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1. 基于 Grassmann 插值的多尺度 PDE 高效局部化模型降阶
Efficient localised model reduction for multiscale PDEs via Grassmannian interpolation
作者: Christian Alber, Markus Bachmayr, Robert Scheichl, Huqing Yang
类别: math.NA
发布日期: 2026-10-06
arXiv ID: 2610.08657v1
摘要翻译: 多尺度、参数相关的偏微分方程 (PDEs) 由于强系数异质性和高维参数空间而带来严峻的计算挑战。我们在多尺度广义有限元方法 (MS-GFEM) 中发展了一种几何插值方法，该方法针对其中最昂贵的组成部分：计算参数相关的最优局部逼近空间。利用 MS-GFEM 的空间局部化，并假设局部参数依赖性，我们将全局问题分解为参数低维的局部子问题。每个参数对应的最优子空间被识别为 Grassmann 流形上的点，并通过稀疏网格上的 Grassmann 插值进行逼近，这在高效处理高维参数空间的同时保持了这些空间的几何结构。由此得到的局部化模型降阶方法继承了 MS-GFEM 的近乎指数阶空间收敛性和稀疏网格的参数收敛速率。针对椭圆问题的数值实验证实了理论收敛结果。
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链接: http://arxiv.org/abs/2610.08657v1
1. 研究对象和背景
本文研究参数依赖的二阶椭圆扩散问题，其强形式为
$$ -\nabla\cdot\big(A(x,z)\nabla u(x,z)\big)=f(x)\quad \text{in }\Omega,\qquad u(x,z)=0\quad \text{on }\partial\Omega, $$其中 $\Omega\subset\mathbb{R}^N$ 是有界 Lipschitz 区域，参数 $z\in[-1,1]^d$，$f\in L^2(\Omega)$，而系数 $A:D\to L^\infty(\Omega;\mathbb{C})$ 是定义在包含 $[-1,1]^d$ 的开集 $D\subset\mathbb{C}^d$ 上的全纯映射，并且对实参数 $z$ 满足 $A(z)\in L^\infty(\Omega;\mathbb{R})$。弱形式为：对给定 $z\in D$，求 $u_z\in H^1_0(\Omega;\mathbb{C})$ 使得
$$ a_{\Omega,z}(u_z,v):=\int_\Omega A(x,z)\nabla u_z(x)\cdot\nabla v(x)\,dx=\int_\Omega f(x)v(x)\,dx,\qquad \forall v\in H^1_0(\Omega;\mathbb{C}). $$在一致椭圆性假设下，即存在 $0 多尺度方法通过把细尺度信息编码到问题自适应的局部近似空间中，使得在粗网格上也能获得高精度。已有方法包括 Heterogeneous Multiscale Method、Two-Scale Finite Element Method、Multiscale Finite Element Method 等依赖周期性或尺度分离假设的方法，也包括 Generalized Multiscale Finite Element Method、Adaptive Local Bases、Localized Orthogonal Decomposition、Gamblets 等对系数假设较弱的方法。本文的工作建立在 Multiscale Spectral Generalized Finite Element Method（MS-GFEM）之上。MS-GFEM 的突出特点是强局部化：其过采样区域通常远小于其他多尺度方法，因此特别适合参数问题，因为在每个子域上系数只依赖少数参数，从而局部问题的有效参数维数较低。 GFEM 的总体框架如下：设 $\{\omega_i\}_{i=1}^M$ 是 $\Omega$ 的一个重叠分解，每点至多属于 $\kappa$ 个子域；取单位分解 $\{\chi_i\}$ 满足 $0\le\chi_i\le 1$、$\operatorname{supp}(\chi_i)\subset\omega_i$、$\sum_i\chi_i\equiv 1$ 且 $\chi_i\in W^{1,\infty}(\Omega)$，并满足 $\max_{x\in\Omega}|\nabla\chi_i|\le C_\chi/\delta_i$。局部近似空间为 $S_{n_i}(\omega_i)\subset H^1(\omega_i)$，全局 GFEM 空间为 论文回忆的 GFEM 误差估计（定理 2.1）表明，若 $\Psi:=u^p+\sum_i\chi_i\phi_i$ 满足 其中 $\omega_i\subset\omega_i^*\subset\Omega$ 是过采样域，则全局误差可控制在局部误差的最大值： 其中 $\kappa_*$ 是过采样域的最大重叠数。这意味着全局参数问题可以约化为每个子域上的独立局部问题，从而为局部模型降阶提供了理论基础。 在参数设置中，MS-GFEM 的最优局部空间由局部特征问题确定。对单个子域 $\omega$ 与过采样域 $\omega^*$，局部特解 $u_i^p=\psi_i|_{\omega_i}$ 由 $a_{\omega_i^*,z}(\psi_i,v)=F(v)$ 在 $H^1_0(\omega_i^*;\mathbb{C})$ 中定义。关键观察是 $u_z|_{\omega^*}-\psi$ 属于 $A(z)$-调和空间 MS-GFEM 使用单位分解算子 $P_z:H_{A(z),0}(\omega^*)\to H^1_0(\omega;\mathbb{C})$，$u\mapsto\chi u|_\omega$，并通过特征问题 取前 $n$ 个最大特征值对应的特征函数张成 $V_n(z)$。该空间在 Kolmogorov $n$-宽意义下最优，并且标准 MS-GFEM 误差分析给出局部误差 $\epsilon=\lambda_{n+1}^{1/2}\le C\exp(-c n^{1/N})$，即空间方向近指数收敛。 然而在参数依赖情形中，直接计算每个参数 $z$ 下的最优局部空间 $V_n(z)$ 极其昂贵，因为需要对每个参数求解局部特征问题。已有参数与随机偏微分方程方法包括随机 Galerkin、随机配点、稀疏网格、张量分解、广义多项式混沌等；模型降阶方法包括 POD 与贪心 reduced basis 方法，它们常与最优局部近似空间结合用于局部模型降阶。但贪心方法通常只保证在有限训练集上准确，而非在整个参数域上一致，且在高维参数空间中构造足够丰富的训练集代价过高，所得局部空间也可能过大。本文试图填补的空白是：不构造单一局部空间来对所有参数都准确，而是构造参数依赖的局部近似空间，并把“参数到最优局部空间”的整映射作为对象来插值。由于这些最优空间是固定 Hilbert 空间中的子空间，它们自然可视为 Grassmann 流形上的点；作者利用 MS-GFEM 的局部化将全局问题约化为参数低维局部子问题，再通过 Grassmann 流形上的稀疏网格插值来近似该映射，从而同时继承 MS-GFEM 的空间近指数收敛与稀疏网格的参数收敛率。 本文的主要结果可分成两部分：参数正则性定理与 Grassmann 流形插值误差分析。首先是 GFEM 的全局误差分解。定理 2.1 给出：若 $\Psi:=u^p+\sum_{i=1}^M\chi_i\phi_i$，$u^p\in H^1_0(\Omega)$，$\phi_i\in S_{n_i}(\omega_i)$，并且 则 其中 $\kappa$ 是子域最大重叠数，$\kappa_*$ 是过采样域最大重叠数。该结果说明全局误差由局部误差的最大值控制，因此可以独立地在每个子域上构造和插值局部空间。 为处理参数依赖，作者引入参数无关的边界迹空间 并定义 $A(z)$-调和延拓算子 $E_z:H^{1/2}_0(\partial\omega^*;\mathbb{C})\to H_{A(z)}(\omega^*)$，以及修正算子 $E^0_z$。