生成时间: 2026年07月03日 10:19:59

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: 22

了解领域论文（摘要翻译）

1. 有限应变弹性的一种统一 CutFEM 格式：能量极小化与角点奇性

A Unified CutFEM Formulation for Finite-Strain Elasticity: Energy Minimisation and Corner Singularities

作者: Michał Tomasz Wichrowski, Ella Godiva Noomen

类别: math.NA

发布日期: 2026-07-02

arXiv ID: 2607.02334v1

链接: http://arxiv.org/abs/2607.02334v1

摘要翻译

中文标题: 有限应变弹性的一种统一 CutFEM 格式：能量极小化与角点奇性

摘要翻译: 我们提出一个全变分、模型无关的有限应变弹性 Cut 有限元方法 (CutFEM) 格式。离散问题是增广能量泛函的驻点条件，该泛函由体超弹性能、弱施加边界条件的 Nitsche 项以及鬼罚稳定化项组成。残量和 (对称化的) 切线由该泛函的逐次变分得到。自动微分 (AD) 直接从标量能量密度生成第一 Piola–Kirchhoff 应力张量和弹性张量，避免在更换超弹性模型时手动重新推导。据我们所知，这是第一个结合纯能量、模型无关的构造与自动微分并在非拟合边界处进行精度分析的非拟合有限应变格式。 对每个 Newton 步中求解的线性化问题进行分析，得到了与切割无关的强制性、连续性和 $O(h^{-2})$ 条件数界，并借助 Brezzi–Rappaz–Raviart 框架得到正则解的拟最优收敛定理。数值上，在光滑测试算例上该方法对线性、二次和三次元均达到最优 $h$-收敛。此外，我们使用 Kolosov–Muskhelishvili 特征方程量化了该方法在 Dirichlet–Neumann 混合连接点处的精度极限。精确解的角点奇性对拟合和非拟合方法均以相同的方式限制收敛阶。我们证明局部网格细化可以消除这一限制，非拟合离散继承恢复的最优收敛率和与切割无关的常数。

2. 调和映射流的线性隐式质量集中投影有限元方法的无条件最优误差估计与能量稳定性

Unconditional Optimal Error Estimates and Energy Stability for a Linearly Implicit Mass-Lumped Projection Finite Element Method for the Harmonic Map Flow

作者: Yongyong Cai, Xingwei Yang

类别: math.NA

发布日期: 2026-07-02

arXiv ID: 2607.02179v1

链接: http://arxiv.org/abs/2607.02179v1

摘要翻译

中文标题: 调和映射流的线性隐式质量集中投影有限元方法的无条件最优误差估计与能量稳定性

摘要翻译: 我们提出并分析了一种用于调和映射到单位球面的热流的线性隐式质量集中有限元方法。该方法由一个线性预测步骤和随后的节点投影组成，因此能在所有有限元节点处精确保持单位长度约束。预测器是从方程的叉积重表述推导出来的，并证明它与原始公式的质量集中离散化加上一个强制节点正交性的修正项等价，也与切平面格式等价。一个关键之处是在质量和刚度项中一致地使用离散内积。这产生了一个节点正交关系，意味着辅助解在每个节点处都位于单位球面上或其外部。因此，投影是良好定义的，且投影误差在离散 (L^2) 范数下满足一个收缩性质。在笛卡尔矩形和立方体的张量积网格上，节点投影在离散狄利克雷能量下也是非膨胀的，这给出了一个无条件的离散能量耗散律。对于足够光滑的解，我们证明了最优误差估计，无需时间步长与网格尺寸之间的任何耦合条件：该方法在 (\ell^\infty(0,T;L^2)) 中以 (O(\Delta t+h^2)) 阶收敛，在 (\ell^2(0,T;H^1)) 中以 (O(\Delta t+h)) 阶收敛。证明结合了投影误差收缩、求积一致性估计、基于边的消去恒等式以及用于控制非线性项的自举论证。数值实验证实了预测的收敛率和离散能量衰减。

3. 四面体Alfeld剖分上具有可杂交应力的低正则性有限元弹性复形

Low-regularity finite element elasticity complexes with hybridizable stresses on tetrahedral Alfeld splits

