Inexact interior-point method for PDE-constrained nonlinear optimization (Q2878952)

The paper deals with optimal control problems constrained by PDEs and nonlinear inequalities on the control variable. The control can appear as a source term in the PDE as well as parameter (boundary data, material coefficients or domain geometry). The problems are often ill-posed and difficult to solve. A numerical approximation of the mathematical model leads to a discrete, finite-dimensional optimization problem with equalities and inequalities for the parameters. The objective function is extended by a Tikhonov regularization term. Following a classical interior-point strategy, slack variables and a barrier term are introduced. The approximated problem is solved through a sequence of barrier subproblems. If the objective function and the constraints are sufficiently smooth, the limit of the solutions of the barrier subproblems satisfies first-order necessary optimality conditions for the finite dimensional problem. The solutions of the barrier subproblems are critical points of the corresponding Lagrange functions and satisfy the Karush-Kuhn-Tucker (KKT) conditions. Premultiplication leads to a primal-dual interior point method. At each Newton step, a very large but sparse linear system is to solve. The authors propose a Schur-complement slack control preconditioning to achieve high parallel scalability. The method solves the KKT-system inexactly. To demonstrate the effectiveness of the method, four PDE-constrained optimization problems are studied.