生成时间: 2026年06月11日 07:19:17

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. 一种三维四阶奇异摄动问题的解耦低阶协调混合有限元方法

A Decoupled Low-Order Conforming Mixed Finite Element Method for a Three-Dimensional Fourth-Order Singularly Perturbed Problem

作者: Yuanchun Tang, Baiju Zhang, Zhimin Zhang

类别: math.NA

发布日期: 2026-06-10

arXiv ID: 2606.12176v1

链接: http://arxiv.org/abs/2606.12176v1

摘要翻译

中文标题: 一种三维四阶奇异摄动问题的解耦低阶协调混合有限元方法

摘要翻译: 本文针对三维空间中的四阶椭圆奇异摄动问题，提出了一种解耦的低阶协调有限元方法。借助广义Helmholtz分解，该问题被化为两个二阶椭圆问题以及一组满足无旋约束的广义奇异摄动Stokes型方程组。前者采用标准线性有限元离散。对于后者，我们使用MINI元，并证明在添加涉及Lagrange乘子的$L^2$项后，所得离散化关于摄动参数具有鲁棒性。我们进一步建立了关于摄动参数一致的阶为$h^{1/2}$的误差估计。数值实验被给出以支持该理论。

2. 完全解耦、线性且保结构的块中心有限差分方法用于交错非均匀网格上的Keller-Segel趋化系统

Fully decoupled, linear and structure-preserving block-centered finite difference methods for the Keller-Segel chemotaxis system on staggered non-uniform grids

作者: Jie Xu, Hongfei Fu

类别: math.NA

发布日期: 2026-06-10

arXiv ID: 2606.12095v1

链接: http://arxiv.org/abs/2606.12095v1

摘要翻译

中文标题: 完全解耦、线性且保结构的块中心有限差分方法用于交错非均匀网格上的Keller-Segel趋化系统

摘要翻译: 本文针对经典Keller-Segel趋化系统，在交错非均匀空间网格上提出了两种完全解耦、线性且保结构的块中心有限差分格式。这两种新格式在空间上均具有二阶精度；一种在时间上为一阶精度，另一种达到时间二阶精度。此外，我们证明了这些格式在离散层面上保持若干固有的物理定律：(i) 细胞密度和化学吸引剂浓度的正值性；(ii) 总细胞质量守恒；(iii) 一阶格式的离散能量耗散性质。特别地，时间一阶格式无条件保持正值性、质量守恒和能量耗散，而二阶格式在充分（但非必要）的时间步长条件下确保正值性。所提出的方法在指定的非均匀空间网格上能够更准确、高效地模拟趋化动力学，尤其是在快速爆破现象存在时。数值实验验证了理论结果，并说明了所提格式的准确性和可靠性。

3. 耦合非牛顿Stokes-Darcy系统的多边形间断Galerkin离散

Polytopal Discontinuous Galerkin Discretizations of Coupled Non-Newtonian Stokes-Darcy Systems

作者: Paola F. Antonietti, Michele Botti, Nicola Parolini, Valentina Pederzoli, Marco Verani

类别: math.NA

发布日期: 2026-06-10

arXiv ID: 2606.11935v1

链接: http://arxiv.org/abs/2606.11935v1

摘要翻译

中文标题: 耦合非牛顿Stokes-Darcy系统的多边形间断Galerkin离散

摘要翻译: 我们提出并分析了一种用于耦合非牛顿Stokes-Darcy系统数值近似的多边形间断Galerkin方法，该系统模拟了非牛顿自由流流体与通过多孔介质的非牛顿流之间的相互作用。由于几何灵活性和任意阶精度，所提出的离散格式非常适合具有复杂几何形状的配置。我们提供了完整的先验分析，分别考虑了自由流和多孔介质区域的剪切依赖和速度依赖的非牛顿粘度模型。在广义inf-sup理论框架下建立了方法的适定性、稳定性和误差界。误差估计通过数值结果得到了确认。

4. 全耦合非线性热-水-力问题的分裂策略

Splitting strategies for the fully-coupled nonlinear thermo-hydro-mechanical problem