对内部子域，$E^0_z u_\partial:=E_z u_\partial-M_z(E_z u_\partial)$，其中 对边界子域，$E^0_z:=E_z$。引理 2.2 证明：对实参数 $z\in[-1,1]^{d_{\mathrm{loc}}}$，$H^{1/2}_0(\partial\omega^*;\mathbb{C})$ 与 $H^1_0(\omega;\mathbb{C})$ 配备由 $a^\partial_{\omega^*,z}(u_\partial,v_\partial):=a_{\omega^*,z}(E_z u_\partial,E_z v_\partial)$ 给出的双线性形式后是 Hilbert 空间；对复参数则一般不 Hermitian。引理 2.3 进一步证明：对实 $z$， 是等距同构。边界子域的逆为迹算子 $\gamma$；内部子域的逆为 由此定义替代参数单位分解算子 $P^\partial_z u_\partial:=\chi E^0_z(u_\partial)$，其定义域不依赖 $z$，且 $P^\partial_z=P_zE^0_z$。引理 2.5 证明对实 $z$，$P^\partial_z$ 是紧算子。 命题 2.6 是核心结构结果：对实 $z\in D$， 其中 $V^\partial_n(z)$ 是特征问题 的前 $n$ 个最大特征值对应的特征空间。对内部子域还有 $E^0_zV|_\omega+\mathbb{R}=E_zV|_\omega+\mathbb{R}$。于是定义参数无关空间上的算子 其中伴随关于 $a^\partial_{\omega^*,z}$ 与 $a_{\omega,z}$ 取。 定理 2.7 证明：算子 $T:[-1,1]^{d_{\mathrm{loc}}}\to L(H^{1/2}_0(\partial\omega^*;\mathbb{C}))$，$z\mapsto T_z$，可以全纯延拓到 $[-1,1]^{d_{\mathrm{loc}}}$ 在 $D\subset\mathbb{C}^{d_{\mathrm{loc}}}$ 中的一个开连通邻域 $\widetilde D$。证明通过将 $T$ 分解为更简单算子完成。为了讨论特征空间对参数的光滑依赖，作者作谱隙假设：假设 2.8 存在 $\delta>0$ 与整数 $n\ge 1$，使得 这里 $T_z$ 紧且关于 $a^\partial_{\omega^*,z}$ 自伴，特征值按降序排列并计重数。记 定义 Riesz 谱投影 其中 $\Lambda(z)$ 是预解集中的闭曲线，分离 $\sigma_1(z)$ 与 $\sigma_2(z)$ 并正向包围 $\sigma_1(z)$。引理 2.9 给出：$\Pi^n_z$ 是投影；对实 $z$ 关于 $a^\partial_{\omega^*,z}$ 自伴；对实 $z$ 有 $\operatorname{range}(\Pi^n_z)=V^\partial_n(z)$；对实 $z$，$\Pi^n_z$ 把 $H^{1/2}_0(\partial\omega^*;\mathbb{R})$ 映到自身。命题 2.10 证明谱投影 $\Pi^n:\widetilde D\to L(H^{1/2}_0(\partial\omega^*;\mathbb{C}))$ 全纯。 在此基础上，作者把最优局部空间视为 Grassmann 流形上的点。设 $V$ 为 Hilbert 空间，Stiefel 流形 $\operatorname{St}_k(V):=\{L\in L(\mathbb{R}^k,V):L^*L=I_k\}$，Grassmann 流形 $\operatorname{Gr}_k(V):=\operatorname{St}_k(V)/O(k)$。固定 $z_0\in[-1,1]^{d_{\mathrm{loc}}}$，取 $V=H^{1/2}_0(\partial\omega^*;\mathbb{R})$ 并赋以 $a^\partial_{z_0,\omega^*}(\cdot,\cdot)$。设 $D_R$ 为 $\widetilde D\cap\mathbb{R}^{d_{\mathrm{loc}}}$ 中包含 $[-1,1]^{d_{\mathrm{loc}}}$ 的开连通分支，定义 并令 定理 2.12 是主要正则性结果：若谱隙假设 2.8 成立，则 $V^\partial_{n,R}$ 与 $V_{n,R}$ 都是实解析映射。证明利用实解析谱投影与固定参考基，构造局部实解析框架。 在插值部分，作者回顾一维 Chebyshev 插值误差估计。引理 3.1 指出：若 $g\in C^0([-1,1];V)$ 可解析延拓到 $\Sigma([-1,1];\tau):=\{z\in\mathbb{C}:\operatorname{dist}(z,[-1,1])\le\tau\}$，则其 Chebyshev 插值 $I_k(g)$ 满足 其中 $1<\rho:=\tau+\sqrt{1+\tau^2}$。随后作者构造各向异性 Smolyak 稀疏网格公式 $I_{w,\alpha}$，其中权重向量 $\alpha=(\alpha_1,\dots,\alpha_d)>0$，水平 $w\in\mathbb{N}$，集合 且 $I_{w,\alpha}=\sum_{w\in\Lambda_{w,\alpha}}\Delta_w$。稀疏网格 $S_{w,\alpha}$ 由各方向 Clenshaw-Curtis 节点 $z^i_j=-\cos(\pi(j-1)/(k_i-1))$ 组成，其中 $k_i=k(w_i)=2^{w_i}+1$（$w_i\ge 1$），$k(0)=1$，$z^1=0$。各向异性权重 $\alpha$ 用于在收敛较慢的方向放置更多节点，以适应系数对参数的强各向异性依赖。 论文声称在定理 3.9 中给出插值局部空间的局部逼近误差界，并组合 MS-GFEM 空间误差估计与 Grassmann 流形上的稀疏网格收敛估计，得到空间与参数方向均近指数收敛的方法。摘要也说明“所得局部模型降阶方法继承 MS-GFEM 的近指数空间收敛与稀疏网格的参数收敛率”，数值实验对椭圆问题确认理论收敛结果。但是，提供的 PDF 摘录仅到第 10 页，未包含定理 3.9 的完整陈述、其假设与精确误差表达式，也未包含数值实验的具体表格或收敛图。因此，从提供内容中无法确认定理 3.9 的精确误差阶、常数依赖以及数值实验的详细设置；只能确认作者明确宣称的组合收敛结论与数值验证方向。 本文的核心技术障碍有两层。第一，MS-GFEM 的最优局部空间由算子 $P_z$ 的特征空间给出，而 $P_z$ 的定义域 $H_{A(z),0}(\omega^*)$ 随参数 $z$ 变化，因此难以直接研究参数到子空间映射的正则性，更难以直接插值。第二，即使忽略定义域变化，对一个参数集合上的最优子空间进行逐点插值，所得的插值子空间一般不属于 $A(z)$-调和空间，因此经典 MS-GFEM 局部误差估计不能直接应用。作者真正的创新点在于：不直接插值 $H_{A(z),0}(\omega^*)$ 中的子空间，而是插值边界迹空间 $H^{1/2}_0(\partial\omega^*;\mathbb{C})$ 中的子空间，然后通过 $A(z)$-调和延拓恢复局部近似空间。由于算子调和函数由其边界迹唯一确定，这一重述使得参数无关空间上的算子 $T_z=(P^\partial_z)^*P^\partial_z$ 成为分析中心，并且插值后的边界迹子空间可以通过 $E^0_z$ 映回局部空间，从而仍能得到局部误差界。 具体地，作者用 $E^0_z$ 把 $H^{1/2}_0(\partial\omega^*;\mathbb{C})$ 等距地映到 $H_{A(z),0}(\omega^*)$。引理 2.3 给出边界子域的逆为迹算子，内部子域的逆为迹算子减去边界平均。这个等距性质是关键：它把参数依赖的算子调和空间问题转化为参数无关的边界迹空间问题，同时保留能量内积结构。定义 $P^\partial_z u_\partial:=\chi E^0_z(u_\partial)$ 后，有 $P^\partial_z=P_zE^0_z$，且 $P^\partial_z$ 的定义域不依赖 $z$。命题 2.6 进一步说明 $V_n(z)=E^0_z(V^\partial_n(z))$，因此插值边界迹空间上的前 $n$ 个特征空间等价于插值最优局部近似空间。 