生成时间: 2026年10月28日 16:30:41
重点关注论文数量: 0
了解领域论文数量: 5
不相关论文数量: 17
了解领域论文（摘要翻译）
1. AMOR：任意网格与不规则三维几何上算子学习的自适应模态算子精化
AMOR: Adaptive Modal Operator Refinement for Operator Learning on Arbitrary Meshes and Irregular 3D Geometries
作者: David Voderholzer, Leonard Vetter, Thomas O’Leary-Roseberry
类别: math.NA, cs.CE, physics.comp-ph
发布日期: 2026-10-27
arXiv ID: 2610.37485v1
链接: http://arxiv.org/abs/2610.37485v1
摘要翻译
中文标题: AMOR：任意网格与不规则三维几何上算子学习的自适应模态算子精化
摘要翻译: 科学与工程中的参数化 PDE 通常定义在不规则网格上，其几何可能包含孔洞或复杂边界，然而诸如傅里叶神经算子（FNO）等流行的算子代理架构依赖结构化几何来实现快速计算核。我们引入自适应模态算子精化（AMOR），它用降基块替代 FNO 基于傅里叶的谱卷积，并通过统一的编码/解码接口支持数据驱动基和预设输入/输出基。因此，AMOR 可未经修改地应用于任意网格；并且由于其谱参数化对空间维度没有张量积依赖，它能有利地扩展到三维，而在三维中 FNO 的谱权重随谱模态数 $m$ 以 $m^3$ 增长。我们证明 AMOR 是连续算子的通用逼近器。在二维结构网格基准上，AMOR 总体上与 FNO 具有竞争力并超过 PCANet：它在非线性扩散和 Helmholtz 问题上优于 FNO，而静态快照基使其在时间相关的 Kolmogorov 流问题上无法达到 FNO 的精度。在三维问题上，它在线性弹性静力学上以更低的推理成本达到或超过 FNO 的精度，而在湍流顶盖驱动方腔流上落后于 FNO，并能有利地扩展到需要高容量的问题。在不规则网格上，FNO 不能直接适用，AMOR 相对于 FNO 的扩展 Geo-FNO 将相对 $L^2$ 误差降低 $31\%$（翼型）和 $42\%$（弹性）。最后，AMOR 在一个拓扑非平凡的三维飞机肋网格上，在三个物理上不同、由实验启发的载荷工况中保持强劲性能，证明其无需修改即可适用于现实工程问题。
2. 椭圆方程的算子 Barron 正则性及其在神经算子逼近中的应用
Operator Barron Regularity for Elliptic Equations and Application to Neural Operator Approximation
作者: Shuhao Cao, Ruchi Guo, Daniel Zhengyu Huang, Na Liu
类别: math.NA
发布日期: 2026-10-27
arXiv ID: 2610.37101v1
链接: http://arxiv.org/abs/2610.37101v1
摘要翻译
中文标题: 椭圆方程的算子 Barron 正则性及其在神经算子逼近中的应用
摘要翻译: 我们建立了一个面向无限维函数空间之间非线性算子的谱 Barron 正则性框架。利用有限维输入截断的取值于 Hilbert 空间的 Fourier 表示，我们建立了关于输入截断维数一致的界，并证明了一类椭圆解算子的任意阶 Barron 正则性。其次，我们利用这种算子 Barron 正则性推导了 FNO 型神经算子的逼近误差界，该界显式依赖于输入截断维数、输出重构维数和网络宽度。对于固定的输入截断维数，在 Barron 正则性阶数上优化神经逼近界，可得到逼近误差随网络宽度呈拉伸指数衰减。我们还给出了一些数值实验来支持所预测的速率。
3. 非均匀声学介质中的运动点源反问题：唯一性、稳定性与重构
An inverse moving point source problem in inhomogeneous acoustic media: uniqueness, stability and reconstruction
作者: Dong Qiu, Xiang Xu, Yue Zhao
类别: math.AP, math.NA
发布日期: 2026-10-27
arXiv ID: 2610.36830v1
链接: http://arxiv.org/abs/2610.36830v1
摘要翻译
中文标题: 非均匀声学介质中的运动点源反问题：唯一性、稳定性与重构
摘要翻译: 本文研究非均匀声学介质中未知运动点源的轨迹与强度的同时反演。我们证明，在有限时间区间上的全边界测量唯一确定该运动源。此外，在适当的传播假设与秩假设下，利用源区域外四个固定接收器测得的波形，得到了确定轨迹与强度的 Lipschitz 稳定性。提出了一种新的 Volterra–Euler 重构格式用于求解该反问题，并给出了该格式的误差估计。数值实验验证了所提数值方法的有效性。
4. 各向异性曲线缩短流的参数有限元逼近的极小化运动框架
A minimizing-movement framework for parametric finite element approximation of anisotropic curve shortening flows
作者: Wei Jiang, Chunmei Su, Ganghui Zhang
类别: math.NA
发布日期: 2026-10-27
arXiv ID: 2610.36753v1
