The following study examines whether the initialisation (multipole expansions at the finest level) and evaluation of the numerical method Fast Multipole Method (FMM) can benefit from implementing SSE instructions. The implementation of SSE-instructions have been studied and compared to the serial case. Moreover, studied parts of the algorithm include arithmetics on complex numbers, and the usage of applying SSE instructions to complex numbers of double precision. In conclusion, the initialisation has not experienced any improvement in terms of throughput by appliying SSE instructions. However, the evaluation reached almost the double speed-up when SSE instructions were applied. The difference in results are most likely due to the structure of the both algorithms. The initialisation is simple, but the evaluation which involves more operations can benefit from SSE instructions. Furthermore, a scheme is proposed for how SSE instructions can be applied to data sets which are not divisable by the unroll factor and to data sets of varying size.

京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第20513号 / 情博第641号 / 新制||情||111(附属図書館) / 京都大学大学院情報学研究科複雑系科学専攻 / (主査)教授 西村 直志, 教授 磯 祐介, 准教授 吉川 仁 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

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[pt] Este trabalho visa à implementação de um programa de elementos de contorno para problemas com milhões de graus de liberdade. Isto é obtido com a implementação do Método Fast Multipole (FMM), que pode reduzir o número de operações, para a solução de um problema com N graus de liberdade, de O(N(2)) para O(NlogN) ou O(N). O uso de memória também é reduzido, por não haver o armazenamento de matrizes de grandes dimensões como no caso de outros métodos numéricos. A implementação proposta é baseada em um desenvolvimento consistente do convencional, Método de colocação dos elementos de contorno (BEM) – com conceitos provenientes do Hibrido BEM – para problemas de potencial e elasticidade de larga escala em 2D e 3D. A formulação é especialmente vantajosa para problemas de topologia complicada ou que requerem soluções fundamentais complicadas. A implementação apresentada, usa um esquema para expansões de soluções fundamentais genéricas em torno de níveis hierárquicos de polos campo e fonte, tornando o FMM diretamente aplicável para diferentes soluções fundamentais. A árvore hierárquica dos polos é construída a partir de um conceito topológico de superelementos dentro de superelementos. A formulação é inicialmente acessada e validada em termos de um problema de potencial 2D. Como resolvedores iterativos não são necessários neste estágio inicial de simulação numérica, podese acessar a eficiência relativa à implementação do FMM. / [en] This work aims to present an implementation of a boundary element solver for problems with millions of degrees of freedom. This is achieved through a Fast Multipole Method (FMM) implementation, which can lower the number of operations for solving a problem, with N degrees of freedom, from O(N(2)) to O(NlogN) or O(N). The memory usage is also very small, as there is no need to store large matrixes such as required by other numerical methods. The proposed implementations are based on a consistent development of the conventional, collocation boundary element method (BEM) - with concepts taken from the variationally-based hybrid BEM - for large-scale 2D and 3D problems of potential and elasticity. The formulation is especially advantageous for problems of complicated topology or requiring complicated fundamental solutions. The FMM implementation presented in this work uses a scheme for expansions of a generic fundamental solution about hierarchical levels of source and field poles. This makes the FMM directly applicable to different kinds of fundamental solutions. The hierarchical tree of poles is built upon a topological concept of superelements inside superelements. The formulation is initially assessed and validated in terms of a simple 2D potential problem. Since iterative solvers are not required in this first step of numerical simulations, an isolated efficiency assessment of the implemented fast multipole technique is possible.

Face à la complexité grandissante des convertisseurs statiques présents dans tout système électrique, les ingénieurs de conception ont besoin d'outils de modélisation électromagnétique de plus en plus performants, notamment en ce qui concerne la Compatibilité ÉlectroMagnétique (CEM). L'objectif de ce travail est de prendre en compte, sous la forme de capacités parasites, les couplages électriques en haute fréquence dans la modélisation CEM de dispositifs d'électronique de puissance. Plusieurs formulations intégrales basées sur la Méthode des Moments, ainsi que l'Adaptive Multi-Level Fast Multipole Method ont été développées et validées pour l'extraction de ces capacités équivalentes. Cette dernière méthode, qui permet d'accélérer les temps de calcul tout en limitant la place mémoire nécessaire (pas de stockage de matrice pleine), a été adaptée au problème pour garantir une meilleure précision des résultats en fonction du maillage. Un prototype de cet algorithme de calcul a été intégré dans le logiciel InCa3D, basée sur la méthode PEEC, permettant ainsi de construire un schéma électrique équivalent à constantes localisées où les effets capacitifs sont couplés au modèle résistif et inductif de la structure. Plusieurs cas tests, issus de la littérature ou d'applications industrielles, ont été simulés par le biais de ces schémas équivalents, soit dans un solveur circuit soit dans InCa3D, afin d'évaluer leurs performances CEM conduites et rayonnées. Enfin, les comparaisons réalisées avec des mesures ont donné de bons résultats et valident ainsi l'approche proposée. Une telle stratégie peut aisément faire partie de toute modélisation de type système, car elle permet de traiter des dispositifs de complexité industrielle sur une large bande de fréquences avec un modèle léger.

[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.