Accelerating Linear System Solutions Using Randomization Techniques

Abstract

We show in this paper how linear algebra calculations can be enhanced by statistical techniques in the case of a square linear system Ax = b. We study a random transformation of A that enables us to avoid pivoting and then to reduce the amount of communication. Numerical experiments show that this randomization can be performed at a very aordable computational price while providing us with a satisfying accuracy when compared to partial pivoting. This random transformation called Partial Random Buttery Transformation (PRBT) is optimized in terms of data storage and ops count. We propose a solver where PRBT and the LU factorization with no pivoting take advantage of the latest generation of hybrid multicore/GPU machines and we compare its Gop/s performance with a solver implemented in a current parallel library.

Using Random Butterfly Transformations to Avoid Pivoting in Sparse Direct Methods

“…In this section we recall the main concepts and definitions related to RBT where the randomization of the matrix is based on a technique initially described in [10] and revisited in [2] for general dense systems. The procedure to solve Ax = b, where A is a general matrix, using a random transformation and the LU factorization is:…”

“…Baboulin et al studied extensively the use of RBT for dense matrices and showed that in practice, d = 1 or 2 is enough; in most cases a few steps of iterative refinement can recover the digits that have been lost. They also showed that random butterfly matrices are cheap to store and to apply (O(nd) and O(dn 2 ) respectively) and they proposed implementations on hybrid multicore/GPU systems for the unsymmetric [2] case. For the symmetric case, they proposed a tiled algorithm for multicore architectures [3] and more recently a distributed solver [4] combined with a runtime system [5].…”

“…In this approach, the Random Butterfly Transformation (RBT) is used to transform the original system into an "easier" one such that, with probability one, the LU factorization of the transformed matrix can be performed without pivoting. This technique was successfully applied and implemented into the dense libraries for LU and LDL T factorizations [2,6]. In this work, we investigate the potential of the RBT method for sparse cases.…”

Reducing the Amount of Pivoting in Symmetric Indefinite Systems

“…This technique was initially proposed in [16] in the context of general linear systems where the randomization is referred to as Random Butterfly Transformation (RBT). Then a modified approach has been described in [5] for the LU factorization of general dense matrices and we propose here to adapt this technique specifically to symmetric indefinite systems. It consists of a multiplicative preconditioning U T AU where the matrix U is chosen among a particular class of random matrices called recursive butterfly matrices.…”

“…These techniques were initially proposed in [16] and modified approaches were studied in [4,5] for the LU factorization. In this context, they are applied to the case of symmetric indefinite systems.…”