subject

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

Spectral Bounds for Kernel Quadrature

A bottleneck in the theory of kernel methods in machine learning is the storage requirement. To ameliorate this, a standard trick is to replace the kernel with an explicit feature map. Perhaps, the most well known example is the Gaussian kernel which can be expressed in terms of the Fourier features. Analytically, the kernel $K$ can be expressed in terms of an integral expression that involves a possibly asymmetric kernel $G$ representing the feature map. Numerically, one needs to approximate this integral by a suitable numerical integration scheme, typically Monte Carlo. In this paper, we demonstrate that the eigenvalues of $K$ are approximated much better by the eigenvalues of the kernel obtained by discretizing the integral using suitable quadrature formulas instead. We illustrate this fact in the case of the Gaussian kernel and neural tangent kernels on the unit sphere of a four-dimensional Euclidean space corresponding to the sigmoid and ReLU activation functions.

math.NA↗

arXiv2026

Comparison of Algebraic Block Multi-Coloring and Leiden Methods for Parallel Preconditioning in the ICCG Method

In the application of incomplete Cholesky preconditioning to the incomplete Cholesky-conjugate gradient (ICCG) method, forward and backward substitutions exhibit sequential dependencies that constitute a major bottleneck for parallelization in multicore environments. To alleviate this bottleneck, the algebraic block multi-coloring (ABMC) method achieves both parallelism and data locality through block-wise coloring. However, ABMC requires the number of blocks to be specified as an input parameter in advance. This study evaluates the Leiden method as an alternative blocking approach for parallel preconditioning in the ICCG method. As a community detection technique that maximizes a quality function for graph partitioning, the Leiden method automatically generates blocks that reflect the matrix structure without requiring the number of blocks a priori. We partition the adjacency graphs of sparse matrices using the Leiden method and utilize the resulting blocks for parallel preconditioning. We implement the Leiden method using modularity and the constant Potts model as quality functions and compare its performance with that of the ABMC method in terms of the number of iterations, execution time, and L2 cache efficiency across eight symmetric positive definite matrices. The experimental results demonstrate that the Leiden method with the constant Potts model achieves performance comparable to that of the ABMC method configured with an optimized number of blocks.

cs.DC↗

arXiv2026

FTU-Seek: Foundation Model-Guided Hard-Negative Learning for Sparse Functional Tissue Unit Segmentation

Functional tissue units (FTUs), including tertiary lymphoid structures (TLSs), blood vessels, and glands, encode localized immune, vascular, and epithelial organization in histopathology. Accurate quantification of these structures is important for studying tissue architecture and disease-associated tissue organization. However, FTUs are frequently sparse, heterogeneous, and surrounded by large amounts of morphologically similar background tissue, making automated segmentation in whole-slide images (WSIs) challenging. We therefore developed FTU-Seek, a pathology foundation model-guided framework that treats morphology-aware negative-patch selection as a key component of sparse FTU segmentation. FTU-Seek uses frozen multi-depth features from the UNI pathology foundation model to train a patch-level classifier that distinguishes FTU-containing from FTU-absent tissue. Target-absent patches are subsequently ranked according to their predicted target-containing probabilities, and the highest-scoring hard negatives are selected through a static Top$K$ strategy to construct compact segmentation training sets. The framework was evaluated using five-fold cross-validation and internal test cohorts across TLS, blood-vessel, and gland segmentation tasks, with an additional independent 30-WSI held-out cohort for TLS. Positive-only, all-tissue, random-negative, and matched random Top$K$ sampling strategies served as comparators. Segmentation-derived phenotypes were further explored in external TCGA cohorts.

cs.CV↗

arXiv2026

Fast computation and convergence analysis of the infinite-product representation of the Schottky--Klein prime function

The Schottky--Klein prime function is a standard tool for boundary-value problems on multiply connected circular domains. Because this function is represented as an infinite product over a Schottky group, numerical evaluation requires truncation to finitely many factors. The standard word-length truncation grows exponentially in cost and becomes inefficient when the boundary circles nearly touch one another or the unit circle. To address this difficulty, we assign to each group element a cross-ratio potential measuring the size of its contribution, and retain only terms below a prescribed threshold. We establish uniform closed-form bounds on the change in this potential when prepending Schottky-group generators, and from these bounds we derive an efficient enumeration algorithm. The resulting relative error decays exponentially with the threshold at a rate determined by the Hausdorff dimension of the limit set of the Schottky group. Numerical experiments demonstrate that the proposed formulation achieves substantial computational speedups over word-length truncation in challenging geometric configurations.