作者: Stefano Bonetti, Michele Botti, Paola F. Antonietti

类别: math.NA

发布日期: 2026-06-10

arXiv ID: 2606.11801v1

链接: http://arxiv.org/abs/2606.11801v1

摘要翻译

中文标题: 全耦合非线性热-水-力问题的分裂策略

摘要翻译: 我们提出了新颖的半解耦和全解耦迭代算法，用于高效求解在多边形网格上采用间断伽辽金方法进行空间离散的全耦合非线性四场热-孔弹性模型。我们给出了模型问题、其四场形式以及用于其空间离散的任意阶加权对称内惩罚格式。该格式对于模型系数中的强非均匀性具有鲁棒性。然后，我们给出了两种求解策略，并证明了在适当条件下两种格式均收敛。通过大量数值模拟来评估所提方法的收敛性和鲁棒性。此外，我们还使用文献中和物理上合理的测试案例对格式进行检验，作为在地球物理背景下概念验证的应用。

5. 保结构神经代理模型及可处理的不确定性量化

Structure-Preserving Neural Surrogates with Tractable Uncertainty Quantification

作者: Handi Zhang, Adrienne M. Propp, Brooks Kinch, Houman Owhadi, Nathaniel Trask

类别: cs.LG, math.NA, physics.comp-ph

发布日期: 2026-06-10

arXiv ID: 2606.11650v1

链接: http://arxiv.org/abs/2606.11650v1

摘要翻译

中文标题: 保结构神经代理模型及可处理的不确定性量化

摘要翻译: 科学机器学习的最新进展为偏微分方程（PDEs）提供了近实时求解的途径，但缺乏支持当代验证与确认的传统模拟器的理论基础。本文构建了数据驱动的降阶模型，用作保结构、实时的代理模型。值得注意的是，施加物理守恒结构的外微分也揭示了拓扑结构，我们利用该拓扑结构为状态-通量关系中的不确定性建立高斯过程（GP）表示，最终得到感兴趣量的Dirichlet-to-Neumann映射，并具有后验不确定性的闭式表达式。我们特别提出了由轻量级transformer所指定的传统Raviart–Thomas元和$dgP_0$元的保结构$H(\mathrm{div})$–$L^2$子空间。与该子空间一致的降阶动力学通过提出一个守恒定律来学习，其中GP描述体积间的通量。本工作的关键在于混合有限元空间与GP回归之间的新颖接口；当训练被表述为最优恢复问题（ORP）时，得到的GP回归可写为具有等式约束的优化问题，这些等式约束施加了守恒结构，这能够采用快速Schur补训练策略。训练后的模型可实时求解，并具有由指定Dirichlet数据驱动的边界通量的闭式估计量。本文包含对线性泛函的RKHS后验误差界，以支持不确定性量化，以及数值实验，演示后验分布作为误差估计代理的准确性。

不相关论文（基本信息）

1. 一种面向化学-力学-热力学接触的耦合有限元列式及其在粘结与脱粘中的应用

A coupled finite element formulation for chemo-mechano-thermodynamical contact and its application to bonding and debonding

作者: Roger A. Sauer

类别: cs.CE, math.NA, physics.comp-ph

发布日期: 2026-06-10

arXiv ID: 2606.12375v1

链接: http://arxiv.org/abs/2606.12375v1

摘要: This work presents a finite element formulation for coupled chemo-mechano-thermodynamical large deformation contact. The formulation is based on the contact theory of Sauer et al. (2022) that contains six coupled (but separate) fields: the deformation and temperature of the two contacting bodies, as well as an interfacial bonding field and interfacial temperature. The latter is governed by the chemical and mechanical energy dissipation at the interface. Here the focus is placed on the evolution of bonding and debonding, and how it is coupled to the mechanical and thermal contact state. Several elementary models are proposed for this based on a quadratic contact potential. The resulting contact formulation becomes very general and versatile, which is illustrated by several challenging examples. They include pressure- and gap- depended bonding, exothermic bonding reactions, thermal hardening and thermal expansion, as well as simultaneous bonding and debonding. They are based on a monolithic finite element implementation using classical and isogeometric shape functions together with implicit time integration. Its full linearization, required for the Newton-Raphson solution method, is also provided. If bonding sites are material points, the bonding variable can be condensed-out locally.

