Spectral approximation with matrices issued from discretized operators

117 Pages·2017·1.66 MB·English

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
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Universidade do Porto
et
Université Jean Monnet
N◦ attribué par la bibliothèque
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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.