生成时间: 2026年06月26日 06:41:08
重点关注论文数量: 1
了解领域论文数量: 2
不相关论文数量: 25
重点关注论文（完整分析）
1. 拟线性非单调椭圆方程的自适应迭代数值均匀化方法
Adaptive Iterative Numerical Homogenization for Quasilinear Nonmonotone Elliptic PDE
作者: Maher Khrais, Barbara Verfürth
类别: math.NA
发布日期: 2026-06-25
arXiv ID: 2606.27256v1
摘要翻译: 我们提出并分析了一种自适应迭代数值均匀化方法，用于逼近一类具有多尺度性质的拟线性非单调椭圆问题的解。该方法基于局部正交分解（LOD）技术，将其应用于Kačanov迭代每一步中的线性问题。在该方法中，多尺度基函数在每次迭代中自适应地重新计算，并利用更新后的多尺度空间求解一个线性问题，在这两个步骤中，非线性项均使用前一次迭代的近似值进行计算。作为所提方法的一个关键组成部分，我们提出了一个局部可计算的误差指示子，它在每次迭代中识别需要更新的基函数，同时尽可能保留先前计算出的基函数。我们给出了先验误差估计，并证明了该方法的收敛性，仅要求右端项具有更高的可积性，而不要求解本身具有更高的可微性。此外，我们讨论了如何在Newton迭代法的框架下调整所提出的自适应迭代LOD方法。数值实验验证了理论并证实了所提方法的适用性。
Comment: 无
链接: http://arxiv.org/abs/2606.27256v1
1. 研究对象和背景
本文研究一类具有多尺度特征的拟线性非单调椭圆型偏微分方程的数值均匀化方法。模型问题为：在 Lipschitz 区域 $\Omega\subset\mathbb{R}^d, d\leq 3$ 上，求 $u$ 满足
$$\begin{cases} -\nabla\cdot(\alpha(x,u)\nabla u) = f, & x\in\Omega, \\ u=0, & x\in\partial\Omega, \end{cases}$$其中右端项 $f\in L^2(\Omega)$，系数 $\alpha(x,s)$ 关于空间变量 $x$ 高度振荡、强非均匀，关于第二个变量 $s$ 满足一致 Lipschitz 连续条件及一致椭圆性与有界性：
$$|\alpha(x,s_1)-\alpha(x,s_2)|\le L_c|s_1-s_2|,\quad \lambda|\psi|^2\le\alpha(x,s)\psi\cdot\psi\le \Lambda|\psi|^2,\quad\forall x\in\Omega,s_1,s_2\in\mathbb{R},\psi\in\mathbb{R}^d.$$这类方程广泛描述非饱和多孔介质流（Richards 方程）、复合热传导以及多种非线性物理过程。弱形式化为求 $u\in H^1_0(\Omega)$ 满足
$$A(u;u,v)=F(v),\quad\forall v\in H^1_0(\Omega),$$其中 $A(z;u,v):=\int_\Omega\alpha(x,z)\nabla u\cdot\nabla v\,dx$，$F(v):=\int_\Omega fv\,dx$。在上述假设下，问题 (5) 的解存在唯一（见 [13,26]）。
传统有限元方法要求网格尺寸 $H$ 充分小以分辨系数 $\alpha$ 的微观变化，导致大规模代数系统，计算代价巨大。数值均匀化方法旨在通过构造包含细尺度信息的多尺度基函数，在粗网格 $V_H$ 上获得高精度的宏观近似。局部正交分解（LOD）方法是其中一类重要途径，它通过正交分解将细尺度空间 $W=\ker I_H$ 的补空间 (1−Qϕ)V_H 作为近似空间，其中 $Q_\phi$ 为基于线性化点 $\phi$ 的椭圓性修正算子，求解局部问题得到。对于非线性问题，LOD 已被用于单调和非单调拟线性方程 [33,46]，但这些工作仅构造一次多尺度空间（基于初始线性化点 $\phi$），随后在该固定空间中迭代求解全局非线性离散问题。数值实验显示，若 $\phi$ 选择不当，可能无法恢复对网格尺寸 $H$ 的一阶收敛；然而若将第一次 LOD 解作为新的线性化点重新构造空间，则最优收敛得以恢复。这启发迭代更新空间的思路，然而反复构造全局修正问题将带来不可忽视的计算开销。
本文试图填补这一空白：提出并分析一种自适应迭代 LOD 方法，在每一步 Kacanov 迭代中根据当前近似解 $u^n_H$ 更新多尺度空间 $(1-Q_{u^n_H})V_H$，并在此更新空间中求解线性化的全局问题，从而避免依赖固定线性化点。同时，为了降低反复构造空间的成本，引入基于局部可计算误差指示器的自适应策略，仅对需要更新的基函数重新计算细尺度修正，保留仍满足精度要求的原有修正基。理论分析表明，仅需右端项 $f$ 具有稍高的可积性（2D 要求 $f\in L^{2+\epsilon}$，1D 无额外要求），而不需要解本身的高阶可微性或系数的光滑性，该方法就能在收缩常数 $\nu<1$ 的条件下保证迭代解的线性收敛，且最终误差具有对 $H$ 的最优阶 $O(H)$。对于 3D，在 $f\in L^3$ 且系数对比度受限时（$C_P(1-\lambda/\Lambda)<1$），同样结论成立。本文进一步讨论了如何将该自适应迭代框架与 Newton 迭代结合，并通过数值实验验证了理论。
2. 主要定理或主要结果
本文的主要理论结果是关于迭代 LOD 方法的先验误差估计与收敛性（定理 4）。记 $\{u^n\}_{n\ge 0}$ 为由连续 Kacanov 迭代 (7) 产生的弱解序列，即 $u^n\in H^1_0(\Omega)$ 满足 $A(u^{n-1};u^n,v)=F(v),\forall v\in H^1_0$；记 $\{u^n_H\}_{n\ge 0}$ 为算法 1 得到的迭代 LOD 逼近解，其中 $u^n_H\in (1-Q_{u^{n-1}_H})V_H$。初始近似 $u^0_H\in H^1_0$ 取为 $\phi$，并满足 $|u^0-\phi|_1\le C_0 H$（例如取 $\phi=u^0$）。在 2D 情形下，假设 $f\in L^{2+\epsilon}(\Omega),\epsilon>0$，定义收缩常数
$$\nu:=\frac{L_c c_{2+\epsilon}C_{q,\Omega}\|f\|_{L^{2+\epsilon}}}{\lambda},$$其中 $c_{2+\epsilon}$ 是由辅助线性问题 (6) 的 $W^{1,2+\epsilon}$ 稳定性估计 (15) 给出的常数，$C_{q,\Omega}$ 为 $H^1_0$ 嵌入 $L^q$ 的常数（$q=2(2+\epsilon)/\epsilon$）。若 $\nu<1$，则迭代 Kaˇ kanov 弱解 $u^n$ 收敛于原问题真解 $u$，且对任意 $n\ge 0$ 有如下误差估计：
$$|u^n-u^n_H|_1\le \nu^n|u^0_H-u^0|_1+\frac{1}{1-\nu}\frac{C_{\mathrm{proj}}}{\lambda} H\|f\|_0.$$在 1D 情形可视为 $\epsilon=0$，此时 $f\in L^2$ 即足，嵌入 $H^1_0\hookrightarrow L^\infty$，类似估计成立。在 3D 情形，若 $f\in L^3(\Omega)$ 且系数对比度满足 $C_P(1-\lambda/\Lambda)<1$（见命题 2），则收缩常数重新定义为
$$\nu:=\frac{L_c c_3 C_{6,\Omega}\|f\|_{L^3}}{\lambda},$$并假设 $\nu<1$，相同的收敛性与误差估计仍成立。
对于自适应迭代 LOD 方法（第 5 节），作者宣称在适当的局部化处理和自适应更新条件下，类似的最优收敛阶可以被保持，但由于所提供的文本未包含完整的自适应算法误差分析定理，具体误差界形式尚无法从当前内容中精确提取。引理 1 给出线性辅助问题的标准 LOD 误差：对任意 $\phi$，有 $|u_\phi-u_{\phi,H}|_1\le \frac{C_{\mathrm{proj}} H}{\lambda}\|f\|_0$，该估计独立于 $\phi$，是后续分析的基础。
3. 研究方法、关键技术和核心工具
