Multiplicateur de Lagrange | EPFL Graph Search

En mathématiques, et plus particulièrement en analyse, la méthode des multiplicateurs de Lagrange permet de trouver les points stationnaires (maximum, minimum…) d'une fonction dérivable d'une ou plusieurs variables, sous contraintes. Dimension finie On cherche à trouver l'extremum, un minimum ou un maximum, d'une fonction φ de n variables à valeurs dans les nombres réels, ou encore d'un espace euclidien de dimension n, parmi les points respectant une contrainte, de type ψ(x) = 0 où ψ est une fonction du même ensemble de départ que φ. La fonction ψ est à valeurs dans un espace euclidien de dimension m. Elle peut encore être vue comme m fonctions à valeurs réelles, décrivant m contraintes. Si l'espace euclidien est de dimension 2 et si la fonction ψ est à valeurs dans ℝ, correspondant à une contrainte mono-dimensionnelle, la situation s'illustre par une figure analogue à celle de droite. La question revient à rechercher le point situé le plus haut, c'est-à-dire

A Euclidean Likelihood Estimator for Bivariate Tail Dependence

Boris Oumow, Michal Warchol

The spectral measure plays a key role in the statistical modeling of multivariate extremes. Estimation of the spectral measure is a complex issue, given the need to obey a certain moment condition. We propose a Euclidean likelihood-based estimator for the spectral measure which is simple and explicitly defined, with its expression being free of Lagrange multipliers. Our estimator is shown to have the same limit distribution as the maximum empirical likelihood estimator of J. H. J. Einmahl and J. Segers, Annals of Statistics 37(5B), 2953–2989 (2009). Numerical experiments suggest an overall good performance and identical behavior to the maximum empirical likelihood estimator. We illustrate the method in an extreme temperature data analysis.

Taylor & Francis Inc2013

Coupling non-conforming discretizations of PDEs by spectral approximation of the Lagrange multiplier space

Simone Deparis, Antonio Iubatti, Luca Pegolotti

This work focuses on the development of a non-conforming method for the coupling of PDEs based on weakly imposed transmission conditions: the continuity of the global solution is enforced by a finite number of Lagrange multipliers defined over the interfaces of adjacent subdomains. The method falls into the class of primal hybrid methods, which include also the well-known mortar method. Differently from the mortar method, we discretize the space of basis functions on the interface by spectral approximation independently of the discretization of the two adjacent domains. In particular, our approach can be regarded as a specialization of the three-field method in which the spaces used to enforce the continuity of the solution and its conormal derivative across the interface are taken equal. One of the possible choices to approximate the interface variational space – which we consider here – is by Fourier basis functions. As we show in the numerical simulations, the method is well-suited for the coupling of problems defined on globally non-conforming meshes or discretized with basis functions of different polynomial degree in each subdomain. We also investigate the possibility of coupling solutions obtained with incompatible numerical methods, namely the finite element method and isogeometric analysis.

2019

Optimized Coding Strategies for Interactive Multiview Video

Ana Karina De Abreu Goes

The natural next step in improving the realistic experience in multimedia services is interactive multiview video (IMV). IMV promises to enable the users to freely navigate through a scene by selecting their preferred viewpoints from any view position for which the corresponding view is generated. A smooth navigation could be achieved with camera views and views synthesized at the decoder. However, the large amount of data required for such navigation experience still represents a challenge for the current systems, which implies the need for new efficient coding strategies that permit to save on storage and transmission resources, while preserving interactivity in the navigation. In this thesis, we focus on the optimization of coding strategies for IMV systems. First, we address the issues related to the coding techniques for IMV in a multiview video plus depth (MVD) scenario, where texture and depth maps are available for view synthesis at the decoder. We propose a low complexity algorithm for the selection of the interview prediction structures (PSs) and associated texture and depth quantization parameters (QPs) for IMV under transmission and storage constraints. Simulation results show that our novel low complexity algorithm has near-optimal compression efficiency while preserving interactivity properties at the decoder. Then, considering the limited and heterogeneous capabilities of current networks and decoding devices, we propose a novel adaptive solution for IMV based on a layered multiview representation where camera views are organized into layered subsets to offer different levels of navigation quality depending on the different client constraints. We propose an optimal and a reduced computational complexity greedy algorithms that jointly select the different view subsets and their encoding rates. Simulation results show the good performance of our novel algorithms compared to a baseline algorithm, proving that an effective IMV adaptive solution should consider the scene content, the client capabilities and their preferences, in building adaptive systems for multiview navigation. Finally, we build on the solution proposed in our second problem and present a general solution to rate allocation problems in multiview video. In particular, we propose a new algorithm to find the optimal Lagrange multiplier in a Lagrangian-based rate allocation problem. We show the performance of our proposed algorithm in both multiview and monoview video scenarios and show that the proposed method is able to compete with complex state-of-the-art rate control techniques. In summary, this thesis addresses important issues for coding multiview video in the design of efficient IMV systems under resource constraints. Our algorithm to select the optimal PS and QPs in a MVD scenario can improve the quality of the rendered views and it can indeed provide new insights for a deeper understanding of specific IMV coding requirements. We show that our algorithm for a layered representation of multiview video provides an effective adaptive streaming solution for IMV systems with users with limited and heterogeneous capabilities. Finally, our proposed Lagrangian-based rate allocation algorithm with an optimized selection of the Lagrange multiplier represents a general contribution that can be used in multiple video scenarios.

EPFL2015

Optimized Coding Strategies for Interactive Multiview Video

Ana Karina De Abreu Goes

EPFL2015