Numerical Analysis
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- [1] arXiv:2509.19488 [cn-pdf, pdf, html, other]
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Title: Stability of high-order Scott-Vogelius elements for 2D non-Newtonian incompressible flowTitle: 高阶Scott-Vogelius元在二维非牛顿不可压缩流中的稳定性Subjects: Numerical Analysis (math.NA)
We consider the stability of high-order Scott-Vogelius elements for 2D non-Newtonian incompressible flow problems. For elements of degree 4 or higher, we construct a right-inverse of the divergence operator that is stable uniformly in the polynomial degree $N$ from $L^p$ to $\boldsymbol{W}^{1,p}$, show that the associated inf-sup constant is bounded below by a constant that decays at worst like $N^{-3\left| \frac{1}{2} - \frac{1}{p}\right|}$, and construct local Fortin operators with stability constants explicit in the polynomial degree. We demonstrate these results with several numerical examples suggesting that the $p$-version method can offer superior convergence rates over the $h$-version method even in the non-Newtonian setting.
我们考虑高阶Scott-Vogelius元在二维非牛顿不可压缩流问题中的稳定性。对于次数为4或更高的元，我们构造了一个右逆散度算子，该算子在多项式次数$N$从$L^p$到$\boldsymbol{W}^{1,p}$的范围内是稳定的，证明了相关的inf-sup常数被一个最坏情况下像$N^{-3\left| \frac{1}{2} - \frac{1}{p}\right|}$那样衰减的常数所下界，并构造了稳定性常数显式依赖于多项式次数的局部Fortin算子。我们通过几个数值例子展示了这些结果，表明即使在非牛顿情况下，$p$版本的方法也可以比$h$版本的方法提供更优的收敛速率。
- [2] arXiv:2509.19544 [cn-pdf, pdf, html, other]
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Title: Spectral theory of matrix-sequences: perspectives of the GLT analysis and beyondTitle: 矩阵序列的谱理论：GLT分析及其以外的前景Comments: 24 pages, 0 figuresSubjects: Numerical Analysis (math.NA)
In recent years there has been a growing attention on distribution results in the sense of Weyl for the collective behavior of eigenvalues and singular values of matrix-sequences. Starting from the work of Szeg\"o regarding the case of Toeplitz matrix-sequences, there has been a wealth of associated results, which culminated in the works of Tilli and of Tyrtyshnikov, Zamarashkin, and of the author for preconditioned and non-preconditioned $r$-block $d$-level Toeplitz matrix-sequences with Lebesgue integrable generating functions. In the latter the use of matrix-valued linear positive operators and related Korovkin theories has been crucial. The subsequent steps induced by the analysis of preconditioning techniques and inspired by the rich world of the (pseudo) differential operators have been studies from the same perspective of matrix-sequences with hidden (asymptotic) structure, widely studied in the literature. The widest generalization is represented by the notion of generalized locally Toeplitz matrix-sequences, which have inherent hidden structure and which include virtually any approximation via local numerical methods of (systems of) integral equations, partial and fractional differential equations, also with nonsmooth variable coefficients and irregular bounded/unbounded domains/manifolds. In the current work, instead of focusing on a specific type of results, starting from the most recent advances on the topic, we describe shortly a series of open problems, challenges to be developed in the future.
近年来，人们对矩阵序列特征值和奇异值集体行为在Weyl意义下的分布结果越来越关注。 从Szegö关于Toeplitz矩阵序列的情况开始，已经出现了大量相关结果，这些结果最终体现在Tilli和Tyrtyshnikov、Zamarashkin以及作者对具有Lebesgue可积生成函数的预处理和非预处理$r$-block $d$-level Toeplitz矩阵序列的研究中。 在后者中，矩阵值线性正算子及其相关的Korovkin理论的使用至关重要。 由预处理技术分析引发的后续步骤，并受到（伪）微分算子丰富世界的启发，是从矩阵序列具有隐藏（渐近）结构的角度进行研究的，这在文献中已被广泛研究。 最广泛的推广是由广义局部Toeplitz矩阵序列的概念所代表，它们具有内在的隐藏结构，并且包括通过局部数值方法对（积分方程）系统、偏微分方程和分数微分方程的几乎任何近似，也包括具有不光滑变量系数和不规则有界/无界域/流形的情况。 在当前的工作中，我们不专注于特定类型的结果，而是从该领域的最新进展出发，简要描述一系列开放问题和未来需要解决的挑战。
- [3] arXiv:2509.19618 [cn-pdf, pdf, html, other]
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Title: HPL-MxP Benchmark: Mixed-Precision Algorithms, Iterative Refinement, and Scalable Data GenerationTitle: HPL-MxP基准：混合精度算法、迭代修正和可扩展数据生成Journal-ref: International Journal of High Performance Computing Applications, 2025Subjects: Numerical Analysis (math.NA) ; Performance (cs.PF)
We present a mixed-precision benchmark called HPL-MxP that uses both a lower-precision LU factorization with a non-stationary iterative refinement based on GMRES. We evaluate the numerical stability of one of the methods of generating the input matrix in a scalable fashion and show how the diagonal scaling affects the solution quality in terms of the backward-error. Some of the performance results at large scale supercomputing installations produced Exascale-level compute throughput numbers thus proving the viability of the proposed benchmark for evaluating such machines. We also present the potential of the benchmark to continue increasing its use with proliferation of hardware accelerators for AI workloads whose reliable evaluation continues to pose a particular challenge for the users.
我们提出了一种混合精度基准测试，称为HPL-MxP，它同时使用了低精度的LU分解和基于GMRES的非平稳迭代修正。 我们评估了以可扩展方式生成输入矩阵的一种方法的数值稳定性，并展示了对角线缩放如何影响解的质量，从后向误差的角度来看。 一些在大型超级计算机设施上的性能结果产生了埃克萨级计算吞吐量数据，从而证明了所提出的基准测试在评估此类机器方面的可行性。 我们还展示了该基准测试的潜力，随着用于AI工作负载的硬件加速器的普及，其使用将继续增加，而这些加速器的可靠评估仍然对用户构成特殊的挑战。
- [4] arXiv:2509.19747 [cn-pdf, pdf, html, other]
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Title: Preconditioning via Randomized Range Deflation (RandRAND)Title: 通过随机范围反向投影的预处理（RandRAND）Subjects: Numerical Analysis (math.NA)
We introduce RandRAND, a new class of randomized preconditioning methods for large-scale linear systems. RandRAND deflates the spectrum via efficient orthogonal projections onto random subspaces, without computing eigenpairs or low-rank approximations. This leads to advantages in computational cost and numerical stability. We establish rigorous condition number bounds that depend only weakly on the problem size and that reduce to a small constant when the dimension of the deflated subspace is comparable to the effective spectral dimension. RandRAND can be employed without explicit operations with the deflation basis, enabling the effective use of fast randomized transforms. In this setting, the costly explicit basis orthogonalization is bypassed by using fast randomized Q-less QR factorizations or iterative methods for computing orthogonal projections. These strategies balance the cost of constructing RandRAND preconditioners and applying them within linear solvers, and can ensure robustness to rounding errors.
