecular generation, we voxelize the molecular structures at multiple resolutions, pretrain a coarse-resolution conditional Variational Autoencoder (CVAE), and initialize a fine-resolution CVAE by transferring shape compatible convolutional parameters from the coarse model. Numerical experiments on receptor-conditioned 3D Ligand generation show that multigrid training accelerates convergence and improves generalization compared to training from scratch.

22. 基于一般乘子的梯度流隐式-显式线性多步方法的能量耗散分析

Energy Dissipation Analysis of Implicit-Explicit Linear Multistep Methods for Gradient Flows Using General Multipliers

作者: Chaoyu Quan, Huaijin Wang, Xuping Wang, Chuanju Xu

类别: math.NA

发布日期: 2026-06-21

arXiv ID: 2606.22290v1

链接: http://arxiv.org/abs/2606.22290v1

摘要: A unified framework is proposed to establish the energy dissipation of implicit-explicit linear multistep methods (IMEX-LMMs) applied to gradient flows. This framework is based on testing the schemes with a general multiplier, defined as a linear combination of first-order differences of the numerical solutions. The Dahlquist’s G-stability theory is generalized to derive conditions on the coefficients of the IMEX-LMM and the multiplier for modified energy dissipation. Given a fixed IMEX-LMM, it is shown that finding an appropriate multiplier for constructing the dissipative energy can be formulated as a feasibility problem over the multiplier coefficients, where the involved positivity inequalities on the unit circle are relaxed to a finite-dimensional linear programming problem after discretization. Within this framework, two specific multipliers are obtained to establish the energy dissipation of the sixth-order IMEX backward differentiation formula (IMEX-BDF6) and a seventh-order IMEX weighted and shifted BDF, and a new eighth-order energy-dissipative IMEX-LMM is developed together with its multiplier. To the best of our knowledge, these are the first energy-dissipation results for the IMEX-BDF6 method and the IMEX-LMMs of order higher than six. It should be mentioned that this framework can also be applied without difficulty to the $L^2$- and $H^1$-stability analysis of general LMMs for linear parabolic problems. Numerical experiments illustrate the temporal accuracy and modified energy dissipation of the schemes studied.

23. 针对错误指定情形的协变量偏移适应中Nyström子抽样的收敛性分析

Convergence Analysis of Nyström Subsampling in Covariate Shift Adaptation for Misspecified case

作者: Hanna Myleiko, Sergei Solodky, Vasyl Semenov

类别: stat.ML, cs.LG, math.NA

发布日期: 2026-06-20

arXiv ID: 2606.22259v1

链接: http://arxiv.org/abs/2606.22259v1

摘要: This paper investigates convergence properties of regularized Nyström subsampling applied to the unsupervised domain adaptation problem under covariate shift. We focus on the low-smoothness (misspecified) case where the target function lies outside the reproducing kernel Hilbert space. By combining Tikhonov regularization with Nyström projection onto a subsampled subspace, we obtain upper bounds on the excess risk that hold with high probability and are expressed in terms of the source condition, the effective dimension, and the sample sizes. We further extend the analysis to the setting where the Radon-Nikodym derivative between the target and source marginal distributions is unknown and must be approximated, and we identify the minimal additional sample sizes required to maintain the same convergence rate as in the oracle case.

24. 广义Lyapunov方程的补偿分裂法

Compensated Splitting For Generalized Lyapunov Equations

作者: Hongjia Chen, Ren-Cang Li

类别: math.NA

发布日期: 2026-06-20

arXiv ID: 2606.22245v1

链接: http://arxiv.org/abs/2606.22245v1

摘要: The generalized Lyapunov equation has a natural splitting that leads to the standard fixed-point iteration (sFPI) for its numerical solution. sFPI is convergent, for an arbitrarily given initial guess, if and only if the spectral radius of the associated linear operator is smaller than $1$. This means that sFPI may diverge if the spectral radius is $1$ or bigger. In this paper, we propose a compensated splitting scheme that aims to reduce the spectral radius so that the resulting compensated fixed-point iteration (cFPI) has a better convergence property, namely, cFPI may still converge even if sFPI does not or cFPI converges faster than sFPI does when the latter is also convergent. Numerical results are presented to demonstrate the superiority of cFPI to sFPI.

25. 用于缓增分数阶耦合相场系统的带有快速历史压缩的分数记忆物理信息神经网络

A Fractional-Memory Physics-Informed Neural Network with Fast History Compression for Tempered Fractional Coupled Phase-Field Systems

