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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

Greedy sampling designs via reduced basis methods: optimal recovery in the uniform norm

We study optimal sampling recovery in reproducing kernel Hilbert spaces (RKHS) in the uniform norm. For every RKHS with bounded kernel, we establish new comparisons between linear sampling widths and Gelfand widths that overcome the known square-root gap, without requiring a measure or a Christoffel-type condition. Our bounds rely on nested sampling designs obtained by kernel interpolation at (weak) P-greedy points. Under additional (polynomial) decay assumptions the decay rate of the Gelfand widths directly transfers to the sampling widths. With either a logarithmic oversampling or passing to the square root of the Gelfand widths we obtain a direct comparison (requiring no decay assumption) between them. This is particularly effective for super-polynomial decay, such as in Paley-Wiener spaces. Our results follow from representations of both widths in terms of kernel translates and yield, in the opposite direction, a new existence result for a sharp reduced basis selection. Numerical experiments for Legendre, mixed-Sobolev, and Paley-Wiener kernels illustrate our findings.

math.NA↗

arXiv2026

Families of relative periodic orbits in the planar three-body problem via consecutive alignments

Relative periodic orbits (RPOs) are solutions of the three-body problem that are periodic in a uniformly rotating reference frame and, in general, quasi-periodic in inertial coordinates. We present a numerical procedure for computing and continuing one-parameter families of RPOs of the planar Newtonian three-body problem. The method exploits consecutive syzygies, understood here as configurations in which the three bodies are aligned and their velocities satisfy the corresponding symmetry conditions. Matching the positions and momenta at two consecutive alignments reduces the computation of RPOs to a low-dimensional nonlinear problem. Its solutions are then numerically continued, and linear stability is determined from the nontrivial eigenvalues of the rotated monodromy matrix after removing the neutral directions associated with conserved quantities and continuous symmetries. The procedure is applied to several mass distributions and initial configurations, producing families of Poincaré, Hill, and binary-type solutions. These families exhibit transitions from nearly circular to highly eccentric motion, changes of stability near resonances and turning points, and absolute periodic solutions when the rotation angle is a rational multiple of 2π. In the Hill families, the continuation connects satellite configurations with circumstellar motion as the smallest body loses its gravitational binding to the intermediate body. Circumbinary and circumstellar configurations are also obtained in the binary regime. The results illustrate the dynamical diversity of RPOs and provide coherent three-body motions that can be used as prescribed trajectories in restricted four-body models.

math.DS↗

arXiv2026

A Double-Adaptivity Solver for Parabolic PDEs

We study minimal residual space-time finite element discretizations of linear parabolic initial value problems in canonical space-time variational form. To deal with the arising dual norm, we introduce the Riesz lift of the residual as an additional variable. Quasi-optimality of the primal variable of the mixed system follows from a uniform inf-sup condition. This condition is known to be satisfied for finite element spaces w.r.t. prismatic partitions of the space-time cylinder that allow for a decomposition into time-slabs. We prove that this condition cannot be expected to hold otherwise. To recover stability for general partitions and the data at hand, we derive an a posteriori condition on the error between the exact Riesz lift of the residual and its Galerkin approximation -- being the secondary variable of our system -- under which the primal variable is quasi-optimal. We derive a posteriori error estimators for both variables, and use them in a double-adaptive loop that alternates test-space with trial-space enrichment. We illustrate our findings with numerical experiments in $1+1$ and $2+1$ dimensions.

math.NA↗

arXiv2026

Learning a general class of admissible multi-species collision operators from molecular dynamics

We develop a structure-preserving, data-driven collision operator for spatially homogeneous multi-species kinetic systems from molecular dynamics (MD). The operator consists of diagonal self-collision blocks and ordered off-diagonal cross-species blocks to describe intra- and inter-species momentum and energy exchange. Within a local and point-wise identifiable kernel class, we develop the necessary and sufficient condition for the admissible kernel class satisfying the conservation laws, the H-theorem, and the frame indifference. Unlike the classical Landau operator, the off-diagonal kernels are not restricted to be symmetric under permutation of the two velocity variables. This unique structural freedom captures the distinct responses of different species to unresolved correlations and many-body effects arising from micro-scale particle interactions. The equivalent parameterizable kernel formalization enables us to learn a generalized data-driven collision operator directly from MD, where the low-rank tensor representations and random sampling are used to achieve efficient kernel training and numerical simulation. Numerical experiments show that the learned operator accurately predicts transport coefficients and the non-equilibrium relaxation, while retaining discrete conservation and entropy production. In particular, it captures plasma kinetics in the moderately coupled regime, where the predictions of both the Landau and the data-driven model restricted to velocity-permutation symmetry show significant discrepancies.

