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At least 19 recordsLinked to original sources

arXiv2026

Accelerated Primal-Dual Proximal Gradient Splitting Methods for Convex-Concave Saddle-Point Problems

In this paper, based a novel primal-dual dynamical model with adaptive scaling parameters and Bregman divergences, we propose new accelerated primal-dual proximal gradient splitting methods for solving bilinear saddle-point problems with optimal nonergodic convergence rates. For the first, using the spectral analysis, we show that a naive extension of acceleration to a quadratic game is unstable. Motivated by this, we present an accelerated primal-dual gradient flow which combines acceleration with careful velocity correction. To work with non-Euclidean distances, we also equip our continuous model with general Bregman divergences and prove the exponential decay of a Lyapunov function. Then, new primal-dual splitting methods are developed based on proper semi-implicit Euler schemes of the continuous model, and the theoretical convergence rates are nonergodic and optimal with respect to the matrix norms, Lipschitz constants and convexity parameters. Moreover, we introduce efficient restart variants to further improve the proposed methods and provide some numerical results to validate the practical performance.

math.OC↗

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

Weighted Laplace Spaces for Spectral Measures and Rational Approximation

We introduce the weighted Laplace space $H_w$, an RKHS of Laplace transforms on $(0,\infty)$, and study spectral measures in its dual space $H'_w$. For conforming FEM discretizations of the Dirichlet Laplacian on bounded Lipschitz domains, we prove the dual-norm inequality $\|μ_h\|_{H'_w} \leq \|μ\|_{H'_w}$, where $μ= \sum_kδ_{λ_k}$ and $μ_h = \sum_k δ_{λ_{k,h}}$. The proof combines min-max monotonicity of FEM eigenvalues with a heat-trace representation of the dual norm. We then analyze $H_w$-adapted rational approximation of shifted symbols $ϕ(x)=(x+κ^2)^{-β}$ and give a conditional transfer principle for estimates proved in the corresponding weighted Laplace pre-image norm. Via dual pairing, the norm inequality yields uniform bounds for finite spectral sums and related transformed observables.

math.NA↗

arXiv2026

Computing Lewis Weights to High Precision by Fixed-Point Iteration

The $\ell_p$-Lewis weights of a matrix are defined by a fixed-point equation. For $p<4$, Cohen and Peng [CP15] showed that iterating an equivalent rearrangement of this equation computes Lewis weights to high precision; for $p\geq4$, prior high-precision methods instead use optimization-based approaches. We show that the direct Lewis fixed-point iteration, appearing in the thesis of Lee [Lee16], computes Lewis weights to high precision for every $p>2$. For a matrix $\mathbf{A}\in\mathbb{R}^{m\times n}$ partitioned into row blocks $\mathbf{A}_{[1]},\ldots,\mathbf{A}_{[k]}$, we compute, for $p>2$, coordinatewise $\varepsilon$-approximate $\ell_p$ block Lewis weights in $O\left(p\log\frac{p\sqrt{\sum_{i=1}^k\operatorname{rank}(\mathbf{A}_{[i]})}}{\varepsilon}\right)$ rounds of exact leverage-score-vector computations. For ordinary Lewis weights, this becomes $O\left(p\log\frac{p\sqrt{m}}{\varepsilon}\right)$, improving the $O\left(p^2\log(m/\varepsilon)\right)$ bound of Gribling, Sidford, and Zhang [GSZ26] for $p\geq4$. Our guarantees also extend to approximate leverage-score computations. Our main observation is that each direct Lewis update contracts the KL divergence to the true weights by a factor of $1-\frac{2}{p}$. We also give an alternate explanation of this contraction through volume sampling and entropic independence. Synthetic experiments closely match our predicted local contraction rates and iteration counts grow approximately linearly with $p$; real-data experiments illustrate the information-concentration tradeoff of finite-$p$ block Lewis designs.

