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math.NA

math.NA: explore 106 source-linked works published from 2024 to 2026, with original documents and citations.

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Includes records with this source-supplied label or an explicit phrase match in their metadata. Matches indicate a mention, not proof that a paper uses a method or tests a material. Source versions are consolidated by DOI.

Sources: arxiv. Collection updated 2026-09-14. Counts describe this index, not the complete source archives.

arXiv2026

Code Generation for Near-Roofline Finite Element Actions on GPUs from Symbolic Variational Forms

We present a novel parallelization strategy for evaluating Finite Element Method (FEM) variational forms on GPUs, focusing on those that are expressible through the Unified Form Language (UFL) on simplex meshes. We base our approach on code transformations, wherein we construct a space of scheduling candidates and rank them via a heuristic cost model to effectively handle the large diversity of computational workloads that can be expressed in this way. We present a design of a search space to which the cost model is applied, along with an associated pruning strategy to limit the number of configurations that need to be empirically evaluated. The goal of our design is to strike a balance between the device's latency-hiding capabilities and the amount of state space, a key factor in attaining near-roofline performance. To make our work widely available, we have prototyped our parallelization strategy within the Firedrake framework, a UFL-based FEM solver. We evaluate the performance of our parallelization scheme on three generations of Nvidia GPUs, specifically the H200, Titan V and Tesla K40c, across a range of operators commonly used in applications, including fluid dynamics, wave propagation, and structural mechanics, in 2D and 3D geometries. Our results demonstrate that our proposed algorithm achieves more than $50\%$ roofline performance in $60\%$ of the test cases.

cs.DC↗

arXiv2026

Easy-to-Implement One-Step Schemes for Stochastic Integration

Convenient, easy to implement stochastic integration methods are developed on the basis of abstract one-step deterministic order $p$ integration techniques. The abstraction as an arbitrary one step map allows the inspection of easy to implement stochastic exponential time differencing Runge-Kutta (SETDRK), stochastic integrating factor Runge-Kutta (SIFRK) and stochastic RK (SRK) schemes. Such schemes require minimal modifications to existing deterministic schemes and converging to the Stratonovich SDE. These schemes capture all symmetric terms in the Stratonovich-Taylor expansion, are order $p$ in the limit of vanishing noise, can attain at least strong order $p/2$ or $p/2-1/2$ (parity dependent) for drift commutative noise, strong order $1$ for commutative noise, and strong order $1/2$ for multidimensional non-commutative noise. Numerical convergence is demonstrated using different bases of noise for 2nd, 3rd and 4th order SETDRK, SIFRK and SRK schemes.

math.NA↗

arXiv2026

A Practical Mode-parallel Implementation of the (H-)Tucker Decomposition via Randomization

In the last decades, tensors have emerged as the right tool to represent multidimensional data in a compact yet informative manner. Moreover, it is well-known that by performing low-rank factorizations of such tensors one is often able to effectively unveil possible hidden structure in data, mainly due to unexpected dependencies among the different variables encoded in the given tensor. However, computing these factorizations is extremely energy-consuming and memory-demanding, especially for high-dimensional tensors, namely those with a large number of modes. In this paper we focus on two state-of-the-art tensor decompositions: the Tucker and H-Tucker decompositions. We propose novel numerical strategies able to perform these factorizations in a mode-parallel fashion, that is the operations required by the algorithm along all modes are performed in parallel. This is in contrast to what is achieved by many procedures available in the literature that parallelize some of the operations along each mode, e.g., tensor-times-matrix steps, while still visiting one mode at the time in a sequential manner. Our strategies make use of cutting-edge randomization techniques comprising fiber sampling and randomized range-finding steps. In case of Tucker decomposition, we provide upper bounds on the expected value of the error provided sufficiently large sampling parameters have been adopted. A panel of numerical results showcases the potential of our approach in reducing both the running time and the storage demand of computing Tucker and H-Tucker decompositions. Moreover, experiments carried out in HPC environments illustrate the good scaling of our mode-parallel approach.

