New papers on Numerical methods
110 new papers on numerical methods in the last 7 days, within Math & statistics. These are the 50 Pipette rates most worth reading, with the main result in the authors' own words.
The best of the week
Infectious behaviour: Simulating the effects of communication and social influence on pathogen transmission in crowds
In a ticket-checkpoint queue of 101 agents, effective communication combined with strong shared social identity with stewards reduces the number of highly exposed agents by 82% for mask wearing.
PreprintReal-world useA Contour Method for Multiparameter Eigenvalue Problems
In this work we develop the first contour method for analytic multiparameter eigenvalue problems.
PreprintBold claims, read criticallyClaims a big stepConvergence analysis of a numerical scheme for the Cahn-Hilliard-Navier-Stokes system with dynamical boundary condition and its application to moving contact line problem
In our knowledge, this work provides a theoretical proof of convergence analysis and error estimate for a numerical scheme to the moving contact line problem, for the first time in the literature.
PreprintClaims a big stepFinite Volume Element Method on Curved-Edge Meshes
Unlike existing theories, which are primarily based on straight-edge meshes, this study is the first to establish an analysis of the stability and optimal convergence of the finite volume method on curved-edge meshes.
PreprintComputation of anisotropic singular sums from high-order derivatives of Epstein zeta functions
We derive a stably computable representation of anisotropic Epstein zeta functions, including the possibility for analytically removing Rayleigh--Wood singularities, and we develop a numerical algorithm for their stable evaluation for any lattice, power-law decay exponent and anisotropy order.
PreprintTangential stability and fully discrete convergence of the classical BGN scheme for curve shortening flow
We prove fully discrete convergence of the classical Barrett--Garcke--N\"urnberg (BGN) scheme for curve-shortening flow of smooth embedded closed planar curves.
PreprintClaims a big stepAn adaptive localized orthogonal decomposition method for Helmholtz problems
The main result is the first wavenumber-robust residual a posteriori error estimate for a Helmholtz LOD discretization within the standard resolution, oversampling, and discrete-stability regime.
PreprintSparsely connected neural network representation of Lagrange finite element function
We construct a mesh-induced sparsely connected neural network framework that exactly reproduces arbitrary-order Lagrange finite element spaces over simplicial meshes.
PreprintBold claims, read criticallyTessellated Isotropic Elastic Lattice Spring Model for Quasi-Brittle Fracture
We propose a tessellated Isotropic Elastic Lattice Spring Model (IELSM) that discretizes continua into polygonal elements with axial springs and a volumetric constraint, achieving isotropic elasticity on arbitrary polygonal tessellations.
PreprintOn degeneration of tetrahedra under longest-edge n-section refinement
For every integer , we construct explicit tetrahedral counterexamples showing that repeated longest-edge (LE) n-section refinement does not, in general, preserve shape regularity, contrary to long-standing conjectural expectations.
PreprintElucidating the Conformal Structure of the Brinkman Penalisation Method for Geometry-Adapted, Structure-Preserving Operator Learning of Hamiltonian PDEs
We show that multi-symplectic Hamiltonian PDEs regularised by Brinkman-type penalisation retain a multi-conformal symplectic structure under a compatibility condition linking the symplectic matrix and the penalisation projection.
PreprintUniform Chebyshev asymptotics for repeated-pole rational approximation of the exponential
For , , the best uniform error has two-sided order ; at the optimal ratio this becomes .
PreprintHutch#: Optimal non-adaptive Frobenius norm estimation
We prove that this estimator yields a multiplicative approximation to when , a quadratic improvement over Girard--Hutchinson.
PreprintFully Discrete Multi-Entropy Stability of High-Order Schemes for Compressible MHD: A Weak-to-Strong Framework
We establish a fully discrete weak-to-strong (W2S) multi-entropy stability theory for arbitrarily high-order finite-volume and discontinuous Galerkin (DG) approximations of the ideal compressible magnetohydrodynamics (MHD) equations on general polytopal meshes, whereby a single numerical update simultaneously satisfies discrete entropy inequalities for any prescribed finite family of convex Harten entropy pairs.
PreprintMIISO: Modal Integrators for Isogeometric Stabilization of Outliers
We introduce MIISO (Modal Integrators for Isogeometric Stabilization of Outliers), a family of exponential Rosenbrock-Krylov integrators that suppress outlier modes temporally while leaving the spatial discretization unmodified.
PreprintIntegral chemical reaction neural networks
In this work, we present integral chemical reaction neural networks (iCRNNs), a framework that combines the interpretable, physics-constrained, architecture of chemical reaction neural networks (CRNNs) with an integral collocation formulation of the governing dynamics.
