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

Greedy recursion parameter selection for one-way spatial integration of hyperbolic equations

Solutions to hyperbolic systems comprise waves propagating at finite speeds. When wave propagation is predominantly unidirectional, one-way wave equations can be used to evolve only the right-going solution by removing support for left-going waves. The One-Way Navier-Stokes (OWNS) approach, which was originally developed for systems of first-order hyperbolic equations, constructs one-way approximations to the linearized Navier-Stokes equations using a recursive filter to remove left-going waves. The computational cost scales with the number of recursion parameters, which must be carefully chosen to ensure accuracy and stability of the resulting one-way equation. Previous work has chosen parameters based on heuristic estimates of key eigenvalues, which requires trial-and-error tuning while also yielding slow error convergence. We propose a greedy algorithm for automatic parameter selection, which we show yields faster convergence and a net decrease in computational cost for linear and nonlinear disturbance evolution in boundary-layer flows. We review the OWNS projection (OWNS-P) and recursive (OWNS-R) methods, comparing their convergence properties, and show through our numerical analysis and experiments that OWNS-P yields superior convergence and stability properties. Although we demonstrate the method for Navier-Stokes equations, we perform our analyses on systems of linear first-order hyperbolic equations and emphasize that the greedy algorithm is applicable to such systems.

math.NA↗

arXiv2026

Interpolation in Polynomial Spaces of p-Degree

We recently introduced the Fast Newton Transform (FNT), an hierarchical algorithm for performing multivariate Newton interpolation in arbitrary downward closed polynomial spaces of spatial dimension $m$. Here, we analyze the FNT in the context of a specific family of downward closed sets $A_{m,n,p}$, defined as all multi-indices with $\ell^p$ norm less than $n$ with $p \in [0,\infty]$. The FNT performs with time complexity $\mathcal{O}(|A_{m,n,p}|mn)$ on the induced downward closed polynomial spaces $Π_{m,n,p}$. We show that the $Π_{m,n,p}$ choice compared to the tensor product spaces $Π_{m,n,\infty}$, reduces time complexity by a factor of $ρ_{m,n,p}$, decaying super exponentially with spatial dimension when $m \lesssim n^p$. We showcase the efficiency of the FNT by computing activity scores in sensitivity analysis.

math.NA↗

arXiv2026

The Bayesian SIAC filter

We propose the Bayesian Smoothness-Increasing Accuracy-Conserving (SIAC) filter---a hierarchical Bayesian generalization of the existing deterministic SIAC filter. The SIAC filter is a powerful numerical tool for removing high-frequency noise from data or numerical solutions without degrading accuracy. However, current SIAC methodology is limited to (i) nodal/modal data (noisy direct function values/coefficients of a piecewise polynomial function approximation) and (ii) deterministic point estimates that do not account for uncertainty propagation of input data to the SIAC reconstruction. The proposed Bayesian SIAC filter overcomes these limitations by (i) supporting general (non-nodal) data models and (ii) enabling rigorous uncertainty quantification (UQ), thereby broadening the applicability of SIAC filtering. We also develop structure-exploiting algorithms for efficient maximum a posteriori (MAP) estimation and Markov chain Monte Carlo (MCMC) sampling, with a focus on linear data models with additive Gaussian noise. Computational experiments demonstrate the effectiveness of the Bayesian SIAC filter across several applications, including signal denoising, image deblurring, and post-processing of numerical solutions to hyperbolic conservation laws. The results show that the Bayesian approach produces point estimates with accuracy comparable to, and in some cases exceeding, that of the deterministic SIAC filter. In addition, it extends naturally to general data models and provides built-in UQ.

