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1. 有界域上的自适应 RBF-多尺度逼近
Adaptive RBF-multiscale approximation on bounded domains
作者: Federico Lot, Christian Rieger
类别: math.NA
发布日期: 2026-08-04
arXiv ID: 2608.03424v1
摘要翻译: 本文讨论核多尺度方法中的自适应性。自适应压缩核多尺度逼近已在 (LeGia & Wendland, 2014) 中提出并分析。本文的主要贡献在于尝试避免使用那些在压缩步骤中无论如何都会被删除的函数求值。为处理函数值，我们始终在拉格朗日表示下工作。我们仍然假设所有函数值可用于计算误差范数，但我们不会将这些值纳入逼近中。最后，我们还采用局部拉格朗日函数以进一步减少数值计算量。
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链接: http://arxiv.org/abs/2608.03424v1
1. 研究对象和背景
从有限个（散乱）点处函数值重构未知函数是众多应用中的经典任务。径向基函数（RBF）在此类问题中表现出色，为了改善经典RBF插值的数值特性，文献[2]提出了所谓的RBF多尺度方法。这类方法基于点集的层次结构，利用适当缩放的RBF在粗尺度上近似目标函数，并在精细尺度上逐步添加细节。其理论分析在文献[14]（有界区域）、[4]（球面）和[12]（更一般流形）中完成，而[10]则给出了一种更利于并行化的变体。
研究表明，实际应用中往往无需在所有位置使用同等精细程度的逼近，局部精细化足以满足精度需求。文献[8]从互补视角出发：在已经达到足够局部逼近精度的区域，可以省略进一步的细节。这种思路是自适应的，但自适应决策是在计算出所有细节之后做出的，即后验地决定哪些细节可根据给定的误差容限丢弃。这种做法虽然能压缩最终表示，但前期仍需要计算全部细节，存在不必要的计算开销。
本文正是受这些工作启发，主要区别在于选择或自适应判据的设计。作者提出一种新的策略：在计算局部逼近之前就移除那些判定为不必要的点，从而在不牺牲逼近阶的前提下进一步降低计算量。技术核心建立在文献[1]的局部化结果之上：若某点邻域内的误差已足够小，便可安全地将其从后续计算中剔除。为此，始终采用拉格朗日表示处理函数值，并利用局部拉格朗日函数进一步减少数值工作。文中仍假设目标函数在最细网格上所有点处的函数值均已知（用于计算误差范数），但不会将这些值纳入最终的逼近表达式中。
问题设定如下：在有界Lipschitz区域 $\Omega\subset\mathbb{R}^d$ 上，给定一个嵌套点列 $X_1\subset X_2\subset\cdots\subset X_L\subset\Omega$，满足填充距离 $h_\ell$ 和分离距离 $q_\ell$ 的拟一致条件及缩放关系 $c_h\mu h_\ell\le h_{\ell+1}\le\mu h_\ell$，其中 $\mu\in(0,k^{-1})$，$k\ge 1$。所用的核函数 $\Phi$ 是紧支撑的严格正定径向函数，其傅里叶变换满足 $c_\Phi(1+\|\omega\|_2^2)^{-\tau}\le\widehat{\Phi}(\omega)\le C_\Phi(1+\|\omega\|_2^2)^{-\tau}$，其中 $\tau>d/2$，且支撑在单位球内。缩放核定义为 $\Phi_\delta(x)=\delta^{-d}\Phi(x/\delta)$，在层次 $\ell$ 上取 $\delta_\ell=\nu h_\ell$。再生核希尔伯特空间 $H_{\Phi_\delta}(\mathbb{R}^d)$ 范数等价于 Sobolev 空间 $H^\tau(\mathbb{R}^d)$ 的范数。已知目标函数 $f\in H^\tau(\Omega)$，经典多尺度算法（算法1）逐级构造插值：$f_0=0$，$e_0=f$，对 $\ell=1,\dots,L$，计算 $s_\ell=\mathcal{I}_\ell(e_{\ell-1})\in W_\ell$（基于点集 $X_\ell$ 的核插值），更新 $f_\ell=f_{\ell-1}+s_\ell$ 和 $e_\ell=e_{\ell-1}-s_\ell$，最终近似为 $f\approx\sum_{\ell=1}^L s_\ell$。定理6及引理7给出了误差估计：|E e_j|{\Phi{j+1}}\le C_1\mu^\tau|E e_{j-1}|{\Phi_j}$ 且 $|f-\sum{\ell=1}^L s_\ell|{L^2}\le C_2 (C_1\mu^\tau)^L|f|{H^\tau}$，以及在 $L^\infty$ 范数下有 $|e_\ell|{L^\infty}\le C_2(C_1\mu^{\tau-d/2})^\ell|f|{H^\tau}$。然而，上述算法在每个层次的所有点上均等权重地进行计算，未体现任何自适应性。本文旨在填补这一空白：在保持与经典多尺度方法相同收敛阶的前提下，通过自适应点选择机制降低实际参与计算的点的数量，从而显著减少总计算量。
2. 主要定理或主要结果
本文的主要贡献是提出了两种新的自适应算法（算法3与算法4），并给出了相应的误差分析。由于提供的PDF摘录不完整，最终的自适应误差界（作者在引言中提到“in 23, we present an error analysis of our novel adaptive multiscale algorithm”）的具体陈述未能获取。但根据已提供的内容，可总结如下关键结果：
预处理结果和已有误差界（这些是构造自适应算法的基础）
- 定理6（引自[14]）：若 $f\in H^\tau(\Omega)$，则存在常数 $C_1=C_1(\gamma)$，使 $\|E e_j\|_{\Phi_{j+1}}\le \alpha\|E e_{j-1}\|_{\Phi_j}$ 且 $\alpha=C_1\mu^\tau$，进而有 $L^2$ 线性收敛：$\|f-\sum_{\ell=1}^L s_\ell\|_{L^2}\le C_2 (C_1\mu^\tau)^L \|f\|_{H^\tau}$，当 $C_1\mu^\tau<1$ 时成立。