作者: Shubham Kumar, Himanshu Kumar Dwivedi, Matthias Ehrhardt, Rajeev

类别: math.NA, math.DS

发布日期: 2026-06-20

arXiv ID: 2606.22191v1

链接: http://arxiv.org/abs/2606.22191v1

摘要: Tempered time-fractional coupled phase-field (tTFCP) systems are used to model interfacial phenomena involving memory-dependent transport and relaxation mechanisms. Numerical solutions to these systems are challenging due to the simultaneous presence of nonlocal temporal operators, weak initial singularities, moving diffuse interfaces, and strongly coupled multiphysics dynamics. In this work, we introduce FM-tfPINN (fractional-memory physics-informed neural network), which is used for forward simulation and inverse parameter identification in tempered fractional coupled phase-field systems. Unlike conventional fractional PINNs, which enforce memory effects solely through residual constraints, our framework incorporates tempered fractional memory directly into the neural representation via latent memory-source functions and a tempered fractional integral operator. We develop a fast shifted residual formulation based on graded temporal meshes and sum-of-exponentials (SOE) history compression to efficiently evaluate the tempered fractional operators. This framework combines interface-aware and residual-adaptive collocation strategies, improving resolution near evolving diffuse interfaces. A unified, physics-informed loss formulation allows for the forward prediction and inverse recovery of unknown physical parameters from sparse observations. We assess the proposed method on a class of tempered fractional corrosion phase-field models, including one-dimensional corrosion-front propagation, activation- and diffusion-controlled regimes, two-dimensional pitting corrosion, and inverse mobility identification problems. The numerical results demonstrate the accurate recovery of coupled phase and concentration fields, the robust prediction of physically relevant interface diagnostics, and the reliable estimation of parameters from limited data.

26. 基于快速平均L1方法的时间分数阶对流-弥散方程的二阶$H^1$范数误差分析

Second-order $H^1$-norm error analysis for time-fractional advection-dispersion equations based on the fast averaged L1 method

作者: Liangcai Huang, Lin Li, Pengcheng Xie

类别: math.NA

发布日期: 2026-06-20

arXiv ID: 2606.22083v1

链接: http://arxiv.org/abs/2606.22083v1

摘要: In this paper, based on the fast averaged L1 method, we present an error analysis for time-fractional advection-dispersion equations with a weak singularity at the initial time. An integrating-factor transformation is introduced to convert the tempered fractional derivative into the standard Caputo derivative, which is more suitable for discretization using the fast averaged L1 method. A sum-of-exponentials approximation is then incorporated into the averaged L1 method to reduce computational cost and storage while preserving the desired accuracy. By deriving error estimates for the discrete coefficients and the accumulated truncation errors, we establish the stability and $H^1$-norm convergence analysis, with a convergence order higher than those in the published literature. Numerical examples are tested to validate our theoretical results. The effects of the fractional parameters $α$ and $λ$ on the solution are discussed. The memory effect and long-time tail phenomenon, which are known to exist in real systems yet cannot be captured by classical integer-order equations, are again found in the current fractional case.

27. 面向大规模牛顿求解器的谱安全神经算子热启动

Spectrally Safe Neural Operator Warm-Starts for Large-Scale Newton Solvers

作者: Jaemin Oh, Youngkyu Lee, Jerome Darbon, George Em Karniadakis

类别: math.NA, cs.LG

发布日期: 2026-06-20

arXiv ID: 2606.21828v1

链接: http://arxiv.org/abs/2606.21828v1

摘要: Neural operators are increasingly used to warm-start Newton solvers for nonlinear PDEs, on the premise that a low test error places the initial guess inside the basin of attraction. We show that this premise is unreliable. An operator trained to the relative (L^2) error (O(10^{-3})) can still produce an initial state in which the discrete Jacobian is indefinite, because the mean-squared training controls error on average while leaving localized pointwise violations of the underlying physics. For a nearly incompressible hyperelasticity problem, we trace this to the predicted volume change: the operator disperses (\mathrm{det} F) well away from one, and the resulting Jacobian acquires negative eigenvalues even when the predicted field is visually indistinguishable from the reference. At a small scale, this is a nuisance; at a multi-million degree-of-freedom scale, it is disqualifying, since the conjugate gradient and other Krylov solvers needed for memory-feasible Newton steps assume a definite spectrum. We then show that a short, label-free fine-tuning phase – penalizing the operator against the discrete energy, with no additional solution data – shifts the Jacobian spectrum back to positive definite. Combined with an inexact outer loop, this gives a warm-started Newton method that converges across the full loading range where the unregularized operator fails, reaching up to 5.4(\times) wall-clock speedup over incremental continuation on a 3D problem with 6.4 million degrees of freedom.

28. 一种用于识别热方程中点源数目之矩–Hankel秩方法

A Moment–Hankel Rank Method for Identifying the Number of Point Sources in the Heat Equation

作者: Zhiliang Deng, Xiaomei Yang, Ailin Qian

类别: math.NA

发布日期: 2026-06-20

arXiv ID: 2606.21815v1

链接: http://arxiv.org/abs/2606.21815v1

摘要: We develop a low-frequency moment–Hankel rank method for identifying the number of time-independent point sources in the heat equation from boundary flux data. The method is formulated in the unit disk, where the Laplace-transformed and normalized boundary flux admits an explicit Fourier moment representation. By taking the low-frequency limit, we obtain a finite exponential-sum moment sequence in which the nodes encode the source locations, the weights encode the source strengths, and the number of terms equals the number of point sources. The associated Hankel matrix admits a Vandermonde factorization, and its rank is exactly the source number under the natural assumptions that the source locations are distinct and the source strengths are nonzero. We also analyze the effect of discrete and noisy boundary data. A uniform moment perturbation bound is propagated to the empirical Hankel matrix, and Weyl’s singular-value perturbation inequality yields a sufficient condition for stable numerical rank recovery in terms of the smallest nonzero singular value of the ideal Hankel matrix. Numerical experiments confirm the exact rank pattern in the noiseless case, validate the stability threshold under moment noise, and illustrate the loss of resolution for close or weak sources. After the source number is identified, the same moment sequence can be used for location and strength recovery through an annihilating-polynomial and Vandermonde reconstruction procedure.