2. 基于MATLAB的分层自适物理信息神经网络在多元耦合高雷诺数Burgers方程中的应用

MATLAB-Based Layerwise Self-Adaptive Physics-Informed Neural Network in Applications to Multidimensional Coupled Burgers’ Equations with High Reynolds Numbers

作者: Harish P. Bhatt, Xi Chen, Jingsai Liang

类别: math.NA

发布日期: 2026-06-10

arXiv ID: 2606.12348v1

链接: http://arxiv.org/abs/2606.12348v1

摘要: This paper presents an improved physics-informed neural network for simulating the spatio-temporal solution profile of the multidimensional coupled Burgers’ equations with high Reynolds numbers. As time evolves, the sharp shock fronts emerge in the solution, creating significant computational challenges for the conventional mesh-based numerical methods. In particular, numerical methods such as finite differences and finite elements suffer from poor stability and strong mesh dependency when resolving the steep solution gradients. To address these challenges, the proposed framework employs a layerwise self-adaptive weighting strategy that dynamically adjusts the penalty weights for the physics residual, initial conditions, and boundary conditions throughout training. Moreover, the framework uses a dual-phase optimization strategy to achieve more stable convergence. To check the effectiveness and accuracy of the proposed framework, a set of numerical experiments is conducted to compare it with the standard Physics-Informed Neural Network (PINN) with and without Limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) optimization. Numerical results exhibit that the proposed framework achieves higher accuracy in terms of relative $L_2-$ error norm than the standard PINN and is able to capture the development of sharp shock fronts as time evolves in the solution.

3. 伴随方法与物理信息神经网络在偏微分方程约束反问题中的比较

Adjoint Method versus Physics-Informed Neural Networks in PDE-Constrained Inverse Problems

作者: Zhen Zhang, Alessandro Alla, George Em Karniadakis

类别: math.NA, cs.LG

发布日期: 2026-06-10

arXiv ID: 2606.12337v1

链接: http://arxiv.org/abs/2606.12337v1

摘要: Inverse problems governed by partial differential equations (PDEs) are central to computational mechanics and are commonly solved by adjoint-based optimization, while physics-informed neural networks (PINNs) have emerged as a flexible alternative. Their relative performance remains difficult to assess because the two approaches are often compared under different formulations, parameterizations, optimizers, and regularization choices. We present a fair comparison of adjoint optimization and PINNs for PDE-constrained inverse problems. From a common abstract formulation, we instantiate both methods on identical domains, governing equations, observation models, and regularization terms, while matching the optimizer, unknown parameterization, and arithmetic precision wherever applicable. The benchmarks include unsteady Burgers, noisy Darcy permeability inversion, three-dimensional Allen–Cahn reaction identification, and unsteady Navier–Stokes viscosity identification. The results show that the representation of the unknown largely determines the preferred method: grid-based fields favor the discrete adjoint, whereas neural representations are native to PINNs and relevant for closure and constitutive modeling. For time-dependent problems, adjoint inversion can be dominated by trajectory storage and differentiation, while PINNs provide satisfactory reconstructions at lower cost. A PINN-warm-started adjoint strategy then recovers adjoint-level accuracy at substantially reduced cost.

4. B样条基条件数的一个改进

An improvement on B-spline basis condition number

作者: Yimin Zhong

类别: math.NA

发布日期: 2026-06-10

arXiv ID: 2606.12270v1

链接: http://arxiv.org/abs/2606.12270v1

摘要: In this short note, we improve the upper bound on the condition number of the univariate B-spline basis of order $k$ from $k 2^k$ to $O(\sqrt{k}\log k\,2^k)$ for all $k\ge 2$.

5. 具有任意密度比的N相不可压缩流体混合物的对称结构保持离散化

Symmetric structure-preserving discretization of N-phase incompressible fluid mixtures with arbitrary density ratios

作者: M. F. P. ten Eikelder, A. Brunk

类别: math.NA, physics.flu-dyn

发布日期: 2026-06-10

arXiv ID: 2606.12256v1

链接: http://arxiv.org/abs/2606.12256v1

摘要: Diffuse-interface models are a widely used framework for interfacial dynamics in complex fluids, in which interfaces are represented through smooth transition layers and capillary effects are encoded by a free-energy functional. For incompressible mixtures with more than two phases, however, robust computation is substantially more difficult because the numerical method should preserve the balance structure of the continuum model, maintain the saturation constraint, dissipate energy, and treat all phases symmetrically even when density ratios are arbitrary. Existing structure-preserving methods are largely developed for binary flows or for formulations that distinguish a reference phase, so a genuinely symmetric N-phase discretization remains lacking. The practical problem is therefore to construct a fully-discrete method for N-phase incompressible Navier–Stokes–Cahn–Hilliard mixture models that retains the key thermodynamic and conservation properties of the continuum equations for arbitrary density ratios. Here we propose a symmetric fully-discrete method for the N-phase incompressible Navier–Stokes–Cahn–Hilliard mixture model with arbitrary density ratios. The method yields a fully-discrete problem in which every solution satisfies exact phase volume conservation, phase mass conservation, total volume conservation, total mass conservation, and a discrete energy-dissipation law. In addition, if the volume-saturation constraint holds for the initial data, then it is preserved at every time step. We numerically verify these structure-preserving properties and demonstrate the robustness of the method in representative multiphase flow problems. The resulting scheme provides a computational framework for incompressible N-phase mixture flows with complex interfacial dynamics and arbitrary density contrasts.