链接: http://arxiv.org/abs/2610.36753v1
摘要翻译
中文标题: 各向异性曲线缩短流的参数有限元逼近的极小化运动框架
摘要翻译: 我们提出了一个极小化运动框架，用于构造各向异性曲线缩短流的参数有限元方法（PFEMs）。其基石是一个容许的修正能量：一个光滑、凸且依赖于迭代的替代泛函，它优超原各向异性能量，并在当前迭代点与之一致。对于任意固定的有限元次数 $n\geq1$ 和任意正表面能密度 $γ\in C^2(\mathbb{S}^1)$，只要当前离散曲线的参数导数在任何网格单元上不为零，无约束单步问题对任意时间步长都有唯一解。它还满足单步离散能量耗散不等式。因此，它保持了底层 $L^2$-梯度流的变分结构。该框架恢复了对 Dziuk 各向同性格式的一种变分解释，并直接适用于弱各向异性情形。对于强各向异性，凸分裂去除了对原始密度的弱各向异性限制。一个标量约束进一步将该框架推广到保面积流。数值实验对光滑的各向同性和弱各向异性算例展示了大致预期的收敛性。它们也证实了能量耗散和面积保持。在强各向异性情形下，计算产生了尖锐的类小平面轮廓，但在网格加密下暴露出网格尺度振荡。
5. 面向网格协调与浸入式近不可压缩弹性分析的具有间断压力的局部稳定化等几何格式
A Local Stabilized Isogeometric Formulation with Discontinuous Pressure for Mesh-Conforming and Immersed Analysis of Nearly Incompressible Elasticity
作者: Ashkan Ali Madadi, Thomas J. R. Hughes
类别: cs.CE, math.NA
发布日期: 2026-10-26
arXiv ID: 2610.36349v1
链接: http://arxiv.org/abs/2610.36349v1
摘要翻译
中文标题: 面向网格协调与浸入式近不可压缩弹性分析的具有间断压力的局部稳定化等几何格式
摘要翻译: 近不可压缩材料出现在软组织、弹性体、凝胶和其他软固体中。然而，数值建模仍然具有挑战性，因为基于位移的格式在接近不可压缩时可能出现体积锁定。混合位移-压力格式解决了这一困难，但需要满足 inf-sup 条件的相容近似空间。在混合格式中，间断压力空间具有吸引力，因为压力自由度可以局部凝聚。然而，在高阶等几何分析中，其 Galerkin 稳定性要求压力网格粗化，且粗化程度随多项式次数增加。本文提出一种变分多尺度（VMS）稳定化混合等几何格式，在保留这种局部性的同时显著放宽该限制。关键观察是：VMS 稳定化控制单元内非常数压力模态，并且只要单元内常数压力分量满足 inf-sup 条件，该稳定化关于其参数无条件稳定。因此，对于最大连续位移样条，粗化因子为 2 的压力网格满足常数模态 inf-sup 判据，且与多项式次数无关。局部凝聚随后得到仅含位移的系统，便于在现有位移求解器中使用。该格式进一步扩展到浸入等几何分析，采用扩展 B-splines、压力单元聚合、基于细分的切割单元求积以及本质边界条件的弱施加。在 Kirsch 和 Cook’s membrane 基准中的离散 inf-sup 研究与收敛性分析证明了稳定性，并表明浸入处理保持了混合稳定性性质。因此，所提出的框架为网格协调和浸入设置下的近不可压缩问题提供了一种局部、高阶且稳健的方法。
不相关论文（基本信息）
1. 强稳定性保持 Runge–Kutta 方法的相位型表示
Phase-type representations for Strong Stability Preserving Runge–Kutta methods
作者: Haijun Yu, Quafan Zhu, Zhanyang Weng
类别: math.NA, math.PR
发布日期: 2026-10-27
arXiv ID: 2610.37425v1
链接: http://arxiv.org/abs/2610.37425v1
摘要: We develop a phase-type (PH) representation of strong stability preserving (SSP) Runge–Kutta (RK) methods. This representation provides a probabilistic interpretation of the RK coefficients and translates the chain order conditions into absorption-time moment identities. Using moment ratio bounds for PH distributions, we prove the conjectured optimal SSP coefficients for general implicit RK methods of orders two and three. The equality characterizations also establish uniqueness of the optimal RK tableaux.
2. 双线性二次输出系统的截断 $\mathcal{H}_2$ 最优模型降阶
Truncated $\mathcal{H}_2$ Optimal Model Order Reduction for Bilinear Quadratic Output Systems
作者: Heike Faßbender, Serkan Gugercin, Till Peters
类别: math.NA