physics.comp-ph↗

arXiv2026

Bounds-Constrained Finite Element Approximation of Time-Dependent Partial Differential Equations

Finite elements provide accurate and efficient methods for the numerical solution of partial differential equations by means of restricting variational problems to finite-dimensional approximating spaces. However, they do not, in general, guarantee enforcement of bounds constraints inherent in the original problem. We propose two approaches to enforcing bounds constraints for time-dependent problems. First, we propose general projective methods which result from a systematic modification of any abstract time-stepping scheme. Second, we present a monolithic technique for which we take a modified formulation of implicit single-stage Runge-Kutta methods and general implicit multistep methods as prototypical examples. By solving a constrained optimization problem, we are able to ensure that the bounds constraints are enforced by the approximate solution at the discrete time levels, obtaining (formally) high order methods both in space and time. Numerical examples for the linear heat and advection equations and nonlinear Allen-Cahn equation are given.

math.NA↗

arXiv2026

Stein's method for marginals on large graphical models

Many spatial models exhibit locality structures that effectively reduce their intrinsic dimensionality, enabling efficient approximation and sampling of high-dimensional distributions. However, existing approximation techniques primarily focus on joint distributions and do not provide precise accuracy control for low-dimensional marginals, which are of primary interest in many practical scenarios. By leveraging the locality structures, we establish a dimension independent uniform error bound for the marginals of approximate distributions. Inspired by the Stein's method, we introduce a novel $δ$-locality condition that quantifies the locality in distributions, and link it to the structural assumptions such as the sparse graphical models. The theoretical guarantee motivates the localization of existing sampling methods, as we illustrate through the localized likelihood-informed subspace method and localized score matching. We show that by leveraging the locality structure, these methods greatly reduce the sample complexity and computational cost via localized and parallel implementations.

stat.ML↗

arXiv2026

Gevrey class regularity for steady-state incompressible Navier-Stokes equations in parametric domains and related models

We investigate parameteric Navier-Stokes equations for a viscous, incompressible flow in bounded domains. The coefficients of the equations are perturbed by high-dimensional random parameters, this fits in particular for modelling flows in domains with uncertain perturbations. Our focus is on deriving bounds for arbitrary high-order derivatives of the pressure and the velocity fields with respect to the random parameters in the context of incompressible Navier-Stokes equation under a small-data assumption. To achieve this, we analyze mixed and saddle-point problems and employ the alternative-to-factorial technique to establish generalized Gevrey-class regularity for the solution pair. Thereby the analytic regularity follows as a special case. In the numerical experiments, we validate and illustrate our theoretical findings using Gauss-Legendre quadrature and Quasi-Monte Carlo methods.

math.NA↗

arXiv2026

Multilevel lattice-based kernel approximation for elliptic PDEs with random coefficients

This paper introduces a multilevel kernel-based approximation method to estimate efficiently solutions to elliptic partial differential equations (PDEs) with periodic random coefficients. Building upon the work of Kaarnioja, Kazashi, Kuo, Nobile, Sloan (Numer. Math., 2022) on kernel interpolation with quasi-Monte Carlo (QMC) lattice point sets, we leverage multilevel techniques to enhance computational efficiency while maintaining a given level of accuracy. In the function space setting with product-type weight parameters, the single-level approximation can achieve an accuracy of $\varepsilon>0$ with cost $\mathcal{O}(\varepsilon^{-η-ν-θ})$ for positive constants $η, ν, θ$ depending on the rates of convergence associated with dimension truncation, kernel approximation, and finite element approximation, respectively. Our multilevel approximation can achieve the same $\varepsilon$ accuracy at a reduced cost $\mathcal{O}(\varepsilon^{-η-\max(ν,θ)})$. Full regularity theory and error analysis are provided, followed by numerical experiments that validate the efficacy of the proposed multilevel approximation in comparison to the single-level approach.

math.NA↗

arXiv2026

Estimating systematic errors in Bayesian inversion using transport maps

In indirect measurements, the sought parameters have to be determined by solving an inverse problem, typically in a Bayesian framework. Often, the accurate numerical simulation of the measuring process is computationally demanding, making it necessary to rely on approximate models. These surrogates, however, introduce an additional model error and thus may distort the resulting parameter distribution. Moreover, even with the additional speed granted by the surrogate, posterior determination through conventional means such as Markov chain Monte Carlo might be cost intensive, specifically for complicated posterior shapes. In this paper, we propose a unified framework that combines Bayesian inference, model error correction and a transport-based sampling scheme to address these issues. To train the transport scheme, we investigate two different losses: one equivalent to the Kullback-Leibler divergence associated to the transport problem and one based on an upper bound of this loss, generally known as the evidence lower bound. We demonstrate that training the transport based on the latter changes the optimisation landscape drastically, potentially introducing an undesired bias in approximating the target posterior. We compare the computational cost of our approach with established methods and underline the theoretical results with numerical examples.