6. 通过矩阵-向量乘积的近乎实例最优稀疏矩阵逼近

Nearly Instance Optimal Sparse Matrix Approximation from Matrix-Vector Products

作者: Christoper Musco, Indu Ramesh

类别: cs.DS, math.NA

发布日期: 2026-06-10

arXiv ID: 2606.12179v1

链接: http://arxiv.org/abs/2606.12179v1

摘要: A large body of work studies the problem of learning an approximation to an implicit matrix $A\in \mathbb{R}^{m\times n}$ that is only accessible implicitly via matrix-vector product queries (matvec queries) of the form ${x} \rightarrow {A}{x}$ or ${x} \rightarrow {A}^T{x}$. Of particular interest are methods that learn a near-optimal approximation with a fixed sparsity pattern. For example, we might want to learn a near-optimal diagonal, banded, or arrow-head approximation to an implicit matrix $A$. Naturally, the number of matvec queries required to solve this problem depends on the sparsity pattern, which can be encoded as a binary matrix ${S}\in \{0,1\}^{m\times n}$. The query complexity of previous algorithms scales with quantities like the total number of ones in ${S}$, its maximum column/row sparsity, or the chromatic number of a its “conflict graph”. These quantities are incomparable: for a given ${S}$, parameterizing by one might yield lower query complexity than another. In this work, we unify and tighten these prior results by providing a nearly sharp characterization of the matvec query complexity of sparse matrix approximation. Generalizing a definition from graph algorithms, let the degeneracy, ${degen}({S})$, denote the smallest number $k$ so that, if we iteratively delete all rows and columns of ${S}$ with $\leq k$ ones, we are left with an empty matrix. We show that a near-optimal approximation to $A$ with sparsity pattern $S$ can be learned with $\tilde{O}({degen}({S}))$ matrix-vector product queries, and $Ω({degen}({S}))$ queries are necessary, for any sparsity pattern ${S}$. Moreover, unlike prior work based on graph coloring, all of our methods run in polynomial time.

7. 基于色散型双曲系统的自适应、高效且可扩展的水波建模

Adaptive, efficient, and scalable water wave modeling with dispersive hyperbolic systems

作者: Carlos Muñoz-Moncayo, David I. Ketcheson

类别: physics.flu-dyn, math.NA

发布日期: 2026-06-10

arXiv ID: 2606.12162v1

链接: http://arxiv.org/abs/2606.12162v1

摘要: Accurate modeling of tsunamis (such as those generated by landslides) requires capturing both wave dispersion in the deep ocean and wave breaking near the shore. The shallow water equations are often preferred for working with tsunamis, but neglect dispersion and may be inaccurate in scenarios where dispersive effects are significant. In this work, we develop an approach that seeks to incorporate the best aspects of both hyperbolic and dispersive models by combining either of two hyperbolic reformulations of the Serre-Green-Naghdi equations away from the shore with the non-dispersive shallow water equations near the shore. The model is discretized and implemented within the GeoClaw software, and incorporates adaptive mesh refinement as well as shared-memory parallelism. We validate it through comparison with benchmarks and real tsunami data. The results and performance compare favorably with the existing dispersive water wave solvers, including a speedup of about 2x relative to GeoClaw’s existing dispersive solver for a large-scale tsunami simulation.

