ML(n)BiCGStab: Reformulation, Analysis and Implementation

TL;DR

This paper systematically reformulates the ML(n)BiCGStab algorithm, introduces multiple variants with different residual definitions, and analyzes their theoretical properties, implementation issues, and connections to other Krylov subspace methods.

Contribution

It provides a new systematic derivation of ML(n)BiCGStab, introduces alternative algorithms with reduced storage, and analyzes their theoretical and practical properties.

Findings

Multiple residual definitions lead to different ML(n)BiCGStab algorithms.

The second algorithm reduces storage requirements.

Theoretical analysis connects ML(n)BiCGStab with Lanczos and Arnoldi methods.

Abstract

With the aid of index functions, we re-derive the ML(n)BiCGStab algorithm in a paper by Yeung and Chan in 1999 in a more systematic way. It turns out that there are n ways to define the ML(n)BiCGStab residual vector. Each definition will lead to a different ML(n)BiCGStab algorithm. We demonstrate this by presenting a second algorithm which requires less storage. In theory, this second algorithm serves as a bridge that connects the Lanczos-based BiCGStab and the Arnoldi-based FOM while ML(n)BiCG a bridge connecting BiCG and FOM. We also analyze the breakdown situations from the probabilistic point of view and summarize some useful properties of ML(n)BiCGStab. Implementation issues are also addressed.