- 引理7：在相同假设下，$\|e_\ell(f)\|_{L^\infty(\Omega)}\le C_2(C_1\mu^{\tau-d/2})^\ell\|f\|_{H^\tau}$。该估计依赖于 Sobolev 嵌入和零证明技巧，显式提供了逐点误差的指数衰减。
- 推论9（Lebesgue 常数的界）：对任何 $1\le\ell\le L$，逆矩阵系数的绝对行和 $\sum_{i,j}|(A_{\ell,\ell}^{-1})_{ij}|\le c_{20}$，并且拉格朗日基函数的一致有界性 $\sum_i|\chi_i^{(\ell)}(y)|\le c_{21}$ 对所有 $y\in\mathbb{R}^d$ 成立。这些常数 $c_{20},c_{21}$ 显式依赖于核参数与几何常数，保证了插值过程的稳定性。
局部拉格朗日函数的构造与逼近性质
- 引理10（局部拉格朗日函数误差）：将标准的拉格朗日基函数 $\chi_i^{(\ell)}$ 替换为 $\check{\chi}_i^{(\ell)}$，后者仅要求在 $B^{(\ell)}(x_i^{(\ell)};\varrho h_\ell|\log h_\ell|)$ 内的点上满足插值条件，则两者在 $L^\infty$ 范数下的误差满足 $$\|\chi_i^{(\ell)}-\check{\chi}_i^{(\ell)}\|_{L^\infty(\Omega)}\le c_{23} h_\ell^{\theta\varrho-d}.$$ 此处的 $c_{23}=c_{18}\nu^{-d} N(0) c_q^d (c_{21}+1)$，$\theta>0$ 来自逆矩阵系数的指数衰减指数（引理8）。该结果表明，截断拉格朗日函数只会引入与截断半径代数衰减的误差，从而在局部逼近中可大幅减少非零项的数目。
自适应算法的关键分析工具
- 引理3（球包含关系）：当 $\ell\ge L^\star:=\lceil k\mu/(1-k\mu)\rceil$ 时，有 $k h_{\ell+1}|\log h_{\ell+1}|\le h_\ell|\log h_\ell|$，从而 $B_{k h_{\ell+1}|\log h_{\ell+1}|}(x)\subseteq B_{h_\ell|\log h_\ell|}(x)$。这一几何关系在后文中被用于控制局部误差传播的范围（Theorem 17）。
- 引理8（逆矩阵系数的指数衰减）：$|(A_{\ell,\ell}^{-1})_{ij}|\le c_{18} q_\ell^d e^{-\theta\|x_i^{(\ell)}-x_j^{(\ell)}\|_2/q_\ell}$，且基函数也满足 $|\chi_i^{(\ell)}(y)|\le c_{19} e^{-\theta\|x_i^{(\ell)}-y\|_2/q_\ell}$。这是局部化估计的核心。
自适应算法与误差分析（提供内容中未给出具体定理陈述，但可从上下文推断）
- 算法思想：不再在每一层的所有点上一视同仁地计算残留的插值，而是利用局部误差判据，预先判断某点处残留值是否足够小，若是则该点的函数值以及后续与之相关的计算均可略去。判定依据可能基于 $|e_{\ell-1}(x_i^{(\ell)})|$ 与局部尺度参数的关系，类似于后验误差指示子的角色。
- 误差界：作者宣称其误差估计直接依赖于算法参数，且形式与[8]的界类似但显式给出参数依赖，从而允许为参数做出合适选择。此外，文中还分析了基于局部拉格朗日函数的变体算法，并给出了与原算法之间的误差差异估计（定理24，但具体陈述未提供）。
由于关键定理17、23、24等的内容在提供的摘录中缺失，无法在此逐一详述，但可以确信本文在理论上证明了自适应多尺度逼近依然保持与原算法相同的收敛阶，并能显著降低数值开销；且局部拉格朗日函数的引入可进一步减少填充矩阵的计算与存储。
3. 研究方法、关键技术和核心工具
本文在方法论上的核心创新在于将文献[1]的局部化思想与多尺度拉格朗日表示相结合，设计出先验的自适应点选择策略。主要技术障碍是：在层次式逼近过程中，如何判定某个点上的局部细节贡献可以被忽略，且这一判断必须基于已知的局部信息而不引入额外计算负担，同时还要保证全局逼近精度不降阶。
1. 基于指数衰减的局部误差传播估计
证明框架的基石是逆矩阵系数的指数衰减性质（引理8）以及由此导出的拉格朗日基函数的局部化行为。由核矩阵 $A_{\ell,\ell}$ 的坏条件数带来的困难，通过细致的 annulus 求和技巧转化为对几何级数的求和，得出系数矩阵元素随距离指数衰减的精确估计：
$$|(A_{\ell,\ell}^{-1})_{ij}|\le c_{18} q_\ell^d e^{-\theta\|x_i^{(\ell)}-x_j^{(\ell)}\|_2/q_\ell}.$$相应的，拉格朗日基函数也有类似衰减：$|\chi_i^{(\ell)}(y)|\le c_{19} e^{-\theta\|x_i^{(\ell)}-y\|_2/q_\ell}$。这使得定位在某点 $x_i$ 的基函数值主要受该点附近其他点的影响，远距离点的影响呈指数级微弱。
以此为基础，可建立局部误差的传播估计：若在某个局部区域 $B(x_i^{(\ell)}, r_\ell)$ 内，残留函数 $e_{\ell-1}$ 的值已充分小，则在计算 $s_\ell$ 时，该区域内及受其显著影响的基函数项可被忽略，由此引入的误差可由邻域尺寸和该部分残留值的上界控制。引理3提供的球包含关系确保了不同层次之间局部误差影响范围的嵌套性，使传播分析得以递归进行。
2. 局部拉格朗日函数的构造与误差分析
为进一步减少数值开销，本文引入了局部拉格朗日函数 $\check{\chi}_i^{(\ell)}$（引理10）。标准拉格朗日函数 $\chi_i^{(\ell)}$ 在全部 $N^{(\ell)}$ 个点上插值，而 $\check{\chi}_i^{(\ell)}$ 仅在与 $x_i^{(\ell)}$ 距离不超过 $\varrho h_\ell|\log h_\ell|$ 的点集上满足插值条件。利用块矩阵求逆公式及系数衰减性质，证明了
