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arXiv2026

Efficient computation of the asymptotics of extensive-rank HCIZ integrals

We study the high-dimensional asymptotics of Harish-Chandra-Itzykson-Zuber (HCIZ) integrals in the extensive-rank regime. The limit of these integrals is governed by a one-dimensional boundary-value hydrodynamical problem originally derived by Matytsin (1994) and rigorously proved by Guionnet and Zeitouni (2002). Despite its wide-ranging applications, explicit solutions to this problem are known only in a few specific cases. In this work, we introduce an efficient numerical scheme based on a particle discretization and prove its convergence to the continuous boundary-value problem for generic boundary densities. We validate our approach against known analytical solutions and apply it to generic densities, uncovering interesting dynamical phenomena. The high-dimensional limit of HCIZ integrals appears in various contexts, from the large deviations of random matrix spectra to the limiting free energy of disordered systems, high-dimensional statistics, and machine learning. As such, our contribution opens the way towards the numerical exploration of a wide range of high-dimensional models that were previously intractable.

math.PR↗
arXiv2026

LoCCA: Localized Chebyshev Cross Approximation for Kernel Matrix Factorization via Nodal Perturbation Stability

We propose Local Chebyshev Cross-Approximation (LoCCA), a data-driven framework that bridges smooth polynomial interpolation with flexible matrix factorizations. LoCCA dynamically maps ideal grid nodes to their nearest physical neighbors within unstructured point sets. We support this framework with a new perturbation theory, proving that displacing Chebyshev nodes onto physical data points retains mathematical stability and accuracy without explosive error growth. Numerically, LoCCA provides strict, reliable error control even on irregular geometries where standard grid-based methods fail, achieving substantial speedups and near-optimal matrix compression.

math.NA↗
arXiv2026

Poisson Image Denoising Using Minimax Concave and Reweighted $\ell_1$ Penalties: Nonblind and Blind Approaches

Images are important tools in various sciences. Despite the development of photo-taking tools, creating clear and image without noise remains challenging in practice. In particular, Poisson noise has an effect on medical and astronomical images, and reduces their quality. Additionally, blur is another factor that has an effect on image quality. The problem of image restoration becomes very complicated when we have no information about the Point Spread Function (PSF). These types of problems are known as blind case. However, in some images, such as some astronomical images, the type of PSF can be specified, and these types of problems are known as nonblind problems. Total Variation (TV) is a widely used method for solving such inverse problems, where the selection of the penalty function is the most critical factor that affects the method's performance. In this paper, to improve edge preservation, we employ a reweighted $\ell_1$-regularization of the fractional order derivative. Furthermore, we propose a nonblind and blind image deblurring approach under Poisson noise using the Minimax Concave Penalty (MCP), which is a continuous, sparsity promoting, and nearly unbiased regularizer. This formulation leads to a nonconvex optimization model. To solve the proposed model, we introduce an efficient numerical algorithm based on the Alternating Direction Method of Multipliers (ADMM) and provide an analysis of its convergence. Finally, the effectiveness of the proposed algorithm are demonstrated through extensive experiments on various images.

cs.CV↗
arXiv2026

Threshold dynamics for subdiffusive grain growth

We propose a new fractional threshold-dynamics algorithm for subdiffusive interface motion and apply it to multiphase grain-growth simulations. The method combines the L1 discretization of the Caputo time derivative with a Helmholtz-resolvent thresholding step. At each time level, the L1 convolution produces a convex history average of the previously thresholded states; this history-dependent field is then propagated by a Helmholtz diffusion solve and projected back to pure phases by thresholding. For $α=1$, the method reduces to a Helmholtz-resolvent analogue of the Merriman-Bence-Osher scheme, whereas for $0<α<1$ it introduces a subdiffusive temporal memory. We establish a variational characterization, maximum and comparison principles, and stability for the continuous Helmholtz diffusion step. We also derive a memory-tail estimate and a finite-dimensional pinning criterion under an explicit resolvent-contraction hypothesis. The equal-tension multiphase scheme is implemented on periodic grids. The numerical experiments investigate the joint influence of the L1 history and the smoothing length. The fixed-step runs show faster coarsening for smaller fractional orders in the tested regime, while a comparison at fixed smoothing length examines the influence of the L1 history on successive updates.

