Forsythe's conjecture resolved for all restart lengths
A 56-year-old open problem about restarted CG convergence is now completely classified.
· “A Complete Resolution of Forsythe's Conjecture for Restarted Conjugate Gradients”
cs.NA
Numerical Analysis
cs.NA is an alias for math.NA. Roughly includes material in ACM Subject Class G.1.
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Spectral type needs two or three limiting stages
For smooth Schrödinger operators, the number of nested limits needed for each spectral part is now established.
· “Sharp Computational Bounds for Spectral Types of Schr\"odinger Operators”
SparseStack embedding matches optimal bounds with explicit constants
New proof via moment method and tensor-product model settles Nelson–Nguyen conjecture.
Pauli principle gives one full order of mixed Barron regularity
Index-set-dependent weights detect smoothness invisible to the isotropic Barron scale; the result is uniformly sharp.
· “Sharp Mixed Spectral Barron Regularity of Coulombic Many-Electron Wave Functions”
Longest-edge refinement adds only a fixed multiple of marked triangles
A closure estimate now bounds new elements by marked triangles across every adaptive step and all tie choices.
· “Closure complexity of longest-edge bisection for triangular meshes”
New transversality rules when boundary sets are orbits
The result fills a gap for three-body trajectory problems, where endpoint orbits resist algebraic equations.
· “Transversality Conditions for Boundary Constraints Defined by Differential Equations”
New SPDE scheme doubles spatial rate to 3/2
Fully discrete method for semilinear SPDEs cuts computational cost from δ⁻³ to δ⁻⁵/³.
· “Overcoming the spatial order barrier for nonlinear SPDEs with additive space-time white noise”
2n²+1 model runs deliver all four response moments
A vine-copula sigma-point scheme matches million-sample Monte Carlo on skewness and kurtosis.
· “Quadratic Point Estimate Method for Uncertainty Quantification with Dependent Non-Gaussian Inputs”
Active subspace sampling cut by RKHS smoothness
A direct projection-error analysis shows sample complexity drops from $\varepsilon^{-2}$ to $\varepsilon^{-2\theta}$ when gradients are…
One condition guarantees unique bridge deflection and fast iteration
The classical Melan suspension-bridge equation has a unique solution under realistic loads, computable by a simple proven algorithm.
CutDG with macro-element limiters preserves maximum principle for conservation laws
Macro-element ghost penalty and flux limiters avoid severe CFL restrictions from small cut cells, achieving robust high-order accuracy.
· “High-Order Discontinuous Cut Finite Element Methods for Scalar Hyperbolic Conservation Laws”
Tamed scheme hits rate 1/2 for hyperbolic SPDEs
First pathwise uniform strong convergence for explicit time-stepping with superlinear drift and multiplicative noise.
· “Strong convergence rates of tamed exponential Euler schemes for superlinear hyperbolic SPDEs”
Pinpoint every scatterer from sparse aperture samples
Geometry-aware Hankel completion fills in missing data; samples needed grow linearly with the number of scatterers.
· “Off-Grid Point-Scatterer Localization from Sparse Limited-Aperture Data via Hankel Completion”
ELM domain decomposition achieves weak scalability for PDEs
FD3 method with GenEO coarse spaces keeps condition number bounded across 4096 subdomains, enabling efficient parallel PDE solving.
Cut tumor-model unknowns to under 3% with adaptive splines
Fourth-order tumor-growth PDEs meet MRI-based breast geometry at reference-level accuracy.
A selector learns when a cheaper boundary law is safe
Paired solves train a network to predict interior and boundary errors, and the expensive full law runs only when those tolerances fail.
· “Selective boundary condition reduction via learned error gating”
Picard iteration finds trapped regions in nonspherical spacetimes
Spectral-element method evolves strongly asymmetric characteristic data, validated by exact solutions and independent residual checks.
· “Picard Iteration for the Characteristic Initial Value Problem in Einstein Equations”
Provably stable scheme computed for damped flexible beams
Splitting sectional forces into elastic and viscous parts removes mixed derivatives and yields a discrete energy law.
· “A Discontinuous Galerkin Method for the Intrinsic Beam Model with Kelvin-Voigt Damping”
Implicit scheme outruns CFL for multicomponent sorption transport
Compact high-resolution method achieves second-order accuracy with Courant numbers up to 12 in Langmuir systems.
