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math.NA: explore 425 source-linked works published from 2021 to 2026, with original documents and citations.

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Sources: arxiv. Collection updated 2026-09-16. Counts describe this index, not the complete source archives.

arXiv2026

A \(3\times 3\) counterexample to Lin and Wimmer's rank-minimization conjecture associated with Roth's similarity theorem

We give a \(3\times 3\) counterexample, valid over every field, to a rank-minimization conjecture of Lin and Wimmer (Bull. Aust. Math. Soc., 84 (3) (2011), 441--443) related to Roth's similarity theorem for the Sylvester matrix equation. We also prove that, over the complex field, no counterexample can occur when one of the two matrix sizes is less than \(3\). Hence the example is dimensionally minimal over the complex field.

math.RA↗
arXiv2026

Structure-Preserving Data-Driven Identification of Port-Hamiltonian Differential-Algebraic Systems

We present a data-driven approach to identifying linear index-1 differential-algebraic pH systems (pH-DAEs) based on input-output measurements. In comparison to the identification of port-Hamiltonian (pH) systems, the algebraic constraint and the index condition pose additional challenges. First, we establish a structure-preserving formulation of the considered pH-DAE class and derive an implicit midpoint discretization that preserves the algebraic constraints and discrete dissipation inequality. We formulate the identification problem as a regularized least-squares minimization problem subject to the pH-DAE dynamics. Exploiting the index-1 structure, we reduce the constrained problem to an unconstrained optimization problem over the system parameters while preserving the port-Hamiltonian structure. Next, we derive an adjoint-based formulation to efficiently evaluate the gradient of the resulting reduced cost functional. This enables us to use gradient-based optimization methods for parameter estimation. Under suitable assumptions on the admissible parameter set, the existence of a minimizer is established. Numerical experiments demonstrate that the proposed approach can identify surrogate pH-DAE systems that accurately reproduce the input-output behavior of reference systems. Further investigations show the approach's potential for identifying reduced-order surrogate models. Cross-validation with independent input signals confirms the predictive capability of the identified models.

math.NA↗
arXiv2026

Analysis of the Ill-Conditioning of the Discrete Inverse Laplace transform in Monte Carlo Simulations of Quantum Many-Body Systems

Analytic continuation of imaginary-time quantum Monte Carlo data to real-frequency spectra requires the inversion of a severely ill-posed two-sided Laplace transform and arises naturally in quantum many-body calculations of dynamic properties. In this work, we distinguish the intrinsic ill-posedness of the continuous inverse two-sided Laplace problem from the conditioning of its finite-dimensional discretization. For equidistant sampling and reconstruction grids, we express the discrete problem via a diagonally scaled monomial Vandermonde matrix with exponentially distributed nodes. Imposing the physical detailed-balance symmetry transforms the discretization into a diagonally scaled Chebyshev-Vandermonde system. Exploiting these structures, we derive explicit lower and upper bounds on the condition numbers in terms of the physical and discretization parameters. For the unconstrained discretization, our bounds reveal super-exponential growth of the condition number with the reconstruction dimension, which cannot be removed by increasing the number of imaginary-time samples alone. Detailed-balance substantially improves the conditioning, especially in the practically relevant pre-asymptotic regime, although the asymptotic super-exponential dependence remains. In both cases, our bounds identify a low-dimensional regime in which the super-exponential contribution is suppressed and stable reconstruction remains feasible. These results provide a mathematical explanation for the effectiveness of low-dimensional spectral representations and detailed-balance in analytic continuation. While they do not remove the fundamental ill-posedness, they substantially mitigate the ill-conditioning of its finite-dimensional discretization, delay the onset of its catastrophic super-exponential growth and thereby enlarge the range of numerically accessible reconstructions.

math.NA↗
arXiv2026

Constitutive State-Space Modeling of Path-Dependent Plasticity: A Resolution-Consistent and Parallelizable Computational Framework