PreprintA structure-preserving staggered semi-implicit four-split finite volume scheme for continuum mechanics on unstructured meshes
We present a new semi-implicit structure-preserving (SP) finite volume discretization for a unified first-order hyperbolic model of continuum mechanics.
PreprintA nonmonotone globalization framework for Anderson acceleration for contractive and nonexpansive fixed point problems
We propose a nonmonotone globalized Anderson acceleration framework that retains the local acceleration of AA while ensuring global convergence.
PreprintBlind error estimation for CUR approximation
We derive a "blind" estimator of the Frobenius error of a CUR decomposition.
PreprintCost-Accuracy Trade-offs: Neural Operator vs Classical Numerical Solver
Neural operators are most competitive at low-to-moderate accuracy requirements.
PreprintFrFNO:Injecting the analytic Mittag-Leffler propagator into a resolution-robust neural operator for space-time fractional PDEs
The central theoretical result is that under zero-shot super-resolution the injected propagator fills the out-of-band modes that a standard Fourier neural operator sets to zero, replacing the out-of-band full-field tail by the smaller residual tail (a -independent constant-factor reduction under weak perturbation).
PreprintBold claims, read criticallyCode availableDomain-decomposed Evolutional Deep Neural Network with Random Features for Transient Pressure Diffusion with Discontinuous and High-Contrast Coefficients
We develop a domain-decomposed random-feature evolutional deep neural network (DD RF-EDNN) that separates spatial approximation from temporal evolution.
PreprintEnergy stability and error estimates for a second-order structure-preserving exponential integrator method for smectic-A liquid crystals
First, to the best of our knowledge, we propose the first integration of the generalized scalar auxiliary variable (GSAV) approach with a second-order exponential time-differencing Runge--Kutta (ETDRK2) discretization, leading to a second-order GSAV--ETD2 scheme.
PreprintBernstein Constraint Complexes for Multivariate Splines on Triangulated Surfaces
We introduce Bernstein constraint complexes, a representation of finite element differential complexes in which every triangle of a triangulation keeps its own Bernstein--B\'ezier coefficients and global continuity is imposed only through smoothness functionals attached to edges and vertices.
PreprintAccurate wall shear stress in immersed flow analysis with application to point cloud-based CFD
In this work, we propose a new method to obtain accurate wall shear stress in immersed flow analysis with application to point cloud-based CFD, using a non-symmetric Nitsche's formulation with near-wall modeling and a patch-based stress recovery approach with traction compatibility.
PreprintReal-world useTensor Decomposition of Transformer Key-Value Caches: Spectral Structure and Format Comparison
Among the four decompositions, Tucker achieves the lowest reconstruction error at every compression ratio from to , because it can leave the full-rank modes untouched.
PreprintOptimal Recovery for Solving Variational Problems
This works presents a two-step procedure for variational energy minimization based on the optimal recovery formulation in a reproducing kernel Hilbert space (RKHS), providing a unified framework for seamlessly incorporating both physical constraints and noisy data.
PreprintAutoencoders vs. Numerical Analysis--Informed Manifold Learning for Navier--Stokes Flows
The resulting nonlinear ROM substantially outperforms POD-based ROMs and achieves reconstruction and prediction accuracy comparable to ---and, in some bifurcating regimes, better than--- that of AE-based ROMs.
PreprintA Second-Order Maximum-Bound-Preserving and Energy-Stable Exponential Time-Differencing Method for Allen--Cahn-Type Gradient Flows
We prove that the proposed scheme unconditionally preserves both the MBP and energy stability.
PreprintAn efficient 0D-space conservative and positivity preserving battery model
We propose a conservative, positivity-preserving model built from the P2D system based on the statement that the Butler-Volmer source term is constant in each subdomain (anode, separator, cathode).
PreprintReal-world useThree-dimensional blind deconvolution by CP-parameterized kernels
The problem is approached by imposing a semiparametric CP decomposition for the kernel and a variational TV penalty for the image.
PreprintConformal Spherical Splines for the Laplace--Beltrami Operator on Genus-Zero Surfaces: Construction, Algorithm, and Experiments
This paper solves the Poisson equation and the eigenvalue problem of the Laplace--Beltrami operator of such a metric with smooth spherical splines, taking the conformal factor as the only description of the geometry.
PreprintHistory-Compatible Energy-Stable Finite Element Schemes for Variable-Density Cahn--Hilliard--Navier--Stokes Flows on Evolving Meshes
We develop decoupled backward Euler (BE) and second-order backward differentiation formula (BDF2) schemes by combining exact physical cross-mesh pairings with history representations compatible with the corresponding phase-energy, kinetic-energy, and pressure-gradient storages.