作者: Johnny Guzmán, Xuehai Huang

类别: math.NA

发布日期: 2026-07-02

arXiv ID: 2607.01933v1

链接: http://arxiv.org/abs/2607.01933v1

摘要翻译

中文标题: 四面体Alfeld剖分上具有可杂交应力的低正则性有限元弹性复形

摘要翻译: 在四面体Alfeld剖分上构造了低正则性的有限元弹性复形。与这类剖分上已有的三维弹性复形相比，此处构造的复形同时降低了Sobolev正则性和多项式次数，并终结于一个无顶点自由度的可杂交$H({\rm div};\mathbb S)$-协调对称应力空间。该构造通过对Alfeld剖分上的多项式de Rham复形应用局部Bernstein–Gelfand–Gelfand论证得到。证明了两类局部多项式弹性复形：一个$H^2$-$H^1({\rm inc})$复形和一个低正则性的$H^1({\rm curl})$-$H({\rm inc}^+)$复形。导出了它们的泡形子复形及维数公式。这些局部正合序列给出了位移空间和不可压缩空间的唯一可解有限元，并给出了相应弹性序列的全局有限元子复形。在最低阶$H^1({\rm curl})$-$H({\rm inc}^+)$有限元复形中，$H({\rm inc}^+;\mathbb S)$-协调张量空间是分片三次的。在同一阶数下，末端应力-位移对恢复了Johnson–Mercier–Křížek元，而该构造涵盖了所有$k\ge1$的高阶可杂交对称应力。第二族给出了所有$k\ge2$的标准弹性序列的低正则性$H^1$-$H({\rm inc})$有限元复形。对两个全局复形均建立了交换插值图。

4. Stokes–Darcy界面问题的流函数–压力虚拟单元法

Stream function – pressure virtual element methods for the Stokes–Darcy interface problem

作者: Franco Dassi, Rekha Khot, David Mora, Andres E. Rubiano, Ricardo Ruiz-Baier

类别: math.NA

发布日期: 2026-07-02

arXiv ID: 2607.01622v1

链接: http://arxiv.org/abs/2607.01622v1

摘要翻译

中文标题: Stokes–Darcy界面问题的流函数–压力虚拟单元法

摘要翻译: 本文提出了一种用于原始-原始形式耦合Stokes–Darcy系统的全新虚拟单元法（VEM）。在自由流Stokes区域，我们采用流函数构式，该构式本身即满足不可压缩约束并降低计算成本。通过界面处强制质量守恒、法向应力平衡及Beavers–Joseph–Saffman滑移条件，将双调和流函数方程与Darcy压力方程相耦合。得益于VEM处理一般多边形网格的能力，所提方法可以自然地适应不规则界面几何，无需重新网格化或自适应细化。方法的精确性通过若干数值模拟加以验证，其中包括死端过滤和生物人工器官中的网络流应用。

5. 针对Vlasov-Maxwell系统的局部宏观守恒(LoMaC)低秩张量方法

A Local Macroscopic Conservative (LoMaC) low rank tensor method for the Vlasov-Maxwell system

作者: Shadi Heenatigala, Wei Guo

类别: math.NA

发布日期: 2026-07-01

arXiv ID: 2607.01381v1

链接: http://arxiv.org/abs/2607.01381v1

摘要翻译

中文标题: 针对Vlasov-Maxwell系统的局部宏观守恒(LoMaC)低秩张量方法

摘要翻译: 求解Vlasov-Maxwell(VM)系统的主要计算挑战包括相空间的高维性、非线性、固有的守恒性质等。在本文中，我们针对VM系统发展了一种新颖的局部宏观守恒(LoMaC)低秩张量方法，作为我们先前工作(arXiv:2207.00518)的延续。该方法利用了Vlasov方程的张量友好结构，并采用低秩分层Tucker分解来逼近高维Vlasov解。因此，可以缓解维数灾难。此外，为了实现LoMaC性质，该算法同时演化质量、动量和能量的守恒律与Vlasov方程，采用带有动力学通量向量分裂的高阶守恒方法。通过守恒正交投影，保证了低秩解具有与由守恒律更新的宏观可观测量相同的值。给出了VM系统的一系列数值试验，以展示所提算法的效率和有效性。

不相关论文（基本信息）

1. 混合网格上$L^2$投影的Sobolev稳定性

Sobolev stability of the $L^2$-projection on hybrid meshes

作者: Lars Diening, Viktoria Lingert, Tabea Tscherpel

类别: math.NA

发布日期: 2026-07-02

arXiv ID: 2607.02362v1

链接: http://arxiv.org/abs/2607.02362v1

摘要: We establish $L^p$- and $W^{1,p}$-stability of the $L^2$-projection onto mapped Lagrange finite elements on hybrid meshes consisting of triangles and convex quadrilaterals arising from adaptive mesh refinement. If $K$ is the (tensor product) degree of polynomials of the discretisation, then we show, in particular, $W^{1,2}$-stability for all $K\geq 2$ for the Q-RG and Q-RB refinements. This extends results by Ali, Funken, and Schmidt (2022) which hold for the range $2 \leq K \leq 9$ for initial meshes consisting of parallelograms. Our proof relies on an extension of the technique by Diening, Storn and Tscherpel (2021) to general convex quadrilaterals.

