A framework for efficient execution of matrix computations

Matrix computations lie at the heart of most scientific computational tasks. The solution of linear systems of equations is a very frequent operation in many fields in science, engineering, surveying, physics and others. Other matrix operations occur frequently in many other fields such as pattern recognition and classification, or multimedia applications. Therefore, it is important to perform matrix operations efficiently. The work in this thesis focuses on the efficient execution on commodity processors of matrix operations which arise frequently in different fields.We study some important operations which appear in the solution of real world problems: some sparse and dense linear algebra codes and a classification algorithm. In particular, we focus our attention on the efficient execution of the following operations: sparse Cholesky factorization; dense matrix multiplication; dense Cholesky factorization; and Nearest Neighbor Classification.A lot of research has been conducted on the efficient parallelization of numerical algorithms. However, the efficiency of a parallel algorithm depends ultimately on the performance obtained from the computations performed on each node. The work presented in this thesis focuses on the sequential execution on a single processor.There exists a number of data structures for sparse computations which can be used in order to avoid the storage of and computation on zero elements. We work with a hierarchical data structure known as hypermatrix. A matrix is subdivided recursively an arbitrary number of times. Several pointer matrices are used to store the location ofsubmatrices at each level. The last level consists of data submatrices which are dealt with as dense submatrices. When the block size of this dense submatrices is small, the number of zeros can be greatly reduced. However, the performance obtained from BLAS3 routines drops heavily. Consequently, there is a trade-off in the size of data submatrices used for a sparse Cholesky factorization with the hypermatrix scheme. Our goal is that of reducing the overhead introduced by the unnecessary operation on zeros when a hypermatrix data structure is used to produce a sparse Cholesky factorization. In this work we study several techniques for reducing such overhead in order to obtain high performance.One of our goals is the creation of codes which work efficiently on different platforms when operating on dense matrices. To obtain high performance, the resources offered by the CPU must be properly utilized. At the same time, the memory hierarchy must be exploited to tolerate increasing memory latencies. To achieve the former, we produce inner kernels which use the CPU very efficiently. To achieve the latter, we investigate nonlinear data layouts. Such data formats can contribute to the effective use of the memory system.The use of highly optimized inner kernels is of paramount importance for obtaining efficient numerical algorithms. Often, such kernels are created by hand. However, we want to create efficient inner kernels for a variety of processors using a general approach and avoiding hand-made codification in assembly language. In this work, we present an alternative way to produce efficient kernels automatically, based on a set of simple codes written in a high level language, which can be parameterized at compilation time. The advantage of our method lies in the ability to generate very efficient inner kernels by means of a good compiler. Working on regular codes for small matrices most of the compilers we used in different platforms were creating very efficient inner kernels for matrix multiplication. Using the resulting kernels we have been able to produce high performance sparse and dense linear algebra codes on a variety of platforms.In this work we also show that techniques used in linear algebra codes can be useful in other fields. We present the work we have done in the optimization of the Nearest Neighbor classification focusing on the speed of the classification process.Tuning several codes for different problems and machines can become a heavy and unbearable task. For this reason we have developed an environment for development and automatic benchmarking of codes which is presented in this thesis.As a practical result of this work, we have been able to create efficient codes for several matrix operations on a variety of platforms. Our codes are highly competitive with other state-of-art codes for some problems

A dependency-aware parallel programming model

Designing parallel codes is hard. One of the most important roadblocks to parallel programming is the presence of data dependencies. These restrict parallelism and, in general, to work them around requires complex analysis and leads to convoluted solutions that decrease the quality of the code. This thesis proposes a solution to parallel programming that incorporates data dependencies into the model. The programming model can handle that information and to dynamically find parallelism that otherwise would be hard to find. This approach improves both programmability and parallelism, and thus performance. While this problem has already been solved in OpenMP 4 at the time of this publication, this research begun before the problem was even being considered for OpenMP 3. In fact, some of the contributions of this thesis have had an influence on the approach taken in OpenMP 4. However, the contributions go beyond that and cover aspects that have not been considered yet in OpenMP 4. The approach we propose is based on function-level dependencies across disjoint blocks of contiguous memory. While finding dependencies under those constraints is simple, it is much harder to do so over strided and possibly partially overlapping sets of data. This thesis also proposes a solution to this problem. By doing so, we increase the range of applicability of the original solution and increase the span of applicability of the programming model. OpenMP4 does not currently cover this aspect. Finally, we present a solution to take advantage of the performance characteristics of Non-Uniform Memory Access architectures. Our proposal is at the programming model level and does not require changes in the code. It automatically distributes the data and does not rely on data migration nor replication. Instead, it is based exclusively on scheduling the computations. While this process is automatic, it can be tuned through minor changes in the code that do not require any change in the programming model. Throughout the thesis, we demonstrate the effectiveness of the proposal through benchmarks that are either hard to program using other paradigms or that have different solutions. In most cases, our solutions perform either on par or better than already existing solutions. This includes the implementations available in well-known high-performance parallel libraries.Dissenyar codis paral·lels es complex. Un dels principals esculls a l'hora de programar aplicacions paral·leles és la presència de dependències. Aquestes constrenyen el paral·lelisme, i en general, per evitar-les es requereix realitzar anàlisis complicades que donen lloc a solucions complexes que redueixen la qualitat del codi. Aquesta tesi proposa una solució a la programació paral·lela que incorpora al model les dependències de dades. El model de programació és capaç d'utilitzar aquesta informació per a trobar paral·lelisme que altrament seria molt difícil de detectar i d'extreure. Aquest enfoc augmenta la programabilitat i el paral·lelisme, i per tant també el rendiment. Tot i que al moment de la publicació d'aquesta tesi, el problema ja ha estat resolt a OpenMP 4, la recerca d'aquesta tesi va començar abans de que el problema s'hagués plantejat en l'àmbit d'OpenMP 3. De fet, algunes de les contribucions de la tesi han influït en la solució emprada a OpenMP 4. Tanmateix, les contribucions van més enllà i cobreixen aspectes que encara no han estat considerats a OpenMP 4. La proposta es basa en dependències a nivell de funció entre blocs de memòria continus i sense intersecció. Tot i que trobar dependències sota aquestes condicions és senzill, fer-ho sobre dades no contínues amb possibles interseccions parcials és molt més complex. Aquesta tesi també proposa una solució a aquest problema. Fent això, es millora el rang d'aplicació de la solució original i per tant el del model de programació. Aquest és un dels aspectes que encara no es contemplen a OpenMP 4. Finalment, es presenta una solució que té en compte les característiques de rendiment de les arquitectures NUMA (Accés No Uniforme a la Memòria). La proposta es planteja a nivell del model de programació i no precisa de canvis al codi ja que les dades es distribueixen automàticament. En lloc de basar-se en la migració i la replicació de les dades, es basa exclusivament en la planificació de l'execució de les computacions. Tot i que aquest procés és automàtic, es pot afinar mitjançant petits canvis en el codi que no arriben a alterar el model de programació. Al llarg d'aquesta tesi es demostra la efectivitat de les propostes a través de bancs de proves que son difícils de programar amb altres paradigmes o que tenen solucions diferents. A la majoria dels casos les nostres solucions tenen un rendiment similar o millor que les solucions preexistents, que inclouen implementacions en ben reconegues biblioteques paral·leles d'alt rendiment