8. 部分观测矩阵-向量积的幂迭代算法分析

Analysis of Power Iteration Algorithm with Partially Observed Matrix-vector Products

作者: Soumyadip Ghosh, Lior Horesh, Vassilis Kalantzis, Yingdong Lu, Tomasz Nowicki, Shashanka Ubaru

类别: math.NA

发布日期: 2026-06-10

arXiv ID: 2606.11956v1

链接: http://arxiv.org/abs/2606.11956v1

摘要: We consider the problem of computing the dominant eigenvector of a symmetric matrix via the power iteration algorithm subject to constraints in the computation of matrix-vector pr ucts. In particular, we focus on scenarios where the entries of matrix-vector products with the input matrix are only partially observed. Such constraints frequently arise on cloud architectures implemented via the controller-worker model where the matrix-vector products are distributed across workers on remote servers. Instead of a prolonged delay incurred by waiting for the slowest workers to return their output to the controller, a phenomenon known as straggling, a set of pre-determined values can replace the values of the delayed workers and allow the power iteration to proceed to the next iteration. In this paper, we develop two algorithms whose expected approximation converges to the true dominant eigenvector. The first algorithm relies on a probabilistic switch between two different approaches to set the omitted entries: either set them to zero or to their previous recorded value. The second algorithm relies on averaging previously generated partial power iteration approximations obtained by ignoring a set of columns of the iteration matrix. several theoretical details are discussed while numerical experiments verify the effectiveness of the two proposed schemes and demonstrate their comparative performance advantage over current state-of-the-art.

9. 稀疏性驱动的层析感知应用中的源定位

Sparsity-Driven Source Localization in Tomographic Sensing Applications

作者: Marco Mattuschka, Noah An der Lan, Arne Ficks, Max von Danwitz, Alexander Popp

类别: math.NA

发布日期: 2026-06-10

arXiv ID: 2606.11840v1

链接: http://arxiv.org/abs/2606.11840v1

摘要: Hyperspectral standoff detection systems such as Focal Plane Array (FPA) Fourier Transform Infrared (FTIR) spectrometers provide high spatial resolution in detecting airborne chemical contaminants that are invisible to the human eye but potentially hazardous. When two such systems are operated simultaneously with a suitable opening angle, they enable tomographic reconstruction of contaminant plumes with improved spatial and temporal accuracy. This work presents a mathematical model of these measurement capabilities and an algorithm to identify, localize, and quantify contaminant release sources. The objective is to develop a a tool that reconstructs release locations and predict the future plume evolution from standoff measurement data, thereby supporting early warning and situational awareness in hazardous material release scenarios. The transport of contaminants is modeled by an advection-diffusion equation, and the corresponding inverse problem for source identification is formulated accordingly. Owing to the severe ill-posedness and underdetermination of the problem, a sparsity-promoting regularization approach is employed together with a high-performance optimization algorithm. To incorporate the tomographic measurement data into the discrete formulation, a level-set description of a threshold concentration is used, allowing the measurements to be represented independently of the computational mesh and avoiding costly remeshing procedures.

10. 加速隐式GDA格式：理论保证及其在邻近增广拉格朗日方法中的应用

Accelerated Implicit GDA Schemes: Theoretical Guarantees and Application to Proximal Augmented Lagrangian Methods

作者: Jiaqi Liu, Bin Shi

类别: math.OC, math.NA

发布日期: 2026-06-10

arXiv ID: 2606.11800v1

链接: http://arxiv.org/abs/2606.11800v1

摘要: Convex optimization problems with linear equality constraints arise ubiquitously in scientific computing, machine learning, and control theory. Classical Krylov methods are effective but rely on problem-specific preconditioners and high memory. Conversely, gradient-based methods like the augmented Lagrangian method (ALM) avoid these issues yet suffer from slow outer iterations. Developing accelerated outer-iteration schemes, therefore, remains a critical research objective. In this study, we demonstrate that incorporating a proximal operation into the augmented Lagrangian framework yields the proximal ALM, where the outer iteration is equivalent to an implicit gradient descent-ascent (GDA) scheme. We further establish that this equivalence extends naturally to the setting of variable step sizes. Through Lyapunov analysis, we show that the underlying potential function must be shifted from the conventional objective gap to a variational inequality measure, signaling a shift in perspective from pure convex optimization to minimax optimization. Motivated by these observations, we first develop an implicit GDA scheme with variable step sizes based on a continuous-time ODE framework, which achieves an $o(1/k)$ last-iterate convergence rate for both the primal-dual objective gap and the gradient norm. Building upon a second-order ODE framework, we then propose a family of Nesterov-type implicit GDA schemes parameterized by $r \geq 0$, which achieves an $o(1/k^{r+1})$ last-iterate convergence rate for the primal-dual objective gap. Furthermore, specializing the second-order ODE formulation to the case $r=0$, we derive a corresponding explicit GDA scheme and prove an $o(1/k)$ last-iterate convergence rate for the primal-dual objective gap. Finally, we present several numerical experiments to validate these theoretical results and demonstrate the effectiveness of the proposed methods.

