Este trabalho que envolve matemática aplicada e processamento paralelo: seu objetivo é avaliar uma estratégia de implementação em paralelo para algoritmos de diferenças finitas que aproximam a solução de equações diferenciais de evolução. A alternativa proposta é a substituição dos produtos matriz-vetor efetuados sequencialmente por multiplicações matriz-matriz aceleradas pelo método de Strassen em paralelo. O trabalho desenvolve testes visando verificar o ganho computacional relacionado a essa estratégia de paralelização, pois as aplicacações computacionais, que empregam a estratégia sequencial, possuem como característica o longo período de computação causado pelo grande volume de cálculo. Inclusive como alternativa, nós usamos o algoritmo em paralelo convencional para solução de algoritmos explícitos para solução de equações diferenciais parciais evolutivas no tempo. Portanto, de acordo com os resultados obtidos, nós observamos as características de cada estratégia em paralelo, tendo como principal objetivo diminuir o esforço computacional despendido. / This work involves parallel processing and applied mathematics: Our goal is to evaluate a strategy for implementing parallel algorithms for finite diference approach,it is the solution of diferential equations of evolution. The alternative proposed is the replacement of the matrix-vector products performed sequentially by matrix-matrix multiplication method accelerated by Strassen in parallel. The work develops tests in order to verify the speedup related to the strategy of parallelization because sequential application have characterized for long periods of computation, this is caused by the large amount of calculation. Even alternatively, we use the algorithm in parallel to conventional explicit solution algorithms for solving partial diferential equations. Therefore, according to the results, we observe the characteristics of each strategy in parallel with the main purpose of reducing the computational effort expended.

Este trabalho que envolve matemática aplicada e processamento paralelo: seu objetivo é avaliar uma estratégia de implementação em paralelo para algoritmos de diferenças finitas que aproximam a solução de equações diferenciais de evolução. A alternativa proposta é a substituição dos produtos matriz-vetor efetuados sequencialmente por multiplicações matriz-matriz aceleradas pelo método de Strassen em paralelo. O trabalho desenvolve testes visando verificar o ganho computacional relacionado a essa estratégia de paralelização, pois as aplicacações computacionais, que empregam a estratégia sequencial, possuem como característica o longo período de computação causado pelo grande volume de cálculo. Inclusive como alternativa, nós usamos o algoritmo em paralelo convencional para solução de algoritmos explícitos para solução de equações diferenciais parciais evolutivas no tempo. Portanto, de acordo com os resultados obtidos, nós observamos as características de cada estratégia em paralelo, tendo como principal objetivo diminuir o esforço computacional despendido. / This work involves parallel processing and applied mathematics: Our goal is to evaluate a strategy for implementing parallel algorithms for finite diference approach,it is the solution of diferential equations of evolution. The alternative proposed is the replacement of the matrix-vector products performed sequentially by matrix-matrix multiplication method accelerated by Strassen in parallel. The work develops tests in order to verify the speedup related to the strategy of parallelization because sequential application have characterized for long periods of computation, this is caused by the large amount of calculation. Even alternatively, we use the algorithm in parallel to conventional explicit solution algorithms for solving partial diferential equations. Therefore, according to the results, we observe the characteristics of each strategy in parallel with the main purpose of reducing the computational effort expended.

