Interior Point Methods and Preconditioning for PDE-Constrained Optimization Problems Involving Sparsity Terms
It addresses the challenge of efficiently solving PDE-constrained optimization with sparsity constraints, which is important for applications like optimal control and inverse problems.
The paper proposes an Interior Point method with tailored preconditioners for PDE-constrained optimization problems with sparsity-promoting L1 terms, achieving robust performance and low iteration counts across various PDE applications.
PDE-constrained optimization problems with control or state constraints are challenging from an analytical as well as numerical perspective. The combination of these constraints with a sparsity-promoting $\rm L^1$ term within the objective function requires sophisticated optimization methods. We propose the use of an Interior Point scheme applied to a smoothed reformulation of the discretized problem, and illustrate that such a scheme exhibits robust performance with respect to parameter changes. To increase the potency of this method we introduce fast and efficient preconditioners which enable us to solve problems from a number of PDE applications in low iteration numbers and CPU times, even when the parameters involved are altered dramatically.