Counting eigenvalues in domains of the complex field

A procedure for counting the number of eigenvalues of a matrix in a region surrounded by a closed curve is presented. It is based on the application of the residual theorem. The quadrature is performed by evaluating the principal argument of the logarithm of a function. A strategy is proposed for selecting a path length that insures that the same branch of the logarithm is followed during the integration. Numerical tests are reported for matrices obtained from conventional matrix test sets.Comment: 21 page

A new approach for regularization of inverse problems in images processing

International audienceOptical flow motion estimation from two images is limited by the aperture problem. A method to deal with this problem is to use regularization techniques. Usually, one adds a regularization term with appriopriate weighting parameter to the optical flow cost funtion. Here, we suggest a new approach to regularization for optical flow motion estimation. In this approach, all the regularization informations are used in the definition of an appropriate norm for the cost function via a trust function to be defined, one does not ever need weighting parameter. A simple derivation of such a trust function from images is proposed and a comparison with usual approaches is presented. These results show the superiority of such approach over usual ones

Some efficient methods for computing the determinant of large sparse matrices

International audienceThe computation of determinants intervenes in many scientific applications, as for example in the localization of eigenvalues of a given matrix A in a domain of the complex plane. When a procedure based on the application of the residual theorem is used, the integration process leads to the evaluation of the principal argument of the complex logarithm of the function g(z) = det((z + h)I - A)/ det(zI - A), and a large number of determinants is computed to insure that the same branch of the complex logarithm is followed during the integration. In this paper, we present some efficient methods for computing the determinant of a large sparse and block structured matrix. Tests conducted using randomly generated matrices show the efficiency and robustness of our methods.Le calcul de déterminants intervient dans certaines applications scientifiques, comme parexemple dans le comptage du nombre de valeurs propres d’une matrice situées dans un domaineborné du plan complexe. Lorsqu’on utilise une approche fondée sur l’application du théorème desrésidus, l’intégration nous ramène à l’évaluation de l’argument principal du logarithme complexe de lafonction g(z) = det((z + h)I − A)/ det(zI − A), en un grand nombre de points, pour ne pas sauterd’une branche à l’autre du logarithme complexe. Nous proposons dans cet article quelques méthodesefficaces pour le calcul du déterminant d’une matrice grande et creuse, et qui peut être transforméesous forme de blocs structurés. Les résultats numériques, issus de tests sur des matrices généréesde façon aléatoire, confirment l’efficacité et la robustesse des méthodes proposées

A Numerical Conformal Mapping Method and the Poisson Equation on Irregular Domains

202 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1984.In this thesis, we propose a numerical conformal mapping procedure for constructing the unique transformation that maps a unit disk onto a simply connected irregular region. All the systems of linear equations that arise in formulating the method, are solved using Rapid Elliptic Solvers. Therefore, this technique may be regarded as a fast mapping technique. Applications of the method include: (i) generation curvilinear coordinate systems, and (ii) solving the Poisson and Laplace equations on two-dimensional simply connected irregular domains.U of I OnlyRestricted to the U of I community idenfinitely during batch ingest of legacy ETD

Some efficient methods for computing the determinant of large sparse matrices

International audienceThe computation of determinants intervenes in many scientific applications, as for example in the localization of eigenvalues of a given matrix A in a domain of the complex plane. When a procedure based on the application of the residual theorem is used, the integration process leads to the evaluation of the principal argument of the complex logarithm of the function g(z) = det((z + h)I - A)/ det(zI - A), and a large number of determinants is computed to insure that the same branch of the complex logarithm is followed during the integration. In this paper, we present some efficient methods for computing the determinant of a large sparse and block structured matrix. Tests conducted using randomly generated matrices show the efficiency and robustness of our methods.Le calcul de déterminants intervient dans certaines applications scientifiques, comme parexemple dans le comptage du nombre de valeurs propres d’une matrice situées dans un domaineborné du plan complexe. Lorsqu’on utilise une approche fondée sur l’application du théorème desrésidus, l’intégration nous ramène à l’évaluation de l’argument principal du logarithme complexe de lafonction g(z) = det((z + h)I − A)/ det(zI − A), en un grand nombre de points, pour ne pas sauterd’une branche à l’autre du logarithme complexe. Nous proposons dans cet article quelques méthodesefficaces pour le calcul du déterminant d’une matrice grande et creuse, et qui peut être transforméesous forme de blocs structurés. Les résultats numériques, issus de tests sur des matrices généréesde façon aléatoire, confirment l’efficacité et la robustesse des méthodes proposées

