Several authors in recent years have proposed discrete analogues to principle component analysis intended to handle discrete or positive only data, for instance suited to analyzing sets of documents. Methods include non-negative matrix factorization, probabilistic latent semantic analysis, and latent Dirichlet allocation. This paperbegins with a review of the basic theory of the variational ext...

Abstract In this article, using critical point theory and variational methods, we investigate the existence of at least three solutions for a class double eigenvalue discrete anisotropic Kirchhoff-type problems. An example is presented to demonstrate applicability our main theoretical findings.

The numerical analysis of variational integrators relies on variational error analysis, which relates the order of accuracy of a variational integrator with the order of approximation of the exact discrete Lagrangian by a computable discrete Lagrangian. The exact discrete Lagrangian can either be characterized variationally, or in terms of Jacobi’s solution of the Hamilton– Jacobi equation. The...

This paper recognizes that in many decision environments in which revenue optimization is attempted, an actual demand curve and its parameters are generally unobservable. Herein, we describe the dynamics of demand as a continuous time differential equation based on an evolutionary game theory perspective. We then observe realized sales data to obtain estimates of parameters that govern the evol...

We consider a variational scheme developed by S. Demoulini, D. M. A. Stuart and A. E. Tzavaras [Arch. Rat. Mech. Anal. 157 (2001)] that approximates the equations of three dimensional elastodynamics with polyconvex stored energy. We establish the convergence of the time-continuous interpolates constructed in the scheme to a solution of polyconvex elastodynamics before shock formation. The proof...

We analyze multigrid convergence rates when elliptic boundary value problems are discretized using finite element methods with numerical integration. The resulting discrete problem does not fall into the standard variational framework for analyzing multigrid methods since the bilinear forms on different grid levels are not suitably related to each other. We first discuss extensions of the stand...

Linear response and variational treatment are formulated for Hartree-Fock (HF) and Kohn-Sham density functional theory (DFT) methods and combined discrete-continuum solvation models that incorporate self-consistently induced dipoles and charges. Due to the variational treatment, analytic nuclear gradients can be evaluated efficiently for these discrete and continuum solvation models. The forces...

The purpose of this paper is to review and further develop the subject of variational integration algorithms as it applies to mechanical systems of engineering interest. In particular, the conservation properties of both synchronous and asynchronous variational integrators (AVIs) are discussed in detail. We present selected numerical examples which demonstrate the excellent accuracy, conservati...

We introduce generalized Galerkin variational integrators, which are a natural generalization of discrete variational mechanics, whereby the discrete action, as opposed to the discrete Lagrangian, is the fundamental object. This is achieved by approximating the action integral with appropriate choices of a finite-dimensional function space that approximate sections of the configuration bundle a...