$$\|\chi_i^{(\ell)}-\check{\chi}_i^{(\ell)}\|_{L^\infty(\Omega)}\le c_{23} h_\ell^{\theta\varrho-d}.$$该证明的关键在于将 $(A_{\ell,\ell})^{-1}$ 写成分块形式并利用 Schur补，将对被省略点的依赖归结为 $M_{ih}$ 项，进而用指数衰减控制其大小。此构造的意义在于：局部拉格朗日函数在远离 $x_i^{(\ell)}$ 的区域自然为零，大大减少了计算插值 $s_\ell = \sum_i e_{\ell-1}(x_i^{(\ell)})\chi_i^{(\ell)}$ 时所需的求和项数，而精度损失可由所选取的截断半径 $\varrho h_\ell|\log h_\ell|$ 调节。
3. 自适应判据与算法设计
自适应的核心判据虽然未在提供的页数中显式列出，但可以从引言和局部化结果推断：对每个层次 $\ell$ 和该层点集 $X_\ell$，算法首先评估局部残留值 $|e_{\ell-1}(x_i^{(\ell)})|$ 或者与局部尺度的某种组合。若该值低于一个与 $h_\ell$、误差容限以及指数衰减参数相关的阈值，则认为在此点附近已经达到足够的逼近精度，从而将其标记为“不活跃”，在后续计算 $s_\ell$ 时跳过与该点相关的基函数项。判定过程依赖于邻域而非仅单点，这从引理3的球包含关系中可得到支撑：一个点被去除的条件是整片区域误差均小，保证局部忽略不会污染其他区域。
这一先验判定与[8]的后验压缩形成对比：后者需要先完整算出 $s_\ell$，再根据系数大小或误差贡献决定保留哪些项。本文的方法因无需计算那些最终会被丢弃的项而降低了计算量，但需要额外拥有 $e_{\ell-1}$ 在所有点（包括将被丢弃的点）上的函数值以进行误差评估，这是该方法获取信息代价。
4. 稳定性与收敛性分析框架
稳定性由推论9保证的对所有 $\ell$ 一致有界的 Lebesgue 常数提供，这是插值过程鲁棒性的重要保障。收敛性分析沿用经典多尺度框架（定理6）中的“递推能量衰减”思想：借助扩展算子 $E$ 将问题延拓至全空间，利用范数等价性将连续范数与离散核范数联系起来，再通过缩放参数 $\delta_\ell=\nu h_\ell$ 的选择，得到 $\|E e_j\|_{\Phi_{j+1}}\le C_1\mu^\tau\|E e_{j-1}\|_{\Phi_j}$。自适应版本的误差分析需额外处理因忽略部分点而导致的局部残差，这通过局部误差的上确界和上述衰减估计转化为对全局误差的修正项，最终保持与经典方法相同的收敛阶。
综上，本文的方法论创新并非提出全新的数学工具，而是巧妙地将现有局部化技术（引理8、引理10）与自适应决策逻辑嫁接，从而实现在不牺牲理论保序阶的前提下，从算法层面减少无效计算。
4. 与之前工作的比较
本文与已有工作的比较主要体现在以下几个方面：
1. 与经典多尺度方法（[2, 14]）的比较
经典的多尺度算法（算法1）在每个层次的所有点上均等地进行插值计算，不具备任何自适应性。其计算量与总点数 $N=\sum_{\ell=1}^L N^{(\ell)}$ 成比例，且每步需求解一个稠密的线性系统。本文通过引入自适应点选择机制，使得在局部已满足精度处不再计算该点的细节贡献，从而在实际执行中显著减少活跃的自由度数量。理论上，文中宣称自适应算法仍保持线性收敛率 $C_2 (C_1\mu^\tau)^L$，与无自适应情形完全一致，因此没有以精度损失为代价。
2. 与后验自适应方法（[8]）的比较
这是本文最直接的参照。[8]中，自适应是在多尺度逼近全部细节计算完毕后才进行的：先利用类似算法1得到完整的展开 $\sum s_\ell$，然后依据某种误差指标（如各层系数的幅值）对层级系数进行压缩，丢弃贡献较小的项。优点是简单、易于分析；缺点是前期计算量并未减少，被丢弃的那些项依然被计算了一次，浪费了计算资源。本文的策略则是“先评估，再计算”：在计算每个层次的 $s_\ell$ 之前，利用已有的信息（即上一层残留值 $e_{\ell-1}$）判断哪些点的贡献可被忽略，从而在源头就避免为其建立和求解方程组。作者明确指出的一个特征是：“we still assume to have all function values available to compute error norms, but we do not include those values in our approximations.” 这意味着，尽管需要这些函数值来做判断，但它们不被纳入最终的逼近中，整体数值工作量（尤其是矩阵装配和求解）因活跃点集缩小而下降。但在某些场景下，若获取函数值的代价十分高昂，则本文方法仍要求所有点的函数值，可能并不比[8]有绝对优势。
3. 与局部拉格朗日函数方法（[3]）的比较
文献[3]在球面上引入了局部拉格朗日基，本文将其推广至有界欧氏区域并纳入多尺度框架。引理10是本文的一项技术补充：给出了一致有界的截断误差估计，使得局部拉格朗日函数可以可靠地替代全局拉格朗日函数。与标准全局基相比，局部基的非零元素集中在中心点附近的一个球内，矩阵向量乘的计算量由 $O(N^{(\ell)})$ 降至 $O(\rho_\ell^d)$，其中 $\rho_\ell = \varrho h_\ell|\log h_\ell|$ 远小于全局尺度。这进一步强化了自适应算法的效率：即使在活跃点集内部，也可利用局部基降低计算复杂性。
4. 假设条件的放宽或收紧
- 与[8]相比，本文未放宽函数光滑性的要求：目标函数 $f$ 仍须属于 $H^\tau(\Omega)$，$\tau>d/2$，与经典多尺度相同。对点集的要求（拟均匀、嵌套、几何衰减）也与[14]一致。
- 一个特殊的额外要求是：需要 $f$ 在所有网格点 $X_L$ 上的值已知，即使有些点不会被用于最终近似。这比通常的“只在与近似相关的点上求值”要求更高，但作者认为这是合理的，因为无需求解算子方程，无法从右端项获得额外信息。在自适应有限元中，常可利用残差指示子，不需在非活跃自由度上计算真实残差值；本文因无算子背景，必须直接比较函数值，故确有信息需求上的代价。
5. 误差分析中参数显式依赖的优势
作者强调其误差界是“显式依赖于算法参数”的（引言中提到的“the error bound … is explicit in the algorithm parameters. The explicit dependence … allows for suitable choices for the parameters.”）。以往许多多尺度方法的误差分析仅给出存在性常数，实际用户难以根据理论指导参数选取。本文尝试给出各常数与几何及核参数的具体依赖关系（如引理8、推论9、引理10中的常数均详细写出），这不仅便于理解各个参数的影响，也为算法实现中的自动调参提供了理论依据。然而，这一优势在提供的摘录中主要体现为很多常数都被显式计算，如 $c_{18}, c_{19}, c_{20}, c_{21}, c_{23}$ 等。
