Multiplicateur et probabilité

Keynes begun his scientifïc career with probability theory. But, he had not the idea, as far as we can know, to give a probabilistic interpretation of his famous multiplier. This article is aimed at showing that probability theory, and especially finite Markov chains theory, gives an easier and even more natural interpretation of the keynesian multiplier than the traditional methods. Multiplier theory may be looked on as old-fashioned today, but it is still at the heart of most of macroeconometric models. So, we define first the relative position of the multiplier, which is linear and actually static, inside these models which are non-linear and dynamic. Secondly, we give a markovian interpretation of the income multiplier in both cases of the simple multiplier and the matrix multiplier. We compare it with the traditional interpretation: in the probabilistic interpretation every kind of economic agents (banks and firms, and not only households) take a part in the process of incomes which leads to the multiplier. Finally, we enlarge our method to the neighbouring analysis of the money multiplier and of the velocity of money. Our conclusion is that the markovian method could also be used for a keynesian crisis analysis

Multiplicateur et probabilité

Keynes begun his scientifïc career with probability theory. But, he had not the idea, as far as we can know, to give a probabilistic interpretation of his famous multiplier. This article is aimed at showing that probability theory, and especially finite Markov chains theory, gives an easier and even more natural interpretation of the keynesian multiplier than the traditional methods.

Scheduling malleable task trees

Solving sparse linear systems can lead to processing tree workflows on a platform of processors. In this study, we use the model of malleable tasks motivated in [Prasanna96,Beaumont07] in order to study tree workflow schedules under two contradictory objectives: makespan minimization and memory minization. First, we give a simpler proof of the result of [Prasanna96] which allows to compute a makespan-optimal schedule for tree workflows. Then, we study a more realistic speed-up function and show that the previous schedules are not optimal in this context. Finally, we give complexity results concerning the objective of minimizing both makespan and memory

Malleable task-graph scheduling with a practical speed-up model

Scientific workloads are often described by Directed Acyclic task Graphs.Indeed, DAGs represent both a model frequently studied in theoretical literature and the structure employed by dynamic runtime schedulers to handle HPC applications. A natural problem is then to compute a makespan-minimizing schedule of a given graph. In this paper, we are motivated by task graphs arising from multifrontal factorizations of sparsematrices and therefore work under the following practical model. We focus on malleable tasks (i.e., a single task can be allotted a time-varying number of processors) and specifically on a simple yet realistic speedup model: each task can be perfectly parallelized, but only up to a limited number of processors. We first prove that the associated decision problem of minimizing the makespan is NP-Complete. Then, we study a widely used algorithm, PropScheduling, under this practical model and propose a new strategy GreedyFilling. Even though both strategies are 2-approximations, experiments on real and synthetic data sets show that GreedyFilling achieves significantly lower makespans

Scheduling and data redistribution strategies on star platforms

In this work we are interested in the problem of scheduling and redistributing data on master-slave platforms. We consider the case were the workers possess initial loads, some of which having to be redistributed in order to balance their completion times. We examine two different scenarios. The first model assumes that the data consists of independent and identical tasks. We prove the NP-completeness in the strong sense for the general case, and we present two optimal algorithms for special platform types. Furthermore we propose three heuristics for the general case. Simulations consolidate the theoretical results. The second data model is based on Divisible Load Theory. This problem can be solved in polynomial time by a combination of linear programming and simple analytical manipulations

Parallel scheduling of task trees with limited memory

This paper investigates the execution of tree-shaped task graphs using multiple processors. Each edge of such a tree represents some large data. A task can only be executed if all input and output data fit into memory, and a data can only be removed from memory after the completion of the task that uses it as an input data. Such trees arise, for instance, in the multifrontal method of sparse matrix factorization. The peak memory needed for the processing of the entire tree depends on the execution order of the tasks. With one processor the objective of the tree traversal is to minimize the required memory. This problem was well studied and optimal polynomial algorithms were proposed. Here, we extend the problem by considering multiple processors, which is of obvious interest in the application area of matrix factorization. With multiple processors comes the additional objective to minimize the time needed to traverse the tree, i.e., to minimize the makespan. Not surprisingly, this problem proves to be much harder than the sequential one. We study the computational complexity of this problem and provide inapproximability results even for unit weight trees. We design a series of practical heuristics achieving different trade-offs between the minimization of peak memory usage and makespan. Some of these heuristics are able to process a tree while keeping the memory usage under a given memory limit. The different heuristics are evaluated in an extensive experimental evaluation using realistic trees.Dans ce rapport, nous nous intéressons au traitement d'arbres de tâches par plusieurs processeurs. Chaque arête d'un tel arbre représente un gros fichier d'entrée/sortie. Une tâche peut être traitée seulement si l'ensemble de ses fichiers d'entrée et de sortie peut résider en mémoire, et un fichier ne peut être retiré de la mémoire que lorsqu'il a été traité. De tels arbres surviennent, par exemple, lors de la factorisation de matrices creuses par des méthodes multifrontales. La quantité de mémoire nécessaire dépend de l'ordre de traitement des tâches. Avec un seul processeur, l'objectif est naturellement de minimiser la quantité de mémoire requise. Ce problème a déjà été étudié et des algorithmes polynomiaux ont été proposés. Nous étendons ce problème en considérant plusieurs processeurs, ce qui est d'un intérêt évident pour le problème de la factorisation de grandes matrices. Avec plusieurs processeurs se pose également le problème de la minimisation du temps nécessaire pour traiter l'arbre. Nous montrons que comme attendu, ce problème est bien plus compliqué que dans le cas séquentiel. Nous étudions la complexité de ce problème et nous fournissons des résultats d'inaproximabilité, même dans le cas de poids unitaires. Nous proposons plusieurs heuristiques qui obtiennent différents compromis entre mémoire et temps d'exécution. Certaines d'entre elles sont capables de traiter l'arbre tout en gardant la consommation mémoire inférieure à une limite donnée. Nous analysons les performances de toutes ces heuristiques par une large campagne de simulations utilisant des arbres réalistes

Computer-assisted Teaching in Class Situation: a High-school Math Lab on Vectors

published as a volume in Lecture Notes in Computer ScienceInternational audienceThis paper presents our design and experiment of a computerassisted class laboratory on vectors in high-school. Our main goal is to improve the acquisition on notions by all pupils, from advanced level to remedial level. Our secondary goal is to improve the motivation of pupils toward maths and sciences. Our approach consists in making the pupils practice the notions included in a given set of pedagogic objectives, and to do it through real worldinspired problems. We draw on MobiNet, a platform that simulate programmable mobiles on screen and through the network. Our class laboratory consists of a set of pre-programmed interactive simulations, allowing the pupils to react on their errors and to validate each exercise in an autonomous way. This class laboratory was conducted in the same conditions and constraints than an ordinary math lab: same duration, and no preparation of the pupils (but the ordinary math course). Still, the autonomy of the exercises and the network ability would also allow the use with no teacher on place. In this paper, we describe our vector class laboratory experiment: objectives, design, conduction, and evaluation

Study of Formation and Decay of Rare-Gas Excimers by Laser- Induced Fluorescence

The aim of this chapter is to review the experimental and numerical techniques for the estimation of the laser-induced fluorescence (LIF) decay in rare gases using time-correlated single-photon counting. The advantages of single-photon counting technique are discussed by means of measurement uncertainty analysis. In addition, this chapter provides information concerning the application of this technique to filamentary dielectric barrier discharges (DBD) and radiation trapping of the resonant transitions