我们引入了RandRAND，一种用于大规模线性系统的新型随机预处理方法。 RandRAND通过将谱分解为随机子空间上的有效正交投影来实现，而无需计算特征对或低秩近似。 这带来了计算成本和数值稳定性的优势。 我们建立了严格的条件数界限，这些界限仅弱依赖于问题规模，并且当被分解子空间的维度与有效谱维数相当时，会减少到一个小常数。 RandRAND可以在不显式操作分解基的情况下使用，从而有效利用快速随机变换。 在这种情况下，通过使用快速随机的无Q的QR分解或用于计算正交投影的迭代方法，可以绕过昂贵的显式基正交化过程。 这些策略平衡了构建RandRAND预处理程序和在线性求解器中应用它们的成本，并可以确保对舍入误差的鲁棒性。
- [5] arXiv:2509.19777 [cn-pdf, pdf, html, other]
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Title: High-order Multiscale Preconditioner for Elasticity of Arbitrary StructuresTitle: 高阶多尺度预条件子用于任意结构的弹性力学Subjects: Numerical Analysis (math.NA) ; Computational Physics (physics.comp-ph)
We present a two-level preconditioner for solving linear systems arising from the discretization of the elliptic, linear-elastic deformation equation, in displacement unknowns, over domains that have arbitrary geometric and topological complexity and heterogeneity in material properties (including fractures). The preconditioner is an algebraic translation of the high-order pore-level multiscale method (hPLMM) proposed recently by the authors, wherein a domain is decomposed into non-overlapping subdomains, and local basis functions are numerically computed over the subdomains to construct a high-quality coarse space (or prolongation matrix). The term "high-order" stands in contrast to the recent low-order PLMM preconditioner, where BCs of local basis problems assume rigidity of all interfaces shared between subdomains. In hPLMM, interfaces are allowed to deform, through the use of suitable mortar spaces, thereby capturing local bending/twisting moments under challenging loading conditions. Benchmarked across a wide range of complex (porous) structures and material heterogeneities, we find hPLMM exhibits superior performance in Krylov solvers than PLMM, as well as state-of-the-art Schwarz and multigrid preconditioners. Applications include risk analysis of subsurface CO2/H2 storage and optimizing porous materials for batteries, prosthetics, and aircraft.
我们提出了一种两层预条件器，用于求解在具有任意几何和拓扑复杂性以及材料属性异质性的域上，由椭圆线性弹性形变方程（位移未知量）离散化产生的线性系统。 该预条件器是作者最近提出的高阶孔级多尺度方法（hPLMM）的代数翻译，在此方法中，一个域被分解为非重叠子域，并在子域上数值计算局部基函数以构建高质量的粗空间（或延拓矩阵）。 “高阶”一词与近期的低阶PLMM预条件器形成对比，在低阶PLMM中，局部基问题的边界条件假设所有子域共享的界面为刚性。 在hPLMM中，通过使用合适的mortar空间允许界面发生变形，从而在具有挑战性的载荷条件下捕捉局部弯曲/扭转变矩。 在广泛的复杂（多孔）结构和材料异质性中进行基准测试，我们发现hPLMM在Krylov求解器中的表现优于PLMM，以及最先进的Schwarz和多重网格预条件器。 应用包括地下CO2/H2储存的风险分析以及优化多孔材料用于电池、假肢和飞机。
- [6] arXiv:2509.19986 [cn-pdf, pdf, other]
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Title: A fast direct solver for two-dimensional transmission problems of elastic wavesTitle: 一种用于二维弹性波传输问题的快速直接求解器Subjects: Numerical Analysis (math.NA)
This paper describes a fast direct boundary element method for elastodynamic transmission problems in two dimensions, which can be used for analyzing elastic wave scattering by an inclusion. This paper reports the development of an efficient solver based on a discretization method that is broadly applicable regardless of the inclusion shape. From the smoothness of the solutions of the Navier--Cauchy equation, it is reasonable that the displacement is approximated by the piecewise linear bases and the traction is approximated by the piecewise constant bases. However, in this mixed bases strategy, Calder\'on preconditioning, that is, an analytical preconditioning with excellent performance, cannot be applied, which means we cannot adopt iterative solvers. To address this challenge, we developed a fast direct solver formulated using both Burton--Miller and Poggio--Miller--Chang--Harrington--Wu--Tsai (PMCHWT) boundary integral equations. Our method uses a technique based on the proxy method for low-rank approximation of the coefficient matrix's off-diagonal blocks. To handle transmission problems, the proposed fast direct solver uses separate binary tree partitions for nodes and elements. Numerical examples demonstrate that our solver achieves linear computational complexity and can efficiently handle problems with multiple right-sides. Notably, the solver based on the Burton--Miller formulation is approximately 20\% faster than the one using the PMCHWT formulation. Our new method provides a versatile, fast solver, whose performance is independent of the shape of inclusions and computational parameters, such as frequency and density, for elastodynamic transmission problems.
本文描述了一种用于二维弹性动力传输问题的快速直接边界元方法，可用于分析包含物对弹性波散射的分析。 本文报告了一种基于离散化方法的高效求解器的开发，该方法无论包含物形状如何都具有广泛的适用性。 从Navier--Cauchy方程解的光滑性来看，将位移近似为分段线性基函数，将应力近似为分段常数基函数是合理的。 然而，在这种混合基函数策略中，Calderón预条件技术，即一种性能优异的解析预条件技术，无法应用，这意味着我们不能采用迭代求解器。 为了解决这一挑战，我们开发了一种基于Burton--Miller和Poggio--Miller--Chang--Harrington--Wu--Tsai（PMCHWT）边界积分方程的快速直接求解器。 我们的方法使用了一种基于代理方法的技术，用于系数矩阵非对角块的低秩近似。 为了处理传输问题，所提出的快速直接求解器对节点和单元使用了独立的二叉树划分。 数值示例表明，我们的求解器实现了线性计算复杂度，并能高效处理多个右端项的问题。 值得注意的是，基于Burton--Miller公式的求解器比使用PMCHWT公式的求解器大约快20%。 我们的新方法提供了一种多功能、快速的求解器，其性能与包含物的形状以及计算参数（如频率和密度）无关，适用于弹性动力传输问题。
- [7] arXiv:2509.19993 [cn-pdf, pdf, html, other]
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Title: Modelling and Analysis of Non-Contacting Mechanical Face Seals with Axial Disturbances and MisalignmentTitle: 非接触式机械端面密封的建模与分析，带有轴向扰动和不对中Comments: 20 pages, 11 figuresSubjects: Numerical Analysis (math.NA)
Advancements in industrial applications are driving developments in non-contacting mechanical seal technology. Key requirements include improvements in efficiency and reliability, which lead to smaller clearances, lower frictional losses, and minimisation of wear, during operation. However, a critical consideration is the effect of external disturbances experience by the seal from the local environment which may cause destabilisation, and lead to premature failures through unanticipated face contact. This work examines the dynamic behaviour of a non-contacting mechanical face seals, where a thin fluid film separates a pair of coaxial discs; a rotor (rotating face) and stator (stationary face). It is assumed that the rotor-stator have an angular misalignment and operation is under conditions involving large axial disturbances, representing external disturbances. A fully coupled unsteady mathematical representation is developed, where the fluid flow is coupled to the structural response of the stator, and the rotor motion is prescribed. The stator is modelled as a spring-mass-damper system, and the fluid film model is based on a lubrication approximation of the Navier-Stokes equations. The governing equations are solved via a numerical technique based on finite element and Runge-Kutta methods. A parameter study reveals the impact of misalignment on the seal dynamics, when experiencing an external disturbance. The angle of misalignment and corresponding amplitude of forcing can be identified when the minimum fluid film thickness becomes less than a given tolerance. This provides insights into safe operating conditions and manufacturing tolerances, with the research aiding to improve the design critera and reliability of non-contacting mechanical face seals.
工业应用的进步正在推动非接触式机械密封技术的发展。 关键要求包括提高效率和可靠性，这会导致运行过程中间隙更小、摩擦损失更低以及磨损最小化。 然而，一个关键的考虑因素是密封件从局部环境经历的外部扰动的影响，这可能导致失稳，并通过意外的端面接触导致过早失效。 本研究考察了非接触式机械端面密封的动态行为，其中一层薄流体膜分隔一对同轴盘；转子（旋转面）和定子（静止面）。 假设转子-定子存在角度不对齐，并且在涉及大轴向扰动的条件下运行，代表外部扰动。 开发了一个完全耦合的非稳态数学模型，其中流体流动与定子的结构响应耦合，而转子运动是规定的。 定子被建模为一个弹簧-质量-阻尼系统，流体膜模型基于纳维-斯托克斯方程的润滑近似。 控制方程通过基于有限元和龙格-库塔方法的数值技术求解。 参数研究揭示了在经历外部扰动时不对齐对密封动态的影响。 当最小流体膜厚度小于给定公差时，可以确定不对齐角度和相应的激励振幅。 这提供了对安全操作条件和制造公差的见解，该研究有助于改进非接触式机械端面密封的设计标准和可靠性。
- [8] arXiv:2509.20039 [cn-pdf, pdf, html, other]
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Title: A note on the compactness properties of discontinuous Galerkin time discretizationsTitle: 关于不连续伽辽金时间离散化的紧性性质的一篇笔记Subjects: Numerical Analysis (math.NA)
This work extends the discrete compactness results of Walkington (SIAM J. Numer. Anal., 47(6):4680-4710, 2010) for high-order discontinuous Galerkin time discretizations of parabolic problems to more general function space settings. In particular, we show a discrete version of the Aubin-Lions-Simon lemma that holds for general Banach spaces $X$, $B$, and $Y$ satisfying $X \hookrightarrow B$ compactly and $B \hookrightarrow Y$ continuously. Our proofs rely on the properties of a time reconstruction operator and remove the need for quasi-uniform time partitions assumed in previous works.