发布日期: 2026-10-27
arXiv ID: 2610.37387v1
链接: http://arxiv.org/abs/2610.37387v1
摘要: We develop an $\mathcal{H}_2$ model reduction framework for polynomial bilinear systems with quadratic outputs. We define the associated $\mathcal{H}_2$ norm and show its consistency with the established norm in the original (non-polynomial) BQO case. We derive first-order necessary conditions for $\mathcal{H}_2$ optimal model reduction of these polynomial systems and use them to construct an efficient iterative algorithm for computing reduced BQO models that approximately satisfy these conditions. Numerical results on two examples demonstrate the accuracy and effectiveness of the proposed reduction approach.
3. 面向无导数 PDE 约束反问题的多保真度变分推断
Multifidelity variational inference for derivative-free PDE-constrained inverse problems
作者: Eric J. Parish
类别: math.NA
发布日期: 2026-10-27
arXiv ID: 2610.37359v1
链接: http://arxiv.org/abs/2610.37359v1
摘要: We develop a multifidelity black-box variational inference (MF-BBVI) approach for PDE-constrained Bayesian inverse problems in which forward solves are expensive and PDE gradients are unavailable. The method is built on score-function estimators for the evidence lower bound (ELBO), which avoid differentiation through the PDE model but are known to suffer from high variance. To address this high variance, we employ multifidelity approximate control-variate estimators that combine a small number of high-fidelity samples with a larger number of low-fidelity samples to reduce estimator variance and the number of high-fidelity evaluations required for optimization. We further develop a PDE-derivative-free second-order optimization strategy based on both gradient and Hessian score-function estimators. For low-fidelity model generation, we propose an on-the-fly strategy that builds and adaptively updates a reduced-order model (ROM) as the variational distribution evolves throughout the optimization. We investigate strategies for adapting the ROM and estimating the approximate control-variate gains, examining tradeoffs between estimator bias, variance reduction, and computational cost. The resulting approach facilitates derivative-free variational inference for large-scale inverse problems through a reduction in overall computational cost. We demonstrate the method on four numerical examples: a linear benchmark with an exact posterior and three PDE-constrained inverse problems.
