[pt] Desde a década de 1990, o Método Fast Multipole (FMM) tem sido usado em conjunto com o Métodos dos Elementos de Contorno (BEM) para a simulação de problemas de grande escala. Este método utiliza expansões em série de Taylor para aglomerar pontos da discretização do contorno, de forma a reduzir o tempo computacional necessário para completar a simulação. Ele se tornou uma ferramenta bastante importante para os BEMs, pois eles apresentam matrizes cheias e assimétricas, o que impossibilita a utilização de técnicas de otimização de solução de sistemas de equação. A aplicação do FMM ao BEM é bastante complexa e requer muita manipulação matemática. Este trabalho apresenta uma formulação do FMM que é independente da solução fundamental utilizada pelo BEM, o Método Fast Multipole Generalizado (GFMM), que se aplica a elementos de contorno curvos e de qualquer ordem. Esta característica é importante, já que os desenvolvimentos de fast multipole encontrados na literatura se restringem apenas a elementos constantes. Todos os aspectos são abordados neste trabalho, partindo da sua base matemática, passando por validação numérica, até a solução de problemas de potencial com muitos milhões de graus de liberdade. A aplicação do GFMM a problemas de potencial e elasticidade é discutida e validada, assim como os desenvolvimentos necessários para a utilização do GFMM com o Método Híbrido Simplificado de Elementos de Contorno (SHBEM). Vários resultados numéricos comprovam a eficiência e precisão do método apresentado. A literatura propõe que o FMM pode reduzir o tempo de execução do algoritmo do BEM de O(N2) para O(N), em que N é o número de graus de liberdade do problema. É demonstrado que esta redução é de fato possível no contexto do GFMM, sem a necessidade da utilização de qualquer técnica de otimização computacional. / [en] The Fast Multipole Method (FMM) has been used since the 1990s with the Boundary Elements Method (BEM) for the simulation of large-scale problems. This method relies on Taylor series expansions of the underlying fundamental solutions to cluster the nodes on the discretised boundary of a domain, aiming to reduce the computational time required to carry out the simulation. It has become an important tool for the BEMs, as they present matrices that are full and nonsymmetric, so that the improvement of storage allocation and execution time is not a simple task. The application of the FMM to the BEM ends up with a very intricate code, and usually changing from one problem s fundamental solution to another is not a simple matter. This work presents a kernel-independent formulation of the FMM, here called the General Fast Multipole Method (GFMM), which is also able to deal with high order, curved boundary elements in a straightforward manner. This is an important feature, as the fast multipole implementations reported in the literature only apply to constant elements. All necessary aspects of this method are presented, starting with the mathematical basics of both FMM and BEM, carrying out some numerical assessments, and ending up with the solution of large potential problems. The application of the GFMM to both potential and elasticity problems is discussed and validated in the context of BEM. Furthermore, the formulation of the GFMM with the Simplified Hybrid Boundary Elements Method (SHBEM) is presented. Several numerical assessments show that the GFMM is highly efficient and may be as accurate as arbitrarily required, for problems with up to many millions of degrees of freedom. The literature proposes that the FMM is capable of reducing the time complexity of the BEM algorithms from O(N2) to O(N), where N is the number of degrees of freedom. In fact, it is shown that the GFMM is able to arrive at such time reduction without resorting to techniques of computational optimisation.

We present algorithms and implementations that overcome obstacles in the migration of the Fast Multipole Method (FMM), one of the most important algorithms in computational science and engineering, to exascale computing. Emerging architectural approaches to exascale computing are all characterized by data movement rates that are slow relative to the demand of aggregate floating point capability, resulting in performance that is bandwidth limited. Practical parallel applications of FMM are impeded in their scaling by irregularity of domains and dominance of collective tree communication, which is known not to scale well. We introduce novel ideas that improve partitioning of the N-body problem with boundary distribution through a sampling-based mechanism that hybridizes two well-known partitioning techniques, Hashed Octree (HOT) and Orthogonal Recursive Bisection (ORB). To reduce communication cost, we employ two methodologies. First, we directly utilize features available in parallel runtime systems to enable asynchronous computing and overlap it with communication. Second, we present Hierarchical Sparse Data Exchange (HSDX), a new all-to-all algorithm that inherently relieves communication by relaying sparse data in a few steps of neighbor exchanges. HSDX exhibits superior scalability and improves relative performance compared to the default MPI alltoall and other relevant literature implementations. We test this algorithm alongside others on a Cray XC40 tightly coupled with the Aries network and on Intel Many Integrated Core Architecture (MIC) represented by Intel Knights Corner (KNC) and Intel Knights Landing (KNL) as modern shared-memory CPU environments. Tests include comparisons of thoroughly tuned handwritten versus auto-vectorization of FMM Particle-to-Particle (P2P) and Multipole-to-Local (M2L) kernels. Scalability of task-based parallelism is assessed with FMM’s tree traversal kernel using different threading libraries. The MIC tests show large performance gains after adopting the prescribed techniques, which are inevitable in a world that is moving towards many-core parallelism.