这项工作将Walkington（SIAM J. Numer. Anal., 47(6):4680-4710, 2010）针对抛物问题高阶不连续Galerkin时间离散化的离散紧性结果扩展到更一般的函数空间设置。特别是，我们展示了一个适用于满足$X \hookrightarrow B$紧嵌入和$B \hookrightarrow Y$连续嵌入的一般巴拿赫空间$X$、$B$和$Y$的离散Aubin-Lions-Simon引理。我们的证明依赖于时间重构算子的性质，并消除了先前工作中假设的准均匀时间划分的需求。
- [9] arXiv:2509.20056 [cn-pdf, pdf, html, other]
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Title: An Overview of Meshfree Collocation MethodsTitle: 一种无网格配点方法的综述Comments: 55 pages, 259 references, Supplementary MaterialSubjects: Numerical Analysis (math.NA) ; Computational Engineering, Finance, and Science (cs.CE)
We provide a comprehensive overview of meshfree collocation methods for numerically approximating differential operators on continuously labeled unstructured point clouds. Meshfree collocation methods do not require a computational grid or mesh. Instead, they approximate smooth functions and their derivatives at potentially irregularly distributed collocation points, often called particles, to a desired order of consistency. We review several meshfree collocation methods from the literature, trace the historical development of key concepts, and propose a classification of methods according to their principle of derivation. Although some of the methods reviewed are similar or identical, there are subtle yet important differences between many, which we highlight and discuss. We present a unifying formulation of meshfree collocation methods that renders these differences apparent and show how each method can be derived from this formulation. Finally, we propose a generalized derivation for meshfree collocation methods going forward.
我们提供了对无网格配点方法的全面概述，用于在连续标记的非结构化点云上数值逼近微分算子。 无网格配点方法不需要计算网格或网格。 相反，它们在可能不规则分布的配点（通常称为粒子）上近似光滑函数及其导数，达到所需的连续性阶数。 我们回顾了文献中几种无网格配点方法，追溯了关键概念的历史发展，并根据其推导原理提出了一种方法分类。 尽管所回顾的一些方法相似或相同，但许多方法之间存在微妙但重要的差异，我们对此进行了强调和讨论。 我们提出了一个统一的无网格配点方法表述，使这些差异明显，并展示了如何从该表述中推导出每种方法。 最后，我们提出了一个向前发展的无网格配点方法的广义推导。
- [10] arXiv:2509.20069 [cn-pdf, pdf, html, other]
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Title: Development of a Model Order Reduced Arbitrary Lagrangian Eulerian (MORALE) formulation for structures subjected to dynamic moving loadsTitle: 动态移动载荷作用下结构的模型降阶任意拉格朗日欧拉（MORALE）公式的发展Subjects: Numerical Analysis (math.NA)
In recent developments, it has been demonstrated that the Arbitrary Lagrangian Eulerian (ALE) formulation can be utilized to improve computational efficiency, when simulating the response of structures subjected to moving loads. It is also well established in literature, that Model Order Reduction (MOR) techniques significantly enhance calculation speed. This contribution details the combination of both these tools into a novel Model Order Reduced Arbitrary Lagrangian Eulerian (MORALE) formulation. Both hyperelastic and viscoelastic material models are considered. Simulations of pavement structures subjected to moving loads are then carried out, which show a significant enhancement in computational speed and efficiency. Such an efficient and fast simulation framework is of vital importance in technologies such as digital twins of roadway infrastructure (like pavements), as it enables engineers to quickly run what-if analyses and make informed decisions about the management of the structure under consideration.
近年来的发展表明，当模拟受移动载荷作用的结构响应时，可以利用任意拉格朗日欧拉（ALE）公式来提高计算效率。文献中也已明确建立，模型降阶（MOR）技术显著提高了计算速度。本研究详细介绍了这两种工具的结合，形成一种新型的模型降阶任意拉格朗日欧拉（MORALE）公式。考虑了超弹性与粘弹性材料模型。随后对受移动载荷作用的路面结构进行了仿真，结果表明计算速度和效率有显著提升。这种高效快速的仿真框架在数字孪生道路基础设施（如路面）等技术中至关重要，因为它使工程师能够快速进行假设分析，并就所考虑结构的管理做出明智决策。
- [11] arXiv:2509.20108 [cn-pdf, pdf, html, other]
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Title: Efficient Long-Time Simulations of Multiscale Systems via High-Order Numerical HomogenizationTitle: 通过高阶数值均质化实现多尺度系统的高效长时间模拟Subjects: Numerical Analysis (math.NA)
By a high-order numerical homogenization method, a heterogeneous multiscale scheme was developed in Jin & Li (2022) for evolving differential equations containing two time scales. In this paper, we further explore the technique to propose an efficient algorithm able to carry out simulations up to a long time which was prohibitive before. The new algorithm is a multigrid-in-time method which combines coarse-grid high-order approximations with fine-grid low-order evaluations. The high efficiency is attained by minimizing the computational cost while the approximation accuracy is guaranteed. A priori error estimates are rigorously established and validated by numerical examples.
通过一种高阶数值均质化方法，Jin & Li (2022) 开发了一种用于包含两个时间尺度的演化微分方程的非均匀多尺度方案。 在本文中，我们进一步探索该技术，提出一种能够进行长时间模拟的高效算法，这在之前是难以实现的。 新算法是一种时间多网格方法，它结合了粗网格的高阶近似与细网格的低阶评估。 通过最小化计算成本实现了高效率，同时保证了近似精度。 先验误差估计得到了严格建立，并通过数值例子进行了验证。
- [12] arXiv:2509.20111 [cn-pdf, pdf, html, other]
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Title: A convergent finite element method for two-phase Stokes flow driven by surface tensionTitle: 一种用于由表面张力驱动的两相斯托克斯流的收敛有限元方法Subjects: Numerical Analysis (math.NA)
We present the first convergence proof for an iso-parametric finite element discretization of two-phase Stokes flow in $\Omega \subset \mathbb{R}^d$, $d=2,3$, with interface dynamics governed by mean curvature. The proof relies on a crucial discrete coupled parabolicity structure of the error system and a powerful iso-parametric framework of convergence analysis where we do not really discriminate consistency and stability. This new mixing idea leads to a non-trivial construction of the bulk mesh in the consistency analysis. The techniques and analysis developed in this paper provide fundamental numerical analysis tools for general curvature-driven free boundary problems.
我们提出了对二维相 Stokes 流在$\Omega \subset \mathbb{R}^d$，$d=2,3$中的等参数有限元离散化的首次收敛性证明，界面动力学由平均曲率控制。 该证明依赖于误差系统的关键离散耦合抛物性结构以及一种强大的等参数收敛分析框架，在此框架中我们并不真正区分一致性和稳定性。 这种新的混合思想导致了在一致性分析中体网格的非平凡构造。 本文开发的技术和分析为一般的曲率驱动自由边界问题提供了基本的数值分析工具。
New submissions (showing 12 of 12 entries )
- [13] arXiv:2509.19467 (cross-list from cs.LG) [cn-pdf, pdf, html, other]
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Title: THINNs: Thermodynamically Informed Neural NetworksTitle: THINNs：热力学信息神经网络Subjects: Machine Learning (cs.LG) ; Numerical Analysis (math.NA)
Physics-Informed Neural Networks (PINNs) are a class of deep learning models aiming to approximate solutions of PDEs by training neural networks to minimize the residual of the equation. Focusing on non-equilibrium fluctuating systems, we propose a physically informed choice of penalization that is consistent with the underlying fluctuation structure, as characterized by a large deviations principle. This approach yields a novel formulation of PINNs in which the penalty term is chosen to penalize improbable deviations, rather than being selected heuristically. The resulting thermodynamically consistent extension of PINNs, termed THINNs, is subsequently analyzed by establishing analytical a posteriori estimates, and providing empirical comparisons to established penalization strategies.