math.NA↗

arXiv2026

Assessment of Numerical Lift Coefficient Data for a Circular Cylinder with Application to Bladeless Turbines

We assess computed lift coefficient data for flow past a circular cylinder to evaluate their suitability for practical applications. Specifically, we consider lift coefficient data for a circular cylinder over Reynolds numbers from 120 to 8000. The results are obtained from two-dimensional finite element simulations of the incompressible Navier-Stokes equations using pressure robust discretizations. We compare the computed lift coefficients with published experimental and numerical results, finding good agreement in some cases but significant disagreement in others. Because lift fluctuations are central to vortex-induced vibration concepts, these data therefore provide input for the analysis and preliminary design of bladeless turbines.

physics.flu-dyn↗

arXiv2026

Sharp Mixed Spectral Barron Regularity of Coulombic Many-Electron Wave Functions

We establish sharp mixed spectral Barron regularity for eigenfunctions of molecular Coulomb Hamiltonians. The mixed norm is a Fourier $L^1$ norm with one isotropic weight and coordinate-product weights, and therefore detects regularity invisible to the isotropic Barron scale. For a nonempty set $I$ of electron indices on which the wave function is antisymmetric, we derive an explicit admissible region for the isotropic order $s$ and the coordinate orders $α,β$. This region is optimal as a uniform statement over the class of clamped-nuclei Coulomb Hamiltonians. For fixed-spin components with two occupied spin blocks, it reduces to $s+α+β<1$; in the fully spin-polarized class it reduces to $s+α<1$. In particular, if $\mathcal I_σ$ denotes the family of occupied same-spin blocks determined by $σ$, then every fixed-spin spatial component $ψ_σ$ satisfies, for every $0\leqα<1$, \[ \left(\sum_{I\in\mathcal I_σ}\prod_{i\in I}\langleξ_i\rangle^α\right)\widehat{ψ_σ}\in L^1(\mathbb{R}^{3N}). \] For a fully spin-polarized state, $\mathcal I_σ=\{\{1,\ldots,N\}\}$.

math.AP↗

arXiv2026

DOFFO_TR: a Decentralized Objective Function-Free Optimization method with Trust-Region

In this paper, we propose a novel objective function-free trust-region method designed to solve optimization problems over decentralized networks. Unlike traditional approaches that often rely on stepsize tuning, our framework employs a function-free trust-region procedure that enables adaptive selection of the step length. Our approach accommodates first- and second-order models and eliminates the need to share local function values and gradients among agents, thereby enhancing privacy and computational efficiency. On the theoretical side, we establish provable iteration complexity guarantees that, for some variants, match those established for classical centralized trust-region methods. Numerical evaluations demonstrate that our approach achieves a favorable trade-off between performance and efficiency, requiring only moderate communication overhead compared to state-of-the-art methods in the literature.

math.OC↗

arXiv2026

Asymptotic behavior of Eckhoff's method for convergence acceleration of Dirac eigenfunction expansions

The current paper considers the problem of recovering a vector-function on $[-1,1]$ from a limited number of coefficients of its expansion into a series of eigenfunctions of a one-dimensional Dirac system. The Krylov--Lanczos--Eckhoff--Gottlieb acceleration method is examined in the situation when the boundary values it requires have to be computed from the generalized Fourier coefficients themselves. This leads to a $2q\times 2q$ linear system whose matrix is a block Vandermonde matrix; its determinant and inverse are computed explicitly, and the asymptotic $L_2$-error constant of the method is found, paralleling the classical trigonometric case.

math.NA↗

arXiv2026

OptiXDE: A fast optical-inspired solver for differential equations

OptiXDE is a matrix-free spectral operator framework for differential equations on uniform grids and embedded domains. Inspired by angular-spectrum propagation in Fourier optics, it maps transform-diagonal spatial operators to analytical modal multipliers and composes them with physical-space operators for nonlinearities, geometry and boundary enforcement. A common transform--operator--inverse-transform backbone is demonstrated across transient diffusion, periodic and embedded-domain Poisson problems, the cubic nonlinear Schr"odinger equation, viscous Burgers dynamics, the two-dimensional Allen--Cahn equation and incompressible flows from the Taylor--Green vortex to embedded-cylinder vortex shedding. Transform-compatible linear problems are recovered near the floating-point limit, whereas errors on the singular L-shaped domain remain localized near the re-entrant corner and regularized interface. Nonlinear benchmarks recover second-order temporal convergence and the expected conservative or dissipative behavior, while incompressibility remains near round-off level during long-time vortex shedding. The matrix-free updates require $\mathcal{O}(N\log N)$ work and $\mathcal{O}(N)$ memory. Device-resident transform workloads reach $94.9\times$ GPU acceleration, and the complete embedded-cylinder solver achieves a $42.1\times$ CPU--GPU speedup under matched numerical settings. These results establish OptiXDE as a deterministic and extensible operator-centric framework for structured and embedded-domain differential equations.