PreprintHigh-dimensional extreme eigenvalue problems: low-rank tensor parametrization and optimization
Instead of tackling the problem in high-dimensional ambient space, we reformulate the problem through low-rank tensor formats, which can significantly reduce the computational cost and storage.
PreprintHigh-order mass-, energy- and momentum-conserving methods for the nonlinear Schr\"odinger equation
This paper introduces a novel formulation and an associated space-time finite element method for simulating solutions to the nonlinear Schr\"odinger equation.
PreprintA Low-rank ADI Algorithm for the Numerical Solution of Large Discrete-time Non-symmetric Algebraic Riccati Equations
A low-rank alternating direction implicit (ADI) method is introduced that recursively constructs a low-rank stabilizing solution without explicitly solving any projected DTNARE.
PreprintFSI modeling of case-specific nonlinear carotid artery mechanics and the role of outlet boundary conditions
The present study demonstrates the feasibility of strain-dependent Young's (elastic) modulus as a means to enhance the capacity of the linear elastic framework to accurately represent the physiologically nonlinear mechanics of arterial walls, striking a balance between implementation effort and physiological fidelity.
PreprintEnhanced attenuation modelling for multispectral computed tomography
With this we both derive new uniqueness conditions for the material decomposition and also offer a three-step reconstruction model.
Preprint with a published versionConvergence of a fully discrete finite element method for the Beris-Edwards system of liquid crystal dynamics
Our main result is that, as the mesh size and the time step tend to zero subject to , the approximations converge along a subsequence to a weak solution of the Beris-Edwards system.
PreprintAn Implementation-Friendly SDG Scheme based on Cartesian Grids for Stokes Equations with Pressure Robustness and Superconvergence
This paper develops a staggered discontinuous Galerkin (SDG) scheme based on Cartesian grids for Stokes equations that is simple to implement, intrinsically pressure-robust, and superconvergent for all variables.
PreprintAnisotropic Kernel-based Multilevel Interpolation of High-Dimensional Functions on Sparse Grids
On this basis we prove error estimates in mixed-regularity Sobolev and continuity norms, extending the existing -theory, and we characterise the full range of anisotropy weights that attain the optimal rate against the number of degrees of freedom.
PreprintA diffusion-free multi-layer neural-network method for multidimensional nonlinear hyperbolic equations
We develop the LeafNet algorithm, which relies on: i) a reformulation of HCL as a coupled system involving smooth solutions, and ii) an accurate physics informed algorithms approximating smooth functions.
PreprintCommunication-Efficient Distributed Training via Ring-Based Coded Approximate All-Reduce
In this work we present a communication-efficient Ring All-Reduce (CERAR) protocol for "approximate" gradient aggregation.
PreprintA Nonlinear Physics-based Reduced Order Model with Convolutional-based Operator Compression
Rather than compressing only the solution field, we learn nonlinear low-dimensional representations of the full-order operators while retaining an explicit reduced system of equations.
PreprintAn Iterative Active Subspace Approach for Model Order Reduction of Parametric Systems with High-Dimensional Parameter Spaces
In this paper, we propose an iterative active subspace (IAS) approach for parametric model order reduction, which, to some extent, addresses the trade-off between accuracy and reduced model size and achieves substantial computational gains compared to the original active subspace method.
PreprintDynamic Ritz projection of mean curvature flow and optimal convergence of parametric FEM
Leveraging these results, optimal-order convergence of parametric finite element methods for mean curvature flow of closed surfaces in the norm is proved, including the convergence of parametric finite element methods with piecewise linear finite elements.
PreprintHDG methods in finite element exterior calculus
We develop and analyze HDG methods for two central problems in finite element exterior calculus, the Hodge-Dirac problem and the Hodge-Laplace problem, in arbitrary dimension .
PreprintApproximating Pi (and other constants) by Radicals in the Footsteps of Rabbi Abraham Ibn Ezra and Guru RSJ Reddy
Inspired by the polymath Abraham Ibn Ezra (1089-1167) who approximated {\pi} by 20/9 times the square-root of 2, and contemporary scholar, RSJ Reddy, who gave (14-sqrt(2))/4, we point out that both of these approximations are optimal in some sense, and we generate many other ones that are even more optimal, albeit not as nice.
PreprintA Regularization Based Computational Method for Quantum Incommensurate Problems
Based on the regularized model recently proposed, this work introduces a regularization framework, rendering physical observables for incommensurate systems mathematically well-defined and computationally accessible with theoretical guarantees.
PreprintTwo-Dimensional Shallow Water Linearized Moment Equations: Hyperbolicity and Well-Balanced Schemes
We construct a rotationally invariant modification and prove its global hyperbolicity for all states with positive water depth.
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