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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

Rate of convergence of a fully discrete structure-preserving midpoint scheme for the stochastic Landau--Lifshitz--Gilbert equation

The stochastic Landau--Lifshitz--Gilbert (sLLG) equation is a strongly nonlinear stochastic PDE with a non-convex pointwise constraint arising in the theory of micromagnetics. We analyse a fully discrete, structure-preserving finite element approximation of the sLLG equation with coloured multiplicative Stratonovich noise on a bounded interval. The method utilises continuous piecewise affine finite elements, mass lumping, and midpoint time discretisation to preserve the unit-length constraint exactly at the finite element nodes. Under suitable regularity assumptions on the initial data and the noise, we establish uniform higher-moment stability and develop an error analysis for the scheme. The analysis exploits the geometric structure of the equation and the stochastic midpoint discretisation. For every $γ\in(0,\frac12)$, we prove first-order spatial convergence and temporal convergence of order $γ$ in the natural discrete energy norm, locally in mean square on events of arbitrarily large probability and, consequently, in probability. To the best of our knowledge, this is the first convergence-rate result for a fully discrete structure-preserving finite element scheme solving the stochastic Landau--Lifshitz--Gilbert equation.

math.NA↗

arXiv2026

Improved Inclusion-Exclusion Eigenvalue Localization Sets for Matrices and Matrix Polynomials

Recent developments have shown that eigenvalue localization regions can often be significantly sharpened by identifying and excluding subsets that are guaranteed to contain no eigenvalues, leading to the concept of inclusion-exclusion localization sets. In this paper, we introduce a new family of inclusion-exclusion eigenvalue localization sets based on the S-strict diagonal dominance framework and the corresponding CKV localization regions. For an arbitrary nonempty subset S of the index set, we construct novel exclusion regions and derive the associated S-SDD inclusion-exclusion localization sets. We prove that every eigenvalue of a complex matrix belongs to the proposed regions and define a refined localization set obtained by intersecting the corresponding inclusion-exclusion regions over all nonempty subsets S. The proposed framework unifies and extends several existing results. In particular, we show that the classical Gershgorin inclusion-exclusion localization set and the Dashnic-Zusmanovich inclusion-exclusion localization set arise naturally as special cases of our construction. Consequently, the new localization set provides, in general, a sharper eigenvalue inclusion region than both of these previously known localization regions. The theory is further extended to matrix polynomials, yielding corresponding inclusion-exclusion localization results for polynomial eigenvalue problems. Several numerical examples illustrate the effectiveness of the proposed approach and demonstrate the improvement achieved over existing localization sets.

math.NA↗

arXiv2026

Introductory Notes on Learning$^2$

Although machine learning can be used to predict the evolution of physical systems from data, a formulation that learns only the system state at each time leaves the temporal and dynamical structure of the solution to be resolved within a broad hypothesis space. We introduce Learning$^2$, a representation-level framework that structures this space by coupling a primary representation to a second representation through a known physical transformation. The resulting cross-representation constraint restricts the effective hypothesis space and provides an ante-hoc, physically interpretable criterion for excluding solutions that satisfy the primary representation alone. We instantiate Learning$^2$ through EuLaNet, an Eulerian--Lagrangian representation for fluid dynamics. Given the velocity state $u(\mathbf{x},t)$, EuLaNet constructs its induced Lagrangian flow map $X(\mathbf{a},t)$ through $\dot{X}(\mathbf{a},t)=u(X(\mathbf{a},t),t)$, from which material transport and finite-time deformation are derived. The resulting representation couples the predicted state to the dynamical consequences it induces, providing a second consistency criterion beyond state-level agreement. We formalize this construction through an effective hypothesis space $\mathcal{H}_{L^2}\subseteq\mathcal{H}$ and define the conditions under which a consequence representation provides discriminative constraints on candidate solutions. EuLaNet is implemented as a model-independent representation module, separating the physical constraint from the downstream learning architecture. This construction provides an ante-hoc mechanism for physically interpretable constraint in scientific learning and offers a basis for developing and evaluating broader classes of Learning$^2$ architectures. The implementation is open-sourced to support the development and extension of the architecture across scientific domains.

math.NA↗

arXiv2026

Halley's Method for Rectangular Matrix Variables and the Matrix Schwarzian Derivative

