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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

Adaptive multigrid for high-order discontinuous Galerkin methods based on the full approximation scheme

We propose an adaptive multigrid (MG) method for discontinuous Galerkin formulations of elliptic problems using Brandt's full approximation scheme (FAS). Unlike common approaches, this method achieves local $hp$-refinement of hexahedral meshes without the need for hanging nodes. The core component of the FAS-MG method is an overlapping Schwarz smoother, which is optionally accelerated by a Krylov method. This smoother is designed for unstructured curvilinear meshes but maintains a tensor-product structure for fast diagonalization. Numerical experiments demonstrate the exceptional efficiency of the FAS-MG method. Dedicated studies confirm its robustness against high aspect ratios, element deformation, and irregular mesh topology. We also verify its capability for dynamic parallel mesh adaptation using the wave-front benchmark of Červený, Dobrev, and Kolev (SIAM J. Sci. Comp. 41, 2019). Finally, we present preliminary results of extending the method to incompressible Navier-Stokes problems.

math.NA↗

arXiv2026

The Minimum Number of Measurements for Almost-Everywhere Complex Phase Retrieval

Let $d\geq 2$ and let $\bf{f}_1,\ldots,\bf{f}_m\in\mathbb C^d$. We prove that if $m\leq 2d-1$, then the intensity measurement map \[ \bf{x}\longmapsto \bigl( |\langle \bf{x},\bf{f}_1\rangle|^2, \ldots, |\langle \bf{x},\bf{f}_m\rangle|^2 \bigr) \] fails to recover almost every signal in $\mathbb C^d$ uniquely up to a global phase factor. Combined with the known generic sufficiency of $2d$ measurements, our result establishes that the minimum number of measurements required for almost-everywhere phase retrieval in $\mathbb C^d$ is exactly $2d$. This resolves an open problem in phase retrieval by determining the exact measurement threshold for almost-everywhere phase retrieval in ${\mathbb C}^d$.

cs.IT↗

arXiv2026

A multi-class kinetic traffic flow model: discrete-velocity formulation and diffusively-corrected macroscopic limits

This paper introduces a multi-class extension of a discrete-velocity kinetic traffic flow model based on a non-local Prigogine-Herman framework. We derive a hyperbolically scaled system of equations from a continuous kinetic formulation describing interactions between different vehicle classes through braking and relaxation terms. The model is then discretized with respect to the velocity variable for an arbitrary number of vehicle classes, and the structural properties of the resulting formulation are analyzed. In particular, we prove hyperbolicity and total linear degeneracy. Due to the non-conservative structure of the model, we employ a path-conservative finite volume scheme for the numerical approximation of the system. Finally, we derive the corresponding diffusively-corrected macroscopic multi-class model, investigate its stability and present numerical simulations on a single-lane road to illustrate the theoretical findings.

math.NA↗

arXiv2026

A Massively Parallel Hybridizable Discontinuous Galerkin Solver for Direct Numerical Simulation of Compressible Flows on GPUs

Direct numerical simulation (DNS) of compressible transitional and turbulent flows requires numerical methods that combine high-order accuracy, robustness, and computational efficiency to resolve a broad range of spatial and temporal scales. This paper presents a massively parallel hybridizable discontinuous Galerkin (HDG) solver for DNS of the compressible Navier-Stokes equations on GPU-accelerated high-performance computing systems. The proposed solver combines high-order HDG discretization with robust shock capturing, diagonally implicit Runge-Kutta (DIRK) time integration, and an efficient Newton-GMRES solution strategy accelerated by additive Schwarz preconditioning and reduced-basis approximation. A distributed implementation of these methods based on GPU-aware MPI, Kokkos, and CUDA/HIP libraries enables scalable execution on heterogeneous computing platforms. The solver is demonstrated on three canonical benchmark problems covering a wide range of Mach-number flow regimes: subsonic transitional flow over the Eppler 387 airfoil, the supersonic Taylor-Green vortex, and hypersonic boundary-layer transition. Numerical results are compared with available experimental measurements and published DNS data, showing good agreement across distinct flow regimes. The results demonstrate the ability of the proposed solver to resolve laminar-turbulent transition, strong compressibility effects, shock-associated flow structures, and fully three-dimensional turbulent dynamics.

math.NA↗

arXiv2026

Optimal error estimates for the half-way bounce-back lattice Boltzmann method for the Stokes equations

