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cs.NA

cs.NA: explore 425 source-linked works published from 2021 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 2026-09-16. Counts describe this index, not the complete source archives.

arXiv2026

Density functions for the overdamped generalized Langevin equation and its Euler--Maruyama method: smoothness and convergence

This paper focuses on studying the convergence rate of the density function of the Euler--Maruyama (EM) method, when applied to the overdamped generalized Langevin equation with fractional noise which serves as an important model in many fields. Firstly, we give an improved upper bound estimate for the total variation distance between random variables by their Malliavin--Sobolev norms. Secondly, we establish the existence and smoothness of the density function for both the exact solution and the numerical one. Based on the above results, the convergence rate of the density function of the numerical solution is obtained, which relies on the regularity of the noise and kernel. This convergence result provides a powerful support for numerically capturing the statistical information of the exact solution through the EM method.

math.NA↗

arXiv2026

Accelerated Primal-Dual Proximal Gradient Splitting Methods for Convex-Concave Saddle-Point Problems

In this paper, based a novel primal-dual dynamical model with adaptive scaling parameters and Bregman divergences, we propose new accelerated primal-dual proximal gradient splitting methods for solving bilinear saddle-point problems with optimal nonergodic convergence rates. For the first, using the spectral analysis, we show that a naive extension of acceleration to a quadratic game is unstable. Motivated by this, we present an accelerated primal-dual gradient flow which combines acceleration with careful velocity correction. To work with non-Euclidean distances, we also equip our continuous model with general Bregman divergences and prove the exponential decay of a Lyapunov function. Then, new primal-dual splitting methods are developed based on proper semi-implicit Euler schemes of the continuous model, and the theoretical convergence rates are nonergodic and optimal with respect to the matrix norms, Lipschitz constants and convexity parameters. Moreover, we introduce efficient restart variants to further improve the proposed methods and provide some numerical results to validate the practical performance.

math.OC↗

arXiv2026

Recursive reduction quadrature for the evaluation of Laplace layer potentials in three dimensions

A high-order quadrature scheme is constructed for the evaluation of Laplace single and double layer potentials and their normal derivatives on smooth surfaces in three dimensions. The construction begins with a harmonic approximation of the density on each patch, which allows for a natural harmonic polynomial extension in a volumetric neighborhood of the patch in the ambient space. Then by the general Stokes theorem, singular and nearly singular surface integrals are reduced to line integrals preserving the singularity of the kernel, instead of the standard origin-centered 1-forms that require expensive adaptive integration. These singularity-preserving line integrals can be semi-analytically evaluated using singularity-swap quadrature. In other words, the evaluation of singular and nearly singular surface integrals is reduced to function evaluations at the vertices on the boundary of each patch. The recursive reduction quadrature largely removes adaptive integration that is needed in most existing high-order quadratures for singular and nearly singular surface integrals, resulting in exceptional performance. The scheme achieves twelve-digit accuracy uniformly for close evaluations and offers a speedup of five times or more in constructing the sparse quadrature-correction matrix compared to previous state-of-the-art quadrature schemes.

math.NA↗

arXiv2026

Geometric Optics Approximation Sampling: A Far-Field Reflector-Induced Transport Framework

We develop a far-field geometric optics approximation sampling (GOAS) framework for constructing direct samplers from target measures. The method exploits the connection between the far-field reflector problem and optimal transport with logarithmic cost, leading to a natural primal--dual transport structure. The associated dual reflector provides a reciprocal backward transport and, in the invertible smooth setting, the inverse of the forward reflector map. For numerical realization, we adopt a supporting hyperparaboloid construction based on a discrete approximation of the target measure. This construction is gradient-free with respect to the target density and naturally accommodates both density-based and sample-based target representations. The resulting piecewise reflector admits two sampling realizations: a primal--dual consistency resampling strategy that operates directly on the nonsmooth reflector, and a softmin-regularized realization yielding an explicit smooth transport map through the physical law of reflection. We establish the well-posedness and stability of the reflector-induced sampling measure and derive Wasserstein error estimates. Numerical experiments on non-Gaussian targets and Bayesian inverse problems demonstrate the accuracy, stability, Wasserstein error behavior, and applicability of the proposed framework, and compare it with MCMC and polynomial transport-map methods.