第二个技术要点是全纯性与实解析性。作者证明 $T_z$ 可全纯延拓到复参数邻域（定理 2.7）。证明不是简单地引用已有结果，而是把 $T$ 分解为更简单的算子，并在复化设置中处理非 Hermitian 双线性形式。随后，在谱隙假设 2.8 下，作者用 Riesz 谱投影 $\Pi^n_z$ 分离前 $n$ 个特征值，并证明 $\Pi^n$ 全纯（命题 2.10）。这里的关键工具是解析微扰理论：谱隙保证特征值簇在参数扰动下不混合，从而可以用围道积分定义局部全纯谱投影。接着，作者把谱投影视为从参数域到 Grassmann 流形的映射，并证明其实解析性（定理 2.12）。为此，作者使用实解析谱投影对固定参考基作用，构造局部实解析框架，并用 Gram 行列式非零保证线性无关。这个实解析性是后续稀疏网格收敛分析的基础，因为标准稀疏网格误差估计要求被插值函数具有解析延拓。 第三个技术要点是 Grassmann 流形上的插值框架。由于 Grassmann 流形没有线性结构，作者采用几何数值积分的思路：在基点 $p\in\operatorname{Gr}_n$ 处取 Riemannian 对数 $\log_p$，将流形值映射 $V^\partial_{n,R}$ 复合为切空间值映射 切空间是 Hilbert 空间，因此可以使用标准各向异性稀疏网格插值。得到 $g^\partial_S$ 后，再用 Riemannian 指数 $\exp_p$ 映回流形，得到 这一构造保持 Grassmann 流形的几何结构，类似已有工作 [6,63] 中的流形插值。作者明确指出，复合 $\log_p\circ V^\partial_{n,R}$ 要求 $V^\partial_{n,R}(z)$ 对所有 $z$ 落在 $\log_p$ 的定义域内，这一条件在定理 3.5 中被精确化；但提供的摘录未包含定理 3.5 的完整陈述。 第四，误差分析分为两步。第一步是把 Grassmann 流形上的插值误差转移为切空间插值误差。作者扩展了 [41] 中有限维流形逼近结果到无限维 Grassmannian。具体而言，需要控制插值子空间与最优子空间之间的测地距离 $d_{\operatorname{Gr}}(\widehat V^\partial_{n,R}(z),V^\partial_{n,R}(z))$，并把它与 $\|g^\partial-g^\partial_S\|$ 联系起来。第二步是控制切空间中的插值误差。由于 $V^\partial_{n,R}$ 是实解析的，$g^\partial=\log_p\circ V^\partial_{n,R}$ 在适当条件下也解析，从而可以应用引理 3.1 的多维推广与各向异性 Smolyak 稀疏网格误差估计。这里各向异性权重 $\alpha$ 起关键作用：它允许在参数依赖较强的方向使用更多节点，在较弱方向使用较少节点，从而缓解高维参数空间中的维数灾难。 第五，局部逼近误差的最终界。作者在摘要和引言中说明，分析的起点是把任意局部子空间的局部逼近误差用最优局部逼近误差与最大主角界来控制。设给定局部子空间与最优子空间之间的最大主角为 $\theta$，则局部误差可由 $\lambda_{n+1}^{1/2}$ 与 $\theta$ 的某种组合界定。由于插值后的空间是通过边界迹空间插值再调和延拓得到，其与最优空间的差异可以由 Grassmann 测地距离控制；而测地距离又由切空间稀疏网格插值误差控制。最后，结合 MS-GFEM 的空间误差估计 $\lambda_{n+1}^{1/2}\le C\exp(-c n^{1/N})$ 与稀疏网格参数误差，得到空间与参数方向均近指数收敛的总体方法。论文声称定理 3.9 给出这些局部逼近误差界，并由此推出全局误差估计；但提供的 PDF 摘录未给出定理 3.9 的完整形式，因此无法确认其中常数、权重重数、参数维数依赖以及是否需要额外的各向异性正则性假设。 此外，作者强调局部插值问题的维数可以通过 MS-GFEM 的子域划分来控制。因为 MS-GFEM 的过采样区域通常较小，每个子域上系数只依赖少数参数，所以局部参数维数 $d_{\mathrm{loc}}$ 远小于全局参数维数 $d$。这解释了为什么该方法能同时处理空间多尺度与高维参数：全局高维问题被分解为许多低维局部问题，每个局部问题的最优子空间作为 Grassmann 流形上的点被插值，而不是在整个全局参数空间上直接构造降阶基。方法的实际计算流程因此是：选择子域划分与单位分解；在少量稀疏网格参数点上求解局部 MS-GFEM 特征问题，得到最优边界迹子空间；在 Grassmann 流形上做对数、稀疏网格插值、指数映射；对任意参数通过调和延拓得到参数依赖局部空间，再组装 GFEM 全局解。该流程保留了 MS-GFEM 的局部化优势，同时用几何插值替代了昂贵的逐参数局部特征求解。 与经典 MS-GFEM 及其后续工作 [8,7,9,53,50,51] 相比，本文并没有改变 MS-GFEM 的空间构造方式，而是针对参数依赖情形中“计算每个参数下最优局部空间”这一最昂贵环节提出加速框架。原始 MS-GFEM 在固定参数下通过局部特征问题得到最优局部空间，并具有近指数空间收敛与强局部化性质；但其直接用于参数问题时，需要对每个参数重复求解局部特征问题，代价随参数采样数增长。本文把参数到最优局部空间的映射视为 Grassmann 流形值映射，并用稀疏网格插值替代逐参数求解。因此，它继承 MS-GFEM 的空间收敛率，同时引入参数方向的稀疏网格收敛率。相比直接 MS-GFEM，本文的代价是在参数域上只选取少量稀疏网格点求解局部特征问题，并对这些子空间进行几何插值；其局限是必须假设局部参数依赖与谱隙等正则性条件。 与随机 Galerkin、随机配点、稀疏网格、张量分解、广义多项式混沌等参数偏微分方程方法 [57,11,25,58,12,13,15,14,68] 相比，本文的插值对象不是解映射本身，而是最优局部近似空间映射。这是一个重要区别：传统参数方法通常在解空间或系数空间上构造多项式、稀疏网格或张量格式，试图直接近似 $z\mapsto u_z$；而本文在 MS-GFEM 的局部化框架下，将问题约化为参数到局部子空间的映射，并在 Grassmann 流形上插值。这样做的好处是，局部空间抓住了多尺度系数的空间结构，并且每个局部问题只依赖少数参数，从而把高维参数问题降为若干低维子问题。代价是需要处理流形值插值和非线性谱问题，且误差分析依赖谱隙与实解析性。 与 POD 和贪心 reduced basis 方法 [18] 以及结合最优局部空间的局部模型降阶工作 [65,23,24,44,1] 相比，本文的推进在于：贪心算法虽然可以达到与 Kolmogorov $n$-宽相当的误差衰减率，但已有结果通常只保证在有限训练集上准确，而不是在整个参数域上一致；在高维参数空间中，构造足够丰富的训练集本身代价极高，且得到的局部降阶空间可能过大。本文则通过解析正则性分析与稀疏网格插值，在参数域上给出可证明的收敛估计，并且不依赖有限训练集。与 [65] 等局部模型降阶工作相比，本文不是用贪心挑选一组局部空间，而是插值整个参数到最优子空间的映射。其代价是需要谱隙假设和局部参数依赖假设，适用范围目前限于二阶椭圆扩散问题；作者在引言中明确表示 MS-GFEM 已被成功用于抛物、Helmholtz 与对流扩散问题，并预期本文结果可在适当假设下推广，但把扩展留作未来工作。 与 Grassmann 流形插值相关工作 [6,63] 相比，本文的创新在于把 Grassmann 流形插值嵌入 MS-GFEM 的局部多尺度框架，并扩展到无限维 Grassmannian。已有流形插值方法多用于有限维降阶模型或动态低秩逼近；本文需要处理无限维 Hilbert 空间中的子空间，因为最优局部空间是 $H^{1/2}_0(\partial\omega^*;\mathbb{R})$ 或 $H^1_D(\omega^*;\mathbb{R})$ 的 $n$ 维子空间。为此，作者扩展了 [41] 的有限维流形逼近结果到无限维 Grassmannian，并把流形上的测地距离与切空间插值误差联系起来。这一扩展不是简单的维数推广，因为需要处理无限维子空间的框架、Riemannian 对数与指数的定义域，以及算子调和延拓与流形插值之间的相互作用。 与参数正则性理论 [33] 相比，本文借用并发展了谱隙假设下的实解析性。作者在定理 2.7 与命题 2.10 中证明算子 $T_z$ 的全纯性与谱投影的全纯性，然后在定理 2.12 中得到参数到 Grassmann 子空间的实解析性。