生成时间: 2026年07月24日 10:00:48
重点关注论文数量: 0
了解领域论文数量: 2
不相关论文数量: 14
了解领域论文（摘要翻译）
1. 一种针对Fisher-Kolmogorov模型的保稳定多胞形间断Galerkin方法及其在神经退行性疾病中的应用
A stability-preserving polytopal discontinuous Galerkin method for the Fisher-Kolmogorov model with applications to neurodegenerative diseases
作者: Paola Francesca Antonietti, Francesca Bonizzoni, Mattia Corti, Nicola De March, Salvatore Di Noto, Francesco Regazzoni
类别: math.NA
发布日期: 2026-07-23
arXiv ID: 2607.21108v1
链接: http://arxiv.org/abs/2607.21108v1
摘要翻译
中文标题: 一种针对Fisher-Kolmogorov模型的保稳定多胞形间断Galerkin方法及其在神经退行性疾病中的应用
摘要翻译: Fisher-Kolmogorov模型是神经退行性疾病研究中最广泛使用的模型之一，原因在于其作为非线性反应扩散方程的简洁结构。特别地，它常被用于描述阿尔茨海默病和帕金森病等蛋白质病。在合适的假设下，在连续层面可以保证解的非负性，这在物理上是相关的，因为解表示相对浓度。然而，这一性质一般在离散层面不能保持，可能产生非物理和不稳定的数值逼近。在本文中，我们分析了Fisher-Kolmogorov模型的一个修正版本，该版本稳定了不稳定平衡点 $c=0$ 附近的动力学。空间离散方面，我们采用多边形和多面体网格上的间断Galerkin方法，时间积分采用Crank-Nicolson格式。我们推导了半离散问题的稳定性和先验误差估计。数值实验支持了理论发现，包括在二维和三维情形下的收敛性研究。最后，我们通过在二维聚集脑切片中模拟 $α$-突触核蛋白扩散来验证模型，展示了所提方法的高阶精度和稳健性。
2. 紧凑李群上的保结构谱动态规划
Structure-Preserving Spectral Dynamic Programming on Compact Lie Groups
作者: Shanqing Liu, Yang Qi
类别: math.OC, math.NA
发布日期: 2026-07-23
arXiv ID: 2607.20854v1
链接: http://arxiv.org/abs/2607.20854v1
摘要翻译
中文标题: 紧凑李群上的保结构谱动态规划
摘要翻译: 我们研究连通紧李群$G$上有限时域最优控制的动态规划半群的谱逼近，以及相关的一阶Hamilton-Jacobi-Bellman方程的谱逼近。Bellman算子在最大值范数下是单调且非扩张的，而$G$上所有Fourier方法所依赖的$L^{2}(G)$的Peter-Weyl分解是正交的，这种不匹配是定量的。Galerkin迭代的自洽上确界范数误差递推在每一步都被谱投影的Lebesgue常数放大，该常数在$S^{1}$上对数增长，在秩为1的紧李群（包括$\mathrm{SO}(3)$）上多项式增长，且在计算中迭代在几步内就违反基本界限。我们通过用Markov类型的谱滤波器替代正交投影，在离散水平上恢复了动态规划结构。一个辅助的热核/消融粘性格式为Lipschitz数据提供了定性的最大值范数收敛性。主要结论是$G$上的一个Fejér型滤波器，它是有限秩、保正性且非扩张的，连同Lipschitz数据下以速率$O(\sqrtδ+\sqrt{ε+1/(δN^2)})$的收敛定理，其中$δ$是时间步长，$N$是谱分辨率，$ε$是粘性系数。粘性系数可为零，此时耦合关系$δ=N^{-1}$给出速率$N^{-1/2}$。证明将滤波器解释为受控动力学的一个小随机扰动，既不需要值函数的先验正则性，也不需要对滤波步骤进行粘性意义下的一致性论证，并推广到基于正容积公式的完全离散实现，在$\mathrm{SO}(3)$上有精确的Wigner传输。数值实验证实了预测的速率和滤波器偏差，并量化了两种基于坐标图的基线方法的标架依赖性。
不相关论文（基本信息）
1. 约当代数、hemiplex数与任意对称矩阵的Cholesky分解
Jordan algebras, hemiplex numbers, and the Cholesky decomposition of arbitrary symmetric matrices
作者: Alan Edelman, Timothy E. Holy
类别: math.NA, math.RA
发布日期: 2026-07-23
arXiv ID: 2607.21383v1
链接: http://arxiv.org/abs/2607.21383v1
摘要: Positive-semidefinite matrices are most efficiently factored using the Cholesky decomposition. For indefinite matrices, the Cholesky factorization does not exist, and the alternatives face greater challenges in achieving numeric stability and preservation of banded structure. Here we pursue an analogy between the requirement for positive-semidefinite matrices and the solution of the quadratic equation x^2 = c for c <= 0. It is shown that a non-associative algebra, called the hemiplex numbers, allows the Cholesky factorization to be computed for arbitrary symmetric matrices. Crucially, the hemiplex Cholesky factorization does not require pivoting for its existence or stability, allowing it to preserve banded structure. For singular matrices it produces a parametrization of the null space, and provides opportunity for truncation of nearly-null directions in a manner similar to common usage of the singular value decomposition. The hemiplex Cholesky factorization may be a practically useful addition to the tools for solving symmetric linear equations.