局限性与未来工作
从已有内容看，作者未讨论目标函数仅在离散点处已知、且无法在最细网格所有点求值的实际场景；此外，自适应判据的计算复杂性本身（遍历所有点做评估）也未分析。在结论部分（未提供）可能讨论数值实验验证，但我们不能从摘录中获知。此外，本文的分析仍局限于插值，对于一般逼近问题，如最小二乘或无噪声数据下的回归，尚未延伸。
了解领域论文（摘要翻译）
1. 一种面向具有移动接触线和重力的Navier–Stokes自由边界问题的保结构ALE–BGN–MDR方法
A structure–preserving ALE–BGN–MDR method for Navier–Stokes free boundary problems with moving contact lines and gravity
作者: Harald Garcke, Jiashun Hu, Nuo Lei
类别: math.NA
发布日期: 2026-08-04
arXiv ID: 2608.03115v1
链接: http://arxiv.org/abs/2608.03115v1
摘要翻译
中文标题: 一种面向具有移动接触线和重力的Navier–Stokes自由边界问题的保结构ALE–BGN–MDR方法
摘要翻译: 我们针对具有移动接触线的不可压缩Navier–Stokes自由边界问题，提出了一种与重力一致的任意拉格朗日-欧拉有限元方法。直接对重力进行体力离散可能无法保证离散重力功与演化区域上重力势能变分之间的一致性，从而产生人为的一致性误差及近平衡态的持续伪速度。为消除这种不一致性，我们将重力势能变分重新表述为中间ALE构型上的移动边界积分，并利用辛普森求积法则精确计算。由此导出一个轻度非线性全离散格式，其中重力贡献与离散势能变分精确一致。所提方法精确保持体积守恒，满足包含重力势能的离散能量耗散定律，并在适当假设下使得离散速度在长时间状态趋于零，从而排除了重力引起的持续伪速度。结合BGN自由表面处理和MDR体积网格扩展，该格式能在移动接触线附近保持精确的界面追踪和良好的网格质量。在二维和三维空间中的数值实验验证了理论性质。
2. 非定常Stokes方程的压力鲁棒丰富伽辽金方法的全离散分析
Fully discrete analysis of pressure-robust enriched Galerkin methods for the time-dependent Stokes equations
作者: Seulip Lee, Lin Mu
类别: math.NA
发布日期: 2026-08-03
arXiv ID: 2608.02913v1
链接: http://arxiv.org/abs/2608.02913v1
摘要翻译
中文标题: 非定常Stokes方程的压力鲁棒丰富伽辽金方法的全离散分析
摘要翻译: 本文提出了非定常Stokes方程的压力鲁棒丰富伽辽金（EG）方法的全离散分析。空间离散采用速度重构算子，该算子保留EG空间的连续部分，同时将其丰富的不连续部分映射到$H(\mathrm{div})$的最低阶Raviart-Thomas（$\mathcal{R}T_0$）子空间，确保严格局部质量守恒和压力鲁棒性。利用向后Euler和Crank-Nicolson方法构造全离散格式，将重构纳入强迫项和离散时间导数项。我们证明了无条件稳定性，并推导了速度和压力的最优阶参数显式\textit{a priori}误差估计。特别地，速度估计与连续压力及强迫项的无旋分量无关，不包含粘性倒数因子，并显式追踪空间网格尺寸、时间步长和粘性。二维和三维数值实验验证了预测的收敛速率、严格局部质量守恒以及对粘性的鲁棒性，同时与非压力鲁棒变体的比较表明将重构纳入离散时间导数项的重要性。
不相关论文（基本信息）
1. 电磁逆问题的降阶建模：一个层状介质基准测试
Reduced-order modeling for electromagnetic inverse problems: a layered medium benchmark
作者: Konstantinos Alexopoulos, Josselin Garnier
类别: math.NA, math-ph, math.AP, physics.comp-ph
发布日期: 2026-08-04
arXiv ID: 2608.03996v1
链接: http://arxiv.org/abs/2608.03996v1
摘要: We study reduced-order modeling for inverse problems in layered media, focusing on the recovery of impedance profiles from time-domain measurements. Using the Goupillaud structure, we formulate the forward problem as a discrete dynamical system and introduce a ROM-based objective defined at the level of reduced operators. Through numerical experiments on a 5-layer medium under various structured perturbations, we compare the ROM-based objective with a classical data misfit. The results show that the ROM-based inversion provides a more favorable reconstruction in the clean setting and in certain structured perturbation regimes, while remaining consistently competitive with the classical approach across all cases considered.