2. 基于偏微分方程的超越经典分数扩散生成建模框架

A PDE-Based Framework for Generative Modeling Beyond Classical Score-Based Diffusion

作者: Horacio Tettamanti, Michael Herty

类别: math.NA, math.PR

发布日期: 2026-07-02

arXiv ID: 2607.02349v1

链接: http://arxiv.org/abs/2607.02349v1

摘要: We introduce an alternative generative framework based on a nonlinear modification of the classical Ornstein–Uhlenbeck dynamics. The proposed dynamics admits both a microscopic description through an interacting particle system and, in the mean-field limit, a macroscopic formulation given by a nonlinear Fokker–Planck equation with a superlinear drift term. We show that, for suitable choices of the model parameters and sufficiently large initial mass, the forward dynamics exhibits condensation phenomena by proving the loss of $L^2$ regularity of the solution in finite time. Building upon this formulation, we derive a stabilized reverse-time partial differential equation that reconstructs the initial distribution from the asymptotic state of the forward dynamics, thereby extending the generative paradigm beyond the classical score-based framework. Furthermore, we introduce numerical discretizations of both the forward and reverse processes that accurately capture the asymptotic behavior of the continuous model while successfully reconstructing the initial distribution. Numerical experiments in one and two spatial dimensions validate the proposed methodology and illustrate its application to density filtering through successive iterations of the generative process.

3. Stokes 问题的位移边界方法离散化的块预处理

Block Preconditioning for Shifted Boundary Method Discretisations of the Stokes Problem

作者: Michał Wichrowski, Ajay Ajith

类别: math.NA

发布日期: 2026-07-02

arXiv ID: 2607.02336v1

链接: http://arxiv.org/abs/2607.02336v1

摘要: The Shifted Boundary Method (SBM) sidesteps body-fitted meshing by shifting boundary conditions onto a surrogate boundary and correcting for the displacement through Taylor expansions. Despite its broad analysis and application, scalable iterative solvers for the incompressible Stokes equations remain underdeveloped. We present a block preconditioner for SBM–Stokes discretisations that uses the velocity block together with a pressure mass matrix as a Schur complement approximation. Because the SBM system is non-symmetric, classical operator preconditioning does not apply directly; a field-of-values analysis instead shows that the non-symmetric SBM contributions act as asymptotically small perturbations of a standard saddle-point operator, yielding mesh-independent GMRES convergence on sufficiently fine meshes. Numerical experiments demonstrate iteration counts under refinement across geometries of increasing complexity. We expose a coarse-mesh regime in which an under-resolved grid produces elevated iteration counts, an artefact of insufficient resolution that vanishes once the mesh captures the geometry.

4. 按单元存储的合并无矩阵有限元

Coalesced Matrix-Free Finite Elements in Cell-Wise Storage

作者: Michał Wichrowski

类别: math.NA

发布日期: 2026-07-02

arXiv ID: 2607.02335v1

链接: http://arxiv.org/abs/2607.02335v1

摘要: We present a GPU-oriented formulation of continuous high-order finite elements in which the redundant, cell-wise (element-local) vector is the persistent primary representation of all field data, rather than a transient stage of matrix-free operator evaluation. We prove that, given a preconditioner whose image is continuous, the entire flexible conjugate gradient iteration can be carried out exactly on this unassembled representation: a simple primal-dual pairing identity shows that all Krylov scalars computed from local data coincide with those of the assembled solve, so inter-element communication is confined entirely to the preconditioner. The required direct stiffness summation (DSS) is then realized without indirect gather-scatter, atomics, or coloring, by a dimensionally-split cascade of one-to-one face exchanges that provably accumulates edge and vertex contributions as a byproduct of sequential axis passes; unstructured macro-block interfaces and $h$-adaptive hanging nodes are handled by disjoint topological kernels and a shadow-cell wrapper that leaves the high-throughput sweeps untouched. The cell-wise storage decouples the memory layout from the mesh topology, and we exploit this freedom to benchmark blocked layouts that trade memory coalescing against element contiguity. Numerical experiments on modern GPUs demonstrate that the resulting operator evaluation and solver outperform state-of-the-art matrix-free implementations, signifficantly exceeding throughput of existing implementations.

5. 一种用于可靠长时间工业声发射的稳定边界元方法

A Stable Boundary Element Method for Reliable Long-Time Industrial Sound Emission

作者: Simon Schneider, Ceyhun Özdemir, Heiko Gimperlein, Karsten Urban, Bernd Graf

类别: math.NA, cs.CE

发布日期: 2026-07-02

arXiv ID: 2607.02308v1

链接: http://arxiv.org/abs/2607.02308v1

摘要: In this paper we investigate a stable space-time formulation for long-time industrial sound emission problems. To this end, we use a well-posed Galerkin formulation in space and time of the acoustic wave equation in $\mathbb{R}^3$, involving a hypersingular boundary integral operator. Our numerical experiments confirm that the resulting time stepping scheme is stable and accurate for complex acoustic problems in industrial geometries, in contrast to alternative well-known schemes. The proposed method is shown to be efficient for real-world problems, and we obtain very good agreement with physical acoustic measurements.