math.NA↗

arXiv2026

Randomized inexact block triangular preconditioners for double saddle-point systems in PDE-constrained optimization

We develop a new class of inexact block triangular preconditioners for double saddle-point systems arising from PDE-constrained optimization. The proposed preconditioners are constructed through matrix factorization techniques while preserving the inherent block structure of the original systems. A comprehensive spectral analysis of the preconditioned matrices is provided, yielding explicit bounds for both real and nonreal eigenvalues. To enable efficient construction of the inexact preconditioners, randomized strategies are introduced to select the required subblocks. We establish high-probability bounds for the expected approximation error, with the error estimates explicitly characterized in terms of the eigenvalues of the associated matrices. Numerical experiments demonstrate the effectiveness, robustness, and scalability of the proposed preconditioners, and validate the efficiency of the randomized construction strategies.

math.NA↗

arXiv2026

Symmetry Inheritance and Symmetry-Reduced Finite Element Analysis for Second-Order Linear Elliptic Problems with Robin Boundary Conditions

A symmetry framework is developed for second-order linear elliptic equations subject to Robin boundary conditions on bounded domains. Orthogonal transformations of the domain are represented through left group actions on scalar, vector, and second-order tensor fields. The corresponding transformation rules for the gradient, divergence, diffusion flux, and conormal boundary term are derived in detail. Based on these relations, symmetry groups are introduced for the principal coefficient tensor, the first-order and zeroth-order coefficients, the differential operator, the Robin coefficient, and the boundary operator. The symmetry properties of the volume and boundary source terms are then incorporated into a common symmetry group for the complete boundary value problem. Under the assumption of unique solvability, it is proved that every element of this common group is also a symmetry of the solution. For reflection symmetries, homogeneous generalized Neumann conditions are obtained on artificial symmetry boundaries, leading to an exact reduction of the computational domain. A corresponding finite element formulation is presented, and numerical results are provided to verify the theoretical symmetry properties and the validity of the resulting domain-reduction strategy. Three examples illustrate radial reduction from multiple dimensions to one dimension, reflection-based domain reduction, and a variable-coefficient problem in which all coefficients and source terms are nonzero.

math.NA↗

arXiv2026

32-point DFT Approximations Based on Minimal Frobenius Error and DFT Symmetries

This work introduces low-complexity, multiplierless approximations for the 32-point discrete Fourier transform. The proposed methods are obtained by minimizing the Frobenius error compared against the DFT matrix over a set of trivial multipliers. A row-wise, symmetry-constrained parameterization is employed to reduce the search space size, rendering the task computationally tractable. The resulting approximations could outperform the reference method in the literature according to energy-based error measurements. A sparse matrix factorization is provided for efficient computation; the arithmetic costs are 152 real additions and 34 bit-shifts only.

eess.SP↗

arXiv2026

On the Gram matrix of standard inner products of asymmetrically-weighted Hermite functions

Let A denote the infinite Gram matrix associated with the standard L2 inner product of asymmetrically-weighted (AW) Hermite functions. We derive an explicit representation of its entries and its Cholesky factorization. We further show that this factorization admits a natural interpretation on a scaled Bargmann-Fock basis. An explicit formula for the inverse of A is also obtained. We then consider the corresponding finite Gram matrix and analyze its asymptotic property, as well as that of its Schur complement. The analysis is motivated by numerical methods for plasma physics, in particular Galerkin spectral methods applied to the Vlasov-Poisson (VP) system. As an application, we demonstrate how the derived Gram matrix formulas and asymptotic results can be exploited in the analysis and implementation of a Galerkin spectral method for the VP system.