4. 带正型记忆项的发展方程有限元逼近的后验误差分析
A Posteriori Error Analysis of Finite Element Approximation to an Evolution Equation with a Positive-Type Memory Term
作者: Sandhya Mel, Amiya Kumar Pani, Gujji Murali Mohan Reddy
类别: math.NA
发布日期: 2026-10-27
arXiv ID: 2610.37264v1
链接: http://arxiv.org/abs/2610.37264v1
摘要: This paper develops reliable a posteriori error estimates for finite element approximation of an evolution equation with a positive-type memory kernels on bounded convex polygonal or polyhedral domains, including both smooth and weakly singular kernels. For the weakly singular kernel $t^{α-1}/Γ(α),\; 0<α<1,$ the model captures the transition from parabolic diffusion to wave propagation, recovering the corresponding limiting equations as $α\to0^+$ and $α\to1^-$, respectively, with the memory term governing the intermediate dynamics. The spatial discretization employs time-dependent finite element spaces, while the backward Euler method combined with numerical quadrature is used for time integration. A unified \emph{a posteriori} analysis is established for both semidiscrete and fully discrete approximations. Elliptic reconstruction yields optimal spatial error estimates, while a memory-lag estimate preserves the essential cancellation in the convolution term, yielding a first-order temporal estimator whose kernel dependence is characterized by its integral. The resulting estimators are robust, with constants independent of the final time, spatial mesh size, and time step. Their kernel dependence involves only computable quantities, avoiding exponential growth in time. Moreover, for the weakly singular kernel, the $α$-dependent factors remain uniformly bounded for $α\in(0,1).$ Numerical experiments on a convex domain validate the theoretical results, and a space-time adaptive algorithm is also demonstrated on an L-shaped domain featuring a corner singularity and two temporal layers.
5. 基于Schrödingerisation的弹性波方程量子线路
Quantum circuits of elastic wave equations via Schrödingerisation
作者: Shi Jin, Chundan Zhang
类别: quant-ph, math.NA
发布日期: 2026-10-27
arXiv ID: 2610.37247v1
链接: http://arxiv.org/abs/2610.37247v1
摘要: We construct explicit quantum circuits for the velocity-stress formulation of elastic wave equations in homogeneous media using Schrödingerisation. After rewriting the equations in symmetric matrix form(SMF), we use Fourier spectral discretization for source-driven periodic problems and staggered-grid finite differences for Dirichlet, Neumann, and impedance boundary conditions. The Schrödingerisation method transforms any linear PDEs into a system of Schrödinger-type PDEs -with unitary evolution- using the warped phase transformation that maps the equations in one higher dimension. For periodic problems, we develop circuits by combining controlled Fourier-mode rotations. For physical boundaries, interleaved encoding swaps the spatial and field-component registers, enabling a more efficient circuit construction using shift operators. We analyze $p$th-order Trotter-Suzuki formulas and derive splitting-error and CNOT-gate estimates, under the assumptions of smooth auxiliary initialization and solution-state recovery. Comparison with classical partitioned splitting methods identifies a polynomial improvement in spatial-resolution dependence for the quantum state-evolution subroutine. Noiseless circuit simulations of a forced periodic problem and an impedance problem illustrate the constructions and quantify their errors against exact solutions and classical reference computations.
6. 通过分层退火降低伪边际序贯蒙特卡洛的复杂度
Reducing Complexity of Pseudo-Marginal Sequential Monte Carlo via Hierarchical Tempering
作者: Josef Martínek, Julian Hofstadler, Alexander M. G. Cox, Robert Scheichl
类别: stat.CO, math.NA, stat.ME
发布日期: 2026-10-27
arXiv ID: 2610.37082v1
链接: http://arxiv.org/abs/2610.37082v1
摘要: The Bayesian approach to inverse problems provides a powerful framework for inferring unknown parameters from noisy observations. Among the computationally most challenging statistical inference problems are problems with intractable likelihoods, especially if no unbiased likelihood estimator is available. We focus on a class of such problems, where a biased likelihood estimator is obtained through a stochastic approximation of a forward map (stochastic solver). Obtaining an empirical posterior approximation in applications is very expensive due to the repeated stochastic solver evaluations required by standard sampling methods. To address this computational bottleneck, we develop a novel algorithm based on a density-tempered marginalized Sequential Monte Carlo (SMC) sampler. Our approach (hierarchical SMC) accelerates inference by leveraging a hierarchy of stochastic solvers with increasing accuracy both along the tempering sequence and within mutation stages of the SMC sampler. We establish, under additional assumptions, that the asymptotic speedup is proportional to the number of tempering stages $T$. We demonstrate computational gains on several examples, including a nonlinear, high-contrast inverse problem arising in photoacoustic medical imaging. On this problem, we demonstrate a speedup by a factor of 14.3, reducing the inference wall-clock time from about 4.6 days to 7.7 hours on an NVIDIA A40 GPU.