Proven stable FEM for submerged plate wave energy models
Rigorous error bounds and open-source Julia code enable reliable simulation of hydroelastic plates for wave energy converters.
· “A discontinuous finite element method for the hydroelastic analysis of submerged structures”
Free-boundary positivity proved under mild time-step condition
Implicit schemes avoid the quadratic time-step restriction of earlier methods, enabling efficient simulation of population spread.
· “Finite difference methods for three kinds of reaction-diffusion equations with free boundaries”
Bounded correction factor: Halley's cubic speed
A positive correction factor stays in (0,2) for every input, so the iteration never reverses or explodes.
· “Third-order Halley-type iterative method with positive and bounded correction function”
A precision bound guarantees CG's backward error within n steps
Exact rounding examples show equivalent solvers can diverge; the same analysis gives a stopping test and a rank audit.
Third-medium contact couples current
A unified FE framework models connectors and switches without explicit surface tracking
· “A third-medium approach for electro-thermo-mechanical contact considering Joule heating”
Lookahead ordering makes Fourier–Motzkin 6–25× faster
A sound combination of redundancy tests plus intelligent variable choice cuts the double-exponential blowup in practice.
Laplace double layer operator defies 30-year-old spectral bound
Computer-assisted construction of a Lipschitz domain shows essential spectral radius exceeds 1/2, disproving Kenig's conjecture.
· “A counterexample to Kenig's conjecture for the Laplace double-layer operator”
Adding fractional memory to threshold dynamics speeds grain coarsening
Each smoothing step acts on a convex average of all past states; quantified bounds control old-state memory and pinning.
MCP plus reweighted ℓ1 boosts Poisson image deblurring
A nonconvex model with closed‑form ADMM updates outperforms BM3D, OGS, and FOTV on several test images.
Nudged Chebyshev nodes keep kernel compression stable
New stability bounds let kernel-matrix factorization run on scattered, nonconvex point clouds where rigid grids fail.
Numerical scheme computes HCIZ integral asymptotics
A convergent particle discretization solves the Matytsin hydrodynamical problem for generic densities.
· “Efficient computation of the asymptotics of extensive-rank HCIZ integrals”
Keep the data, change the representation: a unifying view
Four kernel methods share one invariant – preserve the original samples while transforming the representation – enabling hybrid designs…
· “Mapping for Approximation: A Unified View of Rescaled, Variably Scaled and Rational Kernel Methods”
Model cuts 3D concrete buildability error from 59% to 24%
Friction-aware yield with hardening predicts printed-wall collapse to within one layer.
Spectral method proven optimal for 2D internal wave models
Fourier-Galerkin with implicit midpoint rule converges at theoreticaly optimal rates under smoothness conditions.
Stabilized Crouzeix-Raviart element solves surface Stokes flow
Optimal first-order convergence for velocity and pressure on curved surfaces, proven and tested.
· “Analysis of a Surface Crouzeix-Raviart Element for the Stokes Problem”
Schemes keep reactive sedimentation simulations physically valid
High-order finite volume methods for denitrification in settling tanks preserve nonnegativity and mass balance, proven without diffusion.
· “Invariant-region-preserving high-order schemes for a model of reactive sedimentation”
AI adds 5 percentage points to water saving in smart agriculture
Monte Carlo simulations quantify marginal green contribution of each module; farmer adoption, not accuracy, is the binding constraint
dG scheme provably matches particle fluctuation moments at O(h^p)
High-density regime Nh^d >> 1 makes the extra error exponentially small, replacing costly particle simulation.
· “A discontinuous Galerkin approximation of the Dean--Kawasaki equation”
Two scales identify complex spectra where one fails
Dilation-covariant Hankel pencils enable certifiable recovery of sparse Mellin models from geometric samples.
· “Dilation-Covariant Hankel Pencils for Multiscale Recovery of Sparse Mellin Spectra”
Cubic convergence for matrix Halley method
Generalizes scalar result with explicit matrix Schwarzian derivative, no gradient-field assumption needed.
· “Halley's Method for Rectangular Matrix Variables and the Matrix Schwarzian Derivative”
Link a velocity field to its flow map to shrink the search space
The induced flow map gives fluid models a consistency test that state matching alone cannot provide.