stat.ME↗

arXiv2026

Weighted total variation regularization for inverse problems with significant null spaces

We consider inverse problems with large null spaces, which arise in important applications such as in inverse ECG and EEG procedures. Standard regularization methods typically produce solutions in or near the orthogonal complement of the forward operator's null space. This often leads to inadequate results, where internal sources are mistakenly interpreted as being near the data acquisition sites -- e.g., near or at the body surface in connection with EEG and ECG recordings. To mitigate this, we previously proposed weighting schemes for Tikhonov and sparsity regularization. Here, we extend this approach to total variation (TV) regularization, which is particularly suited for identifying spatially extended regions with approximately constant values. We introduce a weighted TV-regularization method, provide supporting analysis, and demonstrate its performance through numerical experiments. Unlike standard TV regularization, the weighted version successfully recovers the location and size of large, piecewise constant sources away from the boundary, though not their exact shape. Additionally, we explore a hybrid weighted-sparsity and TV regularization approach, which better captures both small and large sources, albeit with somewhat more blurred reconstructions than the weighted TV method alone.

math.NA↗

arXiv2026

Analysis of a finite element method for second order uniformly elliptic PDEs in non-divergence form

We propose one finite element method for both second order linear uniformly elliptic PDE in non-divergence form and the uniformly elliptic Hamilton-Jacobi-Bellman (HJB) equation. For both linear elliptic PDE in non-divergence form and the HJB equation, we prove the well-posedness of strong solution in $W^{2,p}(Ω)$ and optimal convergence in discrete $W^{2,p}$-norm of the finite element approximation to the strong solution for $1<p\leq 2$ on convex polyhedra in $\mathbb{R}^{d}$ ($d=2,3$). If the domain is a two dimensional non-convex polygon, $p$ is valid in a more restricted region. Furthermore, we relax the assumptions on the continuity of coefficients of the HJB equation, which have been widely used in literature.

math.NA↗

arXiv2026

Scalable Pseudospectral Analysis via Low-Rank Approximations of Dynamical Systems

Pseudospectral analysis is fundamental for quantifying the sensitivity and transient behavior of nonnormal matrices, yet its computational cost scales cubically with dimension, rendering it prohibitive for large-scale systems. While existing research on scalable pseudospectral computation has focused on exploiting sparsity structures, common in discretizations of differential operators, these approaches are ill-suited for machine learning and data-driven dynamical systems, where operators are typically dense but approximately low-rank. In this paper, we develop a comprehensive low-rank framework that dramatically reduces this computational burden. Our core theoretical contribution is an exact characterization of the pseudospectrum of arbitrary low-rank matrices, reducing the evaluation of resolvent norms to eigenvalue problems of dimension proportional to the rank. Building on this foundation, we derive rigorous inclusion sets for the pseudospectra of general matrices via truncated and randomized low-rank approximations, with explicit perturbation bounds. These results enable efficient estimators for key stability quantities, including distance to instability and Kreiss constants, at a cost that scales with the effective rank rather than the ambient dimension. We further demonstrate how our framework naturally extends to data-driven settings, providing pseudospectral analysis of transfer operators learned from nonlinear and stochastic dynamical systems. Numerical experiments confirm orders-of-magnitude speedups while preserving accuracy, opening pseudospectral analysis to previously intractable high-dimensional problems in computational PDEs, control theory, and data-driven dynamics.

math.NA↗

arXiv2026

A non-intrusive approach to index-aware learning

We present a non-intrusive version of the index-aware learning framework introduced in arXiv:2309.00958. Index-aware learning itself is an approach for learning the time and parameter dependent solutions of differential-algebraic equations (DAEs), in particular those describing electric circuits. A central feature of the approach is that it ensures the learned solutions to fulfill the inherent constraints of the DAE, such as e.g. Kirchhoff's laws in the case of electric circuits. This is achieved by leveraging a decoupling of the DAE into its differential and algebraic parts, with the non-intrusive version of the approach additionally relying on results from arXiv:2604.20475 and arXiv:2107.07755. We illustrate the approach using a filtered buck converter as an example and compare both the intrusive and non-intrusive versions. The code for the example is openly available.