Alefeld (1981) recast Halley's cubic convergence as Newton's method on $g=f/\sqrt{f'}$, linked to the Schwarzian derivative. We generalize this to matrix gradient fields $F=\nablaϕ$ ($X\in\R^{m\times n}$) via a matrix Schwarzian derivative interpreted through information geometry ($α$-connections). Simplifying Palmore (1994), we construct this operator square-root-free derivative from the third Fréchet derivative of the Newton map. Our main theorem proves local cubic convergence with an explicit error constant, without self-adjointness, commutativity, or gradient-field assumptions. A matrix-free algorithm (Hessian-vector products only) validates the theory. We contrast this with a power-Newton family (whose naive matrix extension fails) and the matrix Laguerre family (which requires an operator square root). A case study on the Oja-type system $\dot X=AXB-XBX^TAX$ reveals that convergence depends on target eigenvalue gaps, cubic gains grow with ill-conditioning, and 70-digit tests confirm exact theoretical orders alongside a working-precision accuracy budget. Finally, we note coupled formulations excel primarily when sequential deflation is inapplicable.

math.NA↗

arXiv2026

Dilation-Covariant Hankel Pencils for Multiscale Recovery of Sparse Mellin Spectra

We study sparse Mellin spectral recovery from geometric samples. Dilation covariance is shown to generate the Hankel structure directly, rather than only after reduction to a classical exponential-sum model. For complex exponents, incommensurate sampling scales remove the logarithmic aliasing that persists at a single scale. A second recovery channel is obtained from the minimal Hankel pencil: contour winding counts spectral nodes regionally, while Rouché-type bounds certify isolated nodes and unresolved clusters under noise. The associated contour margin has an explicit degeneration rate as the sampling ratio approaches one. Numerical experiments show that the auxiliary scale improves both identifiability and resolution, and that certified contour counts can remain reliable beyond the regime of accurate pointwise recovery.

math.NA↗

arXiv2026

A discontinuous Galerkin approximation of the Dean--Kawasaki equation

We introduce and analyse an arbitrary order spatial discontinuous Galerkin (dG) method for the Dean--Kawasaki equation, a highly singular SPDE modelling density fluctuations of $N$ diffusing particles in the regime of large particle number $N \gg 1$. Our starting point is a general procedure for discretising multiplicative, divergence-form noise on finite element spaces whilst preserving its cross-variation structure at the discrete level; the construction is explicit, elementwise, applies to continuous and discontinuous spaces alike, and extends to general mobilities. Using it, we prove weak error estimates of order $O(h^p)$ between fluctuations of the semi-discrete scheme and those of the underlying particle system, together with a correction that is exponentially small in the scaling regime $Nh^d \gg 1$ and arises because the scheme does not preserve positivity. The resulting method is locally and globally conservative and applies on unstructured simplicial meshes. Quantitative numerical experiments support the analysis, and further experiments illustrate the method beyond the scope of the theory: indicator function observables, external and interaction potentials, convection-dominated regimes and reflecting boundary conditions.

math.NA↗

arXiv2026

Simulating the Marginal Green Contribution of AI Modules in a Smart-Agriculture Platform: Evidence from Two Monte Carlo Experiments

Smart agriculture platforms usually bundle AI diagnosis, IoT sensing and decision push into a single package, so the green benefit attributable to each component remains unclear and resource-allocation decisions lack quantitative evidence. Building on a previous platform-level Monte Carlo assessment, this paper makes the components explicit and runs two controlled simulation experiments. Experiment 1 follows the chain from AI capability to farmer behavior to agrochemical input reduction, modeling pesticide/fertilizer reduction as avoidable blind-application share times prescription effectiveness times decision-touch coverage times adoption rate, and compares an experienced-extension mode with the AI mode: the probability of reaching 20% pesticide reduction is essentially zero in the extension mode but 20.7% at baseline, up to 49% with diagnosis accuracy 0.95 and adoption 0.85 under AI; the probability of 15% fertilizer reduction rises from near zero to 52.0%. Experiment 2 compares current practice (P0), IoT engineering retrofit (P1), and P1 plus AI irrigation scheduling (P2): median aggregate water saving rises from 7.8% (P0) to 11.0% (P1) and 16.0% (P2), with AI adding 5.0 percentage points beyond engineering; paddy CH4 reduction reaches 30.5% under AI scheduling versus 19.8% under manual operation, and the rice irrigation-methane subsystem carbon intensity declines 27.9%. Sensitivity analyses of both experiments consistently indicate that the primary bottleneck for meeting green targets is farmer adoption rather than algorithm accuracy, and that AI data fusion is robust to soil-moisture sensing errors. This work provides a reproducible simulation framework for component-level green-value evaluation and promotion-strategy optimization of smart agriculture platforms.

