Schwarz Methods: To Symmetrize or Not to Symmetrize

TL;DR

This paper develops a preconditioning theory for Schwarz algorithms, analyzing when symmetrization is beneficial, and demonstrates that non-symmetric preconditioners can outperform symmetric ones in certain iterative methods.

Contribution

It introduces sufficient conditions for Schwarz preconditioners to be self-adjoint positive definite and explores the effects of symmetrization versus non-symmetrization in iterative methods.

Findings

Non-symmetric preconditioners can be more effective with Bi-CGstab.

Symmetrizing may degrade performance for certain linear methods.

Numerical examples support the theoretical analysis and conjectures.

Abstract

A preconditioning theory is presented which establishes sufficient conditions for multiplicative and additive Schwarz algorithms to yield self-adjoint positive definite preconditioners. It allows for the analysis and use of non-variational and non-convergent linear methods as preconditioners for conjugate gradient methods, and it is applied to domain decomposition and multigrid. It is illustrated why symmetrizing may be a bad idea for linear methods. It is conjectured that enforcing minimal symmetry achieves the best results when combined with conjugate gradient acceleration. Also, it is shown that absence of symmetry in the linear preconditioner is advantageous when the linear method is accelerated by using the Bi-CGstab method. Numerical examples are presented for two test problems which illustrate the theory and conjectures.