6. 针对三维逆声学障碍散射问题的一种新型时域迭代方法

A novel time-domain iterative method for a three-dimensional inverse acoustic obstacle scattering problem

作者: Lu Zhao, Heping Dong, Zhiyong Cheng

类别: math.NA

发布日期: 2026-07-02

arXiv ID: 2607.02286v1

链接: http://arxiv.org/abs/2607.02286v1

摘要: This paper concerns the three-dimensional forward and inverse acoustic obstacle scattering problem in the time domain. For the forward problem, a retarded potential formulation discretized by convolution quadrature and Galerkin methods is introduced. By introducing the retarded boundary integral defined on a homothetic surface, we propose a novel time-domain convolution quadrature based iterative method to reconstruct both the shape and location of a rigid obstacle. The retarded integral in the time domain is reformulated into a system of integrals in the s-domain. The resulting s-domain integrals are very fast to compute, as they only involve non-singular integrals over the homothetic surfaces. Moreover, the Fréchet derivative with respect to the boundary can be derived straightforwardly. We also prove that the scattered field generated by the homothetic surface converges to the exact field in the time domain. To improve the stability of the inversion algorithm, an incremental truncation technique is proposed, and numerical experiments confirm the effectiveness and robustness of our method.

7. 一种用于物理信息神经网络良态训练的优化框架

An Optimisation Framework for the Well-Conditioned Training of Physics-Informed Neural Networks

作者: Joseph Webb, Sadok Jerad, Coralia Cartis

类别: cs.LG, math.NA, math.OC, physics.comp-ph

发布日期: 2026-07-02

arXiv ID: 2607.02194v1

链接: http://arxiv.org/abs/2607.02194v1

摘要: Physics-informed neural networks (PINNs) have emerged as a promising route to solve partial differential equations, yet they have struggled to reach the precision of classical solvers. The obstacle is increasingly understood to be one of optimisation, owing to the severely ill-conditioned loss landscape. We present $\textbf{DSGNAR}$: Doubly-Sketched Gauss-Newton with Adaptive Ratio, a scalable second-order optimisation framework that confronts this ill-conditioning and, in doing so, obtains unprecedented accuracy and speed. $\textbf{DSGNAR}$ couples a doubly-sketched Gauss-Newton model with a novel strategy that carefully controls both regularisation and step length. Across a suite of problems spanning nonlinear, chaotic, multi-scale, high-dimensional, and Navier-Stokes, the framework greatly improves on the state of the art: able to attain relative $\ell_2$ errors as low as $3\times10^{-16}$ in double precision, improve contemporary results by five orders of magnitude on the canonical Burgers’ equation, and as much as eight orders on a high-dimensional Poisson problem, while remaining markedly faster. We further show that, in single precision, solutions at the limit of round-off error can be obtained very quickly: Burgers’ equation to $\ell_2^{\text{rel}} = 4.75 \times 10^{-7}$ in under ten seconds. The framework is also robust to the choice of architecture, arithmetic precision, and initial hyperparameters. The code is available at https://www.github.com/wephy/physics-informed-neural-networks

8. 三维声波障碍物逆散射问题的高效迭代方法

A highly efficient iterative approach for inverse acoustic obstacle scattering problems in three dimensions

作者: Zhiyong Cheng, Heping Dong, Lu Zhao

类别: math.NA

发布日期: 2026-07-02

arXiv ID: 2607.02180v1

链接: http://arxiv.org/abs/2607.02180v1

摘要: This paper concerns a three-dimensional inverse acoustic obstacle scattering problem from scattered field or phased/phaseless far-field data. Based on the boundary integral defined on a homothetic surface, we propose a highly efficient iterative approach for obstacle reconstruction that completely avoids dealing with any singularity. Here, the injectivity and dense-range property of the Fréchet derivative have been proved to ensure the solvability of the linearized equivalent data equation. We also prove that the scattered field generated by the homothetic surface can arbitrarily approximate the exact one. Numerical experiments are presented to verify the superiority and robustness of the proposed approach.

9. 任意网格上的加权导数直方插值：可容许性与精确分解

Weighted Derivative Histopolation on Arbitrary Grids: Admissibility and Exact Factorizations

作者: Allal Guessab, Federico Nudo

类别: math.NA

发布日期: 2026-07-02

arXiv ID: 2607.02105v1

链接: http://arxiv.org/abs/2607.02105v1

摘要: In this paper, we introduce a weighted derivative histopolation framework on families of intervals. The degrees of freedom consist of one scalar normalization and weighted integral moments of the derivative over a prescribed family of subintervals. We prove that the resulting scheme is unisolvent on $Π_N$ when the interval family separates polynomials of degree at most $N-1$ through weighted moments and the normalization is nonzero on constants. Thus, the derivative moments determine the polynomial up to an additive constant, and the scalar normalization fixes this remaining degree of freedom. This gives a sharp criterion for the well-posedness of the interpolation problem and a complete characterization of the admissible scalar normalizations. We then show how admissible families of intervals can be constructed from a fixed grid. When the endpoints of the intervals belong to the grid, admissibility is reduced to the nonsingularity of an interval matrix associated with the family, which depends only on the representation of the intervals in terms of consecutive cells. For Jacobi weights, the associated data matrices have a natural block structure in Jacobi polynomial bases, and the reduced derivative matrix can be expressed in terms of shifted Jacobi moment matrices. We next study Chebyshev configurations in which this structure becomes explicit. For the four classical Chebyshev families, suitable polynomial bases lead to diagonal Gram matrices for the reduced derivative matrices. We show that this diagonal structure depends on the simultaneous choice of the weight, the basis, and the grid. Numerical experiments on equispaced and Chebyshev–Lobatto nodes show the behaviour of the method for different interval families and for different Jacobi parameters.

