International audience

The standard LU factorization-based solution process for linear systems can be enhanced in speed or accuracy by employing mixed-precision iterative refinement. Most recent work has focused on dense systems. We investigate the potential of mixed-precision iterative refinement to enhance methods for sparse systems based on approximate sparse factorizations. In doing so, we first develop a new error analysis for LU- and GMRES-based iterative refinement under a general model of LU factorization that accounts for the approximation methods typically used by modern sparse solvers, such as low-rank approximations or relaxed pivoting strategies. We then provide a detailed performance analysis of both the execution time and memory consumption of different algorithms, based on a selected set of iterative refinement variants and approximate sparse factorizations. Our performance study uses the multifrontal solver MUMPS, which can exploit block low-rank factorization and static pivoting. We evaluate the performance of the algorithms on large, sparse problems coming from a variety of real-life and industrial applications showing that mixed-precision iterative refinement combined with approximate sparse factorization can lead to considerable reductions of both the time and memory consumption.

Matrices coming from elliptic Partial Differential Equations have been shown to have a low-rank property that can be efficiently exploited in multifrontal solvers to provide a substantial reduction of their complexity. Among the possible low-rank formats, the Block Low-Rank format (BLR) is easy to use in a general purpose multifrontal solver and its potential compared to standard (full-rank) solvers has been demonstrated. Recently, new variants have been introduced and it was proved that they can further reduce the complexity but their performance has never been analyzed. In this article, we present a multithreaded BLR factorization and analyze its efficiency and scalability in shared-memory multicore environments. We identify the challenges posed by the use of BLR approximations in multifrontal solvers and put forward several algorithmic variants of the BLR factorization that overcome these challenges by improving its efficiency and scalability. We illustrate the performance analysis of the BLR multifrontal factorization with numerical experiments on a large set of problems coming from a variety of real-life applications.

We are interested in the memory usage of multifrontal methods. Starting from the algorithms introduced by Liu, we propose new schedules to allocate and process tasks that improve memory usage. This generalizes two existing factorization and memory-allocation schedules by allowing a more flexible task allocation together with a specific tree traversal. We present optimal algorithms for this new class of schedules, and demonstrate experimentally their benefit for some real-world matrices from sparse matrix collections where either the active memory or the total memory is minimized.

Summary form only given. The memory usage of sparse direct solvers can be the bottleneck to solve large-scale problems. We describe dynamic scheduling strategies that aim at reducing the memory usage of a parallel direct solver. Combined to static modifications of the tasks dependency graph, experiments show that such techniques have a good potential to improve the memory usage of a parallel multifrontal solver, MUMPS.

This paper provides a comprehensive study and comparison of two state-of-the-art direct solvers for large sparse sets of linear equations on large-scale distributed-memory computers. One is a multifrontal solver called MUMPS, the other is a supernodal solver called superLU. We describe the main algorithmic features of the two solvers and compare their performance characteristics with respect to uniprocessor speed, interprocessor communication, and memory requirements. For both solvers, preorderings for numerical stability and sparsity play an important role in achieving high parallel efficiency. We analyse the results with various ordering algorithms. Our performance analysis is based on data obtained from runs on a 512-processor Cray T3E using a set of matrices from real applications. We also use regular 3D grid problems to study the scalability of the two solvers.