Spectral approximation with matrices issued from discretized operators Ana Luisa Silva Nunes To cite this version: Ana Luisa Silva Nunes. Spectral approximation with matrices issued from discretized operators. General Mathematics [math.GM]. Université Jean Monnet - Saint-Etienne; Universidade do Porto, 2012. English. ￿NNT: 2012STET4006￿. ￿tel-00952977￿ HAL Id: tel-00952977 https://theses.hal.science/tel-00952977 Submitted on 28 Feb 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Universidade do Porto et Université Jean Monnet N◦ attribué par la bibliothèque (cid:116)(cid:116)(cid:116)(cid:116)(cid:116)(cid:116)(cid:116)(cid:116)(cid:116)(cid:116)(cid:116)(cid:116)(cid:116) THÈSE pour obtenir le grade de Docteur de l’Universidade do Porto et Université Jean Monnet Spécialité : Mathématiques Appliquées dans le cadre de l’ Ecole Doctorale Sciences, Ingénierie, Santé : ED SIS 488 présentée et soutenue publiquement par Ana Luísa da Silva Nunes le 11/05/2012 Titre: Approximation spectrale de matrices issues d’opératreurs discrétisés Directeurs de thèse: Paulo Beleza Vasconcelos Co-directeur de thèse: Mario Ahues Blanchait Jury M. Jorge ROCHA, Président de Jury Mme Rekha KULKARNI, Rapporteur Mme Maria Joana SOARES, Examinateur, Rapporteur Mme Laurence GRAMMONT, Examinateur Mme Teresa DIOGO, Examinateur M. Paulo BELEZA VASCONCELOS, Directeur de thèse M. Mario AHUES BLANCHAIT, Co-directeur de thèse To my parents ii Acknowledgments Arriving to this stage, I am glad to complete it by remembering many won- derful people who became involved with this thesis in various ways. First and foremost I want to thank my thesis supervisors. I am extremely grateful for the guidance and energetic support provided by Professor Paulo Vasconcelos and for all the insightful and always fast feedback whenever re- quired, despite his many otheracademic commitments. I deeply acknowledge ProfessorMarioAhuesforallthetimehespentprovidingveryimportantsug- gestions and comments when it was most necessary. I also thank him for his generous and tireless assistance in my stays in Saint-Etienne, as well as in all my registations at the Universit´e Jean Monnet. It has been an honor to work with both of them, and I am deeply grateful to them for their scientific and general support and encouragement throughout my study as a PhD. student. I express my sincere thanks to Professor J. E. Roman from Universidad Politecnica de Valencia (Spain), for all his support in the joint computacional work held in this thesis. I wish to honor the memory of Professor Alain Largillier, with whom I had some scientific profitable talks about my work and helped to make my first stay in Saint-Etienne a pleasant experience. I am very grateful to my colleagues of Universit´e Jean Monnet, specially to Rola Fares and Taline Boyajian, for the hospitality I experienced in my first stay there. I also deeply appreciate all the support shown by my friends and IPCA colleagues. iii iv I wish to express my warm thanks to my beloved husband, Paolo, for his love and understanding in every situation. Last but not least, a very special thank to my wonderful parents for their endless love, patience and encouragement. Without you all, I could not have reached this far. Thank you! Ana Lu´ısa Spectral approximation with matrices issued from discretized operators Abstract In this thesis, we consider the numerical solution of a large eigenvalue problem in which the integral operator comes from a radiative transfer prob- lem. It is considered the use of hierarchical matrices, an efficient data-sparse representation of matrices, especially useful for large dimensional problems. It consists on low-rank subblocks leading to low memory requirements as well as cheap computational costs. We discuss the use of the hierarchical matrix technique in the numerical solution of a large scale eigenvalue problem arising from a finite rank dis- cretization of an integral operator. The operator is of convolution type, it is defined through the first exponential-integral function and hence it is weakly singular. We access HLIB (Hierarchical matrices LIBrary) that provides, among others, routines for the construction of hierarchical matrix structures and arithmetic algorithms to perform approximative matrix operations. More- over, it is incorporated the matrix-vector multiply routines from HLIB, as well as LU factorization for preconditioning, into SLEPc (Scalable Library for Eigenvalue Problem Computations) in order to exploit the available al- gorithms to solve eigenvalue problems.