La complexification de la modélisation multi-physique conduit d’une part à devoir simuler des systèmes d’équations différentielles ordinaires et d’équations différentielles algébriques de plus en plus grands en nombre d’inconnues et sur des temps de simulation longs. D’autre part l’évolution des architectures de calcul parallèle nécessite d’autres voies de parallélisation que la décomposition de système en sous-systèmes. Dans ce travail, nous proposons de concevoir des méthodes de décomposition de domaine pour la résolution d’EDO en temps. Nous reformulons le problème à valeur initiale en un problème aux valeurs frontières sur l’intervalle de temps symétrisé, sous l’hypothèse de réversibilité du flot. Nous développons deux méthodes, la première apparentée à une méthode de complément de Schur, la seconde basée sur une méthode de type Schwarz dont nous montrons la convergence pouvant être accélérée par la méthode d’Aitken dans le cadre linéaire. Afin d’accélérer la convergence de cette dernière dans le cadre non-linéaire, nous introduisons les techniques d’extrapolation et d’accélération de la convergence des suites non-linéaires. Nous montrons les avantages et les limites de ces techniques. Les résultats obtenus nous conduisent à développer l’accélération de la méthode de type Schwarz par une méthode de Newton. Enfin nous nous intéressons à l’étude de conditions de raccord non-linéaires adaptées à la décomposition de domaine de problèmes non-linéaires. Nous nous servons du formalisme hamiltonien à ports, issu du domaine de l’automatique, pour déduire les conditions de raccord dans le cadre l’équation de Saint-Venant et de l’équation de la chaleur non-linéaire. Après une étude analytique de la convergence de la DDM associée à ces conditions de transmission, nous proposons et étudions une formulation de Lagrangien augmenté sous l’hypothèse de séparabilité de la contrainte. / Complexification of multi-physics modeling leads to have to simulate systems of ordinary differential equations and algebraic differential equations with increasingly large numbers of unknowns and over large times of simulation. In addition the evolution of parallel computing architectures requires other ways of parallelization than the decomposition of system in subsystems. In this work, we propose to design domain decomposition methods in time for the resolution of EDO. We reformulate the initial value problem in a boundary values problem on the symmetrized time interval, under the assumption of reversibility of the flow. We develop two methods, the first connected with a Schur complement method, the second based on a Schwarz type method for which we show convergence, being able to be accelerated by the Aitken method within the linear framework. In order to accelerate the convergence of the latter within the non-linear framework, we introduce the techniques of extrapolation and of acceleration of the convergence of non-linear sequences. We show the advantages and the limits of these techniques. The obtained results lead us to develop the acceleration of the method of the type Schwarz by a Newton method. Finally we investigate non-linear matching conditions adapted to the domain decomposition of nonlinear problems. We make use of the port-Hamiltonian formalism, resulting from the control field, to deduce the matching conditions in the framework of the shallow-water equation and the non-linear heat equation. After an analytical study of the convergence of the DDM associated with these conditions of transmission, we propose and study a formulation of augmented Lagrangian under the assumption of separability of the constraint.

Este trabalho apresenta a aplicação de técnicas de Decomposição de Domínio e Processamento Paralelo na solução de grandes sistemas de equações algébricas lineares provenientes da modelagem de fenômenos eletromagnéticos pelo Método de Elementos Finitos. Foram implementadas as técnicas dos tipos Complemento de Schur e o Método Aditivo de Schwarz, adaptadas para a resolução desses sistemas em cluster de computadores do tipo Beowulf e com troca de mensagens através da Biblioteca MPI. A divisão e balanceamento de carga entre os processadores são feitos pelo pacote METIS. Essa metodologia foi testada acoplada a métodos, seja iterativo (ICCG), seja direto (LU) na etapa de resolução dos sistemas referentes aos nós internos de cada partição. Para a resolução do sistema envolvendo os nós de fronteira, no caso do Complemento de Schur, utilizou-se uma implementação paralisada do Método de Gradientes Conjugados (PCG). S~ao discutidos aspectos relacionados ao desempenho dessas técnicas quando aplicadas em sistemas de grande porte. As técnicas foram testadas na solução de problemas de aplicação do Método de Elementos Finitos na Engenharia Elétrica (Magnetostática, Eletrocinética e Magnetodinâmica), sejam eles de natureza bidimensional com malhas não estruturadas, seja tridimensional, com malhas estruturadas. / This work presents the study of Domain Decomposition and Parallel Processing Techniques applied to the solution of systems of algebraic equations issued from the Finite Element Analysis of Electromagnetic Phenomena. Both Schur Complement and Schwarz Additive techniques were implemented. They were adapted to solve the linear systems in Beowulf clusters with the use of MPI library for message exchange. The load balance among processors is made with the aid of METIS package. The methodology was tested in association to either iterative (ICCG) or direct (LU) methods in order to solve the system related to the inner nodes of each partition. In the case of Schur Complement, the solution of the system related to the boundary nodes was performed with a parallelized Conjugated Gradient Method (PCG). Some aspects of the peformance of these techniques when applied to large scale problems have also been discussed. The techniques has been tested in the simulation of a collection of problems of Electrical Engineering, modelled by the Finite Element Method, both in two dimensions with unstructured meshes (Magnetostatics) and three dimensions with structured meshes (Electrokinetics).