cs.AI↗

arXiv2026

Invariant-region-preserving high-order schemes for a model of reactive sedimentation

Reactive sedimentation of mixtures composed by biological and inert solid matter including liquid substrates is modelled by a system of convection-diffusion-reaction equations. In this process, the small solid particles dispersed in the viscous fluid settle due to gravity while simultaneous chemical reactions take place between solid and liquid components (substrates). Among the applications of reactive sedimentation processes, the principal use is in the simulation and control of secondary settling tanks (SSTs) in water resource recovery facilities (WRRFs). The governing equations include a nonlinear strongly degenerate parabolic equation for the total concentration of solids, and transport equations including reaction terms for the solid components and substrates. To accurately approximate the model equations, a high-order finite volume scheme is developed using polynomial reconstructions. Two reconstruction methods are used, a second-order monotonic upstream-centered scheme (MUSCL), and a third-order central weighted essentially non-oscillatory (CWENO3) method. In addition, for the time discretization, we employ strong stability preserving Runge-Kutta (SSPRK) schemes of second- and third-order. The CWENO3 scheme is complemented with limiters, and in the case of zero diffusion, the designed high-order schemes are shown to preserve an invariant-region property. This property is particularly relevant as it ensures that physically relevant numerical solutions are obtained at each time iteration. In addition, high-order approximations for the diffusion terms are proposed to approximate the complete model equations. Simulations of denitrification in SSTs under a continuously operated regime in WRRFs demonstrate the performance of the model and its discretization. Finally, convergence tests show the order of accuracy of the developed scheme and the invariant-region preservation.

math.NA↗

arXiv2026

Numerical approximation to the invariant measure of McKean-Vlasov stochastic differential equations

Inspired by the stochastic particle method, this paper develops an easily implementable explicit scheme for McKean-Vlasov stochastic differential equations (MV-SDEs) with superlinear growth coefficients. We prove that the numerical solution of the interacting particle system (IPS) attains the optimal uniform-in-time strong convergence rate of order 1/2, and that it faithfully captures the long-term dynamics of MV-SDEs, including moment boundedness, stability, and ergodicity. In particular, the existence and uniqueness of an exchangeable numerical invariant probability measure for the IPS are established via an appropriately constructed operator semigroup. Concerning the approximation of the invariant measure, we derive a non-asymptotic error bound between the distribution of the one-particle numerical solution and the marginal distribution of the IPS's invariant measure; By the uniform-in-time propagation of chaos, we further obtain an asymptotic error bound between the one-particle marginal of the IPS's numerical invariant measure and the exact invariant measure of the MV-SDE. Numerical experiments are provided to validate the theoretical results.

math.PR↗

arXiv2026

Multi-Product Hamiltonian Simulation with Explicit Commutator Scaling

The well-conditioned multi-product formula (MPF), proposed by Low-Kliuchnikov-Wiebe (2019), is a high-order, time-independent Hamiltonian simulation algorithm that implements a linear combination of low-order product formulas. Prior work established its well-conditioned algorithmic construction and near-optimal time and precision dependence, but did not simultaneously provide an explicit error bound in terms of nested commutators, which has left its practical advantage uncertain. In this work, we provide a rigorous complexity analysis of the well-conditioned MPF, explicitly establishing both commutator scaling and near-optimal dependence on time and precision through a rigorous Baker-Campbell-Hausdorff (BCH) and variation-of-parameters-based error analysis. Using our improved complexity estimates, we present several physically relevant applications where the second-order-based MPF asymptotically outperforms the second-order product formula in all three aspects of system size, time, and precision. We further demonstrate that, while MPF yields worse scalings than the best post-Trotter methods in most applications, there exist certain parameter regimes in the nonlocal fast-transform Bogoliubov-de Gennes model where MPF can outperform both leading post-Trotter methods and product formulas with fixed or adaptively optimized order.

quant-ph↗

arXiv2026

Tangential-Normal Decompositions of the Second Family of Finite Element Differential Forms

This paper introduces a novel tangential-normal ($t$-$n$) decomposition for the second family of finite element differential forms, presenting a new framework for constructing bases in finite element exterior calculus. The main contribution is the development of a $t$-$n$ basis in which degrees of freedom and shape functions are explicitly dual, a property that streamlines stiffness matrix assembly and enhances the efficiency of interpolation and numerical integration. Additionally, the integration of the well-documented Lagrange element basis supports practical implementation of finite element differential forms in applications.