math.NA↗

arXiv2026

A Variational Method for Conformable Fractional Equations Using Rank-One Updates

We make a complete variational treatment of rank-one Proper Generalised Decomposition for separable fractional partial differential equations with conformable derivatives. The setting is Hilbertian, the energy is induced by a symmetric coercive bilinear form, and the residual is placed in the dual space. A greedy rank-one update is obtained by maximizing an energy Rayleigh quotient over the rank-one manifold, followed by an exact line search. An exact one step energy decrease identity is proved, together with geometric decay of the energy error under a weak greedy condition that measures how well the search captures the Riesz representer of the residual. The alternating least squares realization is analyzed at the level of operators, including well posedness of the alternating subproblems, a characterization of stationary points, and monotonicity of the Rayleigh quotient along the inner iteration. Discretizations based on weighted finite elements and on Grünwald type schemes are described in detail, including assembly, boundary conditions, complexity, and memory. Two model problems, a stationary fractional Poisson problem and a space time fractional diffusion problem, are treated from the continuous level down to matrices.

math.NA↗

arXiv2026

Quantum matrix arithmetics with Hamiltonian evolution

The efficient implementation of matrix arithmetic operations underpins the speedups of many quantum algorithms. We develop a suite of methods to perform matrix arithmetics -- with the result encoded in the off-diagonal blocks of a Hamiltonian -- using Hamiltonian evolutions of input operators. We show how to maintain this $\textit{Hamiltonian block encoding}$, so that matrix operations can be composed one after another, and the entire quantum computation takes $\leq 2$ ancilla qubits. We achieve this for matrix multiplication, matrix addition, matrix inversion, Hermitian conjugation, fractional scaling, integer scaling, complex phase scaling, as well as singular value transformation for both odd and even polynomials. We also present an overlap estimation algorithm to extract classical properties of Hamiltonian block encoded operators, analogous to the well known Hadamard test, at no extra cost of qubit. Our Hamiltonian matrix multiplication uses the Lie group commutator product formula and its higher-order generalizations due to Childs and Wiebe. Our Hamiltonian singular value transformation employs a dominated polynomial approximation, where the approximation holds within the domain of interest, while the constructed polynomial is upper bounded by the target function over the entire unit interval. We describe a circuit for simulating a class of sum-of-squares Hamiltonians, attaining a commutator scaling in step count, while leveraging the power of matrix arithmetics to reduce the cost of each simulation step. In particular, we apply this to the doubly factorized tensor hypercontracted Hamiltonians from recent studies of quantum chemistry, obtaining further improvements for initial states with a fixed number of particles. We achieve this with $1$ ancilla qubit.

quant-ph↗

arXiv2026

An Unconditionally Stable Explicit Robin-Robin Partitioned Scheme for Fluid-Structure Interaction

We propose a parallelizable partitioned (loosely coupled) scheme for fluid structure interaction (FSI) problems, specifically designed for efficient computation in modern engineering simulations. The FSI problem under consideration involves an incompressible viscous fluid, governed by the Navier--Stokes equations, with a thick linear elastic structure. The scheme adopts a Robin--Robin coupling condition, evaluating the right-hand side of the Robin boundary terms at each time step solely from the previous-step solutions. This explicit scheme allows the fluid and structure subproblems to be solved entirely independently within each time step, eliminating the need for staggered coupling or costly sub-iterations, and thus enabling their concurrent computation. Various numerical experiments demonstrate the stability, accuracy, and computational efficiency of the proposed approach, highlighting its potential for large-scale parallel FSI computations in engineering applications.

math.NA↗

arXiv2026

Galerkin-type time discretizations for parabolic and hyperbolic problems: stability and a priori error analysis

Galerkin-type time discretizations are variational time-stepping schemes in which the test and trial spaces consist of piecewise polynomials in time. This paper reviews the schemes from this class available in the literature for parabolic and hyperbolic problems. In particular, we focus on the historical difficulties in their stability and \emph{a priori} error analysis that have limited their applicability, especially for wave propagation problems. We discuss how the limitations of standard energy arguments are behind these theoretical gaps, and present a unified framework that yields stability bounds and \emph{a priori} error estimates in $L^{\infty}(0, T; \mathcal{X})$ norms. Crucially, the stability analysis relies on a suitable choice of nonstandard test functions. This approach is characterized by the absence of Grönwall estimates, applicability to arbitrary approximation degrees, reduced regularity assumptions, and robustness with respect to model parameters.