物理信息神经网络（PINNs）是一类深度学习模型，旨在通过训练神经网络以最小化方程的残差来近似偏微分方程（PDEs）的解。 专注于非平衡涨落系统，我们提出了一种与底层涨落结构一致的物理信息惩罚选择，该结构由大偏差原理表征。 这种方法产生了一种PINNs的新公式，其中惩罚项被选择为惩罚不可能的偏离，而不是被启发式地选择。 随后通过建立分析后验估计并对现有惩罚策略进行实证比较，分析了这种热力学一致的PINNs扩展，称为THINNs。
- [14] arXiv:2509.19474 (cross-list from math.FA) [cn-pdf, pdf, html, other]
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Title: Quantum Harmonic Analysis and the Structure in Data: AugmentationTitle: 量子调和分析与数据结构：增强Comments: 13 pages, 2 figuresSubjects: Functional Analysis (math.FA) ; Machine Learning (cs.LG) ; Numerical Analysis (math.NA)
In this short note, we study the impact of data augmentation on the smoothness of principal components of high-dimensional datasets. Using tools from quantum harmonic analysis, we show that eigenfunctions of operators corresponding to augmented data sets lie in the modulation space $M^1(\mathbb{R}^d)$, guaranteeing smoothness and continuity. Numerical examples on synthetic and audio data confirm the theoretical findings. While interesting in itself, the results suggest that manifold learning and feature extraction algorithms can benefit from systematic and informed augmentation principles.
在本简短的注释中，我们研究数据增强对高维数据集主成分平滑性的影响。 使用量子调和分析的工具，我们证明对应于增强数据集的算子的特征函数位于调制空间$M^1(\mathbb{R}^d)$中，保证了平滑性和连续性。 合成数据和音频数据的数值示例验证了理论结果。 虽然结果本身有趣，但表明流形学习和特征提取算法可以从系统且有根据的增强原则中受益。
- [15] arXiv:2509.19479 (cross-list from math.GR) [cn-pdf, pdf, html, other]
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Title: PySymmetry: A Sage/Python Framework for the Symmetry Reduction of Linear G-Equivariant SystemsTitle: PySymmetry：用于线性G-等变系统的对称约简的Sage/Python框架Subjects: Group Theory (math.GR) ; Symbolic Computation (cs.SC) ; Mathematical Physics (math-ph) ; Numerical Analysis (math.NA) ; Representation Theory (math.RT)
Despite the prevalence of symmetry in scientific linear systems, these structural properties are often underutilized by standard computational software. This paper introduces PySymmetry, an open-source Sage/Python framework that implements classical representation theory to simplify G-equivariant linear systems. PySymmetry uses projection operators to generate symmetry-adapted bases, transforming equivariant operators into a more efficient block-diagonal form. Its functionalities include defining and reducing representations, calculating multiplicities, and obtaining the explicit block structure. We demonstrate PySymmetry's versatility through three case studies: a chemistry application, a numerical benchmark on the non-Hermitian Schr\"odinger equation that achieved a performance increase of over 17x compared to standard methods, and a symbolic investigation that enabled the first complete analytical classification of a challenging problem in celestial mechanics. Designed for seamless integration with libraries like NumPy and SciPy, PySymmetry offers a powerful, user-friendly tool for exploring symmetries in theoretical and applied contexts. ```
尽管对称性在科学线性系统中很普遍，但这些结构特性通常未被标准计算软件充分利用。 本文介绍了 PySymmetry，这是一个开源的 Sage/Python 框架，它实现了经典表示论来简化 G-等变线性系统。 PySymmetry 使用投影算子生成适应对称性的基，将等变算子转换为更高效的块对角形式。 其功能包括定义和约简表示、计算乘数以及获得显式的块结构。 我们通过三个案例研究展示了 PySymmetry 的多功能性：一个化学应用、一个非厄米特薛定谔方程的数值基准测试，其性能相比标准方法提高了 17 倍以上，以及一个符号研究，使得对天体力学中一个具有挑战性问题的首次完整的解析分类成为可能。 设计用于与 NumPy 和 SciPy 等库无缝集成， PySymmetry 为在理论和应用背景下探索对称性提供了一个强大且用户友好的工具。
- [16] arXiv:2509.19528 (cross-list from cs.DS) [cn-pdf, pdf, html, other]
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Title: A Note on Fine-Grained Quantum Reductions for Linear Algebraic ProblemsTitle: 关于线性代数问题的细粒度量子归约的注记Subjects: Data Structures and Algorithms (cs.DS) ; Computational Complexity (cs.CC) ; Numerical Analysis (math.NA)
We observe that any $T(n)$ time algorithm (quantum or classical) for several central linear algebraic problems, such as computing $\det(A)$, $tr(A^3)$, or $tr(A^{-1})$ for an $n \times n$ integer matrix $A$, yields a $O(T(n)) + \tilde O(n^2)$ time \textit{quantum algorithm} for $n \times n$ matrix-matrix multiplication. That is, on quantum computers, the complexity of these problems is essentially equivalent to that of matrix multiplication. Our results follow by first observing that the Bernstein-Vazirani algorithm gives a direct quantum reduction from matrix multiplication to computing $tr(ABC)$ for $n \times n$ inputs $A,B,C$. We can then reduce $tr(ABC)$ to each of our problems of interest. For the above problems, and many others in linear algebra, their fastest known algorithms require $\Theta(n^\omega)$ time, where $\omega \approx 2.37$ is the current exponent of fast matrix multiplication. Our finding shows that any improvements beyond this barrier would lead to faster quantum algorithms for matrix multiplication. Our results complement existing reductions from matrix multiplication in algebraic circuits [BCS13], and reductions that work for standard classical algorithms, but are not tight -- i.e., which roughly show that an $O(n^{3-\delta})$ time algorithm for the problem yields an $O(n^{3-\delta/3})$ matrix multiplication algorithm [WW10].
我们观察到，对于几个中心的线性代数问题，如计算一个$T(n)$时算法（量子或经典）的$\det(A)$，$tr(A^3)$或$tr(A^{-1})$对于一个$n \times n$整数矩阵$A$，将给出一个$O(T(n)) + \tilde O(n^2)$时的\textit{量子算法}对$n \times n$矩阵-矩阵乘法。 也就是说，在量子计算机上，这些问题的复杂度基本上等同于矩阵乘法的复杂度。 我们的结果是通过首先观察到Bernstein-Vazirani算法从矩阵乘法到计算$tr(ABC)$的直接量子归约，对于$n \times n$个输入$A,B,C$。 然后我们可以将$tr(ABC)$减少到我们感兴趣的问题中的每一个。 对于上述问题以及线性代数中的许多其他问题，其已知最快的算法需要$\Theta(n^\omega)$时间，其中$\omega \approx 2.37$是当前快速矩阵乘法的指数。 我们的发现表明，任何超越这一障碍的改进都将导致矩阵乘法的更快量子算法。 我们的结果补充了现有的代数电路中矩阵乘法的归约 [BCS13]，以及适用于标准经典算法的归约，但这些归约并不紧密——即，它们大致表明，该问题的一个$O(n^{3-\delta})$时间算法将产生一个$O(n^{3-\delta/3})$矩阵乘法算法 [WW10]。
- [17] arXiv:2509.19792 (cross-list from math.FA) [cn-pdf, pdf, other]
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Title: Numerical Ranges and Spectral Sets: the unbounded caseTitle: 数值域和谱集：无界情形Michel Crouzeix (IRMAR)Subjects: Functional Analysis (math.FA) ; Numerical Analysis (math.NA)
It is known that, if $\Omega$ $\subset$ C is a convex set containing the numerical range of an operator A, then $\Omega$ is a C $\Omega$ -spectral set for A with C $\Omega$ $\le$ 1+ $\sqrt$ 2. We improve this estimate in unbounded cases.