7. 部分饱和土中弹塑性水-力耦合问题的超缩减策略
A hyper-reduction strategy for elastoplastic hydro-mechanical problems in partially saturated soil
作者: Alba Muixí, Lluís Monforte, Antonio Rodríguez-Ferran, Sergio Zlotnik
类别: cs.CE, math.NA
发布日期: 2026-10-27
arXiv ID: 2610.37060v1
链接: http://arxiv.org/abs/2610.37060v1
摘要: This work proposes a hyper-reduced order model for coupled hydro-mechanical simulations of elastoplastic deformation and water flow in partially saturated soils. These simulations are used to monitor and assess the safety of geotechnical structures such as earthfill dams, where calibration, uncertainty quantification and scenario analysis require solving the same transient model many times for different parameter values. The displacement and the pore pressure are approximated in a field-split reduced basis obtained by proper orthogonal decomposition, and the Galerkin-projected equations are integrated with decomposed cubature rules, one per balance equation, built with the energy-conserving sampling and weighting (ECSW) method. Each residual is split into an affine part and a nonlinear correction: the affine part is computed exactly, at negligible online cost, so that the cubature rule only needs to approximate the correction, which is localized in space. The construction is independent of the constitutive model choices and is implemented in FEniCSx by assembling the original variational forms on the small set of elements selected by ECSW. The method is tested on two- and three-dimensional dams retaining tailings and water, simulating a raise of the dam and the subsequent consolidation phase, with the friction angle, stiffness and hydraulic conductivity of the dam material and the height of the raise as varying parameters. The hyper-reduced model reproduces the displacement and pressure fields with errors below $0.5\%$ with respect to the finite element solution, achieving speed-ups of up to $43\times$ in two dimensions and $77\times$ in three. These performance gains grow with the extent of the plastic yielding, and the selected elements concentrate in the plastified and unsaturated zones, whose size is set by the physics rather than by the mesh resolution.
8. 随机非线性特征值问题的多层蒙特卡罗方法
Multilevel Monte Carlo methods for random nonlinear eigenvalue problems
作者: Jürgen Dölz, Jimena Lozano-Simón
类别: math.NA
发布日期: 2026-10-27
arXiv ID: 2610.36955v1
链接: http://arxiv.org/abs/2610.36955v1
摘要: We develop a multilevel Monte Carlo (MLMC) estimator to compute statistical quantities of interest of solutions to parameter-dependent nonlinear eigenvalue problems. The main difficulty for this problem is that parametrized eigenpairs often lack a globally well-defined parameter-to-solution map due to crossings and bifurcations. These phenomena render eigenvalue-tracking infeasible in multi-dimensional parameter domains. To overcome this issue, we propose both an Eulerian and a Lagrangian perspective on parameter-to-solution maps for nonlinear eigenvalue problems, which remain well-defined even in the presence of local crossings and bifurcations. Their implementation is based on a straightforward modification of Beyn’s contour integral approach. Building on these maps, we propose an MLMC estimator that does not rely on matching eigenvalues or eigenvectors across samples or discretization levels. Nested approximation spaces and inter-level prolongation are only required when statistical quantities of interest related to the eigenspaces are of interest, but not for quantities of interest related to the eigenvalues. We demonstrate the feasibility and validity of our approach by numerical examples using a quasi-Monte Carlo method, including a nonlinear formulation of the Laplace Dirichlet eigenvalue problem on random domains in three dimensions motivated by the work of Steinbach and Unger. To the best of our knowledge, this is the first work considering the approximation of statistical quantities of interest for random nonlinear eigenvalue problems.