math.NA↗

arXiv2026

Data-efficient Kernel Methods for Learning Hamiltonian Systems

Hamiltonian dynamics describe a wide range of physical systems. As such, data-driven simulations of Hamiltonian systems are important for many scientific and engineering problems. In this work, we propose kernel-based methods for identifying and forecasting Hamiltonian systems directly from trajectory data. We present two approaches: a 2-step method that reconstructs trajectories before learning the Hamiltonian, and a 1-step method that jointly infers both. Across several benchmark systems, including mass-spring dynamics, a nonlinear pendulum, and the Henon-Heiles system, we demonstrate that our framework achieves accurate, data-efficient predictions and outperforms 2-step kernel-based baselines, particularly in scarce-data regimes, while preserving the Hamiltonian structure. Moreover, we prove a priori error estimates, ensuring reliability of the learned models. We also provide a more general, problem-agnostic numerical framework that goes beyond Hamiltonian systems and can be used for data-driven learning of arbitrary dynamical systems.

math.NA↗

arXiv2026

Greedy Thiele continued-fraction approximation on continuum domains in the complex plane

We describe an adaptive greedy algorithm for Thiele continued-fraction (TCF) approximation of a function defined on a continuum domain in the complex plane. The algorithm iteratively selects interpolation nodes from an adaptively refined set of sample points on the domain boundary. We also present new algorithms for evaluating Thiele continued fractions and their accessory weights using only a single floating-point division. Numerical experiments comparing the greedy TCF method with the AAA algorithm on several challenging functions defined on the interval $[-1,1]$ and on the unit circle show that continuum TCF is consistently faster than AAA, by factors ranging from 8 to 40.

math.NA↗

arXiv2026

Computing matrix functions associated with a Hermitian-definite pencil

We consider the numerical evaluation of the quantity $Af(A^{-1}B)$, where $A$ is Hermitian positive definite, $B$ is Hermitian, and $f$ is a function defined on the spectrum of $A^{-1}B$. This problem is related to the Hermitian-definite matrix pencil $B-λA$. We study the conditioning of the problem, and we introduce several algorithms that combine the Schur decomposition with either the matrix square root or the Cholesky factorization. We study the numerical behavior of these algorithms in floating-point arithmetic, assess their computational costs, and compare their numerical performance. Our analysis suggests that the algorithms based on the Cholesky factorization will be more accurate and efficient than those based on the matrix square root. This is confirmed by our numerical experiments.

math.NA↗

arXiv2026

Structure-Preserving Transfer of Grad-Shafranov Equilibria to Magnetohydrodynamic Solvers

Magnetohydrodynamic (MHD) solvers used to study dynamic plasmas for magnetic confinement fusion typically rely on initial conditions that describe force balance, which are provided by an equilibrium solver based on the Grad-Shafranov (GS) equation. Transferring such equilibria from the GS discretization to the MHD discretization often introduces errors that lead to unwanted perturbations to the equilibria on the level of the MHD discretization. In this work, we identify and analyze sources of such errors in the context of finite element methods, with a focus on the force balance and divergence-free properties of the loaded equilibria. In particular, we study three main sources of errors: (1) the improper choice of finite element spaces in the MHD scheme relative to the poloidal flux and toroidal field function spaces in the GS scheme, (2) the misalignment of the meshes from two solvers, and (3) possibly under-resolved strong gradients near the separatrix. With this in mind, we study the impact of different choices of finite element spaces, including those based on compatible finite elements. In addition, we also investigate the impact of mesh misalignment and propose to conduct mesh refinement to resolve the strong gradients near the separatrix. Numerical experiments are conducted to demonstrate equilibria errors arising in the transferred initial conditions. Results show that force balance is best preserved when structure-preserving finite element spaces are used and when the MHD and GS meshes are both aligned and refined. Given that the poloidal flux is often computed in continuous Galerkin spaces, we further demonstrate that projecting the magnetic field into divergence-conforming spaces is optimal for preserving force balance, while projection into curl-conforming spaces, although less optimal for force balance, weakly preserves the divergence-free property.