Data-driven constitutive models for path-dependent plasticity are commonly formulated using nonlinear recurrent neural networks, whose sequential state evolution limits parallel training and whose predictions may depend on the discretization of the applied strain path. We introduce a Constitutive State Space (CSS) model that reformulates structured state-space dynamics as an incremental constitutive operator. The strain increment is decomposed into magnitude and direction: the loading direction drives the latent state-space system, while the increment magnitude enters the zero-order-hold discretization of its continuous-time linear recurrence. This mechanics-tailored construction guarantees stationarity under zero increments, strongly reduces sensitivity to strain-path resolution, and retains the parallel-scan structure of S5 for efficient training on long constitutive histories. The CSS and Minimal State Cell (MSC) architectures are compared for four multiaxial path-dependent material models including isotropic J2 plasticity, pressure-sensitive foam plasticity, and combined isotropic-kinematic hardening. CSS matches or exceeds the prediction accuracy of the MSC, including one order of magnitude lower validation losses for the plastically incompressible materials. Importantly, CSS maintains low errors across large changes in strain-path discretization, whereas the MSC error increases substantially when evaluated at coarser resolutions than used for training. CSS trains substantially faster and requires fewer strain-stress pairs to attain comparable or better accuracy. Analysis of the learned state further reveals latent structure consistent with the dimensionality of the underlying physical constitutive models. These results establish mechanics-tailored structured state-space dynamics as a computational framework for efficient and discretization-robust data-driven constitutive modeling.

cs.LG↗
arXiv2026

A new formulation of Discontinuous Galerkin method for interface capturing approaches

In computational fluid dynamics, the numerical simulation of free-surface flows using interface capturing approaches (level set, pseudo-concentration) requires accurate resolution of a transport equation. In the present work, this equation is solved using a new formulation of the Lesaint--Raviart discontinuous Galerkin method, which is particularly suitable for problems without inflow boundaries. The motion is described by the incompressible Navier--Stokes equations with surface tension, discretized using the standard $P2-P1$ Taylor--Hood finite element method in space. Surface tension effects are represented using the so-called Continuum Surface Force model. Finally, the proposed approach is implemented and tested on several benchmarks, including the single rising bubble and the Rayleigh--Taylor instability.

math.NA↗
arXiv2026

Frequency Domain Biot--Allard Equations for Isotropic and Anisotropic Poroelastic Media: Two-field formulations and iterative splitting

We present a frequency-domain formulation of Biot's dynamic poroelastic equations with frequency-dependent dissipation (Biot--Allard) for anisotropic, heterogeneous media with memory effects. Two equivalent two-field representations--a displacement-pressure and a velocity-pressure-rate formulation--enable stabilized iterative splitting. While coupling operators generally lack an adjoint or skew-adjoint relationship at finite frequencies, the velocity--pressure-rate representation restores a skew-adjoint structure in the quasi-static limit. We prove continuity of the coupling operators and coercivity of the diagonal blocks, essential for convergence of the L-stabilized splitting scheme. The frequency-domain setting eliminates convolutional memory terms, incorporates attenuation and dispersion via complex-valued parameters, and reduces the time-dependent problem to a family of elliptic boundary-value problems suited for parallel computation and multi-frequency inversion. A conforming Galerkin finite element discretization preserves block structure, and numerical experiments confirm robustness and capture frequency-dependent attenuation. To illustrate discretization independence, we include a large-scale wave simulation using a pseudo-spectral method. This work provides a rigorous and efficient framework for modeling wave phenomena in complex porous media.

math.NA↗
arXiv2026

Fully discrete stochastic maximal regularity and $H^\infty$-calculus for second-order elliptic operators