已知，如果$\Omega$ $\subset$ C 是一个包含算子 A 的数值域的凸集，则$\Omega$是 A 的一个 C $\Omega$ -谱集，其中 C $\Omega$ $\le$ 1+ $\sqrt$ 2。 我们在无界情况下改进了这一估计。
- [18] arXiv:2509.19887 (cross-list from quant-ph) [cn-pdf, pdf, html, other]
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Title: Dynamically Optimal Unraveling Schemes for Simulating Lindblad EquationsTitle: 动态最优展开方案用于模拟 Lindblad 方程Subjects: Quantum Physics (quant-ph) ; Numerical Analysis (math.NA)
Stochastic unraveling schemes are powerful computational tools for simulating Lindblad equations, offering significant reductions in memory requirements. However, this advantage is accompanied by increased stochastic uncertainty, and the question of optimal unraveling remains open. In this work, we investigate unraveling schemes driven by Brownian motion or Poisson processes and present a comprehensive parametric characterization of these approaches. For the case of a single Lindblad operator and one noise term, this parametric family provides a complete description for unraveling scheme with pathwise norm-preservation. We further analytically derive dynamically optimal quantum state diffusion (DO-QSD) and dynamically optimal quantum jump process (DO-QJP) that minimize the short-time growth of the variance of an observable. Compared to jump process ansatz, DO-QSD offers two notable advantages: firstly, the variance for DO-QSD can be rigorously shown not to exceed that of any jump-process ansatz locally in time; secondly, it has very simple expressions. Numerical results demonstrate that the proposed DO-QSD scheme may achieve substantial reductions in the variance of observables and the resulting simulation error.
随机展开方案是模拟林德布拉德方程的强大计算工具，能够显著减少内存需求。然而，这种优势伴随着增加的随机不确定性，最优展开问题仍然开放。在本工作中，我们研究由布朗运动或泊松过程驱动的展开方案，并提出了这些方法的全面参数化描述。对于单个林德布拉德算符和一个噪声项的情况，这个参数族提供了路径上范数保持的展开方案的完整描述。我们进一步分析推导出动态最优量子态扩散（DO-QSD）和动态最优量子跳跃过程（DO-QJP），它们可以最小化可观测量方差的短时间增长。与跳跃过程假设相比，DO-QSD有两个显著优势：首先，DO-QSD的方差可以严格证明不会超过任何跳跃过程假设在时间上的局部方差；其次，它具有非常简单的表达式。数值结果表明，所提出的DO-QSD方案可能显著降低可观测量的方差和相应的模拟误差。
- [19] arXiv:2509.19888 (cross-list from math.OC) [cn-pdf, pdf, html, other]
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Title: An Alternating Direction Method of Multipliers for Topology OptimizationTitle: 一种用于拓扑优化的交替方向乘子法Subjects: Optimization and Control (math.OC) ; Numerical Analysis (math.NA)
We consider a class of integer-constrained optimization problems governed by partial differential equation (PDE) constraints and regularized via total variation (TV) in the context of topology optimization. The presence of discrete design variables, nonsmooth regularization, and non-convex objective renders the problem computationally challenging. To address this, we adopt the alternating direction method of multipliers (ADMM) framework, which enables a decomposition of the original problem into simpler subproblems that can be solved efficiently. The augmented Lagrangian formulation ensures consistency across variable updates while facilitating convergence under appropriate conditions.
我们考虑一类由偏微分方程（PDE）约束控制的整数约束优化问题，并在拓扑优化的背景下通过总变差（TV）进行正则化。 离散设计变量的存在、非光滑正则化以及非凸目标函数使得该问题计算上具有挑战性。 为了解决这个问题，我们采用交替方向乘子法（ADMM）框架，该框架能够将原始问题分解为可以高效求解的简单子问题。 增广拉格朗日公式确保了变量更新的一致性，同时在适当条件下促进了收敛。
- [20] arXiv:2509.20012 (cross-list from physics.med-ph) [cn-pdf, pdf, other]
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Title: GPU-accelerated FREDopt package for simultaneous dose and LETd proton radiotherapy plan optimization via superiorization methodsTitle: 基于GPU加速的FREDopt软件包，通过超优化方法实现质子放疗计划的剂量和LETd同时优化Damian Borys, Jan Gajewski, Tobias Becher, Yair Censor, Renata Kopeć, Marzena Rydygier, Angelo Schiavi, Tomasz Skóra, Anna Spaleniak, Niklas Wahl, Agnieszka Wochnik, Antoni RucińskiComments: 29 pages. Open Access at: https://iopscience.iop.org/article/10.1088/1361-6560/ade841Journal-ref: Phys. Med. Biol. 70 155011 (2025)Subjects: Medical Physics (physics.med-ph) ; Numerical Analysis (math.NA) ; Optimization and Control (math.OC)
This study presents FREDopt, a newly developed GPU-accelerated open-source optimization software for simultaneous proton dose and dose-averaged LET (LETd) optimization in IMPT treatment planning. FREDopt was implemented entirely in Python, leveraging CuPy for GPU acceleration and incorporating fast Monte Carlo (MC) simulations from the FRED code. The treatment plan optimization workflow includes pre-optimization and optimization, the latter equipped with a novel superiorization of feasibility-seeking algorithms. Feasibility-seeking requires finding a point that satisfies prescribed constraints. Superiorization interlaces computational perturbations into iterative feasibility-seeking steps to steer them toward a superior feasible point, replacing the need for costly full-fledged constrained optimization. The method was validated on two treatment plans of patients treated in a clinical proton therapy center, with dose and LETd distributions compared before and after reoptimization. Simultaneous dose and LETd optimization using FREDopt led to a substantial reduction of LETd and (dose)x(LETd) in organs at risk (OARs) while preserving target dose conformity. Computational performance evaluation showed execution times of 14-50 minutes, depending on the algorithm and target volume size-satisfactory for clinical and research applications while enabling further development of the well-tested, documented open-source software.
本研究介绍了FREDopt，一种新开发的GPU加速的开源优化软件，用于质子治疗计划中同时优化剂量和剂量平均LET（LETd）。 FREDopt完全用Python实现，利用CuPy进行GPU加速，并集成了来自FRED代码的快速蒙特卡罗（MC）模拟。 治疗计划优化流程包括预优化和优化，后者配备了可行性寻找算法的新型优越化方法。 可行性寻找需要找到满足预定约束的点。 优越化将计算扰动交织到迭代的可行性寻找步骤中，以引导它们向更优的可行点发展，从而取代对昂贵的完整约束优化的需要。 该方法在两家临床质子治疗中心的两个治疗计划中进行了验证，在重新优化前后比较了剂量和LETd分布。 使用FREDopt进行同时剂量和LETd优化显著降低了器官风险区（OARs）中的LETd和（剂量）×（LETd），同时保持了靶区剂量的符合性。 计算性能评估显示执行时间为14-50分钟，具体取决于算法和靶区体积大小——对于临床和研究应用来说是令人满意的，同时促进了经过充分测试、文档齐全的开源软件的进一步开发。
- [21] arXiv:2509.20191 (cross-list from physics.comp-ph) [cn-pdf, pdf, html, other]
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Title: Examining the robustness of Physics-Informed Neural Networks to noise for Inverse ProblemsTitle: 检查物理信息神经网络在逆问题中对噪声的鲁棒性Comments: 25 pages without appendix, 22 figures, submitted to a journalSubjects: Computational Physics (physics.comp-ph) ; Machine Learning (cs.LG) ; Numerical Analysis (math.NA)
Approximating solutions to partial differential equations (PDEs) is fundamental for the modeling of dynamical systems in science and engineering. Physics-informed neural networks (PINNs) are a recent machine learning-based approach, for which many properties and limitations remain unknown. PINNs are widely accepted as inferior to traditional methods for solving PDEs, such as the finite element method, both with regard to computation time and accuracy. However, PINNs are commonly claimed to show promise in solving inverse problems and handling noisy or incomplete data. We compare the performance of PINNs in solving inverse problems with that of a traditional approach using the finite element method combined with a numerical optimizer. The models are tested on a series of increasingly difficult fluid mechanics problems, with and without noise. We find that while PINNs may require less human effort and specialized knowledge, they are outperformed by the traditional approach. However, the difference appears to decrease with higher dimensions and more data. We identify common failures during training to be addressed if the performance of PINNs on noisy inverse problems is to become more competitive.