math.NA↗

arXiv2026

Five-Structures Preserving Algorithm for charge dynamics model

This paper develops a family of fast, structure-preserving numerical algorithms for the nonlinear Maxwell-Ampere Nernst-Planck equations. For the first-order scheme, the Slotboom transformation rewrites the Nernst-Planck equation to enable positivity preservation. The backward Euler method and centered finite differences discretize the transformed system. Two correction strategies are introduced: one enforces Gauss's law via a displacement correction, and the other preserves Faraday's law through potential reconstruction. The fully discrete scheme exactly satisfies mass conservation, concentration positivity, energy dissipation, Gauss's law, and Faraday's law, with established error estimates. The second-order scheme adopts BDF2 time discretization while retaining the same structure-preserving strategies, exactly conserving mass, Gauss's law, and Faraday's law. Numerical experiments validate both schemes using analytical solutions, confirming convergence orders and positivity preservation. Simulations of ion transport with fixed charges demonstrate exact preservation of Gauss's and Faraday's laws over long-time evolution, reproducing electrostatic attraction, ion accumulation, and electric field screening. The results fully support the theoretical analysis and the schemes' stability and superior performance.

math.NA↗

arXiv2026

Phase-field digital image correlation for integrated displacement and damage measurements

This work presents a novel digital image correlation (DIC) framework for full-field measurements of displacement, strain, and damage, based on a phase field (PF) approach. The idea is to take advantage of the ability of the PF method to track complex crack morphologies and to provide a natural way in DIC to perform damage and crack measurements from experimental speckle images, in addition to displacement and strain fields. Moreover, incorporating the damage variable into DIC can improve the displacement accuracy near the crack tip, and can avoid the need of user-defined masks when dealing with cracked samples, which is advantageous when cracks become complex and the manual application of masks becomes challenging. The theoretical formulation of the proposed framework, namely PF-DIC, was presented in detail in the paper, along with a finite element implementation. Numerical examples have demonstrated the capability of the proposed PF-DIC in terms of capturing different types of cracks while providing similar measurement accuracy to that of masked DIC. Additionally, it is shown that the PF-DIC can be easily adapted to selectively identify critical cracks under specific loading conditions or mechanisms for damage assessment and diagnostic purposes. The proposed DIC framework can be used to characterize material defects, support structural health monitoring, and enable a potential unification of PF simulations and experimental fracture measurements

math.NA↗

arXiv2026

3D Uncertainty Quantification for Photoacoustic Tomography

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.

math.NA↗

arXiv2026

Why the Kellogg Mesh Is Radial: A Mathematical Explanation of a Classical Computational Benchmark

Kellogg's checkerboard interface problem is a classical benchmark for robust adaptive finite element methods. Its successful adaptive meshes are radial: they refine strongly toward the interface crossing but show no angular structure, despite the large contrast and the asymmetric solution. We explain this by proving that the singular solution $u(r,θ)=r^γμ(θ)$ satisfies the exact identities $κ|\nabla u|^2=Λr^{2γ-2}$ and $κ|\nabla^2 u|_F^2=2(1-γ)^2Λr^{2γ-4}$, with $Λ=γ^2\cos^2(πγ/4)$ and the Hessian taken separately in each quadrant. The point is what has disappeared: the right-hand sides depend on $r$ alone, although $κ$ and $u$ each depend on the angle as well. Combined with equal discretization-error distribution, this shows that the target element density is radial, so a correct mesh should display nothing but refinement toward the center, and a Kellogg mesh that is not radial is visible evidence that the computation is not following the coefficient-weighted local difficulty. The reading is specific to this benchmark: on a second interface problem the same estimator correctly produces a strongly material-biased mesh, with a computed element-count ratio of $3.934$ against the predicted $4$. A byproduct gives the benchmark constants in closed form, so the problem data can be generated from $γ$ alone at any precision.

math.NA↗

arXiv2026

ENPINN: Energy-Norm-Guided Gradient-Enhanced PINNs for Generalized Transport Problems with Sharp Gradients

Physics-informed neural networks (PINNs) have emerged as a meshless alternative to conventional numerical methods for solving partial differential equations (PDEs). However, their limited ability to capture sharp gradients can lead to substantial errors when resolving boundary and interior layers. Here, we introduce an energy-norm-enhanced PINN (ENPINN) that incorporates gradient information and variational structure into the loss function to improve the resolution of layer-dominated solutions. We first examine two related formulations: weak-loss PINNs (WLPINNs), which incorporate test functions into the conventional PINN residual, and gradient-enhanced PINNs (gPINNs), which augment the loss with spatial derivatives of the PDE residual. By analyzing these formulations, we identify their limitations in resolving steep solution gradients and motivate the systematic construction of ENPINN. We establish theoretically how the energy-norm error depends on the ENPINN loss and show that a suitably modified residual-derivative term is essential for accurately capturing boundary layers. We further establish the existence of neural-network approximations with arbitrarily small energy error and derive corresponding derivative bounds, providing a theoretical foundation for the proposed framework. The performance of ENPINN is assessed through systematic comparisons with existing PINN variants for convection-diffusion-reaction problems exhibiting steep gradients. Numerical experiments include a combustion model, a coupled multi-scale system, a two-dimensional Burgers equation with an interior layer, and a three-dimensional time-dependent problem.