这一路线与 [33] 中关于特征值问题参数正则性的思想一致，但本文需要额外处理 $P_z$ 定义域随参数变化的技术困难，并通过边界迹空间 $H^{1/2}_0(\partial\omega^*;\mathbb{C})$ 与算子 $T_z=(P^\partial_z)^*P^\partial_z$ 重新表述。与 [33] 相比，本文的实解析性结果是流形值映射的实解析性，而不仅是特征值或特征向量的正则性；这为后续稀疏网格插值提供了必要输入。 在假设方面，本文相比经典 MS-GFEM 增加了若干条件。首先是局部参数依赖假设：在每个过采样域 $\omega^*$ 上，系数 $A(x,z)$ 只依赖少数参数，且可假设只依赖前 $d_{\mathrm{loc}}$ 个参数。这是 MS-GFEM 局部化优势能够转化为计算优势的关键。其次是谱隙假设 2.8，要求前 $n$ 个特征值与其余谱在参数域上一致分离。没有该假设，特征空间可能随参数剧烈变化甚至交叉，实解析性与插值误差分析不成立。再次是系数 $A$ 的全纯性与一致椭圆性假设。相比一些只要求系数可测或分段常数的多尺度方法，本文对参数依赖的正则性要求更强；但这是为了得到参数方向的收敛率所付出的代价。 在适用范围与代价方面，本文当前处理二阶椭圆扩散方程 (2.1)，但 MS-GFEM 已被用于抛物、Helmholtz、对流扩散等问题；作者明确表示本文结果“预计可扩展”到这些设置，前提是系数对参数全纯等假设成立，但未在本文中证明。因此，从提供内容看，本文严格证明的结论限于二阶椭圆扩散模型。与随机配点或稀疏网格直接逼近解相比，本文方法需要求解局部特征问题，计算量可能更大；但由于只在稀疏网格点求解，且局部问题参数维数低，总体代价可能显著低于逐参数 MS-GFEM。另一方面，本文需要选择各向异性权重 $\alpha$ 和子域划分，以控制局部参数维数与插值误差；这些选择对实际性能的影响在提供的 PDF 摘录中无法确认，因为数值实验细节未给出。 最后，与有限训练集贪心方法相比，本文的误差分析看起来是在整个参数域上一致成立的，这是主要优势。但需要强调：摘要说“数值实验对椭圆问题确认理论收敛结果”，而提供的 PDF 摘录没有包含实验部分，因此无法独立验证实际收敛率、计算代价或与贪心方法的数值对比。此外，定理 3.9 的完整陈述未在摘录中给出，因此无法精确比较其参数误差阶与已有稀疏网格结果。作者在比较中明确提到 [65] 的贪心结果通常只保证有限训练集上的准确性，并提出本文通过几何插值避免该问题；这是本文相对局部模型降阶文献的核心推进。至于本文是否在常数、参数维数依赖或谱隙常数方面优于已有方法，从提供内容中无法确认。 A Cut Finite Element Method for Transient Thermal Simulation in Multi-material Electronic Packaging Structures 作者: Hao Dong 类别: math.NA 发布日期: 2026-10-06 arXiv ID: 2610.08543v1 链接: http://arxiv.org/abs/2610.08543v1 中文标题: 一种用于多材料电子封装结构瞬态热仿真的切割有限元方法 摘要翻译: 先进电子封装结构是延续摩尔定律的一项关键技术途径。电子封装结构由多种性质不同的材料组成，构成具有复杂空间构型的复合结构。本文针对具有复杂材料界面的多材料电子封装结构中的瞬态热传导仿真，提出了一种非贴体切割有限元方法(CutFEM)。在所提出的计算框架中，电子封装结构的非均质空间几何特征被嵌入并在规则背景网格中加以刻画，从而避免了在复杂界面上生成贴体网格。界面温度和热通量传输条件通过系数加权的对称 Nitsche 格式施加。此外，ghost-penalty 稳定化用于控制物理材料与背景网格之间任意小的相交。进一步，提出了半离散和全离散数值格式，并详细推导了显式能量估计。最后，设计了包含基板、塑封料、裸片和焊球的二维与三维电子封装结构，以验证其在模拟具有挑战性的瞬态热问题时的精度、效率和可扩展性。 Learning PDE solution operators with variable initial conditions via Latent Dynamics Networks 作者: Stefano Maria Pizzamiglio, Stefano Pagani, Francesco Regazzoni 类别: cs.LG, math.NA 发布日期: 2026-10-06 arXiv ID: 2610.08475v1 链接: http://arxiv.org/abs/2610.08475v1 中文标题: 通过潜在动力学网络学习具有可变初始条件的 PDE 解算子 摘要翻译: 在许多查询场景中，数据驱动的代理模型为模拟由偏微分方程（PDEs）控制的物理系统提供了一种相对于高保真求解器的高效替代方案。在此背景下，潜在动力学网络（Latent Dynamics Network, LDNet）最近在预测时空系统响应方面表现出显著性能，它将神经常微分方程（Neural Ordinary Differential Equations）与非线性降维相结合。然而，原始表述假设固定的初始条件，限制了其在对从变化初始状态演化的系统具有广泛实际应用中的适用性。在这项工作中，我们克服了这一限制，同时保持原始 LDNet 的端到端训练过程及其无编码器特性，从而保持其本质上不依赖于空间分辨率和网格拓扑。我们直接从一小组早期时间观测中推断初始潜在状态，将潜在状态初始化视为一个适应问题，并研究两种策略：一种自动解码（auto-decoding）表述，以及一种元学习方法，其中初始潜在状态充当任务特定上下文变量。我们在多种物理现象中展示了所提方法的准确性，涵盖对流-扩散、流体动力学和固体力学。元学习显著加速了潜在状态推断，并诱导出更平滑、条件更好的优化景观，且自发地将潜在空间组织为一种结构化表示，该表示反映了底层动力学具有物理意义的特征。基于坐标的解码器使得能够从空间下采样数据中进行训练，同时在推断时恢复高分辨率解场。由此得到的方法为具有变化初始条件的含时 PDEs 的许多查询模拟提供了一个高效且不依赖于分辨率的代理建模框架。 AdHImEx: Adaptively High-Order Implicit-Explicit Transport for Large Time Steps 作者: Amber J. te Winkel, Hilary Weller, Christian Kühnlein, James Kent 类别: math.NA, physics.comp-ph, physics.flu-dyn 发布日期: 2026-10-06 arXiv ID: 2610.08166v1 链接: http://arxiv.org/abs/2610.08166v1 中文标题: AdHImEx：面向大时间步长的自适应高阶隐式-显式输运 摘要翻译: 自适应隐式-显式（AdImEx）时间推进为质量守恒输运提供了大时间步长下的稳定性，使其在天气与气候预测中的平流数值表示方面具有吸引力。现有 AdImEx 输运格式在大时间步长的隐式情形下仅具有一阶精度，并且每个时间步可能需要多次稀疏矩阵求解。本文引入 AdHImEx，一种同时解决上述两个局限的新 AdImEx 格式。AdHImEx 对大 Courant 数具有二阶精度，而每个时间步仅需一次矩阵求解。它是一种 Runge-Kutta 格式，融合了三阶精度的显式时间推进与 Crank-Nicolson 隐式时间推进，从而对高达 100 的 Courant 数提供经数值验证的稳定性。AdHImEx 面向以显式为主的流动设计，仅在出现大 Courant 数的区域激活隐式时间推进。在小 Courant 数区域，它保留了显式时间推进的全部效率和第三阶精度。本文将其与五阶精度的空间有限体积离散相结合，以实现质量守恒。本文的第二个贡献是对散度算子的一种新颖的级相关处理，该处理在隐式和显式 Runge-Kutta 级时间步不同以及隐式性空间变化的情况下仍保持常数性，从而消除虚假散度。这一创新适用于其他 AdImEx 时间推进格式。收敛性、输运和效率测试表明，与先前的一阶 AdImEx 格式相比有显著改进，相位和振幅误差很小，在大 Courant 数的局部加密网格上能够准确输运，并且对大气输运具有有前景的效率特性。 