本文的核心技术障碍在于处理不同迭代步之间的非线性项，特别是形如 $\|(\alpha(x,u^{n-1}_H)-\alpha(x,u^{n-1}))\nabla u^n\|_0$ 的估计，其中 $\nabla u^n$ 在标准假设下仅属于 $L^2$，不足以直接应用系数的 Lipschitz 连续性吸收 $\|u^{n-1}_H-u^{n-1}\|$ 因子。作者的关键创新是通过对右端项 $f$ 附加更高的可积性条件（$f\in L^{2+\epsilon}$ 或 $f\in L^3$），借助扩散椭圆方程的 $W^{1,p}$ 稳定性理论（命题 2），证明线性化辅助问题的解 $u_\phi$ 的梯度具有提升的可积性：$\nabla u_\phi\in L^{2+\epsilon}$（2D）或 $L^3$（3D）。此结论依赖于 [9] 中使用扰动参数建立的推广 Meyers 类型估计，不要求系数 $\alpha$ 的光滑性，仅涉及 $\alpha$ 的 $L^\infty$ 有界性和对比度条件。结合 Sobolev 嵌入 $H^1_0\hookrightarrow L^q$，选取合适的指数对，利用 Hölder 不等式得出
$$\|(\alpha(x,u^{n-1}_H)-\alpha(x,u^{n-1}))\nabla u^n\|_0\le L_c c_{p} C_{q,\Omega}\|f\|_{L^p}|u^{n-1}_H-u^{n-1}|_1,$$其中 $p=2+\epsilon$ 或 $3$，$q=2p/(p-2)$。这一估计将非线性误差转化为前一迭代误差的倍数，从而构造出收缩映射。
证明框架采用固定点论证。关键分解为
$$|u^n-u^n_H|_1\le |u^n-w^n|_1 + |w^n-u^n_H|_1,$$其中 $w^n$ 是以 $u^{n-1}_H$ 为线性化点的辅助问题 (6) 的弱解，即满足 $A(u^{n-1}_H; w^n,v)=F(v)$。对于右端第二项，直接应用引理 1（线性 LOD 误差估计），得到 $|w^n-u^n_H|_1\le \frac{C_{\mathrm{proj}}}{\lambda} H\|f\|_0$。对于第一项，借助椭圆性、系数的 Lipschitz 性质以及上述提高的正则性，推出
$$\lambda|u^n-w^n|_1^2\le A(u^{n-1};u^n-w^n,u^n-w^n) = \big((\alpha(u^{n-1}_H)-\alpha(u^{n-1}))\nabla w^n,\nabla(u^n-w^n)\big) \le \lambda\nu|u^{n-1}_H-u^{n-1}|_1|u^n-w^n|_1.$$化简即得递推关系 $|u^n-w^n|_1\le \nu|u^{n-1}_H-u^{n-1}|_1$。至此由归纳法导出几何衰减和最终的误差界 (21)。常数 $\nu<1$ 的要求为小数据条件，保证了迭代收敛。
自适应策略的实现依赖于局部化修正问题和局部误差指示器。局部化遵循 LOD 标准截断技巧：将区域上的修正限制在 $k$ 层邻域内，利用指数衰减性质控制截断误差。局部误差指示器设计用于比较当前迭代与前一迭代中对应基函数的局部残量，仅标记和更新那些超出给定容限的基函数修正。该方法借鉴了 [24,25] 中线性 LOD 的自适应思想和 [37] 中非线性 Helmholtz 方程的迭代自适应方法，但将其推广到拟线性情形，且需确保整个迭代过程的误差累积可控。完整的自适应误差分析在第 5 节给出，其核心在于证明自适应更新不会破坏整体 $O(H)$ 收敛阶，并保持与标准迭代 LOD 可比的精度。
4. 与之前工作的比较
基于以上分析，本文在数值均匀化领域针对拟线性非单调问题开辟了迭代 LOD 方向，并通过自适应机制使之在计算上更可行，其理论创新在于将高阶可积性条件引入多尺度误差分析，推进了 LOD 方法在低正则非线性问题中能够达到最优收敛的理论保证。
了解领域论文（摘要翻译）
1. Irksome 中的自动化 Galerkin 时间步进
Automated Galerkin time stepping in Irksome
作者: Boris D. Andrews, Pablo Brubeck, Patrick E. Farrell, Robert C. Kirby, Scott P. MacLachlan
类别: math.NA
发布日期: 2026-06-25
arXiv ID: 2606.27300v1
链接: http://arxiv.org/abs/2606.27300v1
摘要翻译
中文标题: Irksome 中的自动化 Galerkin 时间步进
摘要翻译: 随着时、空离散方案研究的不断推进，近期工作集中于 Galerkin 时间离散方案，这类方案能够实现比已知 Runge-Kutta 积分器更广泛的结构保持。尽管此类离散化的前景巨大，但迄今为止其实现通常依赖于定制化的代码，限制了更广泛的使用。本文在 Irksome 中针对半离散变分问题，提出了间断 Galerkin 和连续 Petrov-Galerkin 时间步进的自动化实现。该实现支持辅助变量、灵活的时间积分以及整体代数求解器，并允许用户在 Runge-Kutta 和 Galerkin 时间形式之间切换，而对用户代码的改动极小。数值算例展示了代表性偏微分方程系统的精度、求解器性能和结构保持性。
2. 不依赖TVD时间离散的间断Galerkin方法的TVD与TVB保持
TVD and TVB preservation without TVD time discretization for discontinuous Galerkin methods
作者: Ziyao Xu, Zheng Sun
类别: math.NA
发布日期: 2026-06-25
arXiv ID: 2606.26692v1
链接: http://arxiv.org/abs/2606.26692v1
摘要翻译
中文标题: 不依赖TVD时间离散的间断Galerkin方法的TVD与TVB保持
摘要翻译: 全变差下降(TVD)和全变差有界(TVB)性质对于控制守恒律数值解中的非物理振荡至关重要。在经典的Runge–Kutta (RK) 间断Galerkin (DG)框架中，强制这些性质内在地依赖于TVD时间积分器，后者如今更常被称为强稳定保持(SSP)方法。这种依赖施加了严格的结构限制，包括阶障碍以及与各种全离散DG格式的不兼容性，这些格式涵盖从最近具有紧致模板的RKDG方法(cRKDG)到广泛建立的任意导数(ADER) DG方法。为了绕过这些约束，我们提出了一个新颖的迹限制校正子框架，该框架使用通用的非SSP时间步进保持均值意义下的TVD/TVB性质。基于Harten引理，我们的核心见解是全变差稳定性仅由最终校正阶段中的单元平均值更新所决定。因此，我们仅在最终阶段修改数值通量中的迹，而让中间的预测步阶段不受约束。这一策略将振荡控制与SSP限制解耦，兼容标准RKDG、cRKDG和ADER-DG预测步，并保留了cRKDG框架的紧致性。我们还证明限制器在光滑区域不会被激活，从而保持了本来的精度。最后，给出数值实验以展示该方法的能力和鲁棒性。
不相关论文（基本信息）
1. 误差条件神经求解器
Error-Conditioned Neural Solvers
作者: Haina Jiang, Liam Wang, Peng-Chen Chen, Min Seop Kwak, Seungryong Kim, Brian Bell, Jeong Joon Park
类别: cs.LG, cs.AI, cs.CV, math.NA
发布日期: 2026-06-25
arXiv ID: 2606.27354v1
链接: http://arxiv.org/abs/2606.27354v1
摘要: Neural surrogate models offer fast approximate mappings from PDE parameters to solutions, but they typically treat solving as a purely statistical task: once trained, they struggle to correct their own constraint violations and extrapolate beyond the training distribution. Recent hybrid methods promote physical correctness by targeting the PDE residual via gradient descent or Gauss–Newton steps, but inherit the compute cost and instability of the underlying classical optimizers. We show, theoretically and empirically, that numerically minimizing the PDE residual can be an unreliable proxy for reconstruction