Abstract: Variational inequalities in PDE systems emerge either directly from the physical properties of the system (e.g., friction and contact, plasticity, Bingham flow or rheologies with yielding), or originate from optimization and inverse problems with PDEs due to constraints or non-differentiabilities (e.g., pointwise constraints in PDE-constrained optimization, total variation regularization, sparsity-promoting functionals). I will review algorithms that allow for efficient solution of these variational inequality problems. These methods are based on Lagrange multipliers, complementarity functions and generalized Newton methods, and their performance and analysis depends on the properties of the underlying PDE solution operators. The application of these methods to contact and friction problems in elasticity, to sparsity-promoting regularization and to the solution of flow problems with sheer thickening rheology will be discussed.
Title: Generalized Newton Methods for Variational Inequalities in PDE Problems