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Le travail effectué dans cette thèse a consisté à analyser différents aspects mathématiques et numériques d'une stratégie de résolution des problèmes de propagation d'onde acoustique et électromagnétique en domaine extérieur. Nous nous intéressons plus particulièrement à la méthode de couplage entre éléments finis et représentation intégrale (CEFRI) où nous analysons un algorithme de résolution itérative par analogie avec une méthode de décomposition de domaine ainsi que l'utilisation de la méthode multipôles rapide (FMM). Le système à résoudre fait intervenir des opérateurs intégraux ce qui rend crucial le recours à des méthodes rapides telles que la FMM. L'analogie avec une méthode de décomposition de domaine s'obtient par extension au problème de Maxwell des résultats établis par F. Ben Belgacem et al. pour le problème de Helmholtz posé en domaine non borné. Pour cela, nous avons montré le lien entre la méthode CEFRI et la méthode de Schwarz avec recouvrement total pour la résolution du problème de Maxwell en domaine non borné. Cette relecture de la méthode CEFRI offre également une technique de préconditionnement pour les solveurs de Krylov et nous a permis d'avoir une idée préliminaire sur la convergence de ces méthodes. Ainsi, nous nous intéressons plutôt à des méthodes itératives rapides. Pour cela, nous avons mené une analyse théorique afin de montrer la convergence superlinéaire du GMRES dans une configuration sphérique. La validation de ces aspects a été réalisée par l'enrichissement de nombreux intégrants de la librairie éléments finis Mélina++, en C++. / We are concerned with the study of different aspects of a numerical strategy for the resolution of acoustic and electromagnetic scattering problems. We focus more particu- larly on a coupling of finite element and integral representation (CEFRI) : we study an iterative algorithm by analogy with a domain decomposition method, and consider the use of the Fast Multipole Method (FMM). The system to be solved involves integral operators which requires the use of fast methods such as the FMM. The correspondence with a domain decomposition method is obtained by extending to the exterior Maxwell problem the results derived by F. Ben Belgacem et al. for the Helmholtz problem posed in unbounded domain. To this aim, we show the analogy to the Schwarz method with total overlap. This interpretation of CEFRI suggests a preconditioner for Krylov solvers and enables us to have a preliminary idea of their convergence. We derive in this context an analytical proof of a superlinear convergence of GMRES in a spherical configuration. The validation of these aspects has been achieved by the enrichment of the finite element library Mélina++ in C++.

La méthode multipôle permet d'accélérer les produits matrices-vecteurs, utilisés par les solveurs itératifs pour déterminer le comportement électromagnétique, d'un objet soumis à une onde incidente. Nos travaux ont pour but d'adapter cette méthode pour la rendre efficace sur les architectures hétérogènes contenant des GPU. Pour cela, nous utilisons une ordonnanceur dynamique, StarPU, qui effectuera la distribution des tâches de calcul au sein d'un nœud. Pour la parallélisation en mémoire distribuée, nous effectuerons un ordonnancement statique des boîtes, couplé à un ordonnancement dynamique des interactions proches. / The Fast Multipole Method can speed up matrix-vector products, found in iterative solvers in order to compute the electromagnetics response of an object subject to an incident wave. We have intended to adapt this method to make it effective on heterogeneous architectures with GPUs. For this purpose, we use a dynamic scheduler named StarPU, which distributes the tasks within a node. For the parallelization in distributed memory, we distribute the tasks statically but we distribute the near interactions dynamically..