math.NA↗

arXiv2026

Adjoint DSMC Method for Spatially Inhomogeneous Boltzmann Equation with General Boundary Conditions

We develop adjoint Direct Simulation Monte Carlo (DSMC) formulations for the spatially inhomogeneous Boltzmann equation with periodic, specular reflecting, diffuse thermal, and prescribed inflow boundary conditions. Periodic and specular boundaries are treated using a pathwise particle adjoint conditional on the realized event history. For diffuse thermal boundaries, we introduce a randomized-time regularization of wall-crossing events and use score-function terms to differentiate the resulting boundary probabilities. Reparameterization of the outgoing half-Maxwellian samples provides sensitivities with respect to wall temperatures and tangential wall velocities. Prescribed inflow requires a different construction because perturbations of incoming particles affect subsequent cell populations, local collision frequencies, and collision schedules. We therefore derive an ensemble adjoint based on the locally linearized Boltzmann collision operator and evaluate boundary sensitivities using local inflow-source scores, with injection counts held fixed. For a scalar objective, the dominant adjoint cost is largely independent of the number of parameters. Numerical experiments for Maxwell molecules validate the formulations against centered finite differences for thermal, mixed thermal-specular, two-sided inflow, and high-Mach Couette-flow configurations. The results demonstrate consistent gradient estimates, Monte Carlo convergence, stability with respect to the thermal regularization parameter, and accurate sensitivity calculation in a regime with limited relative statistical noise.

math.NA↗

arXiv2026

Accurate Residues for Floating-Point Debugging

Floating-point arithmetic is error-prone and unintuitive. Floating-point debuggers instrument programs to monitor floating-point arithmetic at run time and flag numerical issues. They estimate residues, i.e., the difference between actual floating-point and ideal real values, for every floating-point value in the program. Prior work explores various approaches for computing these residues accurately and efficiently. Unfortunately, the most efficient methods, based on "error-free transformations", have a high rate of false reports, while the most accurate methods, based on high-precision arithmetic, are very slow. This paper builds on error-free-transformations-based approaches and aims to improve their accuracy while preserving efficiency. To more accurately compute residues, this paper divides residue computation into two steps (rounding error computation and residue function evaluation) and shows how to perform each step accurately via careful improvements to the current state of the art. We evaluate on 44 large scientific computing workloads, focusing on the 14 benchmarks where prior tools produce false reports: our approach eliminates false reports on 10 benchmarks and substantially reduces them on the remaining 3 benchmarks. Moreover, complex numerical issues require additional care due to absorption, where two machine-precision residues cannot both be computed accurately in a single execution. This paper introduces residue override, which re-executes the program multiple times, computing different residues in different executions and assembling a final "patchwork" execution. We evaluate on 169 standard benchmarks drawn from numerical analysis papers and textbooks, requiring only 3.6 re-executions on average. Among 34 benchmarks with false reports in the initial run, residue override is triggered on 29 of them and reduces false reports on 25 of them, averaging 7.1 re-executions.

cs.MS↗

arXiv2026

Cut Finite Element Methods for Convection-Diffusion in Mixed-Dimensional Domains

We develop a cut finite element method (CutFEM) for convection--diffusion problems posed on mixed-dimensional domains, i.e., unions of manifolds of different dimensions arranged in a hierarchical structure where lower-dimensional components form parts of the boundaries of higher-dimensional ones. Such domains arise, for instance, in the modeling of fractured porous media with intersecting fractures. The model problem is formulated in a compact abstract form using mixed-dimensional directional derivative and divergence operators, which allows the problem to be expressed in a way that closely resembles the classical convection--diffusion equation. The proposed CutFEM is based on a fixed background mesh that does not conform to the geometry, with each manifold component represented through its associated active mesh. The method employs continuous piecewise linear elements together with weak enforcement of coupling conditions and suitable stabilization. We prove a priori energy norm error estimates under a global uniform-diffusion assumption, with corresponding extensions to solutions of reduced regularity $u\in H^s$, $1\le s<2$, and derive conditional estimates for the globally pure-convection case. Partially degenerate configurations, in which diffusion is present only on selected components, are explored numerically. The experiments report convergence in both the energy and $L^2$-norms and illustrate the performance of the method.

