Numerical Analysis
Numerical algorithms for problems in analysis and algebra, scientific computation
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HLL flux with entropy-stable speeds plus locally divergence-free projection achieve positivity, entropy stability and divergence control in
· “A positivity preserving and entropy stable nodal discontinuous Galerkin scheme for ideal MHD”
Kronecker-factored transitions cut complexity from exponential to linear while keeping global dynamic programming solutions intact.
When matrices or a matrix and tensor share a common factor their joint decomposition reduces to a single singular value decomposition, with
Euler-Poisson test passes convergence, Jeans-instability, and Sedov blast-wave checks.
· “Multi-Solver Coupling for Parallel Adaptive Multi-Physics Simulations with Trixi.jl and deal.II”
Data-dependent Parseval frames capture localized multiscale features more effectively than fixed global bases while retaining FFT speed.
· “Don't Fix the Basis -- Learn It: Spectral Representation with Adaptive Basis Learning for PDEs”
Two-gradient scheme matches prior convergence guarantees for targets that are merely log-smooth rather than strongly convex.
MgNet keeps recursive skeleton factorization but trains neural networks to replace its coefficient-specific operators, gaining speed and the
· “A Filtered MgNet Solver For Radiative Transfer Equations”
Entropy-stable and mortar-based interface treatments extend high-order accuracy and admissibility to adaptive grids with nonconforming 2:1 h
Proves well-posedness, optimal convergence, and reliable residual-based error estimators for the coupled system.
· “Nitsche method for the Stokes-Poisson-Boltzmann equation with Navier slip boundary condition”
A zero-valued auxiliary variable restores energy laws and second-order H^{2} error bounds for free
· “A Regularized Auxiliary Variable (RAV) Approach for Gradient Flows”
Nesterov-accelerated projection with subspace embeddings preserves accuracy while cutting full-model runtime from over a minute to under 3 s
Fine-tuned models inside a feedback loop outperform larger general models on verified FEniCS scripts for solids, fluids, and multiphysics.
· “ALL-FEM: Agentic Large Language models Fine-tuned for Finite Element Methods”
A kernel filter pulls any iterated integral from a path in parallel time that no longer grows with depth.
Maximal regularity gives strong-norm error bounds and a residual-based estimator that reaches the same order.
Mirror descent and proximal mirror descent with logarithmic barriers achieve a tight O(log k / k) convergence rate when the optimum lies on…
Harmonic-mean weights give the tightest majorizer of the smoothed nuclear norm; one-sided IRLS cannot match its local rate.
· “Tight Majorizations and Convergence Rates of Nuclear Norm Minimization IRLS”
The three-dimensional case settles the 1/6 conjecture for the Neumann–Poincaré operator.
· “A sharp isoperimetric inequality for the Neumann--Poincar\'e operator in every dimension”
Proof shows expected trace error within a 1+ε factor of the optimal rank-r fit in O(r/ε + r√log r) steps.
· “A new analysis of the randomly pivoted Cholesky algorithm”
First rigorous error analysis of an unfitted FEM for fully coupled FSI with a moving interface.
· “Convergence of a CutFEM for fluid--structure interaction with a deforming interface”
The bound is sharp for both BB rules, in Hilbert space, and near nonlinear minimizers.
Auxiliary variables let the continuous area dissipation law hold exactly after arbitrary-order discretization.
· “Arbitrary-order structure-preserving discretizations for geometric curvature flows”
A greedy index-picker for contact forces plus a learned correction keeps agents from overlapping in dense crowds
· “Model order reduction for parametrized variational inequalities: application to crowd motion”
New second-order IMEX method for Cahn-Hilliard-Navier-Stokes needs only linear solves and preserves energy decay.
Offloading finite-element solves of random-material SPDE sampling crosses performance, bringing risk-averse optimization closer to practice.