accuracy in ill-conditioned systems, explaining why these methods often do not make accurate predictions despite achieving low residuals. We propose error-conditioned Neural Solvers (ENS), built on a different principle: rather than an optimization target, the PDE residual field is passed as a direct input to the network at each iteration, enabling it to read the spatial structure of its own errors and learn an update policy to iteratively correct its predictions. Across four PDE families, ENS attains the highest prediction accuracy in the large majority of settings, with gains reaching $10\times$ on turbulent Kolmogorov flow, while avoiding the expensive compute cost of hybrid methods. ENS’s learned correction policy generalizes under distribution shift, including zero-shot parameter changes and cross-equation transfer, where its relative advantage is largest in the ill-conditioned regimes where residual minimization is least reliable. Project website: https://neuralsolver.github.io/.
2. 时间依赖模型中美式期权与灵活远期合约的定价
Valuing American options and Flexible Forwards contracts in time-dependent models
作者: Leif Andersen, Andrey Itkin, Rakhymzhan Kazbek
类别: q-fin.CP, math.NA, q-fin.MF, q-fin.PR
发布日期: 2026-06-25
arXiv ID: 2606.27335v1
链接: http://arxiv.org/abs/2606.27335v1
摘要: A flexible forward (FF) is a customized FX hedging instrument that guarantees a fixed exchange rate while letting the holder choose the delivery date within a pre-agreed window. It is therefore an American-style option on timing, and its valuation must respect the volatility skew of the underlying currency pair. We price FF contracts (and, more generally, American options) under a time-inhomogeneous Heston model which captures the forward-skew term structure while preserving analytical tractability through a recursive (matrix) Riccati solution for the joint characteristic function. Extending the integral-equation (decomposition) approach to time-dependent coefficients, we derive a Volterra equation characterizing the early-exercise surface. The expectation in the decomposition formula is evaluated by two complementary spectral methods: a double cosine (COS) expansion of the transition density, and a damped-Sinc (DSINC) local-basis scheme that is more accurate and stays robust when a low Feller ratio or large vol-of-vol induces Gibbs oscillations in the COS series. Benchmarked against a penalty-iteration MCS-ADI finite-difference solver, both methods price a contract in about 1-2 seconds, roughly an order of magnitude faster than the finest finite-difference grid, while DSINC improves median accuracy over COS by about a factor of twelve. The experiments also show that the early-exercise surface is a substantially nonlinear function of the variance, contrary to the linear-in-variance approximation common in earlier work.
3. 一种模拟一维粘性扩散的修正Euler-Maruyama方法
A modified Euler-Maruyama method to simulate a one-dimensional sticky diffusion
作者: Chenqi Jiang, Miranda Holmes-Cerfon
类别: math.NA, math.PR
发布日期: 2026-06-25
arXiv ID: 2606.27259v1
链接: http://arxiv.org/abs/2606.27259v1
摘要: A sticky diffusion is a process that can stick to and detach from a lower-dimensional boundary. A challenge in simulating such a process is in capturing the change in dimension in a dynamically consistent way. We introduce a numerical algorithm to simulate a one-dimensional sticky diffusion, which sticks to and detaches from a point. Our method is a simple modification of the standard Euler-Maruyama scheme, which chooses with some probability between a reflected Euler-Maruyama update and a jump to the sticky point. We show how to choose this probability to be consistent with the generator of the desired dynamics, and we prove that our scheme converges weakly to a sticky diffusion with order 1.