math.NA↗
arXiv2026

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

Layer potentials provide a classical approach to boundary value problems for Laplace's equation on Lipschitz domains. Kenig's 1994 spectral-radius conjecture for the double-layer operator would ensure operator-norm convergence of the associated Neumann series on mean-zero $L^2$ densities when the boundary is connected. We disprove this conjecture by constructing a bounded simply connected planar Lipschitz domain whose double-layer operator on arclength $L^2$ has essential spectral radius strictly greater than $1/2$. More precisely, for every $t>1/2$ sufficiently close to $1/2$, we obtain such a domain with $\pm i t$ in its Fredholm essential spectrum. The construction starts from smooth graphs whose shapes repeat under translation. In the limit of separated scales, refinement makes solutions of adjoint resolvent equations grow with fixed forcing. The graph slopes remain uniformly bounded. A computer-assisted certificate proves this growth through an inequality for Hermitian $2\times2$ matrices. Its strict margin at $- i/2$ persists at nearby spectral parameters. Normalisation and a Floquet transform then give compactly supported densities with small residuals on the full graphs. We insert rescaled segments of successive graphs into one bounded boundary, where these densities form a weakly null sequence of approximate eigenvectors. The same spectral conclusion holds on a single periodic Lipschitz graph. The certificate combines continuous estimates, exact rational arithmetic and rigorous interval enclosures.

math.AP↗
arXiv2026

Accelerating Fourier--Motzkin elimination: redundancy removal and the choice of variable elimination order

Fourier-Motzkin elimination computes an inequality description of the projection of a polyhedron onto a subset of its coordinates by eliminating one variable at a time. It is used in several areas of optimisation and computer science, and it is a standard way of obtaining the entropic constraints of a causal structure, where the marginalisation over the latent variables produces such a projection. Its limitation is the growth of the intermediate systems of inequalities, which can be doubly exponential in the number of eliminated variables even though the projection itself grows only as a single exponential. In practice the computational overload of the method therefore depends on two choices: how the redundant inequalities are removed after each step, and the order in which the variables are eliminated. We consider both. We first show, by an explicit example, that Imbert's redundancy test cannot be interleaved with redundancy removal by linear programming. We show that the two methods, however, can be combined soundly if the derivation records used by Imbert's test are re-initialised after every step at which linear programming is used. We then propose a rule for choosing the elimination order of the variables that gives a significant computational advantage, however, at the cost of increased resource usage. We demonstrate this advantage on some random polytopes, where the rule reduces the running time by factors of between 6 and 25 compared with the same elimination under a fixed order. For entropic descriptions of causal structures, with more than 250 inequalities and more than 100 variables to eliminate, our rule keeps the number of inequalities handled at each step one to two orders of magnitude lower than a fixed order.

cs.CC↗
arXiv2026

A third-medium approach for electro-thermo-mechanical contact considering Joule heating

Connectors are ubiquitous in technical systems and frequently combine mechanical contact with electric current transfer and heat generation. To capture these interacting processes, an electro-thermo-mechanical contact formulation based on the third-medium concept is proposed. Mechanical contact, electric current flow, and heat conduction are represented within a unified finite element framework without explicit contact-surface tracking. Deformation-dependent switching functions govern the onset of electrical and thermal transport during interface closure. Electrical conduction is coupled to the transient thermal problem through Joule heating, while the temperature dependence of the electrical conductivity provides the corresponding feedback on the current flow. The coupled displacement, electric potential, and temperature fields are solved monolithically and combined with an independently interpolated deformation-gradient-like field for robust third-medium regularization under severe compression. Numerical examples demonstrate contact-induced current transfer, subsequent Joule heating and transient temperature evolution, as well as localized current paths during progressive closure of rough interfaces.

math.NA↗
arXiv2026

A Complete Resolution of Forsythe's Conjecture for Restarted Conjugate Gradients

Forsythe's conjecture, published in 1968, asserts that for each restart length $s$, every exact-arithmetic restarted conjugate-gradient iteration on a real symmetric positive definite problem either terminates or has normalised residuals that converge separately along the even and odd restart subsequences. Apart from the classical steepest-descent case, this asymptotic question remained unresolved in full generality for nearly six decades. We give a complete classification by restart length in the original finite-dimensional setting and identify a sharp threshold. For $s=2$ and $s=3$, every problem either terminates or has convergent even and odd residual directions. For every $s\ge4$, there is a diagonal positive definite counterexample of dimension $s+4$ which never terminates and whose even residual directions do not converge. Together with Akaike's theorem for $s=1$, this shows that the conjectured universal conclusion is true precisely for $s\in\{1,2,3\}$ and false for every $s\ge4$. The positive results follow from a degree-independent double-orthogonality identity and an analysis of the low-degree fixed-point sets. At restart length four, rational interval arithmetic and Sturm sequences certify a transverse Hopf point of the leading vector field of the rescaled squared-weight map. Analytic periodic-orbit and shadowing arguments yield the counterexample at restart length four, and degree elevation extends the construction to every larger restart length. The classification for all $s\ge2$ is also formally verified in Lean.