This paper establishes the fully discrete stochastic maximal $L^p$-regularity and the accompanying sharp maximal estimate for numerical approximations of parabolic stochastic partial differential equations. We consider the spatial finite element discretization $A_h$ of a general second-order elliptic operator $A=-\nabla \cdot a\nabla +b\cdot \nabla +c$ with Dirichlet boundary conditions on a smooth, bounded, convex domain in $\mathbb{R}^3$, coupled with a broad class of temporal schemes, including rational approximations and the exponential Euler method. To obtain these optimal discrete regularity results, we establish a bounded $H^\infty$-calculus for the discrete spatial operator $A_h$, uniformly in the mesh size $h$. As a direct byproduct, we also establish the discrete-in-space stochastic maximal regularity for the corresponding spatial semi-discretizations. As an application, we establish the pathwise uniform convergence of a fully discrete approximation of the three-dimensional stochastic Allen--Cahn equation with variable diffusion coefficients and multiplicative noise.

math.AP↗
arXiv2026

A Systematic Analysis of Automatic Differentiation versus Discretization-based Constraints for Physics-Informed PDE Solvers

Physics-informed neural networks (PINNs) represent a growing frontier in using artificial intelligence to solve partial differential equations (PDEs). Automatic differentiation (AD) plays a central role in this paradigm, which is mesh-free and replaces traditional iterative solvers with gradient-based optimization in continuous space. However, the inherent limitations of AD, particularly in handling higher-order derivatives and discontinuous solutions, pose significant challenges for complex problems. This has motivated a growing number of researchers to explore discretization-based constraints as an alternative path. Yet, the respective applicability of these two paradigms remains largely unexplored. In this work, we conduct systematic experiments across a wide spectrum of problems, from simple linear Poisson to high-Mach hypersonic flows with strong discontinuities. Through a rigorous decomposition of approximation, optimization, and truncation errors, we systematically elucidate the fundamental trade-offs and error-governing mechanisms of both paradigms, as well as two representative network architectures: multi-layer perceptron (MLP) and graph neural network (GNN). Our results reveal a consistent trend: as nonlinearity strengthens, the accuracy advantage of discretization-based constraints becomes increasingly pronounced, with smaller optimization errors compensating for the truncation errors. Moreover, the more complex the nonlinearity and boundary conditions, the greater the advantage of GNN over MLP. These insights offer a robust practical guideline for configuring neural PDE solvers in demanding engineering applications. Our source data and code are available at https://github.com/guoxing0809/neuropde_analysis.

math.NA↗
arXiv2026

A Comparative Study of Finite-Volume-based Coupled and Segregated Reduced-Order Models for Incompressible Flows in Parametrized Domains

This work presents a comparative analysis of Reduced-Order Models (ROMs) applied to incompressible fluid dynamics within geometrically parametrized domains. Two distinct reduced-order solution strategies are investigated and compared: a monolithic coupled solver and a segregated SIMPLE-based algorithm. Their performance is assessed on two steady, two-dimensional benchmark cases: a lid-driven cavity flow and a flow past a cylindrical obstacle. The two algorithms are compared in terms of fields evaluation and aerodynamic coefficients prediction. The computational results highlight a fundamental trade-off between accuracy and numerical efficiency. On the one hand, after an opportune supremizers enrichment, the coupled approach guarantees a faster convergence, despite the need of a larger number of degrees of freedom. On the other hand, the segregated SIMPLE algorithm yields superior reconstruction accuracy, particularly for lower-dimensional reduced spaces, at the cost of a slower convergence rate.

physics.flu-dyn↗
arXiv2026

FPScan: An Automated Constraint-Based Analyzer for Floating-Point Anomaly Detection

Writing error-free floating-point programs is a challenging task, especially for programmers who lack a strong background in numerical analysis and rounding-error propagation. State-of-the-art techniques typically aim to bound such errors using static or dynamic analysis. However, only a few tools explicitly address critical floating-point pitfalls such as absorption and catastrophic cancellation. These anomalies represent situations in which rounding errors are significantly amplified, causing the semantics of the finite-precision computation to deviate substantially from the real-number semantics. In this article, we present FPScan, a novel tool to formally define and detect both catastrophic cancellation and absorption in floating-point programs. Our approach starts with a custom static analyzer based on abstract interpretation to infer the order of magnitude of all program variables. This magnitude information is then used to build a set of first-order constraints that model error propagation and numerical precision within the program. Finally, we employ an off-the-shelf SMT solver to determine whether the program exhibits any of these critical numerical pitfalls. Experiments were conducted on FPBench, a well-known benchmark suite of floating-point programs, to evaluate the effectiveness of our tool. We also present a comparison with state-of-the-art tools regarding soundness and analysis time.

