生成时间: 2026年10月12日 19:43:13
重点关注论文数量: 1
了解领域论文数量: 9
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重点关注论文（完整分析）
1. 有限薄涂层引起的波衍射：渐近展开与有效模型
Diffraction of waves by finite thin coatings: asymptotic expansion and effective models
作者: Cédric Baudet
类别: math.AP, math.NA
发布日期: 2026-10-09
arXiv ID: 2610.13680v1
摘要翻译: 我们考虑位于 Dirichlet 角扇形区域（可能局部扰动）中的 Helmholtz 方程，该区域的一侧被一个薄均匀层覆盖。首先，我们利用匹配渐近展开，在薄层厚度趋于 0 时，对任意角度给出解的任意阶渐近展开。这包括在三个子区域——角点附近、薄层以及区域其余部分——中引入不同的渐近展开，并通过匹配条件将它们联系起来。角点的存在使得这些匹配条件难以推导，因为场具有复杂的奇异行为。我们借助一套新颖的代数工具克服这一困难，其中包括形式级数以及作用在这些级数上的算子。此外，我们通过误差估计证明该渐近展开的合理性，并详细说明展开形式如何依赖于角度。
Comment: PhD thesis, 288 pages, 55 figures
链接: http://arxiv.org/abs/2610.13680v1
1. 研究对象和背景
本文研究二维 Helmholtz 方程在角形区域一侧覆盖有限薄均匀涂层时的波衍射问题。设角 $\Theta\in(0,2\pi)$，引入角形域、层和界面：
$$\Omega=\{(r\cos\theta,r\sin\theta): r\in\mathbb{R}_+^*,\ \theta\in(0,\Theta)\},\quad \Lambda=\mathbb{R}_+^*\times(-1,0),\quad \Gamma=\mathbb{R}_+^*\times\{0\},\quad \Sigma_\Omega=\{(r\cos\Theta,r\sin\Theta): r\in\mathbb{R}_+^*\}.$$进一步设 $\Omega_1\subset\mathbb{R}^2$ 为开集，在圆盘 $B(0,R_c)$ 外与 $\Omega\cup\Gamma\cup\Lambda$ 重合，从而允许角点附近在层厚度尺度上的局部几何扰动。系数 $\mu,\rho\in L^\infty(\Omega_1)$ 满足一致椭圆性，且在 $\Omega\setminus B(0,R_c)$ 中等于常数 $\mu_0,\rho_0$，在 $\Lambda\setminus B(0,R_c)$ 中等于 $\mu_1,\rho_1$。薄层厚度记为 $\varepsilon>0$，物理域为
$$\Omega_\varepsilon=\{(x,y)\in\mathbb{R}^2: (x/\varepsilon,y/\varepsilon)\in\Omega_1\},$$缩放系数为 $\mu_\varepsilon(x,y)=\mu(x/\varepsilon,y/\varepsilon)$，$\rho_\varepsilon(x,y)=\rho(x/\varepsilon,y/\varepsilon)$。设频率参数 $\omega\in\mathbb{C}\setminus\mathbb{R}$，源项 $f_s\in H^{-1}(\Omega)$ 且 $\mathrm{dist}(\mathrm{supp}(f_s),\Gamma)>0$。精确解 $u_\varepsilon\in H^1_0(\Omega_\varepsilon)$ 满足 Dirichlet 边值问题
$$\mathrm{div}(\mu_\varepsilon\nabla u_\varepsilon)+\omega^2\rho_\varepsilon u_\varepsilon=f_s\quad \text{in }\Omega_\varepsilon,\qquad u_\varepsilon=0\quad \text{on }\partial\Omega_\varepsilon.$$由于 $\Im(\omega)\neq0$，Lax-Milgram 给出适定性与关于 $\varepsilon$ 一致的稳定性估计
$$\exists C>0,\ \forall f_s\in H^{-1}(\Omega),\ \forall\varepsilon>0,\quad \|u_\varepsilon\|_{H^1(\Omega_\varepsilon)}\le C\|f_s\|_{H^{-1}(\Omega)}.$$该设定对应半无限直层与角点的学术模型，重点关注薄层边缘或角点附近。动机来自薄涂层在电磁学、弹性、流体等中的广泛应用：直接数值求解需要解析极小厚度，代价高；因此需要渐近展开或有效边界条件。已有文献对完全覆盖的无限平面层和光滑曲层以及异质周期层已有大量结果，但对部分覆盖或有限涂层，尤其带任意角点的情况，研究仍处早期。本文试图填补的空白是：在保留角点奇异性和可能局部扰动的几何下，建立任意阶、任意角的匹配渐近展开，证明误差估计，并构造高阶有效模型。从提供内容无法确认该工作是否处理有界障碍的完整辐射条件；文中说明 $\Im(\omega)\neq0$ 是为半无限层学术设定服务的技术假设，并指出有界障碍可用经典 Sommerfeld 条件。
2. 主要定理或主要结果
从摘要、引言与目录可见，论文的主要结果可分为以下几个层次。第一，作者对精确解 $u_\varepsilon$ 构造了当 $\varepsilon\to0$ 时任意阶、任意角 $\Theta$ 的匹配渐近展开。展开不是在单一尺度上给出，而是在三个子域中分别引入不同渐近展开：角点邻域、薄层内部以及其余区域；这些展开通过匹配条件连接。由于角点处场具有复杂奇异行为，匹配条件的推导是核心困难。作者用新的代数工具系统处理这些匹配条件，包括形式级数以及作用在形式级数上的算子。摘要明确说该方法覆盖任意阶和任意角，并给出与角度有关的展开形式细节。第二，作者用误差估计严格证明该渐近展开，即构造出的近似解与原解之差在适当范数下由高阶项控制。具体误差阶、范数和常数从提供内容中无法完整确认，但目录第 5 章标题为“Construction of the asymptotic expansion and error estimates”，其中 5.4 节分为匹配误差估计以及全局与远场误差估计。第三，论文给出展开的进一步性质。目录第 6 章涉及用 Bessel 函数表达远场奇异性，并讨论依角度的简化，区分有理情形与无理情形；第 7 章给出远场的模态分解，目录中可见定理 7.1.1、定理 7.1.3 与推论 7.1.4。模态分解为后续 DtN 有效模型提供基础。第四，数值验证达到 5 阶。为了计算展开中的角剖面在无穷远处的行为，作者引入一种 Galerkin 型方法，并用“无穷远处的补充项” enrich。这些补充项是已知函数，出现在角剖面无穷远展开中，其系数成为离散系统的未知量。目录第 8 章还给出远场的数值计算。第五，论文提出 3 阶和 5 阶有效模型。远离角点时，薄层被替换为有效 Robin 边界条件（3 阶）或 Ventcel 边界条件（5 阶）；角点附近引入人工边界并施加 Dirichlet-to-Neumann 型条件。该 DtN 映射基于截断到 3 阶或 5 阶的渐近展开的模态分解；模态包含展开项的奇异行为、角剖面以及一些代数匹配系数。作者证明这些有效模型的适定性与误差估计，并给出数值结果。目录第 9.5 节标题为“Final error estimates of order 3 and 5”，因此可推断相应误差阶为 $O(\varepsilon^3)$ 与 $O(\varepsilon^5)$，但具体范数与常数从提供内容中无法确认。第六，关于角奇异性的具体结构，引言给出：在角 $\Theta$ 中，无层或主导远场可写为
$$u_0(r,\theta)=c r^{\pi/\Theta}\sin\left(\frac{\pi}{\Theta}\theta\right)+u_0^{\mathrm{reg}},\qquad u_0^{\mathrm{reg}}\in H^2,$$其中 $c\in\mathbb{C}$。当 $\Theta>\pi$ 且 $c\neq0$ 时，$\partial_n u_0=-\frac{\pi}{\Theta}c r^{\pi/\Theta-1}+\partial_n u_0^{\mathrm{reg}}$ 不属于 $H^{1/2}$，因此不能寻找 $u_1\in H^1$，而必须允许远场 $u_1$ 在角点发散。对于较小角 $\Theta$，$u_1$ 可以有 $H^1$ 意义，但对某些更高阶远场 $u_n$ 仍会出现同类问题。这些奇异结构正是匹配条件复杂和需要形式代数工具的原因。第七，有效边界条件的形状在引言中以无限平面层为例给出：Robin 型条件为
