A 2D/3D Discrete Duality Finite Volume Scheme. Application to ECG simulation

International audienceThis paper presents a 2D/3D discrete duality finite volume method for solving heterogeneous and anisotropic elliptic equations on very general unstructured meshes. The scheme is based on the definition of discrete divergence and gradient operators that fulfill a duality property mimicking the Green formula. As a consequence, the discrete problem is proved to be well-posed, symmetric and positive-definite. Standard numerical tests are performed in 2D and 3D and the results are discussed and compared with P1 finite elements ones. At last, the method is used for the resolution of a problem arising in biomathematics: the electrocardiogram simulation on a 2D mesh obtained from segmented medical images

A domain decomposition strategy for a very high-order finite volumes scheme applied to cardiac electrophysiology

International audienceIn this paper, a domain decomposition technique for a very high-order finite volumes scheme is proposed. The objective is to obtain an efficient way to perform numerical simulations in cardiac electrophysiology. The aim is to extend a very high-order numerical scheme previously designed, where large stencils are used for polynomial reconstructions. Therefore, a particular attention has to be paid to maintain the scalability in parallel. Here, we propose to constrain the stencils inside the subdomains or their first layer of neighbors. The method is shown to remain accurate and to scale perfectly up to the level where there are not enough cells in the subdomains. Hence, these high-order schemes are proved to be efficient tools to perform realistic simulations in cardiac electrophysiology

An Abstract Lyapunov Control Optimizer: Local Stabilization and Global Convergence

Recently, many machine learning optimizers have been analysed considering them as the asymptotic limit of some differential equations when the step size goes to zero. In other words, the optimizers can be seen as a finite difference scheme applied to a continuous dynamical system. But the major part of the results in the literature concerns constant step size algorithms. The main aim of this paper is to investigate the guarantees of the adaptive step size counterpart. In fact, this dynamical point of view can be used to design step size update rules, by choosing a discretization of the continuous equation that preserves its most relevant features. In this work, we analyse this kind of adaptive optimizers and prove their Lyapunov stability and convergence properties for any choice of hyperparameters. At the best of our knowledge, this paper introduces for the first time the use of continuous selection theory from general topology to overcome some of the intrinsic difficulties due to the non constant and non regular step size policies. The general framework developed gives many new results on adaptive and constant step size Momentum/Heavy-Ball and p-GD algorithms

Convergence of the Iterates for Momentum and RMSProp for Local Smooth Functions: Adaptation is the Key

Both accelerated and adaptive gradient methods are among state of the art algorithms to train neural networks. The tuning of hyperparameters is needed to make them work efficiently. For classical gradient descent, a general and efficient way to adapt hyperparameters is the Armijo backtracking. The goal of this work is to generalize the Armijo linesearch to Momentum and RMSProp, two popular optimizers of this family, by means of stability theory of dynamical systems. We establish convergence results, under the Lojasiewicz assumption, for these strategies. As a direct result, we obtain the first guarantee on the convergence of the iterates for RMSProp, in the non-convex setting without the classical bounded assumptions

Обеспечение безопасности ядерных материалов на гипотетическом объекте

Предметом исследования являются инсайдерские угрозы, угрозы со стороны, инсайдер в сговоре с угрозами со стороны, уязвимость атомных электростанций, анализ угроз, беспилотные летательные аппараты (беспилотные летательные аппараты), АЭС ВВЭР, категоризация и анализ наиболее уязвимых районов этого объекта , учет и контроль ядерных материалов, проектирование и функции ППС.The subject of the study are insider threats, outsider threats, insider in collusion with outsider threats, nuclear power plant vulnerabilities, analysis of threats that unmanned aerial vehicles (drones) impose on nuclear power plant, design of hypothetical nuclear facility for VVER NPP, categorization

Modelisation, approximation numerique et applications du transfert radiatif en desequilibre spectral couple avec l'hydrodynamique

In some hypersonic regimes, the effects of radiative transfer can drastically modify the aerodynamics flow. For such applications, it is important to have a model that fully couples hydrodynamics and radiation in order to have a good behaviour of the solution. However, coupling with the full radiative transfer equation is usually very expensive hence it is not reasonable for multidimensionnal unsteady computations. Our choice is to use a moment model for the radiation part, which is way cheaper. This model uses an entropic closure a la Levermore that allows to be consitant with the fundamental physical properties such as energy conservation, entropy dissipation and flux-limitation. We also developped it to be multigroup in order to correctly predict the solution of strongly frequency-dependent problems. The resulting coupled system is hyperbolic and has several interesting properties that are discussed. This radiative model is then coupled to multispecies Navier-Stokes equations. Since in some of our applications radiation has a critical impact on the flow, we use a full-implicit fully coupled approach. Moreover, in order to save memory, we focus on a Jacobian Free algorithm. For these reasons, we choose to develop a preconditionned GMRes (Generalized Minimal Residual) which acts as a JFNK (Jacobian Free Newton Krylov) method. The result proves to be fast enough to perform realistic simulations at a reasonable computation time, which is very hard to get by coupling Navier-Stokes equations with the usual radiation models. Several applications are given to illustrate the good behaviour of the model either in simple academic configurations where comparisons can done or in realistic configurations such as flows around probes in atmospheric superorbital entry.Dans certains regimes hypersoniques, le rayonnement peut enormement modifier l'ecoulement aerodynamique. Pour de telles applications, il est important d'avoir un modele qui realise un couplage fort entre l'hydrodynamique et le transfert radiatif afin d'avoir un bon comportement de la solution. Cependant, le couplage avec l'equation du transfert radiatif est en general extremement couteux et donc peu raisonnable pour des simulations multidimensionnelles instationnaires. Notre choix est d'utiliser un modele aux moments pour la partie rayonnement, ce qui est bien moins couteux. Celui-ci est base sur une fermeture entropique a la Levermore qui permet de conserver les principales proprietes de la physique. On developpe une version multigroupe de ce modele afin de pouvoir traiter des cas realistes tres dependants de la frequence. Le systeme couple resultant est hyperbolique et possede des proprietes interessantes qui sont etudiees. Ce modele radiatif est couple avec les equations de Navier-Stokes avec une approche totalement implicite et fortement couplee. De plus, pour gagner de la place memoire, on choisit d'utiliser une methode sans Jacobienne, en pratique une methode de type GMRes preconditionne. Cette methode se revele assez rapide pour pouvoir simuler des applications realistes a un cout de calcul raisonnable, ce qui n'est pas le cas de la plupart des modeles courament utilises dans la litterature. Plusieurs applications sont donnees pour illustrer le bon comportement du modele a la fois dans des configurations academiques simplifiees ou l'on peut faire des comparaisons et dans des configurations realistes comme l'ecoulement lors de l'entree atmospherique de sondes superorbitales

Modélisation, analyse numérique et simulations de phénomènes complexes pour des systèmes hyperboliques de lois de conservation avec termes sources raides et en électrocardiologie.

Ces travaux de recherche concernent la modélisation, l'analyse numéique et les simulations de phénomènes complexes. Le lien fondamental entre eux est la volonté de pouvoir réaliser des simulations de ces phénomènes complexes décrivant avec le maximum de préision possible des situations physiquement réalistes.Ils peuvent se classer en deux grands thèmes : les systèmes hyperboliques de lois de conservation avec terme source et l'électrocardiologie