2. 三维光声层析成像的不确定性量化
3D Uncertainty Quantification for the Photo-Acoustic Tomography
作者: Babak Maboudi Afkham, Amal Mohammed A Alghami, Hassan Yazdanian, Tanja Tarvainen
类别: math.NA, math.ST
发布日期: 2026-07-23
arXiv ID: 2607.21373v1
链接: http://arxiv.org/abs/2607.21373v1
摘要: Photoacoustic tomography (PAT) is a promising modality for high-resolution biomedical imaging, motivating the need for reliable uncertainty quantification (UQ) of reconstructed images. Bayesian approaches provide a rigorous framework for UQ but remain computationally challenging for realistic three-dimensional PAT and are sensitive to numerical approximations in the governing wave equation. We develop a finite-element Bayesian UQ framework for PAT that accommodates complex computational domains and detector geometries while enabling large-scale three-dimensional inference. The proposed methodology reformulates the randomize-then-optimize (RTO) sampling strategy as a matrix-free algorithm that generates independent posterior samples using only forward and adjoint wave propagations. Particular attention is given to constructing an adjoint discretization that forms an exact transpose pair with the discrete forward operator while remaining consistent with the continuous PAT adjoint, enabling efficient least-squares solvers within the sampling procedure. We investigate the influence of temporal discretization, artificial boundary conditions, and adjoint consistency on posterior uncertainty and identify discretization strategies that avoid numerical artifacts. The framework is validated against exact posterior statistics, existing Bayesian PAT methods, and Hamiltonian Monte Carlo using the No-U-Turn Sampler (NUTS), and is demonstrated on a three-dimensional problem with approximately $2\times 10^5$ unknowns on a general finite-element domain. To the best of our knowledge, this is the first large-scale Bayesian PAT study on general three-dimensional finite-element geometries, and the methodology extends naturally to a broad class of linear PDE-constrained inverse problems.
3. 不规则几何中二维Stokes流的带有Calderón预处理的非拟合边界代数方程方法
An unfitted boundary algebraic equation method with Calderón preconditioning for 2D Stokes flow in irregular geometry
作者: Wenjun Ying, Qing Xia
类别: math.NA, physics.comp-ph, physics.flu-dyn
发布日期: 2026-07-23
arXiv ID: 2607.21295v1
链接: http://arxiv.org/abs/2607.21295v1
摘要: We present an unfitted boundary algebraic equation method for the two-dimensional exterior/interior Stokes equations on a staggered MAC grid. By constructing an explicit free-space pair of velocity and pressure lattice Green’s functions (LGFs) from free-space Laplace LGFs, we represent homogeneous fields using sources supported exclusively on thin staggered boundary layers. This formulation imposes physical Dirichlet data at cut points via local interpolation, while sampled-normal rank updates remove hydrostatic null modes associated with single or multiple obstacles. The workflow parallels that of classical boundary integral formulations and requires no artificial boundary conditions for exterior flows, but follows a discretize-then-represent route and does not require singular/near-singular quadrature. The resulting dense boundary system is solved via GMRES, utilizing a componentwise discrete Calderón preconditioner built from the scalar Laplace kernel and padded FFTs for fast volume convolutions. Extensive numerical validation, including multiply connected domains, narrow gaps, and Moffatt eddies, confirms discrete incompressibility to solver accuracy and recovers the expected Moffatt eddy scaling. We achieve second-order velocity and pressure convergence and bound maximum discrete divergence within numerical accuracy. The discrete Calderón preconditioner reduces the condition number by orders of magnitude and yields nearly mesh-independent conditioning in exterior configurations, while remaining effective—though more demanding—for narrow-gap and fine-grid interior problems.
