Fast Solution of Sparse Linear Systems with Adaptive Choice of Preconditioners Zakariae Jorti
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Fast solution of sparse linear systems with adaptive choice of preconditioners Zakariae Jorti To cite this version: Zakariae Jorti. Fast solution of sparse linear systems with adaptive choice of preconditioners. Math- ematics [math]. Sorbonne Université / Université Pierre et Marie Curie - Paris VI, 2019. English. tel-02425679v1 HAL Id: tel-02425679 https://tel.archives-ouvertes.fr/tel-02425679v1 Submitted on 31 Dec 2019 (v1), last revised 15 Feb 2021 (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 École doctorale de Sciences Mathématiques de Paris Centre Laboratoire Jacques-Louis Lions Résolution rapide de systèmes linéaires creux avec choix adaptatif de préconditionneurs Par Zakariae Jorti Thèse de doctorat de Mathématiques appliquées Dirigée par Laura Grigori Co-supervisée par Ani Anciaux-Sedrakian et Soleiman Yousef Présentée et soutenue publiquement le 03/10/2019 Devant un jury composé de: M. Tromeur-Dervout Damien, Professeur, Université de Lyon, Rapporteur M. Schenk Olaf, Professeur, Università della Svizzera italiana, Rapporteur M. Hecht Frédéric, Professeur, Sorbonne Université, Président du jury Mme. Emad Nahid, Professeur, Université Paris-Saclay, Examinateur M. Vasseur Xavier, Ingénieur de recherche, ISAE-SUPAERO, Examinateur Mme. Grigori Laura, Directrice de recherche, Inria Paris, Directrice de thèse Mme. Anciaux-Sedrakian Ani, Ingénieur de recherche, IFPEN, Co-encadrante de thèse M. Yousef Soleiman, Ingénieur de recherche, IFPEN, Co-Encadrant de thèse Doctoral School Sciences Mathématiques de Paris Centre University Department Laboratoire Jacques-Louis Lions Thesis defended by Zakariae Jorti In order to become Doctor from Sorbonne Université Academic Field Applied Mathematics Fast solution of sparse linear systems with adaptive choice of preconditioners Under the supervision of: Doctoral advisor: Dr. Laura Grigori – INRIA, EPI Alpines, LJLL IFPEN supervisor: Dr. Ani Anciaux-Sedrakian – IFPEN, Digital Sciences & Technologies Division IFPEN supervisor: Dr. Soleiman Yousef – IFPEN, Digital Sciences & Technologies Division Contents Introduction 1 General context and scope of application............................1 State-of-the-art 5 PDE Solving.............................................5 Linear solvers............................................6 Preconditioning the linear systems................................9 Approximate inversion....................................... 14 Error estimators........................................... 15 Background notions......................................... 16 1 Global and local approaches of adaptive preconditioning 21 1.1 Global adaptive preconditioning based on a posteriori error estimates........ 22 1.1.1 Adaptative choice of preconditioners from a posteriori error estimates in a simulation....................................... 23 1.1.2 General parameters................................. 24 1.1.3 Runtime platform.................................. 26 1.1.4 Computing framework............................... 27 1.1.5 Presentation of the study cases........................... 27 1.1.6 Numerical results and comments......................... 27 iv 1.1.7 Adaptive choice with restarted GMRES...................... 28 1.2 Local adaptive preconditioning based on a posteriori error estimates......... 33 1.2.1 Error estimates and conditioning.......................... 34 1.2.2 Partitioning and permuting the matrix...................... 38 1.2.3 Variable block Jacobi-type preconditioning.................... 44 1.2.4 Numerical tests.................................... 45 1.2.5 Conclusion...................................... 52 2 Adaptive solution of linear systems based on a posteriori error estimators 53 2.1 Introduction.......................................... 54 2.2 Model problem........................................ 55 2.3 A posteriori error estimates................................. 56 2.3.1 Basic a posteriori error estimates.......................... 57 2.3.2 Upper bound on the algebraic error........................ 58 2.4 Matrix decomposition and local error reduction..................... 59 2.4.1 Matrix decomposition: sum splitting....................... 60 2.4.2 Matrix decomposition: Block partitioning..................... 60 2.5 Adaptive preconditioner for PCG based on local error indicators........... 63 2.5.1 Partitioned preconditioners suited for error reduction............. 63 2.5.2 Condition number improvement.......................... 