Stabilité numérique

Résumé

En analyse numérique, une branche des mathématiques, la stabilité numérique est une propriété globale d’un algorithme numérique, une qualité nécessaire pour espérer obtenir des résultats ayant du sens. Une définition rigoureuse de la stabilité dépend du contexte. Elle se réfère à la propagation des erreurs au cours des étapes du calcul, à la capacité de l’algorithme de ne pas trop amplifier d’éventuels écarts, à la précision des résultats obtenus. Le concept de stabilité ne se limite pas aux erreurs d’arrondis et à leurs conséquences. Les algorithmiques dédiés à la résolution d’équations différentielles ou d’équations aux dérivées partielles (en particulier la méthode des différences finies et la méthode des éléments finis) se basent sur une discrétisation ou un maillage de l’espace (et du temps) ; dans ce cas, la stabilité se réfère à un comportement numérique robuste lorsque le pas de discrétisation ou la taille des mailles tend vers 0. Un algorithme instable peut être qualifié

Source officielle

https://fr.wikipedia.org/wiki/Stabilit%C3%A9_num%C3%A9rique

À propos de ce résultat

Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.

Publications associées (100)

Chargement

Afficher plus

Publications associées

Chargement

Personnes associées (13)

Personnes associées

Chargement

Unités associées (7)

Unités associées

Chargement

Concepts associés (21)

Concepts associés

Chargement

Cours associés (38)

Cours associés

Chargement

Séances de cours associées (61)

Séances de cours associées

Chargement

Publications associées (100)

Numerical simulation of drug delivery in blood flows

Jonathan Droxler

In this project we consider a simplified model of drug delivery in a stenosed artery. The goal is to simulate the diffusion and transport of drug in blood flows, and to evaluate how physical parameters, such as the drug diffusivity or the geometry of the artery, can influence the process. We model the drug delivery with an unsteady diffusion-advection-reaction equation, where the transport term is given by the blood velocity, obtained as the solution of the Navier-Stokes equations. Since the problem is transport dominated we consider suitable stabilization (namely SUPG and GLS stabilization) in order to control the numerical instabilities.

2014

Chargement

Learning Stable Task Sequences from Demonstration with Linear Parameter Varying Systems and Hidden Markov Models

Aude Billard, José Ramon Medina Hernandez

The problem of acquiring multiple tasks from demonstration is typi- cally divided in two sequential processes: (1) the segmentation or identification of different subgoals/subtasks and (2) a separate learning process that parameterizes a control policy for each subtask. As a result, segmentation criteria typically neglect the characteristics of control policies and rely instead on simplified models. This paper aims for a single model capable of learning sequences of complex time-independent control policies that provide robust and stable behavior. To this end, we first present a novel and efficient approach to learn goal-oriented time independent motion models by estimating both attractor and dynamic behavior from data guaranteeing stability using linear parameter varying (LPV) systems. This method enables learning complex task sequences with hidden Markov models (HMMs), where each state/subtask is given by a stable LPV system and where transitions are most likely around the corresponding attractor. We study the dynamics of the HMM-LPV model and propose a motion generation method that guarantees the stability of task sequences. We validate our approach in two sets of demonstrated human motions

robot-learning2017

Chargement

Options are some of the most traded financial instruments and computing their price is a central task in financial mathematics and in practice. Consequently, the development of numerical algorithms for pricing options is an active field of research. In general, evaluating the price of a specific option relies on the properties of the stochastic model used for the underlying asset price. In this thesis we develop efficient and accurate numerical methods for option pricing in a specific class of models: polynomial models. They are a versatile tool for financial modeling and have useful properties that can be exploited for option pricing. Significant challenges arise when developing option pricing techniques. For instance, the underlying model might have a high-dimensional parameter space. Furthermore, treating multi-asset options yields high-dimensional pricing problems. Therefore, the pricing method should be able to handle high dimensionality. Another important aspect is the efficiency of the algorithm: in real-world applications, option prices need to be delivered within short periods of time, making the algorithmic complexity a potential bottleneck. In this thesis, we address these challenges by developing option pricing techniques that are able to handle low and high-dimensional problems, and we propose complexity reduction techniques. The thesis consists of four parts: First, we present a methodology for European and American option pricing. The method uses the moments of the underlying price process to produce monotone sequences of lower and upper bounds of the option price. The bounds are obtained by solving a sequence of polynomial optimization problems. As the order of the moments increases, the bounds become sharper and eventually converge to the exact price under appropriate assumptions. Second, we develop a fast algorithm for the incremental computation of nested block triangular matrix exponentials. This algorithm allows for an efficient incremental computation of the moment sequence of polynomial jump-diffusions. In other words, moments of order 0, 1, 2, 3... are computed sequentially until a dynamically evaluated criterion tells us to stop. The algorithm is based on the scaling and squaring technique and reduces the complexity of the pricing algorithms that require such an incremental moment computation. Third, we develop a complexity reduction technique for high-dimensional option pricing. To this end, we first consider the option price as a function of model and payoff parameters. Then, the tensorized Chebyshev interpolation is used on the parameter space to increase the efficiency in computing option prices, while maintaining the required accuracy. The high dimensionality of the problem is treated by expressing the tensorized interpolation in the tensor train format and by deriving an efficient way, which is based on tensor completion, to approximate the interpolation coefficients. Lastly, we propose a methodology for pricing single and multi-asset European options. The approach is a combination of Monte Carlo simulation and function approximation. We address the memory limitations that arise when treating very high-dimensional applications by combining the method with optimal sampling strategies and using a randomized algorithm to reduce the storage complexity of the approach. The obtained numerical results show the effectiveness of the algorithms developed in this thesis.

EPFL2019

Chargement

Afficher plus

EPFL2019