cs.CE↗

arXiv2026

Mamba-Assisted Non-Markovian Closure for Reduced-Order Modeling

Reduced-order modeling of high-dimensional dynamical systems is often hindered by closure effects arising from unresolved variables, which can introduce non-Markovian dependence into the resolved dynamics. Motivated by the history-dependent memory term arising in the Mori--Zwanzig formalism, we recast non-Markovian closure modeling as a sequence modeling problem and propose the Mamba-Assisted Closure (MAC) framework. MAC employs a Mamba-based sequence model to predict the closure from the resolved trajectory and couples the learned closure with the reduced-order governing equations through a numerical integrator to advance the resolved variables in time. During training, the selective scan mechanism in Mamba enables efficient parallel sequence processing with linear scaling in sequence length, while autoregressive inference proceeds through recurrent state updates at essentially constant per-step cost. We evaluate MAC on four benchmark systems with complementary characteristics: the viscous Burgers' equation, the chaotic two-scale Lorenz '96 system, the 3-bus DeMarco--Zheng power-grid system, and the dispersive Korteweg--de Vries equation. Across these benchmarks, MAC consistently improves predictive accuracy and long-time rollout stability relative to the comparison models, demonstrating an effective and computationally scalable approach to non-Markovian closure modeling.

cs.LG↗

arXiv2026

An FFT-Accelerated Boundary Integral Equation Method for Wave Scattering by Smooth Surfaces in Three Dimensions

For wave scattering by axisymmetric surfaces, the fast Fourier transform (FFT) method provides an effective tool to accelerate standard boundary integral equation (BIE) solvers. Surface BIEs can be decoupled into a series of curve integral equations on the generating curve, due to the convolution-like integral operators. The Fourier coefficients of the three-dimensional fundamental kernels can be rapidly computed through three-term recurrence relations based on Miller's algorithm. Such well-established techniques break down for nonaxisymmetric surfaces. This paper proposes a novel FFT-accelerated boundary integral method for wave scattering by smooth surfaces of arbitrary shapes. The Fourier coefficients of the singular kernels now satisfy higher-order recurrence relations. Although they can be solved with an optimal linear complexity by the standard Olver's algorithm, it turns out that a singularity swapping approach that rewrites each kernel as the product of a smooth function and an axisymmetric-related singular factor is realistically much faster. Consequently, Miller's algorithm together with the standard FFT convolution yields an ${\cal O}(M\log M)$ approach for evaluating the ${\cal O}(M)$ Fourier coefficients of the kernels, attaining exactly the same order of complexity for axisymmetric surfaces! With such FFT-based efficient procedures, we rewrite the surface BIEs in terms of ${\cal O}(M)$ curve integrals, which are proved to exhibit logarithmic singularities, discretize them by panel-based generalized Gaussian quadratures, and obtain spectrally accurate linear systems to approximate the wavefields. Extensive numerical experiments are carried out to demonstrate the effectiveness and spectral accuracy of the new approach.

math.NA↗

arXiv2026

Entry growth in Gaussian elimination

Gaussian elimination is one of the oldest algorithms in mathematics, and the most popular method for solving an unstructured linear system. Its stability in finite precision is controlled by its growth factor, which measures how large the entries produced during elimination can become. Understanding the worst-case behavior of this quantity has been a central problem in numerical analysis since the 1940s. Here we make a significant leap in that understanding, settling several open problems. In particular, we determine the asymptotic behavior of the maximum growth factor under complete and rook pivoting, proving that both are quasi-polynomial in dimension. We also show that the exponential growth under partial pivoting persists for sparse matrices and that randomized partial pivoting suffers the same instability. In contrast, we show that every non-singular matrix has a row permutation with polynomial growth, though finding the optimal row permutation is NP-hard.