There are several algorithms for solving the linear system of equations that arise from the finite element method with linear or near-linear computational complexity. One way is to find an approximation of the stiffness matrix that is such that it can be used in a preconditioned conjugate residual method, that is, a preconditioner to the stiffness matrix. We have studied two preconditioners for the conjugate residual method, both based on writing the stiffness matrix in block form, factorising it and then approximating the Schur complement block to get a preconditioner. We have studied the stationary reaction-diffusion-advection equation in two dimensions. The mesh is refined adaptively, giving a hierarchy of meshes. In the first method the Schur complement is approximated by the stiffness matrix at one coarser level of the mesh, in the second method it is approximated as the assembly of local Schur complements corresponding to macro triangles. For two levels the theoretical bound of the condition number is 1/(1-C²) for either method, where C is the Cauchy-Bunyakovsky-Schwarz constant. For multiple levels there is less theory. For the first method it is known that the condition number of the preconditioned stiffness matrix is O(l²), where l is the number of levels of the preconditioner, or, equivalently, the number mesh refinements. For the second method the asymptotic behaviour is not known theoretically. In neither case is the dependency of the condition number of C known. We have tested both methods on several problems and found the first method to always give a better condition number, except for very few levels. For all tested problems, using the first method it seems that the condition number is O(l), in fact it is typically not larger than Cl. For the second method the growth seems to be superlinear.

Robust preconditioners on block-triangular and block-factorized form for three types of linear systems of two-by-two block form are studied in this thesis. The first type of linear systems, which are dense, arise from a boundary element type of discretization of crack propagation problems. Numerical experiment show that simple algebraic preconditioning strategies results in iterative schemes that are highly competitive with a direct solution method. The second type of algebraic systems, which are sparse, indefinite and nonsymmetric, arise from a finite element (FE) discretization of the partial differential equations (PDE) that describe (visco)elastic glacial isostatic adjustment (GIA). The Schur complement approximation in the block preconditioners is constructed by assembly of local, exactly computed Schur matrices. The quality of the approximation is verified in numerical experiments. When the block preconditioners for the indefinite problem are combined with an inner iterative scheme preconditioned by a (nearly) optimal multilevel preconditioner, the resulting preconditioner is (nearly) optimal and robust with respect to problem size, material parameters, number of space dimensions, and coefficient jumps. Two approaches to mathematically formulate the PDEs for GIA are compared. In the first approach the equations are formulated in their full complexity, whereas in the second their formulation is confined to the features and restrictions of the employed FE package. Different solution methods for the algebraic problem are used in the two approaches. Analysis and numerical experiments reveal that the first strategy is more accurate and efficient than the latter. The block structure in the third type of algebraic systems is due to a fine-coarse splitting of the unknowns. The inverse of the pivot block is approximated by a sparse matrix which is assembled from local, exactly inverted matrices. Numerical experiments and analysis of the approximation show that it is robust with respect to problem size and coefficient jumps.

Cette thèse présente une méthode de résolution parallèle de systèmes linéaires creux qui combine efficacement les techniques de résolutions directes et itératives en utilisant une approche de type complément de Schur. Nous construisons une décomposition de domaine. L'intérieur des sous-domaines est éliminé de manière directe pour se ramener à un problème sur l'interface. Ce problème est résolu grâce à une méthode itérative préconditionnée par une factorisation incomplète. Un réordonnancement de l'interface permet la construction d'un préconditionneur global du complément de Schur. Des algorithmes minimisant le pic mémoire de la construction du préconditionneur sont proposés. Nous exploitons un schéma d'équilibrage de charge utilisant une répartition de multiples sous-domaines sur les processeurs. Les méthodes sont implémentées dans le solveur HIPS et des résultats expérimentaux parallèles sont présentés sur de grands cas tests industriels. / This thesis presents a parallel resolution method for sparse linear systems which combines effectively techniques of direct and iterative solvers using a Schur complement approach. A domain decomposition is built ; the interiors of the subdomains are eliminated by a direct method in order to use an iterative method only on the interface unknowns. The system on the interface (Schur complement) is solved thanks to an iterative method preconditioned by a global incomplete factorization. A special ordering on the Schur complement allows to build a scalable preconditioner. Algorithms minimizing the memory peak that appears during the construction of the preconditioner are presented. The memory is balanced thanks to a multiple domains per processors parallelization scheme. The methods are implemented in the HIPS solver and parallel experimental results are presented on large industrial test cases.