A SAV Neural Network Method for Allen–Cahn and Cahn–Hilliard Equations 作者: Tao Luo, Guihong Wang, Wei Zhao 类别: math.NA 发布日期: 2026-10-06 arXiv ID: 2610.08088v1 链接: http://arxiv.org/abs/2610.08088v1 中文标题: 一种用于 Allen–Cahn 和 Cahn–Hilliard 方程的 SAV 神经网络方法 摘要翻译: 相场方程是梯度流系统的基本例子，并且需要能够保持内在能量耗散的数值求解器。在这项工作中，我们提出了一种基于神经网络的方法，该方法将 SAV 时间离散与无网格神经表示相结合。我们建立了底层 SAV 半离散的无条件能量稳定性，量化了由求解器残差引起的能量缺陷，并推导了 Allen–Cahn 方程的条件高概率误差估计。数值实验表明，我们的方法始终表现出能量稳定行为，同时优于现有的基于神经网络的求解器。 Implict-Explicit Runge-Kutta schemes for geometric flows 作者: Wei Jiang, Chunmei Su, Kaike Tang 类别: math.NA 发布日期: 2026-10-06 arXiv ID: 2610.08019v1 链接: http://arxiv.org/abs/2610.08019v1 中文标题: 几何流的隐式-显式 Runge-Kutta 格式 摘要翻译: 我们提出一个一步隐式-显式 (IMEX) Runge-Kutta 框架，用于几何流的高阶时间积分。从曲线缩短流的 Dziuk 与 Deckelnick-Dziuk 参数有限元格式出发，我们将空间半离散方程重写为常微分方程组，并构造二阶与三阶格式。每个阶段仅需一次线性求解。关键要素是一种约化系数结构，使得每个阶段的变分问题可以在前一阶段生成的几何上提出。因此，Runge-Kutta 阶段提供了所需的预测几何，而无需单独的预测过程。所得格式不需要额外的启动值。该框架也直接推广到曲线和曲面的平均曲率流与表面扩散的 Barrett-Garcke-Nürnberg 格式。数值实验支持所设计的时间收敛阶，并表明适当的系数选择可产生与相应一阶方法相当的网格质量。 FOSLS-deRhaNN: native de Rham neural classes for H(div) and H(curl) with applications to first-order system least-squares neural network methods for partial differential equations 作者: Shun Zhang 类别: math.NA, cs.LG 发布日期: 2026-10-06 arXiv ID: 2610.08016v1 链接: http://arxiv.org/abs/2610.08016v1 中文标题: FOSLS-deRhaNN：面向 H(div) 与 H(curl) 的原生 de Rham 神经类及其在偏微分方程一阶系统最小二乘神经网络方法中的应用 摘要翻译: 我们构造了面向图空间 H(div) 和 H(curl) 的原生神经逼近类，涵盖二维和三维情形，并且对于 H(div) 涵盖任意维数。每个实现对于所有参数值都位于该空间中；对于带折点的势，例如 ReLU 网络，容许的跳跃在有限宽度处出现。这些类是标量网络和逐分量网络在 de Rham 复形的固定算子下的像，不涉及网格或有限元模拟。对于 R^n 中的 H(div)，在同等基础上给出两个原生类，带一个反对称势 $A$：$\mathrm{Div}\,A+R_nq+\mathbf{h}$，其中散度 $q$ 为显式未知量；以及 $\mathrm{Div}\,A+\mathbf{z}$，其中 $\mathbf{z}$ 为 $H^1$ 场；对于 H(curl)，类似类在二维为 $\mathrm{grad}\,φ+Sr+\mathbf{h}$，在二维和三维为 $\mathrm{grad}\,φ+\mathbf{z}$。在所有这些类中，场的每个界面跳跃都由势项 $\mathrm{Div}\,A$ 或 $\mathrm{grad}\,φ$ 承载，而其余部分没有界面跳跃（在正则分解类中它是 $H^1$ 场）；带有 $\mathbf{z}$ 的类是逐分量方法被这一项丰富后的结果。已知或学习到的界面几何通过具有可训练振幅的因子进入势项；如果散度跳跃，则进入剩余部分。这些类导出了 FOSLS-deRhaNN 方法，即使用 de Rham 神经网络的一阶系统最小二乘，其损失是在弱形式的自然空间中提出的最小二乘泛函；对于椭圆方程，这包括 $H^{-1}$ 右端项和 $H^{1/2}$ Dirichlet 数据。具有间断系数的椭圆方程和 curl-curl 问题作为实例处理，其泛函等价于误差；具有间断解的线性输运和具有激波的守恒律使用相同的通量类。 Efficient Approximation of Structured Linear Operators using Randomized One-Sided Queries 作者: Esther Gallmeier, Samuel E. Otto 类别: math.NA 发布日期: 2026-10-05 arXiv ID: 2610.07513v1 链接: http://arxiv.org/abs/2610.07513v1 中文标题: 使用随机单侧查询高效逼近结构化线性算子 摘要翻译: 仅使用前向算子-向量乘积查询来逼近高维空间上的线性算子，是科学计算与机器学习中一项常见但具有挑战性的任务，当伴随作用在实验上不可获得或计算上难以构造时就会出现。现有的查询高效方法，其查询复杂度关于误差容限为多对数级，依赖于自伴性或对伴随算子的访问，而基于全局低阶正则性的现有无伴随方法具有多项式查询复杂度。我们表明，先验的局部行空间信息可用于查询高效、无伴随的算子逼近，并且该信息可由局部高阶正则性估计提供。我们的方法首先将 (Amsel et al., SIMAX, 2026) 的固定稀疏度矩阵逼近算法推广到用一个行位于指定子空间中的矩阵来逼近一个矩阵，为仅从前向查询利用局部行空间结构提供了一个通用模板。然后，我们将该框架应用于具有渐近光滑核的 Hilbert-Schmidt 积分算子。我们建立了 $d\leq 3$ 维中典型二阶椭圆偏微分方程的 Green 函数的渐近光滑性，包括球中的 Poisson 方程和环面上的常系数方程。在 $d$ 个空间维度中，我们的算法使用 $\mathcal{O}\big(\log(\varepsilon^{-1})\log^{d}(\varepsilon^{-1}δ^{-1/2})\big)$ 次非自适应随机查询，以概率 $1-δ$ 产生 Hilbert-Schmidt 范数下 $\varepsilon$-精确的逼近。这意味着误差在查询次数 $q$ 中按 $\mathcal{O}\big( \exp(- c q^{1/(d+1)}) \big)$ 下降，我们在使用数十到数百次查询来逼近一维变系数二阶椭圆方程的解算子的数值实验中观察到了这一点。 Large Growth Happens: Gaussian Elimination with Partial Pivoting on Random Matrices 作者: Daniel A. Spielman, Xifan Yu 类别: math.NA, cs.DS, math.PR 发布日期: 2026-10-06 arXiv ID: 2610.08701v1 链接: http://arxiv.org/abs/2610.08701v1 摘要: We prove that the probability that Gaussian elimination with partial pivoting on $n \times n$ random matrices has growth $ρ$ is at least inverse quasi-polynomial: $Ω(\exp(-c \log^2 (ρ)\log(n)))$ for some constant $c > 0$.
This lower bound breaks standard conjectures in the smoothed and average-case analysis of Gaussian elimination.