cs.SE↗
arXiv2026

Analysis of a Surface Crouzeix-Raviart Element for the Stokes Problem

Recently, increasing attention has been paid to finite element discretizations of vector-valued flow problems posed on curved surfaces. In this work, we study a surface Stokes system defined on a two-dimensional manifold embedded in three-dimensional space. The surface velocity field is approximated using a nonconforming Crouzeix-Raviart finite element on a polyhedral approximation of the surface, while the pressure is discretized by piecewise constant functions. The governing equations involve the symmetric strain-rate tensor, whose approximation with Crouzeix-Raviart finite elements leads to spurious oscillations in the velocity field because the discrete Korn inequality fails on the nonconforming space. We therefore stabilize the momentum equation by adding an edge-jump penalty term, which restores the coervity of the bilinear term as the discrete Korn inequality holds for this stabilized version. Additionally, we establish that this finite element pair of velocity and pressure spaces satisfies the discrete inf-sup compatibility condition. Furthermore, we derive optimal a-priori error estimates in both the energy norm and the $L^2$-norm. These theoretical results are corroborated by numerical experiments.

math.NA↗
arXiv2026

A numerical method for two-dimensional Boussinesq-Full Dispersion systems modelling internal wave propagation

The paper is concerned with the numerical approximation of some two-dimensional asymptotic models for the propagation of internal waves in a two-layer system in the form of pde's for the interfacial deviation and the velocity. After the study of well-posedness of the periodic initial-value problem, the convergence of a semidiscrete approximation based on a spectral Fourier-Galerkin method is analyzed. The resulting semidiscrete system is then integrated numerically in time with the implicit midpoint rule. Some properties of the fully discrete scheme are described and its performance is illustrated with some numerical experiments.

math.NA↗
arXiv2026

Buildability Assessment of 3D-Printed Concrete Structures Using Smoothed Mohr-Coulomb Plasticity with Isotropic Hardening

Extrusion-based 3D concrete printing builds structural elements layer upon layer without formwork, so freshly deposited material must carry the growing weight of the structure while it is still gaining strength and stiffness. The largest number of layers that can be printed before failure, the buildability, is therefore central to process design. Its prediction rests above all on how accurately the constitutive model represents fresh concrete. Existing finite element frameworks pair gradual pre-failure hardening only with smooth yield surfaces, which leaves the Mohr-Coulomb surface that best describes fresh concrete restricted to perfect plasticity. This work combines a smoothed Mohr-Coulomb yield surface with nonlinear isotropic hardening of the cohesion, formulated in an updated Lagrangian setting with an activation field for layer deposition. Failure is identified from a tangent-stiffness eigenvalue criterion that matches the analytical self-weight buckling load of a slender wall and is insensitive to geometric imperfection. The framework reproduces the measured collapse of a printed straight wall within one layer and predicts the collapse of a hollow cylinder closer to experiment than existing frameworks. A study over six diameters and six hardening rates shows buildability peaking at intermediate diameters and rising with the hardening rate. Recording how much of its strength the material has used at failure, slender cylinders fail with much still in reserve while intermediate ones use nearly all of it. The two failure modes are therefore identified as the limits of one process rather than separate failure mechanisms.

cs.CE↗
arXiv2026

Mapping for Approximation: A Unified View of Rescaled, Variably Scaled and Rational Kernel Methods

