Nonlinear Schwarz Preconditioned Quasi-Newton Methods

Nonlinearity is prevalent in many physics and engineering applications. Newton’s method is widely used to solve nonlinear problems, but it can suffer from slow convergence or divergence when dealing with complex nonlinear behavior. Nonlinear preconditioning has emerged as a promising approach for solving such challenging problems. This research aims to focus on exploring the recent nonlinear preconditioning strategies within the framework of quasi-Newton methods. While some preliminary investigations have been conducted in this direction, there is still significant scope for improvement and exploration. The primary objective of this project is to study the impact of different coarse spaces and trust region-based globalization strategies on the convergence behavior of nonlinearly preconditioned quasi-Newton methods. By developing better coarse spaces and globalization strategies, we aim to improve the efficiency and performance of the overall iterative scheme for solving complex nonlinear problems. This work aims to contribute to the advancement of contemporary nonlinear preconditioning techniques by exploring novel approaches to enhance the convergence of quasi-Newton methods. This research will facilitate a comprehensive exploration of the potential of nonlinear preconditioning in the context of quasi-Newton methods, thereby enabling the development of more robust and efficient solution approaches for nonlinear problems.