Nonlinearly preconditioned semismooth Newton algorithms for nonlinear nonsmooth systems

Abstract

We aim to develop efficient and robust algorithms for nonsmooth nonlinear systems arising from complementarity problems. The semismooth Newton algorithm is popular due to its reliability and efficiency. However, it struggles with issues with imbalanced nonlinearities of the problems, leading to degraded convergence rates or failure despite help from the globalization techniques like linesearch or trust region. We introduce a right nonlinearly preconditioned semismooth Newton algorithm to address this difficulty. The critical success ingredient is that before each global Newton update, a nonlinear preconditioning step implicitly removes the so-called ‘bad components’ causing trouble via nonlinear subspace correction, inspired by Gaussian elimination but adapted nonlinearly to balance system nonlinearities. Additionally, our method integrates with a domain decomposition framework, enhancing parallelism. Numerical results on two classes of problems demonstrate significantly improved convergence over standard semismooth Newton methods.

Keywords:

• Semismooth Newton algorithm
• nonlinear preconditioning
• flow control
• complementarity problems
• parallel computing

2020 Mathematics Subject Classifications:

• 49M40
• 65K10

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author was supported in part by the National Natural Science Foundation of China (No. 12371366 and 12131002), and the Basic and Advanced Research Project of CQ CSTC (No. cstc2021jcyj-msxmX0227) and the third author was supported in part by the National Science and Technology Council, Taiwan, NSTC 110-2115-M-008-005-MY2.