math.NA↗

arXiv2026

Projection-based low-rank assembly in IgA

Isogeometric Analysis (IgA) uses the same spline functions to represent the computational domain and to approximate the solution. This allows exact geometry descriptions, but the resulting mass and stiffness matrices are expensive to assemble and to store, especially in three dimensions. We present a projection-based low-rank approach for assembling the mass and stiffness tensors of orientation-preserving tensor-product B-spline geometries. For the mass tensor, we exploit the polynomial structure of the determinant of the Jacobian of the geometry map and represent it in reduced spline product spaces by univariate coefficient transfer operators; with exact quadrature and without truncation, the resulting low-rank tensor is an exact reformulation of the standard Galerkin mass tensor. For the stiffness tensor, we split the rational weight function into a polynomial numerator, again represented in reduced spline product spaces, and the reciprocal determinant, which is in general not a spline function and is therefore approximated by an $L^2$-projection onto a tensor-product spline space. Both constructions are carried out entirely in the tensor-train (TT) format, with the projection system solved by the alternating minimal energy (AMEn) method, so that full high-order coefficient tensors are never formed and the multidimensional integrals reduce to univariate integrals and contracted products. The method is implemented in MATLAB using GeoPDEs and the TT-Toolbox. Numerical experiments show that it is competitive with full assembly and with the interpolation-based low-rank method, and that it applies in two situations in which interpolation is problematic: a singular interpolation system and nearly singular geometries. The construction is restricted to orientation-preserving tensor-product B-spline geometries and does not cover NURBS.

math.NA↗

arXiv2026

The $α$-Limit Problem: Convergence of a Linear Degenerate Interface Transmission Problem

We study the singular limit of a family of linear degenerate interface transmission problems arising from a regularization procedure in the newly proposed Two-Parameter Diffuse Domain Method (DDM2p). For $α>0$, the regularized problem admits a strictly convex variational formulation on $H^{1}(Ω)$. In the limit $α\to0$, the problem degenerates to a weakly coupled interface system with a nonstandard energy structure. To characterize the limit, we introduce a closed Hilbert subspace $\mathcal{H}\subset H^{1}(Ω)$, defined through an auxiliary Helmholtz problem on an annular subdomain $Ω_2\subset Ω$, and identify the limiting energy functional $\mathcal{E}_{0}$ on $\mathcal{H}$. We prove that the regularized energies $\mathcal{E}_α$ $Γ$-converge to $\mathcal{E}_{0}$ in the strong $L^{2}(Ω)$ topology, using the standard framework. Consequently, minimizers of $\mathcal{E}_α$ converge to the unique minimizer of $\mathcal{E}_{0}$, which is shown to be equivalent to the solution of the limiting interface problem. We further prove strong convergence $u_α\to u_{0}$ in $H^{1}(Ω)$ and establish an $O(α)$ convergence rate. Numerical experiments in one spatial dimension confirm the predicted first-order convergence rate and suggest that this rate is sharp.

math.AP↗

arXiv2026

Accelerating the Improved Arrow--Hurwicz Iteration via the Anderson Algorithm for Steady-State Navier--Stokes Equations

We apply Anderson acceleration to the improved Arrow--Hurwicz (IAH) method for the finite element solution of the steady-state incompressible Navier--Stokes equations. The IAH scheme avoids saddle-point solves by decoupling the velocity and pressure updates, but can require prohibitively many iterations, particularly at high Reynolds numbers. To place the acceleration on a rigorous footing, we reformulate the IAH iteration as a nonlinear fixed-point operator $G$ for the grad-div augmented discrete formulation induced by the scheme and establish its well-definedness, Lipschitz continuity, and Fréchet differentiability, thereby verifying the required smoothness conditions locally near the fixed point. Numerical experiments on problems with known analytical solutions, lid-driven cavity flow up to $Re = 15{,}000$, and channel flow over a full step demonstrate that the resulting Anderson-accelerated improved Arrow--Hurwicz algorithm substantially reduces iteration counts and CPU time while retaining the reported manufactured-solution convergence rates and centerline-velocity agreement.

math.NA↗

arXiv2026

Accelerated primal-dual dynamics and algorithms for convex optimization with nonlinear inequality constraints