1968 var Strassens algoritm en av de stora genombrotten inom matrisanalyser. I denna rapport kommer teorin av Volker Strassens algoritm för matrismultiplikationer tillsammans med teorier om precisioner att presenteras. Även fördelar med att använda denna algoritm jämfört med naiva matrismultiplikation och dess implikationer, samt hur den presterar jämfört med den naiva algoritmen kommer att presenteras. Strassens algoritm kommer också att bli bedömd på hur dess resultat skiljer sig för olika precisioner när matriserna blir större, samt hur dess teoretiska komplexitet skiljer sig gentemot den erhållna komplexiteten. Studier hittade att Strassens algoritm överträffade den naiva algoritmen för matriser av storlek 1024×1024 och större. Den erhållna komplexiteten var lite större än Volker Strassens teoretiska. Den optimala precisionen i detta fall var dubbelprecisionen, Float64. Sättet algoritmen implementeras på i koden påverkar dess prestanda. Ett flertal olika faktorer behövs ha i åtanke för att förbättra Strassens algoritm: optimera dess avbrottsvillkor, sättet som matriserna paddas för att de ska vara mer användbara för rekursiv tillämpning och hur de implementeras t.ex. parallella beräkningar. Även om det kunde bevisas att Strassen algoritm överträffade den naiva efter en viss matrisstorlek så är den inte den mest effektiva; t.ex visades detta med Strassen-Winograd. Man behöver vara uppmärksam på hur undermatriserna allokeras, för att inte ta upp onödigt minne. För fördjupning kan man läsa på om cache-oblivious och cache-aware algoritmer. / Strassen’s algorithm was one of the breakthroughs in matrix analysis in 1968. In this report the thesis of Volker Strassen’s algorithm for matrix multipli- cations along with theories about precisions will be shown. The benefits of using this algorithm compared to naive matrix multiplication and its implica- tions, how its performance compare to the naive algorithm, will be displayed. Strassen’s algorithm will also be assessed on how the output differ when the matrix sizes grow larger, as well as how the theoretical complexity of the al- gorithm differs from the achieved complexity. The studies found that Strassen’s algorithm outperformed the naive matrix multiplication at matrix sizes 1024 1024 and above. The achieved complex- ity was a little higher compared to Volker Strassen’s theoretical. The optimal precision for this case were the double precision, Float64. How the algorithm is implemented in code matters for its performance. A number of techniques need to be considered in order to improve Strassen’s algorithm, optimizing its termination criterion, the manner by which it is padded in order to make it more usable for recursive application and the way it is implemented e.g. parallel computing. Even tough it could be proved that Strassen’s algorithm outperformed the Naive after reaching a certain matrix size, it is still not the most efficient one; e.g. as shown with Strassen-Winograd. One need to be careful of how the sub-matrices are being allocated, to not use unnecessary memory. For further reading one can study cache-oblivious and cache-aware algorithms.

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Nous nous intéressons dans cette thèse à l'estimation non paramétrique de la densité à partir d'un échantillon aléatoire. Nous établissons des propriétés limites d'estimateurs de densité en les déduisant de lois limites fonctionnelles pour le processus empirique local, qui sont démontrées dans un contexte général. L'exposé de thèse, comprenant deux parties, est construit de la manière suivante. La première partie porte sur des lois limites fonctionnelles locales. Elles sont établies pour trois ensembles de suites de fonctions aléatoires, construites à partir: du processus empirique uniforme, du processus empirique de quantiles uniforme et du processus empirique de Kaplan-Meier. Ces lois sont uniformes relativement à la taille des incréments de ces processus empiriques locaux et décrivent le comportement asymptotique de la distance de Hausdorff entre chacun de ces trois ensembles et un ensemble de type Strassen. La deuxième partie porte sur l'estimation non paramétrique de la densité. Nous présentons plusieurs applications des lois limites fonctionnelles locales établies précédemment. Ces résultats comportent, d'une part, la description de lois limites pour des estimateurs non paramétriques de la densité, comprenant les estimateurs à noyau et les estimateurs de la densité par la méthode des plus proches voisins, et d'autre part, des lois limites pour les estimateurs à noyau de la densité des temps de survie et du taux de panne dans un modèle de censure à droite. Ces lois limites ont la particularité d'être établies, dans le cadre de la convergence en probabilité, uniformément relativement aux paramètres de lissage des estimateurs considérés.

This dissertation reviews the theory of fast matrix multiplication from a multilinear-algebraic point of view, as well as recent fast matrix multiplication algorithms based on discrete Fourier transforms over nite groups. To this end, the algebraic approach is described in terms of group algebras over groups satisfying the triple product Property, and the construction of such groups via uniquely solvable puzzles. The higher order singular value decomposition is an important decomposition of tensors that retains some of the properties of the singular value decomposition of matrices. However, we have proven a novel negative result which demonstrates that the higher order singular value decomposition yields a matrix multiplication algorithm that is no better than the standard algorithm. / Mathematical Sciences / M. Sc. (Applied Mathematics)

This dissertation reviews the theory of fast matrix multiplication from a multilinear-algebraic point of view, as well as recent fast matrix multiplication algorithms based on discrete Fourier transforms over nite groups. To this end, the algebraic approach is described in terms of group algebras over groups satisfying the triple product Property, and the construction of such groups via uniquely solvable puzzles. The higher order singular value decomposition is an important decomposition of tensors that retains some of the properties of the singular value decomposition of matrices. However, we have proven a novel negative result which demonstrates that the higher order singular value decomposition yields a matrix multiplication algorithm that is no better than the standard algorithm. / Mathematical Sciences / M. Sc. (Applied Mathematics)