math.NA↗

arXiv2026

Analysis and Approximation of Stochastic Multiscale Subdiffusion Driven by Fractional Gaussian Noise

This paper investigates a stochastic multiscale subdiffusion model driven by fractional Gaussian noise, where the multiscale Abel kernel with variable exponent $α(t)\in(0,1)$ is used to capture multiscale and crossover behavior in anomalous diffusion. The main difficulties of this model lie in the complexity of the multiscale Abel kernel (e.g. non-monotonicity and non-coercivity) and the low regularity caused by the noise. Concerning these issues, we prove the well-posedness and regularity of the mild solutions by means of solution operator approach and a perturbation technique for multiscale Abel kernel. Then both the semidiscrete-in-time and fully-discrete numerical schemes are proposed and analyzed under the low-regularity numerical analysis framework, with proved temporal and spatial convergence rates. Numerical experiments are presented to substantiate the theoretical results.

math.NA↗

arXiv2026

Tensor Orthogonal Subspace Split: Theory and Applications

Tensor representations have emerged as a fundamental paradigm for modeling multidimensional data by preserving intrinsic correlations across multiple modes. This paper proposes a novel theoretical framework, termed Tensor Orthogonal Subspace Split (TOSS), which explicitly splits a tensor, along a prescribed mode, into two orthogonal components: a dominant component lying in a prescribed subspace and a residual component lying in the corresponding orthogonal complement. We first present the general formulation of TOSS and systematically investigate its fundamental properties. As an important and practically meaningful special case, we further introduce the rank-one TOSS, which imposes a separable rank-one structure along the splitting mode and admits a clear geometric interpretation. This formulation naturally captures dominant consistent patterns while effectively isolating orthogonal residual component. The proposed framework establishes a unified theoretical foundation for tensor-domain orthogonal split and opens new avenues for structured tensor modeling across diverse applications. Building upon the developed TOSS theory, hyperspectral image restoration and color video background modeling are considered as two representative tasks, for which corresponding optimization models are formulated. Efficient algorithms are developed to solve the resulting problems. Extensive experimental results validate the effectiveness and superiority of the proposed approaches.

math.NA↗

arXiv2026

Constraint Preserving AFD-WENO Schemes for Relativistic Hydrodynamics with General Equations of State

We develop a high-order physical-constraint-preserving (PCP) alternative finite difference weighted essentially non-oscillatory (AFD-WENO) scheme for the special relativistic hydrodynamics equations with general equations of state. The proposed scheme comprises two key limiters: a state limiter, which acts after the WENO state interpolation step, and a flux limiter, which acts on the final high-order fluxes. The state limiter ensures that the interpolated states are physically admissible, while the flux limiter ensures that the numerical fluxes are physically admissible. The resulting scheme is rigorously proved to satisfy the physical constraints. Incorporating multiple WENO interpolation techniques, including an improved adaptive-order formulation (WENO-AOI), the method is validated through extensive one- and two-dimensional numerical benchmarks with various equations of state. The numerical results demonstrate high-order accuracy, sharp resolution of discontinuities, and robust stability in extreme relativistic regimes.

math.NA↗

arXiv2026

Efficient Hybrid WENO Schemes for Special Relativistic Hydrodynamics with Adaptive Characteristic Reconstruction

Special relativistic hydrodynamics (SRHD) equations arise in the modeling of high-speed fluid flows encountered in astrophysical phenomena such as jets, supernova explosions, and gamma-ray bursts. Owing to their highly nonlinear hyperbolic nature, solutions often develop strong discontinuities, making the design of stable and accurate numerical schemes challenging. Although Weighted Essentially Non-Oscillatory (WENO) schemes are widely used for such problems, component-wise WENO reconstruction may produce spurious oscillations near discontinuities. On the other hand, characteristic-wise WENO reconstruction provides accurate non-oscillatory solutions for systems of conservation laws, but it involves the computation of eigenvectors in each cell, which leads to high computational cost. In this work, we intend to develop hybrid schemes which maintain the non-oscillatory feature of characteristic-wise WENO while being less costly. We propose three hybrid schemes, namely the H1-WENO, H2-WENO, and H3-WENO schemes, based on a new troubled-cell indicator constructed from the smoothness indicators of the WENO scheme. The proposed troubled-cell indicator effectively distinguishes smooth and discontinuous regions, allowing the hybrid schemes to employ inexpensive reconstructions in smooth regions and the characteristic-wise WENO reconstruction only near discontinuities. Numerical experiments demonstrate that the proposed schemes retain the accuracy and robustness of characteristic-wise WENO methods while significantly reducing the computational cost. In particular, the H1-WENO scheme achieves an approximately 30--40% improvement in computational efficiency compared to the standard WENO scheme in 2D test cases.