math.NA↗

arXiv2026

Convergence of a continuous Galerkin method for the Biot-Allard poroelasticity system

We study a space-time finite element method for a system of poromechanics with memory effects that are modeled by a convolution integral. In the literature, the system is referred to as the Biot-Allard model. We recast the model as a first-order system in time, where the memory effects are transformed into an auxiliary differential equation. This allows for a computationally efficient numerical scheme. The system is discretized by continuous Galerkin methods in time and equal-order finite element methods in space. An optimal order error estimate is proved for the norm of the first-order energy of the unknowns of the system. The estimate is confirmed by numerical experiments.

math.NA↗

arXiv2026

A Preliminary Study on the Dimensional Stability Classification of Polynomial Spline Spaces over T-meshes

This paper studies dimensional stability of polynomial spline spaces over T-meshes with highest-order smoothness. Dimensional stability is defined as the invariance of the dimension of the spline space over structurally isomorphic T-meshes, where a structurally isomorphic map preserves edge intersections and the mutual positions of all edges. Absolute stability, a stronger notion, is introduced via structurally similar maps that depend solely on the topology of T-connected components. It is proved that every T-connected component admits a unique decomposition into a diagonalizable part and a completely non-diagonalizable component (CNDC). This reduces stability to the rank stability of the conformality matrix associated with the CNDC. For diagonalizable T-meshes, the conformality vector space decomposes as a direct sum over independent T $l$-edges, supporting basis function construction per T $l$-edge. These results provide a systematic framework for classifying dimensional stability and a rigorous theoretical foundation for constructing basis functions of spline spaces over T-meshes.

math.NA↗

arXiv2026

Convergent adaptive iterative schemes for solving multi-physics problems

In this paper, we derive a practical, general framework for creating adaptive iterative (linearization or splitting) algorithms to solve multi-physics problems. This means that, given an iterative method, we derive \textit{a posteriori} estimators to predict the success or failure of the method. Based on these estimators, we propose adaptive algorithms, including adaptively switching between methods, adaptive time-stepping methods, and the adaptive tuning of stabilization parameters. We apply this framework to two-phase flow in porous media, surfactant transport in porous media, and quasi-static poroelasticity.

math.NA↗

arXiv2026

Adapting the Lanczos algorithm to matrices with almost continuous spectra

We consider approximating $B^T(A+sI)^{-1}B$, where $A\in\mathbb{R}^{n\times n}$ is large, symmetric positive definite, and $B\in\mathbb{R}^{n\times p}$ with $p\ll n$. We focus on the case where $A$ has an almost continuous (dense) spectrum: its eigenvalues fill one or more intervals so densely that Krylov methods cannot resolve them individually. Our target is computing multiple-input multiple-output transfer functions arising from large-scale discretizations of problems with continuous spectral measures, such as linear time-invariant PDEs on unbounded domains. Traditional Krylov methods, such as Lanczos or conjugate gradients, resolve individual eigenvalues of a dense discretization while ignoring the underlying continuous spectral measure these points approximate. We argue it is more efficient to model the operator's inherent branch cut than to exhaustively resolve the artificial point spectrum induced by discretization. We adapt the framework of Kreĭn-Nudelman semi-infinite strings to the block Lanczos algorithm, with parameters chosen adaptively by maximizing the energy absorbed at the string termination relative to the energy stored in the string. This yields a low-rank modification to the block Lanczos matrix, dependent on $\sqrt{s}$, at an additional $O(n)$ cost. We show significant error reductions for large-scale self-adjoint PDE discretizations in unbounded domains, including two- and three-dimensional Maxwell's equations in diffusive regimes. The method is especially advantageous for computing state-space solutions for wave propagation, specifically for 2D wave and 3D Maxwell operators. Replacing the conventional Lanczos spectral decomposition with the continuous Kreĭn-Nudelman spectrum yields a qualitative improvement in finite-difference approximations, transforming standing-wave artifacts into outgoing propagating waves.

math.NA↗

arXiv2026

Hölder-Logarithmic Stability and Convergence Rates for an Inverse Random Source Problem

