Revisiting the Augmented Lagrangian Approach to Saddle-Point Problems

Augmented Lagrangian (AL) methods have a long history in both optimization and numerical PDEs, going back to the work of Powell and Hestenes in the late 1960s and to the work of Glowinski and collaborators starting in the mid-1970s, respectively. Whereas AL-type methods have been long very popular in the field of constrained optimization, where are among the most widely used techniques available (as witnessed, for instance, by the great success enjoyed by methods like ADMM), the track record of these methods in solving PDE-related problems has been until recently somewhat less impressive, due in part to ill-conditioning and implementation difficulties.
In this talk I will describe advances in developing robust and scalable solvers for PDE-related saddle point problems by means of iterative methods and block preconditioners tailored to the AL formulation. The effectiveness of these solvers will be demonstrated on problems from incompressible flow (Oseen problem, coupled Stokes-Darcy problem), liquid crystal directors modeling, and optimal control with PDE constraints. Collaborators over the years include Fatemeh Beik, Chiara Faccio, Patrick Farrell, Santolo Leveque, Maxim Olshanskii and Zhen Wang.

Contatto:
luca.formaggia@polimi.it