68 2.5.3 Context of use..................................... 68 2.6 Numerical results....................................... 69 2.6.1 Some strategies for initiating the adaptive procedure.............. 69 2.6.2 Poisson’s equation.................................. 71 2.6.3 Diffusion equation with inhomogeneous coefficient............... 75 2.7 Conclusions.......................................... 79 3 Approximate adaptive procedure based on a posteriori error estimates 81 3.1 Introduction.......................................... 82 3.2 Preliminaries.......................................... 83 3.3 A low rank approximation on AL .............................. 84 3.4 Some condition number bounds for the exact and the approximate adaptive precon- ditioners............................................ 87 v 3.5 Preconditioning cost..................................... 91 3.6 Possible choices for M2 .................................... 92 3.7 Numerical results....................................... 99 3.7.1 Test case n°1...................................... 100 3.7.2 Test case n°2...................................... 102 3.8 Conclusion........................................... 103 4 Adaptive a posteriori error estimates-based preconditioner for controlling a local alge- braic error norm 105 4.1 Introduction.......................................... 106 4.2 Preliminaries.......................................... 107 4.3 Local error reduction with PCG............................... 108 4.4 Controlling the local algebraic error in fixed-point iteration scheme.......... 111 4.5 Deriving a block partitioning and controlling the corresponding algebraic error norm113 4.6 Link with the adaptive preconditioner for PCG based on local error indicators... 117 4.7 Numerical results....................................... 122 4.8 Conclusion........................................... 127 5 Application to test cases stemming from industrial simulations with finite volume dis- cretization 129 5.1 Introduction.......................................... 130 5.2 Cell-centered finite volume discretization......................... 131 5.3 Matrix decomposition and local error reduction..................... 132 5.4 Adaptive linear solver.................................... 134 5.4.1 Schur complement procedure............................ 135 5.4.2 Error reduction properties of the adaptive procedure.............. 136 5.5 Numerical results....................................... 137 5.5.1 Steady problem: Heterogeneous media and uniform mesh refinement.... 137 5.5.2 Unsteady problem: Heterogeneous media and uniform mesh refinement.. 139 5.6 Conclusion........................................... 144 Conclusion 145 Bibliography 147 vi List of Figures 1 Example of Block-Jacobi preconditionner on 4 subdomains without overlap..... 12 2 Lower triangular matrix L .................................. 12 3 Upper triangular matrix U .................................. 12 1.1 Example of a simple 4 × 4 mesh.............................. 34 1.2 Structure of matrix A ..................................... 35 1.3 Distribution of algebraic EE over main domain during simulation (3DBlackOil test case)............................................... 36 1.4 Distribution of algebraic EE over main domain at the end of simulation t = t38 (SPE10Layer85 test case)................................... 37 1.5 Condition numbers of submatrices VS Error estimates means on subdomains for 2 different sizes of partitions at end of simulation t = t38 (SPE10Layer85 test case).. 37 1.6 Condition numbers of submatrices VS Error estimates means on subdomains for 2 different sizes of partitions at mid-simulation t = t42 (3DBlackOil test case)..... 38 1.7 Constructing and partitioning the graph of the matrix into two parts......... 39 1.8 Row and column permutations after a partitioning.................... 41 1.9 Constructing and partitioning the graph of the matrix into two parts by taking account of the error estimate values............................. 43 1.10 Shape of the Block-Jacobi preconditioning matrix after the merger of the first two subdomains.......................................... 45 2.1 Simple example of the decomposition (2.7) with a 2 × 2 mesh grid........... 56 viii List of Figures 2.2 Splittings of matrix A with local stiffness matrices (left) and the associated algebraic 2 × 2 block splitting (right)................................. 61 (1) 2.3 Galerkin solution uh .................................... 71 2.4 Initial distribution and a posteriori estimation of algebraic error for