math.NA↗

arXiv2026

Universal Approximation of Nonlinear Operators and Their Derivatives

We show that Universal Approximation (UA) of nonlinear operators and their derivatives via Operator Learning (OL) architectures fails in ${C^k_F}$ (Fréchet) compact-open topologies and in Fréchet--Sobolev norms (i.e. under operator norms). We solve this obstruction by restoring UA in natural weaker topologies: $C^k_B$ (Bastiani) compact-open topologies and (novel) weighted Bastiani--Sobolev spaces for general finite input measures. In full Banach-space generality, these are the first complete generalizations of the corresponding influential classical results in [Hornik, 1991] to infinite-dimensional spaces and OL. Based on our UATs, we formulate Bastiani--Sobolev training in DIOL. These results launch Derivative-Informed Operator Learning (DIOL) (i.e. learning nonlinear operators and their derivatives) on general Banach spaces. We parameterize nonlinear operators via Encoder-Decoder Architectures, classical OL architectures available in general Banach spaces; these include DeepONets, Deep-H-ONets, and PCA-Nets, which our UATs cover. A key mathematical result is that our new weighted Bastiani--Sobolev spaces generalize classical Gaussian (Malliavin) Sobolev spaces on Banach spaces. Open frontiers where DIOL and our UATs find applications are: high-order accuracy in OL; fast constrained optimization in Banach spaces (e.g. optimal control of PDEs, inverse problems) via Learn-Then-Optimize; numerical methods for infinite-dimensional PDEs (e.g. HJB PDEs on Banach spaces from infinite-dimensional optimal control via Optimize-Then-Learn, such as optimal control of PDEs, SPDEs, path-dependent systems, partially observed systems, mean-field control).

cs.LG↗

arXiv2026

On the Application of Hybrid Mixed Domain Decomposition Methods to Permanent Magnet Synchronous Machines

In this work, we study the application of a hybrid mixed domain decomposition(HMDD) method for the rotor-stator coupling of a permanent magnet synchronous machine. For this, we derive a variational formulation on the electric machine inspired by hybridized discontinuous Galerkin methods using a mixed magnetostatics problem, an affine material law and boundary conditions respecting the symmetry of the motor. We are then able to locate the resulting finite element method within the HMDD framework. This enables us naturally to transfer the well-posedness results and error estimates for the HMDD method to the finite element method considered in this work. Lastly, as a proof of concept, we consider an academic example and compare the resulting magnetic flux density and potential lines to their counterparts obtained by a well-established in-house code using iso-geometric analysis.

cs.CE↗

arXiv2026

Variation Spaces for Encoder--Decoder Neural Operators: Approximation and Generalization

Inspired by the function-space theory of neural networks, we formulate and analyze a variation space for nonlinear operators between Hilbert spaces, defined through vector-valued Borel measures of bounded variation. We characterize its unit ball as the closed convex hull of a vector-valued single-neuron dictionary in Bochner spaces. For the ReLU activation, the bounded linear operators in this space are precisely the Schatten-$1$ operators, with equivalent norms. For operators in this space, we establish encoder--decoder approximation bounds in the Bochner $L^q$-norm, where the error decomposes into input and output encoding errors and a finite-width term of order $N^{-1/2}$. Under sub-Gaussian assumptions on the input and noise, we further derive high-probability generalization bounds for empirical least squares over path-norm-constrained encoder--decoder networks; the finite-sample contribution to the squared prediction error is of order $K^{-1/2}$ up to logarithmic factors. The finite-width and finite-sample constants are independent of the encoding dimensions and bases, with the latter also independent of the network width. When the encoding errors decay algebraically, these bounds yield algebraic approximation and learning rates, in contrast to the complexity barriers for Lipschitz and Fréchet differentiable operator classes.

