Nonlinear preconditioning is a globalization technique for Newton’s method applied to systems of equations with unbalanced nonlinearities, in which nonlinear residual norm reduction stagnates due to slowly evolving subsets of the degrees of freedom. Even though the Newton corrections may effectively be sparse, a standard Newton method still requires large ill-conditioned linear systems resulting from global linearizations of the nonlinear residual to be solved at each step. Nonlinear preconditioners may enable faster global convergence by shifting work to where it is most strategic, on subsets of the original system. They require additional computation per outer iteration while aiming for many fewer outer iterations and correspondingly fewer global synchronizations. In this work, we improve upon previous nonlinear preconditioning implementations by introducing parameters that allow turning off nonlinear preconditioning during outer Newton iterations where it is not needed. Numerical experiments show that the adaptive nonlinear preconditioning algorithm has performance similar to monolithically applied nonlinear preconditioning, preserving robustness for some challenging problems representative of several PDE-based applications while saving work on nonlinear subproblems.
Bibliographical noteKAUST Repository Item: Exported on 2021-05-01
Acknowledgements: The first and third authors were supported by the SNF – Swiss National Science Foundation through the projects “Parallel multilevel solvers for coupled interface problems” and “Large-scale simulation of pneumatic and hydraulic fracture with a phase-field approach,” and the work of these authors was supported by SCCER-FURIES, Swiss Competence Center for Energy Research–Future Swiss Electrical Infrastructure. The work of the second author was supported by the Extreme Computing Research Center at KAUST. The authors are grateful for the use of a Cray XC40 (“Shah-een”) operated by the Supercomputing Laboratory of the King Abdullah University of Science and Technology.
ASJC Scopus subject areas
- Computational Mathematics
- Applied Mathematics