cs.DS↗

arXiv2026

A multilevel stochastic-gradient neural solver for boundary integral equations

We propose a multilevel stochastic-gradient neural solver (MLSG) for second-kind boundary integral equations. MLSG represents the unknown boundary density using a neural network optimized by stochastic residual minimization over a hierarchy of successively refined Nyström discretizations. Upon transitioning from one level to the next, the network parameters obtained on the previous level initialize training on the current one. This coarse-to-fine strategy retains a continuous, grid-independent density representation and is designed to reduce the total computational effort required to reach a prescribed residual tolerance at the target resolution. The algorithm avoids grid-transfer operators and hierarchical fast-summation machinery, relying instead on batched kernel evaluations and standard network forward and backward passes that map efficiently onto modern GPU architectures. For uniformly stable second-kind discretizations, so strongly nonuniform contraction rates originate in the empirical neural tangent kernel (NTK) rather than in the discretized operator. Within each level, parameter updates can reshape the NTK, while refinement re-samples the tangent kernel on a richer discrete space and reveals directions that were not adequately resolved on coarser levels. A cross-level estimate bounds the warm-start loss in terms of the preceding training tolerance and quadrature error, motivating a tolerance schedule that balances optimization and discretization errors. Experiments on Laplace/Poisson and Helmholtz problems in two and three dimensions, together with an exterior Robin problem on a hypersurface in $\R^4$, demonstrate the method under both parametric and signed-distance surface representations at up to million-scale discretizations.

math.NA↗

arXiv2026

A counterexample to Kenig's conjecture for the Laplace double-layer operator

We disprove Kenig's 1994 spectral-radius conjecture for the double-layer operator in every dimension $d\ge2$. For every $t>1/2$ sufficiently close to $1/2$, we construct a bounded simply connected Lipschitz domain with connected boundary such that the double-layer operator on $L^2$ with respect to the surface measure has $\pm \mathrm{i} t$ in its Fredholm essential spectrum. In particular, the essential spectral radius is greater than $1/2$. In the plane, the boundary may be chosen differentiable everywhere and smooth away from a single point. Our planar construction develops a refinement mechanism that produces unbounded growth of adjoint resolvent solutions with fixed forcing while keeping the graph slopes uniformly bounded. A closed recursion for the density and its image under an auxiliary singular integral reduces the growth estimate to a certified inequality for Hermitian $2\times2$ matrices. The certificate uses continuous estimates, exact rational arithmetic and rigorous interval enclosures. Localisation and rescaled graph insertions then produce a weakly null sequence of normalised approximate eigenvectors for the adjoint double-layer operator on one bounded boundary. A compact intertwining relation on Cartesian products of the planar domain with cubes gives the higher-dimensional examples without changing the spectral parameter.

math.AP↗

arXiv2026

When Does Cheap Physics Help? Gated Multi-Fidelity Physics-Informed Networks with Provable Sample Savings and Conformal Guarantees

High-fidelity (HF) simulations of parametric partial differential equations are expensive, whereas low-fidelity (LF) models such as coarse grids or simplified physics are cheap but biased. Multi-fidelity physics-informed neural networks combine the two, yet it remains unclear when cheap physics actually reduces the number of HF solves, and existing methods silently fail when the LF model is misleading and offer no guaranteed uncertainty. We answer the first question with a sharp sample-complexity bound: learning the HF field through its discrepancy from the LF model reduces the number of HF samples needed for accuracy $\varepsilon$ by the factor $κ^{2γ}$, where $κ$ is the ratio of the complexity of the discrepancy to that of the HF field and $γ$ is the effective-dimension exponent, and we confirm the predicted scaling with exact risk computations in a kernel model. Guided by the analysis, we propose GaMF-PINN, a gated multi-fidelity PINN that learns where to trust the LF model and falls back to physics-only learning where it does not, together with a function-level conformal procedure that returns uncertainty bands containing the entire HF solution with probability at least $1-α$ in finite samples. We release MFPDE-Bench, an open benchmark of paired LF/HF solvers for four parametric PDEs with coarse-grid and simplified-physics LF models. Across MFPDE-Bench, GaMF-PINN reaches the accuracy of a standard HF-PINN with up to 8x fewer HF solves, remains robust when the LF model is deliberately corrupted, and attains the nominal coverage where Bayesian PINNs and deep ensembles under-cover.

