Abstract

We review recent advances in applying the augmented Lagrange multiplier method as a general approach for generating multiplier-free stabilized methods. The augmented Lagrangian method consists of a standard Lagrange multiplier method augmented by a penalty term, penalizing the constraint equations, and is well known as the basis for iterative algorithms for constrained optimization problems. However, its use as a stabilization method in computational mechanics has only recently been appreciated. We consider equality and inequality constraints and apply the framework to several examples in computational mechanics, including inverse problems. Finally, we present tools from machine learning to speed up solutions of inverse problems and to assimilate statistical observations of solutions.