Time Domain Decomposition for Optimal Control Problems

For this project, we will design and develop efficient parallel algorithms for the numerical solution of optimal control problems under partial differential equation (PDE) constraints. In this class of problems, we are given a mechanical or biological system governed by a PDE, and the goal is to find the forcing function with minimal cost that drives the system to the desired target state. The numerical solution of such problems has become an active area of research in the past decade with a growing list of applications, such as the control of fluid flow governed by the Navier-Stokes equations, quantum control and medical applications related to the optimization of radiotherapy administration.

When the constraints are PDEs, standard algorithms for optimal control require the solution of very large systems of linear or nonlinear equations, where the number of degrees of freedom can be in the millions or even billions. The extreme storage and computational requirements impel us to consider distributed computing solutions. We propose to follow the domain decomposition paradigm: we divide the time horizon into subintervals to obtain problems that are small enough to be solved on a single processor. We then assign these smaller problems to separate processors within a large computing cluster, where they are solved in parallel. A consistent global solution is then obtained by iteration.

The novelty in our approach lies in the use of optimized transmission conditions for communication between time intervals. Based on our experience with elliptic problems, where the use of Robin data greatly speeds up the convergence of Schwarz methods, we propose to use a linear combination of state and adjoint variables as interface data for optimal control problems. Preliminary results show that a judicious choice of the Robin parameter leads to significant improvement in the convergence of the method. On the theoretical front, we will establish convergence results for the new method when the PDE is parabolic, paying special attention to the case where the spatial differential operator is non-symmetric, e.g. for the advection-diffusion equation. We will also design a coarse grid correction to make the algorithm scalable when the number of subdomains is large. On the experimental front, we will test our algorithms on parabolic and other types of problems, such as the Stokes equation and nonlinear diffusion problems, in order to confirm our theoretical results and to assess the applicability and efficiency of our method on problems not covered by our analysis.