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arXiv2026

A counterexample to vertex interpolation by planar $C^1$ cubic splines

We construct a nondegenerate conforming straight-line triangulation of a polygonal disk on which some vertex data admit no continuously differentiable piecewise polynomial interpolant of total degree at most three. The triangulation has 41 vertices and 56 triangles, and the graph induced by its interior vertices is a tree with eight arms of length two. Every cubic $C^1$ spline on this triangulation satisfies an explicit linear relation with integer coefficients among its vertex values. We derive the relation by a weighted sum of Bernstein--Bézier smoothness equations and verify it using rational coordinates and weights. This disproves Alfeld's conjecture on vertex interpolation. The nonexistence proof does not require a matrix rank computation. A separate computation in exact arithmetic gives rank 40 for the vertex evaluation map and dimension 107 for the spline space, attaining the classical dimension lower bound for this triangulation.

math.NA↗

arXiv2026

Sharp approximation rates for shallow ReLU neuralnetworks on critical Besov classes

Let $\mathbb D$ be the normalized ridge dictionary generated by $\operatorname{ReLU}^k$ on a bounded Lipschitz domain $Ω\subset\mathbb R^d$. We determine the sharp algebraic rate of finite $n$-term approximation from $\mathbb D$ when the outer $\ell^1$ coefficient budget is independent of $n$. More precisely, let $d\ge3$, $k\in\mathbb N_+$, $0\le m\le k$, $0<p<1$, and $0<q\le1$. For the unit ball of the critical Besov space $B_{p,q}^{k+d/p}(Ω)$, measured in $H^m(Ω)$, the optimal algebraic exponent is \[ \min\left\{\frac{k-m+d/2}{d-1}, \frac{k-m+d/2+1/p-1/2}{d}\right\}. \] We prove two-sided estimates in the three regimes determined by the two branches of this minimum and give the corresponding logarithmic factors at and beyond the transition. The estimates coincide without logarithmic loss in the strict angular regime and at the transition when $q\le[1/2+(k-m+d/2)/(d-1)]^{-1}$. The upper estimate combines critical wavelet sparsity, localized Fourier--Radon representations, and a stable allocation of directions and biases. The lower estimates arise from two distinct obstructions, namely radial ridge approximation on the direction sphere and Gevrey localization in the joint direction--bias space. The critical smoothness is exactly the endpoint at which Besov regularity provides a uniform shallow-network variation bound without additional scale decay. The resulting representation exponent is strictly larger than the degree-limited exponent for fixed-degree isotropic finite elements under a parameter-count comparison. This last statement concerns best approximation, not training or computational complexity.

math.NA↗

arXiv2026

High-Order Structure-Preserving SBP Finite Difference Methods for the Vlasov-Maxwell System on Matrix-Free GPUs

In this paper, we present a high-order, stable summation-by-parts (SBP) finite difference method for solving the Vlasov-Maxwell system in a 2D2V phase space. Central SBP operators for the advection terms are not stable when the solution becomes non-smooth and fine-scale filamentary structures develop, as is typical in high-dimensional Vlasov-Maxwell simulations. To address this issue, the method is stabilized using high-order upwind SBP operators. High-order explicit Runge-Kutta methods are employed for time integration. We prove that the fully discrete scheme exactly conserves mass and preserves momentum up to truncation error. Furthermore, we present a matrix-free implementation of the method on modern GPU architectures. A range of challenging benchmark problems is solved to demonstrate the accuracy, robustness, and performance of the proposed scheme.

math.NA↗

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

Hyperelastic constitutive model discovery with differentiable finite elements and structure-preserving neural networks

The discovery of constitutive laws from experimentally accessible measurements is a central problem in nonlinear computational mechanics. Many data-driven constitutive identification approaches rely either on paired strain-stress data or on full-field displacement measurements, both of which are difficult to obtain in realistic three-dimensional settings. We present a differentiable finite element framework for the discovery of hyperelastic material laws from partial observations, including boundary-only displacement measurements and global reaction forces. The method embeds the nonlinear finite element equilibrium problem directly into the learning loop, so that candidate strain-energy densities are assessed through the deformation fields and reactions they induce. This formulation enforces mechanical equilibrium as a constraint and allows the loss function to be evaluated only at observed locations. To ensure physical admissibility and promote numerical solvability throughout training, the constitutive response is represented by Hyperelastic Neural Networks, a structure-preserving neural class that enforces residual energy and stress-free conditions, frame indifference, isotropic material symmetry, polyconvexity, coercivity, and controlled volumetric growth by construction. The resulting PDE-constrained learning problem is solved using a quasi-Newton strategy combined with continuation and solver-aware backtracking. Numerical experiments in two- and three-dimensional finite elasticity demonstrate accurate recovery of hyperelastic isotropic responses from boundary-only data, robustness to measurement noise, and generalization across geometries, loading conditions, and boundary conditions.

math.NA↗

arXiv2026

Strong convergence of finite element schemes for the stochastic Landau--Lifshitz--Bloch equation

The dynamics of magnetisation in a bounded ferromagnet in $\mathbb{R}^d$ ($d=1,2$) at high temperatures can be described by the stochastic Landau--Lifshitz--Bloch (sLLB) equation, which is a vector-valued quasilinear stochastic partial differential equation. In this paper, assuming adequate regularity of the initial data, we establish strong convergence in $L^2(Ω)$ of several semi-implicit and implicit fully discrete finite element schemes for the sLLB equation, together with explicit convergence rates. The analysis relies on localised error estimates and new exponential moment bounds for the exact solution. As a by-product, these moment bounds yield mean-square exponential stability of solutions and uniqueness of the invariant measure in one spatial dimension under a small noise assumption. We also sharpen existing convergence-in-probability results for the numerical schemes. Numerical experiments are presented to illustrate and support the theoretical findings.

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

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↗