Scalable and Portable LU Factorization with Partial Pivoting on Top of Runtime Systems

Task-based runtime systems have demonstrated efficiency in leveraging the capabilities of large, heterogeneous architectures. Many linear algebra algorithms and applications have been implemented on top of runtime systems to increase their performance. However, the High Performance Linpack (HPL) benchmark, used by the TOP500 to rank supercomputers, has not yet been successfully implemented using task-based runtime systems. In this paper, we explore solutions to implement efficient LU factorization with partial pivoting using the sequential task-flow programming model. We show that, due to the pivoting strategy, this algorithm generates a large number of very small tasks, which usually overload the runtime system and make it inefficient. We propose two solutions to improve the efficiency and reduce the number of tasks. First, we apply well-known blocking strategies in the context of task-based algorithms. Secondly, we explore batching techniques to reduce the number of tasks submitted to the runtime system. Moreover, in distributed architectures, partial pivoting generates many reductions on the critical path throughout the factorization which needs to be carefully handled to reach high performance. Two task-based reduction algorithms are proposed to express these operations and improve the runtime reactivity on the critical path. These proposals have been implemented in the dense linear algebra library Chameleon on top of the STARPU runtime system. Experiments conducted on our cluster with these optimizations show that our LU with partial pivoting asymptotically reaches the performance of the non-pivoting algorithm.