stat.ML↗

arXiv2026

Analysis, thermodynamics, and a numerical solver for a pressure-temperature equilibrium closure of the four-equation model

We analyze an often used closure model for multi-material hydrodynamics where pressure-temperature equilibrium (PTE) is assumed for every state; emphasis is placed on tabular equations of state. This multi-material model is often referred to as the four-equation model. The identification of the admissible set is presented and is proven to be convex, setting the foundation for development of invariant-domain preserving methods for this model. A novel numerical method is presented for solving the highly nonlinear system for the equilibrated pressure and temperature with an arbitrary number of materials. This new method is compared with some traditional iterative solvers through a collection of different tests. Additionally, we provide a detailed analysis of the thermodynamics of the mixture model for general equations of state and prove existence and uniqueness of the pressure-temperature equilibrium solution under some thermodynamic assumptions.

math.NA↗

arXiv2026

Fourier--Hankel Moment Recovery in Acoustic Scattering: Multichannel Stabilization and Radial Interface Resolution

We study direct recovery of visible phase centers and concentric radial interfaces from full-aperture acoustic far-field data under the Born approximation. Two complementary moment structures are extracted from the two-angle Fourier matrix. At fixed positive total Fourier order, the nonnegative-order channels share the same leading phase-center moment, and a generalized least-squares combination yields an exact variance gain over a single Fourier row. The corresponding Hankel rank and shifted pencil recover distinct phase centers. Signed moments reveal off-center cavities but become degenerate when material and cavity centers coincide. Zero-total-order coefficients retain complementary radial Bessel moments. For a piecewise-constant radial average, low-frequency extrapolation produces a second finite exponential sequence whose nodes are the squared interface radii. We establish rank and perturbation results for both reductions. Numerical experiments verify multichannel stabilization, concentric-cavity resolution, and recovery of multiple radial interfaces. A final full-wave Helmholtz experiment, generated without the Born substitution, assesses the Born-derived reconstruction under model mismatch and shows how nonlinear scattering eventually appears as an additional Hankel tail.

math.NA↗

arXiv2026

Computational Oncology of Chemotaxis-Driven Tumour--Immune Spatial Patterning and Stability

We develop a reaction--diffusion--chemotaxis model for spatial tumour--immune--chemokine dynamics that couples logistic tumour growth, immune-mediated killing, chemokine-dependent immune recruitment, chemotactic migration, and signal production. For the nondimensional system, we establish local classical solvability, nonnegativity, a uniform tumour-density bound, and global mass estimates for the immune and chemokine components. The tumour-free equilibrium is stable precisely when the baseline immune-control index satisfies $σ_0/δ>1$, whereas positive homogeneous coexistence is characterized by a scalar nonlinear equation. Linearization in the Neumann Laplacian eigenbasis yields a mode-dependent cubic dispersion relation, showing that chemotaxis does not alter the tumour-invasion threshold but can destabilize homogeneous coexistence through a finite-wavelength oscillatory instability above a critical sensitivity $ξ_c$. A conservative finite-volume discretization with upwind chemotactic fluxes and implicit backward differentiation formula time integration is used to test these predictions. Numerical experiments recover the analytical equilibria and growth rates, identify the dominant unstable mode, reproduce the transition to spatial heterogeneity, and quantify the effects of immune recruitment, decay, and diffusion on the stability boundary. Grid-refinement, mass-balance, residual, and nonnegativity diagnostics support the computational reliability of the results.

math.AP↗

arXiv2026

Windowed thinning and query complexity for the bouncy particle and Zigzag samplers