4. 为实数线上抛物型微分方程的解提供快速收敛的切尔诺夫近似的拟费曼公式
Quasi-Feynman formulas that provide fast converging Chernoff approximations to solution of parabolic differential equation on the real line
作者: Ivan D. Remizov, Alexandr V. Vedenin
类别: math.NA, math.FA
发布日期: 2026-06-25
arXiv ID: 2606.27232v1
链接: http://arxiv.org/abs/2606.27232v1
摘要: We construct explicit approximations to the solution of a second-order parabolic partial differential equation on the real line with variable coefficients. The method is based on Chernoff’s product formula and uses a new operator-valued function defined through proper Riemann integrals over a bounded interval, which makes the approach readily usable in numerical practice. For sufficiently smooth initial data and coefficients, we prove that the resulting Chernoff approximations converge uniformly in space and time with a quadratic rate, improving the standard first-order estimate. The construction yields a new class of quasi-Feynman formulas that are neither grid-based nor Galerkin-type, but instead rely on semigroup theory and multiple bounded integrals. The theoretical findings are validated by symbolic computation, and the paper contributes both refined error bounds and a practical analytical tool for parabolic problems.
5. 分层Muon：用于高效Muon优化的分块Newton-Schulz更新
Hierarchical Muon: Tiled Newton-Schulz Updates for Efficient Muon Optimization
作者: Ziyuan Tang, Tianshi Xu, Yousef Saad, Yuanzhe Xi
类别: math.NA, cs.LG
发布日期: 2026-06-25
arXiv ID: 2606.27216v1
链接: http://arxiv.org/abs/2606.27216v1
摘要: Muon-type optimizers construct update directions for dense neural-network weights by applying a finite Newton-Schulz map to momentum-gradient matrices. For an $H \times W$ matrix, with $r=\min\{H,W\}$ and $s=\max\{H,W\}$, $K$ steps of the full-matrix Newton-Schulz update require $O(r^2 s K)$ work and couple all rows and columns through repeated Gram matrix products. We introduce Hierarchical Muon (HiMuon), a tiled Newton-Schulz scheme for Muon-type optimization. HiMuon partitions each momentum-gradient matrix into $T \times T$ tiles, applies the same finite Newton-Schulz map independently to each tile, and reassembles the results. For finite $T$ below the matrix dimensions, HiMuon defines a local matrix-function map rather than a convergent approximation to the full-matrix update: spectral interactions are preserved within tiles and discarded across tile boundaries. For fixed finite $T$, the leading Newton-Schulz work decreases to $O(H W T K)$, and the computation decomposes into independent small dense matrix operations. This structure enables tile-size-dependent GPU kernels, cross-layer batching, memory-bounded chunking, and runtime tile-size schedules. Experiments on transformer training and controlled matrix-function diagnostics show that HiMuon improves optimizer-step efficiency while keeping training behavior close to full-matrix Muon in the tested regimes.
6. 路径依赖McKean-Vlasov控制的数值逼近与非渐近误差估计
Numerical Approximation for Path-Dependent McKean-Vlasov Control with Non-Asymptotic Error Estimates
作者: Olivier Bokanowski, Jean-Francois Chassagneux, Xinyu Li, Christoph Reisinger
类别: math.OC, math.NA
发布日期: 2026-06-25
arXiv ID: 2606.27181v1
链接: http://arxiv.org/abs/2606.27181v1
摘要: Path-dependent McKean–Vlasov (MKV) control models large interacting populations with history-dependent dynamics and costs. This paper develops a unified approximation-and-learning framework for continuous time path-dependent MKV problem under open-loop controls. First, an Euler discretization scheme with piecewise-constant controls is shown to achieve a non-asymptotic error of $O(h^{1/4})$. Second, we establish a discrete dynamic programming principle and prove value equivalence between open-loop and history-dependent feedback controls, enabling optimization on a reduced filtration. Third, an interacting particle system is introduced to approximate the continuous-time value, yielding an overall error bound of $O(h^{1/4}) + O(M^{-γ})$ for $M$ particles and an explicitly given $γ> 0$. Finally, we propose a fully implementable neural-network policy-gradient method using pathwise features. Numerical experiments, including a path-dependent linear-quadratic benchmark, demonstrate the effectiveness of the algorithm.
7. De la Vallée Poussin型逼近求解某些弗雷德霍姆积分方程
De la Vallée Poussin type approximation for solving some Fredholm integral equations
作者: Domenico Mezzanotte, Donatella Occorsio, Mario Pezzella, Woula Themistoclakis
类别: math.NA, math.FA
发布日期: 2026-06-25
arXiv ID: 2606.27178v1
链接: http://arxiv.org/abs/2606.27178v1
摘要: In the present paper, we introduce a numerical method for second-kind Fredholm integral equations (FIEs) based on de la Vallée Poussin-type (VP) polynomial approximations at Jacobi zeros. This class of approximations offers several advantages over classical Lagrange interpolation at the same nodes. In particular, it guarantees uniformly bounded Lebesgue constants in suitable weighted function spaces and provides near-best uniform approximation for functions in these spaces, while also significantly mitigating the Gibbs phenomenon. We show how these properties can be exploited in the numerical solution of FIEs. In particular, the proposed approach effectively handles functions with possible algebraic endpoint singularities and kernel functions featuring weak singularities or highly oscillatory behavior. Under suitable assumptions, we prove stability and convergence of the method in weighted uniform spaces. Furthermore, we develop an efficient implementation based on the solution of a well-conditioned linear system. Numerical results confirm the theoretical error estimates and show that the proposed method achieves higher local accuracy than the corresponding Lagrange-based projection method.
8. 关于基于微分强度测量的傅里叶相位恢复及其在波前传感中的应用
On Fourier Phase Retrieval from Differential Intensity Measurements with Applications to Wavefront Sensing
作者: Simon Hubmer, Lukas Weissinger, Ronny Ramlau, Otmar Scherzer
类别: math.NA
发布日期: 2026-06-25
arXiv ID: 2606.27176v1
链接: http://arxiv.org/abs/2606.27176v1
摘要: In this paper, we consider Fourier phase retrieval from differential intensity measurements, i.e., the problem of determining the phase of a complex-valued function from a series of intensity measurements differing only by slight modulations in Fourier domain. These modulations may be induced by optical elements such as prisms or phase plates, or via spatial-light modulators. Generalizing the principles behind the transport of intensity equation, we show that given such differential intensity measurements, the phase can be determined as the solution of certain partial differential or integro-differential equations. This is then used to design efficient reconstruction algorithms for a number of Fourier-type wavefront sensors, such as the pyramid wavefront sensor, commonly used in adaptive optics. Numerical experiments illustrate the usefulness of our proposed approach.