$$\mu_0\partial_n v_\varepsilon=-\frac{\mu_1}{\varepsilon}v_\varepsilon,$$Ventcel 型条件为
$$\mu_0\partial_n v_\varepsilon=-\frac{\mu_1}{\varepsilon}v_\varepsilon+\frac{\varepsilon}{3}(\mu_1\partial_x^2+\omega^2\rho_1)v_\varepsilon.$$本文在远离角点处采用同类条件，并在角点附近用 DtN 条件处理截断的奇异展开。具体本文模型的完整变分形式和边界条件从提供内容中无法逐项确认。
3. 研究方法、关键技术和核心工具
技术障碍首先来自角点。对于完全覆盖的无限平面层或光滑曲层，渐近展开可写成外层 $\sum_{n\ge0}\varepsilon^n u_n(s,\nu)$ 与内层 $\sum_{n\ge0}\varepsilon^n U_n(s,\nu/\varepsilon)$，并通过归纳计算。但在角形区域中，主导远场含角奇异 $r^{\pi/\Theta}\sin(\pi\theta/\Theta)$，其法向导数在 $\Theta>\pi$ 时失去 $H^{1/2}$ 正则性。于是某些远场项必须在角点发散，匹配条件不能简单在 $H^1$ 或 $H^{1/2}$ 框架中推导。作者的关键创新是建立一套代数形式级数工具，用来系统处理带角奇异性的匹配条件。目录第 3 章定义形式级数集合，并扩展到指标在 $\mathbb{R}$ 和 $\mathbb{R}^2$（涉及 $\varepsilon$）中的形式级数，引入具有“度”的算子以及算子的几何级数。这些工具使作者能够把匹配条件写成形式级数方程，并在形式层面求解、消元或逆运算。第 4 章将这套工具用于准齐次角奇异空间 $\mathcal{A}$，在该空间中求解 Poisson 与 Helmholtz 模型问题，并研究角奇异形式级数的 $r$ 渐近行为及其按 $\varepsilon$ 的缩放。随后第 4.3 节证明匹配假设与匹配条件等价，第 4.4 节总结方法，第 4.5 节数值计算场行为与匹配系数。第二个技术点是三分区匹配渐近展开的构造。作者在角点邻域、层内和其余区域分别设 ansatz，并利用匹配区条件连接。匹配区位于距层 $y\in(\varepsilon^a,2\varepsilon^a)$，$a\in(0,1)$，从远场看趋于界面，从近场看趋于无穷。因此需要精确描述近场在无穷远处的渐近行为，包括可能的多项式、对数与角奇异模态。第 5 章建立 far-field-like 与 corner-field-like 模型问题的存在唯一性，证明其 $r$ 渐近行为，然后归纳构造各阶场，并估计匹配误差、全局误差和远场误差。第三，远场模态分解和第 6、7 章的进一步性质是构造透明边界条件的基础。第 6 章用 Bessel 函数表示远场奇异性，并按角 $\Theta$ 是否为有理数讨论简化。第 7 章证明远场模态展开的收敛性，区分有理情形与一般情形，并对单个模态与形式级数给出估计。目录中定理 7.1.1、7.1.3 与推论 7.1.4 是这一部分的核心。第四，数值方法方面，为了验证 5 阶展开，需要计算角剖面在无穷远处的行为。作者提出一种 Galerkin 型方法，并用“无穷远处的补充项”丰富试探空间。这些补充项是已知函数，出现在角剖面无穷远渐近展开中，其系数作为额外未知量与离散解一起求解。这样可以把无穷远条件以稳定方式纳入有限计算。第 8 章还说明如何数值计算远场。第五，有效模型的技术核心是 DtN 映射。第 9 章先构造精确解的近似模态分解，再重新表达远场的角点条件，得到最终分解。然后在角点附近引入人工边界，定义 DtN 算子；DtN 的模式由截断到 3 阶或 5 阶的渐近展开给出，包含展开项的奇异行为、角剖面和代数匹配系数。作者研究 DtN 算子当 $\varepsilon$ 小时的渐近行为，利用椭圆正则性工具证明有效模型的适定性与稳定性，最后给出 3 阶和 5 阶误差估计。真正的新意不在于泛泛使用能量估计或紧性，而在于把角点奇异、薄层微观尺度和匹配条件统一到形式代数级数框架中，并用该框架生成可证明误差的高阶有效边界条件。
4. 与之前工作的比较
与已有文献相比，本文的推进点首先是几何设定。90 年代 [BL96; EN93] 对无限平面层和光滑曲层建立了形如
$$\sum_{n\ge0}\varepsilon^n u_n(s,\nu)\quad(\nu>0),\qquad \sum_{n\ge0}\varepsilon^n U_n(s,\nu/\varepsilon)\quad(-\varepsilon<\nu<0)$$的渐近展开，并导出 Robin 型或 Ventcel 型有效条件；例如在无限平面 Dirichlet 层情形，Robin 条件
$$\mu_0\partial_n v_\varepsilon=-\frac{\mu_1}{\varepsilon}v_\varepsilon$$对应 $O(\varepsilon^3)$ 误差，而 Ventcel 条件
$$\mu_0\partial_n v_\varepsilon=-\frac{\mu_1}{\varepsilon}v_\varepsilon+\frac{\varepsilon}{3}(\mu_1\partial_x^2+\omega^2\rho_1)v_\varepsilon$$对应 $O(\varepsilon^5)$ 误差。本文在远离角点处采用同类 Robin/Ventcel 条件，但对象是有限薄涂层且存在角点和局部扰动。已有工作主要处理完全覆盖的无限或光滑曲层，均匀同质层可直接用 Fourier 变换求 DtN 并 Taylor 展开；异质和周期层在 [AH97; AH98; AL98; MV07; DHJ12; DC13; Del14; DH20; Bou23] 中研究，此时微观变化会泄漏到涂层外的薄边界层，需要多尺度或匹配渐近展开。本文属于部分覆盖或有限涂层方向，引言明确指出该方向“仍处早期”。第二个推进是角点奇异性。带角无层 Poisson 问题的多尺度渐近展开已在 [Cal+06; Via03] 中建立，形式为
$$\sum_{p\in\mathbb{N}+\frac{\pi}{\Theta}\mathbb{N}}\sum_{\ell=0}^{\infty}\varepsilon^p\ln^\ell\varepsilon\,u_{p,\ell}(x,y).$$本文把类似角奇异结构与薄层匹配结合，并允许任意角 $\Theta\in(0,2\pi)$。与 [TVD06] 中角无层的多尺度与匹配方法比较不同，本文需要同时处理薄层内微观尺度与角点准齐次奇异。第三，方法层面，已有工作分别使用多尺度渐近展开和匹配渐近展开；本文明确采用匹配渐近展开，并在角点邻域、层和远场三部分分别展开。与仅在单一尺度或简单几何可做 Fourier 展开的情形相比，本文必须引入形式级数与算子工具来推导匹配条件。这是本文最显著的技术推进。第四，有效模型层面，已有高阶模型大多在远处用 Robin/Ventcel 条件，不显式处理角点附近的透明条件。本文在 3 阶和 5 阶模型中，在远离角点处使用 Robin/Ventcel，在角点附近引入人工边界并施加 DtN 型条件。DtN 映射基于截断渐近展开的模态分解，包含角奇异、角剖面和匹配系数。这比单纯局部边界条件更能捕捉角点衍射，但代价是模型更复杂，需要预先计算角剖面、匹配系数和 DtN 模态。第五，适用范围与代价。本文设定为半无限直层并假设 $\Im(\omega)\neq0$，作者说明这是为半无限层选取合适辐射条件的技术假设，对有界障碍可用经典 Sommerfeld 条件。因此本文不是直接处理实际有界障碍的全部辐射问题，而是先解决角点与薄层相互作用的学术模型。从提供内容看，作者未显式展开与所有已有有效模型的数值代价比较；关于三维、随机层、均匀化边界层等文献主要作为相关技术背景被提及，本文是否覆盖这些情形从提供内容无法确认。总体而言，本文把有限薄涂层、任意角点和局部扰动下的渐近展开推进到任意阶，并给出带角点 DtN 的高阶有效模型，其代价是形式代数工具与角剖面计算的额外复杂性。
了解领域论文（摘要翻译）
1. 三维不精确 Schwarz 区域分解方法的分析
Analysis of an inexact Schwarz domain decomposition method in 3D
作者: Nibedita Ghosh
类别: math.NA