math.NA↗
arXiv2026

Isogeometric $C^1$ mortar method

We present an isogeometric mortar method for the discretization of the biharmonic equation posed on multi-patch domains. We assume only $C^0$-conformity at interfaces and employs a mortar approach to weakly enforce $C^1$-continuity across patch interfaces. Discrete inf-sup stability is ensured by selecting a Lagrange multiplier space consisting of splines of degree reduced by two compared to the primal space, with increased smoothness or merged elements near vertices. We prove optimal a priori error estimates and confirm the theoretical findings with a series of numerical experiments.

math.NA↗
arXiv2026

On the computation of the cumulative distribution function of the Normal Inverse Gaussian distribution

In this paper, we obtain various series and asymptotic expansions involving the modified Bessel function of the second kind for the normal inverse Gaussian cumulative distribution function. The new expansions accelerate computations, complementing the numerical integration methods implemented in statistical software packages. We also provide a detailed description of the algorithm and its corresponding implementation in C++. The performance and accuracy of the algorithm are extensively tested and benchmarked with open-source implementations, offering superior accuracy and speed-ups of a factor from 5 to 60.

math.NA↗
arXiv2026

An unfitted HDG discretization for a model problem in shape optimization

We apply an unfitted HDG discretization to a model problem in shape optimization. The method proposed uses a fixed, shape regular, non-geometry conforming mesh and a high order transfer technique to deal with the curved boundaries arising in the optimization process. The use of this strategy avoids the need for constant remeshing and enables a highly accurate description of the domain using a coarse computational mesh. We develop a rigorous analysis of the well-posedness of the problems that arise from the optimality conditions, and provide an a priori error analysis for the resulting discrete schemes. Numerical examples with manufactured problems are provided demonstrating the convergence of the scheme and the efficiency of the transfer path method. The approach proposed yields high resolution approximations of the boundary using grids with coarse mesh parameters.

math.NA↗
arXiv2026

Characteristic Based Physics Informed Neural Networks for Advection Equations with a Focus on Discontinuous Solutions

This paper presents a characteristic-based loss function for physics-informed neural networks (PINNs) that circumvents automatic differentiation and significantly reduces training cost for advection equations. However, standard PINNs often fail to accurately resolve discontinuous solutions, as their global smoothness prior and spectral bias lead to smeared shocks or spurious oscillations near sharp gradients. To accurately capture these discontinuities, the density of sampling points in their vicinity must be sufficiently high, and an appropriate sampling strategy must be employed. For problems involving discontinuous initial and boundary conditions, several complementary techniques are introduced: an adaptive sampling strategy that concentrates collocation points along characteristic paths, a Fourier feature mapping with two-stage training and adaptive weighting to mitigate spectral bias, and an adaptive median filter applied to the spatial data at each time instant, which suppresses spurious oscillations while preserving sharp features such as corners and peaks. The effectiveness of the proposed framework is demonstrated through numerical experiments, showing improved accuracy and reduced computational effort in approximating discontinuous solutions.

math.NA↗
arXiv2026

Multi-Fidelity Physics-Informed Neural Networks with Bayesian Uncertainty Quantification and Adaptive Residual Learning for Efficient Solution of Parametric Partial Differential Equations

Physics-informed neural networks (PINNs) have emerged as a powerful paradigm for solving partial differential equations (PDEs) by embedding physical laws directly into neural network training. However, solving high-fidelity PDEs remains computationally prohibitive, particularly for parametric systems requiring multiple evaluations across varying parameter configurations. This paper presents MF-BPINN, a novel multi-fidelity framework that synergistically combines physics-informed neural networks with Bayesian uncertainty quantification and adaptive residual learning. Our approach leverages abundant low-fidelity simulations alongside sparse high-fidelity data through a hierarchical neural architecture that learns nonlinear correlations across fidelity levels. We introduce an adaptive residual network with learnable gating mechanisms that dynamically balances linear and nonlinear fidelity discrepancies. Furthermore, we develop a rigorous Bayesian framework employing Hamiltonian Monte Carlo.

cs.LG↗
arXiv2026

A Unified CutFEM Formulation for Finite-Strain Elasticity: Energy Minimisation and Corner Singularities

