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arXiv2026

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

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

eess.SP↗
arXiv2026

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

Nonconforming virtual element method for the Monge-Ampère equation

In this article, we develop the $C^1$-nonconforming $C^0$-conforming virtual element method (VEM) for the vanishing moment approximation of the second-order fully nonlinear Monge-Ampère equation in two dimensions. In the vanishing moment equation an artificial biharmonic term is introduced which produces a quasilinear fourth order problem. We derive optimal a priori error estimates in the $H^2$-, $H^1$- and $L^2$-norms for the virtual element method, and show the existence and uniqueness of the virtual element solution. We perform several numerical experiments to validate the convergence rate of the error with respect to the mesh size.

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

Convergence rates of randomly pivoted methods for low-rank approximation

Randomly pivoted Cholesky, QR, and LU are iterative algorithms that form structured low-rank approximations of a matrix by sampling columns, or rows and columns, from the residual. We give convergence rates for these methods that depend on the decay of the singular values of the original matrix. The rates hold, up to a constant, when pivots are drawn from an approximation to the exact distribution. This extra degree of freedom can be used to trade exact sampling for cheaper approximations. We use it to establish rates for variants of randomly pivoted LU, including one that picks pivots using a sketch of the input matrix.

math.NA↗
arXiv2026

Positivity loss in bandlimited spectral reproduction on spheres

How small can the positivity loss be for an $N$-bandlimited spectral operator that exactly reproduces all modes up to degree $L$? For spherical polynomial approximation on $\mathbb S^d$, we prove that the smallest possible excess of the uniform operator norm above 1, equivalently the least positivity loss, is of sharp order $\left({L}/{(N+1)}\right)^2$ when $1\le L< N$. The lower bound follows from a Fejér peak test and a concentration estimate for bandlimited kernels, while a matching upper bound is obtained by correcting a positive Jackson operator with a smooth filter. We illustrate the result in three settings. On the circle, taking $N=sL-1$, this determines the sharp order of the generalized-projection constant above 1 and identifies the gap between the $s^{-1}$ excess of delayed de la Vallée--Poussin means and the optimal $s^{-2}$ order. For filtered hyperinterpolation, whose operator norm has long been known to be uniformly bounded, we give a quantitative lower bound on its separation from the positivity threshold 1. Finally, we identify an operator-level obstruction to maximum principles.

math.NA↗
arXiv2026

Solving the Offline and Online Min-Max Problem of Non-smooth Submodular-Concave Functions: A Zeroth-Order Approach

We consider max-min and min-max problems with objective functions that are possibly non-smooth, submodular with respect to the minimiser and concave with respect to the maximiser. We investigate the performance of a zeroth-order method applied to this problem. The method is based on the subgradient of the Lovász extension of the objective function with respect to the minimiser and based on Gaussian smoothing to estimate the smoothed function gradient with respect to the maximiser. In expectation sense, we prove the convergence of the algorithm to an $ε$-saddle point in the offline case. Moreover, we show that, in the expectation sense, in the online setting, the algorithm achieves $O(\sqrt{N(1+\bar{P}_N)})$ online duality gap, where $N$ is the number of iterations and $\bar{P}_N$ is the path length of the sequence of optimal decisions. The complexity analysis and hyperparameter selection are presented for all the cases. The theoretical results are illustrated via numerical examples.

math.OC↗
arXiv2021

Least-squares spectral methods for ODE eigenvalue problems

We develop spectral methods for ODEs and operator eigenvalue problems that are based on a least-squares formulation of the problem. The key tool is a method for rectangular generalized eigenvalue problems, which we extend to quasimatrices and objects combining quasimatrices and matrices. The strength of the approach is its flexibility that lies in the quasimatrix formulation allowing the basis functions to be chosen arbitrarily (e.g. those obtained by solving nearby problems), and often giving high accuracy. We also show how our algorithm can easily be modified to solve problems with eigenvalue-dependent boundary conditions, and discuss reformulations as an integral equation, which often improves the accuracy.

math.NA↗
arXiv2026

An improved estimate of the intermediate internal energy in the energy-consistent HLLD scheme

The robustness of approximate Riemann solutions has been a crucial topic in computational magnetohydrodynamics, from both theoretical and practical perspectives. Recently, the widely used HLLD approximate Riemann solution was revised, becoming significantly more robust under strong magnetic fields. Yet, simplifications were needed as the compressible slow magnetoacoustic mode is not included in the HLLD scheme, and as a result non-physical density distribution has been found when having strong slow shocks. In this note, a simple fix is introduced for the estimated intermediate internal energy, providing robust results in several representative test cases, preserving pressure-positivity when the magnetic field is enhanced by a factor of 1000.

math.NA↗
arXiv2026

A structure-preserving implicit-explicit method for a hyperbolic approximation of fourth-order PDEs

We introduce a novel structure-preserving numerical method for a first-order hyperbolic approximation system, which approximates the solutions of general fourth-order partial differential equations. By employing an implicit-explicit (IMEX) splitting between the stiff and non-stiff terms, we rigorously prove that the proposed scheme is energy-consistent and positivity-preserving under a CFL-type condition. Furthermore, we show that the method remains robustly stable in asymptotic regimes, with a time-step restriction that is entirely independent of the relaxation parameters. Finally, we present a series of numerical examples for thin film equations to validate the theoretical properties and efficacy of the scheme.

math.NA↗
arXiv2026

The Levin Method for the Summation of One-dimensional and Multidimensional Infinite Series