math.NA↗

arXiv2026

Wigner-Eckart Factorization of the Polyatomic Boltzmann Collision Operator

We extend the Wigner-Eckart factorization of the spectral Boltzmann collision operator to polyatomic gases with continuous internal energy. Because internal energies are invariant under spatial rotations, the SO(3) reduction survives the Borgnakke-Larsen energy exchange, and the twelve-dimensional collision integral collapses onto a nine-dimensional kinematic core. The core splits into a sparse geometric tensor, evaluated exactly, and a dense physical tensor, integrated by singularity-resolving Gauss rules with an auxiliary Laplace representation of the fractional energy couplings. The quadrature attains near machine precision at the fractional exponents of real gases. The collision invariants are embedded exactly, preserving the translational-internal energy exchange. The factorization compresses the operator by three to nearly four orders of magnitude and accelerates its evaluation 40-fold over dense formulations. The method is validated against the exact monatomic limit, Landau-Teller relaxation, and an analytic frozen-channel Prandtl number, and it matches a published calibration of the same kernel for N2, CO, and H2.

math.NA↗

arXiv2026

High-order energy-stable BGN parametric finite element methods for geometric flows

We construct high-order Runge--Kutta extensions of Barrett--Garcke--Nürnberg (BGN) parametric finite element methods for curve-shortening flow and curve diffusion of planar curves and for mean curvature flow and surface diffusion of closed genus-$0$ surfaces. On each time slab $I_m=(t_m,t_{m+1}]$, the continuous equations are posed on the left-endpoint surface $Γ^m=Γ^{t_m}$ through the map $\X^{m,t}:Γ^m\toΓ^t$, whose target is the evolving surface at time $t$. For curves, this formulation follows by harmonic pullback from $Γ^0$ to $Γ^m$. For surfaces, an orientation-preserving harmonic diffeomorphism is posed separately on each time slab, and conformality yields the weight $\frac12|\nabla_{Γ^m}\X^{m,t}|^2$. Evaluating the time-slab equations at the Runge--Kutta internal times and applying mass-lumped parametric finite elements yields systems on the common domain $Γ^m$, with $Γ^{t_m+c_iτ_m}$ as the intermediate target geometry. This internal-time discretization constructs high-order BGN-structured extensions for the four flows. For an algebraically stable tableau with nonnegative weights, every exact stage solution with nondegenerate intermediate configurations satisfies monotone decay of the discrete curve length or surface area for every positive time step for which that solution exists. Radau IIA experiments exhibit energy decay for all four flows and BGN-type mesh redistribution in the curve tests. The Hausdorff self-convergence results exhibit high-order behavior consistent with the corresponding design orders.

math.NA↗

arXiv2026

Advancements in Spectral Collocation Methods for High-Order Eigenvalue Problems

This paper focuses on computing spectral solutions for high-order eigenvalue problems using an efficient discretization method based on Chebfun spectral discretization algorithms and domain truncation. We solve several numerical eigenvalue problems, demonstrating both the accuracy and computational efficiency of the proposed approach.

math.NA↗

arXiv2026

A variant of the block preconditioner for indefinite complex symmetric linear systems

In this paper, we propose an efficient preconditioner for solving indefinite complex symmetric linear systems within a block preconditioning framework. We analyze the convergence of the corresponding iterative method and investigate several spectral properties of the preconditioned matrix, including eigenvalue distributions and eigenvector structures. The new preconditioner is used to accelerate the convergence of the flexible version of GMRES. Numerical experiments are presented to illustrate the effectiveness of the proposed preconditioner, and comparisons with existing block preconditioners demonstrate its superior performance.

math.NA↗

arXiv2026

Second-Order Accuracy from Large Local Errors in Interface Problems: A Discrete Green's Function Analysis