cs.LG↗

arXiv2026

Dynamical Low-Rank Filters for Data Assimilation

We propose dynamical low-rank (DLRA) filters for data-assimilation problems based on stochastic differential equations (SDEs). We first consider a Kalman filter applied to a Euler-Maruyama discretization of the SDE and observation processes, and derive a DLRA Kalman-type filter, named DLRA-JMCO, by minimizing jointly the error on the mean and covariance. We then obtain a time continuous DLRA-JMCO filter as the continuous limit of the time discrete one, and introduce a corresponding particle approximation. We also propose a strategy to enrich the DLRA-JMCO filter with a simple Gaussian process to capture the neglected dynamics in the orthogonal to the DLRA subspace. This correcting term allows the DLRA subspace evolution to be directly influenced by the observation operator. We finally propose a particle filter version of the DLRA-JMCO filter. Numerical experiments assess the performance of the proposed filters.

math.NA↗

arXiv2026

Fully discrete stochastic maximal regularity and $H^\infty$-calculus for second-order elliptic operators

This paper establishes the fully discrete stochastic maximal $L^p$-regularity and the accompanying sharp maximal estimate for numerical approximations of parabolic stochastic partial differential equations. We consider the spatial finite element discretization $A_h$ of a general second-order elliptic operator $A=-\nabla \cdot a\nabla +b\cdot \nabla +c$ with Dirichlet boundary conditions on a smooth, bounded, convex domain in $\mathbb{R}^3$, coupled with a broad class of temporal schemes, including rational approximations and the exponential Euler method. To obtain these optimal discrete regularity results, we establish a bounded $H^\infty$-calculus for the discrete spatial operator $A_h$, uniformly in the mesh size $h$. As a direct byproduct, we also establish the discrete-in-space stochastic maximal regularity for the corresponding spatial semi-discretizations. As an application, we establish the pathwise uniform convergence of a fully discrete approximation of the three-dimensional stochastic Allen--Cahn equation with variable diffusion coefficients and multiplicative noise.

math.AP↗

arXiv2026

Differentiable Hybrid Modelling for Learning and Optimising Chemical Transport Processes from Experimental Data

Transport models underpin the design and optimisation of chemical processes, yet their constitutive laws are typically hand-picked, their experimental initial conditions are assumed exact, and their computational cost restricts optimisation to a handful of process variables. Here we introduce a differentiable hybrid modelling framework that embeds neural networks within a GPU-accelerated population balance equation solver to discover constitutive laws and assimilate initial conditions directly from experimental crystallisation data. Using thirteen real experiments with multi-dimensional size and shape measurements, we show that treating measured initial conditions as exact yields a physically inconsistent rate law despite improved predictive accuracy, whereas assimilating them recovers a physically consistent, non-linear law and reduces trajectory error sevenfold relative to conventional modelling. End-to-end differentiability further enables gradient-based optimisation of high-dimensional operating profiles, identifying a 300-variable temperature profile that reduces crystal aspect ratio by 25% at constant mass, facilitating superior product attributes. Our framework augments, rather than replaces, mechanistic modelling, offering a general route to learning transport models from data and optimising processes with them.

cs.CE↗

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↗

arXiv2026

Second-Order Accuracy from Large Local Errors in Interface Problems: A Discrete Green's Function Analysis

Finite difference methods for interface problems often exhibit large local truncation errors near the interface, particularly when discontinuities in coefficients or solution derivatives are present. Nevertheless, many such schemes achieve second-order accuracy, a phenomenon not fully explained by standard pointwise consistency arguments. In this work, we provide a precise explanation of this behavior through the structure of the discrete Green's function associated with the underlying conservative difference operator. An explicit representation of the Green's function reveals a two-plateau structure in its weighted increments. By expressing the numerical error in terms of this Green kernel, we show that the dominant interface truncation errors, although individually of order $O(1)$, possess a cancellation structure that reduces their effective contribution to the global error. The exact Green's-function analysis leads to a class of conservative interface flux--balance schemes for which the cancellation mechanism yields second-order accuracy in the maximum norm. The weighted harmonic discretization is included as a particular member, and its cancellation property is established directly. For Cartesian grids with a flat interface parallel to a coordinate direction, the same conservative cancellation mechanism persists along grid lines crossing the interface. Numerical experiments in one and two dimensions directly illustrate the predicted cancellation behavior and the resulting second-order accuracy. These results demonstrate that second-order accuracy in interface problems can arise from the interaction between localized truncation errors and the global structure of the discrete operator, rather than from pointwise consistency alone.

