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Vector Quantization (VQ) is a method for discretizing latent representations and has become a major part of the deep learning toolkit. It has been theoretically and empirically shown that discretization of representations leads to improved…
We hybridize the methods of finite element exterior calculus for the Hodge-Laplace problem on differential $k$-forms in $\mathbb{R}^n$. In the cases $k = 0$ and $k = n$, we recover well-known primal and mixed hybrid methods for the scalar…
We apply a model for the study of quark-hadron duality in inclusive electron scattering to the calculation of spin observables. The model is based on solving the Dirac equation numerically for a scalar confining linear potential and a…
In this work, we design and analyze semi/fully-discrete virtual element approximations for the time-dependent Navier--Stokes-Cahn--Hilliard equations, modeling the dynamics of two-phase incompressible fluid flows with diffuse interfaces. A…
We propose a new approach to the combinatorial interpretations of linearization coefficient problem of orthogonal polynomials. We first establish a difference system and then solve it combinatorially and analytically using the method of…
In solid-state physics, energies of molecular systems are usually computed with a plane-wave discretization of Kohn-Sham equations. A priori estimates of plane-wave convergence for periodic Kohn-Sham calculations with pseudopotentials have…
On the basis of the canonical quantization procedure of a system defined on a cubic lattice, we propose a new method, in which resolutions of unity expressed in terms of eigenvectors are naturally provided, to find eigenvectors of field…
Port-Hamiltonian (PH) systems provide a framework for modeling, analysis and control of complex dynamical systems, where the complexity might result from multi-physical couplings, non-trivial domains and diverse nonlinearities. A major…
We adapt the compression algorithm of Weinstein, Auerbach, and Chandra from eigenvectors of spin lattice Hamiltonians to eigenvectors of light-front field-theoretic Hamiltonians. The latter are approximated by the standard discrete…
The Kahan discretization of the Lotka-Volterra system, associated with any skew-symmetric graph $\Gamma$, leads to a family of rational maps, parametrized by the step size. When these maps are Poisson maps with respect to the quadratic…
We present a novel numerical method for solving ODEs while preserving polynomial first integrals. The method is based on introducing multiple quadratic auxiliary variables to reformulate the ODE as an equivalent but higher-dimensional ODE…
A mimetic spectral element discretization, utilizing a novel Galerkin projection Hodge star operator, of the macroscopic Maxwell equations in Hamiltonian form is presented. The idea of splitting purely topological and metric dependent…
This paper presents a method for learning Hamiltonian dynamics from a limited set of data points. The Hamiltonian vector field is found by regularized optimization over a reproducing kernel Hilbert space of vector fields that are inherently…
R. Hirota and K. Kimura discovered integrable discretizations of the Euler and the Lagrange tops, given by birational maps. Their method is a specialization to the integrable context of a general discretization scheme introduced by W. Kahan…
We give here a covariant definition of the path integral formalism for the Lagrangian, which leaves a freedom to choose anyone of many possible quantum systems that correspond to the same classical limit without adding new potential terms…
Diffeomorphic matching (only one of several names for this technique) is a technique for non-rigid registration of curves and surfaces in which the curve or surface is embedded in the flow of a time-series of vector fields. One seeks the…
Vector quantization via random projection followed by scalar quantization is a fundamental primitive in machine learning, with applications ranging from similarity search to federated learning and KV cache compression. While dense random…
The accurate and efficient representation of atmospheric dynamics remains a central challenge in numerical weather prediction. A particular difficulty arises from the strong anisotropy of the atmosphere, in which horizontal and vertical…
The problem of finding superintegrable Hamiltonians and their integrals of motion can be reduced to solving a series of compatibility equations that result from the overdetermination of the commutator or Poisson bracket relations. The…
In this work, merging ideas from compatible discretisations and polyhedral methods, we construct novel fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra. The spaces and operators that appear in these…