Safe localization of eigenvalues

International audienceLocalizing some eigenvalues of a given large sparse matrix in a domain of the complex plane is a hard task when the matrix is non symmetric, especially when it is highly non normal. For taking into account, possible perturbations of the matrix, the notion of the of \epsilon-spectrum or pseudospectrum of a matrix $A \in \mathbb{R}^{n \times n}$ was separately defined by Godunov and Trefethen. Determining an $\epsilon$-spectrum consists of determining a level curve of the 2-norm of the resolvent $R(z) = (zI-A)^{-1}$. A dual approach can be considered: given some curve $(\Gamma)$ in the complex plane, count the number of eigenvalues of the matrix $A$ that are surrounded by $(\Gamma)$. The number of surrounded eigenvalues is determined by evaluating the integral $\frac{1}{2i\pi} \int_{\Gamma}{\frac{d}{dz}\log \det (zI-A) dz}$. This problem was considered in [Bertrand and Philippe, 2001] where several procedures were proposed and more recently in [Kamgnia and Philippe, to appear] where the stepsize control in the quadrature is deeply studied. The present goal is to combine the two approaches: (i) consider the method PAT [Mezher and Philippe, Numer. Algorithms, 2002, Mezher and Philippe, Parallel Comput, 2002] which is a path following method that determines a level curve of the function $s(z)=\sigma_{\min}(zI-A)$; (ii) apply the method EIGENCNT of [Kamgnia and Philippe, to appear] for computing the number of eigenvalues included. The combined procedure will be based on a computing kernel which provides the two numbers (\sigma_{min}(zI - A), det(zI - A)) for any complex number $z \in \mathbb{C}$. These two numbers are obtained through a common LU factorization of $(zI - A)$. In order to introduce a second level of parallelism, we consider a preprocessing transformation similar to the approach developed in SPIKE [Polizzi and Sameh, 2006]

Localisation robuste et dénombrement de valeurs propres

International audienceThis article deals with the localization of eigenvalues of a large sparse and not necessarilysymmetric matrix in a domain of the complex plane. It combines two studies carried out earlier.The first work deals with the effect of applying small perturbations on a matrix, and referred to ase -spectrum or pseudospectrum. The second study describes a procedure for counting the numberof eigenvalues of a matrix in a region of the complex plain surrounded by a closed curve. The twomethods are combined in order to share the LU factorization of the resolvent, that intervenes in thetwo methods, so as to reduce the cost. The codes obtained are parallelized.L’article se consacre à la localisation de valeurs propres pour une grande matrice creuse, apriori non symétrique, dans un domaine du plan complexe. Il combine deux notions déjà étudiées. Lapremière précise l’effet de perturbations sur la matrice par la définition de e -spectre ou pseudospectre.La deuxième consiste à dénombrer les valeurs propres entourées par une courbe a priori donnéedans le plan complexe. A partir de travaux antérieurs, on combine ici les deux approches avec l’objectifde mettre en commun les factorisations LU de la résolvante nécessaire aux deux approches et d’endiminuer le nombre. Les codes obtenus sont parallélisés

Parallélisation de GMRES préconditionné par une itération de Schwarz multiplicatif

Cette thèse propose une alternative à la parallélisation de GMRES préconditionné par Schwarz multiplicatif par la technique de coloriage de graphe adjacent à la matrice. Cette parallélisation suppose que le graphe adjacent à la matrice est partitionné selon une direction. A partir de ce partionnement on peut dériver une forme explicite pour l'itération de Schwarz multiplicatif. On utilise cette forme explicite dans un pipeline pour la construction de l'espace de Krylov. On conserve les qualités du pipeline, en évitant d'inserrer les points de synchronisation comme les produits scalaires globaux dans le procédé d'Arnoldi. Pour cela, on utilise une version de GMRES qui découple la construction de l'espace de Krylov et la factorisation QR dans le procédé d'Arnoldi. Tous ces algorithmes sont implémentés sur le standard PETSc et portent le nom de GPREMS (Gmres PREconditionned by multiplicatif Schwarz). Les tests sont réalisés sur des problèmes issus de la simulation de semiconducteurs et de la mécanique des fluides. Cette validation numérique confirme les qualités parallèles de notre code, mais aussi sa compétitivité par rapport aux autres préconditionneurs du type décomposition de domaine comme Schwarz additif ou le complément de Schur.This thesis proposes an alternative to the parallelization of GMRES preconditioned by multiplicative Schwarz by the technique of coloring adjacent graph to the matrix. This parallelization implies that the adjacent graph to the matrix is partitioned according to one direction. From this partitioning we can derive an explicit form of spliting of multiplicatif Schwarz. We use this explicite form in a pipeline for the construction of Krylov subspace basis. The qualities of the pipeline are prevent by avoiding a synchronization points due to the dot product in overall process Arnoldi. For this reason, we use a version of GMRES which decouples the construction of the space Krylov and QR factorization in the process Arnoldi. All these algorithms are implemented on standard PETSc and bears the name of GPREMS (GMRES PREconditoned by multiplicative Schwarz). The tests are performed on problems arising from the simulation of semiconductors and fluid mechanics. This validation confirms the parallel qualities of our code, but also its competitiveness with other preconditioner type domain decomposition as Schwarz additive or additional Schur.RENNES1-BU Sciences Philo (352382102) / SudocRENNES-INRIA Rennes Irisa (352382340) / SudocSudocFranceF

An explicit formulation of the multiplicative Schwarz preconditionner

We provide an explicit formulation of the splitting associated with the Multiplicative Schwarz iteration. We show the advantage of considering the explicit formulation, when the iteration is used as a preconditioner of a Krylov method. // A partir d'une expression explicite du splitting défini par l'itération multiplicative de Schwarz, nous étudions son utilisation comme précondtionnement d'une méthode de Krylov