9. 用 GMRES 加速子空间上的交替投影法
Acceleration of the Method of Alternating Projections on subspaces by GMRES
作者: Luis Manuel Hernández-Ramos, Maryam Calles-Cabriles, Gorka Emaldi-Porco
类别: math.NA
发布日期: 2026-10-27
arXiv ID: 2610.36938v1
链接: http://arxiv.org/abs/2610.36938v1
摘要: In this paper we propose a GMRES-accelerated version of the Method of Alternating Projections (MAP) for subspaces. It is proved that the accelerated version not only converges to a feasible solution, but also solves a best approximation problem. Comparisons are made with other accelerations proposed in the literature, for solving saddle point problems.
10. 从随机游走到骨架：面向交互式蒙特卡罗偏微分方程求解器的可复用输运架构
From Walks to Skeletons: A Reusable Transport Architecture for Interactive Monte Carlo PDE Solvers
作者: Yunfeng Liao, Yucheng Wu, Xiucheng Li
类别: cs.GR, math.NA
发布日期: 2026-10-27
arXiv ID: 2610.36931v1
链接: http://arxiv.org/abs/2610.36931v1
摘要: Walk-on-Spheres (WoS) enables mesh-free solutions of elliptic PDEs, but conventional methods repeatedly discard sampled trajectories, limiting their efficiency in interactive applications. We introduce reusable transport skeletons, a persistent representation that organizes stochastic walks into a shared computational graph. Our framework enables efficient reuse across spatial queries and boundary conditions, while supporting progressive refinement, structure-preserving compression, and local adaptation to dynamic geometry. It further enables interactive editing of transport between selected regions. Experiments demonstrate substantial computational savings while preserving solution accuracy. By transforming transient stochastic trajectories into reusable transport structures, our method provides a flexible foundation for interactive Monte Carlo PDE solving.
11. 全隐式形式的随机传染病模型数值解的动力学性质
Dynamic Properties of Numerical Solution for Stochastic Epidemic Models with a Fully Implicit Form
作者: Zeyao Jin, Zhiming Li, Shenglong Chen
类别: math.NA, math.PR, stat.AP
发布日期: 2026-10-27
arXiv ID: 2610.36924v1
链接: http://arxiv.org/abs/2610.36924v1
摘要: This paper establishes a fully implicit model to investigate the spread of epidemics and studies its dynamic properties, extending the semi-implicit model in Yang et al.(Commun. Nonlinear Sci. Numer. Simul., 152 (2026) 109140 ). We first employ the split-step backward Euler method to address the non-adaptability arising from backward Euler discretization and ensure the global convergence of the numerical solution. By linearizing the model, the computational cost is reduced at the expense of a slight increase in error, effectively avoiding complex iterative solution processes. Then, we derive the existence, uniqueness, positivity, and boundedness of the numerical solution, and obtain its global convergence order. On this basis, the extinction and persistence of the disease are investigated under three different scenarios. The results indicate that environmental noise is a key factor leading to disease extinction. Finally, numerical simulations visually verify the obtained theoretical conclusions.