Random pivot selection matches optimal low-rank rates
The core result is a bound on the expected residual trace after k iterations of γ-approximate RPCholesky: E[tr $A^{{(k-1)}}$] ≤ $γ^{{-1}}$…
· “Convergence rates of randomly pivoted methods for low-rank approximation”
Initial-state overlap sets tQPE sampling limit
Numerical study shows that once Trotter error is low, more resources don't improve the phase-sampling rate—the initial state is the…
One inequality controls both FEM spectra and rational approximation
Weighted Laplace spaces give a simple comparison of continuous and discrete spectra, with a conditional path to rational covariance…
· “Weighted Laplace Spaces for Spectral Measures and Rational Approximation”
Breaks the curse of dimensionality: densities recovered at 150 assets
Cost grows linearly in the number of assets, and basket-option prices stay accurate all the way up.
· “Fourier-cosine Tensor Trains for Density Recovery and Expectation Calculation”
Finite convolution transform for hypergeometric series
Divided differences yield a new identity that expresses an r+2F_{r+1}(1) series as a finite sum of r+1F_r(1) series.
· “Terminating Zero-Balanced Hypergeometric Series Using Divided Differences”
Levin collocation method sums infinite oscillatory series
Transforms the series into a functional equation on [0,1], recovers the sum from a collocation approximation at the origin.
· “The Levin Method for the Summation of One-dimensional and Multidimensional Infinite Series”
Top-mounted porous baffles cut sloshing by 50 percent
A scaled-boundary finite element study ranks baffle shapes for partially filled swaying tanks.
Second-order scheme for Cahn-Hilliard preserves positivity and energy law
Rigorous convergence analysis for a model with variable mobility and singular logarithmic free energy
Sharp h in 2D, h^{1/2} in 3D for polytopic DG of white-noise SPDEs
New analysis proves symmetric interior-penalty method achieves optimal first-order strong error on general polygon and polyhedron meshes…
· “Polytopic symmetric interior-penalty approximation of elliptic SPDEs driven by spatial white noise”
One eigenvalue bounds Poisson L2 errors on anisotropic meshes
The same mesh constant controls conforming and Crouzeix-Raviart errors, with no global convexity or H2 assumption.
· “Crouzeix--Raviart--Marini realisation of computable a priori L²-error bounds on anisotropic meshes”
Farthest-point sampling needs N log N candidates for quasi-uniformity
Proof shows logarithmic oversampling is unavoidable when selecting points from iid candidates.
A median of scrambled Sobol' nets tames high-dimensional integration
Coarse scrambling shrinks the gain coefficient; the median then beats Monte Carlo's rate in every dimension.
· “Robust high-dimensional integration using medians of coarsely scrambled Sobol' sequences”
New IMEX scheme keeps energy and positivity for thin-film flows
A numerical method for a hyperbolic relaxation system that remains stable and consistent as relaxation vanishes, validated on…
New smoothing conditions tame singular memory kernels
High-order collocation reaches rate m once the nonlinear exponent vanishes at the initial value.
Wave solver cuts per-patch unknowns from quadratic to linear growth
Enforcing smoothness first keeps high-frequency, variable-material waves accurate with far fewer unknowns.
300-fold speedup: one trained network replaces repeated cell solves
Trained once, the cell response stays differentiable in temperature and gradient, so multiscale Newton steps need no re-solves.
· “Nonlinear elliptic homogenization with the parametric Deep Ritz method”
New boundary conditions slash DNS cost for flame instability simulations
Truncated-domain method reproduces the full instability cascade—including cellular parametric flames—at 80–98% lower cost.
This paper finds the smallest possible rounding error when computing a matrix product…
A gauge-invariant lower bound on quantized matrix multiplication error is established, and almost optimal transforms that achieve it are…
· “A Nuclear-Norm Lower Bound for Dithered Scalar Quantization of Matrix Products”
Shallow ReLU^k networks achieve optimal mixed-smoothness rates
Worst-case L2 error decays as n^{-min(α,k+1)} up to log factors, with matching lower bound for any dimension.