11. 超弹性中的曲率诱导力场

Curvature-Induced Force Fields in Hyperelasticity

作者: Victor Dods

类别: math.NA, math-ph, math.DG

发布日期: 2026-06-10

arXiv ID: 2606.11772v1

链接: http://arxiv.org/abs/2606.11772v1

摘要: Originally motivated by creating first-person computer visualizations within Riemannian manifolds – the author was led to study deformable-body mechanics, as rigid-body mechanics is not available in a generic Riemannian manifold due to its lack of nontrivial isometry group. Hyperelasticity is a particularly nice sub-category of continuum mechanics in which a deformable, elastic body’s behavior is determined by a stored energy density function. This allows problems to be posed variationally, and powerful tools brought to bear on studying and solving them. This article presents numerical simulations of static solutions to a particular class of problems in hyperelastic mechanics in 2-dimensional Riemannian manifolds in which a flat hyperelastic body $B$ is embedded into a region $Ω$ in a nowhere-flat surface $S$ of revolution $z=z\left(r\right)$ such that $\left|K\left(r\right)\right|$ decreases as $r\to\infty$, where $K$ denotes the Gaussian curvature of $S$. For example, the funnel $z=-r^{-1}$ or the paraboloid $z=\frac{1}{2}r^{2}$. Because $B$ is flat, the body can’t achieve a zero-stored-energy configuration, and restorative forces arise in the body to move it toward a region of lower stored energy – meaning, toward a flatter configuration. With the addition of a gravitational potential $U\left(r\right)=z\left(r\right)$ on $S$, forces act on the body to pull it toward $r=0$. If the body has sufficient stiffness and remains within the region $Ω$, then the body has an equilibrium configuration in which the body’s deformation-response forces perfectly cancel the gravitational forces. Such a configuration represents a kind of “levitation” phenomenon within this surface. The numerical implementation of this problem will be detailed and the resulting numerical solutions and various consequences discussed.

12. 用于导数非线性薛定谔方程的高阶多结构保持指数积分器

High-order multi-structures-preserving exponential integrators for the derivative nonlinear Schrödinger equation

作者: Liping Wu, Li Yang, Chaolong Jiang

类别: math.NA

发布日期: 2026-06-10

arXiv ID: 2606.11734v1

链接: http://arxiv.org/abs/2606.11734v1

摘要: This paper presents a novel class of high-order mass-, energy- and momentum-preserving exponential integrators for solving the derivative nonlinear Schrödinger equation. Firstly, we reformulate the original system into an exponential supplementary variable system based on the idea of the exponential supplementary variable approach, and then the reformulated system is discretized by using the standard Fourier pseudo-spectral method in space and the high-order prediction and correction Lawson Runge-Kutta method in time, respectively. The proposed method is highly efficient, temporally high-order accurate, and simultaneously preserves the mass, energy and momentum in the discrete setting. Finally, numerical experiments validate the accuracy and energy-preserving properties.

13. 一种用于低秩张量列车近似的双边素描算法

A Two-Sided Sketching Algorithm for Low-rank Tensor Train Approximation

作者: Gaohang Yu, Yihao Pan, Ailun Jian, Xiaohao Cai

类别: math.NA

发布日期: 2026-06-10

arXiv ID: 2606.11603v1

链接: http://arxiv.org/abs/2606.11603v1

摘要: Tensor train (TT) decomposition is a powerful method to acquire low-rank tensors. However, the computational process is frequently obstructed by the large-scale matrix singular value decomposition (SVD). The sketching algorithm serves as an efficient data compression technique that can quickly derive low-rank matrix approximations. In this paper, we propose a randomized algorithm to obtain the TT approximation of tensors using a one-pass sketching algorithm and subspace iteration, and offer thorough error-bound and robustness analysis. Numerical experiments on synthetic and real-world datasets demonstrate the effectiveness and efficiency of the proposed algorithm.

