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The classical approach for solving evolution Partial Differential Equations (PDEs) using a parallel computer consists in first partitioning the spatial domain and assigning each subdomain to a processor to achieve space-parallelism, then advancing the solution sequentially. However, enabling parallelism along the time dimension, despite its intrinsic difficulty, can be of paramount importance to fast computations when space-parallelism is unfeasible, cannot fully exploit a massively parallel machine or when near-real-time prediction is desired. The aforementioned objective can be achieved by applying classical domain decomposition principles to the time axis. The latter is first partitioned into time-slices to be processed independently. Starting with approximate seed information that provides a set of initial conditions, the response is then advanced in parallel in each time-slice using a standard time-stepping integrator. This decomposed solution exhibits discontinuities or jumps at the time-slice boundaries if the initial guess is not accurate. Applying a Newton-like approach to the time-dependent system, a correction function is then computed to improve the accuracy of the seed values and the process is repeated until convergence is reached. Methods based on the above concept have been successfully applied to various problems but none was found to be competitive for even for the simplest of second-order hyperbolic PDEs, a class of equations that covers the field of structural dynamics among others. To overcome this difficulty, a key idea is to improve the sequential propagator used for correcting the seed values, observing that the original evolution problem and the derived corrective one are closely related. The present work first demonstrates how this insight can be brought to fruition in the context of linear oscillators, with numerical examples featuring structural models ranging from academic to more challenging large-scale ones. An extension of this method to nonlinear equations is then developed and its concrete application to geometrically nonlinear transient dynamics is presented. Finally, it is shown how the time-reversibility property that characterizes some of the above problems can be exploited to develop a new framework that provides an increased speed-up factor.
This volume contains a selection of papers presented at the 21st international conference on domain decomposition methods in science and engineering held in Rennes, France, June 25-29, 2012. Domain decomposition is an active and interdisciplinary research discipline, focusing on the development, analysis and implementation of numerical methods for massively parallel computers. Domain decomposition methods are among the most efficient solvers for large scale applications in science and engineering. They are based on a solid theoretical foundation and shown to be scalable for many important applications. Domain decomposition techniques can also naturally take into account multiscale phenomena. This book contains the most recent results in this important field of research, both mathematically and algorithmically and allows the reader to get an overview of this exciting branch of numerical analysis and scientific computing.
The main goal of this book is to provide an overview of some of the most recent developments in the field of Domain Decomposition Methods. Domain decomposition relates to the construction of preconditioners for the large algebraic systems of equations which often arise in applications, by solving smaller instances of the same problem. It also relates to the construction of approximation methods built from different discretizations in different subdomains. The resulting methods are among the most successful parallel solvers for many large scale problems in computational science and engineering. The papers in this collection reflect some of the most active research areas in domain decomposition such as novel FETI, Neumann-Neumann, overlapping Schwarz and Mortar methods.
L'EXPLOITATION DE SATELLITES EN ORBITE IMPLIQUE UNE EVALUATION PRECISE ET FREQUENTE DE LEUR POSITION. POUR CALCULER LA TRAJECTOIRE DU SATELLITE, IL FAUT RESOUDRE UN SYSTEME D'EQUATIONS DIFFERENTIELLES DU SECOND ORDRE. LA RESOLUTION DE TELS SYSTEMES EST REALISEE A L'AIDE DE METHODES D'INTEGRATION NUMERIQUES QUI PERMETTENT D'ATTEINDRE LA PRECISION SOUHAITEE. LE BUT DE NOTRE ETUDE EST DE PROPOSER UNE METHODE NUMERIQUE DE CALCUL D'ORBITE QUI SOIT PRECISE ET RAPIDE. POUR REDUIRE LE TEMPS DE CALCUL, NOUS PROPOSONS DES ALGORITHMES PARALLELES DE CALCUL D'ORBITES. LE PARALLELISME REPOSE SUR UNE PARTITION DE L'INTERVALLE DE TEMPS ET DES CALCULS SIMULTANES DANS CHAQUE SOUS-INTERVALLE. UN PROCEDE ITERATIF DE CORRECTION PERMET D'OBTENIR LA SOLUTION FINALE. NOTRE PREMIER ALGORITHME EST BASE SUR LA REPARTITION DES SOUS-INTERVALLES EN BLOCS SUR CHAQUE PROCESSEUR. PUIS NOUS CONSTRUISONS UN DEUXIEME ALGORITHME PARALLELE QUI CONSISTE A CALCULER AUTANT DE SOUS-INTERVALLES QUE DE PROCESSEURS PUIS A DECALER LA FENETRE DES SOUS-INTERVALLES JUSQU'A OBTENIR LA CONVERGENCE GLOBALE. UNE ETUDE SUR LA CONVERGENCE DE LA METHODE, AINSI QUE LA PRISE EN COMPTE DE LA GEOMETRIE DU PROBLEME, NOUS PERMETTENT ENSUITE D'OPTIMISER LES PERFORMANCES DE NOTRE METHODE. NOUS PROPOSONS UN ALGORITHME DE DECOUPAGE AUTOMATIQUE DE L'INTERVALLE DE TEMPS QUI MINIMISE LA DUREE DES CALCULS DANS NOTRE ALGORITHME. ENFIN, NOUS PRESENTONS LES RESULTATS NUMERIQUES DES DIFFERENTS ALGORITHMES EXECUTES SUR UNE MACHINE DE 64 PROCESSEURS.
This book explains deep learning concepts and derives semi-supervised learning and nuclear learning frameworks based on cognition mechanism and Lie group theory. Lie group machine learning is a theoretical basis for brain intelligence, Neuromorphic learning (NL), advanced machine learning, and advanced artifi cial intelligence. The book further discusses algorithms and applications in tensor learning, spectrum estimation learning, Finsler geometry learning, Homology boundary learning, and prototype theory. With abundant case studies, this book can be used as a reference book for senior college students and graduate students as well as college teachers and scientific and technical personnel involved in computer science, artifi cial intelligence, machine learning, automation, mathematics, management science, cognitive science, financial management, and data analysis. In addition, this text can be used as the basis for teaching the principles of machine learning. Li Fanzhang is professor at the Soochow University, China. He is director of network security engineering laboratory in Jiangsu Province and is also the director of the Soochow Institute of industrial large data. He published more than 200 papers, 7 academic monographs, and 4 textbooks. Zhang Li is professor at the School of Computer Science and Technology of the Soochow University. She published more than 100 papers in journals and conferences, and holds 23 patents. Zhang Zhao is currently an associate professor at the School of Computer Science and Technology of the Soochow University. He has authored and co-authored more than 60 technical papers.
The book is a self-contained introduction to the results and methods in classical invariant theory.
Understanding the behavior of basic sampling techniques and intrinsic geometric attributes of data is an invaluable skill that is in high demand for both graduate students and researchers in mathematics, machine learning, and theoretical computer science. The last ten years have seen significant progress in this area, with many open problems having been resolved during this time. These include optimal lower bounds for epsilon-nets for many geometric set systems, the use of shallow-cell complexity to unify proofs, simpler and more efficient algorithms, and the use of epsilon-approximations for construction of coresets, to name a few. This book presents a thorough treatment of these probabilistic, combinatorial, and geometric methods, as well as their combinatorial and algorithmic applications. It also revisits classical results, but with new and more elegant proofs. While mathematical maturity will certainly help in appreciating the ideas presented here, only a basic familiarity with discrete mathematics, probability, and combinatorics is required to understand the material.