29. 分层Tucker格式下线性与非线性系统的多层自适应秩方法

Multilevel Adaptive-Rank Methods for Linear and Nonlinear Systems in the Hierarchical Tucker Format

作者: William A. Sands, Pierson T. Guthrey, Nathan V. Roberts

类别: math.NA, physics.comp-ph

发布日期: 2026-06-19

arXiv ID: 2606.21750v1

链接: http://arxiv.org/abs/2606.21750v1

摘要: We develop multilevel adaptive-rank iterative methods for the solution of linear and nonlinear systems arising from high-dimensional partial differential equations. Our contributions are threefold. First, we extend the projection method of Ballani and Grasedyck [6] to enable flexible preconditioning of high-dimensional linear systems in low-rank tensor formats. Second, we construct multilevel preconditioning strategies by adapting geometric multigrid methods to the low-rank setting. In contrast to prior work, which primarily employs multigrid as a standalone solver, we emphasize its role as an efficient and robust preconditioner. Third, we integrate these techniques within an inexact Newton framework for the solution of nonlinear systems. The proposed methods are evaluated on a range of model problems, including both linear and nonlinear equations, to assess their convergence behavior and computational efficiency. The results demonstrate that multilevel adaptive-rank strategies yield robust and scalable preconditioners, providing effective solvers for high-dimensional problems in low-rank formats.

30. 用于计算临界悬链面莫尔斯指数的物理信息神经网络

Physics-Informed Neural Networks for Computing the Morse Index of the Critical Catenoid

作者: Miraj Samarakkody

类别: math.DG, cs.LG, math.NA, math.SP

发布日期: 2026-06-19

arXiv ID: 2606.21725v1

链接: http://arxiv.org/abs/2606.21725v1

摘要: The Morse index of a free boundary minimal surface is encoded in its Jacobi-Steklov spectrum, and we test how faithfully a physics-informed neural network (PINN) reproduces that spectrum on a problem whose answer is already known in closed form. The benchmark is the critical catenoid in the unit ball $\mathbb{B}^3$, where it is well known that the Morse index equals $4$ and the nullity equals $2$. Separating the angular variable reduces the eigenvalue problem to a family of one-dimensional Robin problems on $[-T,T]$, one for each Fourier mode. A network that enforces the parity of each mode by construction, and carries the eigenvalue as a trainable parameter, returns the three eigenvalues below the stability threshold to within $10^{-6}$ to $10^{-4}$ of their exact values, with PDE residuals of order $10^{-4}$; assembling them recovers the index $4$ and the nullity $2$. We then track the spectrum along a one-parameter homotopy joining a flat reference operator to the catenoid Jacobi operator and identify the crossings at which the index changes. Since the critical catenoid is rigid, a fact we prove, this homotopy deforms operators rather than surfaces. We close by explaining how the same pipeline, with its one-dimensional solver replaced by a two-dimensional one, is poised to address genuinely geometric families in ellipsoidal balls, where the boundary curvature is no longer constant, and the Morse index is not yet known.

31. GPU上亥姆霍兹方程的高效区域分解

Efficient Domain Decomposition for the Helmholtz Equation on GPUs

作者: Amit Rotem

类别: cs.DC, math.NA

发布日期: 2026-06-19

arXiv ID: 2606.21695v1

链接: http://arxiv.org/abs/2606.21695v1

摘要: The Helmholtz equation governs wave propagation in acoustics, electromagnetics, and seismology, but its indefinite nature makes it difficult to solve with iterative methods. Domain decomposition methods are a natural fit for massively parallel architectures, yet mapping efficient Helmholtz solvers onto modern GPUs remains a challenge. We address both with two key contributions: (1) a block-level domain decomposition scheme, in which each subdomain is assigned to a single thread block and all solves run concurrently in a single kernel launch, and (2) WaveHoltz as the subdomain solver. WaveHoltz is a fixed-point iteration that is uniquely well-suited to the GPU execution model due to its minimal memory footprint and no reduction operations. Together, these eliminate device-level synchronizations and replace global memory traffic with shared memory and register-level operations, keeping subdomain data largely resident in L1 and L2 cache. We explore two threading strategies: one degree of freedom per thread for small subdomains, and multiple degrees of freedom per thread for larger ones. Benchmarks of our CUDA based implementation on a NVIDIA A100 show that WaveHoltz achieves 2x-25x speedup over MINRES, with the advantage growing with subdomain size. Crucially, evaluating the subdomain solver in single rather than double precision yields an additional 2x-10x speedup–a benefit largely unattainable by MINRES due to loss of Krylov vector orthogonality under reduced precision.