To the best of our knowledge, it is the first non-trivial lower bound on the probability of large growth for Gaussian random matrices. Effective computation of Moore-Penrose inverses over fields of rational functions by specializations 作者: Juana Sendra 类别: math.NA, math.RA 发布日期: 2026-10-06 arXiv ID: 2610.08676v1 链接: http://arxiv.org/abs/2610.08676v1 摘要: In this paper we consider matrices whose entries are rational functions of several parameters over a Moore-Penrose field, that is, a field with an involutory automorphism over which every matrix has Moore-Penrose inverse. We prove that over any such field the Penrose conditions can be solved with linear algebra alone, even though they form a polynomial system of degree two. We then bound the degrees of the numerator and of the denominator of the pseudoinverse in terms of the degree of the entries and of the rank, and not of the dimensions of the matrix. Specialization is the thread: the closed form is computed once over the field of rational functions, and the question is at which parameter values it still returns the pseudoinverse of the specialized matrix. We determine the values where it does not, from the matrix alone and before computing the pseudoinverse. They are the zeros of a polynomial built from the maximal minors, and they are the values at which the rank decreases. This turns previous sufficient conditions into an exact characterization. The theory also gives symbolic algorithms for the real, the complex and the parametric case, which we implement in Maple and test on 972 timed replications. We also apply them to a Leontief economic model, where the excluded value is the point at which the economy stops being viable. Subset selection for matrices by volume sampling 作者: Ivan Kozyrev, Alexander Osinsky 类别: math.NA 发布日期: 2026-10-06 arXiv ID: 2610.08576v1 链接: http://arxiv.org/abs/2610.08576v1 摘要: We address the Subset selection problem for matrices, where the goal is to select a subset $\mathcal{S}$ of $k$ column indices from a \enquote{short-and-fat} matrix $X \in \mathbb{R}^{m \times n}$, such that the sampled submatrix $X_{\mathcal{S}}$ has $\|X_{\mathcal{S}}^†X\|_F$ as small as possible. Our approach is centered on volume sampling, which attains the tightest known bound on this objective in expectation. As our primary contribution, we propose a new deterministic algorithm, Forward derandomized volume sampling (FDVS), which provably attains this bound and has asymptotic complexity $O(nkm)$. In contrast, the complexity of all previously known algorithms with this guarantee scales at least quadratically in $n$ when $k \ll n$, making FDVS particularly attractive for very wide matrices. In addition, we systematically structure the landscape of volume sampling methods: we propose a simple $O(nm^2)$ algorithm for exact forward volume sampling, show that a known deterministic method is in fact a derandomization of Reverse iterative volume sampling, and derive a modification of the latter that avoids repeated SVD downdating, along with a fast greedy forward variant. The algorithms are verified and compared in numerical experiments involving optimal experimental design, sensor placement, and DLRA-DEIM. Strong Convergence of Continuous Median Filter Scheme 作者: Fabius Krämer, Tim Laux 类别: math.AP, math.DG, math.NA 发布日期: 2026-10-06 arXiv ID: 2610.08484v1 链接: http://arxiv.org/abs/2610.08484v1 摘要: We introduce a continuous-time counterpart of the median filter that gradually denoises a given image. In the limit of vanishing stencil size, our results show that the evolution converges to level-set mean curvature flow. Surprisingly, and for the first time for any median-type filter scheme, we can prove the convergence of energies in this limit. Such strong convergence results are of major interest as they often have to be assumed in the literature to prove convergence of related schemes. Moreover, this result carries crucial geometric information, namely the unit multiplicity of interfaces. The proof relies on a compensated compactness argument inspired by the work of Evans and Spruck. A symmetric counterexample to Strang’s conjecture for bivariate $C^1$ cubic splines on triangulations 作者: Pratyush Potu 类别: math.NA, math.CO 发布日期: 2026-10-06 arXiv ID: 2610.08470v1 链接: http://arxiv.org/abs/2610.08470v1 摘要: We exhibit a triangulation of an equilateral triangle for which the space of bivariate $C^1$ cubic splines has dimension larger than the dimension formula conjectured by Strang. Notably, the triangulation is such that no two edges sharing a vertex are collinear, and the triangulation is invariant under the action of the isometry group ($D_3$) of the equilateral triangle which is triangulated. Analysis of split exponential integrators for semilinear problems with noncommuting operators 作者: Luca Battestini, Marco Caliari, Fabio Cassini 类别: math.NA 发布日期: 2026-10-06 arXiv ID: 2610.07989v1 链接: http://arxiv.org/abs/2610.07989v1 摘要: Splitting the exponential and exponential-like functions can significantly reduce the overall computational cost of exponential integrators, providing a strong motivation for their rigorous analysis. In this paper, we analyze splitting errors for two noncommuting unbounded operators and derive new representations that include the popular Lie–Trotter and Strang cases. Moreover, within the abstract framework of strongly continuous semigroups, we prove the convergence of a split version of the exponential Euler method and a second-order exponential Runge–Kutta method for semilinear problems. Numerical experiments on a nonlinear dispersive equation and a diffusion–reaction equation demonstrate the theoretical findings, stability properties, and efficiency of the proposed split exponential integrators compared to the state of the art. Efficient subsampling-rate selection for practical Multilevel Markov chain Monte Carlo 作者: Lise Guilliams, Pieter Vanmechelen, Giovanni Samaey 类别: math.NA, stat.CO 发布日期: 2026-10-06 arXiv ID: 2610.07956v1 链接: http://arxiv.org/abs/2610.07956v1 摘要: The Multilevel Markov chain Monte Carlo algorithm accelerates Bayesian inversion by exploiting correlations between discretized quantities of interest across a hierarchy of model resolutions. A key practical parameter is the subsampling rate used to generate coarse-level proposals for the finer-level chains. Existing practice sets this rate equal to an integrated autocorrelation time as a proxy for proposal decorrelation, but this choice can be overly conservative and lead to substantial computational overhead. In this work we treat finite subsampling as an error-budget question. We derive a cost-error criterion for selecting a computationally efficient subsampling rate using estimates of sample variances, integrated autocorrelation times, and per-level costs. The criterion balances multilevel sampling errors across levels rather than enforcing approximate independence of the coarse proposals. We clarify the theoretical role of independence through an ideal reset-reference kernel, for which independent coarse proposals may be replaced by proposals generated from a reversible coarse-level kernel. We derive a Poisson-equation representation of the perturbation bias resulting from finite subsampling. Under a state-averaged lower-level forgetting assumption, this yields a conditional geometric perturbation-bias bound; in computation, the corresponding projected diagnostic is used as an empirical error-budget check. Numerical experiments on a highly autocorrelated elasticity beam benchmark show substantial reductions in computational cost compared with IAT-based subsampling. A second Darcy-flow inverse problem gives supporting evidence of qualitatively similar finite-subsampling behaviour in another elliptic benchmark. Learning a generalized Navier-Stokes model beyond the continuum regime 作者: Sihong Shao, Yanli Wang, Zhongwei Xu 类别: math.NA, physics.flu-dyn 发布日期: 2026-10-06 arXiv ID: 2610.07952v1 链接: http://arxiv.org/abs/2610.07952v1 摘要: Navier-Stokes (NS) equations lose accuracy in the transition regime when the Knudsen number is large, while the Boltzmann equation provides an adequate kinetic description with a substantially higher computational cost. In this work, a data-driven generalized Navier-Stokes (GNS) model is proposed to extend the applicability of the NS equations to the transition regime. A new relationship between the non-equilibrium variables, such as the stress tensor and the heat flux, and the equilibrium variables, including the density, macroscopic velocity, and the temperature, together with the Knudsen number, is learned from the data formed by the numerical solution to the Boltzmann equation. The standard end-to-end learning is utilized to construct the loss function, based on which, a long-term end-to-end learning method is proposed for the loss function to reduce accumulated error and improve long-term predictive accuracy. Several numerical examples, including one-dimensional wave and Riemann problems as well as two-dimensional isentropic vortex and Taylor-Green vortex problems, are studied to validate the efficiency of this new GNS model. Learned Adaptive Multiresolution Diffusion Imaging 作者: Christian Tantardini, Stig Rune Jensen, Roberto Di Remigio Eikås, Joakim Henrik Beck 类别: math.NA, cs.LG, eess.IV 发布日期: 2026-10-06 arXiv ID: 2610.07884v1 链接: http://arxiv.org/abs/2610.07884v1 摘要: Adaptive multiresolution methods reduce representation cost by concentrating fine-scale degrees of freedom where needed, but their tree updates are usually governed by fixed local criteria. We introduce Learned Adaptive Multiresolution Diffusion Imaging (Learned AMDI), which preserves the AMDI fixed-tree propagator and hierarchy constraints while replacing the post-propagation selector with a shared local policy trained by proximal