The Levin method transforms the evaluation of a highly oscillatory integral into the solution of a first-order linear ODE for a slowly varying auxiliary function. This ODE is typically approximated by collocation, after which the integral value is recovered from the auxiliary function at the endpoints. The present work develops a new extension of the Levin method for the summation of one-dimensional and multidimensional infinite oscillatory series. The summation problem is transformed into the solution of a functional equation involving transformed arguments of the unknown function. The resulting approach is particularly attractive in the multidimensional setting, where the range of existing numerical methods is relatively limited.

math.NA↗
arXiv2026

A uniform elliptic reduction, an order-matching criterion, and precision benchmarks for the strong-coupling Birman-Schwinger analysis of the lattice three-boson trimer

We present a rigorous strong-coupling Birman-Schwinger analysis of the three-boson lattice Schroedinger operator on Z^2 at the exceptional quasimomentum K = pi. First, we provide an exact closed-form benchmark for the fiber Fredholm determinant, valid for every quasimomentum K, obtained via a uniform elliptic reduction. Second, we formulate and prove a general order-matching criterion determining whether a leading-order Fredholm determinant asymptotic, with relative error O(1/mu), suffices to fix the constant-order additive energy correction, or if the next-order refinement is required. Applying the criterion to the formal branch z = -2mu + d, we identify an algebraic crossing -2mu + 6 + 8/mu + O(mu^{-2}), but demonstrate that this crossing does not correspond to a true eigenvalue of the full Hamiltonian. The actual ground state obeys the rigorous variational bounds -3mu <= z_1^{pi,s}(mu) <= -3mu + 6, so that z_1^{pi,s}(mu) = -3mu + O(1), the same leading branch as at K = 0; direct finite-volume diagonalisation confirms the refined asymptotic -3mu + 6 + O(mu^{-1}) and spectral gap 2mu - 2 + O(mu^{-1}). We independently confirm that the known K = 0 constant C approximately 3.96458 requires no analogous refinement. Finally, we compare the asymptotic precision levels achieved across recent lattice few-body models and draw a structural parallel with the parity-based classification of topological band insulators at time-reversal-invariant momenta.

math-ph↗
arXiv2026

Feasible approximation of matching equilibria for large-scale matching for teams problems

We propose a numerical algorithm for computing feasible and approximately optimal solutions of the matching for teams problem. Specifically, we introduce the notion of approximate matching equilibrium as a feasible approximation of a matching equilibrium with relaxed rationality, and we show that a true equilibrium is recovered in the limit of a sequence of approximate matching equilibria with sub-optimality approaching 0. In our approximation scheme, we parametrize the so-called transfer functions, and we show that tackling the resulting parametric primal and dual optimization problems yields two approximate matching equilibria as well as provable and computable lower and upper bounds for the optimal social welfare. Under a flexible Euclidean setting, we show that the approximation error of our scheme can be controlled to be arbitrarily close to 0, we derive an explicit computational complexity bound, and we develop an algorithm for computing approximate matching equilibria that is efficient for large-scale problems involving a large number of agent populations. We study three problems in our numerical experiments: a retail business problem, the Wasserstein barycenter problem, and a large-scale problem involving up to 1000 agent populations. We show that the proposed algorithm can produce nearly optimal approximate matching equilibria to provide quantitative managerial insights for policymakers, and that the computed sub-optimality estimates are much less conservative than theoretical estimates.

math.OC↗
arXiv2026

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

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

math.NA↗
arXiv2026

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

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

Families of relative periodic orbits in the planar three-body problem via consecutive alignments

Relative periodic orbits (RPOs) are solutions of the three-body problem that are periodic in a uniformly rotating reference frame and, in general, quasi-periodic in inertial coordinates. We present a numerical procedure for computing and continuing one-parameter families of RPOs of the planar Newtonian three-body problem. The method exploits consecutive syzygies, understood here as configurations in which the three bodies are aligned and their velocities satisfy the corresponding symmetry conditions. Matching the positions and momenta at two consecutive alignments reduces the computation of RPOs to a low-dimensional nonlinear problem. Its solutions are then numerically continued, and linear stability is determined from the nontrivial eigenvalues of the rotated monodromy matrix after removing the neutral directions associated with conserved quantities and continuous symmetries. The procedure is applied to several mass distributions and initial configurations, producing families of Poincaré, Hill, and binary-type solutions. These families exhibit transitions from nearly circular to highly eccentric motion, changes of stability near resonances and turning points, and absolute periodic solutions when the rotation angle is a rational multiple of 2π. In the Hill families, the continuation connects satellite configurations with circumstellar motion as the smallest body loses its gravitational binding to the intermediate body. Circumbinary and circumstellar configurations are also obtained in the binary regime. The results illustrate the dynamical diversity of RPOs and provide coherent three-body motions that can be used as prescribed trajectories in restricted four-body models.

math.DS↗
arXiv2026

Stabilizing the Rayleigh--Ritz procedure by randomization

Extracting approximate eigenpairs from a prescribed subspace is of fundamental importance in eigenvalue computation. While projecting the target eigenvector onto the subspace yields satisfactory accuracy, extracting an approximate eigenpair that attains a comparable convergence rate has remained a long-standing open problem. Although the standard Rayleigh--Ritz procedure is widely used for this purpose, it may suffer from deteriorated convergence of Ritz values and may even fail to produce convergent Ritz vectors. In this paper, we address this long-standing open problem by introducing a randomized Rayleigh--Ritz procedure whose output converges at a rate similar to the ideal projection. Our analysis requires only the simplicity of the target eigenvalue and extends naturally to nonlinear eigenvalue problems.

math.NA↗