Fine-Grained Multithreading for the Multifrontal QR Factorization of Sparse Matrices

International audienceThe advent of multicore processors represents a disruptive event in the history of computer science as conventional parallel programming paradigms are proving incapable of fully exploiting their potential for concurrent computations. The need for different or new programming models clearly arises from recent studies which identify fine-granularity and dynamic execution as the keys to achieving high efficiency on multicore systems. This work presents an approach to the parallelization of the multifrontal method for the $QR$ factorization of sparse matrices specifically designed for multicore based systems. High efficiency is achieved through a fine-grained partitioning of data and a dynamic scheduling of computational tasks relying on a dataflow parallel programming model. Experimental results show that an implementation of the proposed approach achieves higher performance and better scalability than existing equivalent software

A Class of Parallel Tiled Linear Algebra Algorithms for Multicore Architectures

As multicore systems continue to gain ground in the High Performance Computing world, linear algebra algorithms have to be reformulated or new algorithms have to be developed in order to take advantage of the architectural features on these new processors. Fine grain parallelism becomes a major requirement and introduces the necessity of loose synchronization in the parallel execution of an operation. This paper presents an algorithm for the Cholesky, LU and QR factorization where the operations can be represented as a sequence of small tasks that operate on square blocks of data. These tasks can be dynamically scheduled for execution based on the dependencies among them and on the availability of computational resources. This may result in an out of order execution of the tasks which will completely hide the presence of intrinsically sequential tasks in the factorization. Performance comparisons are presented with the LAPACK algorithms where parallelism can only be exploited at the level of the BLAS operations and vendor implementations

Implementing multifrontal sparse solvers for multicore architectures with Sequential Task Flow runtime systems

International audienceTo face the advent of multicore processors and the ever increasing complexity of hardware architectures, programming models based on DAG parallelism regained popularity in the high performance, scientific computing community. Modern runtime systems offer a programming interface that complies with this paradigm and powerful engines for scheduling the tasks into which the application is decomposed. These tools have already proved their effectiveness on a number of dense linear algebra applications. This paper evaluates the usability and effectiveness of runtime systems based on the Sequential Task Flow model for complex applications , namely, sparse matrix multifrontal factorizations which feature extremely irregular workloads, with tasks of different granularities and characteristics and with a variable memory consumption. Most importantly, it shows how this parallel programming model eases the development of complex features that benefit the performance of sparse, direct solvers as well as their memory consumption. We illustrate our discussion with the multifrontal QR factorization running on top of the StarPU runtime system. ACM Reference Format: Emmanuel Agullo, Alfredo Buttari, Abdou Guermouche and Florent Lopez, 2014. Implementing multifrontal sparse solvers for multicore architectures with Sequential Task Flow runtime system

Multifrontal QR Factorization for Multicore Architectures over Runtime Systems

International audienceTo face the advent of multicore processors and the ever increasing complexity of hardware architectures, programming models based on DAG parallelism regained popularity in the high performance, scientific computing community. Modern runtime systems offer a programming interface that complies with this paradigm and powerful engines for scheduling the tasks into which the application is decomposed. These tools have already proved their effectiveness on a number of dense linear algebra applications. This paper evaluates the usability of runtime systems for complex applications, namely, sparse matrix multifrontal factorizations which constitute extremely irregular workloads, with tasks of different granularities and characteristics and with a variable memory consumption. Experimental results on real-life matrices show that it is possible to achieve the same efficiency as with an ad hoc scheduler which relies on the knowledge of the algorithm. A detailed analysis shows the performance behavior of the resulting code and possible ways of improving the effectiveness of runtime systems

Exploiting a Parametrized Task Graph model for the parallelization of a sparse direct multifrontal solver

International audienceThe advent of multicore processors requires to reconsider the design of high performance computing libraries to embrace portable and effective techniques of parallel software engineering. One of the most promising approaches consists in abstracting an application as a directed acyclic graph (DAG) of tasks. While this approach has been popularized for shared memory environments by the OpenMP 4.0 standard where dependencies between tasks are automatically inferred, we investigate an alternative approach, capable of describing the DAG of task in a distributed setting, where task dependencies are explicitly encoded. So far this approach has been mostly used in the case of algorithms with a regular data access pattern and we show in this study that it can be efficiently applied to a higly irregular numerical algorithm such as a sparse multifrontal QR method. We present the resulting implementation and discuss the potential and limits of this approach in terms of productivity and effectiveness in comparison with more common parallelization techniques. Although at an early stage of development, preliminary results show the potential of the parallel programming model that we investigate in this work

Improving multifrontal methods by means of block low-rank representations

Submitted for publication to SIAMMatrices coming from elliptic Partial Differential Equations (PDEs) have been shown to have a low-rank property: well defined off-diagonal blocks of their Schur complements can be approximated by low-rank products. Given a suitable ordering of the matrix which gives to the blocks a geometrical meaning, such approximations can be computed using an SVD or a rank-revealing QR factorization. The resulting representation offers a substantial reduction of the memory requirement and gives efficient ways to perform many of the basic dense algebra operations. Several strategies have been proposed to exploit this property. We propose a low-rank format called Block Low-Rank (BLR), and explain how it can be used to reduce the memory footprint and the complexity of direct solvers for sparse matrices based on the multifrontal method. We present experimental results that show how the BLR format delivers gains that are comparable to those obtained with hierarchical formats such as Hierarchical matrices (H matrices) and Hierarchically Semi-Separable (HSS matrices) but provides much greater flexibility and ease of use which are essential in the context of a general purpose, algebraic solver

Grouping variables in Frontal Matrices to improve Low-Rank Approximations in a Multifrontal Solver

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