4. 关于与旋转矩阵相关的酉算子群的逼近及其在抽象双曲型方程中的应用
On the Approximation of the Unitary Operator Group Associated with a Rotation Matrix and Its Applications to Abstract Hyperbolic Equations
作者: Jemal Rogava, Zurab Vashakidze
类别: math.NA, math.FA
发布日期: 2026-07-23
arXiv ID: 2607.21293v1
链接: http://arxiv.org/abs/2607.21293v1
摘要: The solution of the Cauchy problem for homogeneous abstract hyperbolic equations, together with its derivative, admits a vector representation in terms of a unitary operator group associated with a rotation matrix. A rational approximation of this unitary group is constructed and shown to possess optimal fourth-order convergence. The order of convergence is determined in accordance with the smoothness scale. Based on this rational approximation, a two-layer semi-discrete scheme is constructed for the approximate solution of Cauchy problems for nonhomogeneous abstract hyperbolic equations in both the linear and semi-linear settings. The convergence properties of the scheme are examined in relation to the regularity of the solution.
5. 用于带间断散乱数据插值的自适应非线性单位分解方法
Adaptive Non-Linear Partition of Unity Methods for Scattered Data Interpolation with Discontinuities
作者: Adeeba Haider, Roberto Cavoretto, Juan Ruiz-Álvarez, Dionisio F. Yáñez
类别: math.NA
发布日期: 2026-07-23
arXiv ID: 2607.21287v1
链接: http://arxiv.org/abs/2607.21287v1
摘要: Scattered data approximation with discontinuities is challenging due to the Gibbs phenomenon, which significantly reduces accuracy near interfaces. The recently introduced Non-Linear Partition of Unity Method (NL-PUM) addresses this by combining Radial Basis Function (RBF) interpolation with a non-linear Weighted Essentially Non-Oscillatory (WENO) strategy. While effective, NL-PUM’s performance relies heavily on two fixed hyperparameters: the RBF shape parameter and the patch radius. This work extends NL-PUM by adapting both hyperparameters locally using Leave-One-Out Cross-Validation (LOOCV) minimized via Global Optimization with Optimistic Improvement (GOOI). Our main innovation is a smoothness indicator linking a discontinuity-aware shrinkage process to LOOCV-based radius selection: patches in smooth regions remain unchanged, while those near discontinuities automatically shrink to avoid crossing the interface. The resulting method, LOOCV-NL-PUM-GOOI, requires no prior knowledge of interface geometry and introduces no extra cost beyond standard adaptive shape parameter selection. Numerical experiments on synthetic test functions with jump discontinuities and a real-data application to Norwegian Fjords elevation data confirm that this approach substantially reduces approximation error near discontinuities while preserving full accuracy in smooth regions.
