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Solving Linear Systems Arising from Reservoirs Modeling Hussam Al Daas

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Solving Linear Systems Arising from Reservoirs Modeling Hussam Al Daas Solving linear systems arising from reservoirs modeling Hussam Al Daas To cite this version: Hussam Al Daas. Solving linear systems arising from reservoirs modeling. Mathematics [math]. Inria Paris; Sorbonne Université, UPMC University of Paris 6, Laboratoire Jacques-Louis Lions, 2018. English. tel-01984047v1 HAL Id: tel-01984047 https://hal.inria.fr/tel-01984047v1 Submitted on 16 Jan 2019 (v1), last revised 11 Jun 2020 (v2) 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. Sorbonne Université Inria Doctoral School École Doctorale Sciences Mathématiques de Paris Centre University Department Inria Thesis defended by Hussam Al Daas Defended on 13th December, 2018 In order to become Doctor from Sorbonne Université Academic Field Applied Mathematics Speciality Numerical Linear Algebra Solving linear systems arising from reservoirs modelling Thesis supervised by Laura Grigori Supervisor Pascal Hénon Co-Supervisor Committee members Referees Serge Gratton Professor at INP ENSEEIHT Julien Langou Professor at UC Denver Examiners Frédéric Nataf Senior Researcher at CNRS Committee President Philippe Ricoux Professor at MIT Daniel Ruiz Associate Professor at INP EN- SEEIHT Supervisors Laura Grigori Senior Researcher at Inria Pascal Hénon Senior Researcher at TOTAL Sorbonne Université Inria Doctoral School École Doctorale Sciences Mathématiques de Paris Centre University Department Inria Thesis defended by Hussam Al Daas Defended on 13th December, 2018 In order to become Doctor from Sorbonne Université Academic Field Applied Mathematics Speciality Numerical Linear Algebra Solving linear systems arising from reservoirs modelling Thesis supervised by Laura Grigori Supervisor Pascal Hénon Co-Supervisor Committee members Referees Serge Gratton Professor at INP ENSEEIHT Julien Langou Professor at UC Denver Examiners Frédéric Nataf Senior Researcher at CNRS Committee President Philippe Ricoux Professor at MIT Daniel Ruiz Associate Professor at INP EN- SEEIHT Supervisors Laura Grigori Senior Researcher at Inria Pascal Hénon Senior Researcher at TOTAL Sorbonne Université Inria École doctorale École Doctorale Sciences Mathématiques de Paris Centre Unité de recherche Inria Thèse présentée par Hussam Al Daas Soutenue le 13 décembre 2018 En vue de l’obtention du grade de docteur de l’Sorbonne Université Discipline Mathématiques appliquées Spécialité Algèbre linéaire numérique Résolution de systèmes linéaires issus de la modélisation des réservoirs Thèse dirigée par Laura Grigori directrice Pascal Hénon co-directeur Composition du jury Rapporteurs Serge Gratton professeur à l’INP ENSEEIHT Julien Langou professeur à l’UC Denver Examinateurs Frédéric Nataf directeur de recherche au CNRS président du jury Philippe Ricoux professeur au MIT Daniel Ruiz mcf à l’INP ENSEEIHT Directeurs de thèse Laura Grigori directrice de recherche à l’Inria Pascal Hénon directeur de recherche au TO- TAL Keywords: krylov, block methods, inexact breakdown, deflation, recycling, domain decomposition Mots clés : krylov, méthodes par bloc, inexact breakdown, déflation, recyclage, décomposition de domaine This thesis has been prepared at the following research units. Inria 2 rue Simone Iff 75012 Paris France T +33 1 44 27 42 98 Web Site http://www.inria.fr/ Laboratoire Jacques-Louis Lions 4 place Jussieu 75005 Paris France T +33 1 44 27 42 98 Web Site http://ljll.math.upmc.fr/ ACKNOWLEDGEMENTS No matter how the eloquence harmonizes and combines words to communicate my gratitude to everyone who contributed to this work, it will be difficult to fulfil the goal. For all of you and from the bottom of my heart, I say thank you! Foremost, I would like to express my sincere and limitless gratitude to my advisor, Laura Grigori, thank you very much for the opportunities you gave me. My work would not have succeeded without you. I deeply admire your perfectionism, patience, and knowledge. What I learned from you over the last four years at both professional and scientific levels push me to give my best. I am grateful to Pascal Hénon, who also supervised my thesis, he smoothed this work from its beginning and supported until its very end. The moments I spent in Pau were enriching and unforgettable. Next, I would like to thank Julien Langou and Serge Gratton for reporting my manuscript. It is a great honour that both of you as well as Frédéric Nataf, Daniel Ruiz, and Philippe Ricoux participate in the jury of this thesis. I also pronounce my acknowledgements to Philippe Ricoux who believed in this work and subsidized it to be achieved. I will not forget Frédéric Nataf and the many discussions about domain decomposition that enriched my knowledge considerably. Many thanks to James Demmel for the research stay offered at UC