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1. 非单调半线性扩散-反应方程的迭代有限元逼近

Iterative finite element approximation of non-monotone semilinear diffusion-reaction equations

作者: Florian Spicher, Thomas P. Wihler

类别: math.NA

发布日期: 2026-07-08

arXiv ID: 2607.07327v1

摘要翻译: 我们研究用于具有渐近线性增长的非线性反应项的半线性椭圆边值问题数值逼近的迭代线性化有限元方法。我们的方法显著超越了经典理论，允许非线性项不必是单调的。我们考察了一种稳定的隐显式（IMEX）迭代与可解析多边形区域角奇性的分级网格上一阶有限元离散的结合。为顾及解析解降低的正则性，我们在角加权 Sobolev 空间中建立了正则性估计。本文的主要创新在于对迭代过程产生的离散 Galerkin 极限的逼近性质进行了非标准分析，由此得到最优的先验收敛估计。数值实验验证了理论结果。

Comment: 无

链接: http://arxiv.org/abs/2607.07327v1

1. 研究对象和背景

本文研究二维有界多边形区域 $\Omega\subset\mathbb{R}^2$ 上半线性椭圆边值问题

$$-\Delta u = f(\cdot, u)\;\;\text{在}\;\Omega\;\text{内},\qquad u = 0\;\;\text{在}\;\partial\Omega\;\text{上}, \eqno(1.1)$$

的数值逼近, 其中非线性反应项 $f:\Omega\times\mathbb{R}\to\mathbb{R}$ 满足渐近线性增长条件 $|f(\cdot,s)|\leq C(1+|s|)$, 但不施加任何单调性假设. 该问题的弱形式为: 求 $u\in V:=H^1_0(\Omega)$ 使得

$$\langle E'(u), v\rangle = \int_\Omega (\nabla u\cdot\nabla v - f(x,u)v)\,dx = 0 \quad \forall v\in V, \eqno(2.14)$$

其中 $E:V\to\mathbb{R}$ 是相应的能量泛函

$$E(u) = \frac12\int_\Omega |\nabla u|^2\,dx - \int_\Omega F(x,u)\,dx,\quad F(x,t)=\int_0^t f(x,s)\,ds. \eqno(2.11),(2.12)$$

对于单调非线性问题, 已有经典有限元理论和压缩迭代方法(如 Banach 不动点定理)可以建立最优误差估计. 然而, 当 $f$ 非单调时, 标准工具(例如 Céa 引理或基于强单调性的迭代)不再直接适用, 这给数值分析带来本质困难. 最近, [AHW23] 提出了一种基于能量框架的分析途径: 通过引入 $L^2$ 稳定化项(即考虑算子 $-\Delta + \lambda^{-1}$)并利用伪时间迭代

$$-\Delta u^{n+1} + \frac{1}{\Delta t} u^{n+1} = \frac{1}{\Delta t} u^n + f(\cdot, u^n), \eqno(1.3)$$

(可视为 IMEX 格式), 在 $\Delta t$ 充分小且初始猜测合适时, 能证明迭代在闭子空间(包括有限维 Galerkin 子空间)中弱收敛至弱解; 并且在分级稠密子空间序列上, 极限趋于原问题的一个弱解. 这一框架的关键在于将迭代解释为能量泛函的局部极小化过程, 并利用泛函的弱强制性与一阶变分的 Lipschitz 连续性质.

然而, [AHW23] 并未涉及先验误差分析, 也未给出任何收敛速度. 本文的目标正是填补这一空白: 在非单调、半线性椭圆问题中, 为分级网格上 $P_1$ 有限元逼近建立完整的先验误差分析, 并证明最优收敛率 $O(h)$ (即 $\dim(V_N)^{-1/2}$). 此外, 由于区域 $\Omega$ 为多边形, 可能有凹角导致解在角点附近失去 $H^2$ 正则性, 这需要额外的正则性理论和网格分级策略. 为此, 本文首先在 Kondrat’ev 型加权 Sobolev 空间 $H^2_\beta(\Omega)$ 中建立了解的正则性 (Thm 2.4), 从而自然引向在角点附近采用分级网格(源于 [BKP79])以恢复最优逼近阶. 总之, 本文试图将 [AHW23] 的能量迭代框架与多边形区域奇点分析和最优误差估计结合起来, 得到可用于实际计算的理论保证.

2. 主要定理或主要结果

在假设 $f$ 对第二变量可微且导数满足 $ -\rho-\mu_f \le f_u(x,u) < \rho $ (对于某个 $\rho>0$ 和参数 $\mu_f\ge 0$) 以及迭代参数 $\Delta t\in\Lambda_f(\rho)$ 满足稳定性条件 (Asm 3.1) 的前提下, 本文证明了以下主要结果.

  1. 加权正则性 (定理 2.4): 设 $\Omega$ 的角点 $c_j$ 处内角为 $\omega_j$, 若权重指数满足 $1 - \pi/\omega_j < \beta_j < 1$, 则问题 (1.1) 的任一弱解 $u$ 均属于 $V\cap H^2_\beta(\Omega)$, 其中 $H^2_\beta(\Omega)$ 是以 $\Phi_\beta(x)=\prod_j \mathrm{dist}(x,c_j)^{\beta_j}$ 加权的 Sobolev 空间.

  2. 迭代 Galerkin 解的收敛性 (命题 3.4 和定理 3.2): 在一串满足稠密性假设 (Asm 3.3) 的有限元空间 $V_N\subset V$ 上, 对每个 $N$ 由迭代 (3.7) 生成的序列强收敛(若 $\dim V_N<\infty$)到离散 Galerkin 解 $u_N^\star\in V_N$, 即

$$\int_\Omega \nabla u_N^\star\cdot\nabla v \,dx = \int_\Omega f(x,u_N^\star) v\,dx \qquad \forall v\in V_N. \eqno(3.11)$$

进而, 将 $u_N^\star$ 作为下一级 $V_{N+1}$ 迭代的初值, 可提取子列使得 $u_N^\star$ 强收敛于 $L^2(\Omega)$ 中并弱收敛于 $V$ 中某个极限 $u$, 且 $u$ 是原问题的弱解.