We give a mathematical proof of the optimal convergence rates for the D2Q9 BGK lattice Boltzmann method with the half-way bounce-back rule for the incompressible Stokes equations in a flat channel. The convergence rates are second-order for the velocity and first-order for the pressure as the lattice spacing $h$ tends to zero, in agreement with formal analyses and numerical experiments, whereas the available rigorous convergence theorems only yield an $O(h^{1/2})$ bound for the velocity error. A key step in the proof is a decomposition of the leading boundary consistency error into macroscopic and kinetic components. These components are absorbed by suitably constructed Stokes and discrete Knudsen layer correctors, respectively. Incorporating these correctors into the prediction function used in previous rigorous analyses, we obtain a refined prediction function with consistency errors of sufficiently high order. Combined with the known weighted $L^2$-stability estimate, this gives the optimal convergence rates.

math.NA↗

arXiv2026

A Two-Level Preconditioner Based on Dominant Components for Time-Dependent Multiscale High-Contrast Problem

In this work, we develop a two-level overlapping preconditioner for time-dependent problems in high-contrast multiscale media. We present a coarse-space construction based on multiscale methods, with emphasis on the relaxed nonlocal multicontinuum (NLMC) method. The NLMC space can be separated into components representing the high-permeability regions and the low-permeability background. We show that the complete NLMC space effectively preconditions the heterogeneous stiffness operator, whereas the high-permeability component alone captures the contrast-dependent global modes. For suitable small time-step sizes, the mass matrix controls the low-permeability contribution and only the high-permeability component in NLMC space is required for the global coarse correction in the two-level preconditioner. For general time-step sizes, performance can be maintained by using the full NLMC space or augmenting the high-permeability component with a standard multiscale space. The proposed coarse-space construction lowers the computational cost while preserving robustness with respect to coefficient contrast and fine-scale resolution. To further improve efficiency, the multiscale basis functions can be constructed by iteratively solving the relaxed energy-minimizing formulation. We demonstrate the robustness, efficiency and scalability of the proposed method through several numerical experiments.

math.NA↗

arXiv2026

Quantile Randomized Kaczmarz for Streaming Linear Systems with Massart Noise

Quantile randomized Kaczmarz (QRK) has proven to be an efficient solver for corrupted linear systems and has received much attention. It was recently shown by Cai et al. (SIAM J. Matrix Anal. Appl. 47(2):802-823, 2026) that using $O(\log T/\log(1/β))$ samples for computing the quantile is necessary and sufficient for QRK to converge linearly over $T$ iterations when solving linear systems with a $β$-fraction of arbitrary corruptions, as long as $β$ is small enough. However, it remains unclear how large the corruption level $β$ can be, and how to compute the required subsample size $D$ explicitly, without hidden constants. This paper studies streaming linear systems with Massart noise via QRK using an order-optimal batch size $D=O(\log T)$ in each update. The independence of samples from previous iterations in the streaming setting enables a sharper analysis, yielding explicit, computable bounds on both the tolerable corruption level and the required subsample size. In particular, we establish linear convergence for corruption levels of up to approximately 7%. We also discuss how the constants improve under oblivious noise.

math.NA↗

arXiv2026

Transport-Matched Penalties for Diffusion Synthetic Acceleration of Polytopic Discontinuous Galerkin Discretisations

Diffusion synthetic acceleration is most effective when its diffusion correction reflects the transport discretisation that generates the iteration error. We develop this principle for high-order upwind discontinuous Galerkin discretisations of discrete-ordinates transport on polytopic meshes. From the discrete transport sweep, we derive the exact scalar correction that removes the source-iteration scalar error in one step. We prove that the associated scalar response is positive and self-adjoint, obtain an exact expression for the source-iteration convergence factor, and quantify the additional damping produced by vacuum leakage. Using the exact correction as a reference, we construct a transport-matched modified interior penalty correction whose boundary terms are inherited directly from homogeneous vacuum inflow. In the optically thick regime, the resulting MIP form approximates the exact correction with relative error proportional to the effective cell Knudsen number. This gives a strict acceleration of source iteration, with bounds uniform in mesh size, polynomial degree, and element face count for admissible polytopic meshes. Numerical experiments on Cartesian and centroidal Voronoi meshes confirm the predicted convergence and correction-operator scaling.

math.NA↗

arXiv2026

Conditioning and interpolation error bounds for second-order Stiefel retractions with closed-form inverses

