A model of a homogenized cavity corresponding to a multinozzle droplet generator for continuous ink-jet printers

We modelize the behavior of a vibrating fluid cavity. Small nozzles are uniformly distributed in one direction on a side of the cavity. By means of asymptotic expansion in powers of the smallest dimension of the cavity, including boundary layer terms, we get the convergence of the solution of the three-dimensional problem, as well as the convergence of the solution of the “homogenized” three-dimensional problem towards the solution of the same two-dimensional problem. Numerical experiments have been carried out in order to illustrate the previous theoretical results. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 821–842, 1998     

Computing small solutions of Burgers' equation backwards in time

We construct and analyze an algorithm for the numerical computation of Burgers' equation for preceding times, given an a priori bound for the solution and an approximation to the terminal data. The method is based on the “backward beam equation” coupled with an iterative procedure for the solution of the nonlinear problem via a sequence of linear problems. We also present the results of several numerical experiments. It turns out that the procedure converges “asymptotically,” i.e., in the same manner in which an asymptotic expansion converges. This phenomenon seems related to the “destruction of information,” at t = 0, which is typical in backwards dissipative equations. We derive a priori stability estimates for the analytic backwards problem, and we observe that in many numerical experiments, the distance backwards in time where significant accuracy can be attained is much larger than would be expected on the basis of such estimates. The method is useful for small solutions. Problems where steep gradients occur require considerably more precision in measurement. The algorithm is applicable to other semilinear problems.     

Etude de l'algorithme de Rémès en l'absence de condition de Haar

Summary Using a geometric interpretation, the unicity of a best approximation and the convergence of the Remes-algorithm without the Haar-condition are investigated (in the case of the Tchebycheff-approximation of a continuous function by a linear combination of two continuous functions). Replacing the Haar-condition by a weaker one which is generally satisfied, one obtains a convergence theorem that is weaker than the classical convergence theorem but is sufficient for the applicatons. A generalization of these results is indicated.     

Observation of chaotic and regular dynamics in atom-optics billiards

We report on experimental observations of chaotic and regular motion of ultracold atoms confined by a billiard-shaped optical dipole potential induced by a rapidly scanning laser beam. To investigate the dynamics of the atoms confined by such an "atom-optics" billiard we measure the decay of the number of trapped atoms through a hole on the boundary. A fast and purely exponential decay, the clear signature of chaotic motion, is found for a stadium billiard, but not for a circular or an elliptical billiard, in agreement with theory. We also investigated the effects of decoherence, velocity spread, and gravity on regular and chaotic motion.     

Approximate computation of the permeability tensor of a periodic porous medium

The permeability tensor K of an infinite periodic porous medium, obtained using the homogenization theory, is considered. The solutions of an optimal control problem for the Dirichlet or Neumann equation are used to obtain optimal upper bounds for K. The test functions used for the estimations are simpler than those obtained by other authors. Some possibilities are given to obtain also lower bounds.     

Reconstructing the past from imprecise knowledge of the present: Effective non‐uniqueness in solving parabolic equations backward in time

Identifying sources of ground water pollution and deblurring astronomical galaxy images are two important applications generating growing interest in the numerical computation of parabolic equations backward in time. However, while backward uniqueness typically prevails in parabolic equations, the precise data needed for the existence of a particular backward solution is seldom available. This paper discusses previously unexplored non‐uniqueness issues, originating from trying to reconstruct a particular solution from imprecise data. Explicit 1D examples of linear and nonlinear parabolic equations are presented, in which there is strong computational evidence for the existence of distinct solutions w^red(x,t) and w^green(x,t), on 0 ≤ t ≤ 1. These solutions have the property that the traces w^red(x,1) and w^green(x,1) at time t = 1 are close enough to be visually indistinguishable, while the corresponding initial values w^red(x,0) and w^green(x,0) are vastly different, well‐behaved, physically plausible functions, with comparable L² norms. This implies effective non‐uniqueness in the recovery of w^red(x,0) from approximate data for w^red(x,1). In all these examples, the Van Cittert iterative procedure is used as a tool to discover unsuspected, valid, additional solutions w^green(x,0). This methodology can generate numerous other examples and indicates that multidimensional problems are likely to be a rich source of striking non‐uniqueness phenomena. Published 2012. This article is a US Government work and is in the public domain in the USA.