Let $μ(d x)\propto e^{-U(x)} d x$ on $\R^d$, where $U$ is $m$-strongly convex and $L$-smooth, and denote by $κ=L/m$ the condition number. We consider windowed thinning, an exact simulation method for the bouncy particle sampler and the coordinate Zigzag process. The method divides a trajectory into deterministic windows and uses a gradient evaluation at the beginning of each window to construct a tractable local envelope for the event rate. Combining this construction with quantitative mixing estimates and finite-time bounds on the expected numbers of bounces and flips yields query complexity guarantees from a Gaussian cold start. For total-variation error $\varepsilon$, the expected query counts are $O(κ^{1/2}d\,(d\logκ+\log\frac1\varepsilon))$ gradient queries for the bouncy particle sampler and $O(κd^{1/4}(d\logκ+\log\frac1\varepsilon))$ full-gradient equivalents for Zigzag, where $d$ coordinate-partial queries count as one equivalent.

math.NA↗

arXiv2026

The Frame Kernel Method for Multiscale Operator Learning

We present a natively multiscale operator learning method for the surrogate modeling of (numerical solvers for) multiscale partial differential equations (PDEs). The primary novelty of our method lies in a novel multiscale kernel frame function approximation technique. Leveraging this new kernel frame technique, we cast the operator learning problem as one of learning frame coefficients of output functions as a function of frame coefficients of input functions. The generalization step then automatically allows for a multiscale decomposition of the output functions. Our method is applicable to both tensor-product grids and point clouds. We present interpolation proofs, error estimates, and numerical convergence rates for our frame approximation. We the demonstrate the applicability of our method for the surrogate modeling of inherently multiscale PDEs. The new multiscale frame kernel method is significantly more accurate than popular neural operators on challenging problems from the literature, while simultaneously admitting an a posteriori multiscale decomposition upon generalization.

cs.LG↗

Compare source metadata on this page

Work	Published	Source identifier	Source
Greedy recursion parameter selection for one-way spatial integration of hyperbolic equations	2026-09-01	2506.02320	arxiv
Interpolation in Polynomial Spaces of p-Degree	2026-09-01	2507.13640	arxiv
The Bayesian SIAC filter	2026-09-01	2509.14771	arxiv
A Variational Method for Conformable Fractional Equations Using Rank-One Updates	2026-09-01	2510.03778	arxiv
Quantum matrix arithmetics with Hamiltonian evolution	2026-09-01	2510.06316	arxiv
An Unconditionally Stable Explicit Robin-Robin Partitioned Scheme for Fluid-Structure Interaction	2026-09-01	2510.13096	arxiv
Galerkin-type time discretizations for parabolic and hyperbolic problems: stability and a priori error analysis	2026-09-01	2601.19828	arxiv
Adjoint DSMC Method for Spatially Inhomogeneous Boltzmann Equation with General Boundary Conditions	2026-09-01	2603.20946	arxiv
Accurate Residues for Floating-Point Debugging	2026-09-01	2604.06258	arxiv
Cut Finite Element Methods for Convection-Diffusion in Mixed-Dimensional Domains	2026-09-01	2604.07162	arxiv
Universal Approximation of Nonlinear Operators and Their Derivatives	2026-10-01	2605.15285	arxiv
On the Application of Hybrid Mixed Domain Decomposition Methods to Permanent Magnet Synchronous Machines	2026-09-01	2605.31032	arxiv
Variation Spaces for Encoder--Decoder Neural Operators: Approximation and Generalization	2026-09-01	2606.01244	arxiv
Analysis, thermodynamics, and a numerical solver for a pressure-temperature equilibrium closure of the four-equation model	2026-09-01	2606.27726	arxiv
Fourier--Hankel Moment Recovery in Acoustic Scattering: Multichannel Stabilization and Radial Interface Resolution	2026-09-01	2606.29678	arxiv
Computational Oncology of Chemotaxis-Driven Tumour--Immune Spatial Patterning and Stability	2026-09-01	2607.03813	arxiv
Windowed thinning and query complexity for the bouncy particle and Zigzag samplers	2026-09-01	2607.28413	arxiv
The Frame Kernel Method for Multiscale Operator Learning	2026-09-01	2608.25084	arxiv

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