Retractions provide a computationally efficient alternative to the Riemannian exponential and logarithm maps for practical data-processing tasks on manifolds. In particular, second-order retractions with closed-form inverse are well-suited for interpolation problems on manifolds. On the Stiefel manifold of orthogonal frames, there are only two retractions of this type: the Cayley retraction, which is second-order accurate under the canonical metric, and the recently proposed polar-light retraction, which is second-order accurate under the Euclidean metric. In this paper, we study the properties of these maps in the context of interpolation on the Stiefel manifold. To obtain explicit interpolation error bounds, we examine the conditioning of the retraction maps and their inverses. We show that the retractions are well-conditioned, and we derive interpolation error bounds similar to those of classical Euclidean interpolation. The inverse retractions are not well-conditioned in general, and we discuss how data can be mapped via an isometric group action to ensure stable computations. As with all retractions on compact manifolds, the inverse canonical Cayley retraction and the inverse polar-light retraction exist only locally, and we construct normal neighborhoods around any point in which either the inverse Cayley retraction or the invese polar-light retraction are guaranteed to be computable. As an application of the retraction maps, we consider Hermite interpolation, where the objective is to reproduce both sampled function values and derivative information. A numerical example demonstrates that retraction-based interpolation is competitive with classical methods based on Riemannian normal coordinates.

math.NA↗

arXiv2026

Performance Evaluation of Fast Fourier Transforms on Emerging RISC-V Hardware with Vector Extension Support

This manuscript presents a performance evaluation of Fast Fourier Transform (FFT) implementations on emerging processors supporting the RISC-V Vector Extension (RVV 1.0). By introducing juFFTe, a light-weight high-performance library for discrete Fourier transforms, it is demonstrated how effective vectorization of performance-critical FFT kernels can be achieved on RVV-enabled hardware. Comprehensive benchmarks on three RVV 1.0-ready processors, the SiFive X280, the X100 core of the SpacemiT K3 and the C920v2 core of the Sophon SG2044, reveal substantial performance improvements of juFFTe (https://github.com/FZJ-JSC/juFFTe) over the widely used FFTW3 library. Although RVV-enabled platforms show promising results at this stage of development, a comparison with AMD's Zen 5 architecture indicates that RISC-V needs further maturing to reach the performance of established micro-architectures.

cs.MS↗

arXiv2026

Inverse Source Problem for a Time-Fractional Diffusion-Wave Equation with a Singular Inverse-Square Potential

This paper investigates an inverse source problem for a time-fractional diffusion-wave equation with a singular inverse-square potential. The source term is assumed to consist of a known temporal factor and an unknown spatial component, which is to be recovered from terminal-state measurements. The well-posedness and regularity of the forward problem are established within an appropriate energy framework by exploiting Hardy-type inequalities and the spectral properties of the associated singular elliptic operator. The terminal observation operator is then shown to be compact, and uniqueness of the spatial source is established under a suitable nondegeneracy condition on the temporal factor. To stabilize the resulting ill-posed inverse problem, a Tikhonov regularization approach is introduced. The gradient of the regularized functional is derived through an adjoint problem involving a right-sided fractional derivative, leading to an adjoint-based conjugate gradient method with an exact line search for the numerical reconstruction of the unknown source. Numerical experiments are conducted on both one-and two-dimensional spatial domains, using both exact and noisy terminal data, to demonstrate the effectiveness and stability of the proposed source reconstruction method.

math.NA↗

arXiv2026

A high-order polynomial-corrected shifted boundary method for simulating fully nonlinear water waves

We present a novel unfitted computational framework for simulating fully nonlinear potential flow-based water waves. Focusing on wave propagation, we describe the core methodology, which involves a high-order polynomial-corrected shifted-boundary approximation on unfitted spectral elements. This approach allows for the simulation of a curved, highly time-dependent (moving and deforming) free surface affected by bathymetric changes. All that on a simple Cartesian background mesh without re-meshing or further approximations of non-affine geometric features. In addition, we highlight the importance of proper gradient recovery using a polynomial-preserving technique to accurately capture the vertical free-surface velocity. Ultimately, the goal is to develop a high-order convergent numerical scheme capable of simulating highly nonlinear waves over long periods. This is achieved through an arbitrary-order finite-difference approximation of the free surface combined with added hyperviscosity for numerical stability. We present verification and validation test cases for wave propagation in both periodic and finite domains. Emphasis is placed on convergence studies, the justification for using high-order approximations, the importance of optimal gradient recovery, long-time simulation of highly nonlinear waves, and nonlinear wave interactions with both affine and curved bathymetry changes, as well as with vertical walls, for highly nonlinear stream function waves and high-amplitude solitons. The novel computational model offers a comprehensive, all-in-one framework for simulating ocean waves and their interactions with offshore structures. It provides significant geometric flexibility, enabling boundaries such as the free surface to move and deform over time without the need to re-mesh a boundary-fitted mesh.