6. 超缩减神经网络增强的半光滑牛顿法求解非线性参数变分不等式
A Hyper-Reduced Neural Network-Augmented Semi-smooth Newton Method for Nonlinear Parametric Variational Inequalities
作者: Sofiane Ezzehi, Virginie Ehrlacher, Guillaume Enchéry, Thibault Faney
类别: math.NA
发布日期: 2026-07-23
arXiv ID: 2607.21242v1
链接: http://arxiv.org/abs/2607.21242v1
摘要: We propose a model order reduction framework for nonlinear parametrized variational inequalities arising in computational mechanics. The high-dimensional model is written in mixed primal-dual form with projection-based complementarity conditions, leading to nonlinear nonsmooth algebraic systems solved by a semi-smooth Newton method in primal-dual form. On this basis, reduced models are constructed by proper orthogonal decomposition (POD) of both primal and dual solution snapshots, and the resulting reduced systems are solved by semi-smooth Newton iterations in the reduced space. To address cases where low-dimensional linear spaces provide limited approximation efficiency, we introduce a neural-network-augmented reduced model. Two feedforward networks learn corrections in the truncated POD coordinates of the primal and dual variables, defining a nonlinear manifold approximation that is embedded directly in the semi-smooth Newton iterations. The online cost associated with high-dimensional residual evaluations is reduced through hyper-reduction, using a sparse cubature approach based on greedy nonnegative least squares. Particular attention is paid to the interaction between hyper-reduction and the learned nonlinear manifold. The proposed methodology is assessed on two nonlinear variational inequalities with distinct sources of nonlinearity: a two-dimensional obstacle problem with a cubic nonlinearity in the state equation, and a three-dimensional frictional contact problem in which the Coulomb law induces a nonlinear projection in the constraint equation. Numerical results compare the high-dimensional model, the linear reduced model, the neural-network-augmented reduced model, and their hyper-reduced variants, demonstrating accurate approximations with substantial reductions in online computational cost.
7. 一种混合离散科瑟拉杆格式
A Mixed Discrete Cosserat Rod Formulation
作者: Tianxiang Dai, Marco Herrmann, Jonas Breuling, Remco I. Leine, Simon R. Eugster
类别: math.NA, math-ph
发布日期: 2026-07-23
arXiv ID: 2607.21164v1
链接: http://arxiv.org/abs/2607.21164v1
摘要: In this communication we propose a discrete Cosserat rod formulation in which a slender elastic rod is represented as a chain of rigid bodies (nodes) coupled by compliant elastic forces and moments acting between adjacent node pairs. Discrete dilatation, shear, torsion and curvature strain measures are evaluated from the relative kinematics of each node pair, while the constitutive behavior is expressed in compliance form through independent stress degrees of freedom. We show that the resulting model arises rigorously from a mixed Petrov–Galerkin Cosserat rod finite element formulation (FEM) at linear kinematic interpolation order when the internal virtual work is integrated by the midpoint rule and the external and inertial contributions by the trapezoidal rule. The proposed formulation inherits the robustness and the absence of locking from the underlying mixed FEM while simultaneously exposing a two-node coupling structure that mirrors discrete rod models from the computer graphics community. This is in sharp contrast to the dense coupling of strain-parameterized reduced-order models often used in soft robotic applications. Three numerical examples involving piecewise-varying cross sections, tendon-driven actuation under different spatial discretizations, and coupled longitudinal-torsional dynamics confirm the accuracy, robustness, and convergence behavior of the presented approach.
8. 一种用于欧式期权定价的阻尼SWIFT方法：系数衰减、截断与误差分析
A Damped SWIFT Method for European Option Pricing: Coefficients Decay, Truncation, and Error Analysis
作者: Davide Trevisani, José Germán López Salas, Chiheb Ben Hammouda, Cornelis W. Oosterlee
类别: math.NA
发布日期: 2026-07-23
arXiv ID: 2607.21109v1
链接: http://arxiv.org/abs/2607.21109v1
摘要: We introduce a damped variant of the Shannon Wavelet Inverse Fourier Technique (SWIFT) for pricing European options when the characteristic function of the underlying model is available. The key idea is to apply an exponential damping transformation to the payoff, which enables the direct computation of Fourier coefficients in the frequency domain without introducing an additional physical-domain truncation parameter. We provide a rigorous analysis of the decay of these coefficients by exploiting the singularity structure of the associated Fourier transforms. For light-tailed models, we obtain Gaussian-type decay estimates, while for semi-heavy and heavy-tailed models whose singularities are poles, algebraic branch points, or logarithmic branch points, we derive exponential decay bounds with explicit polynomial prefactors. The resulting sharp bounds make it possible to truncate the Fourier series without relying on the cumulants of the underlying density, which are often unavailable or difficult to compute in practice. We further derive an error decomposition separating projection, truncation, and quadrature errors, and translate the analysis into practical rules for selecting the damping parameter, resolution level, and truncation range. Numerical experiments demonstrate that the proposed approach consistently improves the accuracy of the original SWIFT method while requiring a significantly smaller number of Fourier coefficients and remaining stable in cases where the undamped method deteriorates.