  1. 最优误差估计 (定理 4.10 及前述讨论): 考虑分级网格 $V_N=S^1(\Omega,\mathcal{T}_N)$, 其网格尺寸满足 $h(N)\sim \dim(V_N)^{-1/2}$. 根据加权插值估计 (4.3), 有 $\|\nabla (v - I_h v)\|_{L^2(\Omega)} \le C_I \dim(V_N)^{-1/2} \|v\|_{H^2_\beta(\Omega)}$. 利用此性质, 本文在三种不同的渐近范式下分析了误差 $\|u - u_N^\star\|_{\Delta t}$.
    • 压缩情形 ($\rho\le 1/C_P$): 迭代映射为压缩, 得到拟最优估计 $$\|u - u_N^\star\|_{\Delta t} \le C(\Delta t) \inf_{v\in V_N}\|u - v\|_{\Delta t}. \eqno(4.7)$$
    • 一般情形: 引入参数 $\varsigma$ (由 $f_u$ 的渐近行为与区域特征决定, 见 (4.14)) 来刻画扩散项与反应项的主导性.
      • 当 $\varsigma < 0$ 时, 同样有拟最优估计;
      • 当 $\varsigma = 0$ 时, 反应项渐近可忽略;
      • 当 $\varsigma > 0$ 时, 经典方法失效. 此时借助半线性 Aubin-Nitsche 对偶技巧, 在加权空间中证明新存在性与稳定性结果 (利用 Fredholm 理论), 最终确立最优收敛率 $$\|u - u_N^\star\|_{\Delta t} \le C \dim(V_N)^{-1/2}, \eqno(4.1)$$ 其中常数 $C$ 与 $N$ 无关.
  2. 迭代步数的控制 (第5节): 若迭代序列收敛到目标 Galerkin 解, 则在任何固定有限元空间 $V_N$ 上, 可通过有限次迭代使迭代误差小于逼近误差, 从而对任意给定精度达到最优收敛率. 特别地, 在压缩情况 ($\rho\le 1/C_P$) 下, 所需迭代次数仅以对数形式依赖于 $\dim(V_N)$, 即 $n\sim |\log (\dim(V_N))|$.

3. 研究方法、关键技术和核心工具

本文的数值分析建立在 [AHW23] 的能量框架之上, 但将其大幅度推进至完整的误差估计. 其核心障碍在于: 非线性反应项不满足单调性, 使得标准 Galerkin 正交性失效, 且线性对偶问题(即 Aubin-Nitsche 技巧中用到的)的适定性以及解的正则性并非显然. 作者的主要创新点在于:

  1. 参数化范数与能量极小化. 引入内积 $B_\lambda(u,v)=\int_\Omega (\lambda\nabla u\cdot\nabla v + uv)\,dx$ 和对应的范数 $\|\cdot\|_\lambda$, 将迭代改写为 $B_{\Delta t}(u^{n+1},v) = \ell_{\Delta t}(u^n; v)$ (3.2), 其中 $\ell_\lambda$ 由 $g_\lambda(x,u)=f(x,u)+\lambda^{-1}u$ 生成. 当 $\lambda\in\Lambda_f(\rho)$ 时, $g_\lambda$ 关于第二变量是 Lipschitz 连续的, Lipschitz 常数为 $\sigma_f(\lambda) < \rho + 1/\lambda$, 这为稳定性提供了关键控制. 进一步, 能量泛函的一阶变分 $E'$ 也满足 Lipschitz 估计 (引理 2.3): 对 $\forall u,v,w\in V$,
$$|\langle E'(u)-E'(v), w\rangle| \le L_{E'}(\lambda) \|u-v\|_\lambda \|w\|_\lambda, \qquad L_{E'}(\lambda)=\lambda^{-1} + \theta(\lambda)\sigma_f(\lambda),$$

其中 $\theta(\lambda)=C_P/(C_P+\lambda)$ 来自 Poincaré 不等式 (2.3). 这一 Lipschitz 性质是整个分析的基础, 它不要求单调性, 仅依赖于导数 $f_u$ 的有界性 (2.9).

  1. 加权 Sobolev 正则性 (定理 2.4). 由于区域 $\Omega$ 带凹角, 解通常不在标准 $H^2(\Omega)$ 中. 作者利用线性椭圆方程在 Kondrat’ev 权函数下的正则性理论 [BG88, BKP79], 并通过非线性项 $f(\cdot,u)$ 的有界增长 (由 (2.9) 保证 $f(\cdot,u)\in L_\beta(\Omega)$), 将解的正则性提升到 $H^2_\beta(\Omega)$. 这是后续分级网格插值估计的前提.

  2. 分级网格与插值. 针对 $H^2_\beta(\Omega)$ 函数, 文献 [BKP79] 设计了角点附近加密的分级网格, 使得线性插值 $I_h$ 在 $H^1$ 半范意义下的误差达到最优阶 $\|\nabla(u-I_h u)\|_{L^2} \le C_I h \|u\|_{H^2_\beta}$, 同时保持总自由度仍为 $O(h^{-2})$. 该技术被直接用来处理由奇性引起的逼近精度损失.

  3. 区分渐近范式的误差分析. 本文最关键的创新是引入一个有限参数 $\varsigma$ 来度量扩散与反应的相对强度 (具体定义见 (4.14), 源自能量二阶变分的符号). 根据 $\varsigma$ 的符号, 分为三类:

    • $\varsigma\le 0$: 扩散项占优或平衡, 可沿用类似于线性问题的拟最优估计;
    • $\varsigma > 0$: 反应项占优, 此时 Galerkin 正交性严重缺失. 作者发展了一套 半线性 Aubin-Nitsche 对偶论证: 考虑线性化对偶问题 $$-\Delta w + \bar{q} w = u - u_N^\star \;\;\text{在}\;\Omega\;\text{内},\quad w=0\;\text{在}\;\partial\Omega,$$ 其中 $\bar{q}$ 依赖于 $u$ 和 $u_N^\star$ 的某种平均值. 在 $\varsigma>0$ 时, $\bar{q}$ 可能使得该对偶问题有接近零的特征值, 需用 Fredholm 交替定理证明其适定性 (唯一可解) 并建立加权 $H^2_\beta$ 估计. 这是本文理论难度最高的部分, 文中为此证明了新的存在性与稳定性结果. 一旦对偶解 $w$ 存在且具有足够的正则性, 便可结合分级网格的插值估计将 $L^2$ 误差转化为 $H^1$ 误差, 最终导出最优阶.