发布日期: 2026-10-09
arXiv ID: 2610.13793v1
链接: http://arxiv.org/abs/2610.13793v1
摘要翻译
中文标题: 三维不精确 Schwarz 区域分解方法的分析
摘要翻译: 我们考虑 $3$D 中 Schwarz 区域分解方法的一个不精确版本，用以求解一个区域上的 Laplace 问题，该区域是两个重叠球体的并集。我们通过将其边界数据 $L^2$-正交投影到次数至多为 $N$ 的球谐函数空间上，来近似求解每个子问题。该模型问题来源于计算化学中用于隐式溶剂化的 ddCOSMO 求解器。我们通过建立最大值原理的一个新变体来分析 Schwarz-球面区域分解方法。
2. 用于确定性和随机旋转浅水方程的能量守恒相容时空有限元法
Energy conserving compatible space–time FEM for deterministic and stochastic rotating shallow water
作者: Maneesh Kumar Singh, Colin J. Cotter
类别: math.NA
发布日期: 2026-10-09
arXiv ID: 2610.13624v1
链接: http://arxiv.org/abs/2610.13624v1
摘要翻译
中文标题: 用于确定性和随机旋转浅水方程的能量守恒相容时空有限元法
摘要翻译: 我们提出一种旋转浅水方程的全离散能量守恒离散化，其在空间和时间上具有任意阶。采用时空有限元离散，空间部分由相容有限元方法构建，时间部分由连续 Petrov-Galerkin 格式构建。为精确能量守恒，需要针对 Hamiltonian 导数的投影辅助变量。我们提供一种基于整体式 patch 光滑子的迭代求解器方法，使我们能够高效地将 Newton 法充分收敛，从而使由求解器引起的能量误差非常小。我们将该构造推广到能量守恒的 Lie 输运随机强迫（Stochastic Forcing by Lie Transport）格式。随后，我们在数值测试中展示了收敛性、守恒性和鲁棒性。
3. 关于球谐函数离散化中的一个最大值原理
On a maximum principle in discretization with spherical harmonics
作者: Arnold Reusken
类别: math.NA
发布日期: 2026-10-09
arXiv ID: 2610.13540v1
链接: http://arxiv.org/abs/2610.13540v1
摘要翻译
中文标题: 关于球谐函数离散化中的一个最大值原理
摘要翻译: 球谐函数已被广泛研究，并应用于许多领域。我们考虑球谐函数很可能是最早的应用，即用球谐函数基表示边界数据，来表示 $d$ 维单位球面上 Laplace 方程的解。我们考虑一般的 $d \geq 2$。对于 Laplace 方程的解 $u$，基本最大值原理成立。如果在由次数至多为 $N$ 的所有球谐函数张成的有限维空间中逼近边界数据，则所得的离散解，记作 $u_N$，一般不满足最大值原理。在本文中，我们推导出 $r_N^{\rm max}$ 的尖锐估计，使得对于所有满足 $\|x\| \leq r_N^{\rm max}$ 的 $x \in \Bbb{R}^d$，最大值原理对 $u_N(x)$ 成立。我们证明，当 $d=3$ 时，有 $r_N^{\rm max} \gtrsim 1- 3 N^{-1} \ln\big( 2^\frac13 N \big) $。对于 $d \neq 3$，也推导出了类似的估计。我们还研究了这些估计的尖锐性。
4. 关于带有键断裂准则的非线性键基近场动力学模型的无网格离散的收敛性
On the Convergence of the Mesh-Free Discretization of a Non-Linear Bond-Based Peridynamic Model with a Bond-Breakage Criterion
作者: Lukas Pflug, Michael Stingl, Max Zetzmann
类别: math.NA
发布日期: 2026-10-09
arXiv ID: 2610.13355v1
链接: http://arxiv.org/abs/2610.13355v1
摘要翻译
中文标题: 关于带有键断裂准则的非线性键基近场动力学模型的无网格离散的收敛性
摘要翻译: 我们证明了针对固定视界 $δ>0$ 的、通过键断裂准则实现不可逆损伤的动态非线性键基近场动力学模型的一种常用无网格离散的收敛性。这里选取的模型是文献中常见的不连续近场动力学算子的一个数学上适定的正则化，该算子假设断裂仅由拉伸应变引起。我们在分析中显式包含了该格式所采用的一点求积。收敛性是相对于由 $C^1([0,T],L^\infty(Ω)^d)$ 给出的解空间的自然范数而言的，因此在空间坐标中为 $L^\infty(Ω)^d$。该结果极大地约束了连续解的空间正则性，我们随后证明这一点。更具体地说，对于局部 $L$-Lipschitz 连续的输入数据，无论损伤如何，都推导出位移的局部 Lipschitz 界。这与将此处使用的近场动力学模型解释为对具有由视界 $δ>0$ 给出的长度尺度参数的一个真正不连续断裂模型的近似是一致的。我们注意到，我们不处理当 $δ\searrow 0$ 时解的极限。
5. ADER-ZIST——一种用于笛卡尔切割单元网格的混合显式 ADER-DG 与分区隐式时空 DG 方法
ADER-ZIST - a mixed explicit ADER-DG and zonally implicit space-time DG approach for Cartesian cut cell meshes
作者: Michael Dumbser, Christian Engwer, Sandra May, Peter Munch
类别: math.NA
发布日期: 2026-10-09
arXiv ID: 2610.13245v1
链接: http://arxiv.org/abs/2610.13245v1
摘要翻译
中文标题: ADER-ZIST——一种用于笛卡尔切割单元网格的混合显式 ADER-DG 与分区隐式时空 DG 方法
摘要翻译: 我们提出用于求解笛卡尔切割单元网格上双曲守恒律系统的高阶精确混合显隐式间断 Galerkin (DG) 格式。这类网格可能包含任意小的切割单元，这导致众所周知的小单元问题：显式格式的 CFL 条件会严重限制允许的时间步长。为克服这一困难，我们仅在远离嵌入边界的笛卡尔单元上使用高阶显式全离散单步 ADER-DG 格式。靠近嵌入边界的区域（由切割单元及其直接面相邻单元组成）采用隐式时空 DG 格式离散，该格式通过数值通量与显式格式直接耦合。所得格式称为 ADER-ZIST（具有分区隐式时空 DG 的 ADER-DG），其构造是守恒的，并且其时间步长仅由笛卡尔背景单元的大小决定。弯曲嵌入边界由切割单元内部的高阶等参子三角形表示，隐式单元的非线性系统通过非精确无矩阵 Newton-Krylov 方法求解。对于含激波的问题，我们添加人工黏性项，其强度由消除振荡的光滑性传感器控制。ADER-ZIST 在可压缩 Euler 方程的一组综合测试问题上进行了评估，范围从直线和弯曲嵌入边界周围的平滑流动到声散射和激波主导的流动。对于光滑解以及高达 N=5 的多项式次数，在最小切割单元体积分数低至 10^-7 量级的网格上，该格式基本达到其标称精度阶 N+1。
6. 非等温液-汽流动的动态 van der Waals 模型的线性熵稳定格式
Linear and entropy-stable scheme for the dynamic van der Waals model of non-isothermal liquid-vapor flows
作者: Yunzhuo Guo, Zhonghua Qiao, Xianmin Xu
类别: math.NA
发布日期: 2026-10-09
arXiv ID: 2610.13029v1
链接: http://arxiv.org/abs/2610.13029v1
摘要翻译
中文标题: 非等温液-汽流动的动态 van der Waals 模型的线性熵稳定格式
摘要翻译: 本文提出了一种线性且解耦的全离散格式，该格式保持了动态 van der Waals 模型的熵产性质。该模型为描述单组分液-汽系统中的非等温相变提供了一个热力学一致的扩散界面框架，包含质量、动量和能量的耦合守恒律。我们首先引入控制方程的一种等价重述，以避免使用链式法则恒等式，因为这类恒等式在离散微分下通常不成立。进一步，采用逐分量凸分裂方法来计算离散化学势和温度。这种处理给出了一个无时间步长限制的离散熵稳定不等式。基于显式处理的速度，该格式允许三个守恒方程以解耦方式线性且顺序求解。由于动量方程不直接影响熵稳定性分析，我们采用经典线性半隐格式求解它。数值实验验证了预期的数值精度。液滴蒸发以及二维和三维 Leidenfrost 效应的模拟进一步表明，所提方法能够捕捉复杂的非等温相变动力学，同时在数值上重现离散熵产行为。