9. 用于垂直强磁场中带电粒子的具有改进一致精度的高阶指数积分器
High-Order Exponential Integrators with Improved Uniform Accuracy for Charged-Particle in a Perpendicular Strong Magnetic Field
作者: Zhihao Qi, Weibing Deng
类别: math.NA
发布日期: 2026-07-23
arXiv ID: 2607.21096v1
链接: http://arxiv.org/abs/2607.21096v1
摘要: This paper considers a class of charged-particle dynamics problems in which the particle is subjected to a magnetic force, with a magnetic flux density inversely proportional to a small parameter $0<\varepsilon\ll 1$, and a nonlinear electric force. The resulting highly oscillatory behavior poses significant challenges for numerical computation. To enhance the performance of exponential integrators (EIs), this paper employs a technique that linearizes the ordinary differential equation through a dimension-raising approach. Based on this technique, a new family of EIs is developed that achieves arbitrarily high order. For short-time simulations on the interval $[0,T]$, it is rigorously proved that the proposed method–which employs auxiliary polynomials of degree $k$ and a time step $Δt$–satisfies two distinct error bounds: $O(\varepsilon Δt^{k+1})$ and $O(\varepsilon^{k+2})$. The latter bound guarantees that the algorithm stays accurate even when the step size is of order $O(1)$. Furthermore, when a large step size $\varepsilon^{-1}Δt$ is used to simulate the long-term dynamics over $[0,\varepsilon^{-1}T]$, the numerical scheme attains a uniform convergence rate of $O(Δt^{k+1})$. Several numerical experiments confirm these theoretical results.
10. yancc: 托卡马克和仿星器中新经典输运的GPU加速可微求解器
yancc: A GPU-accelerated, differentiable solver for neoclassical transport in tokamaks and stellarators
作者: Rory Conlin, Matt Landreman
类别: physics.plasm-ph, math.NA
发布日期: 2026-07-23
arXiv ID: 2607.20861v1
链接: http://arxiv.org/abs/2607.20861v1
摘要: We present yancc, a new GPU-accelerated solver for the drift kinetic equation that computes neoclassical transport fluxes, flows, and currents in tokamaks and stellarators. The drift kinetic equation is challenging to solve numerically due to strong advection-dominance, recirculating flows, internal boundary layers, severe anisotropy, and high dimensionality. The code solves both the full four-dimensional drift kinetic equation (retaining speed-dependent collisions, energy scattering, and full interspecies coupling), and the reduced monoenergetic form. The discretization combines a Maxwell polynomial collocation grid in speed with finite differences in pitch angle and the flux surface coordinates, using a modified upwind stencil designed to improve diagonal dominance for multigrid efficiency. The resulting linear system is solved with a multigrid-preconditioned Krylov method. Built in JAX, yancc is fully differentiable, enabling gradient-based optimization and adjoint sensitivity analysis. Benchmarks against MONKES and SFINCS show agreement within 1% across a range of collisionalities, geometries, and multi-species configurations. yancc achieves roughly an order of magnitude speedup over SFINCS on a per-scan basis while using an order of magnitude less memory, with runtime remaining nearly flat across the full range of collisionality. The combination of speed, low memory footprint, and differentiability makes yancc well suited for integration into stellarator optimization workflows, uncertainty quantification, and profile prediction.