policy optimization. Regression tests reproduce deterministic AMDI trajectories to machine precision when identical trees are used. In the Haar implementation studied here, the deterministic one-step selector accepts no refinements in 54 decisions. Across nine held-out cases, Learned AMDI executes 393 refinements and reduces the mean terminal reference discrepancy from $0.17496$ to $0.13657$, while occupancy rises from $0.13737$ to $0.26660$. Step-resolved diagnostics reveal occasional small adaptation-energy increases; fixed-tree energy stability therefore does not guarantee monotonicity of the learned outer iteration. At comparable occupancy, a validation-tuned observed-detail threshold reaches a discrepancy of $0.13792$ with slightly better RMSE and SSIM, placing both methods on essentially the same accuracy–occupancy tradeoff. A decision-1-only control reaches $0.13742$, indicating that most of the improvement on this static benchmark arises from the initial allocation. The shared actor transfers without retraining to $64\times64$ and $128\times128$ images, improving reference discrepancy, RMSE, and SSIM relative to deterministic AMDI, while the frozen threshold rule remains competitive. Learned AMDI thus provides a hierarchy-constrained, resolution-transferable mechanism for adaptive allocation and clarifies the contribution of sequential decisions. Sampling discretization of the uniform norm for hyperbolic-cross trigonometric polynomials 作者: Feng Dai, Andriy Prymak 类别: math.NA, math.CA, math.FA, math.PR 发布日期: 2026-10-06 arXiv ID: 2610.07679v1 链接: http://arxiv.org/abs/2610.07679v1 摘要: We study sampling discretization of the uniform norm for trigonometric polynomials with frequencies in a hyperbolic cross of level $N$ on $\T^d$. For every $d,N\ge2$ and $\varepsilon\in(0,1)$, we construct an explicit norming set with at most $\bigl(C_1(1+\varepsilon^{-1})\bigr)^{d-1}N^{1+\varepsilon}$ points and norming constant at most $\bigl(C_2(1+\varepsilon^{-1})\bigr)^{d-1}$, where $C_1,C_2$ are absolute constants. We also prove that independent Haar-distributed points achieve the same exponent $1+\varepsilon$: a sample of size at least $C(d,\varepsilon)N^{1+\varepsilon}\log(2/η)$ norms the entire space simultaneously with probability at least $1-η$, with a norming constant bounded by $\exp(C_d\varepsilon^{-d})$ and independent of $N$. The deterministic construction combines uniformly stable de la Vallée Poussin sampling operators with a Smolyak-type combination identity. The random result follows from an abstract norming theorem for sums of spaces associated with commuting regular partitions, together with a multiscale approximation of hyperbolic-cross polynomials. Finally, we show that every norming set of constant $B>1$ for the trigonometric polynomials with frequencies in $\{-1,0,1\}^d$ has at least $\bigl(πd/(2e\log B)\bigr)^{d/2}$ points. This rules out quadratic bounds in the dimension of the polynomial space with a prefactor growing only polynomially in $d$ when the norming constant is fixed. Low-rank SI-DSA and GMRES-DSA for Radiative Transfer Equation with Adaptive Accuracy Control 作者: Chenxi Han, Lukas Einkemmer, Wei Guo, Zhichao Peng 类别: math.NA 发布日期: 2026-10-06 arXiv ID: 2610.07653v1 链接: http://arxiv.org/abs/2610.07653v1 摘要: Recently, efficient sweep-based low-rank iterative solvers have been developed to reduce the computational costs of steady-state solves and implicit time integration of the radiative transfer equation (RTE). In particular, angular sampling based on the discrete empirical interpolation method (DEIM) or on error-indicated randomized greedy selection enables efficient, non-intrusive reuse of existing transport sweep implementations. However, randomized-greedy low-rank source iteration with diffusion synthetic acceleration (SI–DSA) enforces uniformly tight accuracy tolerances across all source iterations, thereby oversolving the early iterations, and it does not explicitly reuse angular bases from preceding iterations. More critically, like its full-rank counterpart, this approach may lose effectiveness in multiscale problems with strong material discontinuities. We address these limitations through two complementary developments. First, we introduce adaptive accuracy control into randomized-greedy low-rank SI–DSA, reducing intermediate ranks and computational cost while maintaining the accuracy required for convergence. A two-stage hybrid sampling strategy can provide further acceleration by switching to DEIM-based sampling near convergence, once the preceding angular basis becomes sufficiently representative. Second, we develop an inexact flexible GMRES–DSA solver for the integral scalar-flux reformulation, in which sweep-based low-rank approximations are confined to the construction of the right-hand side and the matrix–vector products. The scalar-flux Krylov basis is not compressed, thereby preserving Arnoldi orthogonality in exact arithmetic and allowing established inexact-GMRES analysis to guide adaptive accuracy control of low-rank operations. We present numerical results on benchmark problems that demonstrate the effectiveness of the proposed methods. Endpoint-Stable Interval Inverse Inequalities for Multiscale Kernels with Positive Exponential Spectra 作者: Chengming Li, Qiming Wang 类别: math.NA, math.ST 发布日期: 2026-10-06 arXiv ID: 2610.07612v1 链接: http://arxiv.org/abs/2610.07612v1 摘要: Inverse inequalities are essential for controlling derivative growth and ensuring stable sampling in kernel approximation. Existing whole-space and single-scale estimates provide an established analytical basis, but their extension to multiscale analytic spaces on finite intervals requires uniform control of boundary effects and cross-scale cancellation. This paper establishes unweighted interval (L^2) inverse inequalities for spaces generated by primitives of kernels with positive exponential spectra. By retaining exterior diagonal energy and controlling off-diagonal interactions through one-sided estimates with endpoint-distance weights, we derive uniform coefficient bounds for normalized high derivatives. A synthesis-operator perturbation argument extends these bounds to positive spectral mixtures, while adjacent-derivative comparison and interval interpolation yield an inverse bound of order (h_^{-1}), where (h_) is the smallest scale. Under geometric scale separation and within-scale centre separation, the constant is independent of the number of scales and centres and their distances from the endpoints, allowing endpoint centres and coincident centres across scales. The coefficient lower bound approaches the optimal limit (1/2). Two-centre constructions establish the sharpness of the scale exponent for every fixed kernel in the class and, for a single exponential spectrum, a sharp deficit of order (q^{-1/2}) from (1/2) for the best uniform coefficient lower bound, where (q) is the derivative order. Numerical illustrations for Cauchy and logistic kernels demonstrate coefficient stability in several multiscale configurations. These results extend interval inverse estimates to multiscale analytic kernel spaces with endpoint centres and provide quantitative conditions for deterministic sampling stability. A Neural JKO Scheme for Hellinger-Kantorovich Gradient Flows via Monge-Growth Pairs 作者: Geuntaek Seo, Cheolhyeong Kim, Hwijae Son, Hyung Ju Hwang 类别: math.NA, cs.LG, math.AP, math.OC 发布日期: 2026-10-06 arXiv ID: 2610.07602v1 链接: http://arxiv.org/abs/2610.07602v1 摘要: We develop a mesh-free neural JKO scheme for advection-reaction-diffusion equations with a gradient-flow structure in the Hellinger-Kantorovich (HK) geometry of unbalanced optimal transport. Each update is parametrized by a spatial map and a mass-changing factor, allowing spatial redistribution and local mass creation or loss to be treated jointly within a single variational step. Their cone action bounds the squared HK distance from above, yielding a sufficient condition for discrete energy dissipation through comparison with the identity pair. Minimizing the pair objective over all admissible pairs recovers the exact JKO minimum when the source and a minimizer have positive densities. We establish existence and mass bounds for JKO minimizers and, under additional assumptions, obtain positivity and regularity together with a discrete Euler-Lagrange equation and a metric-dissipation identity. The self-consistent chemical potential is then nonincreasing along an optimal map. There exist parametric pairs whose endpoint densities and objective values converge to those of an exact JKO minimizer, provided a regular-pair approximation hypothesis holds. Finally, we show that a primal-dual gap controls objective suboptimality and, for Boltzmann entropy, the $L^1$ density error, assuming exact-step regularity, positive-semidefinite interactions, and global dual feasibility. Numerical experiments examine pointwise agreement with the PDE, energy dissipation, and the roles of transport, reaction, and fully implicit interactions. Sharp conditioning and quantitative stability of optimal transport 作者: Yuanlong Ruan 类别: math.OC, math.NA, math.PR 发布日期: 2026-10-06 arXiv ID: 2610.07577v1 链接: http://arxiv.org/abs/2610.07577v1 摘要: When a feasible plan is obtained whose quadratic transport cost is known to be close to the optimal cost, we try to determine how close the feasible plan is to the true optimal map under mild density and moment conditions. The source and target may be unbounded or have non-compact supports. We show that for a source with density in $L^p$ and an $n$-th moment, let $s=1-1/p$ and $η=sn/[n+(d-1)s]$. Under the target constraint $\int |y|^m[\log(e+|y|)]^β\,dν\leqslant1$, the worst-case class-uniform conditioning has a sharp modulus comparable to [
t^{η(m-2)/[m(1+η)-η]}