  4. 迭代与逼近误差的平衡. 由于 Galerkin 解 $u_N^\star$ 本身并不直接计算, 而是通过迭代极限得到, 实际计算中只能取有限步迭代的近似解. 本文证明了在每一步网格细化时, 只需将上一级 Galerkin 解作为初值 (3.12), 并执行适当多步迭代, 就能使迭代误差小于当前网格的逼近误差, 从而整体误差仍保持最优收敛率. 特别地, 在压缩情形下迭代次数仅以对数增长, 保证了算法的高效性.

4. 与之前工作的比较

本文在非单调半线性问题的有限元误差分析方面取得了显著推进, 与已有工作相比, 其贡献和限制可概括如下:

  1. 与 [AHW23] 的比较: [AHW23] 提出了基于能量的迭代框架, 并证明了在闭子空间中迭代的弱收敛、在分级稠密子空间序列下极限的弱解性质, 但完全没有涉及误差估计或收敛速度. 本文是该框架的自然延续和重大发展: 首次给出了先验误差分析, 并在一般非单调情况下获得了最优收敛率. 此外, [AHW23] 仅讨论了区域足够光滑(未出现奇性)的情形, 而本文更现实地考虑了多边形区域及其角奇性, 引入了加权 Sobolev 空间和分级网格, 使得理论可以覆盖实际计算中常见的几何奇异问题.

  2. 与单调半线性问题理论的比较: 经典方法(如 [SW25] 或更早的著作)假设反应项的导数 $f_u\ge 0$ 或有界单调, 从而可用 Céa 引理型估计或强制性质导出拟最优误差界. 本文去除了单调性假设, 允许 $f_u$ 取值在区间 $(-\rho-\mu_f, \rho)$ 内, 这意味着非线性项可以是非单调、非强制甚至局部有负的斜率. 这种一般性带来了实质性挑战, 本文通过稳定化迭代和新的对偶论证克服了这些困难. 相应地, 代价是迭代参数 $\Delta t$ 必须满足稳定性条件 $\Delta t\in\Lambda_f(\rho)$, 这在一定程度上限制了伪时间步长的选择, 但与问题的非单调本质相符.

  3. 与 [SW25] 的具体对比: [SW25] 研究了单调半线性椭圆问题在分级网格上的有限元方法, 得到了最优收敛率. 本文可视为该工作的非单调推广, 然而技术路线迥异: [SW25] 直接离散非线性方程并利用单调性分析, 而本文采用迭代线性化策略, 将非线性求解转化为线性步的序列, 从而避免了直接解非线性 Galerkin 方程的困难. 在误差分析上, [SW25] 仍使用某种单调性导出准正交性, 本文则必须发展全新的半线性 Aubin-Nitsche 技巧. 此外, 本文还给出了迭代次数的控制, 这在 [SW25] 中不需考虑.

  4. Aubin-Nitsche 技巧的新应用: Aubin-Nitsche 对偶技巧在线性椭圆问题中是标准工具, 用于从 $H^1$ 误差提升到 $L^2$ 误差. 对于非线性问题, 该技巧通常用于某些特殊结构, 例如在 [Har17] 中用于半线性问题, 但在那里假设解具有完全的 $H^2$ 正则性且非线性项满足较强的条件. 本文将这一技巧移植到加权 Sobolev 空间并结合 Fredholm 理论, 首次处理了多边形角奇性带来的正则性不足, 同时非线性仅要求渐近线性有界且导数有界. 这拓展了该方法的适用范围.

  5. 与直接迭代法的比较: 许多非单调问题可通过 Newton 法或 Picard 迭代求解, 但这些迭代常要求好的初值, 且其收敛性分析复杂, 往往仅给出局部收敛. 本文的 IMEX 型迭代 (1.3) 具有全局收敛性质(在弱强制意义下), 且无须计算雅可比矩阵, 实现简单. 但在 $\varsigma>0$ 的非压缩区域, 收敛速度可能减慢, 需要额外的迭代步骤, 这与 Newton 法的二次收敛形成对比. 作者明确假定了迭代序列必须收敛到 Galerkin 解才能在有限步内达成精度, 但并未为一般情况提供迭代收缩因子, 这是实用中的一个局限.

  6. 对分级网格理论的贡献: 分级网格的思想起源于 [BKP79] 等, 已广泛用于线性椭圆问题. 本文的贡献在于证明了: 即使迭代解是在加权空间 $H^2_\beta$ 中, 且非线性是非单调的, 分级网格依然能保证 $P_1$ 元的最优逼近阶; 并且, 通过将能量框架与加权范数结合, 完成了完整的一致性分析. 这为在复杂几何上求解非单调半线性问题提供了一套可靠的理论方案.

  7. 未解决的问题与代价: 本文假设 $f$ 的导数有界 (2.9) 且满足 $\rho>0$ 的特定约束, 这排除了某些超线性增长或更奇异的情况. 同时, 分析依赖弱强制 (Asm 3.1(ii)), 对于非强制情形(如 $f$ 导致能量无下界) 本文结果不再适用. 此外, 参数 $\varsigma>0$ 时的对偶论证虽给出最优率, 但常数 $C$ 可能随 $\varsigma$ 增长而变得巨大, 文中并未定量讨论. 最后, 关于迭代步数的估计仅在压缩情况给出了显式控制, 一般情形仍属于定性保证.

了解领域论文（摘要翻译）

1. 一个近似有界上链投影

An Approximate Bounded Cochain Projection

作者: Marc Gerritsma, Suyash Shrestha

类别: math.NA

发布日期: 2026-07-08

arXiv ID: 2607.07457v1

链接: http://arxiv.org/abs/2607.07457v1

摘要翻译

中文标题: 一个近似有界上链投影

摘要翻译: 本文给出了一个从微分 $k$-形式的无限维希尔伯特复形到有限维分片多项式子复形的投影算子的构造。我们证明，在可缩区域上，所提出的投影算子达到有界上链投影算子的三个性质，即投影算子是幂等的、在索伯列夫范数下一致有界，且与外微分交换。在非可缩区域上，投影算子仍然幂等且一致有界，而交换性质可以满足到任意精度。

2. 面向参数化系统的紧致有理 Krylov 方法及其在边界元频率扫描中的应用

Compact Rational Krylov for Parametrized Systems with Application to BEM Frequency Sweeping

