Colloquium

I will introduce approaches to solve time domain decomposition (TDD) formulations of large-scale optimal control problems. Optimal control problems arise in many science and engineering applications, but their numerical solution is expensive, both in terms of computing time and memory requirements. TDD or direct multiple shooting formulations of optimal control problems address these challenges by decomposing the underlying differential equations into equations on shorter time subintervals and couple these at the time interval boundaries. These coupling conditions must be satisfied at the solution, but not during the iteration of an optimization algorithm. This is exploited to improve the numerical solution of such problems through superior stability properties of sub-problems, introduction of parallelism, and reduction of permanent memory requirements. However, TDD formulations have a price: The auxiliary initial data at time interval boundaries are additional optimization variables and the coupling conditions are additional constraints. For problems governed by (discretized) PDEs this leads to huge increases in optimization variables and constraints. I will discuss methods to solve TDD formulations, present convergence results, illustrate their performance on applications, and sketch several open problems.