· “Risk-averse Optimization in Random Materials: Algorithmic Advances and HPC Acceleration”
The paper reports numerical demonstrations of budgeted multilevel Monte Carlo and multilevel stochastic gradient descent for…
· “Optimized Multilevel Sampling Methods under Resource Constraints”
Lower bounds from tight frames match known constructions, settling L2 Marcinkiewicz–Zygmund sample complexity.
· “Required Number of Points in L₂ Marcinkiewicz-Zygmund Inequalities”
Total-variation penalties localize inclusions most sharply; smooth penalties blur them, sparse ones distort them.
Carleman linearization plus a dissipative lifted system keeps final-state preparation cost independent of T.
Symbol-based circulant blocks keep iteration counts fixed as the space-time mesh shrinks.
Low-order DeC iterations run on fewer subtimenodes; an adaptive rule picks the order automatically.
· “New Efficient Implicit-Explicit Deferred Correction methods”
The paper turns the discrete sequence into chains and co-chains whose maps are algebraic and read the mesh geometry.
· “A Vector-Valued Co-Chain/Chain Complex Associated to the Elasticity Complex”
On every mesh edge, PWDG jump terms equal half the UWVF transmission defect, explaining a classical equivalence.
Randomized vortices converge in Wasserstein distance to a measure solving the Euler statistical hierarchy.
· “Computing statistical Euler limits of the Navier--Stokes equations in three dimensions”
An unfitted RBF partition-of-unity method computes deformation of real diaphragms with a provable algebraic error rate.
· “The least squares RBF-PU method for linear elasticity in the diaphragm geometry”
A neural operator that splits solutions into a manifold coordinate and a residual beats FNO over long rollouts.
· “Inertial Manifold Neural Operator for Dissipative Time-Dependent Partial Differential Equations”
A new proof gives explicit remainder terms and uniform eigenfunction bounds for every fractional order between 0 and 2.
Compressed tensor-train and hierarchical Tucker factors replace full tensors, so high-order multilinear algebra can scale.
· “Transform-Based Multilinear Algebra via Tensor Decomposition”
Linear-relation machinery puts Eckart–Young, pseudoinverse, and restricted low-rank problems under one theorem.
· “Least-Squares and Low-Rank Approximation for Linear Relations Using a Diagrammatic Language”
Two valid bisections reproduce the shape at half scale, forcing one dihedral angle to zero and another to pi.
· “The longest-edge bisection algorithm may produce degenerating tetrahedra”
Dual-horizon equations fall out of the action principle; asynchronous stepping cuts force evaluations by 30%.
One fixed configuration solves 3D wave equations across smooth, discontinuous, and geophysical media—no retuning
· “A Massively Parallel Three-Grid Preconditioner for the High-Frequency Helmholtz Equation”
A factorization-free three-grid preconditioner solves 10.89-billion-unknown time-harmonic Maxwell systems on 64 A100 GPUs in under 72…
A particle method for the Boltzmann equation samples the finite-time transition of a lifted linear operator, learning the needed Green's…
Every fully semimonotone matrix with positive determinant is a P_0-matrix, for arbitrary order n.
The paper proves that symmetric Taylor-Hood finite elements are inf-sup stable for the linear stress-gradient elasticity problem in 2D and…
· “Symmetric Taylor--Hood elements for the linear stress gradient problem”
A positivity criterion for rational time-stepping maps of eventually positive generators selects Padé(0,2) over Crank-Nicolson and yields a…
Pairwise split-coupling variance at the path-count cost of common reaction paths, applied to three estimation settings.
A half-bin heat mismatch and fixed Burgers endpoints explain the floors; particle error still falls at Monte Carlo rate.
Partitioning the graph and adding a coarse level lets AdaGrad reach the same accuracy with far fewer full-graph steps.
· “Two-level domain-decomposition AdaGrad method for scalable training of graph neural networks”
A two-stage compression keeps optimal L2 rates while cutting cost to O(N), even in high dimensions.
A finite-sample formula separates target and kernel geometry from their noise levels; smoothness sets the rate.
· “Non-asymptotic Analysis of Mat\'ern Regression: The Roles of Target and Kernel Lengthscales”
Point loads on plates reach optimal adaptive error rates via a companion operator.