9. DMK 框架下矩形长方体上带任意周期性的快速求和
Fast summation on rectangular cuboids with arbitrary periodicity in the DMK framework
作者: David Krantz, Ludvig af Klinteberg, Anna-Karin Tornberg
类别: math.NA
发布日期: 2026-06-25
arXiv ID: 2606.27134v1
链接: http://arxiv.org/abs/2606.27134v1
摘要: Dual-space multilevel kernel-splitting (DMK) is a fast summation framework that combines ideas from the fast multipole method, Ewald summation, and multilevel summation. Originally formulated for free-space problems, and later extended to fully periodic problems on a cube, it decomposes the kernel interaction into a smooth global contribution and a hierarchy of localized interactions evaluated on an octree. We extend DMK to problems on rectangular cuboids with periodic boundary conditions in one, two, or three coordinate directions. The periodization leverages the fact that interactions on all tree levels below the root are localized, allowing for their evaluation with minimal modification on a cubical tiling of the domain. The remaining smooth root-level far-field contribution is evaluated in Fourier space, with Fourier series in the periodic directions and Fourier integrals in the free directions. For reduced periodicity, truncated kernels are used to regularize singular and near-singular Fourier kernels, yielding rapidly convergent trapezoidal discretizations and a unified treatment of all periodicities. For large-aspect-ratio cuboids, the root-level sum can be accelerated using the fast Fourier transform. We validate the method for the electrostatic potential and Stokeslet, stresslet and rotlet potentials, for all periodicities and a wide range of aspect ratios. Numerical experiments show that the periodization adds only a small overhead to the original free-space DMK algorithm, also for high-aspect-ratio cuboids. The resulting method provides a framework for applying DMK to problems with mixed periodicity on rectangular cuboids, and extends naturally to other non-oscillatory kernels for which a kernel split is available.
10. 用于相对论爆轰波的任意拉格朗日-欧拉求解器
An Arbitrary-Lagrangian-Eulerian solver for relativistic detonation waves
作者: Sara Rinaldi, Olindo Zanotti, Michael Dumbser
类别: math.NA, gr-qc, physics.flu-dyn
发布日期: 2026-06-25
arXiv ID: 2606.27113v1
链接: http://arxiv.org/abs/2606.27113v1
摘要: In this paper we study the dynamics of relativistic detonation waves theoretically and numerically. The reaction is physically accounted for by an extra term in the definition of the total energy density and by an additional equation for the evolution of the mass fraction of the reactant, while leaving formally unmodified the equations of mass and energy-momentum conservation. In this way, the Rankine-Hugoniot relations maintain the same formal structure of the inert version. For the numerical solution we use a second order finite volume ALE scheme with TVD reconstruction, where the mesh velocity is chosen equal to the shock speed. We also adopt a locally implicit algorithm for the treatment of potentially stiff reaction source terms that arise in the equation of the reactant. We furthermore propose a particularly efficient algorithm for the conversion from the conserved to the primitive variables, which for the relativistic Euler equations is known to be nontrivial. Following this approach, we can successfully solve the Zel’dovich-von Neumann-Doering profile of a relativistsic detonation wave, up to Lorentz factors of the shock front $γ_S\sim 7$. Our analysis allowed us to highlight a new special relativistic effect, which has remained unnoticed so far. While in Newtonian detonations the Zel’dovich pressure jump decreases monotonically with the mass flux through the shock front, in the relativistic case it shows a minimum and then rises monotonically as a function of the mass flux. This may have interesting physical implications on the amount of energy that can be extracted from a relativistic detonation wave.
11. 整体多方差分解：在低秩张量近似中自适应模态间相互关系
Holistic Multivariance Decomposition: Adapting Mode Interrelations in Low-Rank Tensor Approximations
作者: Süha Tuna
类别: cs.CE, math.NA
发布日期: 2026-06-25
arXiv ID: 2606.26965v1
链接: http://arxiv.org/abs/2606.26965v1
摘要: Low-rank tensor approximation is a foundational tool for multidimensional data analysis in scientific computing, classically dominated by Tucker and Canonical Polyadic (CP) decompositions. While widely adopted, these standard approximation schemes represent data as sums of rank-1 tensors formed via mode-wise outer products. This inherent mathematical structure captures the independent variations of individual modes but systematically neglects the mutual interactions and coupled dimensional interdependencies natively embedded within the tensor. To overcome this structural limitation, we introduce the Holistic Multivariance Decomposition (HMD) framework. HMD provides a novel tensor decomposition algorithm that explicitly models both isolated mode effects and higher order mutual relationships through specialized projection operators. Numerical evaluations focusing on three distinct benchmarks from various fields demonstrate that the proposed HMD framework consistently yields significantly lower reconstruction errors compared to both Tucker and CP decomposition. These results establish HMD as a robust, high fidelity computational method for resolving complex, deeply coupled multidimensional data structures in science and engineering applications.
12. 求解拟可分矩阵的位移系统
Solving Shifted Systems for Quasiseparable Matrices
作者: Luca Gemignani
类别: math.NA
发布日期: 2026-06-25
arXiv ID: 2606.26952v1
链接: http://arxiv.org/abs/2606.26952v1
摘要: In this paper we develop fast numerical algorithms for solving shifted linear systems with semidefinite quasiseparable matrices. A combination of Givens and hyperbolic plane rotations is used to update the Cholesky-type factorization of the input quasiseparable matrix by determining a factorization of its shifted version of the form $LDL^T$, where $L$ is lower triangular and $D$ is a signature matrix. If the shifted matrix is also definite then the Cholesky factorization of the shifted matrix is computed in a stable way by using orthogonal transformations. Since quasiseparability is maintained under diagonal shifting, a fast variant of the updating procedure using computations with generators is also devised. Numerical experiments show the effectiveness and robustness of the proposed algorithm.