10. NLF：用于凸网络流均衡的电阻网络框架和线性时间求解器

NLF: A Resistor-Network Framework and Linear-Time Solver for Convex Network-Flow Equilibria

作者: Oren E. Livne

类别: math.NA

发布日期: 2026-07-02

arXiv ID: 2607.02041v1

链接: http://arxiv.org/abs/2607.02041v1

摘要: We present NLF (Nonlinear Laplacian Flow), a unified framework and linear-time solver for convex network-flow equilibria. Congestion routing, minimum-delay routing, and maximum flow share one form: the nonlinear graph Laplacian $Bρ(B^Tφ)=αd$, where a monotone edge law $ρ_e$ encodes the physics (undirected graphs; directed variants are future work). NLF solves it by a damped chord-Newton iteration whose frozen linearization – a weighted graph Laplacian – is inverted by a near-linear Laplacian solver (default: approximate Cholesky, LAMG+ interchangeable). The nonlinear solve costs $2$–$4$ linear Laplacian solves, making the wall-clock empirically $O(m)$ in the edge count $m$ (not a proved bound). On single-commodity congestion (BPR cost), NLF converges on all 2,003 SuiteSparse corpus graphs up to $1.8\times10^7$ edges. Against a state-of-the-art interior-point method, NLF is a median $2.6\times$ faster where both converge and $>45\times$ on poorly-separable graphs where the IPM’s direct core is superlinear; against L-BFGS, a median $4.2\times$ faster and the only solver to finish on the 90 hardest instances. A multicommodity extension routes $K$ commodities through one shared hierarchy at $O(Km)$ per step. The same machinery recovers the exact max-flow as a short sequence of Laplacian solves, with the cut potential as a by-product. Code: https://github.com/orenlivne/nlf

11. Biot方程与摩擦接触力学耦合的数值分析

Numerical analysis of the Biot equations coupled to frictional contact mechanics

作者: Marius Nevland, Kundan Kumar, Inga Berre, Jakub Wiktor Both, Eirik Keilegavlen

类别: math.NA

发布日期: 2026-07-02

arXiv ID: 2607.02030v1

链接: http://arxiv.org/abs/2607.02030v1

摘要: We consider a mathematical model of a poro-visco-elastic medium subject to frictional contact with a rigid obstacle, and study its numerical approximation. This model couples the Biot equations and contact conditions in the form of normal compliance and Coulomb friction. The resulting variational problem consists of a linear partial differential equation coupled to a nonlinear variational inequality. We propose and analyze a fully discrete numerical scheme for this problem, using conformal finite elements in space and the implicit Euler method in time. Existence and uniqueness of the discrete solution is established, and stability and a priori error estimates are derived. A numerical experiment is performed in which numerical error estimates are computed and compared to the theoretical results.

12. 弹性固体中动态断裂的变分非局部相场模型

A Variational Nonlocal Phase-Field Model for Dynamic Fracture in Elastic Solids

作者: Qing Cheng, Yuqi Sun, Xuejun Xu

类别: math.NA

发布日期: 2026-07-02

arXiv ID: 2607.01881v1

链接: http://arxiv.org/abs/2607.01881v1

摘要: We develop a variational nonlocal phase-field model for dynamic fracture in elastic solids. The proposed formulation is distinguished by three main features. First, the model is formulated through nonlocal kinematics and kernel-dependent function spaces, allowing weaker regularity requirements while recovering the classical local theory as the nonlocal interaction domain vanishes. Second, a nonlocal crack-surface functional is introduced as an integral counterpart of the Ambrosio–Tortorelli regularization, so that the characteristic length of the diffusive crack is implicitly determined by the nonlocal interaction domain rather than by a prescribed length scale. Third, the degraded nonlocal elastic energy and the nonlocal crack-surface functional are combined into a variationally consistent dynamic fracture system, consisting of a nonlocal momentum balance and an irreversible nonlocal gradient-flow evolution law for the phase field. The coupled system is solved using two temporal discretization strategies: a structure-preserving scalar auxiliary-variable scheme and a staggered alternating scheme, both combined with finite element discretization in space. Numerical examples involving Mode-I fracture, dynamic crack branching, Kalthoff–Winkler-type shear fracture, and fragmentation show that the proposed model captures complex crack initiation, propagation, branching, and interaction without explicit crack tracking. Quantitatively, the predicted crack-tip velocities remain below $0.6c_R$ in the dynamic branching and shear-loading tests, and the shear-loading benchmark gives an inclined crack path of approximately $48^\circ$, consistent with the characteristic Kalthoff–Winkler fracture pattern.