[\log(e/t)]^{-β(2+η)/[m(1+η)-η]}, ] whenever $t>0$ is small. This holds for $m\geqslant2$ and $β\geqslant0$. When $m=2$, every $β>0$ gives a sharp logarithmic modulus, whereas the $m=2,\,β=0$ extreme has no vanishing modulus. The matching lower bound is verified for both the squared map error and barycentric projection error. The sharp map error bounds allow us to directly derive quantitative stabilities of common source Brenier maps without passing through potential estimates, thereby avoiding loss of information. These stabilities strictly improve the corresponding results of Delalande-Merigot \cite{delalande2023quantitative} and Letrouit-Mérigot \cite{letrouit2026gluing} under identical or weaker settings, particularly no convexity of the source support or bounds of the source density are assumed. A B-Spline Galerkin Method for Option Pricing under the CGMY Tempered Stable Process 作者: Davood Damircheli 类别: math.NA 发布日期: 2026-10-06 arXiv ID: 2610.07554v1 链接: http://arxiv.org/abs/2610.07554v1 摘要: We develop and analyze a B-spline scaling-function Galerkin method for the tempered fractional partial differential equation governing European and American option prices under the CGMY process. The spatial generator couples exponentially tempered left- and right-sided Riemann–Liouville derivatives of order $Y\in(0,2)$, whose nonlocality and asymmetry present discretization challenges. We derive a corrected Fourier-symbol construction of the tempered connection-coefficient matrices whose interior blocks are Toeplitz, maintain exact fractional scaling, and enable $\mathcal{O}(N\log N)$ matrix–vector products via the fast Fourier transform. In the regime $Y\in(1,2)$, we prove full-line and finite-domain Gårding inequalities, coercive whenever the risk-free rate is positive, via zero-extension to the full-line symbol. On this foundation, we establish elliptic-projection, semidiscrete, and fully discrete error estimates with spatial rate $\mathcal{O}(h^{p-Y/2})$, alongside an unconditional shifted-energy stability estimate for the forced Crank–Nicolson scheme with mesh-independent constants. For positive rates the admissible shift vanishes, governing the plain scheme exactly as implemented. A single backward solve prices an entire strike strip at fixed maturity with smooth, analytically differentiated Greeks. The same discretization prices American puts through a projected linear complementarity solve, with early-exercise premiums and exercise boundaries validated against a resolved deterministic reference and a Longstaff–Schwartz Monte Carlo benchmark. Calibration to S&P~500 option surfaces across low-volatility, crisis, and recovery regimes yields implied-volatility errors near two vol-points on well-identified surfaces, while an accuracy-matched comparison with a stabilized Grünwald–Letnikov baseline delineates precisely when the high-order variational structure pays off. Numerical Construction of Quasi-Periodic Solutions for Nonlinear PDEs: I. Bounded Perturbation 作者: Mingwei Fu, Bin Shi 类别: math.NA 发布日期: 2026-10-05 arXiv ID: 2610.07262v1 链接: http://arxiv.org/abs/2610.07262v1 摘要: In this paper, we propose an alternating numerical scheme for constructing quasi-periodic solutions to a class of nonlinear partial differential equations (PDEs) with bounded perturbations, including the one-dimensional nonlinear Schrödinger (NLS) equations and nonlinear wave (NLW) equations under periodic and Dirichlet boundary conditions. A key observation concerning the resonant $Q$-equations allows us to establish directly a local diffeomorphism between the amplitudes and the drifted frequencies. This renders the proof considerably cleaner, since it avoids the Birkhoff normal form and the associated coordinate transformations required in classical KAM theory for PDEs.Unlike in the finite-dimensional setting, the lattice boxes must also include the spatial index, so that the multiscale induction has to be carried out in one additional dimension. This introduces two difficulties. First, the measure estimate for the excluded set of Well-posed by Design: Learning Constitutive Laws from Velocity Data using Convex Neural Network Potentials 作者: Gonzalo G. de Diego, Georg Stadler 类别: physics.flu-dyn, math.NA, math.OC 发布日期: 2026-10-05 arXiv ID: 2610.07236v1 链接: http://arxiv.org/abs/2610.07236v1 摘要: Learning constitutive laws of complex fluids from velocity data (i.e. indirect observations) is a PDE-constrained inverse problem in which expressive neural parameterizations risk breaking the well-posedness of the forward physics model. We address this tension by learning, rather than the constitutive law itself, the dissipation potential: a scalar function whose convexity, frame-indifference, and dissipativity propagate into the underlying continuum mechanics and guarantee the structural properties needed for a well-posed and generalizable forward PDE. We introduce ICNNE, an input-convex neural architecture that exactly enforces convexity, frame-indifference, and evenness in the second strain-rate invariant via symmetrization, alleviating the numerical instabilities that occur at small strain rates in standard formulations. By weakly enforcing zero gradients in the origin of the potential via penalization, ICNNE also satisfies the dissipativity property. The loss objective and its gradient are computed using a combination of finite element and neural network methods, coupling the Firedrake and PyTorch libraries. We evaluate on four problems spanning compressible and incompressible regimes: compressible Navier-Stokes, Herschel-Bulkley yield-stress flow, Hibler’s viscous-plastic sea-ice model, and discrete-element-method data with no known constitutive law. We show that the learned potentials (i) recover ground-truth physics where it is known, (ii) transfer accurately to geometries unseen during training, and (iii) succeed in regimes where unstructured methods diverge. These results suggest that embedding mathematical structure into the parameterization, rather than into the loss, is a robust path to learning physics from indirect data. An overview of machine learning-enhanced iterative methods for systems of linear and nonlinear equations 作者: Yuhuang Meng, Jing Zhao, Alexander Heinlein 类别: math.NA, cs.LG 发布日期: 2026-10-05 arXiv ID: 2610.07211v1 链接: http://arxiv.org/abs/2610.07211v1 摘要: Systems of equations arise in a wide range of scientific and engineering applications. The present work focuses on solvers for general systems of equations, including but not limited to those arising from partial differential equations. These systems can be broadly categorized into linear and nonlinear problems. For large linear systems, iterative solvers are generally preferred over direct methods due to the latter’s superlinear growth of computational costs. Although convergence theory is well-developed under certain assumptions on the coefficient matrix, many classes of systems still pose open challenges. These difficulties become even more severe for systems of nonlinear equations, where nonlinear solvers typically rely on repeated linearization. For example, Newton’s method may even converge quadratically near the solution; it can also converge slowly or diverge when the initial guess is not chosen appropriately. A wide range of solvers with diverse variants and hyperparameter settings exists, and the development of efficient and robust iterative methods remains an active area of research. Recently, machine learning (ML) techniques have been applied to enhance the efficiency of classical iterative methods while preserving their interpretability and reliability. We refer to these ML-enhanced iterative methods as hybrid iterative methods, in the sense that they combine classical iterative methods with ML. This paper provides a comprehensive overview of state-of-the-art approaches to constructing hybrid iterative methods for systems of both linear and nonlinear equations, while also discussing open challenges and outlining potential directions for future research.2. 主要定理或主要结果
3. 研究方法、关键技术和核心工具
4. 与之前工作的比较
了解领域论文（摘要翻译）
1. 一种用于多材料电子封装结构瞬态热仿真的切割有限元方法
摘要翻译
2. 通过潜在动力学网络学习具有可变初始条件的 PDE 解算子
摘要翻译
3. AdHImEx：面向大时间步长的自适应高阶隐式-显式输运
摘要翻译
4. 一种用于 Allen–Cahn 和 Cahn–Hilliard 方程的 SAV 神经网络方法
摘要翻译
5. 几何流的隐式-显式 Runge-Kutta 格式
摘要翻译
6. FOSLS-deRhaNN：面向 H(div) 与 H(curl) 的原生 de Rham 神经类及其在偏微分方程一阶系统最小二乘神经网络方法中的应用
摘要翻译
7. 使用随机单侧查询高效逼近结构化线性算子
摘要翻译
不相关论文（基本信息）
1. 大增长会发生：随机矩阵上的部分主元高斯消去法
2. 通过特殊化有效计算有理函数域上的 Moore-Penrose 逆
3. 基于体积采样的矩阵子集选择
4. 连续中值滤波格式的强收敛性
5. 三角剖分上双变量 $C^1$ 三次样条的 Strang 猜想的一个对称反例
6. 具有非交换算子的半线性问题的分裂指数积分器分析
7. 实用多层马尔可夫链蒙特卡洛的高效子采样率选择
8. 学习连续介质区域之外的广义 Navier-Stokes 模型
9. 学习型自适应多分辨率扩散成像
10. 双曲交叉三角多项式一致范数的采样离散化
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13. 基于 Monge-Growth 对的 Hellinger-Kantorovich 梯度流的神经 JKO 格式
14. 最优传输的尖锐条件性与定量稳定性
15. CGMY 缓和稳定过程下期权定价的 B 样条 Galerkin 方法
16. 非线性偏微分方程拟周期解的数值构造：I. 有界扰动
bad'' frequencies must account for the spatial index. For both the NLS and NLW equations, the linear frequencies and their differences are integers or close to integers; this allows the spatial dependence to be controlled through the modulus of the temporal index, so that the excluded measure remains under control. Second, a singular set may contain sites with the same temporal index but widely separated spatial indices, which obstructs the inversion implies localization’’ process. For the one-dimensional NLS and NLW equations, however, such sites occur only in pairs with opposite spatial indices. They can therefore be separated by small boxes, after which the global inverse is obtained from the local inverses via the resolvent identity. Numerical experiments illustrate the scheme for nonlinear traveling and standing waves, exhibiting both the computed quasi-periodic solutions and the convergence of the iteration.17. 设计即适定：利用凸神经网络势从速度数据学习本构律
18. 机器学习增强的线性与非线性方程组迭代方法综述