A new training schedule drives smooth networks to the entropy solution with a proved error rate.
· “A Jin--Xin Relaxation Gradual Convergence Method for Conservation-Law PINNs”
One checkable number certifies two splitting families, up to ninth order, for incompressible Navier-Stokes.
Explicit advection cancels exactly in the energy balance, so persistent forcing can't pile up error over long runs.
New implicit-function theorem pins down explicit stability exponents for solution mappings of generalized equations.
Its smallest and largest eigenvalues control convergence and spectral bias; now they come with explicit formulas.
Rank-one Gaussian sensors attain it; a convex recovery rule stays stable at optimal sampling scale.
· “Optimal Condition Numbers in Low-Rank Positive Semidefinite Matrix Sensing”
Generalized LRCF-ADI enables frequency-limited Gramians and data-driven balanced truncation from samples alone.
· “A New Generalized Low-Rank Cholesky Factor ADI Algorithm for Large-Scale Stein Equations”
PINN learns the interface derivative once; elastic, piezoelectric and piezomagnetic half-spaces reuse it.
A Gaussian-process derivative match seeds a multi-shooting refinement that enforces global dynamical consistency.
· “Gaussian process learning with flow map refinement for parameter estimation in dynamical systems”
A palindromic splitting maximizes either its stable time step or its phase accuracy; real coefficients cannot do both.
A Legendre antiderivative fixes the exact superconvergence points for every B-spline degree.
New mean-convergence bounds cover RK, REK, and block variants with no initial-condition assumption.
The endpoint rate is exact: 32πZ|φ3(0,0)|/ε² for two-electron atoms, so the quadratic growth is sharp
· “Sharp Barron Regularity Results for Coulombic Many-Electron Wave Functions”
Boundary values inherit the global quasiperiodic field, so finite-domain runs avoid the rational-approximation stall.
A residual-iteration scheme on Gauss nodes claims exponential error decay and long-time stability without solving linear systems.
· “An hp-version time stepping spectral Monte Carlo method for semi-linear parabolic equations”
Estimators bound the error of this parabolic-equation integrator from above and below, enabling adaptive steps.
Four spectral fingerprints separate fixable conditioning from saddles Adam cannot escape.
Per-tensor adaptive cubic steps keep exact curvature at 91.4M parameters and escape the saddles that trap Adam.
· “Blockwise Stabilized Adaptive Cubic Regularization with Subsolvers via Recurrence”
New virtual element analysis covers distributed and boundary controls with parameter-uniform errors.
Reusing f'(x0) cuts per-step derivative cost, but quadratic speed appears only when the slope stays nearly constant.
· “On Convergence of an Accelerated Modified Newton Method for Nonlinear Equations”
A few dozen modes capture the density in all scattering regimes, with certified error bars.
· “Data-driven reduced-order models for the radiative transfer equation”
DySCo keeps climate extremes coherent with the driving model while adding 6x spatial detail.
· “DySCo: Dynamically consistent data-driven downscaling of extremes in climate projections”
A diffusion time-changed SIS epidemic model is shown to be well-posed and to exhibit extinction or persistence according to R0, and a…
Weighted Laplacian flow moves mass across barriers and converges in L∞ and KL divergence.
· “Weighted Laplacian Flow: A Deterministic Particle Flow with Provable Convergence”
FFT convolution on 64 GPUs cuts the source-to-waveform map to two matvecs, making 10,000-scenario ensembles a 4-minute job.
A Wasserstein gradient flow turns moment observations into particle corrections for plasma PIC runs.
· “Wasserstein Moment Nudging for Vlasov-Poisson Data Assimilation”
Choosing blocks from the largest residuals beats static-partition tensor Kaczmarz solvers in iterations and time.
Prompted with benchmark data, an LLM reproduces SmartSolve's LU algorithm-selection tree.
· “Data-Driven Dynamic Algorithm Dispatch with Large Language Models”