physics.flu-dyn↗

arXiv2026

Stochastic trace estimation for positive trace-class operators

Implicit trace estimation aims to approximate the trace of a matrix or linear operator accessible only through matrix-vector or operator-vector products. In the matrix setting, the Girard-Hutchinson estimator typically requires $\mathcal{O}(\varepsilon^{-2})$ products to achieve accuracy $\varepsilon$, while the variance-reduced Hutch++ algorithm reduces this sample complexity to $\mathcal{O}(\varepsilon^{-1})$ for positive semidefinite matrices. We develop infinite-dimensional analogues of these estimators for positive trace-class operators on separable Hilbert spaces. The idealized estimators use Gaussian random elements whose covariance is determined by the target operator, leading to unbiased operator versions of Girard-Hutchinson and Hutch++. We prove high-probability error bounds analogous to the finite-dimensional matrix results; in particular, idealized infHutch++ achieves $\mathcal{O}(\varepsilon^{-1})$ sample complexity. For practical computation, we introduce truncated implementations that restrict the random samples to finite-dimensional subspaces; for fixed sample budget, we show that truncated infHutch++ converges in distribution to its idealized counterpart as the truncation dimension tends to infinity. Numerical experiments with integral operators, density-of-states approximations, and spectral filtering for a radial Dirac operator show that these truncated estimators can achieve accuracy comparable to the ContHutch++ algorithm by Zvonek, Horning & Townsend while using lower-degree function representations and smaller internal discretizations in chebfun.

math.NA↗

arXiv2026

Well-posedness and numerical reconstruction of a source term for linear parabolic problems with an integral constraint

This work investigates a time-dependent source identification problem for linear parabolic equations subject to an integral constraint and Neumann boundary conditions in a domain of $\mathbb{R}^d$, $d\ge 1$. We establish well-posedness and higher regularity of the solution pair in parabolic Hölder spaces. A numerical algorithm based on a finite element discretization in space and an implicit time-stepping scheme is then developed for the reconstruction of the unknown source. The resulting discrete inverse problem involves a well-conditioned operator. We show that identity Tikhonov regularization provides only uniform, nonselective shrinkage in this setting, whereas Tikhonov regularization with derivative-based penalties, analyzed through the generalized singular value decomposition, provides an effective denoising strategy. The regularization parameter is selected using the Morozov discrepancy principle. Numerical errors are evaluated using full parabolic Hölder norms, which provide a more comprehensive assessment of the reconstruction by incorporating errors in the solution, its derivatives, and the associated Hölder seminorms. Numerical experiments for smooth and piecewise constant sources demonstrate accurate and robust reconstructions under increasing levels of noise.

math.AP↗

arXiv2026

Conformal Uncertainty Quantification Guarantees for Neural Operators

Neural operators provide fast surrogate models for approximating operators between function spaces, but their predictions often lack uncertainty quantification. We develop a split conformal framework to guarantee that a calibrated pointwise band around the neural operator output contains the true solution on at least a $1-γ$ fraction of the evaluation domain, with probability at least $1-α$ over test and calibration inputs, where $α,γ\in(0,1)$. Our method reduces a normalized residual field to its spatial $(1-γ)$-quantile and computes a scaling factor using a held-out calibration dataset. We prove marginal coverage guarantees for measurable residual fields defined on arbitrary probability spaces, covering both continuum domains and fixed discretizations. Under mild assumptions on the data distribution, we show that the coverage conditional on the calibration set follows a Beta distribution, which we verify with numerical experiments on Darcy flow and Navier--Stokes equations, where our calibration yields bands consistently tighter than existing corrections while retaining the target coverage.

math.NA↗

arXiv2026

Closest Normal Matrix Found Again Using Riemannian Optimization

We propose an approach based on Riemannian optimization to compute a nearest normal matrix to a given one. The problem can be formulated as the minimization of a smooth function either on the manifold $U(n)$ of unitary matrices of size n or on the flag manifold $U (n)/U (1)^n$. The flag manifold is particularly suitable for theoretical analysis; we characterize the global maximum of the objective function and prove that, for generic inputs, its local minimizers are finitely many and isolated; in turn, this implies the original nearest normal matrix problem generically has finitely many local minimizers, all with distinct eigenvalues. We also develop a Riemannian trust-region method that improves substantially on classical algorithms and can handle considerably larger matrices, as well as a variant for computing the nearest real normal matrix. The paper is complemented by extensive numerical experiments.

math.NA↗

arXiv2026

QGPINNs: A Physics-Informed Neural Network Framework for Nonlocal Differential Equations on Quantum Graphs