7. 演化方程的数值光滑性与误差分析
Numerical Smoothness and Error Analysis for Evolution Equations
作者: Tong Sun
类别: math.NA
发布日期: 2026-10-09
arXiv ID: 2610.12888v1
链接: http://arxiv.org/abs/2610.12888v1
摘要翻译
中文标题: 演化方程的数值光滑性与误差分析
摘要翻译: 这是一份讲义，旨在介绍数值光滑性这一创新概念及其在演化方程数值解误差分析中的应用。我们从两种根本不同的误差分裂出发。一种分裂很常用，它把我们引向沿原解定义的局部误差以及由数值格式导致的误差传播。众所周知的 Lax 等价定理正是基于这种误差分裂。另一种分裂只能在少数数值 ODE 教材中见到，它把我们引向沿数值解定义的局部误差以及由演化方程的解导致的误差传播。数值光滑性这一创新概念源自第二种误差分裂。尽管在非线性演化方程的误差分析中使用数值稳定性存在许多困难，但我们将表明，数值光滑性在误差分析/估计/控制中自然地发挥作用，并在若干方面具有优势。我们将定义时间与空间数值光滑性指标，证明这些指标的有界性对于获得最优收敛阶是必要的，并给出作为这些光滑性指标函数的可计算误差估计。此外，这些误差估计在近似解的收敛阶方面是最优的，在演化方程的误差传播方面也是最优的。与 Lax 等价定理的误差分析框架相比，基于数值光滑性框架中证明的必要性结果和误差估计，我们得到一个新的等价理论：Consistency + Numerical Smoothness <=> optimal error estimates。与 Lax 等价定理不同，新的等价理论适用于非线性问题、非线性格式以及问题的许多其他复杂情形。
8. 聚集-扩散方程的变分神经输运
Variational neural transport for aggregation-diffusion equations
作者: Qianyi Li, Hailiang Liu
类别: math.NA, math.OC
发布日期: 2026-10-08
arXiv ID: 2610.12745v1
链接: http://arxiv.org/abs/2610.12745v1
摘要翻译
中文标题: 聚集-扩散方程的变分神经输运
摘要翻译: 我们提出了一个神经输运框架，用于处理来源于趋化性、聚合物动力学及相关应用中的非线性、非局部聚集-扩散方程。基于 Onsager 变分原理，我们将控制方程重新表述为一个约束最优控制问题，以速度场作为控制，并通过相应的连续性方程演化密度。速度场由神经网络参数化，该神经网络在沿粒子轨迹计算的变分目标上进行训练。通过将密度方程和广义得分方程耦合为一个用 Euler 方法求解的单一系统，得分函数能够高效传播以用于损失评估。此外，我们建立了一个严格的误差分析，将总误差分解为逼近、优化、Monte Carlo 采样和时间离散化分量，并证明了相应的误差界。在若干代表性模型上的数值实验证明了所提方法的精度和有效性。
9. 基于 Maxwell 方程解析解的任意阶电磁场逼近框架
A Framework for Arbitrary-Order Electromagnetic Field Approximation Based on Analytical Solutions of Maxwell’s Equations
作者: David Wei Ge
类别: math.AP, math.NA
发布日期: 2026-10-08
arXiv ID: 2610.12682v1
链接: http://arxiv.org/abs/2610.12682v1
摘要翻译
中文标题: 基于 Maxwell 方程解析解的任意阶电磁场逼近框架
摘要翻译: 标准的 Yee 有限差分时域 (FDTD) 算法利用 Maxwell 方程将一阶时间导数替换为空间旋度。本文将这一关系推广到任意阶：一个时间-空间定理建立了高阶时间导数与逐次高阶空间旋度算子之间的对应关系。基于这一关系，发展了用于电磁场逼近的任意阶框架。该框架通过更高阶的旋度构造更高阶的时间阶，并允许不同的空间离散和旋度估计技术，从而为发展任意阶数值算法提供了通用基础。作为一种实现，利用交错时间与空间步进以及递归旋度估计构造了任意阶 FDTD 算法。在空间和时间步长保持固定时，对一个初值问题进行了时间逼近阶从二阶到十阶的数值模拟。与解析解的比较表明，随着逼近阶的提高，误差迅速且单调地减小。这些结果展示了该框架在高精度电磁计算中的实际潜力，并为进一步发展任意阶数值方法提供了基础。
不相关论文（基本信息）
1. 快速且稳定的任意精度近似矩阵乘法
Fast and Stable Arbitrary Precision Approximate Matrix Multiplication
作者: Ioana Dumitriu, Olga Holtz, Abraham Hsu, Nicholas Liskij, Yoav Moran, Ryan Schneider, Oded Schwartz, Shuncheng Yuan
类别: math.NA
发布日期: 2026-10-09
arXiv ID: 2610.13905v1
链接: http://arxiv.org/abs/2610.13905v1
摘要: We present several new and surprising results on the theory and practice of Arbitrary Precision Approximate (APA) matrix multiplication, which refers to algorithms that output merely approximations to matrix multiplication even in exact arithmetic. While APA algorithms have been around for nearly half a century, and despite the fact that they have potentially lower (asymptotic) cost than many fast exact matrix multiplication schemes, they have not been studied as closely, to the extent that relatively little is known about their numerical properties. In this paper, we shine a spotlight on these algorithms by presenting three important results. First, we provide a new search algorithm for APA routines that is easy to use, automatically generates encoding matrices with functional entries, and can match or beat the best known APA algorithms for several sizes. In particular, we improve the upper bound on the border rank of $\langle 2,4,5 \rangle$ matrix multiplication. Second, we combine this insight with alternative bases to minimize the number of additions in an APA algorithm, thereby improving its leading coefficient. Finally, we prove the same kind of stability that characterizes fast exact matrix multiplication for an algorithm of Bini [Calcolo 1980] that computes an exact product of two matrices using a modified APA scheme (with or without alternative bases).