12. ODAT-SE：一个用于基于模拟的反问题的模块化并行框架
ODAT-SE: A modular and parallel framework for simulation-based inverse problems
作者: Tatsumi Aoyama, Kazuyoshi Yoshimi, Yuichi Motoyama, Akito Nakano, Hengyu Li, Yusuke Konishi, Takeo Hoshi
类别: math.NA, cond-mat.mtrl-sci
发布日期: 2026-10-27
arXiv ID: 2610.36746v1
链接: http://arxiv.org/abs/2610.36746v1
摘要: Open Data Analysis Tool for Science and Engineering (ODAT-SE) is an open-source Python framework for inverse problems that are solved by repeated forward simulations. It succeeds 2DMAT, which was developed for surface-structure analysis with quantum-beam diffraction and has since been applied to other fields. In version 3, the search algorithms were separated from the application-specific solvers, which became independent packages, and the software was renamed; this paper describes the current version 4.0. The framework connects optimization and sampling algorithms to solvers through a common interface for parameter mapping, constraints, and objective evaluation, so that an existing simulation code can be used with any of the algorithms. Relative to the previously published 2DMAT, it provides two-level MPI parallelism, which distributes candidate evaluations over processes and each evaluation over a parallel solver, checkpoint and continuation facilities, random search, global optimizers from SciPy, tensor-train optimization, and a Bayesian workflow that estimates partition-function ratios, model evidence, effective noise levels, and marginal posterior distributions. We validate the Bayesian workflow against closed-form results for a linear regression, benchmark tensor-train optimization on continuous, binary, and mixed-integer problems, and verify the two-level parallelization by the same optimum for all process decompositions. Taking up the positron-diffraction example of the 2DMAT paper, the workflow identifies the correlated structural solutions and an effective noise scale close to the imposed perturbation, while exposing incomplete convergence of symmetry-related mode weights. Solver packages, templates, and a gallery of examples support the use of the framework as a general platform for inverse-problem analysis.
13. 用于高维反射布朗运动中高效平稳尾部计算的全纯Cauchy网络
Holomorphic Cauchy Networks for Efficient Stationary Tail Computation in High-Dimensional Reflected Brownian Motions
作者: Kehan Wang
类别: cs.LG, math.NA, math.PR
发布日期: 2026-10-27
arXiv ID: 2610.36558v1
链接: http://arxiv.org/abs/2610.36558v1
摘要: Stationary tail computation for high-dimensional reflected Brownian motions (RBMs) remains computationally challenging. Building on NN4MGF’s shared-coordinate basic-adjoint-relationship (BAR) framework, we develop Holomorphic Cauchy Networks (HCNs). Native complex-valued rational Cauchy subnetworks enforce holomorphicity on pole-free domains, removing sampled Cauchy-Riemann penalties and their input-derivative graphs. Exact origin anchoring and boundary-coordinate exclusion enforce structural identities, while anchored interactions represent dependent transforms. Under holomorphic-log assumptions, we establish simultaneous value and first-derivative approximation for the constrained architectures and a mixed-derivative lower bound for coordinate-only representations under nonzero cross-covariance. Closed-form real-axis derivatives and sequential backpropagation further control memory. On the same NVIDIA H100 NVL, the tested 30D HCN configuration uses approximately tenfold less peak allocated memory and has 9.7-fold lower mean update time than NN4MGF; extrapolation to the respective scheduled update counts gives a projected full-schedule update-time ratio of 58.2. With 64 million sampled base inputs per high-dimensional run, HCN reaches 50 dimensions on a laptop GPU with 6.367% mean relative error over thresholds 1 through 8. Degree-matched comparisons yield lower reported mean relative errors over the primary evaluation ranges, including 1.422% versus 6.684% on the 2D Harrison non-product benchmark. An exploratory 10D correlated example illustrates the interaction extension.
14. QT-乘积下四元数张量奇异值分解的快速算法及其在彩色视频处理中的应用
A Fast Algorithm for Quaternion Tensor Singular Value Decomposition under the QT-Product with Applications to Color Video Processing
作者: Daochang Zhang, Yue Zhao, Jingqian Li
类别: math.NA
发布日期: 2026-10-27
arXiv ID: 2610.36550v1
链接: http://arxiv.org/abs/2610.36550v1
摘要: Based on the complex representation of quaternion matrices, this paper revisits the quaternion tensor singular value decomposition (QT-SVD) under the QT-product and establishes the corresponding $z$-block circulant matrix decomposition. We develop an efficient implementation of the QT-SVD. Numerical experiments show that this implementation is substantially faster than QT-SVD based on the quaternion matrix SVD approach \cite[Theorem~2.3]{AMLzhang2022} while maintaining high numerical accuracy. We further apply the QT-SVD to color video encryption and reconstruction. Experiments on five DAVIS video sequences show that the QT-SVD achieves higher reconstruction quality and shorter decomposition times than the quaternion T-SVD \cite{tsvdarxiv} under the same processing procedure. Both methods exhibit comparable ciphertext statistics and differential sensitivity, while the QT-SVD introduces less additional distortion under the tested input perturbations.