作者: Kobe Bruyninckx, Daan Huybrechs, Karl Meerbergen

类别: math.NA, cs.MS

发布日期: 2026-07-08

arXiv ID: 2607.07440v1

链接: http://arxiv.org/abs/2607.07440v1

摘要翻译

中文标题: 面向参数化系统的紧致有理 Krylov 方法及其在边界元频率扫描中的应用

摘要翻译: 在参数化线性系统 $\mathsf{P}(\mu)\mathsf{x}=\mathsf{b}$ 中，系统矩阵 $\mathsf{P}$ 非线性地依赖于参数 $\mu$，需对此参数的许多值求解。我们证明，最初为求解非线性特征值问题而引入的紧致有理 Krylov (CORK) 框架，可用来一次性有效地为该参数化系统在许多参数值下产生近似解。在该方法中，参数化系统首先被线性化，得到一个大型移位线性系统 $(\boldsymbol{\mathsf{A}}-\mu\boldsymbol{\mathsf{B}})\boldsymbol{\mathsf{y}}=\boldsymbol{\mathsf{d}}$。我们针对移位线性系统构造了左、右预条件的有理 Krylov GMRES 方法，并展示了如何在参数化线性系统的背景下利用 CORK 框架加速这些方法。此外，我们展示了如何纳入也依赖于参数的右端项 $\mathsf{b}(\mu)$，如何选择移位以控制收敛性，以及如何在迭代过程中允许这些移位处的不精确求解。作为应用，我们考虑通过边界元方法 (BEM) 进行的 Helmholtz 散射问题的“频率扫描”，该方法通过对密集但数据稀疏的波数依赖系统矩阵进行高效表示来实现。

不相关论文（基本信息）

1. 超越条件数的快速量子线性系统求解器

Faster quantum linear system solver beyond the condition number

作者: Alexander M. Dalzell, Jianqiang Li, Yuan Su

类别: quant-ph, cs.DS, math.NA

发布日期: 2026-07-08

arXiv ID: 2607.07691v1

链接: http://arxiv.org/abs/2607.07691v1

摘要: The spectral condition number is a widely adopted measure of worst-case cost for quantum linear system solvers. Yet it can significantly overestimate the actual runtime for a typical problem instance. We present two quantum algorithms that produce the normalized solution $|x\rangle$ of linear system $Ax=| b \rangle$ to accuracy $ε$ with complexity independent of the condition number $κ=\lVert A^{-1}\rVert$. We focus on the standard input model where $A$ is accessed through a block encoding and $| b \rangle$ is prepared by a unitary. But we also introduce an affine dilation model that encodes $A$ and $| b \rangle$ jointly, allowing further refinements of the query complexity. Our truncation-based solver makes an optimal number of queries to $| b \rangle$ and $\operatorname{\mathbf{O}}\left(κ_{\mathrm{eff}}\operatorname{polylog}\left(\frac{κ_{\mathrm{eff}}}ε\right)\right)$ queries to $A$. We prove a family of upper bounds on the effective condition number, including $κ_{\mathrm{eff}}\leq\frac{\lVert(A^\dagger A)^{-t/2}|x\rangle\rVert^{1/t}}{ε^{1/t}}$ for positive even integer $t$ and $κ_{\mathrm{eff}}\leq\frac{\lVert A^{-1\dagger}(A^\dagger A)^{-(t-1)/2}|x\rangle\rVert^{1/t}}{ε^{1/t}}$ for positive odd $t$, overcoming the $κ$-barrier. Our filtering-based solver is extremely simple with a favorable runtime prefactor. In particular, the solver has query complexity $6\frac{\lVert A^{-1\dagger}|x\rangle\rVert}ε\ln\left(\frac{1}ε\right)$ to leading order when the solution norm is known. We then present a similarly simple solution norm estimator with the same asymptotic cost up to logarithmic factors. Our quantum linear system solvers thus substantially improve a recent algorithm of Li, enabling faster quantum linear system solving beyond the condition number.

2. 引导破坏拟合算子：无分类器引导的终端拟合修复

Guidance Breaks the Fitted Operator: A Terminal-Fitted Repair for Classifier-Free Guidance

作者: Shiheng Zhang

类别: cs.LG, math.NA

发布日期: 2026-07-08

arXiv ID: 2607.07665v1

链接: http://arxiv.org/abs/2607.07665v1

摘要: Classifier-free guidance (CFG) is the standard way to strengthen class-conditioning in diffusion and flow-matching samplers, yet at large guidance it oversaturates and destabilizes, symptoms practitioners suppress with more steps or limited-interval schedules. We analyze CFG through an asymptotic-preserving, numerical-analysis lens. Building on a recent result that the deterministic DDIM step is the unique fitted operator for the unguided terminal layer, exact on the final small-sigma stretch of sampling, we show that guidance re-stiffens exactly the discriminative subspace to an anomalous exponent 1+w. DDIM is therefore no longer fitted there, and on coarse meshes its guided residual diverges as sigma_min goes to zero. We prove a guided clock barrier with three ordered step-size thresholds, and read one-step oversaturation as its endpoint: a solver artifact on the calibration model rather than the continuous guided law. The same analysis yields a one-coefficient, zero-extra-NFE repair: replace CFG’s w(r-1) by r^(1+w)-r on the guidance direction. On the calibration model’s discriminative crossover, this removes CFG’s sigma_min-divergent blow-up and is first-order accurate against the exact guided flow as sigma_min goes to zero. On learned CIFAR-10 checkpoints, and as a cross-domain smoke test on Stable Diffusion 1.5 DDIM, it acts as a high-guidance stabilizer at no extra cost rather than a universal quality knob: it cuts residual amplification and saturation, gives 9/9 point-FID wins over CFG on the tested grid, and preserves classifier-proxy target accuracy in the hard-cell blocks. We report the limits alongside: it is not a universal image-quality win, and against a dense vanilla-CFG reference it is not a uniformly better integrator of that field.

3. 一种基于最优控制的神经网络架构自适应方法及后验误差估计

An optimal control approach for neural network architecture adaptation with a posteriori error estimation

作者: C G Krishnanunni, Thomas Scott, Tan Bui-Thanh

类别: cs.LG, math.NA, math.OC

发布日期: 2026-07-08

arXiv ID: 2607.07637v1

链接: http://arxiv.org/abs/2607.07637v1

摘要: This work presents a novel approach for adapting neural network architecture along the depth based on a posteriori error estimation. By formulating neural network training as a continuous-time optimal control problem, we derive rigorous error estimates that quantify how approximation error distributes across network layers. This error decomposition enables a principled depth adaptation strategy: new layers are inserted at locations of maximum estimated error, allowing the network to efficiently capture complex, nonlinear variations in the underlying problem. Our framework introduces a novel network architecture that treats weights and biases as piecewise linear functions varying across layers, with the error estimator bounding the discrepancy between this discrete representation and the true continuous optimal control solution. The approach leverages dual weighted residual methodology from finite element analysis to derive computable upper bounds on the functional error. A key theoretical contribution is the derivation of explicit error bounds that decompose the total approximation error into interval-wise contributions, providing a rigorous basis for targeted architecture refinement. We demonstrate the effectiveness of our method on scientific datasets, including learning the observable-to-parameter map for the Navier-Stokes equation. Numerical results reveal that our approach consistently outperforms existing architecture adaptation methods in terms of generalization performance.