Dans cette thèse, nous nous intéressons à la résolution parallèle de grands systèmes linéaires creux. Nous nous focalisons plus particulièrement sur les solveurs linéaires creux hybrides directs itératifs tels que HIPS, MaPHyS, PDSLIN ou ShyLU, qui sont basés sur une décomposition de domaine et une approche « complément de Schur ». Bien que ces solveurs soient moins coûteux en temps et en mémoire que leurs homologues directs, ils ne sont néanmoins pas exempts de surcoûts. Dans une première partie, nous présentons les différentes méthodes de réduction de la consommation mémoire déjà existantes et en proposons une nouvelle qui n’impacte pas la robustesse numérique du précondionneur construit. Cette technique se base sur une atténuation du pic mémoire par un ordonnancement spécifique des tâches de calcul, d’allocation et de désallocation des blocs, notamment ceux se trouvant dans les parties « couplage » des domaines.Dans une seconde partie, nous nous intéressons à la question de l’équilibrage de la charge que pose la décomposition de domaine pour le calcul parallèle. Ce problème revient à partitionner le graphe d’adjacence de la matrice en autant de parties que de domaines désirés. Nous mettons en évidence le fait que pour avoir un équilibrage correct des temps de calcul lors des phases les plus coûteuses d’un solveur hybride tel que MaPHyS, il faut à la fois équilibrer les domaines en termes de nombre de noeuds et de taille d’interface locale. Jusqu’à aujourd’hui, les partitionneurs de graphes tels que Scotch et MeTiS ne s’intéressaient toutefois qu’au premier critère (la taille des domaines) dans le contexte de la renumérotation des matrices creuses. Nous proposons plusieurs variantes des algorithmes existants afin de prendre également en compte l’équilibrage des interfaces locales. Toutes nos modifications sont implémentées dans le partitionneur Scotch, et nous présentons des résultats sur de grands cas de tests industriels. / In this thesis, we focus on the parallel solving of large sparse linear systems. Our main interestis on direct-iterative hybrid solvers such as HIPS, MaPHyS, PDSLIN or ShyLU, whichrely on domain decomposition and Schur complement approaches. Althrough these solvers arenot as time and space consuming as direct methods, they still suffer from serious overheads. Ina first part, we thus present the existing techniques for reducing the memory consumption, andwe present a new method which does not impact the numerical robustness of the preconditioner.This technique reduces the memory peak by doing a special scheduling of computation, allocation,and freeing tasks in particular in the Schur coupling blocks of the matrix. In a second part,we focus on the load balancing of the domain decomposition in a parallel context. This problemconsists in partitioning the adjacency graph of the matrix in as many domains as desired. Wepoint out that a good load balancing for the most expensive steps of an hybrid solver such asMaPHyS relies on the balancing of both interior nodes and interface nodes of the domains.Through, until now, graph partitioners such as MeTiS or Scotch used to optimize only thefirst criteria (i.e., the balancing of interior nodes) in the context of sparse matrix ordering. Wepropose different variations of the existing algorithms to improve the balancing of interface nodesand interior nodes simultaneously. All our changes are implemented in the Scotch partitioner.We present our results on large collection of matrices coming from real industrial cases.

<p>This thesis studies subclasses of the class of H-matrices and their applications, with<br />emphasis on the investigation of the Schur complement properties. The contributions<br />of the thesis are new nonsingularity results, bounds for the maximum norm of the<br />inverse matrix, closure properties of some matrix classes under taking Schur<br />complements, as well as results on localization and separation of the eigenvalues of<br />the Schur complement based on the entries of the original matrix.</p> / <p>Докторска дисертација изучава поткласе класе Х-матрица и њихове примене,<br />првенствено у истраживању својстава Шуровог комплемента. Оригиналан допринос<br />тезе представљају нови услови за регуларност матрица, оцене максимум норме<br />инверзне матрице, резултати о затворености појединих класа матрица на Шуров<br />комплемент, као и резултати о локализацији и сепарацији карактеристичних<br />корена Шуровог комплемента на основу елемената полазне матрице.</p> / <p>Doktorska disertacija izučava potklase klase H-matrica i njihove primene,<br />prvenstveno u istraživanju svojstava Šurovog komplementa. Originalan doprinos<br />teze predstavljaju novi uslovi za regularnost matrica, ocene maksimum norme<br />inverzne matrice, rezultati o zatvorenosti pojedinih klasa matrica na Šurov<br />komplement, kao i rezultati o lokalizaciji i separaciji karakterističnih<br />korena Šurovog komplementa na osnovu elemenata polazne matrice.</p>