2. 约束概率数值方法：线性不等式约束下线性偏微分方程的高斯过程求解器
Constrained Probabilistic Numerics: Gaussian Process Solvers for Linear PDEs under Linear Inequality Constraints
作者: Boyuan Deng, Kshitiz Upadhyay, Michael Shields
类别: stat.ML, cs.LG, math.NA
发布日期: 2026-10-09
arXiv ID: 2610.13804v1
链接: http://arxiv.org/abs/2610.13804v1
摘要: Gaussian process solvers for linear partial differential equations (PDEs) return posteriors rather than point estimates, but they condition only on equality information: the PDE residual, boundary data, and measurements. Many solutions of interest also obey inequality constraints, including nonnegativity, monotonicity, and absence of arbitrage, which equality conditioning cannot enforce. We prove that conditioning a Gaussian process prior jointly on linear equality observations and a finite-dimensional inequality event is equivalent to conditioning sequentially in terms of Borel measures on the solution space. The physics-informed posterior is therefore a valid prior for the inequality update at virtual points. The combined law is non-Gaussian yet computable through truncated-Gaussian moments, and neither the PDE conditioning nor the inequality update requires modification. We evaluated the construction on a Poisson benchmark, option-price surfaces, and air-quality fields. Under incomplete boundary data, a monotonicity constraint lowered the error of a two-dimensional Poisson reconstruction by $63\%$. On a real S&P 500 option chain thinned to $20$ retained quotes, no-arbitrage constraints reduced held-out extrapolation error by $22\%$ and cut the calendar-arbitrage rate from $10\%$ to $1\%$. On four real air-quality snapshots, nonnegativity reduced a Gaussian moment-based sub-zero-tail proxy by $74\%$ at the sparsest monitor count. Changes in mean accuracy and constraint diagnostics differed across data regimes. The probability of the noisy virtual constraint event under the equality posterior measures agreement with the specified bounds before the update. The construction turns admissibility from a post-processing step into conditioning information for probabilistic PDE solvers.
3. 边界上具有变化 Robin 参数的波动方程的一个逆谱问题
An inverse spectral problem for the wave equation with a varying Robin parameter at the boundary
作者: Antti Hannukainen, Petr Kulikov, Matti Lassas, Lauri Oksanen
类别: math.AP, math.NA, math.OC, math.SP
发布日期: 2026-10-09
arXiv ID: 2610.13629v1
链接: http://arxiv.org/abs/2610.13629v1
摘要: An inverse spectral problem for a $(1+1)$-dimensional wave equation with wave speed $c(x)$ is considered. The spectral data consist of the eigenvalues of the Neumann–Dirichlet problem for the operator $-c^2(x)\partial_x^2$ together with their derivatives with respect to a Robin parameter at the boundary. We propose a two-step recovery procedure. In the first step, the Neumann-to-Dirichlet map $Λ$ is reconstructed from the spectral data using the results of arXiv/0406582. In the second step, the reconstructed $Λ$ is used within the regularization strategy for the boundary control method presented in arXiv:1509.04478. As an improvement, the regularization of the control problems employs a weighted combination of $L^2$ and $H^1$ penalty terms, and we prove a uniform convergence result for the family of optimal controls, thereby allowing the regularization parameters to be tuned simultaneously. We also prove that the forward spectral map, which assigns to a wave speed its spectral data, is Lipschitz continuous, and that the inverse spectral map underlying the reconstruction procedure is Hölder continuous on a subclass of wave speeds. The method is implemented numerically and validated on both smooth and piecewise smooth wave speed profiles. Numerical experiments demonstrate that the recovery of the wave speed exhibits high accuracy and robustness.
