Decompose-then-optimize: a new approach to design domain decomposition methods for optimal control problems

For optimal control problems there is a classical discussion of whether one should first optimize the problem and then discretize it, or the other way round. We are interested in exploring a similar question related to domain decomposition methods for optimal control problems which have received substantial attention over the past two decades, but new methods were mostly developed using the optimize-then-decompose approach. After a detailed introduction to this subject, we present and analyze a new domain decomposition method for optimal control problems that comes from the decompose-then-optimize strategy which is less common. We use as our model problem a linear quadratic optimal control problem which we decompose and then solve using an augmented Lagrangian optimization technique. This leads to a new domain decomposition algorithm for such problems that has very good scalability properties. We prove that, when the algorithm converges, it necessarily converges to an optimal point of the original, non-decomposed problem. We illustrate the efficiency of our new domain decomposition method with numerical examples from which we obtain very desirable properties for domain decomposition methods, namely that the convergence is independent of the meshsize and the number of subdomain.