Berkeley where I discovered the inner world of numerical computation. My two dear friends, Olivier Tissot and Sebastien Cayrols, without you, these years would have been much harder. You are the best co-bureau ever. Our friendship is some- thing I really appreciate. Oliver, where do I begin? With the scientific, the philosophical, or the linguistic discussions? Or should I start by thanking you for replacing me in the conferences that I could not attend? I could write pages thanking you for all the great (the not so great and, especially, the hardest) times where you were constantly supportive. Thank you, my friend. Sebastien, you do not know how the smile lost its way to our office after you left. I will always remember "the two minutes left" to finish the TSQR implementation. (For more than a week, they seemed endless.) Thank xi xii Acknowledgements you for the Five minutes of C lectures, they played an important role of finalizing the implementation part of this work. Furthermore, I cannot forget to thank Jan, Léa, Fabien R, Fabien W, Mathieu, Martin, Simplice, Alan, Bilal, Frédéric A, Noémie, Antoine, Chloé, Ani, Mehdi, and all my colleagues at LJLL and Inria-Paris especially who are at building A. I also thank all the members of the team ALPINES whome I have not got to mention yet, Frédéric Hecht, Xavier Claeys, Pierre Marchand, Pierre-Henri Tournier, Marcela Bonazzoli, Axel Fourmont, and Ryadh Haferssas. I thank Pierre Jolivet as well for his advices and the discussions about Krylov methods. To all those who prepared and facilitated my administrative papers, I thank you very much for your efforts, Laurence, Marion, and Ortenca from Inria; Catherine and Salima from LJLL; Monique from Total. Of course there are my friends who were supportive on the social as well as the psychological levels, Soleiman, Racha, Bayram, Gontran, Stephan, and Chadi. Nevertheless, if I go back in time to look for the reasons why I am here, I have no doubt that I will find it in my family home. There, where Science is sacred. That home contained unlimited love although it was small. My father, I do not forget my first lesson of numerical computing. It took place in the car on the way Damascus to Palmyra about eleven years ago. You are my inspiring idol and the first reason for my passion in science. I wish you were alive, I really miss you. My mother, whatever I do, I will never be able to give you back "—of course for an arbitrary small " > 0—of what you offered and still do. Although I wanted to be in your class at school, you did not want to. After many years, I realized that you were my first teacher in life. Thank you for your unconditional giving! Hazem, Mawea, Ahmad, Sarah, Abboudeh, my lovely brothers and sisters, thank you for being always available and supportive despite the long distance between us. It has been more than eight years since the last time we were all together. I hope we can meet all soon. Maria, Mazen, Julia, Reem, and Walid, the next generation, you are the sweetest angels I have ever seen. I dedicate this thesis to my father, grandmother, grandfather, aunt and her son. RIP. Finally, I would like to say thank you for Laura Mendoza who was next to me and supported me infinitely and unconditionally. You could stand my thesis symptoms of bad moods over the last two years. The long and cold winter night is etched in my memory. Without you the previous paragraphs would have been full of mistakes. Gracias por todo mi amor! Abstract xiii Solving linear systems arising from reservoirs modelling Abstract This thesis presents a work on iterative methods for solving linear systems that aim at reducing the communication in parallel computing. The main type of linear systems in which we are interested arises from a real-life reservoir simulation. Both schemes, implicit and explicit, of modelling the system are taken into account. Three approaches are studied separately. We consider non-symmetric (resp. symmetric) linear systems. This corresponds to the explicit (resp. implicit) formulation of the model problem. We start by presenting an approach that adds multiple search directions per iteration rather than one as in the classic iterative methods. Then, we discuss different strategies of recycling search subspaces. These strategies reduce the global iteration count of a considerable factor during a sequence of linear systems. We review different existing strategies and present a new one. We discuss the parallel implementation of these methods using a low-level language. Numerical experiments for both sequential and parallel implementations are presented. We also consider the algebraic domain decomposition approach. In an algebraic framework, we study the two-level additive Schwarz preconditioner. We provide the algebraic explicit form of a class of local coarse spaces that bounds the spectral condition number of the preconditioned matrix by a number pre-defined.
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