Composing Scalable Nonlinear Algebraic Solvers

TL;DR

This paper introduces a framework for composing nonlinear algebraic solvers, demonstrating that such compositions can significantly outperform traditional Newton-Krylov methods in solving nonlinear PDEs.

Contribution

It presents the concept of nonlinear solver composition, a software framework for exploring combinations, and evidence of substantial performance improvements.

Findings

Composed nonlinear solvers outperform standard methods.

Software framework enables flexible exploration of solver combinations.

Performance gains are significant in nonlinear PDEs.

Abstract

Most efficient linear solvers use composable algorithmic components, with the most common model being the combination of a Krylov accelerator and one or more preconditioners. A similar set of concepts may be used for nonlinear algebraic systems, where nonlinear composition of different nonlinear solvers may significantly improve the time to solution. We describe the basic concepts of nonlinear composition and preconditioning and present a number of solvers applicable to nonlinear partial differential equations. We have developed a software framework in order to easily explore the possible combinations of solvers. We show that the performance gains from using composed solvers can be substantial compared with gains from standard Newton-Krylov methods.