4. 基于黎曼法坐标的Levenberg-Marquardt方法的高阶几何更新

Higher-Order Geometric Updates for Levenberg-Marquardt Method via Riemann Normal Coordinates

作者: Jianing Liu, Dong H. Zhang

类别: cs.LG, math.NA, physics.chem-ph, physics.comp-ph, physics.data-an

发布日期: 2026-07-08

arXiv ID: 2607.07623v1

链接: http://arxiv.org/abs/2607.07623v1

摘要: Nonlinear least-squares optimization is central to regression, physics-informed neural networks, and other machine-learning tasks. Such problems have a natural geometric interpretation, model predictions form a manifold in data space, while the chosen parameterization can introduce parameter-effects curvature that becomes a dominant source of nonlinearity. This exposes a limitation of the Levenberg-Marquardt (LM) method, its tangent-space step is applied as a straight update in parameter coordinates. Geodesic acceleration gives a second-order correction, but its removal of parameter-effect curvature is exact only in the infinitesimal-step limit. We propose a Riemann-normal-coordinate Levenberg-Marquardt method (RNC-LM) to improve this consistency for finite optimization steps. By reformulating the geodesic equation, RNC-LM extends geodesic acceleration to arbitrary-order corrections and constructs finite-step updates with progressively higher reparameterization consistency. A line search along the resulting RNC curve controls the traveled distance while keeping the cost close to standard LM. The method eliminates the tangential component of residual acceleration order by order in a moving tangent frame, making the actual objective reduction more consistent with the linear model prediction of LM. On classical nonlinear least-squares benchmarks, RNC-LM improves convergence and robustness in curved valleys and rank-deficient problems. On a reaction-diffusion PINN failure-mode benchmark, it reduces the relative L2 error to the order of 1e-3 and recovers a physically meaningful solution. On a large-scale machine-learning potential-energy-surface fitting task, it achieves a 34-fold speedup over standard LM.

5. 结构引导的高斯网络方法：线性对流-反应方程

Structure-Guided Gauss-Network Method: Linear Advection-Reaction Equation

作者: Zhiqiang Cai, César Herrera

类别: math.NA

发布日期: 2026-07-08

arXiv ID: 2607.07506v1

链接: http://arxiv.org/abs/2607.07506v1

摘要: The least-squares neural network (LSNN) method introduced in [5] for linear advection-reaction equations is capable of accurately approximating discontinuous solutions without a priori knowledge of the interface location. However, the resulting discretization is a non-convex optimization problem that is computationally intensive and complex. In this paper, we propose a structure-guided Gauss-Newton (SgGN) method that alternates between the linear (output) and the nonlinear (hidden layer) parameters. At each outer iteration, the linear parameters are computed by a linear solver, and the nonlinear parameters are updated by a modified Gauss-Newton (GN) method that explicitly removes the singularities of the GN matrix. Numerical experiments for all test problems presented in [5] show that the SgGN method is superior to the Adam optimizer [13], the commonly used first-order optimization algorithm, not only in computational cost but, more importantly, in accuracy

6. 单对角隐式 Runge-Kutta 方法的显式稳定化实现

Explicit stabilized implementation of singly diagonally implicit Runge-Kutta methods

作者: Ibrahim Almuslimani, Gilles Vilmart, Konstantinos Zygalakis

类别: math.NA

发布日期: 2026-07-08

arXiv ID: 2607.07497v1

链接: http://arxiv.org/abs/2607.07497v1

摘要: Implicit methods are a natural approach for the integration of stiff differential equations, to avoid time-step restrictions faced by standard explicit integrators. Explicit stabilised integrators are an alternative to implicit methods, which can be particularly efficient in high-dimensional applications with diffusive terms. Towards the best of both worlds, we introduce a new explicit stabilised implementation of a class of diagonally implicit Runge-Kutta methods. This allows us to implement high-order singly-diagonally-implicit Runge-Kutta methods for advection-diffusion-reaction PDEs with a provable computational cost analogous to that of standard explicit stabilised methods. The main ingredient is to recast the implicit Runge-Kutta update as the steady state of a modified auxiliary system, which is then computed using a partitioned Runge-Kutta-Chebyshev method inspired by optimisation techniques.

7. 协调有限元的黑盒多层代数预处理框架

A black-box, multilevel algebraic preconditioning framework for conforming finite elements

作者: O. A. Krzysik, B. S. Southworth, G. A. Wimmer

类别: math.NA

发布日期: 2026-07-08

arXiv ID: 2607.07485v1

链接: http://arxiv.org/abs/2607.07485v1

摘要: Recently we introduced the least-squares algebraic-multigrid domain-decomposition (LS-AMG-DD) method as a multilevel, algebraic preconditioner for sparse symmetric positive definite (SPD) matrices that admit a Gram representation (A=G^{\top}G) \cite{southworth2026lsamgdd}. The factor (G) induces a local symmetric positive semidefinite (SPSD) splitting of (A) used to define local spectral problems from which an interpolation $P$ is built, and a coarse-level Gram operator induced under Galerkin coarsening, (A_c=G_c^\top G_c), for (G_c:=GP). This paper clarifies when this Gram structure arises, showing that, on a prescribed degree-of-freedom cover ({\cal C}), a ({\cal C})-local Gram representation of $A$ exists if and only if (A) admits a ({\cal C})-local SPSD splitting. We then connect this viewpoint to conforming finite-element discretizations, where bilinear forms are naturally assembled from elementwise SPSD energies and therefore admit element-local Gram representations after choosing local factors (e.g., via algebraic factorizations of element blocks). Taken together, these observations provide an essentially black-box route for applying LS-AMG-DD to conforming finite-element problems. Numerical tests illustrate the robustness of the method on several problems for which classical AMG methods require more than $10^5$ iterations to converge, including high-order discretizations of grad–div in (\hdiv), anisotropic hyperdiffusion in $H^2$, and linear elasticity in vector (H^1). Moreover, in some comparisons with existing AMG methods, LS-AMG-DD produces errors that are 2–5 orders of magnitude smaller, even when all methods are stopped at the same relative residual tolerance.

