Domain decomposition based nonlinear preconditioning for optimal control problems

The focus of this project is on designing efficient and scalable iterative methods for solving optimal control problems, where the system is governed by semilinear partial differential equations (PDEs), and with the possible presence of a non-smooth objective function and/or control constraints. Here, the goal is to find the forcing function (i.e., the control) with minimal cost that drives the system to the desired target state. When the constraints are PDEs, standard algorithms for optimal control require the solution of very large systems of nonlinear equations, where the number of degrees of freedom can be in the millions or even billions. The extreme storage and computational requirements impels us to consider distributed computing solutions.

There are two key ideas in our proposed approach. The first is to use an optimized Schwarz-like approach for decomposing the optimal control problem into a collection of smaller subdomain problems in space, with transmission conditions designed to optimize convergence of the resulting fixed-point method. This allows the subdomain problems to be solved independently and in parallel on large computing clusters. We will also consider two-level variants, where a coarse grid correction is used improve the scalability of the method, so that convergence does not deteriorate as the number of subdomains increases. The second idea is to use the above one- and two-level methods as nonlinear preconditioners to the first order optimality system, and apply a semi- smooth Newton or quasi-Newton method to solve the resulting nonlinear problem.

The advantage in our approach is twofold. First, based on our experience with elliptic problems, we expect that the use of optimized transmission conditions (i.e., a linear combination of the state/adjoint and their derivatives) can lead to significant improvements to the convergence behaviour of the method also for optimal control problems. Indeed, preliminary results show that a judicious choice of the Robin parameter leads to significant improvement in the convergence of the method. Secondly, because the strongest nonlinearity and non-smoothness in these problems tend to be local in nature, by breaking up the global problem into many subproblems, we expect that the preconditioned problem would have less severe nonlinearities in the coupling of the subdomains, which in turn leads to faster convergence by (quasi-)Newton. On the experimental front, we will test our algorithms on the simplified superconductivity problem, as well as the semilinear heat equation.