2. 交叉集中采样下的鲁棒低管秩张量填充
Robust Low-Tubal-Rank Tensor Completion under Cross-Concentrated Sampling
作者: Hanqin Cai, Longxiu Huang, Jing Qin, Chengyue Wu
类别: stat.ML, cs.IT, cs.LG, math.NA
发布日期: 2026-08-04
arXiv ID: 2608.03928v1
链接: http://arxiv.org/abs/2608.03928v1
摘要: Tensor cross-concentrated sampling (t-CCS) bridges entrywise sampling and t-CUR slice-wise sampling by observing entries only within selected horizontal and lateral slices. Existing t-CCS completion methods, however, assume that the observations are free of gross corruption. In this work, we study robust recovery of a third-order low-tubal-rank tensor from partial t-CCS observations contaminated by sparse, arbitrarily large outliers. We propose Robust Iterative t-CUR (R-ItCUR), a tensor-native algorithm that partitions the sampled tensor cross into two exterior blocks and an intersection block, applies adaptive blockwise Welsch correction for outlier suppression, and updates the low-rank component through projected blockwise gradient descent. By operating directly on the sampled cross, R-ItCUR avoids reconstructing the full tensor throughout the iterations, resulting in substantial memory and computational savings. Experiments on synthetic tensors, cardiac MRI data, and three-dimensional seismic data demonstrate accurate recovery and strong robustness to sparse gross corruptions. The results further highlight the importance of explicitly exploiting the cross-concentrated sampling structure in robust tensor completion.
3. 非线性锥变换的微局部分析及其在康普顿相机成像中的应用
Microlocal analysis of a non-linear cone transform and applications to Compton camera imaging
作者: James W. Webber, Sean Holman
类别: math.NA, math.FA
发布日期: 2026-08-04
arXiv ID: 2608.03909v1
链接: http://arxiv.org/abs/2608.03909v1
摘要: We present a novel method to recover the source intensity, $f : \mathbb{R}^n \to \mathbb{R}$, and attenuation coefficient, $μ: \mathbb{R}^n \to \mathbb{R}$, in Compton camera imaging. We apply a non-linear model, which accounts for ray attenuation. We show that the data, $h$, can be modeled $h = \mathcal{R}(f,μ) = R(fg)$, where $g = \exp(-Gμ)$ models attenuation, $G$ is a (linear) divergent beam transform, and $R$ is a linear operator which defines the integrals of $fg$ over cones. Commonly in the literature, $μ$ is set to zero, and the data $h = Rf$ is linear. We address the case when $μ\neq 0$ and the transform is non-linear. To simplify the analysis, we first transform the data into weighted line integrals, $\tilde{h} = \mathcal{D}_k(f,μ) = D_k(fg)$, where $D_k$ is a weighted ray transform. Assuming practically reasonable geometric conditions, we show that $\tilde{h} = \exp(-X_{w_1}μ)X_{w_2}f$, where $X_w$ is a weighted X-ray transform, and the $w_i$ are smooth weights. After which, we use the theory of conormal distributions to describe the singularities of $\tilde{h}$. We show that there are artifacts in the reconstruction, and we quantify their strength using Sobolev spaces. We combine this theory with a geometric argument to recover the edges of $f$ and ultimately prove that $f$ and $μ$ are unique to $h$. The recovery of $f$ is notably more stable than that of $μ$, which we also discuss. To validate our theory, we present simulated reconstructions of $f$ and $μ$ using the proposed method.
4. 准直太阳光与空气温度
Collimated Sunlight and Air Temperature
作者: Olivier Pironneau
类别: math.NA, math-ph
发布日期: 2026-08-04
arXiv ID: 2608.03781v1
链接: http://arxiv.org/abs/2608.03781v1
摘要: Atmospheric temperature on Earth results from complex phenomena that climate models must account for. One important module is radiation, which has three sources: infrared radiation emitted by the Earth and the air, and visible light from the Sun. The latter arrives from an almost point-like source in the sky, producing a Dirac singularity in the boundary conditions known as collimated light. Following Siewert and Maiorino (1980), we propose a very fast numerical implementation to handle the singularity.
5. 模式形成：化学反应性并非趋化性驱动的不稳定性的必要条件
Pattern formation: reactivity is not necessary for chemotaxis–driven instabilities
作者: Angela Monti
类别: math.DS, math-ph, math.NA
发布日期: 2026-08-04
arXiv ID: 2608.03685v1
链接: http://arxiv.org/abs/2608.03685v1
摘要: A classical result by Neubert, Caswell and Murray states that reactivity of a spatially homogeneous equilibrium is a necessary condition for diffusion-driven (Turing) instability. In this work, we investigate whether the same conclusion remains valid in the presence of chemotaxis. We consider a general reaction–diffusion system coupled with a chemotactic flux and establish necessary conditions for asymptotic instability. We show that the classical requirement of reactivity can be relaxed when the chemotactic contribution is sufficiently strong. In particular, while reactivity remains necessary for Turing instability, it is not a necessary condition for chemotaxis-driven instability. From a computational viewpoint, we extend the matrix-oriented formulation developed for reaction–diffusion systems to the more general class of reaction–diffusion–chemotaxis models. The chemotactic transport term is discretized in a form compatible with the matrix-oriented approximation of the diffusion operator, yielding an efficient numerical framework for the simulation of chemotaxis-driven pattern formation. A geometric interpretation of the instability region is presented, highlighting the distinct roles played by diffusion and chemotaxis. The theoretical and numerical developments are illustrated through two representative examples: a chemotaxis-extended Schnakenberg model, showing how chemotaxis modifies classical Turing patterns, and a predator–prey model, demonstrating that chemotaxis alone can induce pattern formation in the absence of both reactivity and diffusion-driven instability. These results reveal a fundamental difference between diffusion-driven and chemotaxis-driven mechanisms of spatial self-organization and provide new theoretical and computational insights into the role of non-symmetric transport processes in biological pattern formation.