11. 固定D3Q125速度集上的分层对数高斯松弛
Hierarchical Log-Gaussian Relaxation on a Fixed D3Q125 Velocity Set
作者: Bjørn Wu
类别: math.NA, physics.comp-ph
发布日期: 2026-07-23
arXiv ID: 2607.20846v1
链接: http://arxiv.org/abs/2607.20846v1
摘要: We develop a hierarchical order-resolved relaxation model for a fixed D3Q125 discrete-velocity kinetic formulation. Conventional adaptive collision models often use one scalar rarefaction or nonequilibrium indicator for all retained moment orders, thereby coupling distinct kinetic sectors. Here, a shared macroscopic-gradient background is combined separately with second-, third-, and fourth-order thermodynamic nonequilibrium indicators to define effective measures K2, K3, and K4, each driving its own log-Gaussian relaxation spectrum. Pure-order perturbation tests verify selective activation, with nonmatching sectors remaining at roundoff level. Homogeneous mixed-order, amplitude, and composition tests show lower residual nonequilibrium than a common-sensor model while preserving positive populations. In a smooth periodic compression wave at the stated reference discretization and in the TNE-only sensor limit, the peak total nonequilibrium intensity is reduced by 6.565%, with reductions throughout the domain and in all retained moment sectors. Additional timestep, transport-discretization, relaxation-spectrum, uniform-boost, long-time, and shear-wave studies show that the sign of the hierarchical correction is robust over the tested configurations, while its magnitude depends on timestep, transport scheme, sensor frame, and relaxation spectrum. The periodic benchmarks preserve the principal global invariants to floating-point accuracy and remain positive. These results establish the mechanism and numerical behavior of order-resolved activation on a fixed velocity set; independent kinetic-reference validation is still required before claiming universal accuracy improvement.
12. 一种用于与热方程耦合的达西问题的虚拟元方法
A virtual element method for Darcy’s problem coupled with a heat equation
作者: Danilo Amigo, Felipe Lepe
类别: math.NA
发布日期: 2026-07-22
arXiv ID: 2607.20755v1
链接: http://arxiv.org/abs/2607.20755v1
摘要: In two dimensions, we develop a virtual element method to solve the Darcy’s problem coupled with a nonlinear heat equation. This coupled model may allow for thermal diffusion and viscosity as a function of temperature. Under standard discretization assumptions and appropriate assumptions on the data, we prove the well posedness of the proposed numerical scheme. We also derive optimal error estimates under suitable regularity assumptions for the solution and appropriate assumptions on the data. We conclude with a series of numerical tests performed on different families of meshes that complement the theoretical findings.
13. 朴素 Jacobi 特征值算法的收敛性与混合精度预处理
Convergence and mixed-precision preconditioning for the naive Jacobi eigenvalue algorithm
作者: Erna Begovic, Marija Miloloza Pandur, Ana Perkovic
类别: math.NA
发布日期: 2026-07-22
arXiv ID: 2607.20749v1
链接: http://arxiv.org/abs/2607.20749v1
摘要: The paper studies a Jacobi-type method for the eigenvalue problem of general complex matrices with simple eigenvalues. The method applies elementary triangular similarity transformations in order to annihilate selected off-diagonal elements and, when convergent, produces highly accurate eigenvalues. We give a new proof of its asymptotic quadratic convergence and derive an explicit, verifiable bound that describes the region in which this convergence is guaranteed. To make the method applicable well beyond matrices already close to the diagonal form, we introduce a preconditioning strategy. We use two types of preconditioners, both based on theoretical convergence results. The preconditioner is computed at lower precision to reduce computational cost, the associated similarity transformation is applied either at working or at higher precision, to preserve spectral information, while the main algorithm performs at working precision. Numerical experiments demonstrate that the resulting algorithm is robust and produces very accurate eigenvalues.
14. librla：随机线性代数库
librla: Randomized Linear Algebra Library
作者: Adrianna Gillman, Zydrunas Gimbutas
类别: cs.MS, math.NA
发布日期: 2026-07-22
arXiv ID: 2607.20732v1
链接: http://arxiv.org/abs/2607.20732v1
摘要: The library \texttt{librla} is a randomized linear algebra library that is specifically designed for the intermediate-sized matrices (of dimension up to roughly 10,000) that arise in applications such as reduced order modeling, fast direct solvers, least squares solves and, in some settings, data compression. \texttt{librla} is the first software package that is both stable and efficient in several high-level languages: MATLAB, Python and Julia. It also provides increased functionality over existing software. Specifically, it allows the user to choose to create a factorization based on a fixed rank or a desired tolerance. The factorization options include QR, SVD and the interpolative decomposition. Additionally, the factorization can be generated either with access to the matrix or access to a matrix-vector multiplication routine. Numerical results compare the Python implementation with the available PyTorch and SciPy randomized factorizations. Performance of \texttt{librla} in the three languages is comparable.