Parallel symmetric eigenvalue problem solvers

Sparse symmetric eigenvalue problems arise in many computational science and engineering applications: in structural mechanics, nanoelectronics, and spectral reordering, for example. Often, the large size of these problems requires the development of eigensolvers that scale well on parallel computing platforms. In this dissertation, we describe two such eigensolvers, TraceMin and TraceMin-Davidson. These methods are different from many other eigensolvers in that they do not require accurate linear solves to be performed at each iteration in order to find the smallest eigenvalues and their associated eigenvectors. After introducing these closely related eigensolvers, we discuss alternative methods for solving the saddle point problems arising at each iteration, which can improve the overall running time. Additionally, we present TraceMin-Multisectioning, a new TraceMin implementation geared towards finding large numbers of eigenpairs in any given interval of the spectrum. We conclude with numerical experiments comparing our trace-minimization solvers to other popular eigensolvers (such as Krylov-Schur, LOBPCG, Jacobi-Davidson, and FEAST), establishing the competitiveness of our methods

Book of Abstracts of the Sixth SIAM Workshop on Combinatorial Scientific Computing

Book of Abstracts of CSC14 edited by Bora UçarInternational audienceThe Sixth SIAM Workshop on Combinatorial Scientific Computing, CSC14, was organized at the Ecole Normale Supérieure de Lyon, France on 21st to 23rd July, 2014. This two and a half day event marked the sixth in a series that started ten years ago in San Francisco, USA. The CSC14 Workshop's focus was on combinatorial mathematics and algorithms in high performance computing, broadly interpreted. The workshop featured three invited talks, 27 contributed talks and eight poster presentations. All three invited talks were focused on two interesting fields of research specifically: randomized algorithms for numerical linear algebra and network analysis. The contributed talks and the posters targeted modeling, analysis, bisection, clustering, and partitioning of graphs, applied in the context of networks, sparse matrix factorizations, iterative solvers, fast multi-pole methods, automatic differentiation, high-performance computing, and linear programming. The workshop was held at the premises of the LIP laboratory of ENS Lyon and was generously supported by the LABEX MILYON (ANR-10-LABX-0070, Université de Lyon, within the program ''Investissements d'Avenir'' ANR-11-IDEX-0007 operated by the French National Research Agency), and by SIAM

Performance and Energy Optimization of the Iterative Solution of Sparse Linear Systems on Multicore Processors

En esta tesis doctoral se aborda la solución de sistemas dispersos de ecuaciones lineales utilizando métodos iterativos precondicionados basados en subespacios de Krylov. En concreto, se centra en ILUPACK, una biblioteca que implementa precondicionadores de tipo ILU multinivel para la solución eficiente de sistemas lineales dispersos. El incremento en el número de ecuaciones, y la aparición de nuevas arquitecturas, motiva el desarrollo de una versión paralela de ILUPACK que optimice tanto el tiempo de ejecución como el consumo energético en arquitecturas multinúcleo actuales y en clusters de nodos construidos con esta tecnología. El objetivo principal de la tesis es el diseño, implementación y valuación de resolutores paralelos energéticamente eficientes para sistemas lineales dispersos orientados a procesadores multinúcleo así como aceleradores hardware como el Intel Xeon Phi. Para lograr este objetivo, se aprovecha el paralelismo de tareas mediante OmpSs y MPI, y se desarrolla un entorno automático para detectar ineficiencias energéticas.In this dissertation we target the solution of large sparse systems of linear equations using preconditioned iterative methods based on Krylov subspaces. Specifically, we focus on ILUPACK, a library that offers multi-level ILU preconditioners for the effective solution of sparse linear systems. The increase of the number of equations and the introduction of new HPC architectures motivates us to develop a parallel version of ILUPACK which optimizes both execution time and energy consumption on current multicore architectures and clusters of nodes built from this type of technology. Thus, the main goal of this thesis is the design, implementation and evaluation of parallel and energy-efficient iterative sparse linear system solvers for multicore processors as well as recent manycore accelerators such as the Intel Xeon Phi. To fulfill the general objective, we optimize ILUPACK exploiting task parallelism via OmpSs and MPI, and also develope an automatic framework to detect energy inefficiencies

Hydrodynamics of Suspensions of Passive and Active Rigid Particles: A Rigid Multiblob Approach

We develop a rigid multiblob method for numerically solving the mobility problem for suspensions of passive and active rigid particles of complex shape in Stokes flow in unconfined, partially confined, and fully confined geometries. As in a number of existing methods, we discretize rigid bodies using a collection of minimally-resolved spherical blobs constrained to move as a rigid body, to arrive at a potentially large linear system of equations for the unknown Lagrange multipliers and rigid-body motions. Here we develop a block-diagonal preconditioner for this linear system and show that a standard Krylov solver converges in a modest number of iterations that is essentially independent of the number of particles. For unbounded suspensions and suspensions sedimented against a single no-slip boundary, we rely on existing analytical expressions for the Rotne-Prager tensor combined with a fast multipole method or a direct summation on a Graphical Processing Unit to obtain an simple yet efficient and scalable implementation. For fully confined domains, such as periodic suspensions or suspensions confined in slit and square channels, we extend a recently-developed rigid-body immersed boundary method to suspensions of freely-moving passive or active rigid particles at zero Reynolds number. We demonstrate that the iterative solver for the coupled fluid and rigid body equations converges in a bounded number of iterations regardless of the system size. We optimize a number of parameters in the iterative solvers and apply our method to a variety of benchmark problems to carefully assess the accuracy of the rigid multiblob approach as a function of the resolution. We also model the dynamics of colloidal particles studied in recent experiments, such as passive boomerangs in a slit channel, as well as a pair of non-Brownian active nanorods sedimented against a wall.Comment: Under revision in CAMCOS, Nov 201