math.NA↗

arXiv2026

Waveguiding in systems of high contrast resonators: Theory and fast computations

In this work, we study guided modes in systems of high-contrast resonators near nonzero interior Neumann frequencies, beyond the subwavelength regime. In the regular exterior regime, where the exterior Dirichlet problem is well-posed at the reference wavenumber, we introduce an infinite-dimensional frequency-dependent capacitance operator obtained by compressing the exterior Helmholtz Dirichlet-to-Neumann map to the traces of the interior resonant Neumann eigenspaces. We prove the norm-resolvent convergence of the continuous problem to this discrete effective operator as the contrast $δ\to0$, and derive first-order asymptotic formulas for compact-defect frequencies and line-defect band functions. We then establish exponential off-diagonal decay of the capacitance coefficients by a Combes--Thomas argument, yielding an exponentially accurate truncation of the discrete operator, and show that its retained coefficients can be computed from local Helmholtz problems. At the physical frequency, this local approximation converges exponentially under a uniform stability assumption for the growing finite-cluster problems. The stability assumption can be removed by introducing a vanishing complex absorption together with a Hermitian symmetrization. In particular, an absorption parameter of order $\sqrtδ$, together with interaction truncation and patch radii of order $|\logδ|$, suffices to preserve the $O(δ^2)$ accuracy of the first-order high-contrast expansion of the defect eigenfrequencies, yielding a fast computational method. Numerical experiments for dipole and quadrupole resonances illustrate the accuracy, exponential locality, and applicability of the discrete model to straight and bent waveguides generated by material or geometric detuning.

math.NA↗

arXiv2026

Structure-Preserving Detailed-Balance Master-Equation Discretizations for Fokker--Planck Equations

We develop a variational--Markov construction of detailed-balance master-equation discretizations for Fokker--Planck equations directly from their energy--dissipation laws. Rather than discretizing the differential operator, we represent mass transfer between neighboring grid points by two directional jump rates. On each edge, the discrete energy--dissipation law determines the net flux, while detailed balance fixes the rate ratio; a local reconstruction of edge quantities from neighboring grid-point values then uniquely determines both rates. A logarithmic-mean reconstruction recovers the classical Scharfetter--Gummel/Wang--Peskin--Elston rates, while alternative reconstructions yield other reversible schemes. The resulting semi-discrete systems conserve mass, preserve nonnegativity, satisfy detailed balance, and dissipate a discrete free energy. The same edgewise construction extends to state-dependent and degenerate mobilities, nonlocal interaction energies and higher-dimensional problems. The corresponding implicit and semi-implicit fully discrete schemes are linear, mass-conservative, positivity-preserving, and energy-stable, with a time-step restriction required only for nonlocal interaction energies whose kernel is not negative semidefinite. Numerical experiments confirm the expected spatial convergence rates of all detailed-balance schemes and verify their equilibrium accuracy, positivity preservation, and free-energy decay with discontinuous potentials, saturation, nonlocal interactions, and two-dimensional problems.

math.NA↗

Compare source metadata on this page

Work	Published	Source identifier	Source
Greedy sampling designs via reduced basis methods: optimal recovery in the uniform norm	2026-09-01	2609.01578	arxiv
Families of relative periodic orbits in the planar three-body problem via consecutive alignments	2026-09-01	2609.01585	arxiv
A Double-Adaptivity Solver for Parabolic PDEs	2026-09-01	2609.01748	arxiv
Learning a general class of admissible multi-species collision operators from molecular dynamics	2026-09-01	2609.01845	arxiv
Bounds-Constrained Finite Element Approximation of Time-Dependent Partial Differential Equations	2026-09-01	2609.01915	arxiv
Stein's method for marginals on large graphical models	2026-08-31	2410.11771	arxiv
Gevrey class regularity for steady-state incompressible Navier-Stokes equations in parametric domains and related models	2026-08-31	2504.13753	arxiv
Multilevel lattice-based kernel approximation for elliptic PDEs with random coefficients	2026-08-31	2504.15810	arxiv
Estimating systematic errors in Bayesian inversion using transport maps	2026-08-31	2509.16116	arxiv
Weighted total variation regularization for inverse problems with significant null spaces	2026-08-31	2512.04729	arxiv
Analysis of a finite element method for second order uniformly elliptic PDEs in non-divergence form	2026-08-31	2512.14219	arxiv
Scalable Pseudospectral Analysis via Low-Rank Approximations of Dynamical Systems	2026-08-31	2602.01721	arxiv
A non-intrusive approach to index-aware learning	2026-08-31	2605.30955	arxiv
Mamba-Assisted Non-Markovian Closure for Reduced-Order Modeling	2026-08-31	2606.05371	arxiv
An FFT-Accelerated Boundary Integral Equation Method for Wave Scattering by Smooth Surfaces in Three Dimensions	2026-08-31	2608.16208	arxiv
Entry growth in Gaussian elimination	2026-08-31	2608.19189	arxiv
Waveguiding in systems of high contrast resonators: Theory and fast computations	2026-08-31	2608.26906	arxiv
Structure-Preserving Detailed-Balance Master-Equation Discretizations for Fokker--Planck Equations	2026-08-31	2608.30121	arxiv

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