近似求解偏微分方程（PDEs）对于科学和工程中动态系统的建模是基础性的。 物理信息神经网络（PINNs）是一种基于机器学习的最新方法，其许多特性和局限性仍未知。 PINNs通常被认为在计算时间和准确性方面不如传统的求解PDEs的方法，例如有限元法。 然而，PINNs常被声称在求解反问题和处理噪声或不完整数据方面具有前景。 我们比较了PINNs在求解反问题时的性能与传统方法的性能，该传统方法结合了有限元法和数值优化器。 这些模型在一系列越来越复杂的流体力学问题上进行了测试，包括有噪声和无噪声的情况。 我们发现，尽管PINNs可能需要更少的人工努力和专业知识，但它们的表现仍不如传统方法。 然而，随着维度的增加和数据量的增多，这种差异似乎在减小。 我们识别出训练过程中常见的失败情况，如果要使PINNs在噪声反问题上的性能更具竞争力，这些问题需要得到解决。
Cross submissions (showing 9 of 9 entries )
- [22] arXiv:2410.00678 (replaced) [cn-pdf, pdf, html, other]
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Title: On high-order/low-order and micro-macro methods for implicit time-stepping of the BGK modelTitle: 关于BGK模型隐式时间推进的高阶/低阶和微观-宏观方法Subjects: Numerical Analysis (math.NA)
In this paper, a high-order/low-order (HOLO) method is combined with a micro-macro (MM) decomposition to accelerate iterative solvers in fully implicit time-stepping of the BGK equation for gas dynamics. The MM formulation represents a kinetic distribution as the sum of a local Maxwellian and a perturbation. In highly collisional regimes, the perturbation away from initial and boundary layers is small and can be compressed to reduce the overall storage cost of the distribution. The convergence behavior of the MM methods, the usual HOLO method, and the standard source iteration method is analyzed on a linear BGK model. Both the HOLO and MM methods are implemented using a discontinuous Galerkin (DG) discretization in phase space, which naturally preserves the consistency between high- and low-order models required by the HOLO approach. The accuracy and performance of these methods are compared on the Sod shock tube problem and a sudden wall heating boundary layer problem. Overall, the results demonstrate the robustness of the MM and HOLO approaches and illustrate the compression benefits enabled by the MM formulation when the kinetic distribution is near equilibrium.
在本文中，将高阶/低阶（HOLO）方法与微观-宏观（MM）分解相结合，以加速气体动力学BGK方程全隐式时间步进的迭代求解器。 MM公式将动力学分布表示为局部麦克斯韦分布和扰动的总和。 在高度碰撞区域，初始和边界层以外的扰动较小，可以被压缩以减少分布的整体存储成本。 在一种线性BGK模型上分析了MM方法、常规HOLO方法和标准源迭代方法的收敛行为。 HOLO和MM方法均使用相空间中的不连续伽辽金（DG）离散化实现，这自然保持了HOLO方法所需的高阶和低阶模型之间的一致性。 在Sod激波管问题和突然壁面加热边界层问题上比较了这些方法的准确性和性能。 总体而言，结果展示了MM和HOLO方法的鲁棒性，并说明了当动力学分布接近平衡时，MM公式所实现的压缩优势。
- [23] arXiv:2410.21824 (replaced) [cn-pdf, pdf, html, other]
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Title: Secure numerical simulations using fully homomorphic encryptionTitle: 使用全同态加密的安全数值模拟Comments: accepted manuscriptJournal-ref: Comput. Phys. Commun. 318 (2026) 109868Subjects: Numerical Analysis (math.NA) ; Cryptography and Security (cs.CR) ; Computational Physics (physics.comp-ph)
Data privacy is a significant concern when using numerical simulations for sensitive information such as medical, financial, or engineering data -- especially in untrusted environments like public cloud infrastructures. Fully homomorphic encryption (FHE) offers a promising solution for achieving data privacy by enabling secure computations directly on encrypted data. Aimed at computational scientists, this work explores the viability of FHE-based, privacy-preserving numerical simulations of partial differential equations. The presented approach utilizes the Cheon-Kim-Kim-Song (CKKS) scheme, a widely used FHE method for approximate arithmetic on real numbers. Two Julia packages are introduced, OpenFHE$.$jl and SecureArithmetic$.$jl, which wrap the OpenFHE C++ library to provide a convenient interface for secure arithmetic operations. With these tools, the accuracy and performance of key FHE operations in OpenFHE are evaluated, and implementations of finite difference schemes for solving the linear advection equation with encrypted data are demonstrated. The results show that cryptographically secure numerical simulations are possible, but that careful consideration must be given to the computational overhead and the numerical errors introduced by using FHE. An analysis of the algorithmic restrictions imposed by FHE highlights potential challenges and solutions for extending the approach to other models and methods. While it remains uncertain how broadly the approach can be generalized to more complex algorithms due to CKKS limitations, these findings lay the groundwork for further research on privacy-preserving scientific computing.
数据隐私在使用数值模拟处理敏感信息（如医疗、金融或工程数据）时是一个重要的关注点——尤其是在不受信任的环境中，如公共云基础设施。全同态加密（FHE）通过允许在加密数据上直接进行安全计算，为实现数据隐私提供了一个有前景的解决方案。针对计算科学家，这项工作探讨了基于FHE的、保护隐私的偏微分方程数值模拟的可行性。所提出的方法利用了Cheon-Kim-Kim-Song（CKKS）方案，这是一种广泛用于实数近似运算的FHE方法。介绍了两个Julia包，OpenFHE$.$jl和SecureArithmetic$.$jl，它们封装了OpenFHE C++库，以提供安全算术运算的便捷接口。借助这些工具，评估了OpenFHE中关键FHE操作的精度和性能，并展示了使用加密数据求解线性对流方程的有限差分格式的实现。结果表明，密码学安全的数值模拟是可行的，但必须仔细考虑由使用FHE带来的计算开销和数值误差。对FHE施加的算法限制的分析突出了将该方法扩展到其他模型和方法的潜在挑战和解决方案。尽管由于CKKS的限制，这种方法在更复杂算法中的推广程度尚不确定，但这些发现为隐私保护科学计算的进一步研究奠定了基础。
- [24] arXiv:2411.12467 (replaced) [cn-pdf, pdf, html, other]
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Title: On Multilevel Energy-Based Fragmentation MethodsTitle: 多级基于能量的碎片化方法Subjects: Numerical Analysis (math.NA)
Energy-based fragmentation methods approximate the potential energy of a molecular system as a sum of contribution terms built from the energies of particular subsystems. Some such methods reduce to truncations of the many-body expansion (MBE); others combine subsystem energies in a manner inspired by the principle of inclusion/exclusion (PIE). The combinatorial technique of M\"obius inversion of sums over partially ordered sets, which generalizes the PIE, is known to provide a non-recursive expression for the MBE contribution terms, and has also been connected to related cluster expansion methods. We build from these ideas a very general framework for decomposing potential functions into energetic contribution terms associated with elements of particular partially ordered sets (posets) and direct products thereof. Specific choices immediately reproduce not only the MBE, but also a number of other existing decomposition forms, including, e.g., the multilevel ML-BOSSANOVA schema. Furthermore, a different choice of poset product leads to a setup familiar from the combination technique for high-dimensional approximation, which has a known connection to quantum-chemical composite methods. We present the ML-SUPANOVA decomposition form, which allows the further refinement of the terms of an MBE-like expansion of the Born-Oppenheimer potential according to systematic hierarchies of ab initio methods and of basis sets. We outline an adaptive algorithm for the a posteori construction of quasi-optimal truncations of this decomposition. Some initial experiments are reported and discussed.
