A note on a collectively compact approximation for weakly singular integral operators

AbstractWe prove the collectively compact convergence of the Nyström method when applied on a truncated version of a weakly singular integral operator. As a consequence, we get the quasi compact convergence of finite rank projection operators derived from the Kantorovitch singularity subtraction approximation

On ε-spectra and stability radii

AbstractTechniques of Krylov subspace iterations play an important role in computing ε-spectra of large matrices. To obtain results about the reliability of this kind of approximations, we propose to compare the position of the ε-spectrum of A with those of its diagonal submatrices. We give theoretical results which are valid for any block decomposition in four blocks, A11,A12,A21,A22. We then illustrate our results by numerical experiments. The same kind of problem arises when we compute the stability radius of a large matrix. In that context, we propose a new sufficient condition for the stability of a matrix involving quantities readily computable such as stability radius of small submatrices

Echec imminent : à la recherche d'un profil

Devant le constat, tant en Belgique qu'en France, des faibles taux de réussite en premiêre année universi­taire, nous avons cherché à  dégager le profil des étudiants qui risquent de se trouver en situation d'échec en fin d'année. Ainsi ils pourront bénéficier d'actions ciblées menées par l'université en vue d'augmenter leur probabilité de réussite et ce avant les premiêres évaluations. Un questionnaire a été élaboré et distribué en début d'année, et ce durant deux années, auprês d'étudiants de la Faculté de Sciences et Techniques d'une université française, d'un IUT[1] de la même université et de la Faculté des Sciences d'une université belge. Chaque étudiant a pu être décrit au moyen de 375 varia­bles. Nous présentons dans cet article l'analyse des résultats obtenus.   [1] IUT : Institut Universitaire Technologique

Modélisation mathématique et simulation numérique par un schéma volumes-finis de la synthèse du carbure de titane par procédé d'autocombustion SHS

Dans ce travail, nous considérons le problème de la synthèse du carbure de titane par procédé SHS d'un point de vue mathématique et numérique. Après avoir modélisé ce procédé de combustion par une équation parabolique nonlinéaire couplée à une équation différentielle, nous prouvons l'existence et l'unicité de la solution du problème faible couplé linéarisé et montrons qu'elle satisfait à un principe du maximum. Une discrétisation volumes-finis du problème couplé est établie ainsi qu'une estimation d'erreur, des propriétés de stabilité et de principe du maximum relatif à la solution discrète. Le schéma numérique est implémenté dans notre code de calcul Héphaïstos écrit en C++. Une analyse de sensibilité de la solution discrète au raffinement de maillage est conduite. Nous étudions en une dimension d'espace la sensibilité du temps d'induction aux paramètres thermophysiques et cinétiques ainsi que la nature de la propagation du front de combustion en une et deux dimensions d'espaceIn this work, we consider the titanium carbide synthesis by SHS process from a mathematical and numerical point of view. The modelling of this combustion process is based upon a coupling between a nonlinear parabolic equation and a differential equation. We prove the existence and uniqueness of the solution of the weak, linearised, coupled problem and show that the solution satisfy a maximum principle property. A finite-volume discretization of the coupled problem is established with an error estimate, stability and discrete maximum principle properties of the discrete solution. The numerical scheme is implemented into our C++ software Héphaïstos. A sensibility analysis of the discrete solution with respect to mesh refinement is conducted. We investigate in one spatial dimension the sensibility of the induction time with respect to thermophysical and kinetics parameters and analyse the nature of the propagation of the combustion front in one and two spatial dimensionsST ETIENNE-BU Sciences (422182103) / SudocSudocFranceF

Superconvergence of some projection approximations for weakly singular integral equations using general grids

International audienc

Newton Method for Refining Schur and Gauss Triangular Forms

International audienc

Spectral computations for bounded operators

Exact eigenvalues, eigenvectors, and principal vectors of operators with infinite dimensional ranges can rarely be found. Therefore, one must approximate such operators by finite rank operators, then solve the original eigenvalue problem approximately. Serving as both an outstanding text for graduate students and as a source of current results for research scientists, Spectral Computations for Bounded Operators addresses the issue of solving eigenvalue problems for operators on infinite dimensional spaces. From a review of classical spectral theory through concrete approximation techniques to finite dimensional situations that can be implemented on a computer, this volume illustrates the marriage of pure and applied mathematics. It contains a variety of recent developments, including a new type of approximation that encompasses a variety of approximation methods but is simple to verify in practice. It also suggests a new stopping criterion for the QR Method and outlines advances in both the iterative refinement and acceleration techniques for improving the accuracy of approximations. The authors illustrate all definitions and results with elementary examples and include numerous exercises.Spectral Computations for Bounded Operators thus serves as both an outstanding text for second-year graduate students and as a source of current results for research scientists

Iterative refinement for invariant subspaces of matrices with application to the Promethee-Gaia method

Recent results in Operational Research have provided decision makers with a highly adaptable tool that is able to quickly synthesize the performance measures of several solutions to be compared in a lim- ited time. In the context of multicriteria limited-time decision making problems, two of the authors have developed hybrid models composed of two mathematical models, a set of dedicated heuristics, a stochastic local search, meta-heuristic and a simulation model. According to the decision makers, the solutions are ranked on the basis of several criteria whose importance determines this ranking. A multicriteria method is incoporated into the hybrid. In order to summarize the huge amount of resulting data/information, we have embeded the Promethee II multicriteria method and the GAIA plane. This extension requires to compute eigenelements on the output produced by Promethee. Eigenelement computation is proposed here in two steps: 1) Get some rough approximation to the desired part of the spectrum and the cor- responding maximal invariant subspace, and 2) Refine this approximation with an iterative scheme. A Newton-based scheme is proposed in this paper and applied to a matrix issued from the Promethee-Gaia method