math.NA↗

arXiv2026

Pressure-Robust Enriched Galerkin Methods for Navier-Stokes Equations

This paper presents an enriched Galerkin (EG) finite element method for the incompressible Navier--Stokes equations. The method augments continuous piecewise linear velocity spaces with elementwise bubble functions, yielding a locally conservative velocity approximation while retaining the efficiency of low-order continuous elements. The viscous term is discretized using a symmetric interior penalty formulation, and the divergence constraint is imposed through a stable pressure space. To enhance the robustness of the velocity approximation with respect to the pressure, a reconstruction operator is introduced in the convective and coupling terms, resulting in a pressure-robust scheme whose accuracy does not deteriorate for small viscosities. Both Picard and Newton linearizations are formulated in a fully discrete manner, and the corresponding linear systems are assembled efficiently at each iteration. Optimal a~priori error estimates are established for the velocity in the mesh-dependent energy norm and for the pressure in the $L^2$ norm. Two representative numerical experiments are presented: a smooth manufactured solution and the lid-driven cavity flow. The numerical results confirm the theoretical convergence rates, demonstrating first-order convergence of the velocity in the energy norm, second-order convergence in the $L^2$ norm, and first-order convergence of the pressure. The proposed EG scheme accurately captures characteristic flow structures, illustrating its effectiveness and robustness for incompressible flow simulation.

math.NA↗

arXiv2026

Tree-cotree gauging for two-dimensional hierarchical splines

In magnetostatics and eddy current problems, formulated in terms of the magnetic vector potential, the solution is not unique, because the addition of an irrotational function to the solution remains a valid solution. The tree-cotree decomposition is a gauging technique to recover uniqueness when using finite elements, which consists in considering the mesh as a graph, and building a spanning tree on that graph. The idea has been recently extended to isogeometric analysis, applying the construction of the spanning tree on the control mesh, or equivalently, on the Greville grid. In the present paper we extend the construction to hierarchical splines, a set of splines with multi-level structure for adaptive refinement, by constructing a spanning tree for each single level. Since for degree $p=1$ the spaces of finite elements and hierarchical splines coincide, the presented construction is also valid for quadrilateral finite element meshes with hanging nodes. To assess the correctness of the method, we present numerical results for Maxwell eigenvalue problem.

math.NA↗

arXiv2026

Active learning for data-driven reduced models of parametric differential systems with Bayesian operator inference

This work develops an active learning framework to intelligently enrich data-driven reduced-order models (ROMs) of parametric dynamical systems, which can serve as the foundation of virtual assets in a digital twin. Data-driven ROMs are explainable, computationally efficient scientific machine learning models that aim to preserve the underlying physics of complex dynamical simulations. Since the quality of data-driven ROMs is sensitive to the quality of the limited training data, we seek to identify training parameters for which using the associated training data results in the best possible parametric ROM. Our approach uses the operator inference methodology, a regression-based strategy which can be tailored to particular parametric structure for a large class of problems. We establish a probabilistic version of parametric operator inference, casting the learning problem as a Bayesian linear regression. Prediction uncertainties stemming from the resulting probabilistic ROM solutions are used to design a sequential adaptive sampling scheme to select new training parameter vectors that promote ROM stability and accuracy globally in the parameter domain. We conduct numerical experiments for several nonlinear parametric systems of partial differential equations and compare the results to ROMs trained on random parameter samples. The results demonstrate that the proposed adaptive sampling strategy consistently yields more stable and accurate ROMs than random sampling does under the same computational budget.

stat.ML↗

arXiv2026

An Iterative Method with Asymptotic Orthogonality for Simultaneous Eigenpair Computation

The simultaneous computation of a cluster of eigenpairs with mutually orthogonal eigenvectors is a basic task in scientific computing. We develop a predictor--corrector discretization of the quasi-Grassmannian gradient flow for simultaneous eigenpair computation. The proposed iteration requires neither orthogonal initial data nor any orthogonalization operation: the predictor preserves the current Gram matrix, whereas the corrector reduces the orthogonality error, so that the iterates approach orthogonality asymptotically. We establish well-posedness of the discrete scheme and prove invariant-subspace nonexpansion, asymptotic orthogonality, energy decrease, and convergence to the target eigenspace. Numerical experiments with the discrete equations solved approximately show that the numerical iterates converge to eigenpair approximations whose eigenvectors are numerically orthogonal to high accuracy, while the energy and gradient norm decrease over the iterations.

math.NA↗

arXiv2026

Regularity of Second-Order Elliptic PDEs in Spectral Barron Spaces

We establish a regularity theorem for second-order elliptic PDEs on $\mathbb{R}^{d}$ in spectral Barron spaces. Under mild ellipticity and smallness assumptions, the solution gains two additional orders of Barron regularity. As a corollary, we identify a class of PDEs whose solutions can be approximated by two-layer neural networks with cosine activation functions, where the width of the neural network is independent of the spatial dimension.