Two computationally fast fluid mechanical models for vertical axis turbines are the streamtube and the vortex model. The streamtube model is the fastest, allowing three-dimensional modeling of the turbine, but lacks a proper time-dependent description of the flow through the turbine. The vortex model used is two-dimensional, but gives a more complete time-dependent description of the flow. Effects of a velocity profile and the inclusion of struts have been investigated with the streamtube model. Simulations with an inhomogeneous velocity profile predict that the power coefficient of a vertical axis turbine is relatively insensitive to the velocity profile. For the struts, structural mechanic loads have been computed and the calculations show that if turbines are designed for high flow velocities, additional struts are required, reducing the efficiency for lower flow velocities.Turbines in channels and turbine arrays have been studied with the vortex model. The channel study shows that smaller channels give higher power coefficients and convergence is obtained in fewer time steps. Simulations on a turbine array were performed on five turbines in a row and in a zigzag configuration, where better performance is predicted for the row configuration. The row configuration was extended to ten turbines and it has been shown that the turbine spacing needs to be increased if the misalignment in flow direction is large.A control system for the turbine with only the rotational velocity as input has been studied using the vortex model coupled with an electrical model. According to simulations, this system can obtain power coefficients close to the theoretical peak values. This control system study has been extended to a turbine farm. Individual control of each turbine has been compared to a less costly control system where all turbines are connected to a mutual DC bus through passive rectifiers. The individual control performs best for aerodynamically independent turbines, but for aerodynamically coupled turbines, the results show that a mutual DC bus can be a viable option.Finally, an implementation of the fast multipole method has been made on a graphics processing unit (GPU) and the performance gain from this platform is demonstrated.

GPGPU Computing using CUDA is rapidly gaining ground today. GPGPU has been brought to the masses through the ease of use of CUDA and ubiquity of graphics cards supporting the same. Although CUDA has a low learning curve for programmers familiar with standard programming languages like C, extracting optimum performance from it, through optimizations and hand tuning is not a trivial task. This is because, in case of GPGPU, an optimization strategy rarely affects the functioning in an isolated manner. Many optimizations affect different aspects for better or worse, establishing a tradeoff situation between them, which needs to be carefully handled to achieve good performance. Thus optimizing an application for CUDA is tough and the performance gain might not be commensurate to the coding effort put in. I propose to simplify the process of optimizing CUDA programs using a CUDA Performance Analyzer. The analyzer is based on analytical modeling of CUDA compatible GPUs. The model characterizes the different aspects of GPU compute unified architecture and can make prediction about expected performance of a CUDA program. It would also give an insight into the performance bottlenecks of the CUDA implementation. This would hint towards, what optimizations need to be applied to improve performance. Based on the model, one would also be able to make a prediction about the performance of the application if the optimizations are applied to the CUDA implementation. This enables a CUDA programmer to test out different optimization strategies without putting in a lot of coding effort.

The numerical simulation of elastic wave propagation in unbounded media is a topical issue. This need arises in a variety of real life engineering problems, from the modelling of railway- or machinery-induced vibrations to the analysis of seismic wave propagation and soil-structure interaction problems. Due to the complexity of the involved geometries and materials behavior, modelling such situations requires sophisticated numerical methods. The Boundary Element method (BEM) is a very effective approach for dynamical problems in spatially-extended regions (idealized as unbounded), especially since the advent of fast BEMs such as the Fast Multipole Method (FMM) used in this work. The BEM is based on a boundary integral formulation which requires the discretization of the only domain boundary (i.e. a surface in 3-D) and accounts implicitly for the radiation conditions at infinity. As a main disadvantage, the BEM leads a priori to a fully-populated and (using the collocation approach) non-symmetrical coefficient matrix, which make the traditional implementation of this method prohibitive for large problems (say O(106) boundary DoFs). Applied to the BEM, the Multi-Level Fast Multipole Method (ML-FMM) strongly lowers the complexity in computational work and memory that hinder the classical formulation, making the ML-FMBEM very competitive in modelling elastic wave propagation. The elastodynamic version of the Fast Multipole BEM (FMBEM), in a form enabling piecewise-homogeneous media, has for instance been successfully used to solve seismic wave propagation problems in a previous work (thesis dissertation of S. Chaillat, ENPC, 2008). This thesis aims at extending the capabilities of the existing frequency-domain elastodynamic FMBEM in two directions. Firstly, the time-harmonic elastodynamic ML-FMBEM formulation has been extended to the case of weakly dissipative viscoelastic media. Secondly, the FMBEM and the Finite Element Method (FEM) have been coupled to take advantage of the versatility of the FEM to model complex geometries and non-linearities while the FM-BEM accounts for wave propagation in the surrounding unbounded medium. In this thesis, we consider two strategies for coupling the FMBEM and the FEM to solve three-dimensional time-harmonic wave propagation problems in unbounded domains. The main idea is to separate one or more bounded subdomains (modelled by the FEM) from the complementary semi-infinite viscoelastic propagation medium (modelled by the FMBEM) through a non-overlapping domain decomposition. Two coupling strategies have been implemented and their performances assessed and compared on several examples

