Séminaire et groupe de travail d’Approximation et Analyse matricielle

Le séminaire et groupe de travail d’algèbre réunit les membres de l’équipe Approximation (et toutes personnes intéressées).

Responsable : Khalide Jbilou

Liste des exposés de 2019

• Mardi 25 juin 2019

  Stefano Pozza (Charles University, Prague)

  Lanczos-like method for the time-ordered exponential (⊕ résumé)

  Let A(t) be a time-dependent matrix with t in an interval. The time-ordered exponential of A(t) is defined as the unique solution U(t) of the system of coupled linear differential equations A(t)U(t)=d/dt U(t) with initial condition U(0)=I. In the general case (when A does not commute with itself at all times), the ordered exponential has no known explicit form in terms of A. The problem of evaluating U(t) is a central question in the field of system dynamics, in particular in quantum physics where A is the quantum Hamiltonian. Until now, few methods have been proposed to approximate the ordered exponential, but a satisfactory answer to this problem is still missing. In 2015, P.-L. Giscard proposed a method to obtain ordered exponentials using graph theory and necessitating only the entries A(t) to be bounded functions of time. While this approach provides exact solutions and is always convergent, it suffers from computational drawbacks. The talk will describe a model-reduction strategy that solves such computational issue by a Lanczos-like algorithm, giving a converging and computable (in term of complexity) strategy for getting U(t). Such a technique is based on the connections between the Lanczos-like algorithm and the moment problem, graph approximations, and continued fractions.

  13:30 - 14:30 C106

• Mardi 25 juin 2019

  Lothar Reichel (Kent State University - USA)

  Linearized Krylov subspace Bregman iteration with nonnegativity constraint (⊕ résumé)

  Bregman-type iterative methods have attracted considerable attention in recent years due to their ease of implementation and the high quality of the computed solutions they deliver. However, these iterative methods may require a large number of iterations and this reduces their attractiveness. This talk describes a linearized Bregman algorithm defined by projecting the problem to be solved into an appropriately chosen low-dimensional Krylov subspace. The projection reduces both the number of iterations and the computational effort required for each iteration. A variant of this solution method, in which nonnegativity of each computed iterate is imposed, also is described. The talk presents joint work with A. Buccini and M. Pasha.

  14:30 - 15:30 C106

• Jeudi 20 juin 2019

  Marilena Mitrouli (Université d'Athènes)

  On the estimation of the tuning parameter in regularized linear regression models (⊕ résumé)

  In the regularized linear regression models the appropriate choice of the tuning parameter plays a dominant role in the selection of the correct model. Most statistical methods employ the tool of the generalized cross-validation (GCV) for the selection of values of this parameter. In this work, we utilize extrapolation estimates for the GCV function whose minimizers can lead to the determination of the tuning parameter. The selection of an efficient estimate depends on an appropriately defined index of proximity. Bounds and specific values are derived for this index and a thorough study proves that the proposed one-term estimate suits perfectly to statistical models with high correlated variables. This is confirmed through simulation tests for several datasets.

  11:00 - 12:00 B014

• Vendredi 14 juin 2019

  Abderrahim Messaoudi (Université Mohammed V - Maroc)

  RMVPIA : A new algorithm for computing Lagrange multivariate-interpolation polynomials (⊕ résumé)

  Recently Messaoudi et al. presented a new algorithm for computing the Hermite interpolation algorithm, for a general case, called Generalized Recursive Polynomial Interpolation Algorithm (GRPIA). In this work we will give a new formulation of the Lagrange twovariate polynomial interpolation, a result of the existence and the uniqueness of this polynomial will be given. We will show that this polynomial can be expressed as a Schur complement . So using the Sylvester’s identity, we will give a new algorithm for computing this polynomial. Some properties of this algorithm will also be discussed and a generalization of this algorithm will be proposed for the Lagrange multivariate-interpolation polynomial.

  10:00 - 11:00 B014

• Vendredi 14 juin 2019

  Hassane Sadok (LMPA - ULCO)

  A unified approach of Krylov subspace methods for solving nonsymmetric linear systems (⊕ résumé)

  By using a recursive method for computing a left inverse of a Krylov matrix we derive a general algorithm for the subspaces Krylov methods. This technique allows us to give all the parameters of the algorithms from the Krylov vectors. In particulier we give the condition of the existence of the algorithms based on these vectors.

  11:00 - 12:00 B014

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