Finite difference methods for interface problems often exhibit large local truncation errors near the interface, particularly when discontinuities in coefficients or solution derivatives are present. Nevertheless, many such schemes achieve second-order accuracy, a phenomenon not fully explained by standard pointwise consistency arguments. In this work, we provide a precise explanation of this behavior through the structure of the discrete Green's function associated with the underlying conservative difference operator. An explicit representation of the Green's function reveals a two-plateau structure in its weighted increments. By expressing the numerical error in terms of this Green kernel, we show that the dominant interface truncation errors, although individually of order $O(1)$, possess a cancellation structure that reduces their effective contribution to the global error. The exact Green's-function analysis leads to a class of conservative interface flux--balance schemes for which the cancellation mechanism yields second-order accuracy in the maximum norm. The weighted harmonic discretization is included as a particular member, and its cancellation property is established directly. For Cartesian grids with a flat interface parallel to a coordinate direction, the same conservative cancellation mechanism persists along grid lines crossing the interface. Numerical experiments in one and two dimensions directly illustrate the predicted cancellation behavior and the resulting second-order accuracy. These results demonstrate that second-order accuracy in interface problems can arise from the interaction between localized truncation errors and the global structure of the discrete operator, rather than from pointwise consistency alone.

math.NA↗

arXiv2026

Fortin operators for DPG advection discretizations

We construct Fortin operators for discontinuous Petrov-Galerkin (DPG) discretizations of the advection equation with a piecewise constant divergence-free advection vector $β$, on simplicial meshes of any spatial dimension $N$ and for any polynomial degree $k \ge 1$ of the trial space. A minimal test space is built on each element from facet and interior bubbles and later augmented. The Fortin operator is shown to be bounded, uniformly over shape-regular mesh families, in the natural $β$-weighted broken test graph norm built on $L_q$ for every $1 < q < \infty$, where $q$ is the exponent conjugate to the trial exponent $p$. A non-characteristic facet condition is assumed when $k \ge 2$, while the lowest-order case requires no such condition and admits characteristic facets. As applications we prove that the practical fully discrete residual minimization method is quasioptimal in the DPG energy norm for every $1 < p < \infty$, with a quasioptimality constant governed solely by the Fortin operator, that its computable residual is a globally reliable and efficient a posteriori error estimator, and that augmenting the test space and changing the test norm improves the results in the lowest-order case.

math.NA↗

Compare source metadata on this page

Work	Published	Source identifier	Source
Code Generation for Near-Roofline Finite Element Actions on GPUs from Symbolic Variational Forms	2026-08-30	2506.17471	arxiv
Easy-to-Implement One-Step Schemes for Stochastic Integration	2026-08-30	2510.12583	arxiv
A Practical Mode-parallel Implementation of the (H-)Tucker Decomposition via Randomization	2026-08-30	2603.21379	arxiv
Five-Structures Preserving Algorithm for charge dynamics model	2026-08-30	2604.08228	arxiv
Phase-field digital image correlation for integrated displacement and damage measurements	2026-08-30	2606.16850	arxiv
3D Uncertainty Quantification for Photoacoustic Tomography	2026-08-30	2607.21373	arxiv
Why the Kellogg Mesh Is Radial: A Mathematical Explanation of a Classical Computational Benchmark	2026-08-30	2608.22991	arxiv
ENPINN: Energy-Norm-Guided Gradient-Enhanced PINNs for Generalized Transport Problems with Sharp Gradients	2026-08-30	2608.29499	arxiv
Analysis and Approximation of Stochastic Multiscale Subdiffusion Driven by Fractional Gaussian Noise	2026-08-30	2608.29533	arxiv
Tensor Orthogonal Subspace Split: Theory and Applications	2026-08-30	2608.29638	arxiv
Constraint Preserving AFD-WENO Schemes for Relativistic Hydrodynamics with General Equations of State	2026-08-30	2608.29654	arxiv
Efficient Hybrid WENO Schemes for Special Relativistic Hydrodynamics with Adaptive Characteristic Reconstruction	2026-08-30	2608.29776	arxiv
Wigner-Eckart Factorization of the Polyatomic Boltzmann Collision Operator	2026-08-30	2608.29825	arxiv
High-order energy-stable BGN parametric finite element methods for geometric flows	2026-08-30	2608.29877	arxiv
Advancements in Spectral Collocation Methods for High-Order Eigenvalue Problems	2026-08-30	2608.29932	arxiv
A variant of the block preconditioner for indefinite complex symmetric linear systems	2026-08-30	2608.29985	arxiv
Second-Order Accuracy from Large Local Errors in Interface Problems: A Discrete Green's Function Analysis	2026-08-30	2608.30020	arxiv
Fortin operators for DPG advection discretizations	2026-08-30	2608.30043	arxiv

These are bibliographic comparisons, not experimental rankings. Follow the original document for methods and conditions.