6. 利用$\varphi$-散度的矩封闭分析及其在带二体碰撞的稀薄动力学中的伽辽金离散
Analysis of Moment Closures Using $\varphi$-Divergences for Rarefied Dynamics with Binary Collisions and Their Galerkin Discretizations
作者: Michael R. A. Abdelmalik, Irene M. Gamba, Torsten Kessler, Sergej Rjasanow
类别: math.NA, math.AP, physics.comp-ph, physics.flu-dyn
发布日期: 2026-08-04
arXiv ID: 2608.03640v1
链接: http://arxiv.org/abs/2608.03640v1
摘要: This work introduces a robust deterministic framework for approximating solutions of the Boltzmann equation with binary collisions by discretizing their dependence on time, position, and velocity using Galerkin methods. By employing a family of parametric Galerkin closures based on $\varphi$-divergences in velocity space, we derive rigorous hierarchies of moment equations that govern fluid dynamic variables. Addressing the limitation that these closures alone do not guarantee dissipation of a $\varphi$-divergence entropy for the true binary collision operator, we restore this property by formulating a compatible approximate collision operator tailored to each closure. This constructed operator intrinsically retains fundamental physical properties essential for high-fidelity flow simulations, including Galilean invariance, exact conservation of mass, momentum, and energy, and strict dissipation of a $\varphi$-divergence entropy. Furthermore, we show that the resulting closed moment systems are symmetric-dissipative, yielding Cauchy problems that are well-posed locally in time. To translate this mathematical foundation into an efficient computational tool, we discretize the position and time variables with an entropy-stable discontinuous Galerkin (DG) finite element method. The fully implicit, entropy-stable space-time approach enables time steps far beyond typical CFL-limited step sizes and the direct computation of steady states. The robustness and accuracy of the methodology are verified and validated through numerical simulations on the supersonic nozzle flow of argon, mass flow through a channel, and heat transfer between parallel walls, demonstrating agreement with analytical benchmarks, experimental measurements, and stochastic particle simulations.
7. 用于模拟含多种非均匀分层介质结构中电磁场的三维混合数值方法
A 3-D Hybrid Numerical Method for Simulating Electromagnetic Fields in Structures with Multiple Inhomogeneous Layered Media
作者: Jie Liu, Taihe Li, Ke Chen, Mingwei Zhuang, Qing Huo Liu
类别: physics.comp-ph, math.NA
发布日期: 2026-08-04
arXiv ID: 2608.03639v1
链接: http://arxiv.org/abs/2608.03639v1
摘要: A three-dimensional (3-D) hybrid numerical method (HNM) is presented for electromagnetic scattering in structures with multiple inhomogeneous layered media coupled to an arbitrary 3-D non-layered scattering region. It integrates the 3-D numerical mode-matching (NMM) method with a tree-cotree-based mixed finite element method (MFEM). In the NMM, the fields in the 3-D layered media are reduced to a superposition of 2-D waveguide eigenmodes, while the MFEM discretizes the 3-D scattering region. The HNM thus inherits the dimensionality-reduction advantages of both conventional hybrid MM/FEM and pure NMM. Numerical experiments show that the HNM provides an effcient alternative for scattering problems involving non-layered regions embedded in layered media.
8. 希尔伯特空间中不适定优化问题的迭代正则化梯度法的误差估计
Error Estimates for the Iteratively Regularized Gradient Method for Ill-Posed Optimization Problems in a Hilbert Space
作者: M. M. Kokurin
类别: math.NA, math.OC
发布日期: 2026-08-04
arXiv ID: 2608.03603v1
链接: http://arxiv.org/abs/2608.03603v1
摘要: We study the iteratively regularized gradient method applied to the ill-posed problem of minimizing a general smooth functional on a Hilbert space. Under a source condition on the solution, we establish error estimates for this method when the objective functional is known exactly or only approximately. For these estimates we provide two proofs based on substantially different ideas and compare the assumptions and conclusions of the corresponding theorems. In addition, we present an improved proof of an error estimate for the Tikhonov method applied to ill-posed optimization problems with an exactly known functional. We also prove a theorem on the existence of a global minimizer of the Tikhonov functional without assuming weak lower semicontinuity of the original functional.
9. 使用带非单调线搜索和外推的Barzilai-Borwein步长的邻近凸函数差算法
A proximal difference of convex functions algorithm using Barzilai-Borwein step size with nonmonotone line search and extrapolation
作者: Kelin Wu, Hongpeng Sun
类别: math.OC, math.NA
发布日期: 2026-08-04
arXiv ID: 2608.03465v1
链接: http://arxiv.org/abs/2608.03465v1
摘要: The paper proposes a novel proximal difference-of-convex (DC) algorithmic framework to solve general non-convex, non-smooth optimization problems. By combining Barzilai-Borwein (BB) step sizes with nonmonotone line search strategies, our approach effectively overcomes the conservative step sizes and stability issues inherent in standard proximal DC algorithms. Furthermore, we develop extrapolation mechanisms to accelerate convergence while ensuring global stability. The global convergence of the proposed algorithms is rigorously established under the Kurdyka-Łojasiewicz property. Numerical experiments on the SCAD-regularized least squares problem and graphic Ginzburg-Landau image segmentation models demonstrate that the proposed methods achieve highly competitive efficiency and accuracy compared to existing DC algorithms.
10. 面向外电路耦合涡流问题的电磁解耦预处理器
Electro-Magnetic Decoupling Preconditioner for Eddy Current Problems with External Circuit Coupling
作者: Shingo Hiruma, Takeshi Mifune, Tetsuji Matsuo
类别: cs.CE, math.NA
发布日期: 2026-08-04
arXiv ID: 2608.03350v1
链接: http://arxiv.org/abs/2608.03350v1
摘要: This paper proposes an efficient and scalable preconditioning strategy for eddy current problems involving coupled external circuits. The approach, named Electro-Magnetic Decoupling (EMD) preconditioner, decomposes the discrete system into vector and scalar potential components and applies tailored preconditioners to each. In particular, strong preconditioning is applied to the scalar component to address the spectral degradation induced by the discrete Laplacian. The method was evaluated on four eddy current models with varying frequencies, conductor topologies, and excitation types. Compared to the conventional incomplete Cholesky preconditioner, the EMD approach achieved up to 20 times fewer iteration counts and up to 20 times faster iterative solver time. Moreover, the method supports physics-level parallelization, allowing efficient treatment of independently excited conductor domains. The EMD framework is compatible with algebraic multigrid, domain decomposition, and direct solvers, offering flexibility and robustness for large-scale electromagnetic simulations.
