Electro-muscular disruption (EMD) devices such as TASER M26 andX26 have been used as a less-than-lethal weapon. Such EMD devicesshoot a pair of darts toward an intended target to generate anincapacitating electrical shock. In the use of the EMD device,there have been controversial questions about its safety andeffectiveness. To address these questions, we need to investigatethe distribution of the current density J inside the targetproduced by the EMD device. One approach is to develop acomputational model providing a quantitative and reliable analysisabout the distribution of J. In this paper, we set up amathematical model of a typical EMD shock, bearing in mind that weare aiming to compute the current density distribution inside thehuman body with a pair of inserted darts. The safety issue ofTASER is directly related to the magnitude of |J| at the regionof the darts where the current density J is highlyconcentrated. Hence, fine computation of J near the dart isessential. For such numerical simulations, serious computationaldifficulties are encountered in dealing with the darts having twodifferent very sharp corners, tip of needle and tip of barb. Theboundary of a small fishhook-shaped dart inside a largecomputational domain and the presence of corner singularitiesrequire a very fine mesh leading to a formidable amount ofnumerical computations. To circumvent these difficulties, wedeveloped a multiple point source method of computing J. It hasa potential to provide effective analysis and more accurateestimate of J near fishhook-shaped darts. Numerical experimentsshow that the MPSM is just fit for the study of EMD shocks.

In this work we deal with the numerical solution of aHamilton-Jacobi-Bellman (HJB) equation with infinitely manysolutions. To compute the maximal solution – the optimalcost of the original optimal control problem – we present acomplete discrete method based on the use of some finite elementsand penalization techniques.

Motivated by the pricing of American options for baskets weconsider a parabolic variational inequality in a boundedpolyhedral domain $\Omega\subset\mathbb{R}^d$ with a continuous piecewisesmooth obstacle. We formulate a fully discrete method by usingpiecewise linear finite elements in space and the backward Eulermethod in time. We define an a posteriori error estimator and showthat it gives an upper bound for the error inL2(0,T;H1 (Ω)). The error estimator is localized in thesense that the size of the elliptic residual is only relevant inthe approximate non-contact region, and the approximability of theobstacle is only relevant in the approximate contact region. Wealso obtain lower bound results for the space error indicators inthe non-contact region, and for the time error estimator.Numerical results for d=1,2 show that the error estimator decayswith the same rate as the actual error when the space meshsize hand the time step τ tend to zero. Also, the error indicatorscapture the correct behavior of the errors in both the contact andthe non-contact regions.

Partant du principe de conservation de la masse et du principefondamental de la dynamique, on retrouvel'équation d'Euler nous permettant de décrire les modèlesasymptotiques de propagation d'ondes dans des eaux peu profondesen dimension 1. Pour décrire la propagation des ondes en dimension2, Kadomtsev et Petviashvili [ 15 (1970) 539] utilisent une perturbationlinéaire de l'équation de KdV. Mais cela ne précise pas si leséquations ainsi obtenues dérivent de l'équation d'Euler, c'est ceque montrent Ablowitz et Segur dans l'article [J. Fluid Mech.92 (1979) 691–715]. Oninsistera, de la même manière, sur le fait que les équations deKP-BBM peuvent être aussi obtenues à partir de l'équation d'Euler,et dans quelle mesure elles décrivent le modèle physique. Dans unsecond temps, on reprend la méthode introduite dans l'article deBona et al. [Lect. Appl. Math.20 (1983) 235–267] dans lequel ils comparent lessolutions d'ondes longues en dimension 1, à savoir les solutionsdes équations KdV et BBM, pour montrer ici que les solutions deséquations KP-II et KP-BBM-II sont proches sur un intervalle de tempsinversement proportionnel à l'amplitude des ondes. Du point de vue de la modélisation,il sera clair, d'après la première partie, que seul le modèledécrit par KP-BBM-II est bien posé, et comme du point de vuephysique, KP-II et KP-BBM-II décrivent les ondes longues de faibleamplitude lorsque la tension de surface est négligeable, il est intéressant de les comparer.De plus, on verra que la méthode utilisée ici reste valable pourles problèmes périodiques.

This paper is concerned with optimal design problems with aspecial assumption on the coefficients of the state equation.Namely we assume that the variations of these coefficientshave a small amplitude. Then, making an asymptotic expansionup to second order with respect to the aspect ratio of thecoefficients allows us to greatly simplify the optimal designproblem. By using the notion of H-measures we are able toprove general existence theorems for small amplitudeoptimal design and to provide simple and efficient numericalalgorithms for their computation. A key feature of thistype of problems is that the optimal microstructures arealways simple laminates.

In this paper, we extend the reduced-basis approximations developed earlier for linear elliptic and parabolic partial differential equations with affine parameterdependence to problems involving (a) nonaffine dependence on theparameter, and (b) nonlinear dependence on the field variable.The method replaces the nonaffine and nonlinear terms with a coefficient function approximation which then permits an efficient offline-online computationaldecomposition. We first review the coefficient function approximation procedure: the essential ingredients are (i) a good collateralreduced-basis approximation space, and (ii) a stable and inexpensiveinterpolation procedure. We then apply this approach to linear nonaffine and nonlinear elliptic and parabolic equations; in eachinstance, we discuss the reduced-basis approximation and the associated offline-online computationalprocedures. Numerical results are presented to assess our approach.

The aim of this article is to propose a new method for the grey-level image classification problem. We first present the classicalvariational approach without and with a regularization term in order tosmooth the contours of the classified image. Then we present the generaltopological asymptotic analysis, and we finally introduce its application tothe grey-level image classification problem.

The primary objective of this work is to develop coarse-grainingschemes for stochastic many-body microscopic models and quantify theireffectiveness in terms of a priori and a posteriori error analysis. Inthis paper we focus on stochastic lattice systems ofinteracting particles at equilibrium. The proposed algorithms are derived from an initial coarse-grainedapproximation that is directly computable by Monte Carlo simulations, and the corresponding numerical error is calculated using the specific relative entropy between the exact and approximate coarse-grained equilibrium measures. Subsequently we carry out a cluster expansion around this first – and often inadequate – approximation and obtain more accurate coarse-graining schemes.The cluster expansions yield also sharp a posteriori error estimates forthe coarse-grained approximations that can be used for the construction ofadaptive coarse-graining methods.
We present a number of numerical examples that demonstrate that thecoarse-graining schemes developed here allow for accurate predictions of critical behavior and hysteresis in systems with intermediate and long-range interactions. We also present examples where they substantially improvepredictions of earlier coarse-graining schemes for short-range interactions.