13. 从流式TT格式数据增量压缩张量火车：在降阶建模中的应用
Incremental Tensor-Train Compression from Streaming TT-Formatted Data: Applications to Reduced-Order Modeling
作者: Wei Guo, Zhichao Peng
类别: math.NA
发布日期: 2026-06-25
arXiv ID: 2606.26900v1
链接: http://arxiv.org/abs/2606.26900v1
摘要: High-dimensional tensor data streams arise naturally in scientific and engineering applications, such as simulations of kinetic equations and quantum systems, where samples become available sequentially and are often already represented in compressed low-rank tensor formats. Existing streaming tensor-train (TT) algorithms typically construct or update representations from dense tensor data or randomized sketches. However, when high-dimensional data are generated directly in TT or related low-rank formats, reconstructing dense tensors solely for the purpose of compression is unnecessary and computationally prohibitive. We develop a deterministic incremental TT compression algorithm that operates directly on streaming TT-formatted data. Given a new TT tensor, the proposed method updates an accumulated TT representation through core-wise projection, residual orthogonalization, and adaptive enrichment, retaining only the complementary information that cannot be represented within a prescribed tolerance. By operating entirely at the level of TT cores, the algorithm avoids reconstructing either the incoming tensor or the accumulated full tensor. We establish approximation error bounds for the proposed incremental approach. Moreover, we show that the accumulated TT representation corresponds to a compressed analogue of standard proper orthogonal decomposition for full-order snapshot data, enabling reduced-order models to be constructed directly from streaming low-rank solution data through operations on TT cores, without first reconstructing full snapshots. Numerical experiments on parametric radiative transfer equations demonstrate that the proposed method achieves comparable reconstruction accuracy with substantially reduced wall time and yields efficient and accurate ROMs directly from compressed low-rank data.
14. 使用SamAdams变时间步长和位置自适应朗之万动力学的加速采样
Accelerated sampling using SamAdams variable timesteps and position-adaptive Langevin dynamics
作者: Benedict Leimkuhler, Peter A. Whalley
类别: math.NA, cs.LG, stat.CO
发布日期: 2026-06-25
arXiv ID: 2606.26881v1
链接: http://arxiv.org/abs/2606.26881v1
摘要: We introduce an accelerated Langevin-based sampling method that is based on two complementary devices: \emph{SamAdams} adaptive timestepping, which automatically shrinks the effective integration step in stiff regions of phase space using a relaxed stiffness monitor, and \emph{position-adaptive Langevin} (PAL) dynamics, which concentrates friction along the local force direction while preserving the canonical distribution as the exact invariant measure. The resulting combined scheme (SA-PAL) is implemented in a palindromic integrator which requires only one force evaluation per iteration through suitable organisation of the integration steps and by exploiting the rank-one-plus-scalar structure of the PAL friction tensor. We test the method on various model problems: the Rosenbrock function, a thin entropic channel, the Mueller-Brown potential, and a Bayesian parameterisation problem with a sparsity-inducing shrinkage prior. On the Rosenbrock and Mueller-Brown potentials mixing rates are improved by 1.5-3 times compared to fixed stepsize integration. Efficiency gains of more than an order of magnitude are documented in the other examples.
15. 位移单调性下一阶平均场博弈的数值分析
Numerical analysis of first-order mean field games under displacement monotonicity
作者: Alpár R. Mészáros, Yohance A. P. Osborne
类别: math.NA, math.AP, math.OC
发布日期: 2026-06-25
arXiv ID: 2606.26853v1
链接: http://arxiv.org/abs/2606.26853v1
摘要: We introduce a particle method for the numerical approximation of time-dependent first-order Mean Field Games (MFGs) systems with non-separable, displacement monotone Hamiltonians and terminal costs, for arbitrary time-horizons and (possibly) singular initial player distributions in $\mathcal{P}_2(\mathbb{R}^d)$. The numerical scheme is based on an implicit Euler discretization in time and sampling in space of the characteristic Hamiltonian/Pontryagin system associated with the continuous MFGs system. We prove convergence of the approximations of the player distribution in the $L^{\infty}(\mathcal{W}_2)$-metric and the approximations for the gradient of the value function along optimal trajectories in the $L^{\infty}{(L^2)}$-norm as the number of spatial samples tends to infinity jointly with the temporal time-step vanishing. The error bound that we establish for this convergence further implies rates of convergence of the scheme for a range of spatial sampling techniques. Provided that the Lagrangian and terminal costs are additionally locally Lipschitz continuous, we also establish an asymptotic error bound in the $L^{\infty}(L^1)$-norm for the approximations of the value function along optimal trajectories. This is the first work in the literature on rigorous numerical approximation and analysis of first-order MFG systems that handles non-separable Hamiltonians and potentially singular initial agent distributions for arbitrary long time horizons. We illustrate the performance of the scheme in numerical experiments for a range of initial agent distributions, time horizons and space dimensions.
16. 伪谱计算中的奇异子空间与投影子空间重用
Recycling singular and projection subspaces for pseudospectra computation
作者: Kuan Deng, Kuan Xu
类别: math.NA
发布日期: 2026-06-25
arXiv ID: 2606.26785v1
链接: http://arxiv.org/abs/2606.26785v1
摘要: Computing matrix pseudospectra over a prescribed region requires evaluating the smallest singular value of $C-zI$ at a large number of grid points, which can be prohibitively expensive for large-scale matrices. We develop a recycling-based framework for accelerating such computations for both dense and sparse matrices. The main idea is to exploit the correlation between singular value problems at neighboring grid points by adaptively recycling singular subspaces computed at previously visited points by an iterative SVD solver. We develop fast Rayleigh-Ritz-SVD procedures for extracting Ritz singular pairs from the recycled singular subspaces, together with fast residual evaluation procedures, with an overall cost that scales linearly with the number of recycled samples. When the iterative SVD solver admits preconditioning, we propose using a two-level preconditioner whose projection subspaces are recycled. Numerical experiments demonstrate that the proposed recycling strategies yield substantial speedups over existing methods while maintaining the accuracy of the computed pseudospectra.