math.NA↗

arXiv2026

Shallow neural network approximation in mixed Sobolev spaces

We investigate the best $L_2$ approximation of mixed Sobolev spaces by shallow neural networks with $n$ neurons and general activation functions. We first establish an activation-independent Fourier-block principle: if an activation has univariate approximation order $ρ$ in the sense of the Fourier-block property, then the global approximation rate has algebraic order $\min\{α,ρ\}$ for target functions of mixed smoothness $α$, up to explicit logarithmic factors. To verify this property for concrete activations, we introduce a structured univariate approximation condition that implies the Fourier-block property with explicit parameters. For $\mathrm{ReLU}^k$, a matching algebraic lower bound identifies $\min\{α,k+1\}$ as the optimal algebraic approximation exponent in any dimension, up to logarithmic factors in the upper bound. The framework also yields the exponent $\min\{α,k+1\}$ for cardinal B-splines and soft-$\mathrm{ReLU}^k$, and the full mixed-smoothness exponent $α$ for ELU and cosine activations, again up to logarithmic~factors.

math.NA↗

arXiv2026

The 2M Multiplication Algorithm for Complex Matrices

Complex matrix multiplication is typically computed using 4 real matrix multiplications (GEMMs) of the same size. The well-known 3M multiplication algorithm reduces this cost to 3 real GEMMs, together with quadratic time pre- and post-processing steps. In this paper, we reduce 3M to 2M for matrices with integer real and imaginary parts, performing complex GEMM with only 2 real GEMMs of the same size, along with quadratic time pre- and post-processing. For floating-point matrices, 2M multiplication combines naturally with the Ozaki-II scheme, yielding a practical, high-performance algorithm for computing a complex floating-point GEMM in roughly twice the time of a real GEMM of the same size. As corollaries, we derive new algorithms for symmetric rank-$k$ updates (SYRK/HERK) that internally use full rectangular GEMMs.

cs.DC↗

arXiv2026

Computing matrix functions associated with a Hermitian-definite pencil

We consider the numerical evaluation of the quantity $Af(A^{-1}B)$, where $A$ is Hermitian positive definite, $B$ is Hermitian, and $f$ is a function defined on the spectrum of $A^{-1}B$. This problem is related to the Hermitian-definite matrix pencil $B-λA$. We study the conditioning of the problem, and we introduce several algorithms that combine the Schur decomposition with either the matrix square root or the Cholesky factorization. We study the numerical behavior of these algorithms in floating-point arithmetic, assess their computational costs, and compare their numerical performance. Our analysis suggests that the algorithms based on the Cholesky factorization will be more accurate and efficient than those based on the matrix square root. This is confirmed by our numerical experiments.

math.NA↗

arXiv2026

A Sharp Unitarily Invariant Norm Bound for the Off-Diagonal Block Perturbation of a Hermitian Matrix