math.AP↗

arXiv2026

Exposing Finite-Depth, Finite-Shot Guarantees for Constrained Quantum Optimization via Fejér Filtering

Constrained quantum optimization algorithms need quantitative guarantees that connect circuit resources to the probability of actually sampling feasible or optimal solutions in finitely many shots. We establish such a connection by exposing a positive sampling law in which mixer-driven exploration and spectral selection can be controlled separately. We show that after removing interference between distinct cost eigenspaces as an analytic device, the measurement distribution becomes the normalized product of a mixer-induced exploration envelope and a Fejér spectral weight, with the former describing how the mixer spreads probability over the encoded manifold and the latter enhancing the target cost phase while suppressing spectrally separated nontarget phases. In this model, finite-shot success becomes a tractable competition between target weight and off-target leakage, yielding an explicit lower bound on the probability of sampling an optimum. For the primary bound, we rescale the cost Hamiltonian to an integer-valued spectrum, placing the wrapped cost phases on a controlled lattice for Fejér filtering. We then define $δ$ as the minimum circular separation between the optimal phase and every nontarget phase. The single-shot success probability $q_0$ satisfies \[ q_0 \ge \frac{x}{1+x}, \qquad x = (p+1)^2 \sin^2\!\left(\fracδ{2}\right) C_β, \] where $p$ is the filter order and $C_β$ is the mixer-envelope mass on the optimal set, exposing a finite-resource compensation law in which weaker phase separation or smaller envelope mass can be compensated by increased filter order and additional shots. The same filtering principle exposes a feasibility guarantee when applied to penalty phases. We further prove analogous bounds for nonlattice spectra through off-target suppression, extending our results beyond exact lattice normalization.

quant-ph↗

arXiv2026

h-Adaptive FV Subcell Shock-Capturing for DGSEM on Heterogeneous Curvilinear Meshes

High-order methods offer superior dispersion and dissipation properties compared to low-order schemes but require robust stabilization for discontinuities. To ensure stability, local artificial viscosity is common, but often degrades sub-element resolution. Conversely, subcell resolution preserving limiting strategies such as the finite volume subcell method are typically restricted to uniform topologies, such as purely hexahedral or simplex meshes, or linear elements. This leaves a significant gap in treating the hybrid-element topologies necessary for complex engineering geometries. To bridge this gap, we introduce a robust shock-capturing approach for the discontinuous Galerkin spectral element method on mixed curvilinear meshes containing hexahedral, prismatic, tetrahedral, and pyramid elements. Non-hexahedral elements are handled via collapsed coordinate transformations. The proposed method utilizes an $h$-adaptive finite volume subcell scheme with an arbitrary subcell resolution up to $2\mathcal{N}+1$. Special care is taken to ensure conforming subcell distributions across different element types. Crucially, discrete conservation is proven theoretically and verified numerically, alongside validations for high-order convergence and robust shock capturing. Finally, the method's applicability to complex configurations is demonstrated through a simulation of the flow around a NACA 0012 airfoil.

math.NA↗

arXiv2026

Dynamical Low-Rank Filters for Data Assimilation

We propose dynamical low-rank (DLRA) filters for data-assimilation problems based on stochastic differential equations (SDEs). We first consider a Kalman filter applied to a Euler-Maruyama discretization of the SDE and observation processes, and derive a DLRA Kalman-type filter, named DLRA-JMCO, by minimizing jointly the error on the mean and covariance. We then obtain a time continuous DLRA-JMCO filter as the continuous limit of the time discrete one, and introduce a corresponding particle approximation. We also propose a strategy to enrich the DLRA-JMCO filter with a simple Gaussian process to capture the neglected dynamics in the orthogonal to the DLRA subspace. This correcting term allows the DLRA subspace evolution to be directly influenced by the observation operator. We finally propose a particle filter version of the DLRA-JMCO filter. Numerical experiments assess the performance of the proposed filters.