8. 基于外推法的单辅助量子比特量子微分方程求解器的电路深度缩减

Circuit Depth Reduction of One-Ancilla Quantum Differential Equation Solver via Extrapolation

作者: Di Fang, Justin Park

类别: quant-ph, math.NA

发布日期: 2026-07-08

arXiv ID: 2607.07389v1

链接: http://arxiv.org/abs/2607.07389v1

摘要: Solving linear differential equations is a fundamental task in scientific computing and an important primitive for quantum computing. A recent one-ancilla quantum differential equation solver provides a hardware-friendly and locality-preserving approach with provable performance guarantees, making it highly suitable for the early fault-tolerant and near-term regimes. Its simple circuit structure comes with a natural trade-off: the maximum single-run circuit depth scales as $O (1/ε)$ in the target accuracy $ε$. In this work, we reduce this depth by combining the solver with classical step-size postprocessing. By running the one-ancilla solver at a logarithmic number of finite time step sizes and using classical post-processing to cancel leading discretization errors, we reduce the maximum single-run circuit depth to $O(\mathrm{polylog}(1/ε))$ without adding quantum ancillae or sacrificing locality. Technically, extending extrapolation ideas beyond Hamiltonian and Lindbladian dynamics requires regularity estimates for observable maps under nonunitary evolution, which we obtain through a holomorphic extension of the adjoint evolution. Numerical experiments on the Hatano-Nelson model (ODE) and the convection-diffusion equation (PDE) demonstrate the effectiveness of the approach.

9. JAX-FVM: 一种用于可压缩流的非结构网格上可微、熵稳定的有限体积求解器

JAX-FVM: A differentiable, entropy-stable finite volume solver on unstructured meshes for compressible flows

作者: Guillaume de Romémont

类别: math.NA, cs.MS

发布日期: 2026-07-08

arXiv ID: 2607.07385v1

链接: http://arxiv.org/abs/2607.07385v1

摘要: We present JAX-FVM, an open-source, fully differentiable finite volume method (FVM) for the two-dimensional compressible Euler and Navier-Stokes equations on unstructured triangular meshes. The solver is written entirely in JAX, so that every operation : mesh connectivity, flux evaluation, slope limiting, and time integration is just-in-time compiled, vectorised, and end-to-end differentiable through automatic differentiation (AD), and runs transparently on CPU or GPU. On the numerical side, JAX-FVM is built around an entropy-conservative Tadmor/Ismail-Roe two-point flux supplemented with entropy-variable Rusanov or Roe dissipation, second-order MUSCL reconstruction of primitive variables with least-squares gradients and Venkatakrishnan limiting, and a family of explicit (RK2-4) and matrix-free implicit (Newton, SDIRK2) time integrators whose Jacobian actions are obtained by AD. The combination of an unstructured-mesh compressible FVM with end-to-end differentiability fills a gap left by existing differentiable CFD frameworks, which are almost exclusively restricted to structured grids or spectral discretisations. We describe the governing equations, the discretisation, the software architecture, and a set of standard verification cases. The code is openly available at https://github.com/guigzair/jax_fvm.

10. POO-LPSP：用于层次分析法优先级推导的并行鱼鹰优化最小惩罚平方优先化方法

POO-LPSP: Parallel Osprey Optimized Least Penalty-Squared Prioritization Methods for Priority Derivation in the Analytic Hierarchy Process

作者: Kevin Kam Fung Yuen

类别: math.OC, cs.AI, cs.DC, math.NA

发布日期: 2026-07-08

arXiv ID: 2607.07313v1

链接: http://arxiv.org/abs/2607.07313v1

摘要: Pairwise comparison (PC) via pairwise reciprocal matrices (PRMs) is central to the Analytic Hierarchy Process (AHP). Although the traditional eigenvector method is widely applied to derive priorities, its theoretical robustness in reflecting true priority vectors remains debated. Building upon a previous iteration of this study, this research develops the revised Least Penalty-Squared Prioritization (LPSP) optimization models, including the revised Least Product of Penalty and Direct Squares (LPPDS) and revised Weighted Squares (LPPWS), to minimize the revised Root Mean Penalty-Squared Variance (RMPSV) and the revised Root Mean Penalty-Weighted Square Variance (RMPSWV). However, solving these non-linear formulations is computationally complex for decision-makers. To overcome these limitations, this study proposes the Parallel Osprey Optimized Least Penalty-Squared Prioritization (POO-LPSP) method. By integrating an improved bio-inspired metaheuristic Parallel Osprey Optimization Algorithm (POOA), this framework efficiently solves complex LPSP models to minimize RMPSV and RMPSWV, thereby enhancing prioritization reliability. The practical utility and computational efficiency of the POO-LPSP method are validated through a numerical application focusing on a Generative AI (GAI) vendor selection problem. To extend, POO-LPSP can serve as a robust alternative to Saaty’s Eigen system method for AHP applications.

11. 基于Carleman估计的衰减介质中光声成像向后问题的稳定性

The Stability of the Backward Problem for Photoacoustic Imaging in Attenuating Media via Carleman Estimates

作者: Qihang Chen, Zhiyuan Li, Song Xu

类别: math.AP, math.NA

发布日期: 2026-07-08

arXiv ID: 2607.07244v1

链接: http://arxiv.org/abs/2607.07244v1

摘要: This paper investigates the backward problem in time for photoacoustic tomography (PAT) in attenuating media. It is well-established that photoacoustic imaging in attenuating media can be accurately modeled by spatial fractional-order damping. This inverse problem is ill-posed in the sense of Hadamard. In this work, we construct a novel class of Carleman estimates independent of spatial variables, and by virtue of these estimates, we establish conditional stability estimates for this problem for the first time. Building upon this, we propose a Tikhonov-type regularization functional and derive its associated adjoint system. Furthermore, leveraging the established conditional stability results, we derive the convergence rate of the proposed regularization approach. Finally, we validate the effectiveness of our theoretical findings through extensive numerical experiments.