14. 恢复细胞侵袭非线性偏微分方程模型中的初始条件与物理系数

Recovering the initial condition and physical coefficients in a nonlinear PDE model of cell invasion

作者: Beiji Chen, Kui Ren

类别: math.AP, math.NA

发布日期: 2026-06-10

arXiv ID: 2606.11554v1

链接: http://arxiv.org/abs/2606.11554v1

摘要: This paper investigates an inverse problem for the simultaneous reconstruction of two spatially varying reaction coefficients, the local proliferation rate and the competition (saturation) coefficient, together with the unknown initial condition, in a nonlinear, density-dependent reaction-diffusion model motivated by cell invasion and tumor growth dynamics. Using Carleman estimates, we establish a global uniqueness result together with a Lipschitz-type stability estimate for the reaction coefficients and a weaker, logarithmic stability estimate for the initial condition. For the numerical reconstructions, we develop a two-stage algorithm employing a time-shift strategy to decouple the coefficient and the initial condition. Numerical experiments are presented to illustrate the feasibility, accuracy, and robustness of the proposed inversion method.

15. PHASE：细分单元上的泡利分层装配用于量子兼容算子合成

PHASE: Pauli Hierarchical Assembly on Subdivided Elements for Quantum-Compatible Operator Synthesis

作者: Tillman Philo, Caglar Oskay

类别: quant-ph, math.NA

发布日期: 2026-06-09

arXiv ID: 2606.11478v1

链接: http://arxiv.org/abs/2606.11478v1

摘要: Efficiently decomposing finite element stiffness matrices into the Pauli basis is challenging due to the exponential growth of Pauli strings with problem size. A naive Pauli expansion requires $Θ(8^{\lceil \log_2 N \rceil})$ operations, where $N$ denotes the number of degrees of freedom, rendering direct decomposition infeasible for large systems. Existing approaches exploit algebraic sparsity or operator structure but do not incorporate the geometric organization intrinsic to finite element discretizations, and consequently exhibit poor scaling for stiffness matrices. To address this problem, we introduce PHASE, a hierarchical, geometry-aware Pauli decomposition algorithm that leverages recursive mesh partitioning to organize element contributions across multiple spatial scales. PHASE employs a hybrid strategy that combines full- and reduced-space Tensorized Pauli Decomposition with Fast Walsh-Hadamard Transform-based aggregation to assemble global Pauli coefficients efficiently. We show that this approach yields a dimension-dependent reduction in the exponential scaling exponent of Pauli assembly asymptotic complexity relative to existing methods, reducing the cost from $2^{2{\lceil \log_2 N \rceil}}$ to $2^{γ_d{\lceil \log_2 N \rceil}}$ with $γ_d < 2$ under standard mesh regularity and balanced partition assumptions. These results substantially improve the feasibility of quantum-compatible operator synthesis for large-scale finite element models.

16. 具有交换子缩放的哈密顿模拟的线性组合

Linear Combination of Hamiltonian Simulation with Commutator Scaling

作者: Junaid Aftab, Dong An, Konstantina Trivisa

类别: quant-ph, math.NA

发布日期: 2026-06-09

arXiv ID: 2606.11475v1

链接: http://arxiv.org/abs/2606.11475v1

摘要: The Linear Combination of Hamiltonian Simulation (LCHS) framework simulates dissipative linear dynamics by representing time evolution as an integral over unitary operators, which is discretized by quadrature and implemented via Hamiltonian simulation. While existing analyses achieve near-optimal scaling in time and precision using norm-based quantities of the dissipative generator, we show that implementing the Hamiltonian simulation steps with Multi-Product Formulas (MPFs) yields commutator-sensitive error and complexity bounds. We demonstrate that the quadrature rule affects not only discretization error but also commutator structure and query complexity. This dependence is quantified through post-quadrature analysis for abstract MPF error profiles and for general time-independent and local Hamiltonians using known commutator-sensitive MPF error estimates. We compare uniform trapezoidal and free-scale sinh–sinh quadrature, showing improved quadrature-cardinality scaling for the latter, and illustrate the framework with applications to fractional diffusion, advection–diffusion, and open quantum systems.

17. 对偶 Gauss–Legendre 多项式

Dual Gauss–Legendre polynomials

作者: Paweł Woźny

类别: math.NA

发布日期: 2026-06-09

arXiv ID: 2606.11355v1

链接: http://arxiv.org/abs/2606.11355v1

摘要: We define and investigate two families of dual polynomials associated with the Gauss–Legendre polynomials, which have recently found interesting applications in computer graphics. Using the presented results, one can derive representations of the Gauss–Legendre polynomials, construct the dual bases for Lagrange bases and solve certain approximation problems arising, for example, in CAGD.