In this paper, we investigate an inverse random source problem concerned with recovering the strength of a random, uncorrelated acoustic source from correlation measurements of emitted time-harmonic acoustic waves. Such problems arise in applications including aeroacoustics and seismic imaging. Unlike their deterministic counterparts, inverse random source problems are known to be uniquely solvable in the absence of noise. Nevertheless, due to their inherent ill-posedness, regularization is required to stably reconstruct the source strength. We derive conditional Hölder-logarithmic stability estimates under Sobolev smoothness assumptions by employing complex geometrical optics solutions. Moreover, by establishing a variational source condition, we obtain Hölder-logarithmic convergence rates for spectral regularization methods. At fixed frequency, the exponents in the logarithmic stability and convergence estimates grow unboundedly as the Sobolev regularity of the source increases. Finally, we present numerical experiments supporting our theoretical findings.

math.NA↗

arXiv2026

A counterexample to Kusner's conjecture on equilateral sets

We disprove Kusner's 1983 conjecture that every equilateral set in $\ell_p^n$ with $2 57$. This is the first equilateral set of more than $n+1$ points in $\ell_p^n$ for any finite $p\ge2$. The construction persists on an open interval of exponents around $5$; since Ge, Xu and Zhou recently proved the conjecture for $2\le p\le4$, the infimum of exponents at which it fails lies in $[4,5)$. The configuration is the unique solution of an explicit polynomial system with rational coefficients in a rational box, established in exact arithmetic.

math.MG↗

arXiv2026

Robust High-Order Projector-Splitting Integrators

We develop a general framework for constructing robust high-order projector-splitting integrators for dynamical low-rank approximation. For a prescribed matrix increment, we show that the standard K-S-L projector-splitting step is equivalent to a reduced K-L step, thereby eliminating the explicit backward S-step. We then establish a central relaxed exactness property of the standard projector-splitting integrator: the rank-$r$ approximation inherits the accuracy of the numerical matrix increment, with an error bound independent of small singular values. The resulting schemes evolve fixed rank-$r$ factors and require neither basis augmentation nor rank truncation. As concrete examples, we combine the framework with selected second- and third-order Runge--Kutta methods to obtain robust high-order projector-splitting integrators. Numerical experiments confirm the predicted uniform convergence rates with respect to small singular values.

math.NA↗

Compare source metadata on this page

Work	Published	Source identifier	Source
Rate of convergence of a fully discrete structure-preserving midpoint scheme for the stochastic Landau--Lifshitz--Gilbert equation	2026-09-06	2609.06459	arxiv
Improved Inclusion-Exclusion Eigenvalue Localization Sets for Matrices and Matrix Polynomials	2026-09-06	2609.06538	arxiv
Introductory Notes on Learning$^2$	2026-09-06	2609.06546	arxiv
Halley's Method for Rectangular Matrix Variables and the Matrix Schwarzian Derivative	2026-09-06	2609.06592	arxiv
Dilation-Covariant Hankel Pencils for Multiscale Recovery of Sparse Mellin Spectra	2026-09-06	2609.06661	arxiv
A discontinuous Galerkin approximation of the Dean--Kawasaki equation	2026-09-06	2609.06681	arxiv
Simulating the Marginal Green Contribution of AI Modules in a Smart-Agriculture Platform: Evidence from Two Monte Carlo Experiments	2026-09-06	2609.06740	arxiv
Invariant-region-preserving high-order schemes for a model of reactive sedimentation	2026-09-06	2609.06846	arxiv
Numerical approximation to the invariant measure of McKean-Vlasov stochastic differential equations	2026-09-05	2401.02878	arxiv
Multi-Product Hamiltonian Simulation with Explicit Commutator Scaling	2026-09-05	2403.08922	arxiv
Tangential-Normal Decompositions of the Second Family of Finite Element Differential Forms	2026-09-05	2410.20408	arxiv
Convergence of a continuous Galerkin method for the Biot-Allard poroelasticity system	2026-09-05	2504.06763	arxiv
A Preliminary Study on the Dimensional Stability Classification of Polynomial Spline Spaces over T-meshes	2026-09-05	2508.06217	arxiv
Convergent adaptive iterative schemes for solving multi-physics problems	2026-09-05	2601.16640	arxiv
Adapting the Lanczos algorithm to matrices with almost continuous spectra	2026-09-05	2602.11449	arxiv
Hölder-Logarithmic Stability and Convergence Rates for an Inverse Random Source Problem	2026-09-05	2602.20822	arxiv
A counterexample to Kusner's conjecture on equilateral sets	2026-09-05	2608.14013	arxiv
Robust High-Order Projector-Splitting Integrators	2026-09-05	2608.17157	arxiv

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