4. 预处理近端 Galerkin：变分不等式的网格与阶数稳健求解器
Preconditioning proximal Galerkin: mesh and degree robust solvers for variational inequalities
作者: Ioannis P. A. Papadopoulos, Pablo D. Brubeck
类别: math.NA
发布日期: 2026-10-09
arXiv ID: 2610.13591v1
链接: http://arxiv.org/abs/2610.13591v1
摘要: The proximal Galerkin algorithm is a structure-preserving solver for variational inequalities that results in discretization-independent nonlinear iteration counts. However, in large three-dimensional simulations, the arising Newton saddle point systems cannot be feasibly solved by direct methods. To leverage an iterative linear solver, we propose a robust and scalable block preconditioner. Instead of using the usual dual Schur complement that acts on the latent variable, we design a preconditioner for the primal Schur complement. This leads to a block preconditioner that requires two approximate inverses of the latent block and only one of the primal Schur complement. The latent block represents an algebraic equation which, after a small perturbation, can be cheaply inverted. We then introduce two choices of approximations for the primal Schur complement: a PDE-operator-based and an algebraic approach. Either approximation can be efficiently inverted by geometric multigrid. We investigate the effectiveness of these strategies when proximal Galerkin is applied to obstacle, Signorini, and strain-constrained elasticity problems and also compare with a monolithic multigrid approach. Choices of standard relaxation methods, in the block preconditioner route, lead to robust iteration counts with respect to the mesh size and discretization degree. Moreover, in the context of the Signorini, specialized relaxation and prolongation also result in effective solvers for nearly-incompressible elastic materials. The linear solves are parameter-robust in 2D, with only mild growth in 3D. This allows for the solution of a three-dimensional contact problem, with a Poisson’s ratio of 0.499, discretized with over ten million degrees of freedom.
5. 复合线性反问题的正则化与参数选择规则
Regularisation and Parameter Choice Rules for Composite Linear Inverse Problems
作者: Christina Brandt, Patrick Horn
类别: math.NA
发布日期: 2026-10-09
arXiv ID: 2610.13483v1
链接: http://arxiv.org/abs/2610.13483v1
摘要: In this paper we propose a novel regularisation strategy for structured linear inverse problems of the form $Ax=F \circ G x = z$. Instead of using the operator $A$ or a complex scheme involving both $F$ and $G$ simultaneously, combining already existing and simpler regularisations for $F$ and $G$ allows the use of established and optimised programs for high-speed computations. As classical parameter choice rules fail in the scenario when no source conditions are known because intermediate error bounds may be unavailable, we provide both a-priori and a-posteriori rules with a convergence speed analysis given source conditions. We test the proposed method on the pre-clinical medical imaging technique of magnetic particle imaging with different choices of regularisation and on a convolution problem for an analysis of the convergence speed.
6. 用于抑制尾隆不稳定性的束流剖面全局优化
Global Optimization of Beam Profiles for Suppressing the Bump-On-Tail Instability
作者: Sandro Mignelli, Lukas Einkemmer
类别: math.NA, physics.acc-ph, physics.comp-ph
发布日期: 2026-10-09
arXiv ID: 2610.13391v1
链接: http://arxiv.org/abs/2610.13391v1
摘要: A particle beam with constant intensity that propagates into a plasma triggers a bump-on-tail instability. It has been demonstrated that beam modulation can be used to significantly reduce this instability. However, finding the optimal beam profiles that accomplish this is a complex optimization problem that suffers from an extremely rugged optimization landscape and a relatively high-dimensional parameter space. In this paper, we evaluate a number of optimization algorithms and find that population based methods (in particular, CMA-ES and Differential Evolution) perform best. In particular, we investigate beam profiles where the beam is either completely turned ON or OFF. This configuration is easier to accomplish experimentally, but leads to a higher-dimensional and thus more challenging optimization problem. We obtain ON/OFF beam profiles that reduce the instability by more than 80% and thus perform significantly better than what has been reported in the literature.
7. 面向智能建筑的可持续云-雾-边缘系统：内存约束下的算法自适应与动态编排策略
Sustainable Cloud-Fog-Edge Systems for Smart Buildings: Algorithmic Adaptation under Memory Constraints and Dynamic Orchestration Strategies
作者: Mamadou Sow, Christophe Cerin
类别: cs.LG, cs.DC, math.NA
发布日期: 2026-10-09
arXiv ID: 2610.13307v1
链接: http://arxiv.org/abs/2610.13307v1
摘要: The escalating environmental footprint of the Information and Communication Technol- ogy (ICT) sectorwith global Data Center electric- ity usage potentially reaching 3% by 2030 and the imperative for real-time responsiveness in smart buildings demand a foundational shift towards Edge Computing. This article proposes a holis- tic architectural and algorithmic approach for cre- ating autonomous, resilient, and privacy-preserving learning systems within smart buildings, advancing the core concept of the building itself as the data center. Our contributions, demonstrating originality compared to conventional centralized systems, are structured around three main pillars: 1) Validation of resilient data acquisition chains using the af- fordable Arduino ecosystem (Uno R4 WiFi/BME280) and secure persistence solutions including centralized archiving (Node-RED/MySQL) and secure remote transfer (Python/rsync/SSH). 2). The proposal of a lightweight, KOptim orchestration framework, inter- operable with the FIWARE standard, for the dynamic management of Service Level Agreements (SLA). 3) The adaptation and experimental valida- tion of the Generalized Hebbian Algorithm (GHA) for Principal Component Analysis (PCA) on low- power microcontrollers,We introduce and compare an iterative learning strategy using data buffers, featuring sorting and explicit preservation of extreme samples to maintain diversity. Experimental results on the high-dimensional Tour Perret sensor data confirm that this novel method, despite introducing a necessary temporal complexity of O(W log W ) due to sorting, achieves convergence and final result quality that is visually equivalent to processing the entire dataset in full memory, reliably identifying two distinct operational states.