17. 巴拿赫空间中度量广义逆的正则化与二分现象
Regularization of the metric generalized inverse in Banach spaces and the dichotomy phenomenon
作者: Peter Mathé, Bernd Hofmann
类别: math.NA, math.FA
发布日期: 2026-06-25
arXiv ID: 2606.26730v1
链接: http://arxiv.org/abs/2606.26730v1
摘要: For a bounded linear operator acting between Banach spaces, its metric generalized inverse is the analog to the prominent Moore-Penrose inverse for operators acting between Hilbert spaces. This generalized inverse is well-defined for Banach spaces that are strictly convex and reflexive. Previous studies had been restricted to closed range situations, where the metric generalized inverse constitutes a continuous homogeneous mapping. The focus of the present study is on the ill-posed situation in the sense of Nashed, when the governing operator has a non-closed range. We define and analyze iterative schemes as the Landweber and the Schulz-Newton method, as well as parametric schemes with focus on a specific Tikhonov method. Both types are called regularizations aimed at approximating the metric generalized inverse. As a fundamental feature of such schemes we observe a dichotomy. This emphasizes that these schemes, when applied to elements of the domain of the metric generalized inverse, approximate the corresponding best approximate solutions well, whereas the resulting approximations will be asymptotically unbounded if they are applied to elements that do not belong to the domain of the metric generalized inverse.
18. 深度学习的算法基础：复杂度理论速率与通用逼近的刻画
Algorithmic Foundations of Deep Learning: Complexity-Theoretic Rates and a Characterization of Universal Approximation
作者: Anastasis Kratsios, Simone Brugiapaglia, Bum Jun Kim, Gregory Cousins, Haitz Sáez de Ocáriz Borde
类别: cs.LG, cs.AI, cs.LO, math.NA
发布日期: 2026-06-25
arXiv ID: 2606.26705v1
链接: http://arxiv.org/abs/2606.26705v1
摘要: Feedforward neural network (NN) expressivity is typically studied by emulating optimal basis-expansion schemes. While powerful, this perspective is incomplete: it primarily captures complexity through regularity, and therefore does not distinguish intuitively simple and complicated objects with comparable regularity, such as the square-root function and a typical Brownian path. The guiding message is that neural networks should be viewed not only as flexible basis functions, but also as models of computation. If a function is computable by a real-valued circuit over a prescribed elementary gate language, then it can be computed to comparable accuracy by an NN with explicit depth, width, and non-zero-parameter bounds controlled by the depth, width, gate count, and gate structure. Thus, neural-network complexity is not governed by regularity alone, but also by algorithmic complexity. We then show that any definable NN model satisfying a natural parallelization condition, allowing possibly multivariate non-linearities such as attention or layer normalization, is a universal approximator if and only if it contains a non-affine nonlinearity. The scope of our theory is illustrated by deducing universal approximation guarantees for continuous functions, minimax-optimal approximation guarantees for Besov classes, logarithmic-error complexity for holomorphic functions, and by showing that NNs can emulate numerical algorithms such as Newton-Raphson root finding and power iteration without architecture-specific arguments. Its precision is illustrated by shortest-path computation on $k$-vertex graphs: compiling the tropical dynamic-programming circuit yields NNs with O(log(1/ε)) non-zero parameters, exponentially improving in 1/ε over the generic $O(ε^{-c k^2})$ Lipschitz-approximation scale, for a constant c>0.
19. Kohn-Sham密度泛函理论中磁系统的预条件
Preconditioning Magnetic Systems in Kohn-Sham Density Functional Theory
作者: Clémentine Barat, Antoine Levitt, Marc Torrent
类别: cond-mat.mtrl-sci, math.NA
发布日期: 2026-06-25
arXiv ID: 2606.26693v1
链接: http://arxiv.org/abs/2606.26693v1
摘要: The convergence of the self-consistent field iterations in Kohn-Sham density functional theory can be significantly hindered by the presence of small eigenvalues in the dielectric matrix, which are often associated with electronic phase transitions in magnetic systems. In this work, we study this type of convergence issues and propose a new preconditioning scheme to mitigate them. Our preconditioning scheme is inspired by the Stoner model and based on a non-interacting susceptibility that neglects orbital variations. We demonstrate the effectiveness of our approach on a range of ferromagnetic systems, showing that it can significantly reduce the number of iterations required to achieve convergence in the vicinity of magnetic phase transitions.
20. 张量奇异值分解模型的自适应随机主元选取
Adaptive Randomized Pivoting for Tensor Singular Value Decomposition Model
作者: Ahmadsho Akdodshoev, Valentin Leplat, Salman Ahmadi-Asl
类别: math.NA
发布日期: 2026-06-25
arXiv ID: 2606.26688v1
链接: http://arxiv.org/abs/2606.26688v1
摘要: This paper studies how adaptive randomized pivoting (ARP), recently introduced for matrix column subset selection, can be extended to tensors in the t-product framework. We propose two constructions. The first one, called ARP-T-CUR, applies matrix ARP independently to the frontal slices of the tensor in the Fourier domain. This gives a Fourier-slicewise CUR approximation and leads to a direct expected-error bound inherited from the matrix theory. The second construction, called T-ARP, selects common lateral and horizontal slices for the whole tensor. This produces a genuine tensor cross approximation in the t-product sense, but also introduces a new difficulty: the same pivot indices must be used across all Fourier slices. We make this coupling explicit and prove an expected-error bound under a frequency-alignment condition measuring how far the common tensor-level sampling rule is from the slice-wise ARP sampling rules. This condition recovers the usual $r+1$-type factor when the leverage-score distributions are aligned across frequencies. We also discuss the resulting tensor cross approximation and its connection with t-DEIM. Numerical experiments on synthetic tensors, images, and videos illustrate the behavior of the proposed methods and show the benefit of common-index tensor sampling over standard tensor cross baselines.
21. 稀疏随机图上神经ODE的零样本尺寸迁移：图子极限与伴随收敛
Zero-Shot Size Transfer for Neural ODEs on Sparse Random Graphs: Graphon Limits and Adjoint Convergence
作者: Mingsong Yan, Zhida Wang, Sui Tang
类别: cs.LG, cs.AI, math.DS, math.NA, math.OC
发布日期: 2026-06-25
arXiv ID: 2606.26662v1
链接: http://arxiv.org/abs/2606.26662v1
摘要: Graph Neural Differential Equations (GNDEs) model continuous-time graph dynamics by parameterizing Neural ODE velocity fields with Graph Neural Networks. Their local, size-independent filters suggest a zero-shot size-transfer principle: train on a small graph and deploy on larger, similar graphs without retraining. We develop a quantitative theory for this principle on sparse random graphs sampled from graphons. We consider Graphon Neural Differential Equations (Graphon-NDEs) and adjoint Graphon-NDEs as the infinite-node limits of the forward and adjoint GNDE systems, and establish well-posedness. For an $n$-node random graph with sparsity parameter $α_n$, we prove trajectory-wise convergence of GNDE solutions to Graphon-NDE solutions at rate $O((α_n n)^{-1/2})$, up to logarithmic factors, with high probability. We also establish uniform-in-time convergence bounds for adjoint systems governing hidden-state and parameter gradients. We further study discretize-then-optimize (DTO) and optimize-then-discretize (OTD) training. Under explicit Euler discretization with $M$ steps, we show that DTO and OTD are asymptotically consistent, with hidden-state and local parameter-gradient discrepancies of orders $O(1/M)$ and $O(1/M^2)$, respectively, up to sparsity and logarithmic factors. Experiments on HSBM and tent graphons support the theoretical rates, while zero-shot transfer experiments across four graphon classes demonstrate accurate deployment of learned GNDEs on larger independently sampled graphs.
