Using Random Butterfly Transformations to Avoid Pivoting in Sparse Direct Methods

Abstract

Abstract. We consider the solution of sparse linear systems using direct methods via LU factorization. Unless the matrix is positive definite, numerical pivoting is usually needed to ensure stability, which is costly to implement especially in the sparse case. The Random Butterfly Transformations (RBT) technique provides an alternative to pivoting and is easily parallelizable. The RBT transforms the original matrix into another one that can be factorized without pivoting with probability one. This approach has been successful for dense matrices; in this work, we investigate the sparse case. In particular, we address the issue of fill-in in the transformed system.

Using Random Butterfly Transformations in Parallel Schur Complement-Based Preconditioning

“…It has also been applied recently to a sparse direct solver in a preliminary paper [21]. The procedure to solve Ax = b, where A is a general matrix, using a random transformation and the LU factorization is summarized in Algorithm 1.…”

Randomized numerical linear algebra: Foundations and algorithms

“…Parker also observed that structured random matrices (such as the randomized trigonometric transforms from Section 9.3) allow us to perform the preconditioning step at lower cost than the subsequent Gaussian elimination procedure. Parker (1995) inspired many subsequent papers, including (Baboulin, Li and Rouet 2014, Trogdon 2017, Demmel et al 2012, Baboulin, Dongarra, Rémy, Tomov and Yamazaki 2017 and (Pan and Zhao 2017). Another related direction concerns the smoothed analysis of Gaussian elimination undertaken in (Sankar, Spielman and Teng 2006).…”

An improved analysis and unified perspective on deterministic and randomized low rank matrix approximations

“…However, this sheds light on the strategy of using conditioners to avoid pivoting during Gaussian Elimination. This has been popularized in work such as [1]. The theoretical support of such work has been lacking.…”