8. 半可分矩阵通过其生成元的块编码
Block encoding of semiseparable matrices via their generators
作者: Giacomo Antonioli, Gianna M. Del Corso, Roel Van Beeumen
类别: quant-ph, math.NA
发布日期: 2026-10-09
arXiv ID: 2610.13228v1
链接: http://arxiv.org/abs/2610.13228v1
摘要: We present an explicit quantum circuit that constructs a block encoding of a generator-representable semiseparable (one-pair) matrix directly from its two generators. For an $N\times N$ semiseparable matrix with $N=2^{n}$, the circuit uses $2n+2$ ancilla qubits and has $O({\rm polylog} N)$ depth, and it realizes a $(2\sqrt{N},\,2n+2,\,2 \sqrt{2}\ \varepsilon)$-block encoding, where $\varepsilon$ is controlled by the accuracy of the state-preparation routines that load the generators. The building blocks of the circuit are cyclic shifts, swaps, and Hadamard gates, each admitting an efficient implementation through the binary decomposition of the relevant indices.
9. 扩展域谱配点法的数值稳定性：浮点正则化与框架常数的平衡律
Numerical Stability of Extended-Domain Spectral Collocation: Floating-Point Regularization and an Equilibrium Law for the Frame Constants
作者: Po-Yi Wu
类别: math.NA
发布日期: 2026-10-09
arXiv ID: 2610.13222v1
链接: http://arxiv.org/abs/2610.13222v1
摘要: Extended-domain spectral collocation expands the solution of a boundary value problem in the Laplacian eigenbasis of an enclosing box and collocates at as many nodes as modes. The square system is exponentially ill-conditioned, and the impossibility theorem for equispaced data rules out stable geometric convergence; yet such solvers converge geometrically to a few orders of magnitude above the unit roundoff. We explain why in one dimension. The collocation solution is the solution operator applied to the interpolant of the data in an operator-dependent test space whose Lebesgue constant must grow exponentially for geometric convergence, so exact solutions can amplify data errors exponentially. In floating-point arithmetic the error of any coefficient vector is bounded by its nodal residual, its distance to a competitor and two frame constants, a leakage and a restricted Marcinkiewicz–Zygmund constant, bounded uniformly in the number of modes at a fixed positive threshold; for a backward-stable solve, under a condition on the perturbed matrix, the bound is uniform in the conditioning for suitably analytic data and, with computed constants, within a factor of about ten of the error floor of a direct solver. For the Poisson operator the rate of the two constants along the tail of the spectrum is given by a constrained equilibrium problem of logarithmic potential theory. For an interval of length $L$ extended by $δ$ at each end, the problems for $δ$ and $L^2/(4δ)$ are dual, the rate is linear for $δ= L/2$, and it starts with slope minus the fraction of the extended domain occupied by the extension, so for $δ< L/2$ the leakage constant obeys no power law with the exponent of the end of the spectrum. In experiments, convection–diffusion operators behave like the Poisson operator at moderate Péclet numbers.
10. Gross–Pitaevskii 基态问题的 Sobolev 梯度流的黎曼 Hessian 分析
Riemannian Hessian analysis of Sobolev gradient flows for the Gross–Pitaevskii ground state problem
作者: Shixin Zheng, Xiangxiong Zhang
类别: math.NA, math.OC, physics.comp-ph
发布日期: 2026-10-09
arXiv ID: 2610.12833v1
链接: http://arxiv.org/abs/2610.12833v1
摘要: Sobolev gradient flows for the Gross–Pitaevskii ground state problem are Riemannian gradient descent methods on the sphere under a metric induced by a discrete elliptic operator. We analyze their convergence through the condition number of the Riemannian Hessian at the ground state. Bounds of the Rayleigh quotient of the Riemannian Hessian give a uniform analysis of these methods for different metrics. The uniform analysis applies to the classical metrics, such as the shifted Laplacian and the energy-adaptive metrics. Inspired by this analysis, we also consider three new metrics. One of them is a Thomas–Fermi metric, whose condition number is bounded uniformly in the interaction strength for the one-dimensional harmonic trap and whose iteration count does not grow in the Thomas–Fermi regime in the numerical tests. The performance suggested by our analysis is consistent with three-dimensional numerical tests with up to one billion unknowns, for both Riemannian gradient descent and Riemannian conjugate gradient methods, whose practical performance is also related to the Hessian condition number. The Thomas–Fermi flow needs the fewest iterations, while the metrics inverted exactly by tensor-product solvers are the fastest in computational time on one GPU card.
11. 一种核无关的预校正快速傅里叶变换方法
A kernel-independent precorrected-FFT method
作者: Tristan Goodwill, Owen Melia
类别: math.NA
发布日期: 2026-10-09
arXiv ID: 2610.12821v1
链接: http://arxiv.org/abs/2610.12821v1
摘要: The precorrected-FFT has been used to rapidly evaluate $N$-body sums arising in electromagnetics and acoustics settings. It has, however, been historically limited to kernels that satisfy a Green’s identity. In this work, we demonstrate that, using an improved version of the method of proxy points, the precorrected-FFT can be used to efficiently compute $N$-body sums with a wide variety of kernels. This algorithm requires no analytic information beyond pointwise access to the kernel. Our examples include kernels from non-local PDEs, Matern kernels, anisotropic kernels, and the Green’s functions of high order PDEs. We also provide a detailed algorithm for choosing the proxy points and spreading grid, as implemented in the new pcfft package. We show these algorithmic choices result in favorable scaling when compared to other kernel-independent fast summation methods. Finally, we use the algorithm to accelerate a wide variety of integral equations solvers involving non-standard kernels, with applications in surface scattering, geophysics, and microbial community formation.
12. 面向各向异性超弹性均匀化的具有学习配对方向特征的多凸神经网络代理模型
Polyconvex neural-network surrogates with learned paired directional features for anisotropic hyperelastic homogenization
作者: S. Ares de Parga, A. Cornejo, J. A. Hernández, R. Rossi
类别: physics.comp-ph, cs.LG, math.NA
发布日期: 2026-10-08
arXiv ID: 2610.12801v1
链接: http://arxiv.org/abs/2610.12801v1
摘要: Approximating the effective anisotropic hyperelastic response of a representative volume element (RVE) requires constitutive expressiveness, mechanical consistency, and computational efficiency. This work develops a polyconvex physics-augmented neural network (PANN) construction based on paired directional features whose orientations, stretch powers, and determinant exponents adapt to the RVE response. The features are composed with convex, componentwise nondecreasing cores, realized as an input-convex neural network (ICNN) or an input-convex Kolmogorov-Arnold network (ICKAN). The complete energies satisfy objectivity, an exactly stress-free reference, two-dimensional polyconvexity, and prescribed growth conditions by construction, and nonnegative energy is certified separately for the selected models. On a multicavity and a single-cavity periodic cell, these energies are compared with an Unconstrained energy, trained identically without the curvature and growth constraints, and with three hyperreduced projection-based reduced-order models (HPROMs), which serve as physics-based references. On the first cell, learning the features lowers the stress error sevenfold, and six features suffice on both. The constrained energies remain below one percent in stress error and in the displacement error of a tensile-coupon FE$^2$ application with a full-order reference, whereas the Unconstrained energy and the HPROMs are more accurate inside the data. Beyond the data, however, the Unconstrained energy loses rank-one convexity, and under confined compression it stops below half of the 500 MPa that the constrained energies reach. These results quantify what the guarantees cost in accuracy and what they secure beyond the data.
