A Note on Adaptive Nonlinear Preconditioning Techniques

​Nonlinear preconditioning is a globalization technique for Newton’s method appliedto  systems  of  equations  with  unbalanced  nonlinearities,  in  which  nonlinear  residual  norm  reduc-tion stagnates due to slowly evolving subsets of the degrees of freedom.  Even though the Newtoncorrections may effectively be sparse, a standard Newton method still requires large ill-conditionedlinear 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 moststrategic, on subsets of the original system.  They require additional computation per outer iterationwhile aiming for many fewer outer iterations and correspondingly fewer global synchronizations.  Inthis work, we improve upon previous nonlinear preconditioning implementations by introducing pa-rameters that allow turning off nonlinear preconditioning during outer Newton iterations where itis not needed.  Numerical experiments show that the adaptive nonlinear preconditioning algorithmhas performance similar to monolithically applied nonlinear preconditioning, preserving robustnessfor some challenging problems representative of several PDE-based applications while saving workon nonlinear subproblems.