An introduction to ML(n)BiCGStab

TL;DR

ML(n)BiCGStab is a Krylov subspace method bridging BiCGStab and GMRES, offering improved stability and convergence for large sparse non-symmetric systems, with demonstrated computational efficiency and lower storage needs.

Contribution

The paper introduces ML(n)BiCGStab, a new Krylov method that unifies and extends BiCGStab and GMRES, including a novel algorithm with A-transpose and practical implementation details.

Findings

ML(n)BiCGStab reduces computational time by over 60% compared to BiCGStab.

It offers better stability and convergence for ill-conditioned problems.

Requires only O(nN) storage, avoiding restart strategies needed in GMRES.

Abstract

ML(n)BiCGStab is a Krylov subspace method for the solution of large, sparse and non-symmetric linear systems. In theory, it is a method that lies between the well-known BiCGStab and GMRES/FOM. In fact, when n = 1, ML(1)BiCGStab is BiCGStab and when n = N, ML(N)BiCGStab is GMRES/FOM where N is the size of the linear system. Therefore, ML(n)BiCGStab is a bridge that connects the Lanczos-based BiCGStab and the Arnoldi-based GMRES/FOM. In computation, ML(n)BiCGStab can be much more stable and converge much faster than BiCGStab when a problem with ill-condition is solved. We have tested ML(n)BiCGStab on the standard oil reservoir simulation test data called SPE9 and found that ML(n)BiCGStab reduced the total computational time by more than 60% when compared to BiCGStab. Tests made on the data from Matrix Market also support the superiority of ML(n)BiCGStab over BiCGStab. Because of the O(N^2) storage requirement in the full GMRES, one has to adopt a restart strategy to get the storage under control when GMRES is implemented. In comparison, ML(n)BiCGStab is a method with only O(nN) storage requirement and therefore it does not need a restart strategy. In this paper, we introduce ML(n)BiCGStab (in particular, a new algorithm involving A-transpose), its relations to some existing methods and its implementations.