We consider convex optimization with nonlinear inequality constraints and develop a primal-dual multiplier framework that is consistent in continuous and discrete time. We first propose continuous-time dynamics with Nesterov-type vanishing damping $α/t$, together with compatible extrapolations of the dual variable and the nonlinear constraint mapping. Under convexity assumptions and $α\geq3$, we establish $\mathcal O(t^{-2})$ convergence rates for both nonlinear feasibility and the objective residual. In the noncritical regime $α>3$, with an admissible choice of the extrapolation parameter, we further prove that the entire primal-dual trajectory converges to a KKT pair and sharpen both continuous-time estimates to $o(t^{-2})$. We then derive an inexact accelerated primal-dual algorithm through a compatible discretization of a perturbed version of the dynamic. For composite convex objectives, a weighted summability condition on the primal inexactness yields $\mathcal O(k^{-2})$ rates for feasibility and the objective residual. In the corresponding noncritical regime, the discrete primal-dual sequence converges to a KKT pair and both residual estimates improve to $o(k^{-2})$. Thus the continuous and discrete results exhibit matching accelerated rates and matching asymptotic improvements. To the best of our knowledge, this is the first Nesterov-type primal-dual multiplier framework for convex optimization with nonlinear inequality constraints.

math.OC↗

arXiv2026

Score-Based Generative Data Assimilation for Integrating Aggregated Surveillance Data into Agent-Based Models in Epidemic Tracking

Reliable epidemic monitoring often requires inferring regional infection burden and transmission heterogeneity from noisy, spatially aggregated, and potentially sparse surveillance data. Agent-based models (ABMs) are attractive for this task because they represent individual behavior, contact heterogeneity, and localized interventions, but these same features make them difficult to calibrate online. We develop a generative AI-based data-assimilation (GenDA) framework for partially observed epidemic ABMs that estimates both the epidemic state and a heterogeneous parameter field while respecting the gap between observable macrostates and latent agent-level microstates. GenDA combines a training-free, score-based generative update for macrostate correction with a direct parameter update based on macrostate discrepancies, followed by a macro-micro reassignment step that restores consistency with the ABM. In controlled and geographically explicit synthetic experiments, the framework recovers regional epidemic burden, dominant hotspot structures, and effective transmission heterogeneity from aggregated observations, while improving post-assimilation forecasts relative to state-only assimilation.

math.NA↗

Compare source metadata on this page

Work	Published	Source identifier	Source
Spectral Bounds for Kernel Quadrature	2026-09-01	2609.00553	arxiv
Comparison of Algebraic Block Multi-Coloring and Leiden Methods for Parallel Preconditioning in the ICCG Method	2026-09-18	2609.00561	arxiv
FTU-Seek: Foundation Model-Guided Hard-Negative Learning for Sparse Functional Tissue Unit Segmentation	2026-09-01	2609.00704	arxiv
Fast computation and convergence analysis of the infinite-product representation of the Schottky--Klein prime function	2026-09-01	2609.00785	arxiv
Assessment of Numerical Lift Coefficient Data for a Circular Cylinder with Application to Bladeless Turbines	2026-09-01	2609.00850	arxiv
Sharp Mixed Spectral Barron Regularity of Coulombic Many-Electron Wave Functions	2026-09-01	2609.00872	arxiv
DOFFO_TR: a Decentralized Objective Function-Free Optimization method with Trust-Region	2026-09-01	2609.00878	arxiv
Asymptotic behavior of Eckhoff's method for convergence acceleration of Dirac eigenfunction expansions	2026-09-01	2609.00895	arxiv
OptiXDE: A fast optical-inspired solver for differential equations	2026-09-01	2609.01009	arxiv
Randomized inexact block triangular preconditioners for double saddle-point systems in PDE-constrained optimization	2026-09-01	2609.01031	arxiv
Symmetry Inheritance and Symmetry-Reduced Finite Element Analysis for Second-Order Linear Elliptic Problems with Robin Boundary Conditions	2026-09-01	2609.01107	arxiv
32-point DFT Approximations Based on Minimal Frobenius Error and DFT Symmetries	2026-09-01	2609.01115	arxiv
On the Gram matrix of standard inner products of asymmetrically-weighted Hermite functions	2026-09-01	2609.01144	arxiv
Projection-based low-rank assembly in IgA	2026-09-01	2609.01218	arxiv
The $α$-Limit Problem: Convergence of a Linear Degenerate Interface Transmission Problem	2026-09-01	2609.01237	arxiv
Accelerating the Improved Arrow--Hurwicz Iteration via the Anderson Algorithm for Steady-State Navier--Stokes Equations	2026-09-01	2609.01288	arxiv
Accelerated primal-dual dynamics and algorithms for convex optimization with nonlinear inequality constraints	2026-09-22	2609.01415	arxiv
Score-Based Generative Data Assimilation for Integrating Aggregated Surveillance Data into Agent-Based Models in Epidemic Tracking	2026-09-01	2609.01434	arxiv

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