Classical approximation methods are usually improved by changing the sampling set, increasing the number of data points, or selecting a different basis. We advocate a complementary viewpoint: keep the sampled values fixed and modify the representation through a suitable mapping. This viewpoint unifies several constructions that were originally introduced for different purposes. Rescaled radial basis function interpolation maps the interpolation operator through a normalization that enforces exact reproduction of constants. Variably scaled kernels map the geometry by lifting the data to a higherdimensional manifold determined by a scale function. Mapped bases and fake nodes map the approximation space without resampling the data, while rational kernel expansions can be interpreted as nonlinear mappings of the trial space. We develop a common notation for these mechanisms, summarize their approximation and stability properties, and explain how discontinuous and rational variants fit the same framework. Reproducible numerical experiments illustrate constant reproduction, error reduction through data-mimicking scale functions, and the suppression of the Runge phenomenon by mapped polynomial bases. The results support a general principle: mappings provide a flexible way to adapt approximation spaces to data geometry and regularity without modifying the original observations

math.NA↗
arXiv2026

Efficient computation of the asymptotics of extensive-rank HCIZ integrals

We study the high-dimensional asymptotics of Harish-Chandra-Itzykson-Zuber (HCIZ) integrals in the extensive-rank regime. The limit of these integrals is governed by a one-dimensional boundary-value hydrodynamical problem originally derived by Matytsin (1994) and rigorously proved by Guionnet and Zeitouni (2002). Despite its wide-ranging applications, explicit solutions to this problem are known only in a few specific cases. In this work, we introduce an efficient numerical scheme based on a particle discretization and prove its convergence to the continuous boundary-value problem for generic boundary densities. We validate our approach against known analytical solutions and apply it to generic densities, uncovering interesting dynamical phenomena. The high-dimensional limit of HCIZ integrals appears in various contexts, from the large deviations of random matrix spectra to the limiting free energy of disordered systems, high-dimensional statistics, and machine learning. As such, our contribution opens the way towards the numerical exploration of a wide range of high-dimensional models that were previously intractable.

math.PR↗
arXiv2026

LoCCA: Localized Chebyshev Cross Approximation for Kernel Matrix Factorization via Nodal Perturbation Stability

We propose Local Chebyshev Cross-Approximation (LoCCA), a data-driven framework that bridges smooth polynomial interpolation with flexible matrix factorizations. LoCCA dynamically maps ideal grid nodes to their nearest physical neighbors within unstructured point sets. We support this framework with a new perturbation theory, proving that displacing Chebyshev nodes onto physical data points retains mathematical stability and accuracy without explosive error growth. Numerically, LoCCA provides strict, reliable error control even on irregular geometries where standard grid-based methods fail, achieving substantial speedups and near-optimal matrix compression.

math.NA↗
arXiv2026

Poisson Image Denoising Using Minimax Concave and Reweighted $\ell_1$ Penalties: Nonblind and Blind Approaches

Images are important tools in various sciences. Despite the development of photo-taking tools, creating clear and image without noise remains challenging in practice. In particular, Poisson noise has an effect on medical and astronomical images, and reduces their quality. Additionally, blur is another factor that has an effect on image quality. The problem of image restoration becomes very complicated when we have no information about the Point Spread Function (PSF). These types of problems are known as blind case. However, in some images, such as some astronomical images, the type of PSF can be specified, and these types of problems are known as nonblind problems. Total Variation (TV) is a widely used method for solving such inverse problems, where the selection of the penalty function is the most critical factor that affects the method's performance. In this paper, to improve edge preservation, we employ a reweighted $\ell_1$-regularization of the fractional order derivative. Furthermore, we propose a nonblind and blind image deblurring approach under Poisson noise using the Minimax Concave Penalty (MCP), which is a continuous, sparsity promoting, and nearly unbiased regularizer. This formulation leads to a nonconvex optimization model. To solve the proposed model, we introduce an efficient numerical algorithm based on the Alternating Direction Method of Multipliers (ADMM) and provide an analysis of its convergence. Finally, the effectiveness of the proposed algorithm are demonstrated through extensive experiments on various images.