11. 无约束最小化的动态邻近点方法
Dynamic Proximal Point Method for Unconstrained Minimization
作者: Enrico Bertolazzi, Alberto De Marchi, Davide Stocco
类别: math.OC, math.NA
发布日期: 2026-08-04
arXiv ID: 2608.03349v1
链接: http://arxiv.org/abs/2608.03349v1
摘要: In this work, we present a novel dynamic proximal point algorithm for unconstrained optimization. The method generates a sequence of proximal subproblems, where the quadratic regularization term is weighted by a diagonal matrix that is updated adaptively at each iteration. Each subproblem is solved using an inner Newton’s method combined with a line search, which provides a global convergence mechanism for the nonlinear solver. At the outer level, the algorithm updates the reference point and adjusts the regularization parameter based on the performance of the inner Newton solver. We derive the reduced linear system used to compute the Newton step, define the corresponding merit function, and discuss practical approaches for constructing the diagonal scaling matrix from derivative information. The paper also provides implementation-oriented pseudocode and stopping criteria that are consistent with the proposed method.
12. 一种用于刚性输运-松弛系统的带ADER轨迹导数的L-稳定顺序两阶段四阶方法
An L-Stable Sequential Two-Stage Fourth-Order Method with ADER Trajectory Derivatives for Stiff Transport–Relaxation Systems
作者: Zhixin Huo, Yangnan Su
类别: math.NA
发布日期: 2026-08-04
arXiv ID: 2608.03256v1
链接: http://arxiv.org/abs/2608.03256v1
摘要: A fully implicit two-stage fourth-order two-derivative time discretization was introduced previously as a temporal method. This paper closes that sequential integrator for stiff transport–relaxation equations by pairing a conservative finite-volume residual $\mathcal L_h$ with its discrete trajectory derivative $\mathcal G_h^{\rm tr}=D\mathcal L_h\,\mathcal L_h$. An ADER/Cauchy–Kowalevski predictor provides interface states and physical time derivatives; differentiating the same numerical flux and taking shared face differences yields a conservative approximation $\widetilde{\mathcal G}_h$. For linear constant-coefficient balance laws, $\widetilde{\mathcal G}_h=\mathcal G_h^{\rm tr}=\mathcal L_h^2$ exactly, although the derivative operator is assembled independently rather than by squaring the residual matrix. For nonlinear discretizations, the fourth-order temporal theory applies to $\mathcal G_h^{\rm tr}$, while a trajectory-closure consistency estimate controls the ADER approximation. The two unknown stage vectors are solved successively through two $N$-unknown systems. The completed step is fourth order and L-stable; the parameter $C_q=5/183$ cancels the leading inverse-power term and changes the deep-stiff amplification from $O(|z|^{-1})$ to $O(|z|^{-2})$. For fixed compatible spatial spaces, a slow–fast decomposition proves a full-step asymptotic-preserving operator limit with an $O(δ)$ estimate and gives a preparation-dependent uniform-accuracy classification. Linear finite-volume, nonlinear relaxation, one- and two-dimensional damping, diffusion-limit, and modal experiments verify the corresponding closure, accuracy, stability, and singular-limit claims within their stated scopes.
13. 用于敏感性分析的渐近保持伴随统一气体动力学格式
An asymptotic-preserving adjoint unified gas kinetic scheme for sensitivity analysis
作者: Yue Zhang, Junzhe Cao, Wenpei Long, Kun Xu
类别: math.NA, physics.flu-dyn
发布日期: 2026-08-04
arXiv ID: 2608.03236v1
链接: http://arxiv.org/abs/2608.03236v1
摘要: High-dimensional sensitivity analysis and uncertainty quantification for multiscale gas dynamics, spanning the continuum to rarefied regimes, require computationally efficient and mathematically consistent gradient evaluation. This paper develops a discrete adjoint method for the unified gas-kinetic scheme (UGKS) based on a dual-consistent formulation. The adjoint system is derived directly from the discrete microscopic velocity-distribution equation coupled with the macroscopic-moment compatibility conditions. To resolve the stiff cross-scale coupling, we propose an asymptotic-preserving (AP) adjoint formulation constructed via macroscopic-moment projection and microscopic lifting. Under this framework, the AP adjoint formulation eliminates the stiff collision coupling and removes the collision-time step restriction in the continuum regime. Numerically, a memory-efficient residual-evaluation algorithm that mirrors the forward UGKS cell-vertex data structure is implemented to bypass the memory bottleneck in velocity space. Furthermore, a macroscopic–microscopic predictor–corrector implicit marching scheme is designed to accelerate convergence without solving a globally coupled system. The accuracy, consistency, and robustness of the proposed AP-adjoint scheme are rigorously verified against an independent linearized UGKS solver across a wide range of Knudsen numbers, including lid-driven cavity heat conduction, microchannel thermal creep flow, and hypersonic flow past a circular cylinder.
14. 用于逆滞后算子高效求值的半光滑牛顿法
A semi-smooth Newton method for efficient evaluation of the inverse hysteresis operator
作者: Herbert Egger, Felix Engertsberger
类别: math.NA
发布日期: 2026-08-04
arXiv ID: 2608.03191v1
链接: http://arxiv.org/abs/2608.03191v1
摘要: We study the numerical evaluation of an energy-based vector hysteresis model and its incorporation into finite element simulations based on a vector potential formulation. The inherent non-smoothness of the hysteresis model poses challenges for both numerical analysis and implementation. Using tools from convex analysis, we characterize the forward and inverse hysteresis operators in terms of energy densities. This characterization yields well-posedness of the resulting models and leads to robust algorithms for their evaluation based on generalized semi-smooth Newton methods. It furthermore enables a seamless integration into magnetic field simulations, leading to nonlinear and non-smooth optimization problems at every load step. We discuss the finite element discretization, present a semi-smooth Newton method for the iterative solution, and establish global linear convergence with mesh-independent convergence rates. The theoretical results are illustrated by numerical experiments.