15. 基于轨迹优化的最优控制平衡活跃子空间
Balanced Active Subspaces for Optimal Control via Trajectory Optimization
作者: Emmanuel S. Ameh, Samuel E. Otto
类别: math.OC, cs.LG, math.DS, math.NA
发布日期: 2026-10-26
arXiv ID: 2610.36342v1
链接: http://arxiv.org/abs/2610.36342v1
摘要: The curse of dimensionality associated with solving the Hamilton-Jacobi-Bellman equation for the value function remains a key obstacle for real-time optimal feedback control of high-dimensional nonlinear dynamical systems. This is especially severe in aerospace applications such as relaminarizing unsteady turbulent flows and recovering from aerodynamic stall where nonlinear dynamics play a significant role. We introduce Balanced Active Subspaces for Optimal Control (BASOC), a model reduction framework which seeks to find directions in state space along which the value function is most sensitive, and the distribution of states exhibits maximum variance along optimal trajectories. BASOC leverages trajectory optimization and the Pontryagin minimum principle to obtain state and co-state (value function gradient) data along locally optimal trajectories for constructing state and gradient covariance matrices. Balancing these matrices provides an oblique projection for approximating the value function, which can also be used to construct reduced-order models and surrogate feedback control laws. We show that for linear-quadratic optimal control problems, BASOC becomes equivalent to a snapshot method for initial condition balancing of the optimally controlled closed-loop realization. The framework is demonstrated on a nonlinear three-dimensional system where POD and balanced truncation are known to perform poorly, and on the nonlinear complex Ginzburg-Landau equation to demonstrate feasibility for larger-scale systems.
16. 蒙特卡洛幂迭代算法：收敛性、分析及其在反应堆临界计算中的应用
Monte Carlo Power Iteration Algorithms: Convergence, Analysis and Application to Reactor Criticality Calculations
作者: Alexander M. G. Cox, Tom Davis, Julian Hofstadler, Andreas E. Kyprianou
类别: math.NA, math.PR
发布日期: 2026-10-26
arXiv ID: 2610.36266v1
链接: http://arxiv.org/abs/2610.36266v1
摘要: We study Monte Carlo power iteration algorithms in the context of reactor criticality simulations. Under mild conditions on the underlying physical system and the algorithmic structure, we derive explicit error bounds, which are to the best of our knowledge the first rigorous results for such methods which are ubiquitous in nuclear engineering. This is done by exploiting connections to Sequential Monte Carlo algorithms, and extending recent developments about stochastic neutron transport. We demonstrate the applicability of our results by considering a concrete version of the Monte Carlo power iteration algorithm as well as variance reduction schemes.
17. Wigner–Fokker–Planck 方程的动态张量列近似及其在开放量子系统中的应用
Dynamical Tensor Train Approximation for Wigner–Fokker–Planck Equation with Applications to Open Quantum Systems
作者: Geshuo Wang, Jingwei Hu
类别: math.NA, quant-ph
发布日期: 2026-10-26
arXiv ID: 2610.36257v1
链接: http://arxiv.org/abs/2610.36257v1
摘要: We develop a dynamical tensor-train method for the Wigner–Fokker–Planck equation arising in continuous-variable open quantum systems. The Wigner function is evolved directly in tensor-train form using a projector-splitting integrator, while the nonlocal Wigner pseudo-differential operator is treated through low-rank representations of the potential difference and matrix-free exponential actions. The resulting method avoids full tensor-product computations in high-dimensional phase space. Numerical experiments in one, two, and three spatial dimensions demonstrate the accuracy and scalability of the method, including simulations in a six-dimensional phase space.