13. 利用基于任务的并行化实现二维网格上的红黑高斯-赛德尔方法

Exploiting Task-Based Parallelism for the Red-Black Gauss-Seidel Method on 2D Grids

作者: Shiting Long, Gustavo Ramirez-Hidalgo, Andreas Frommer, Dirk Pleiter

类别: cs.DC, math.NA

发布日期: 2026-07-02

arXiv ID: 2607.01735v1

链接: http://arxiv.org/abs/2607.01735v1

摘要: Gauss-Seidel is a well-established iterative method for the solution of linear systems, and multicoloring has been widely used to increase parallelism in iterative solution techniques. Implementing multi-color Gauss-Seidel with conventional divide-and-conquer parallelization strategies, however, may be inefficient due to global synchronization requirements and load imbalances. Task-based programming models can mitigate these issues by enabling fine-grained parallelism, removing global barriers and allowing updates of different colors to partially overlap in time. In this work, we implement the red-black Gauss-Seidel method using two task-based programming models and compare them with a classical divide-and-conquer parallel implementation to evaluate the impact of fine-grained parallelism on execution efficiency. The red-black scheme serves as a representative example, as task-based approaches naturally extend to more general multi-color schemes arising from unstructured grids and wider stencils. Using the solve of the 2D Poisson equation as benchmark, our results show that task-based implementations can achieve performance comparable to conventional divide-and-conquer parallelization while providing greater resilience to hardware-level asynchronicity.

14. 通过低秩输运特征的保结构降阶建模

Structure-Preserving Reduced-Order Modeling via Low-Rank Transport Signatures

作者: Jiajia Yu, Jingwei Hu, Fengyan Li, Shanyin Tong, Yunan Yang, Zhaiming Shen

类别: math.NA

发布日期: 2026-07-02

arXiv ID: 2607.01696v1

链接: http://arxiv.org/abs/2607.01696v1

摘要: Parametrized PDEs with density-valued solutions are often difficult to approximate with classical linear reduced-order models, especially in transport-dominated regimes. We introduce an optimal-transport-based reduced-order modeling that represents each density by the Kantorovich potential transporting a fixed reference density to the target density, and then maps these potentials to transport signatures using a weighted Laplacian associated with the reference measure. This embeds the density-valued solution map in a Hilbert space while preserving control of the induced transport maps and Wasserstein error. We treat the signature map as a continuous matrix indexed by parameters and space, construct a low-rank skeleton decomposition using a maximal-volume criterion, and learn the parameter-to-coefficient map with a neural network for efficient non-intrusive online evaluation. The reconstructed solution is obtained by pushing forward the reference density, so mass preservation is built into the method. We prove a mean-squared Wasserstein error bound separating low-rank approximation, discretization, sampling, and learning errors, and demonstrate the method on a two-dimensional continuity equation, where transport signatures yield substantially lower-rank structure than the original density snapshots.

15. 电阻型浸入表面法与浸入边界法在主动脉瓣模拟中的比较分析

Comparative analysis of resistive immersed surface and immersed boundary methods for aortic valve simulation

作者: Han Zhao, Alexander D. Kaiser, Fanwei Kong, Aaron L. Brown, Zinan Hu, David Codoni, Sujal Dave, Alison L. Marsden

类别: physics.flu-dyn, math.NA

发布日期: 2026-07-02

arXiv ID: 2607.01662v1

链接: http://arxiv.org/abs/2607.01662v1

摘要: Numerical modeling of aortic valve dynamics is essential for understanding the complex fluid-structure interaction (FSI) governing valve biomechanics in health and disease. Immersed methods provide a flexible computational framework for simulating the large deformations of valve leaflets and associated blood flow without requiring body-fitted meshes. Among these approaches, the Resistive Immersed Surface (RIS) and Immersed Boundary (IB) methods are widely used. However, systematic comparative analysis of these methods for realistic aortic valve simulations has not been performed. In this work, we compare a prescribed-kinematics RIS workflow implemented in SimVascular’s svMultiPhysics solver with a fully coupled IB workflow using IBAMR for trileaflet and bicuspid aortic valve configurations. The RIS method represents the valve as a surface with prescribed kinematics embedded in the fluid domain and introduces a penalty force that drives the surrounding fluid velocity toward the prescribed leaflet velocity. This formulation reduces modeling complexity and provides useful hemodynamic predictions when representative leaflet kinematics are available. In contrast, the IB method models the leaflets as elastic structures fully immersed in the fluid domain and resolves leaflet deformation through fully coupled two-way FSI. The study focuses on the extent to which RIS reproduces bulk hemodynamic features and transvalvular pressure gradients. Results show that the RIS method captures the large-scale flow structures and predicts the mean transvalvular pressure gradient with a relative error within 15% of the fully coupled IB simulation, improving to within 5% when inlet boundary conditions are matched, while reducing computational cost by approximately 60%.

