Randomized Sparse Direct Solvers

Abstract

We propose randomized direct solvers for large sparse linear systems, which integrate randomization into rank structured multifrontal methods. The use of randomization highly simplifies various essential steps in structured solutions, where fast operations on skinny matrix-vector products replace traditional complex ones on dense or structured matrices. The new methods thus significantly enhance the flexibility and efficiency of using structured methods in sparse solutions. We also consider a variety of techniques, such as some graph methods, the inclusion of additional structures, the concept of reduced matrices, information reuse, and adaptive schemes. The methods are applicable to various sparse matrices with certain rank structures. Particularly, for discretized matrices whose factorizations yield dense fill-in with some off-diagonal rank patterns, the factorizations cost about O(n) flops in two dimensions (2D), and about O(n) to O(n 4/3) flops in three dimensions (3D). The solution costs and memory sizes are nearly O(n) in both 2D and 3D. These counts are obtained based on two optimization strategies and a sparse rank relaxation idea. The methods are especially useful for approximate solutions and preconditioning. Numerical tests on both discretized PDEs and more general problems are used to demonstrate the efficiency and accuracy. The ideas here also have the potential to be generalized to matrix-free sparse direct solvers based on matrix-vector multiplications in future developments.

A Distributed-Memory Randomized Structured Multifrontal Method for Sparse Direct Solutions

“…The class of structured multifrontal methods [30,31,33] is based on the idea that, for certain matrices A, the local dense frontal and update matrices are rank structured. For some cases, F i and U i may be approximated by rank structured forms such as HSS forms.…”

“…A simplified version in [28] uses dense U i instead. To enhance the efficiency and flexibility, randomized HSS construction [24,35] is used in [31]. The basic idea is to compress the off-diagonal blocks of F i via randomized sampling [21].…”

“…In the RSMF method [31], the product (3.5) and selected entries of F i are used to construct an HSS approximation to F i . A partial ULV factorization [6,34] of the HSS form is then computed.…”

“…Unlike other multifrontal methods, here the update matrix U i does not directly participate in the extend-add operation. Instead, its products with random vectors are used in a skinny extend-add [31] that only involves vectors and their row permutation and addition. This significantly enhances the flexibility and efficiency of extend-add.…”

“…This significantly enhances the flexibility and efficiency of extend-add. We sketch the main framework of the method, which slightly generalizes the method in [31] to nonsymmetric matrices. A pictorial illustration is given in Figure 2.…”