Relaxation techniques for PDE-constrained optimization in inverse problems

Tristan van Leeuwen, Universiteit Utrecht, The Netherlands - 11:15

PDE-constrained optimization problems arise in many applications, including inverse problems and optimal control. As optimization over both the control and state parameters is not feasible for large-scale problems, one often resorts to a reduced formulation by eliminating the constraints. The resulting optimization problem is often highly non-linear, which may cause local descent methods to stall at stationary points away from the global minimizer. Another issue is that it may not even be possible to eliminate the constraints as the PDE may be ill-posed (e.g., due to missing boundary conditions). In this talk, I will discuss ways to relax the constraints and reduce the problem implicitly. The resulting reduced optimization problem can in some cases be much less non-linear than the original reduced problem. I will discuss numerical methods to solve the reduced relaxed problem and illustrate the approach with a variety of numerical examples.


invited by Sebastian Reich