· “Shallow neural network approximation in mixed Sobolev spaces”
Five tweaks let a general LLM beat Bitcoin forecasters
Systematic study shows fine-tuning, prompting, and decoding choices matter more than model specialization for short-horizon crypto price…
· “PRICE: A Systematic Study of LLM Adaptation Choices for Bitcoin Price Forecasting”
Two exponentials hold a shallow net's error flat as layers thin to 1e-11
Putting known boundary-layer shapes into the trial space keeps errors flat as parameters fall from 1e-3 to 1e-11.
· “A simple shallow neural network for emulating the solution to singularly perturbed problems”
Mobile ions slow-shift long-lived KEEN waves
A multi-map flow solver shows ions leave a measurable late-time imprint on the electron phase space.
High-order multiscale FEM cuts cost for high-frequency heterogeneous scattering
New error analysis shows increasing polynomial degree reduces the pollution term, making high-frequency simulations affordable.
Local Fortin operators gain trace preservation for Stokes elements
A unifying correction framework yields divergence-free, trace-preserving projections for Bernardi-Raugel, Taylor-Hood, and more.
Two tweaks turn flash attention into exact Gauss sums
The kernel sums behind MMD and SVGD flows become an attention call with linear memory and faster fp16 results for D>8.
Structure-preserving splittings match energy to order h^{p+1}
Order-two splits keep the power balance when every subproblem stays port-Hamiltonian; the diagonal decomposition stays fast without losing…
· “Energy-Consistent Splitting and Decomposition Approaches for Coupled port-Hamiltonian ODEs”
Computation backs Hardy's lattice-point conjecture to 10^18
Numerical exponent <0.261 supports E(t)=O(t^{1/4+ε}) and suggests log-factor refinements.
· “Numerical experiments on the Hardy conjecture for the Gauss circle problem”
Operator DG method delivers infinite entropy conditions
High‑order schemes proven to satisfy infinitely many entropy inequalities while preserving optimal accuracy and strong convergence for…
Eigenmodes of a fake structure give compact airfoil design variables
A tunable pseudo-structure lets engineers embed geometric constraints before optimisation, cutting variable count to single digits while…
· “Geometry Parameterisation via a Structural Modal Basis for Aerodynamic Shape Optimisation”
Shadow cells make adaptive refinement run like a uniform GPU mesh
A proven identity replaces subface quadrature with one uniform face kernel; iterations stay flat as refinement deepens.
· “Adaptive Multilevel Discontinuous Galerkin Methods on GPUs”
Adding rotation to learned spin dynamics cuts rollout error 17-fold
Energy-gradient neural ODEs need an antisymmetric term to match long magnetization rollouts; damping alone drifts.
Matrix inversion jumps to sixth order at fourth-order cost
A scalar norm-based accelerator cuts per-step matrix products from six to four, lifting the efficiency index.
· “Improving the efficiency of a Hyperpower method for approximating inverse matrices”
New analysis cuts iteration count from O(p²) to O(p) for Lewis weights
Key insight: each update contracts KL divergence by factor 1 − 2/p, enabling high-precision guarantee for all p > 2.
· “Computing Lewis Weights to High Precision by Fixed-Point Iteration”
Posterior for Bayesian PDE solver is Gaussian at √N in strong norm
A Bernstein–von Mises theorem shows the infinite-dimensional posterior converges in Sobolev spaces, enabling uncertainty quantification at…
· “Bernstein--von Mises theorems for Bayesian probabilistic numerics”
Cut-mesh multigrid converges uniformly in mesh size and geometry
Two-level proof with vertex-patch smoothing holds however the boundary slices the grid; degree cost is explicit
Well-conditioned inputs can cause ill-conditioned Sylvester solution
The condition number of the solution is not controlled by condition numbers of A, B, or C, as shown by a 3×3 example.
Filament shape shifts3D concrete buildability by up to80 percent
Elliptical layer models match real collapse height within one layer; rectangles can mislead by tens of percent.
One framework learns hidden dissolution laws and cuts prediction error sevenfold
A differentiable hybrid model for crystal dissolution discovers physically consistent kinetics and optimizes temperature profiles in a…
Flux-corrected scheme keeps second-order accuracy and no oscillations
It provably reaches second-order accuracy for nonsmooth data while obeying the discrete maximum principle.
Mini element escapes locking in nearly incompressible elasticity
Errors stay flat as λ grows to 10⁶, and the new eigensolver's convergence rate never depends on λ.
· “Mini mixed finite element method for nearly incompressible linear elasticity problems”