13. 面向加速潮流分析的基于符号置换的稀疏雅可比矩阵直接装配
Symbolic-Permutation-Based Direct Sparse Jacobian Assembly for Accelerated Power-Flow Analysis
作者: Tianshi Cheng, Venkata Dinavahi
类别: eess.SY, cs.PF, math.NA
发布日期: 2026-10-08
arXiv ID: 2610.12654v1
链接: http://arxiv.org/abs/2610.12654v1
摘要: The repeated assembly of the sparse Jacobian matrix constitutes a primary computational bottleneck in the Newton-Raphson power-flow analysis. Through an algebraic equivariance analysis of the Jacobian matrix, this paper proposes a direct sparse Jacobian assembly method based on symbolic permutation. By pre-applying the permutation to the admittance matrix during a one-time symbolic phase, the runtime bus-type projection overhead is entirely eliminated. Consequently, complex Jacobian differentiation and global matrix assembly fuse into a single operation, yielding a single-pass, branchless algorithm that streams values directly into contiguous memory. The framework is realized in the open-source RustPower solver. Benchmarks on IEEE test systems and PEGASE 9241 demonstrate that the proposed assembly is 24.4-30.1x faster than the MATPOWER- style baseline and 6.0-7.8x faster than the Schafer-Braun optimized assembly. End-to-end, the grid-level solve time is 11.7x-154.3x faster than pandapower 3.5.4 with Numba, while leaving numerical results and convergence behavior unchanged. Compared with the native C++ LightSim2Grid 1.0.0, grid-level performance is comparable on IEEE 39 and 1.3x-1.8x faster on IEEE 118 and the PEGASE systems. The method extends directly to general problems where a sparse Jacobian inherits its structure from a static network operator, including transient stability analysis, EMT simulation, and OPF Hessian assembly.
14. 用于嵌套积分的中位数随机化拟蒙特卡洛
Median randomized quasi-Monte Carlo for nested integration
作者: Arved Bartuska, Yang Liu, Raúl Tempone
类别: math.NA
发布日期: 2026-10-08
arXiv ID: 2610.12643v1
链接: http://arxiv.org/abs/2610.12643v1
摘要: Nested integrals $\int f\left(\int g(\boldsymbol{y},\boldsymbol{x})\mathrm{d}{}\boldsymbol{x}\right)\mathrm{d}{}\boldsymbol{y}$ where $\boldsymbol{y}\in[0,1]^{d_1}$ and $\boldsymbol{x}\in[0,1]^{d_2}$ frequently arise in Bayesian experimental design, medical decision making, and computational finance. Statistical estimators for nested integrals are more computationally demanding than for standard integrals. Recent studies demonstrate that the median randomized quasi-Monte Carlo (mRQMC) method achieves super-polynomial convergence rates for analytic integrands. We extend mRQMC methods to nested integrations and propose a novel double-loop mRQMC estimator with super-polynomial convergence rate if both the inner and outer integrands are analytic. Established estimators for nested integrals, such as double-loop or multilevel Monte Carlo, are by contrast limited to algebraic convergence rates. However, expected information gain (EIG) applications from experimental design typically do not fulfill this condition due to a lack of regularity in the outer integrand. We address this through a theoretically-grounded importance sampling regularization scheme. Although super-polynomial convergence rates remain achievable for the inner integral, only the algebraic rate $α+3/2-ε$ for any $ε>0$ can be theoretically guaranteed for the outer integral in the EIG, where $α\in\mathbb{N}\cup\{0\}$ denotes the order of the mixed derivatives of the regularized outer integrand. Our approach shows substantial improvements over double-loop randomized quasi-Monte Carlo methods based on the sample mean in moderately high-dimensional settings for a simple polynomial example. Moreover, we demonstrate improved convergence behavior of the median-based estimator for the more challenging setting of EIG estimation.
15. 网格原生标记化：将基础模型改造以适配非结构化数据
Mesh-Native Tokenisation: Retrofitting Foundation Models to Unstructured Data
作者: Georg Maierhofer, Miles Cranmer, Payel Mukhopadhyay
类别: cs.CE, cs.AI, math.NA
发布日期: 2026-10-08
arXiv ID: 2610.12618v1
链接: http://arxiv.org/abs/2610.12618v1
摘要: Physics foundation models are an increasingly popular approach for building versatile emulators for spatiotemporal systems. They require tokenisation layers that lift physical field data into a latent representation suitable for dynamic processing. A typical choice for this layer is a convolutional encoder, which lifts field information in a locality-aware way and can be trained very efficiently. However, this choice cannot be applied to a large class of scientific computing problems where data are represented on unstructured meshes, for example simulations on domains with curved boundaries. Such data typically require graph neural network architectures, which are more expensive to train and, in the form currently used, encode information differently from their CNN counterparts, thus limiting the portability of physics emulators between these data regimes. In this work we introduce Mesh-Native Convolution (MNC), a locality-aware tokenisation layer that encodes unstructured mesh-based data in a way that is analogous to CNNs. This approach leads to models that can be trained once on mesh data and then deployed to arbitrary unstructured meshes, even if these are irregular, with minimal retraining. We show that this tokenisation approach can be used to retrofit grid-pretrained foundation models efficiently on unstructured meshes and provide theoretical results to confirm the consistency of this approach with classical CNN encoders. We demonstrate the favourable performance of this retrofitting approach, which in many cases reaches acceptable accuracy even in zero-shot deployment, and outperforms existing baselines after only a few epochs of fine-tuning, thus providing an efficient bridge between the two data regimes.
16. 基于傅里叶变换的双电子四中心库仑排斥积分计算
Fourier Transform-Based Computation of Two-Electron Four-Center Coulomb Repulsion Integrals
作者: William Chen, Lior Silberman, Ury Segal, Chen Greif
类别: math.NA, physics.chem-ph, physics.comp-ph
发布日期: 2026-10-08
arXiv ID: 2610.12608v1
链接: http://arxiv.org/abs/2610.12608v1
摘要: We present a Fourier-based method for evaluating two-electron four-center Coulomb repulsion integrals over Slater-type orbitals. Spherical Fourier-Bessel coefficients of the convolution factors are precomputed and reused. Since the spherical part of the integral can be evaluated directly, there remains only a one-dimensional quadrature problem in the radial direction. For the ten reference integrals reported here, the largest absolute discrepancy is $1.7\times10^{-9}$. A preliminary resident-cache benchmark computes approximately $4.2\times10^5$ unique integrals per second on an NVIDIA V100 at the contraction stage.