16. 用于压缩感知光片显微镜的即插即用体积重建

Plug-and-Play Volumetric Reconstruction for Compressive Sensing Light-Sheet Microscopy

作者: Jianqing Jia, Yi Gong, Xinyuan Zhang, Jichen Chai, Yichen Ding, Yifei Lou

类别: cs.CV, math.NA

发布日期: 2026-07-02

arXiv ID: 2607.01654v1

链接: http://arxiv.org/abs/2607.01654v1

摘要: We investigate volumetric reconstruction for compressive sensing light-sheet microscopy (CS-LSM), where fast volumetric imaging is achieved by encoding multiple axial planes into each camera exposure. To recover the underlying volume from highly multiplexed measurements, we propose a plug-and-play (PnP) framework that flexibly incorporates any user-specified denoiser into the reconstruction process. Building on a slice-based formulation, we further introduce an axial-coupled model that exploits correlations between adjacent slices to improve volumetric continuity. For efficient computation, we derive a Woodbury-based update for the data-consistency step in both the slice-based and axial-coupled formulations, and employ a Gauss-Seidel sweep for the denoising step in the axial-coupled model. Under a weakly convex regularization assumption, we establish subsequential convergence of the proposed algorithm. Experiments on synthetic and real zebrafish-heart data demonstrate that the proposed framework successfully recovers cellular structures from compressed measurements, and provide practical insights into the comparative performance of commonly used denoisers within the PnP framework under the CS-LSM setup.

17. Fourier对角化的自然梯度与Sobolev镜像下降

Fourier-Diagonalized Natural Gradients and Sobolev Mirror Descent

作者: Jeongbin Jo

类别: math.NA

发布日期: 2026-07-02

arXiv ID: 2607.01634v1

链接: http://arxiv.org/abs/2607.01634v1

摘要: We study natural-gradient updates whose metric operators are diagonalized by the Fourier transform and relate them to Sobolev mirror descent. Translation-invariant Fisher geometries and Sobolev mirror geometries share a common inverse-map structure in the spectral domain. The Fisher metric is represented by a positive Fourier symbol, while Sobolev mirror geometry corresponds to the specific Bessel-potential symbol associated with the Sobolev norm. When these symbols coincide, the natural-gradient and mirror-descent updates are identical; otherwise, Sobolev mirror descent provides a canonical spectral preconditioner for the Fisher inverse geometry. This gives a mathematical lens through which spectral filtering and truncation techniques in PDE and operator learning can be viewed as natural actions of inverse metric geometry. We introduce Spectral Natural Gradient, an FFT-based implementation of these geometric updates.

18. 基于Gegenbauer重构的输运主导问题降阶模型后处理

Post-Processing Reduced-Order Models for Transport-Dominated Problems by Gegenbauer Reconstruction

作者: Lei Yan, Yan Jiang, Chi-Wang Shu

类别: math.NA

发布日期: 2026-07-02

arXiv ID: 2607.01619v1

链接: http://arxiv.org/abs/2607.01619v1

摘要: In this paper, we develop a physics-based post-processing technique for data-driven reduced-order models (ROMs) of transport-dominated problems. Besides the slow decay of the Kolmogorov n-width, ROMs based on globally supported bases often produce unphysical oscillations when approximating solutions with shocks or sharp gradients, a phenomenon analogous to Gibbs oscillations in spectral approximations. To address this issue, we introduce a post-processing framework based on Gegenbauer polynomial reconstruction. The key idea is to re-project the ROM solution onto a Gegenbauer polynomial basis over each interval of analyticity. Originally developed for spectral approximations, Gegenbauer reconstruction achieves spectral accuracy while effectively suppressing Gibbs oscillations. We extend this technique to data-driven ROMs and consider three representative approaches: Proper Orthogonal Decomposition (POD)-Galerkin ROM, Operator Inference (OpInf), and nonlinear manifold ROMs based on convolutional autoencoders (CAE). Numerical results show that the proposed post-processing consistently removes spurious oscillations and substantially improves solution quality for all three ROMs. For one-dimensional problems, the method is straightforward to implement once discontinuities are detected. We further develop a practical extension to two-dimensional problems using line-by-line reconstruction in each coordinate direction. Extensive numerical experiments demonstrate that the proposed method reduces errors by up to one or two orders of magnitude for inviscid transport problems and significantly outperforms total variation regularization in both numerical accuracy and the sharp resolution of discontinuities.

19. 一种用于有限快照数据参数化偏微分方程的谱子空间增强POD-Galerkin方法

A spectral-subspace-augmented POD-Galerkin method for parametrized PDEs with limited snapshot data