12. 一种具有多重离散层的非协调三角网格生成框架

A framework for generating nonconforming triangular meshes with multiple discretization layers

作者: I. L. Semenov, M. M. Becker

类别: math.NA

发布日期: 2026-07-08

arXiv ID: 2607.07242v1

链接: http://arxiv.org/abs/2607.07242v1

摘要: We present a framework for generating nonconforming triangular meshes with multiple discretization layers. The framework exploits characteristic structural properties of meshes produced by a frontal Delaunay algorithm with uniform element size. The bulk region of such meshes exhibits a structured pattern resembling a regular triangular lattice. Owing to this structure, the bulk region can be coarsened by grouping elements into connected subsets of larger composite elements. When applied repeatedly, this procedure produces a mesh with multiple discretization layers. For complex geometries, the framework can be used to create composite multidomain meshes, where multiple discretization layers are generated within each subdomain. We present examples of meshes generated by the proposed framework and discuss postprocessing strategies for their refinement and coarsening. The resulting meshes are well suited for the application of adaptive mesh refinement techniques. The proposed framework can be readily integrated with existing mesh generators and finite-element solvers that support nonconforming triangular meshes.

13. 广义有限差分法的优化模板策略：应用于稳态非线性问题

Optimized Stencil Strategy for the Generalized Finite Difference Method: Application to Steady-State Non-Linear Problems

作者: Stéphane Clain, Jorge Figueiredo

类别: math.NA

发布日期: 2026-07-08

arXiv ID: 2607.07165v1

链接: http://arxiv.org/abs/2607.07165v1

摘要: We propose an optimized stencil strategy for the Generalized Finite Difference Method (GFDM) applied to non-linear problems. We take advantage of the flexibility of GFDM to engineer specific stencils by agglomerating nodes and balancing size with numerical accuracy. Previous work, focusing on the stencil construction for the linear convection-diffusion problem, showed that optimizing stencils and scaling parameters improves the scheme. The present study aims to investigate similar benefits for non-linear problems such as the Burgers’ equations and the weakly compressible Navier-Stokes system.

14. 常微分方程的混合精度显式数值方法

Mixed precision explicit numerical methods for ordinary differential equations

作者: M Al Sayed Ali, S Bernard, A Marzorati, J Rouzaud-Cornabas

类别: math.NA

发布日期: 2026-07-08

arXiv ID: 2607.07080v1

链接: http://arxiv.org/abs/2607.07080v1

摘要: Our objective is to solve large systems of ordinary differential equations (ODEs) commonly used to model biological processes. These equations are typically nonlinear, complex, and high-dimensional. In computational biology, such ODEs are generally solved using numerical methods. In this work, we focus on explicit numerical methods because of their flexibility. However, their limited stability regions may result in high computational costs. To mitigate this issue, we investigate mixed precision algorithms designed to reduce computational effort by performing selected parts of the numerical method in lower arithmetic precision. We develop several mixed precision explicit methods and assess their performance on two large scale biological benchmark ODE models. Our theoretical analysis highlights the effectiveness of partially reducing arithmetic precision within explicit methods. Numerical experiments demonstrate that our mixed methods-implemented in both sequential and parallel versions using MPI-combining single (float) and double precision arithmetic can achieve up to twice the speed of a fully double precision implementation while preserving the same level of accuracy. Furthermore, the results indicate that decreasing the timestep improves the performance and robustness of our mixed methods, while the single precision method fails to converge.

15. 多孔介质中扩散释放的拓扑特征

Topological Signatures of Diffusive Release in Porous Media

作者: Donghan Kim

类别: cs.CE, math.NA

发布日期: 2026-07-08

arXiv ID: 2607.07061v1

链接: http://arxiv.org/abs/2607.07061v1

摘要: We used persistent homology to quantify the multiscale topological and geometric organization of porous media, including solid connectivity and the formation of loop-like and cavity-like structures across spatial scales. Through statistical analysis, we show that these topological and geometric features are closely associated with diffusion-driven release behavior in porous media. In particular, even within each target-porosity level, samples with richer topological features tend to exhibit long-tailed release, indicating that release behavior depends not only on the amount of pore space but also on the multiscale organization of the solid phase. We further show that persistent homology-based features can classify release-curve regimes using a simple classification model. Notably, feature extraction is substantially faster than finite element diffusion simulations. Together, these results suggest that persistent homology provides a lightweight, interpretable, and geometry-based descriptor for screening diffusive release behavior in porous media.

16. 不可压缩电阻性MHD自由表面流精确扩散界面模型的数值解

Numerical solutions of an accurate diffuse interface model of the incompressible resistive MHD free surface flow

作者: Maojun Li, Jiancheng Wang, Zeyu Xia, Liwei Xu

类别: math.NA

发布日期: 2026-07-08

arXiv ID: 2607.07025v1

链接: http://arxiv.org/abs/2607.07025v1

摘要: In this paper, we derive a new model to simulate the incompressible resistive magnetohydrodynamic (MHD) free surface flow. A thermodynamically consistent diffuse interface method is adopted to characterize the moving interface in the modeling process. The formal convergence of the proposed MHD free surface flow model to the sharp interface model is established via a matched asymptotic argument, and the model can be solved without the need for sophisticated free surface capturing schemes. We design a fully decoupled linear finite element scheme that preserves the divergence-free constraint of the magnetic field at a discrete level. The reliability and robustness of the proposed model and algorithm are validated through numerical investigations of the magnetic damping effect on bubble dynamics. In particular, we provide a quantitative numerical comparison of the present results with those obtained from an inductionless MHD model and a sharp interface arbitrary Lagrangian–Eulerian model.

17. 基于半张量积与t-乘积的张量方程 $\mathcal{A} \ltimes \mathcal{X} \ltimes \mathcal{B}= \mathcal{C}$ 的解分析

Solution Analysis of Tensor Equation $\mathcal{A} \ltimes \mathcal{X} \ltimes \mathcal{B}= \mathcal{C}$ via Semi Tensor Product with t-product

作者: Bhawna Garg, Ranjan Kumar Das

类别: math.NA

发布日期: 2026-07-08

arXiv ID: 2607.07006v1

链接: http://arxiv.org/abs/2607.07006v1

摘要: This paper focuses on the analysis of the tensor equation $\mathcal{A\ltimes X\ltimes B=C}$, formulated via the semi tensor product with t-product. For the unknown vector $\mathcal{X}$, we establish a necessary and sufficient condition that provides an equivalence criterion for the existence of solutions. For matrix valued and higher-order tensor valued unknown $\mathcal{X}$, solvability is determined by corresponding compatibility requirements. Moreover, the explicit structure (Toeplitz and Circulant) of $\mathcal{C}$ is characterized. The derived results are supported by several illustrative examples.

