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.
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.
Achraf Badahmane (ULCO-LMPA)
Informations : 11:00 - 12:00 B014
In this talk, we propose the preconditioned global MINRES as a new strategy to solve problems $AX=B$ with several right-hand sides.
The preconditioner is obtained by replacing the block (2,2) by another block of the matrix A.
We apply the global MINRES method for this problem with several right hand sides and we give new convergence results and analyze the eigenvalue-distribution and the eigenvectors of the preconditioner.
Finally, numerical results show that our preconditioned global MINRES method, is very efficient for solving problem with several right hand sides.
Séminaire et groupe de travail d’Approximation et Analyse matricielle du 9 février 2018Yassine Kaouane (ULCO)
Informations : 11:00 - 12:00 Salle B014
We present a new approach for model order reduction in large-scale dynamical systems, with multiple inputs and multiple outputs (MIMO). This approach will be named : Adaptive Block Tangential Arnoldi Algorithm (ABTAA) and is based on interpolation via block tangential Krylov subspaces requiring the selection of shifts and tangent directions via an adaptive procedure. We give some algebraic properties and present some numerical examples to show the effectiveness of the proposed method.
A. Messaoudi (ENS, Université Mohammed VI, Rabat)
Informations : 11:00 - 12:00 B014
Le résumé est diponible ici pdf
Marcos Raydan (Universidad Simón Bolívar)
Informations : 11:00 - 12:00 B014
The Quadratic Finite Element Model Updating Problem (QFEMUP) concerns
with updating a symmetric second-order finite element model so that it
remains symmetric and the updated model reproduces a given set of
desired eigenvalues and eigenvectors by replacing the corresponding
ones from the original model. Taking advantage of the special
structure of the constraint set, it is first shown that the QFEMUP can
be formulated as a suitable constrained nonlinear programming
problem. Using this formulation, we present and analyze two different
methods based on successive optimizations. To avoid that spurious
modes (eigenvectors) appear in the frequency range of interest
(eigenvalues) after the model has been updated, additional constraints
based on a quadratic Rayleigh quotient are dynamically included in the
constraint set. The results of our numerical experiments on
illustrative problems show that the algorithms work well in practice.
Achraf Badahmane (ULCO)
Informations : 14:30 - 15:30 B014
In some applications, we have to solve large linear saddle
point problems with multiple right-hand sides. Instead of applying a
standard iterative process to the solution of each saddle point
problem indepentely, it’s more efﬁcient to apply a global process. We
use different techniques of preconditioning ( Diagonal preconditioner,
Triangular preconditioner, P0 preconditioner ,.. ) to improve spectral
proprieties of the saddle point matrix and to accelerate the
Yassine Kaouane (LMPA, ULCO)
Adaptive tangential Computational Krylov subspaces methods for model reduction in large-scale dynamical systemsInformations : 14:30 - 15:30 B014, Mi-voix
In this talk, we present two new approaches for model order reduction
problem, with multiple inputs and multiple outputs (MIMO). The
Adaptive Global Tangentiel Arlondi Algorithms (AGTAA), and the
Adaptive Global Tangentiel Lanczos Algorithms (AGTLA).These methods
are based on a generalization of the global Arnoldi and the global
Laczos algorithms. The selection of the shifts and the tangent
directions is done with an adaptive procedure. We give some algebraic
properties for the global case. Finally, some numerical examples are
presented to show the effectiveness of the proposed algorithms.
Key words : Global, Arnoldi, Lanczos, Model reduction, Tangential directions.
Hassane Sadok (ULCO)
Informations : 13:30 - 14:30 B014
Krylov subspace methods are widely used for the iterative solution of
a large variety of linear systems of equations with one or several
right hand sides or for solving nonsymmetric eigenvalue problems. The
solution of linear systems of equations with several right-hand sides
is considered. Approximate solutions are conveniently computed by
block GMRES methods. We describe and study three variants of block
GMRES. These methods are based on three implementations of the block
Arnoldi method, which differ in their choice of inner product.. The
Block GMRES is classically implemented by first applying the Arnoldi
iteration as a block orthogonalization process to create a basis of
the block Krylov space generated by the matrix of the system from the
initial residual. Next, the method is solving a block least-squares
problem, which is equivalent to solving several least squares problems
implying the same Hessenberg matrix. These laters are usually solved
by using a block updating procedure for the QR decomposition of the
Hessenberg matrix based on Givens rotations. A more effective
alternative was given by M. H. Gutknecht and T. Schmelzer which uses
the Householder reflectors. We propose a new and simple implementation
of the block GMRES algorithm, based on a generalization of Ayachour’s
method given for the GMRES with a single right-hand side. Several
numerical experiments are provided to illustrate the performance of
the new implementation.