The Method of Moments is a numerical technique for solving electromagnetic problems with integral equations. The method discretizes a surface in three dimensions, which reduces the dimension of the problem with one. A drawback of the method is that it yields a dense system of linear equations. This effectively prohibits the solution of large scale problems. Papers I-III describe the Fast Multipole Method. It reduces the cost of computing a dense matrix vector multiplication. This implies that large scale problems can be solved on personal computers. In Paper I the error introduced by the Fast Multipole Method is analyzed. Paper II and Paper III describe the implementation of the Fast Multipole Method. The problem of computing monostatic Radar Cross Section involves many right hand sides. Since the Fast Multipole Method computes a matrix times a vector, iterative techniques are used to solve the linear systems. It is important that the solution time for each system is as low as possible. Otherwise the total solution time becomes too large. Different techniques for reducing the work in the iterative solver are described in Paper IV-VI. Paper IV describes a block Quasi Minimal Residual method for several right hand sides and Sparse Approximate Inverse preconditioner that reduce the number of iterations significantly. In Paper V and Paper VI a method based on linear algebra called the Minimal Residual Interpolation method is described. It reduces the work in an iterative solver by accurately computing an initial guess for the iterative method. In Paper VII a hybrid method between Physical Optics and the Fast Multipole Method is described. It can handle large problems that are out of reach for the Fast Multipole Method.

Les techniques avancées pour l’approximation de rang faible des matrices sont des outils de réduction de dimension fondamentaux pour un grand nombre de domaines du calcul scientifique. Les approches hiérarchiques comme les matrices H2, en particulier la méthode multipôle rapide (FMM), bénéficient de la structure de rang faible par bloc de certaines matrices pour réduire le coût de calcul de problèmes d’interactions à n-corps en O(n) opérations au lieu de O(n2). Afin de mieux traiter des noyaux d’interaction complexes de plusieurs natures, des formulations FMM dites ”kernel-independent” ont récemment vu le jour, telles que les FMM basées sur l’interpolation polynomiale. Cependant elles deviennent très coûteuses pour les noyaux tensoriels à fortes dimensions, c’est pourquoi nous avons développé une nouvelle formulation FMM efficace basée sur l’interpolation polynomiale, appelée Uniform FMM. Cette méthode a été implémentée dans la bibliothèque parallèle ScalFMM et repose sur une grille d’interpolation régulière et la transformée de Fourier rapide (FFT). Ses performances et sa précision ont été comparées à celles de la FMM par interpolation de Chebyshev. Des simulations numériques sur des cas tests artificiels ont montré que la perte de précision induite par le schéma d’interpolation était largement compensées par le gain de performance apporté par la FFT. Dans un premier temps, nous avons étendu les FMM basées sur grille de Chebyshev et sur grille régulière au calcul des champs élastiques isotropes mis en jeu dans des simulations de Dynamique des Dislocations (DD). Dans un second temps, nous avons utilisé notre nouvelle FMM pour accélérer une factorisation SVD de rang r par projection aléatoire et ainsi permettre de générer efficacement des champs Gaussiens aléatoires sur de grandes grilles hétérogènes. Pour finir, nous avons développé un algorithme de réduction de dimension basé sur la projection aléatoire dense afin d’étudier de nouvelles façons de caractériser la biodiversité, à savoir d’un point de vue géométrique. / Advanced techniques for the low-rank approximation of matrices are crucial dimension reduction tools in many domains of modern scientific computing. Hierarchical approaches like H2-matrices, in particular the Fast Multipole Method (FMM), benefit from the block low-rank structure of certain matrices to reduce the cost of computing n-body problems to O(n) operations instead of O(n2). In order to better deal with kernels of various kinds, kernel independent FMM formulations have recently arisen such as polynomial interpolation based FMM. However, they are hardly tractable to high dimensional tensorial kernels, therefore we designed a new highly efficient interpolation based FMM, called the Uniform FMM, and implemented it in the parallel library ScalFMM. The method relies on an equispaced interpolation grid and the Fast Fourier Transform (FFT). Performance and accuracy were compared with the Chebyshev interpolation based FMM. Numerical experiments on artificial benchmarks showed that the loss of accuracy induced by the interpolation scheme was largely compensated by the FFT optimization. First of all, we extended both interpolation based FMM to the computation of the isotropic elastic fields involved in Dislocation Dynamics (DD) simulations. Second of all, we used our new FMM algorithm to accelerate a rank-r Randomized SVD and thus efficiently generate multivariate Gaussian random variables on large heterogeneous grids in O(n) operations. Finally, we designed a new efficient dimensionality reduction algorithm based on dense random projection in order to investigate new ways of characterizing the biodiversity, namely from a geometric point of view.