We propose QGPINNs, a physics-informed neural network framework developed in PyTorch for the numerical solution of nonlocal differential equations on quantum graphs. The framework is designed as a general computational implementation in which the solution on each edge of the graph is approximated by a neural network, while a unified graph-based loss function enforces the governing equations together with initial, boundary, and vertex transmission conditions. In particular, the formulation incorporates standard continuity and Kirchhoff-Neumann vertex conditions and Dirichlet boundary conditions into the learning process to couple the local edge-wise neural approximations into a global solution on the graph. The framework is developed for two representative classes of nonlinear models: multi-order fractional elliptic problems and time-fractional evolution equations on quantum graphs. To improve accuracy and training stability, QGPINNs integrates several graph-adapted learning strategies, including soft and hard constraint enforcement, dynamic loss balancing, Fourier feature embeddings, and a learnable singularity-capturing feature for weakly singular solutions arising in the considered problems. The framework also extends naturally to inverse problems, including the identification of the orders of fractional operators and physical parameters from noisy observational data. We validate the accuracy, computational efficiency, and physical consistency of the proposed framework through numerical experiments on benchmark graph structures and real-world networks, including the IEEE 14-bus system and an open-channel agricultural drainage network.

cs.LG↗

arXiv2026

Overcoming the spatial order barrier for nonlinear SPDEs with additive space-time white noise

We introduce a fully discrete numerical scheme for semilinear SPDEs with additive space-time white noise that overcomes the previous order barrier for the spatial convergence rate. The scheme, which we refer to as the doubly accelerated exponential Euler scheme, achieves a strong convergence rate of $M^{-1+ε}$ in time and $N^{-3/2+ε}$ in space for any $ε>0$, where $M^{-1}$ and $N^{-1}$ are the temporal, respectively the spatial, meshsizes. This substantially improves the standard spatial error bounds of order $N^{-1/2}$ in the literature. Numerical simulations support the findings.

math.NA↗

Compare source metadata on this page

Work	Published	Source identifier	Source
Adaptive multigrid for high-order discontinuous Galerkin methods based on the full approximation scheme	2026-08-28	2608.14123	arxiv
The Minimum Number of Measurements for Almost-Everywhere Complex Phase Retrieval	2026-08-28	2608.15003	arxiv
A multi-class kinetic traffic flow model: discrete-velocity formulation and diffusively-corrected macroscopic limits	2026-08-28	2608.27159	arxiv
A Massively Parallel Hybridizable Discontinuous Galerkin Solver for Direct Numerical Simulation of Compressible Flows on GPUs	2026-08-28	2608.27863	arxiv
Optimal error estimates for the half-way bounce-back lattice Boltzmann method for the Stokes equations	2026-08-28	2608.27926	arxiv
A Two-Level Preconditioner Based on Dominant Components for Time-Dependent Multiscale High-Contrast Problem	2026-08-28	2608.27958	arxiv
Quantile Randomized Kaczmarz for Streaming Linear Systems with Massart Noise	2026-08-28	2608.27968	arxiv
Transport-Matched Penalties for Diffusion Synthetic Acceleration of Polytopic Discontinuous Galerkin Discretisations	2026-08-28	2608.28022	arxiv
Conditioning and interpolation error bounds for second-order Stiefel retractions with closed-form inverses	2026-08-28	2608.28073	arxiv
Performance Evaluation of Fast Fourier Transforms on Emerging RISC-V Hardware with Vector Extension Support	2026-08-28	2608.28076	arxiv
Inverse Source Problem for a Time-Fractional Diffusion-Wave Equation with a Singular Inverse-Square Potential	2026-08-28	2608.28103	arxiv
A high-order polynomial-corrected shifted boundary method for simulating fully nonlinear water waves	2026-08-28	2608.28123	arxiv
Stochastic trace estimation for positive trace-class operators	2026-08-28	2608.28135	arxiv
Well-posedness and numerical reconstruction of a source term for linear parabolic problems with an integral constraint	2026-08-28	2608.28171	arxiv
Conformal Uncertainty Quantification Guarantees for Neural Operators	2026-08-28	2608.28515	arxiv
Closest Normal Matrix Found Again Using Riemannian Optimization	2026-08-28	2608.28545	arxiv
QGPINNs: A Physics-Informed Neural Network Framework for Nonlocal Differential Equations on Quantum Graphs	2026-08-28	2608.28589	arxiv
Overcoming the spatial order barrier for nonlinear SPDEs with additive space-time white noise	2026-09-30	2608.28727	arxiv

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