We present a fully variational, model-independent formulation of the Cut Finite Element Method (CutFEM) for finite-strain elasticity. The discrete problem is derived from a single augmented energy functional consisting of the bulk hyperelastic energy, the Nitsche terms that impose the boundary conditions weakly, and the ghost-penalty stabilisation. At each nonlinear iterate, the residual is the exact first variation of this functional with the adaptive Nitsche weight frozen, while the correction uses a symmetric, coercive approximation of its Hessian. Automatic differentiation (AD) generates the first Piola--Kirchhoff stress tensor and the elasticity tensor directly from the scalar energy density, avoiding manual re-derivation when exchanging hyperelastic models. To our knowledge, this is the first unfitted finite-strain scheme combining an energy-only, model-independent construction with AD and an accuracy analysis at unfitted boundaries. Analysis of the linearised problem solved at each quasi-Newton step establishes cut-independent coercivity, continuity, and an $O(h^{-2})$ condition number bound, yielding a quasi-optimal convergence theorem for regular solutions through the Brezzi--Rappaz--Raviart framework. Numerically, the method attains optimal $h$-convergence for quadratic and cubic elements on a smooth test case. Furthermore, we quantify the method's accuracy limit at mixed Dirichlet--Neumann junctions using the Kolosov--Muskhelishvili characteristic equation. The exact solution's corner singularity caps the convergence rate identically for fitted and unfitted methods. Local mesh refinement removes this bound: we verify the recovery of optimal rates numerically for first-order elements, and prove that the unfitted discretisation inherits the rate the underlying refinement attains, with cut-independent constants.

math.NA↗
arXiv2026

A review of weakly enforced Dirichlet boundary conditions in computational flow analysis

Strongly enforced Dirichlet boundary conditions require highly refined near-wall meshes to resolve steep velocity and thermal gradients. This introduces high computational costs, especially for practical flow simulations. Weakly enforced boundary conditions alleviate this burden by acting as a variationally consistent near-wall model. By allowing a controlled slip at the solid wall, weak enforcement recovers accurate flow quantities on coarse boundary-layer meshes across both incompressible and compressible regimes. Furthermore, weak boundary conditions serve as the fundamental enabling technology for immersogeometric analysis. Because the weak operator evaluates boundary integrals independently of the background mesh, high-fidelity flow analysis can be performed directly on complex geometries without fitting a mesh to the surface. This capability has facilitated direct geometry-to-analysis workflows for boundary-representation CAD models, raw point clouds, photogrammetric reconstructions, and segmented medical images. This review examines the unified mathematical development of the weak boundary condition framework from scalar advection-diffusion equations to the full Navier-Stokes equations, illustrating its versatility and robustness across incompressible and compressible flows, whether in traditional boundary-fitted, sliding-interface, or advanced immersogeometric applications.

math.NA↗
arXiv2026

An Inverse Problem for Determining the Piston Speed from a Given Lipschitz Leading Shock

We analyze an inverse problem for determining the piston speed and the associated flow field from a prescribed leading shock and the initial data in a shock tube. The gas flow is described by the isentropic Euler equations (i.e., the $p$-system), while the trajectory of the leading shock is prescribed as a given Lipschitz curve. Under an Oleĭnik-type entropy condition on the leading shock, we develop a modified wavefront tracking scheme to construct the flow field behind the shock. This construction enables us to determine the corresponding piston speed and the associated flow field.

math.AP↗
arXiv2026

Rough variational principles and applications to adjoint systems

We consider a Type-II variational principle driven by geometric rough path with split boundary conditions naturally suited to adjoint systems. From this rough variational principle we derive rough Hamilton's equations, establish their pathwise conservation laws and associated Hamilton--Jacobi equation. We then specialise the framework to rough adjoint systems, obtaining pathwise conservation and quasi-conservation laws that underpin adjoint sensitivity analysis with respect to initial conditions and parameters. On the discrete side, we construct a rough Galerkin discretisation of the rough Type-II variational principle and show that it generates a symplectic flow with discrete analogues of the continuous conservation laws. We establish it's equivalence to a class of Rough Symplectic Partitioned Runge--Kutta (RSPRK) methods and analyse its convergence and naturality properties. Lastly, we perform numerical experiments to validate the predicted convergence rates and demonstrate that RSPRK methods preserve the adjoint conservation laws to machine precision, yielding more accurate and stable gradients in optimisation problems than non-symplectic alternatives.

math.NA↗