Let $$ A=\begin{bmatrix} H_1 & E^* \\ E & H_2 \end{bmatrix} \quad\text{and}\quad \widetilde A=\begin{bmatrix} H_1 & 0 \\ 0 & H_2 \end{bmatrix} $$ be partitioned Hermitian matrices, with $\widetilde A$ obtained by dropping the off-diagonal blocks of $A$. Let $η$ be the minimum distance between the spectra of $H_1$ and $H_2$; these spectra may interlace. Define, for $δ,ε\ge0$, $$ ϕ(δ,ε)= \begin{cases} 2ε/(δ+\sqrt{δ^2+4ε^2}), &\quad\mbox{if $(δ,ε)\ne (0,0)$}, 1, &\quad\mbox{if $(δ,ε) = (0,0)$}, \end{cases} $$ and let $\widetilde E=A-\widetilde A$ and $ε_2=\|E\|_2=\|\widetilde E\|_2$, the matrix spectral norm. Li and Li [\emph{Linear Algebra Appl.}, 395:183--190, 2005] established a sharp spectral-norm bound on the changes in the eigenvalues of $A$: $$ \big\|\mathrm{diag}\big(\pmbλ(A)-\pmbλ(\widetilde A)\big)\big\|_2 \le ϕ(η,ε_2)\,\|E\|_2, $$ where $\pmbλ(A)$ is the vector whose components are the eigenvalues of $A$ in descending order and similarly for $\pmbλ(\widetilde A)$. In this paper, this bound is extended to every unitarily invariant norm $\|\cdot\|_\mathrm{ui}$: $$ \big\|\mathrm{diag}\big(\pmbλ(A)-\pmbλ(\widetilde A)\big)\big\|_\mathrm{ui} \le ϕ(η,ε_2)\,\|A-\widetilde A\|_\mathrm{ui}. $$ A $2\times2$ example attains equality for every such norm. An analogous bound for singular values is also established when the off-diagonal blocks of a rectangular matrix are dropped.

math.NA↗

arXiv2026

Geometric Ergodicity of Affine Invariant Ensemble Langevin and its Discrete Time Variants

Affine-invariant ensemble samplers are widely used in Bayesian applications. However, their quantitative convergence theory, in particular geometric ergodicity, remains a basic open question. We study the affine invariant ensemble Langevin dynamics, an interacting particle system that uses the empirical covariance of the whole ensemble as a preconditioner. While effective in practice, theoretical understanding of this method is not available beyond plain qualitative convergence in total variation, with a central difficulty being that the empirical covariance can become degenerate. For potentials with bounded Hessian that are strongly convex outside a ball, we prove geometric ergodicity using a novel Lyapunov function that combines the inverse covariance with a coercive exponential energy. We then show that directly applying the Euler--Maruyama scheme can diverge with positive probability, even for a one-dimensional Gaussian target. This motivates a covariance-trace regularization in continuous time. We prove geometric ergodicity of the regularized diffusion and, for sufficiently small step size, of its unadjusted Euler--Maruyama discretization. We also show that the invariant distributions of the discretization converge weakly to the product target distribution as the step size tends to zero. Lastly, we provide an explicit discretization that is both affine-invariant and geometrically ergodic, based on an adaptive step size clamping.

math.ST↗

arXiv2026

Annular liquid jets: exact constraint absorption and finite-time loss of global admissibility

An annular liquid jet may enclose a volume of gas and, when surface tension dominates over inertia, close on the symmetry axis at a finite distance. In the one-dimensional membrane model, closure provides an algebraic relation among the state variables at the free end. Earlier domain-adaptive schemes differentiate this relation to obtain an evolution equation for the domain length. We instead absorb the endpoint closure identically in the dependent variable and discretise the reduced system by Chebyshev collocation. Convergence is geometric, published steady data are recovered, and the unsteady calculations are checked by componentwise off-grid residuals, temporal refinement, and an independent upwind finite-difference discretisation. For jets of finite thickness, the convergence length decreases with nozzle thickness when the Froude number, Weber number, nozzle angle, and pressure coefficient are held fixed and the terminal zero remains transversal; this is a parameter-specific model result rather than a universal experimental thickness law. The forced response has a pressure-coefficient maximum near a Strouhal number of 0.093, whereas the convergence-length gain increases monotonically over the globally admissible frequency interval studied. Most importantly, a global positivity check changes the interpretation of the first finite-time event. Under body-force forcing, the mean radius first reaches zero at an interior point while the tracked endpoint remains transversal. The interior contact is quadratic, and the independent finite-difference scheme converges to the same first-contact time. After this event, the tracked endpoint is no longer the first physical closure point, so the single-interval annular-jet representation is no longer globally admissible.

cs.CE↗