Adaptive preconditioners for nonlinear systems of equations

The use of preconditioned Krylov methods is in many applications mandatory for computing efficiently the solution of large sparse nonlinear systems of equations. However, the available preconditioners are often sub-optimal, due to the changing nature of the linearized operator. In this work we introduce and analyse an adaptive preconditioning technique based on the Krylov subspace information generated at previous steps in the nonlinear iteration. In particular, we use an adaptive technique suggested in [J. Baglama, D. Calvetti, G.H. Golub, L. Reichel, Adaptively preconditioned GMRES algorithms, SIAM J. Sci. Comput. 20(1) (1998) 243-269] for restarted GMRES to enhance existing preconditioners with information available from previous stages in the nonlinear iteration. Numerical experiments drawn from domain decomposition techniques and fluid flow applications are used to validate the increased efficiency of our approach.