基于能量的碎片化方法将分子系统的势能近似表示为由特定子系统能量构建的贡献项之和。 一些此类方法退化为多体展开（MBE）的截断；其他方法则以包含/排除原理（PIE）为灵感，以某种方式组合子系统能量。 对部分有序集上的求和进行莫比乌斯反演的组合技术，它推广了PIE，已知可以为MBE贡献项提供非递归表达式，并且也与相关的簇展开方法有关。 我们从这些思想出发，构建了一个非常通用的框架，用于将势函数分解为与特定部分有序集（posets）及其直积元素相关联的能量贡献项。 具体的选择立即不仅再现了MBE，还再现了许多其他现有的分解形式，包括例如多级ML-BOSSANOVA方案。 此外，选择不同的poset乘积会导致一种从高维逼近的组合技术中熟悉的设置，该设置已知与量子化学组合方法有关。 我们提出了ML-SUPANOVA分解形式，该形式允许根据从头算方法和基组的系统层次进一步细化Born-Oppenheimer势的类似MBE展开中的项。 我们概述了一种自适应算法，用于事后构建该分解的准最优截断。 报告并讨论了一些初步实验。
- [25] arXiv:2503.09021 (replaced) [cn-pdf, pdf, html, other]
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Title: A deep learning approach to inverse medium scattering: Learning regularizers from a direct imaging methodTitle: 一种用于逆介质散射的深度学习方法：从直接成像方法中学习正则化器Subjects: Numerical Analysis (math.NA)
This paper aims to solve numerically the two-dimensional inverse medium scattering problem with far-field data. This is a challenging task due to the severe ill-posedness and strong nonlinearity of the inverse problem. As already known, it is necessary but also difficult numerically to employ an appropriate regularization strategy which effectively incorporates certain a priori information of the unknown scatterer to overcome the severe ill-posedness of the inverse problem. In this paper, we propose to use a deep learning approach to learn the a priori information of the support of the unknown scatterer from a direct imaging method. Based on the learned a priori information, we propose two inversion algorithms for solving the inverse problem. In the first one, the learned a priori information is incorporated into the projected Landweber method. In the second one, the learned a priori information is used to design the regularization functional for the regularized variational formulation of the inverse problem which is then solved with a traditional iteration algorithm. Extensive numerical experiments show that our inversion algorithms provide good reconstruction results even for the high contrast case and have a satisfactory generalization ability.
本文旨在数值求解具有远场数据的二维反向介质散射问题。 由于逆问题的严重不适定性和强非线性，这是一个具有挑战性的任务。 众所周知，为了克服逆问题的严重不适定性，从数值上采用一种有效的正则化策略来结合未知散射体的某些先验信息是必要且困难的。 在本文中，我们提出使用深度学习方法，从一种直接成像方法中学习未知散射体支撑的先验信息。 基于学习到的先验信息，我们提出了两种用于求解逆问题的反演算法。 在第一种算法中，学习到的先验信息被结合到投影Landweber方法中。 在第二种算法中，学习到的先验信息用于设计逆问题的正则化变分形式的正则化泛函，然后使用传统的迭代算法进行求解。 大量的数值实验表明，即使对于高对比度的情况，我们的反演算法也能提供良好的重建结果，并具有令人满意的泛化能力。
- [26] arXiv:2509.06744 (replaced) [cn-pdf, pdf, other]
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Title: Chebyshev smoothing with adaptive block-FSAI preconditioners for the multilevel solution of higher-order problemsTitle: 带有自适应块-FSAI预条件器的切比雪夫平滑用于高阶问题的多级求解Subjects: Numerical Analysis (math.NA)
In this paper, we assess the performance of adaptive and nested factorized sparse approximate inverses as smoothers in multilevel V-cycles, when smoothing is performed following the Chebyshev iteration of the fourth kind. For our test problems, we rely on the partition of unity method to discretize the biharmonic and triharmonic equations in a multilevel manner. Inspired by existing algorithms, we introduce a new adaptive algorithm for the construction of sparse approximate inverses, based on the block structure of matrices arising in the partition of unity method. Additionally, we also present a new (and arguably simpler) formulation of the Chebyshev iteration of the fourth kind.
在本文中，我们评估自适应和嵌套的因子化稀疏近似逆矩阵作为多级V循环中的平滑器的性能，当平滑操作是在第四种切比雪夫迭代之后进行时。 对于我们的测试问题，我们依赖于单位分解方法以多级方式离散双调和方程和三调和方程。 受现有算法的启发，我们引入了一种新的自适应算法，用于构建稀疏近似逆矩阵，该算法基于单位分解方法中出现的矩阵的块结构。 此外，我们还提出了一种新的（且可能更简单的）第四种切比雪夫迭代的表述。
- [27] arXiv:2509.08563 (replaced) [cn-pdf, pdf, html, other]
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Title: Error Analysis of Krylov Subspace approximation Based on IDR($s$) Method for Matrix Function Bilinear FormsTitle: 基于IDR($s$)方法的Krylov子空间近似在矩阵函数双线性形式中的误差分析Subjects: Numerical Analysis (math.NA)
The bilinear form u^\top f(A) v of matrix functions appears in many application problems, where u, v \in R^n\), A \in R^{n * n}\), and f(z) is a given analytic function.The IDR(s) method effectively reduces computational complexity and storage requirements by introducing dimension reduction techniques, while maintaining the numerical stability of the algorithm. This paper studies the numerical algorithm and posterior error estimation for the matrix function bilinear form u^{\top} f(A) v based on the IDR(s) method. Through the error analysis of the IDR(s) algorithm, the corresponding error expansion is derived, and it is verified that the leading term of the error expansion serves as a reliable posterior error estimate. Based on this, in this paper a corresponding stopping criterion is proposed. This approach is dedicated to improving computational efficiency, especially by showing excellent performance in handling ill-posed and large-scale problems.
矩阵函数的双线性形式 u^\top f(A) v 出现在许多应用问题中，其中 u, v \in R^n\), A \in R^{n乘n}\), 并且 f(z) 是一个给定的解析函数。IDR(s) 方法通过引入降维技术有效地降低了计算复杂度和存储需求，同时保持了算法的数值稳定性。 本文研究基于 IDR(s) 方法的矩阵函数双线性形式 u^{\top } f(A) v 的数值算法和后验误差估计。 通过 IDR(s) 算法的误差分析，推导出相应的误差展开式，并验证了误差展开式的主项作为可靠的后验误差估计。 基于此，在本文中提出了一种相应的停止准则。 这种方法旨在提高计算效率，特别是在处理不适定和大规模问题时表现出色。
- [28] arXiv:2509.17713 (replaced) [cn-pdf, pdf, html, other]
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Title: Schrodingerization based quantum algorithms for the time-fractional heat equationTitle: 基于薛定谔化的求解时间分数阶热方程的量子算法Comments: Quantum algorithms for time-fractional equationsSubjects: Numerical Analysis (math.NA)
We develop a quantum algorithm for solving high-dimensional time-fractional heat equations. By applying the dimension extension technique from [FKW23], the $d+1$-dimensional time-fractional equation is reformulated as a local partial differential equation in $d+2$ dimensions. Through discretization along both the extended and spatial domains, a stable system of ordinary differential equations is obtained by a simple change of variables. We propose a quantum algorithm for the resulting semi-discrete problem using the Schrodingerization approach from [JLY24a,JLY23,JL24a]. The Schrodingerization technique transforms general linear partial and ordinary differential equations into Schrodinger-type systems--with unitary evolution, making them suitable for quantum simulation. This is accomplished via the warped phase transformation, which maps the equation into a higher-dimensional space. We provide detailed implementations of this method and conduct a comprehensive complexity analysis, demonstrating up to exponential advantage--with respect to the inverse of the mesh size in high dimensions~--~compared to its classical counterparts. Specifically, to compute the solution to time $T$, while the classical method requires at least $\mathcal{O}(N_t d h^{-(d+0.5)})$ matrix-vector multiplications, where $N_t $ is the number of time steps (which is, for example, $\mathcal{O}(Tdh^{-2})$ for the forward Euler method), our quantum algorithms requires $\widetilde{\mathcal{O}}(T^2d^4 h^{-8})$ queries to the block-encoding input models, with the quantum complexity being independent of the dimension $d$ in terms of the inverse mesh size $h^{-1}$. Numerical experiments are performed to validate our formulation.