cs.CV↗
arXiv2026

Threshold dynamics for subdiffusive grain growth

We propose a new fractional threshold-dynamics algorithm for subdiffusive interface motion and apply it to multiphase grain-growth simulations. The method combines the L1 discretization of the Caputo time derivative with a Helmholtz-resolvent thresholding step. At each time level, the L1 convolution produces a convex history average of the previously thresholded states; this history-dependent field is then propagated by a Helmholtz diffusion solve and projected back to pure phases by thresholding. For $α=1$, the method reduces to a Helmholtz-resolvent analogue of the Merriman-Bence-Osher scheme, whereas for $0<α<1$ it introduces a subdiffusive temporal memory. We establish a variational characterization, maximum and comparison principles, and stability for the continuous Helmholtz diffusion step. We also derive a memory-tail estimate and a finite-dimensional pinning criterion under an explicit resolvent-contraction hypothesis. The equal-tension multiphase scheme is implemented on periodic grids. The numerical experiments investigate the joint influence of the L1 history and the smoothing length. The fixed-step runs show faster coarsening for smaller fractional orders in the tested regime, while a comparison at fixed smoothing length examines the influence of the L1 history on successive updates.

math.NA↗
Compare source metadata on this page
WorkPublishedSource identifierSource
A \(3\times 3\) counterexample to Lin and Wimmer's rank-minimization conjecture associated with Roth's similarity theorem2026-09-072609.07113arxiv
Structure-Preserving Data-Driven Identification of Port-Hamiltonian Differential-Algebraic Systems2026-09-072609.07187arxiv
Analysis of the Ill-Conditioning of the Discrete Inverse Laplace transform in Monte Carlo Simulations of Quantum Many-Body Systems2026-09-072609.07208arxiv
Constitutive State-Space Modeling of Path-Dependent Plasticity: A Resolution-Consistent and Parallelizable Computational Framework2026-09-072609.07294arxiv
A new formulation of Discontinuous Galerkin method for interface capturing approaches2026-09-072609.07327arxiv
Frequency Domain Biot--Allard Equations for Isotropic and Anisotropic Poroelastic Media: Two-field formulations and iterative splitting2026-09-072609.07388arxiv
Fully discrete stochastic maximal regularity and $H^\infty$-calculus for second-order elliptic operators2026-10-122609.07436arxiv
A Systematic Analysis of Automatic Differentiation versus Discretization-based Constraints for Physics-Informed PDE Solvers2026-09-072609.07437arxiv
A Comparative Study of Finite-Volume-based Coupled and Segregated Reduced-Order Models for Incompressible Flows in Parametrized Domains2026-09-072609.07472arxiv
FPScan: An Automated Constraint-Based Analyzer for Floating-Point Anomaly Detection2026-09-072609.07492arxiv
Analysis of a Surface Crouzeix-Raviart Element for the Stokes Problem2026-09-072609.07527arxiv
A numerical method for two-dimensional Boussinesq-Full Dispersion systems modelling internal wave propagation2026-09-072609.07659arxiv
Buildability Assessment of 3D-Printed Concrete Structures Using Smoothed Mohr-Coulomb Plasticity with Isotropic Hardening2026-09-072609.07679arxiv
Mapping for Approximation: A Unified View of Rescaled, Variably Scaled and Rational Kernel Methods2026-09-072609.07743arxiv
Efficient computation of the asymptotics of extensive-rank HCIZ integrals2026-09-072609.07771arxiv
LoCCA: Localized Chebyshev Cross Approximation for Kernel Matrix Factorization via Nodal Perturbation Stability2026-09-072609.07882arxiv
Poisson Image Denoising Using Minimax Concave and Reweighted $\ell_1$ Penalties: Nonblind and Blind Approaches2026-09-072609.07916arxiv
Threshold dynamics for subdiffusive grain growth2026-09-072609.07938arxiv

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