22. 基于图的心电图合成：激活一致性认证与诊断感知的形态学整理
Graph-Based ECG Synthesis with Activation-Consistency Certification and Diagnostics-Aware Morphology Curation
作者: Yongjun Yoo
类别: math.NA, eess.SP
发布日期: 2026-06-25
arXiv ID: 2606.26580v1
链接: http://arxiv.org/abs/2606.26580v1
摘要: Synthetic electrocardiogram (ECG) generation can support algorithm development and robustness evaluation, but simulated signals must preserve interpretable activation, recovery, and morphology properties. We present a graph-based ECG synthesis framework that combines activation-consistency certification with diagnostics-aware morphology curation. A unified heart graph supports an eikonal-template backend (ET) and a pseudo-diffusion reaction–eikonal backend (RE). We formulate graph Eikonal activation as a Bellman fixed-point problem and use the Bellman residual as a computable certificate for activation-time consistency. Each simulated ECG is evaluated by a two-stage diagnostics pipeline that separates metric computation from experiment-specific acceptance policies. On the cardiac graph, RE-derived activation times showed near-millisecond agreement with the Eikonal backbone and achieved $R^2=0.99876$ after causal predecessor filtering. Recovery experiments showed that endo-epicardial APD gradients determined the main T-wave morphology window, whereas the diffusion strength $κ$ provided secondary repolarization smoothing. In final balanced multi-lead curation, RE accepted 658/2000 samples versus 578/2000 for ET and increased per-model morphology coverage from 0.09248 to 0.09888. The framework provides a conservative basis for controllable and curated synthetic ECG generation.
23. 魔鬼在（尾）细节中：稀疏逼近的改进恢复保证
The devil in the (de)tails: an improved recovery guarantee for sparse approximation
作者: Ben Adcock, Simone Brugiapaglia, Avi Gupta
类别: math.NA, cs.IT
发布日期: 2026-06-24
arXiv ID: 2606.26459v1
链接: http://arxiv.org/abs/2606.26459v1
摘要: Many functions exhibit approximate sparsity in their coefficients with respect to a given dictionary. In recent literature, sparse approximation in such a dictionary from i.i.d. pointwise samples, underpinned by compressed sensing, has become a powerful tool for high-dimensional function approximation. A key step in this framework is truncating the (typically countably-infinite) dictionary to a finite index set of size $n$, so that compressed sensing tools can be used to approximate the function by a sparse combination of these truncated dictionary elements. This introduces a discrete $L^2$-truncation error over the sample points, which in standard approaches, is bounded by the continuous $L^\infty$-norm. Such a deterministic, worst-case bound ignores the randomness of the sample points entirely. As a result, $n$ must be taken unnecessarily large to keep the truncation error under control, which directly inflates the size of the matrix involved in the sparse recovery algorithm and increases computational cost. In this paper, we show that by exploiting the i.i.d. structure of the sample points, the discrete $L^2$ truncation error admits a bound that instead reflects the faster decay behaviour of the continuous $L^2$-norm truncation error and yields significantly smaller truncation sets and decreased computational cost. We demonstrate this through applications to weighted Wiener spaces and anisotropic Sobolev spaces, in each case obtaining significantly smaller truncation sets than recent works. In addition, we also present an improved bound of independent interest for sparse approximation in bounded Riesz systems, where the measurement condition exhibits a smaller (and scale-invariant) dependence on the Riesz constants than in previous works.
24. 基于Halley方法与谱混沌方法的随机特征值问题快速收敛求解
A Fast-Convergence Resolution of the Stochastic Eigenproblem Using Halley’s Method and the Spectral-Chaos Approach
作者: Hugo Esquivel, Kabir Oluwatobi Idowu, Guang Lin
类别: math.NA
发布日期: 2026-06-24
arXiv ID: 2606.26375v1
链接: http://arxiv.org/abs/2606.26375v1
摘要: Solving stochastic eigenvalue problems has long been essential for informed decision-making, advancing scientific knowledge, and ensuring the reliability of engineering designs and applications. This paper underscores the need to continue enhancing existing numerical methods for solving the stochastic eigenproblem in order to improve convergence rates, computational efficiency, and robustness. Specifically, we propose a novel spectral-chaos method for solving the stochastic (linear) eigenvalue problem, employing Halley’s method as the root-finding algorithm to leverage its cubic convergence properties. Our method achieves maximal convergence in solving stochastic eigenvalue problems since its rate cannot be further improved using a higher-order Householder method due to the quadratic nature of the resulting system of equations. Additionally, due to the complexity of the resulting system of equations, a tensorial approach was developed to tackle the challenges associated with the dimensional multiplicity of the stochastic eigenvalue problem, without which the solution would have been intractable. The method is derived rigorously, with a detailed error analysis that highlights the benefit of using our approach when the eigenvector components are nearly known, the computational cost of the method is also rigorously presented, and an illustrative example is provided to demonstrate the implementation of the method. Subsequently, a case study is demoed to analyze the results and validate the advantages of using Halley’s method over Newton’s method and Monte Carlo simulations.
25. 基于混合随机动力学的采样
Sampling Using Hybrid Stochastic Dynamics
作者: Björn Engquist, Kui Ren, Yunan Yang
类别: math.NA, stat.CO
发布日期: 2026-06-24
arXiv ID: 2606.26314v1
链接: http://arxiv.org/abs/2606.26314v1
摘要: This work proposes a framework for sampling from the Gibbs distribution of a given potential using hybrid stochastic dynamics. In this framework, two distinct sampling dynamics are run in different regions of the state space. The two dynamics are coupled across the interface through natural transmission conditions that preserve the target distribution. Using a specially constructed regularization scheme, we establish an exponential rate of convergence for the hybrid dynamics to equilibrium. We also analyze the metastability properties of the hybrid dynamics in a radially symmetric landscape, showing that the hybrid scheme can improve the mean exit time. This advantage is further confirmed by the numerical experiments.