math.NA↗

arXiv2026

Dynamical Low-Rank Smoothing

Computational costs often make smoothing procedures prohibitive for high-dimensional data assimilation problems. To address this challenge, we propose a dynamical low-rank approximation (DLRA) methodology for smoothing concerning frameworks based on stochastic differential equations. We extend the previously developed joint mean-and-covariance optimization (JMCO) filtering setting to derive a reduced-order smoother via the Rauch--Tung--Striebel recursion and establish the corresponding Kalman--Bucy smoothing for affine drift dynamics. The resulting algorithms retain the adaptive nature of DLRA while significantly reducing the computational time and storage of the whole smoothing procedure.

math.NA↗

arXiv2026

Analysis of Moment Closures Using $φ$-Divergences for Rarefied Dynamics with Binary Collisions and Their Galerkin Discretizations

This work introduces a robust deterministic framework for approximating solutions of the Boltzmann equation with binary collisions by discretizing their dependence on time, position, and velocity using Galerkin methods. By employing a family of parametric Galerkin closures based on $φ$-divergences in velocity space, we derive rigorous hierarchies of moment equations that govern fluid dynamic variables. Addressing the limitation that these closures alone do not guarantee dissipation of a $φ$-divergence entropy for the true binary collision operator, we restore this property by formulating a compatible approximate collision operator tailored to each closure. This constructed operator intrinsically retains fundamental physical properties essential for high-fidelity flow simulations, including Galilean invariance, exact conservation of mass, momentum, and energy, and strict dissipation of a $φ$-divergence entropy. Furthermore, we show that the resulting closed moment systems are symmetric-dissipative, yielding Cauchy problems that are well-posed locally in time. To translate this mathematical foundation into an efficient computational tool, we discretize the position and time variables with an entropy-stable discontinuous Galerkin (DG) finite element method. The fully implicit, entropy-stable space-time approach enables time steps far beyond typical CFL-limited step sizes and the direct computation of steady states. The robustness and accuracy of the methodology are verified and validated through numerical simulations on the supersonic nozzle flow of argon, mass flow through a channel, and heat transfer between parallel walls, demonstrating agreement with analytical benchmarks, experimental measurements, and stochastic particle simulations.

math.NA↗

Compare source metadata on this page

Work	Published	Source identifier	Source
Density functions for the overdamped generalized Langevin equation and its Euler--Maruyama method: smoothness and convergence	2026-09-03	2405.11744	arxiv
Accelerated Primal-Dual Proximal Gradient Splitting Methods for Convex-Concave Saddle-Point Problems	2026-09-28	2407.20195	arxiv
Recursive reduction quadrature for the evaluation of Laplace layer potentials in three dimensions	2026-09-03	2411.08342	arxiv
Geometric Optics Approximation Sampling: A Far-Field Reflector-Induced Transport Framework	2026-09-03	2411.09964	arxiv
Data-efficient Kernel Methods for Learning Hamiltonian Systems	2026-09-03	2509.17154	arxiv
Greedy Thiele continued-fraction approximation on continuum domains in the complex plane	2026-09-03	2510.07295	arxiv
Computing matrix functions associated with a Hermitian-definite pencil	2026-10-05	2510.16473	arxiv
Structure-Preserving Transfer of Grad-Shafranov Equilibria to Magnetohydrodynamic Solvers	2026-09-03	2511.07763	arxiv
Pressure-Robust Enriched Galerkin Methods for Navier-Stokes Equations	2026-09-03	2511.20240	arxiv
Tree-cotree gauging for two-dimensional hierarchical splines	2026-09-03	2512.01580	arxiv
Active learning for data-driven reduced models of parametric differential systems with Bayesian operator inference	2026-09-03	2601.00038	arxiv
An Iterative Method with Asymptotic Orthogonality for Simultaneous Eigenpair Computation	2026-09-03	2601.02108	arxiv
Regularity of Second-Order Elliptic PDEs in Spectral Barron Spaces	2026-09-03	2602.19381	arxiv
Exposing Finite-Depth, Finite-Shot Guarantees for Constrained Quantum Optimization via Fejér Filtering	2026-09-03	2603.01809	arxiv
h-Adaptive FV Subcell Shock-Capturing for DGSEM on Heterogeneous Curvilinear Meshes	2026-09-03	2604.27735	arxiv
Dynamical Low-Rank Filters for Data Assimilation	2026-10-13	2607.27432	arxiv
Dynamical Low-Rank Smoothing	2026-09-03	2607.27438	arxiv
Analysis of Moment Closures Using $φ$-Divergences for Rarefied Dynamics with Binary Collisions and Their Galerkin Discretizations	2026-09-03	2608.03640	arxiv

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