作者: Tianhao Hu, Zecheng Gan

类别: math.NA

发布日期: 2026-07-01

arXiv ID: 2607.01534v1

链接: http://arxiv.org/abs/2607.01534v1

摘要: Parametrized partial differential equations (PDEs) arise in many-query simulation, optimization, control, and uncertainty quantification, where repeated full-order solves restrict the number of high-fidelity snapshots that can be generated. This limitation is particularly pronounced in computational energy science, where multiscale models of porous-media flow, transport, and energy materials often make large snapshot datasets impractical. Proper orthogonal decomposition (POD) constructs compact reduced bases from solution snapshots, but it may exhibit limited out-of-sample predictive capability when the snapshots insufficiently sample the solution manifold. To address this limitation, we propose a spectral-subspace-augmented POD-Galerkin method (SS-POD) tailored to limited-data regimes. SS-POD starts from a problem-adapted spectral approximation space, partitions it into orthogonal subspaces, and performs POD locally on the projected snapshot matrices. An energy-balancing rule determines the spectral partition so that the resulting local POD problems are assigned comparable amounts of snapshot energy. For nonlinear parametrized PDEs, SS-POD is coupled with the discrete empirical interpolation method (DEIM). Numerical experiments show that SS-POD improves out-of-sample accuracy over standard POD-Galerkin while retaining compact reduced bases in limited-snapshot regimes. In particular, for a Laplace-Beltrami problem on the unit sphere with only 5 snapshots, SS-POD achieves a relative error of $3.9*10^{-8}$ using 91 basis functions, whereas the standard POD error saturates at $7.8*10^{-4}$ and the spectral-Galerkin method requires 256 basis functions for comparable accuracy. These results indicate that SS-POD provides an effective strategy for high-fidelity reduced-order modeling from limited snapshot data.

20. 几何感知的 R-结构 Kolmogorov-Arnold 网络

Geometry-Aware R-Structured Kolmogorov-Arnold Networks

作者: Sergei Kucherenko, Nilay Shah

类别: cs.LG, math.NA

发布日期: 2026-07-01

arXiv ID: 2607.01449v1

链接: http://arxiv.org/abs/2607.01449v1

摘要: We propose a novel hybrid neural architecture, the Geometry-aware R-Structured Kolmogorov-Arnold Network (GRS-KAN), which integrates V.L.Rvachev’s R-functions into the Kolmogorov-Arnold Network (KAN) framework. The proposed approach combines two complementary modeling mechanisms: smooth nonlinear structure is learned by KAN branches, while known geometric or logical constraints are encoded analytically using differentiable R-functions. This enables explicit representation of discontinuities, feasible regions, and implicit geometric boundaries within a trainable neural architecture. The framework implements differentiable logical operations through R-conjunctions and R-disjunctions, allowing complex geometric supports to be represented analytically and incorporated directly into regression models. Several GRS-KAN variants are introduced, including additive, multiplicative, and agnostic branch-weighted architectures. The method is demonstrated on regression problems involving discontinuities with circular and rectangular supports. Numerical experiments show that explicit geometric encoding substantially improves predictive accuracy and boundary localization compared with standard KANs. In the considered benchmarks, geometry-aware GRS-KAN models reduce test RMSE by up to 67% while simultaneously improving interpretability through explicit analytical representation of the learned geometric structure. The agnostic variant further demonstrates the ability to automatically determine whether geometric priors are beneficial for a given learning task.

21. Fredholm–留数选择非定常Kutta振幅

Fredholm–residue selection of the unsteady Kutta amplitude

作者: Jiguang Yu, Louis Shuo Wang

类别: math.NA

发布日期: 2026-07-01

arXiv ID: 2607.01403v1

链接: http://arxiv.org/abs/2607.01403v1

摘要: We give an operator-theoretic interpretation of unsteady Kutta selection in trailing-edge acoustic receptivity. The inviscid acoustic–wake problem leaves one outgoing wake amplitude undetermined. We show that, under explicit structural hypotheses, this amplitude is the same scalar obtained from three representations: cancellation of the inverse-square-root edge singularity, Fredholm compatibility of the viscous lower-deck problem, and the residue of the Kutta-normalized transform solution at the downstream wake pole: $\displaystyle A = -\frac{C_-^{(0)}}{C_-^{(KH)}} = -\frac{\langle \mathbf F_{\rm inc},Ψ^\ast\rangle}{\langle \mathbf F_{KH},Ψ^\ast\rangle} = i\operatorname*{Res}_{α=α_{KH}}\mathcal M(α)$. The inner Fredholm–edge mechanism is verified exactly in a linear-shear lower-deck model, where the primal shear and adjoint velocity are Airy fields and the edge concomitant is nonzero outside a discrete resonance set.

22. 基于宏观和微观分量解分解的混合双层输运方法

Hybrid Two-Level Transport Method with Solution Decomposition in Macro and Micro Components

作者: Caleb A. Shaw, Dmitriy Y. Anistratov

类别: math.NA, physics.comp-ph

发布日期: 2026-07-01

arXiv ID: 2607.01346v1

链接: http://arxiv.org/abs/2607.01346v1

摘要: This paper presents a new hybrid MC/deterministic method for solving the one-group steady-state Boltzmann transport equation based on decomposition of solution in macro and micro components. The macro component captures the large-scale structure of the solution. It is represented by angular moments of the high-order transport solution. The $P_1$ approximation is applied to define the macro component. The first two angular moments are obtained as a solution of hybrid low-order moment equations with exact closures. The equation for the micro component is solved using a MC simulation. The hybrid two-level system of equations for macro and micro components is solved by fixed-point iteration scheme. Numerical results are presented to demonstrate variance reduction of stochastic numerical solution and improvement in computational efficiency.