Application of HPC in eddy current electromagnetic problem solution

As engineering problems are becoming more and more advanced, the size of an average model solved by partial differential equations is rapidly growing and, in order to keep simulation times within reasonable bounds, both faster computers and more efficient software implementations are needed. In the first part of this thesis, the full potential of simulation software has been exploited through high performance parallel computing techniques. In particular, the simulation of induction heating processes is accomplished within reasonable solution times, by implementing different parallel direct solvers for large sparse linear system, in the solution process of a commercial software. The performance of such library on shared memory systems has been remarkably improved by implementing a multithreaded version of MUMPS (MUltifrontal Massively Parallel Solver) library, which have been tested on benchmark matrices arising from typical induction heating process simulations. A new multithreading approach and a low rank approximation technique have been implemented and developed by MUMPS team in Lyon and Toulouse. In the context of a collaboration between MUMPS team and DII-University of Padova, a preliminary version of such functionalities could be tested on induction heating benchmark problems, and a substantial reduction of the computational cost and memory requirements could be achieved. In the second part of this thesis, some examples of design methodology by virtual prototyping have been described. Complex multiphysics simulations involving electromagnetic, circuital, thermal and mechanical problems have been performed by exploiting parallel solvers, as developed in the first part of this thesis. Finally, multiobjective stochastic optimization algorithms have been applied to multiphysics 3D model simulations in search of a set of improved induction heating device configurations

Sur la conception de solveurs linéaires hybrides pour les architectures parallèles modernes

In the context of this thesis, our focus is on numerical linear algebra, more precisely on solution of large sparse systems of linear equations. We focus on designing efficient parallel implementations of MaPHyS, an hybrid linear solver based on domain decomposition techniques. First we investigate the MPI+threads approach. In MaPHyS, the first level of parallelism arises from the independent treatment of the various subdomains. The second level is exploited thanks to the use of multi-threaded dense and sparse linear algebra kernels involved at the subdomain level. Such an hybrid implementation of an hybrid linear solver suitably matches the hierarchical structure of modern supercomputers and enables a trade-off between the numerical and parallel performances of the solver. We demonstrate the flexibility of our parallel implementation on a set of test examples. Secondly, we follow a more disruptive approach where the algorithms are described as sets of tasks with data inter-dependencies that leads to a directed acyclic graph (DAG) representation. The tasks are handled by a runtime system. We illustrate how a first task-based parallel implementation can be obtained by composing task-based parallel libraries within MPI processes throught a preliminary prototype implementation of our hybrid solver. We then show how a task-based approach fully abstracting the hardware architecture can successfully exploit a wide range of modern hardware architectures. We implemented a full task-based Conjugate Gradient algorithm and showed that the proposed approach leads to very high performance on multi-GPU, multicore and heterogeneous architectures.Dans le contexte de cette thèse, nous nous focalisons sur des algorithmes pour l’algèbre linéaire numérique, plus précisément sur la résolution de grands systèmes linéaires creux. Nous mettons au point des méthodes de parallélisation pour le solveur linéaire hybride MaPHyS. Premièrement nous considerons l'aproche MPI+threads. Dans MaPHyS, le premier niveau de parallélisme consiste au traitement indépendant des sous-domaines. Le second niveau est exploité grâce à l’utilisation de noyaux multithreadés denses et creux au sein des sous-domaines. Une telle implémentation correspond bien à la structure hiérarchique des supercalculateurs modernes et permet un compromis entre les performances numériques et parallèles du solveur. Nous démontrons la flexibilité de notre implémentation parallèle sur un ensemble de cas tests. Deuxièmement nous considérons un approche plus innovante, où les algorithmes sont décrits comme des ensembles de tâches avec des inter-dépendances, i.e., un graphe de tâches orienté sans cycle (DAG). Nous illustrons d’abord comment une première parallélisation à base de tâches peut être obtenue en composant des librairies à base de tâches au sein des processus MPI illustrer par un prototype d’implémentation préliminaire de notre solveur hybride. Nous montrons ensuite comment une approche à base de tâches abstrayant entièrement le matériel peut exploiter avec succès une large gamme d’architectures matérielles. À cet effet, nous avons implanté une version à base de tâches de l’algorithme du Gradient Conjugué et nous montrons que l’approche proposée permet d’atteindre une très haute performance sur des architectures multi-GPU, multicoeur ainsi qu’hétérogène

Software for Exascale Computing - SPPEXA 2016-2019

This open access book summarizes the research done and results obtained in the second funding phase of the Priority Program 1648 "Software for Exascale Computing" (SPPEXA) of the German Research Foundation (DFG) presented at the SPPEXA Symposium in Dresden during October 21-23, 2019. In that respect, it both represents a continuation of Vol. 113 in Springer’s series Lecture Notes in Computational Science and Engineering, the corresponding report of SPPEXA’s first funding phase, and provides an overview of SPPEXA’s contributions towards exascale computing in today's sumpercomputer technology. The individual chapters address one or more of the research directions (1) computational algorithms, (2) system software, (3) application software, (4) data management and exploration, (5) programming, and (6) software tools. The book has an interdisciplinary appeal: scholars from computational sub-fields in computer science, mathematics, physics, or engineering will find it of particular interest