我们开发了一种用于求解高维时间分数阶热方程的量子算法。 通过应用[FKW23]中的维度扩展技术，将$d+1$维的时间分数阶方程重新表述为$d+2$维的局部偏微分方程。 通过对扩展域和空间域进行离散化，通过简单的变量变换得到一个稳定的常微分方程组。 我们提出了一种用于所得半离散问题的量子算法，该算法使用了[JLY24a,JLY23,JL24a]中的薛定谔化方法。 薛定谔化技术将一般的线性偏微分方程和常微分方程转化为具有单位演化形式的薛定谔型系统，使其适用于量子模拟。 这是通过扭曲相位变换实现的，该变换将方程映射到更高维空间。 我们提供了该方法的详细实现，并进行了全面的复杂度分析，与经典方法相比，在高维情况下显示出指数级的优势——相对于网格尺寸的倒数而言。 具体来说，为了计算时间$T$的解，而经典方法至少需要$\mathcal{O}(N_t d h^{-(d+0.5)})$次矩阵-向量乘法，其中$N_t $是时间步数（例如，前向欧拉方法中为$\mathcal{O}(Tdh^{-2})$），我们的量子算法需要$\widetilde{\mathcal{O}}(T^2d^4 h^{-8})$次对块编码输入模型的查询，量子复杂度在逆网格尺寸$h^{-1}$的意义上与维度$d$无关。进行了数值实验以验证我们的公式。
- [29] arXiv:2409.09245 (replaced) [cn-pdf, pdf, html, other]
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Title: Robust Training of Neural Networks at Arbitrary Precision and SparsityTitle: 神经网络在任意精度和稀疏性下的鲁棒训练Subjects: Machine Learning (cs.LG) ; Artificial Intelligence (cs.AI) ; Computation and Language (cs.CL) ; Computer Vision and Pattern Recognition (cs.CV) ; Numerical Analysis (math.NA)
The discontinuous operations inherent in quantization and sparsification introduce a long-standing obstacle to backpropagation, particularly in ultra-low precision and sparse regimes. The standard Straight-Through Estimator (STE) is widely used to address this, but the well-understood mismatch between its quantization-aware forward pass and quantization-oblivious backward pass leads to unmanaged error that can corrupt the learning process. We solve this by introducing a denoising dequantization transform derived from a principled ridge regression objective. This transform makes the entire learning process aware of and robust to the quantization error that STE's surrogate gradient bypasses, by creating an explicit, corrective gradient path. We extend this principle to sparsification by viewing it as a special form of quantization that maps insignificant values to zero. Our unified framework allows existing models to be trained at a wide spectrum of precisions and sparsity levels with off-the-shelf recipes, achieving stable training of fully binary (A1W1) and sparse sub-1-bit networks where other methods falter. This approach yields state-of-the-art results and provides a theoretically-grounded path to hyper-efficient neural networks.
量化和稀疏化中固有的不连续操作给反向传播带来了一个长期存在的障碍，尤其是在超低精度和稀疏情况下。标准的直通估计器(STE)被广泛用于解决这个问题，但其量化感知的前向传递与量化无关的反向传递之间存在广为人知的不匹配，这会导致未管理的误差，可能破坏学习过程。我们通过引入一种从原理性岭回归目标派生的去噪解量化变换来解决这个问题。该变换通过创建一个显式的校正梯度路径，使整个学习过程意识到并抵御STE的替代梯度所绕过的量化误差。我们通过将稀疏化视为一种将不显著值映射为零的特殊量化形式，将这一原则扩展到稀疏化。我们的统一框架允许现有模型使用现成的方案在广泛的精度和稀疏性水平上进行训练，在其他方法失败的情况下实现了全二进制(A1W1)和稀疏子1位网络的稳定训练。这种方法取得了最先进的结果，并提供了一条理论基础明确的超高效神经网络路径。
- [30] arXiv:2502.08998 (replaced) [cn-pdf, pdf, html, other]
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Title: Generic Structural Stability for $2 \times 2$ Systems of Hyperbolic Conservation LawsTitle: 泛结构稳定性对于$2 \times 2$系统的双曲守恒定律Comments: 38 pages, 6 figures, link to github code: https://github.com/HK-Tan/Generic-Structural-Stability---Numerical-Simulations-for-Particle-Laden-FlowSubjects: Analysis of PDEs (math.AP) ; Differential Geometry (math.DG) ; Numerical Analysis (math.NA)
This paper presents a proof of generic structural stability for Riemann solutions to $2 \times 2$ system of hyperbolic conservation laws in one spatial variable, without diffusive terms. This means that for almost every left and right state, shocks and rarefaction solutions of the same type are preserved via perturbations of the flux functions, the left state, and the right state. The main assumptions for this proof involve standard assumptions on strict hyperbolicity and genuine non-linearity, a technical assumption on directionality of rarefaction curves, and the regular manifold (submersion) assumption motivated by concepts in differential topology. We show that the structural stability of the Riemann solutions is related to the transversality of the Hugoniot loci and rarefaction curves in the state space. The regular manifold assumption is required to invoke a variant of a theorem from differential topology, Thom's parametric transversality theorem, to show the genericity of transversality of these curves. This in turn implies the genericity of structural stability. We then apply this theorem to two examples: the p-system and a $2 \times 2$ system governing the evolution of gravity-driven monodisperse particle-laden thin films. In particular, we illustrate how one can verify all the above assumptions for the former, and apply the theorem to different numerical and physical aspects of the system governing the latter.
本文提出了对一维空间变量的双曲守恒律系统$2 \times 2$的 Riemann 解的通用结构稳定性的证明，该系统不包含扩散项。 这意味着对于几乎所有的左状态和右状态，通过通量函数、左状态和右状态的扰动，相同类型的激波和稀疏解得以保持。 此证明的主要假设包括对严格双曲性和真实非线性的标准假设，对稀疏曲线方向性的技术假设，以及由微分拓扑概念所激发的正则流形（子mersion）假设。 我们表明，Riemann 解的结构稳定性与状态空间中 Hugoniot 簇和稀疏曲线的横截性有关。 正则流形假设用于调用微分拓扑中的一个定理变体——Thom 的参数横截性定理，以证明这些曲线横截性的普遍性。 这进而意味着结构稳定性的普遍性。 随后，我们将这个定理应用于两个例子：p 系统和一个描述重力驱动单分散颗粒载薄膜演化的$2 \times 2$系统。 特别是，我们展示了如何验证前者的所有上述假设，并将定理应用于后者的不同数值和物理方面。
- [31] arXiv:2502.13105 (replaced) [cn-pdf, pdf, html, other]
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Title: Enhanced uncertainty quantification variational autoencoders for the solution of Bayesian inverse problemsTitle: 用于贝叶斯反问题求解的增强不确定性量化变分自编码器Comments: 23 pages, 9 figuresSubjects: Machine Learning (cs.LG) ; Numerical Analysis (math.NA)
Among other uses, neural networks are a powerful tool for solving deterministic and Bayesian inverse problems in real-time, where variational autoencoders, a specialized type of neural network, enable the Bayesian estimation of model parameters and their distribution from observational data allowing real-time inverse uncertainty quantification. In this work, we build upon existing research [Goh, H. et al., Proceedings of Machine Learning Research, 2022] by proposing a novel loss function to train variational autoencoders for Bayesian inverse problems. When the forward map is affine, we provide a theoretical proof of the convergence of the latent states of variational autoencoders to the posterior distribution of the model parameters. We validate this theoretical result through numerical tests and we compare the proposed variational autoencoder with the existing one in the literature both in terms of accuracy and generalization properties. Finally, we test the proposed variational autoencoder on a Laplace equation, with comparison to the original one and Markov Chains Monte Carlo.
在其他用途中，神经网络是解决实时确定性和贝叶斯反问题的强大工具，其中变分自编码器作为一种特殊的神经网络，能够从观测数据中实现模型参数及其分布的贝叶斯估计，从而实现实时反问题的不确定性量化。 在本工作中，我们基于现有的研究[Goh, H. 等，机器学习研究会议论文集，2022]，提出了一种新颖的损失函数，用于训练变分自编码器解决贝叶斯反问题。 当正向映射是仿射的时候，我们提供了变分自编码器潜在状态收敛到模型参数后验分布的理论证明。 我们通过数值测试验证了这一理论结果，并在准确性和泛化性能方面将所提出的变分自编码器与文献中的现有方法进行了比较。 最后，我们将所提出的变分自编码器应用于拉普拉斯方程，并与原始方法和马尔可夫链蒙特卡洛方法进行了比较。