18. 用于MIONets的混合最小二乘/梯度下降方法

Hybrid Least Squares/Gradient Descent Methods for MIONets

作者: Jun Choi, Chang-Ock Lee, Minam Moon

类别: cs.LG, cs.AI, math.NA

发布日期: 2026-07-08

arXiv ID: 2607.06976v1

链接: http://arxiv.org/abs/2607.06976v1

摘要: In this paper, we propose an efficient hybrid least squares/gradient descent (LSGD) method for MIONets to accelerate training. This method generalizes the LSGD method for DeepONets. Since MIONet is the sum of the entrywise product of multiple branch networks and a trunk network, it can be viewed as a multilinear function with respect to the last layer parameters of each branch network. These sets of parameters can be optimized using the alternating least squares method, where we solve the LS system for a single branch network in turn. To handle the large-sized system matrix, we introduce Kronecker and Khatri-Rao products and tensor permutation matrices to factor the large matrix into small ones. Our method is compatible with a general type of $L^2$ loss with regularization terms for the last layer parameters of each branch, where linear operators can be applied to the MIONet output in each loss term.

19. RPLSS: 一种随机投影线性方程组求解器

RPLSS: A randomized projected linear systems solver

作者: Meng-Long Xiao, Tao Li, Deanna Needell

类别: math.NA

发布日期: 2026-07-08

arXiv ID: 2607.06917v1

链接: http://arxiv.org/abs/2607.06917v1

摘要: The projected linear system solver (PLSS), by incrementally appending columns to a random or deterministic sketching matrix, provides an attractive finite termination property for consistent linear systems. Nevertheless, a critical computational bottleneck of PLSS is accessing the whole coefficient matrix per iteration, making it prohibitive for extremely large-scale problems or applications with missing data. To alleviate this limitation, we propose a unified randomized PLSS (RPLSS) framework, built upon the tailored randomized row or column selection strategies that require only partial matrix information per iteration, for solving a general linear system, whether it is under- or overdetermined, and whether it is consistent or not. Within this framework, we develop a randomized Gaussian Kaczmarz method and its extended variant as row-action solvers, and randomized coordinate descent variants as column-action solvers. Theoretically, we prove that our methods inherit the finite termination property of PLSS, while achieving an exponential convergence rate, overcoming the sluggish convergence inherent in conventional randomized Kaczmarz and coordinate descent methods. Numerical experiments demonstrate the superiority of our method against state-of-the-art randomized methods, particularly in scenarios with large missing data.

20. 基于纯伪应力格式的弹性特征值问题免锁混合有限元法

A locking free mixed FEM based on a pure pseudostress based formulation for the elasticity eigenproblem

作者: Arbaz Khan, Felipe Lepe, Jesus Vellojin

类别: math.NA

发布日期: 2026-07-08

arXiv ID: 2607.06890v1

链接: http://arxiv.org/abs/2607.06890v1

摘要: We analyze a novel locking-free mixed formulation for the elasticity eigenvalue problem in both two and three dimensions, expressed exclusively in terms of the pseudostress tensor. An important feature of this formulation is that it does not require the enforcement of symmetry, either in a weak or strong sense. The displacement of the structure is recovered via a postprocess of the computed pseudostress. We introduce a mixed finite element method based in the tensorial version of the standard families of finite elements to discretize the space $\boldsymbol{\mathcal{H}}(\bdiv)$. We prove convergence and a priori error estimates under the theory of non-compact operators. Additionally, we perform an a posteriori error analysis for the problem, proving reliability and efficiency of the proposed indicator. We validate our theoretical results with numerical tests on different geometrical and physical configurations.

21. NVIDIA Tensor Cores上的多重双精度算术

Multiple Double Arithmetic on NVIDIA Tensor Cores

作者: Howard Chen, Jan Verschelde

类别: cs.MS, cs.DC, math.NA

发布日期: 2026-07-08

arXiv ID: 2607.06881v1

链接: http://arxiv.org/abs/2607.06881v1

摘要: A multiple double is an unevaluated sum of doubles. An NVIDIA tensor core is a specialized high performance compute core for matrix multiplication. The Ampere A100, released in 2020, introduced tensor cores capable of 64-bit floating-point arithmetic. Every multiple double arithmetical operation requires renormalization, which involves branching, for which tensor cores are unsuited. To solve this problem caused by renormalization, we apply a solution similar to the Ozaki scheme [Ozaki et al, Numerical Algorithms, 2012]. Our software is available under the GPU GPL license on github.

22. 关于重用最后一级的Runge–Kutta方法的阶

On the order of Runge Kutta methods reusing last stage

作者: Manuel Calvo, Juan I. Montijano, Luis Rández

类别: math.NA

发布日期: 2026-07-07

arXiv ID: 2607.06788v1

链接: http://arxiv.org/abs/2607.06788v1

摘要: In this paper we consider explicit Runge–Kutta (RK) methods for the numerical solution of Initial Value Problems (IVPs) in differential equations in which the last function evaluation in a step is reused, substituting the first evaluation for the next step. This requirement implies that, except for the first step, the computational cost of an $s$–stage explicit RK method reduces from $s$ to $(s-1)$ function evaluations per step. It will be seen that, in general, if a RK method has order $p$, when introducing this reused stage the new method has in general order $

23. 自适应无矩阵低秩近似

Adaptive, Matrix-Free Low-Rank Approximation

作者: Arnel I. Smith, Elly Do, Chao Chen

类别: math.NA

发布日期: 2026-07-07

arXiv ID: 2607.06758v1

链接: http://arxiv.org/abs/2607.06758v1

摘要: We study fixed-tolerance low-rank approximation in the matrix-free setting, where a matrix or linear operator $\mathbf{A}$ is accessible only through matrix-vector products and its rank must be determined adaptively to meet a prescribed error tolerance. We introduce a family of adaptive, matrix-free randomized QB algorithms. A randomized error indicator estimates the residual norm – in either the Frobenius or the spectral norm – directly from a random sketch, remaining accurate down to machine precision. A matrix-free rank-pruning step decouples the computational block size from the final rank, so that large, BLAS-3-friendly blocks can be used without over-estimating the rank, and an adjoint-free variant returns the orthonormal basis using only the forward operator. Across test matrices with diverse singular-value decays, the proposed methods attain ranks close to the truncated-SVD optimum while meeting the prescribed tolerance with high probability.