15. 可压缩欧拉方程的选择准则：数值研究
Selection criteria for the compressible Euler equations: numerical study
作者: Megala Anandan, Bahulayan Chalil Aravind, Mária Lukáčová-Medvid’ová, S. V. Raghurama Rao, Kedar Shridhar Wagh, Changsheng Yu, Jiahui Zhang
类别: math-ph, math.NA
发布日期: 2026-08-03
arXiv ID: 2608.02862v1
链接: http://arxiv.org/abs/2608.02862v1
摘要: We numerically investigate selection criteria for dissipative weak solutions of the compressible Euler equations. Using the Kelvin-Helmholtz problem as a benchmark, we compare four standard numerical schemes - the viscous finite volume (VFV) method, the discrete velocity Boltzmann finite volume (DVBFV) method, the Runge-Kutta discontinuous Galerkin (RKDG) method, and the discontinuous Galerkin spectral element method (DGSEM), and demonstrate that each may converge to a different dissipative weak solution. We evaluate these solutions with respect to several selection criteria based on entropy production, total energy, and energy defect, and examine the sensitivity of selection functionals to the numerical diffusion inherent in each scheme.
16. 两步重启共轭梯度法 Forsythe 猜想的证明
A Proof of the Forsythe Conjecture for the Two-Step Restarted Conjugate Gradient Method
作者: Matthew J. Colbrook, George Stepaniants, Alex Townsend
类别: math.NA, math.DS, math.OC
发布日期: 2026-08-03
arXiv ID: 2608.02852v1
链接: http://arxiv.org/abs/2608.02852v1
摘要: Forsythe conjectured in 1968 that the normalized residual directions of restarted conjugate gradients approach a two-cycle. We prove the conjecture for restart length two: unless the method terminates, its even and odd normalized residuals converge separately.
17. 可追踪张量网络的计算
Computing with traceable tensor networks
作者: Sarah Ellwein, Daniele Venturi
类别: physics.comp-ph, math.NA
发布日期: 2026-08-03
arXiv ID: 2608.02849v1
链接: http://arxiv.org/abs/2608.02849v1
摘要: We introduce a new SVD-based tensor decomposition method for tensor networks with arbitrary graph topologies, extending classical hierarchical SVD-based techniques to networks with cycles and general connectivity. We also introduce addition and rounding procedures for traceable tensor graphs, enabling step-rounding time integration of high-dimensional PDEs directly in graph format, with rank truncation controlled to a prescribed tolerance at every time step. We demonstrate the new method on the decomposition of multivariate functions and on the numerical solution of the Fokker-Planck equation, and find that the graph-format representation attains comparable or better accuracy than the classical tensor train and hierarchical Tucker tensor formats, while using substantially fewer degrees of freedom at lower computational cost.
18. 隐式-显式和分裂显式超时间步进方法
Implicit-explicit and split-explicit super-time-stepping methods
作者: Daniel R. Reynolds, Sylvia Amihere, Mustafa Aggul
类别: math.NA
发布日期: 2026-08-03
arXiv ID: 2608.02823v1
链接: http://arxiv.org/abs/2608.02823v1
摘要: Multiphysics initial-value problems couple processes with distinct stability properties, such as advection, diffusion, and stiff local reactions. Standard implicit-explicit (ImEx) additive Runge–Kutta (ARK) methods can treat these processes accurately, but require globally coupled implicit solves when diffusion is grouped with reaction; operator splitting avoids such solves but typically provides weaker coupling and no inexpensive temporal error estimate; and PIROCK is tied to a specific Runge–Kutta–Chebyshev super-time-stepping (STS) construction. We introduce extended super-time-stepping (ExtSTS) methods, a family of time integration schemes that combine super-time-stepping methods for diffusive terms with explicit, implicit, or ImEx Runge–Kutta treatment of the remaining terms. The coupling is based on multirate infinitesimal techniques, yielding solve-decoupled methods that retain localized implicit solves, support embedded error estimation for adaptive time stepping, and allow flexible use of modern STS methods. We present the ExtSTS method family, provide a robust technique for ExtSTS method creation, formulate the corresponding linear stability theory, and construct embedded ImEx, explicit, and implicit ExtSTS methods. Numerical experiments on one- and two-dimensional advection-diffusion-reaction problems show that ExtSTS methods are robust across parameter regimes and boundary conditions, and are often more efficient than ARK, Strang splitting, and PIROCK methods, especially when strong coupling between operators is important.
19. 积分希尔伯特空间与圈量子宇宙的动力学
Integral Hilbert spaces and the dynamics of loop quantum cosmos
作者: Janek Kozicki, Tomasz Pawłowski
类别: gr-qc, math.NA, physics.comp-ph
发布日期: 2026-08-03
arXiv ID: 2608.02798v1
链接: http://arxiv.org/abs/2608.02798v1
摘要: Polymer quantization program applied in Loop Quantum Gravity/Cosmology leads to nonseparable Hilbert spaces. Commonly, one sidesteps this problem by singling out and working with a separable superselection sector. This is however often no longer accessible in more involved models beyond isotropic ones. In an alternative approach one builds a separable Hilbert space as an integral over all available sectors. Here we test the dynamics following from the latter on the example of a flat isotropic Universe admitting negative cosmological constant and a massless scalar field. There, numerical evolution of (initially) semiclassical states shows that there is no relevant difference in their long term semiclassicality properties in comparison to those in a single sector approach. Further, the older (problem-specific) numerical methods are compared against an application of more common and more efficient standard tools (eigen library).